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T¥, h D
(q•, k ) L
(a) Under conditions for which a uniform surface temperature Ts is maintained around the circum- ference of the heater and the temperature Too and convection coefficient h of the airflow are known, obtain an expression for the rate of heat transfer per unit length to the air. Evaluate the
heat rate for Ts = 300°C, D = 20 mm, an alu- minum sleeve (ks = 240 W/m · K), w = 40 mm, N = 16, t = 4 mm, L = 20 mm, Too = 50°C, and h = 500 W/m2 · K.
(b) For the foregoing heat rate and a copper heater of thermal conductivity kh = 400 W/m · K, what is the required volumetric heat generation within the heater
Aluminum block, Tb
Pin fins (30), D = 1.5 mm
L = 15 mm
Copper, 5-mm thickness
and its corresponding centerline temperature?
(c) With all other quantities unchanged, explore the effect of variations in the fin parameters (N, L, t) on the heat rate, subject to the constraint that the fin thickness and the spacing between fins cannot be less than 2 mm.
4.37 For a small heat source attached to a large substrate, the spreading resistance associated with multidimensional conduction in the substrate may be approximated by the
(a) Calculate the temperature the device will reach, assuming that all the power generated by the device must be transferred by conduction to the block.
(b) To operate the device at a higher power level, a
expression [Yovanovich, M. M., and V. W. Antonetti, in Adv. Thermal Modeling Elec. Comp. and Systems, Vol. 1, A. Bar-Cohen and A. D. Kraus, Eds., Hemisphere, NY, 79–128, 1988]
3 5 7
circuit designer proposes to attach a finned heat sink
1 – 1.410 Ar + 0.344 Ar + 0.043 Ar + 0.034 Ar
to the top of the device. The pin fins and base mate- rial are fabricated from copper (k = 400 W/m · K)
and are exposed to an airstream at 27 C for which the convection coefficient is 1000 W/m2 · K. For the device temperature computed in part (a), what is the permissible operating power?
where Ar = As,h/As,sub is the ratio of the heat source area to the substrate area. Consider application of the expres- sion to an in-line array of square chips of width Lh = 5 mm on a side and pitch Sh = 10 mm. The interface
between the chips and a large substrate of thermal con- ductivity ksub = 80 W/m · K is characterized by a ther- mal contact resistance of Rt”,c = 0.5 X 10-4 m2 · K/W.
Substrate, ksub Chip, Th
the sketch, the boundary condition changes from specified heat flux qs” (into the domain) to convection, at the location of the node (m, n). Write the steady- state, two-dimensional finite difference equation at this node.
If a convection heat transfer coefficient of h = 100 W/m2 · K is associated with airflow (Too = 15°C) over the chips and substrate, what is the maximum allowable chip power dissipation if the chip tempera- ture is not to exceed Th = 85°C?
4.38 Consider nodal configuration 2 of Table 4.2. Derive the finite-difference equations under steady-state conditions for the following situations.
(a) The horizontal boundary of the internal corner is perfectly insulated and the vertical boundary is sub- jected to the convection process (Too, h).
(b) Both boundaries of the internal corner are perfectly insulated. How does this result compare with Equation 4.41?
4.39 Consider nodal configuration 3 of Table 4.2. Derive the finite-difference equations under steady-state conditions for the following situations.
(a) The boundary is insulated. Explain how Equation
4.42 can be modified to agree with your result.
(b) The boundary is subjected to a constant heat flux.
4.40 Consider nodal configuration 4 of Table 4.2. Derive the finite-difference equations under steady-state condi- tions for the following situations.
(a) The upper boundary of the external corner is per- fectly insulated and the side boundary is subjected to the convection process (Too, h).
(b) Both boundaries of the external corner are perfectly insulated. How does this result compare with Equa- tion 4.43?
4.41 One of the strengths of numerical methods is their ability to handle complex boundary conditions. In
4.42 Determine expressions for q(m-1,n) ® (m,n), q(m+1,n) ® (m,n), q(m,n+1) ® (m,n) and q(m,n-1) ® (m,n) for conduction associated with a control volume that spans two different materials. There is no contact resistance at the interface between the materials. The control volumes are L units long into the page. Write the finite difference equation under steady- state conditions for node (m, n).
4.43 Consider heat transfer in a one-dimensional (radial) cylindrical coordinate system under steady-state condi- tions with volumetric heat generation.
(a) Derive the finite-difference equation for any inte- rior node m.
(b) Derive the finite-difference equation for the node n located at the external boundary subjected to the convection process (Too, h).
4.44 In a two-dimensional cylindrical configuration, the radial (l1r) and angular (l1) spacings of the nodes are uniform. The boundary at r = ri is of uniform temperature Ti. The boundaries in the radial direction are adiabatic (insulated) and exposed to surface convection (Too, h), as illustrated. Derive the finite-difference equations for (i) node 2,
(ii) node 3, and (iii) node 1.
(b) Node (m, n) at the tip of a cutting tool with the upper surface exposed to a constant heat flux qo”, and the diagonal surface exposed to a convection cooling process with the fluid at Too and a heat transfer coefficient h. Assume l1x = l1y.
m + 1, n
4.45 Upper and lower surfaces of a bus bar are convectively cooled by air at Too, with hu -:f hl. The sides are cooled by maintaining contact with heat sinks at To, through a thermal contact resistance of Rt”,c. The bar is of thermal conductivity k, and its width is twice its thickness L.
Consider steady-state conditions for which heat is uni- formly generated at a volumetric rate . due to passage
of an electric current. Using the energy balance method, derive finite-difference equations for nodes 1 and 13.
4.47 Consider the nodal point 0 located on the boundary between materials of thermal conductivity kA and kB.
Derive the finite-difference equation, assuming no internal generation.
4.48 Consider the two-dimensional grid (l1x = l1y) represent- ing steady-state conditions with no internal volumetric generation for a system with thermal conductivity k. One of the boundaries is maintained at a constant temperature Ts while the others are adiabatic.
12 11 10 9 8
4.46 Derive the nodal finite-difference equations for the fol- lowing configurations.
(a) Node (m, n) on a diagonal boundary subjected to convection with a fluid at Too and a heat transfer coefficient h. Assume l1x = l1y.
13 4 5 6 7
x 16 1
Derive an expression for the heat rate per unit length nor- mal to the page crossing the isothermal boundary (Ts).
4.49 Consider a one-dimensional fin of uniform cross- sectional area, insulated at its tip, x = L. (See Table 3.4,
case B). The temperature at the base of the fin Tb and of the adjoining fluid Too, as well as the heat transfer coef- ficient h and the thermal conductivity k, are known.
(a) Derive the finite-difference equation for any inte- rior node m.
(b) Derive the finite-difference equation for a node n
located at the insulated tip.
Consider the network for a two-dimensional system with- out internal volumetric generation having nodal tempera- tures shown below. If the grid spacing is 125 mm and the thermal conductivity of the material is 50 W/m · K, calcu- late the heat rate per unit length normal to the page from the isothermal surface (Ts).
difference method with l1x = l1y = 100 mm and treating the wood as a two-dimensional extended surface (Figure 3.17a), enlighten the students as to whether location A or location B will be more effective in igniting the wood by determining the maximum local steady-state temperature.
(b) Some students wonder whether the same technique can be used to melt a stainless steel hull. Repeat part (a) considering a stainless steel mockup of the same dimensions with k = 15 W/m · K and s = 0.2. The value of the absorbed irradiation is the same as in part (a).
4.52 Consider the square channel shown in the sketch oper- ating under steady-state conditions. The inner surface of the channel is at a uniform temperature of 600 K, while the outer surface is exposed to convection with a fluid at 300 K and a convection coefficient of 50 W/m2 · K. From a symmetrical element of the channel, a two-
dimensional grid has been constructed and the nodes labeled. The temperatures for nodes 1, 3, 6, 8, and 9 are identified.
4.51 An ancient myth describes how a wooden ship was destroyed by soldiers who reflected sunlight from their polished bronze shields onto its hull, setting the ship
T¥ = 300 K
h = 50 W/m2• K
1 2 3 4
5 6 7
D x = Dy = 0.01 m
ablaze. To test the validity of the myth, a group of col-
lege students are given mirrors and they reflect sunlight onto a 100 mm X 100 mm area of a t = 10-mm-thick
k = 1 W/m•K
T = 600 K x
plywood mockup characterized by k = 0.8 W/m · K. The
T1 = 430 K
T8 = T9
= 600 K
bottom of the mockup is in water at Tw = 20 C, while
the air temperature is Too = 25 C. The surroundings are
T3 = 394 K
T6 = 492 K
at Tsur = 23 C. The wood has an emissivity of s= 0.90;
both the front and back surfaces of the plywood are char- acterized by h = 5 W/m2 · K. The absorbed irradiation from the N students’ mirrors is GS,N = 70,000 W/m2 on the front surface of the mockup.
Tsur = 23°C
(a) Beginning with properly defined control volumes, derive the finite-difference equations for nodes 2, 4, and 7 and determine the temperatures T2, T4, and T7 (K).
(b) Calculate the heat loss per unit length from the channel.
L2 = 800 mm
T¥ = 25°C
h = 5 W/m2·K
4.53 A long conducting rod of rectangular cross section
• • • • • • • • •
• • • • • • • •
Irradiation location A
• • • • • • •
Irradiation location B
• • • • • •
L1 = 500 mm
H = 300 mm
Tw = 20°C
(20 mm X 30 mm) and thermal conductivity k =
20 W/m · K experiences uniform heat generation at a rate . = 5 X 107 W/m3, while its surfaces are main-
tained at 300 K.
(a) Using a finite-difference method with a grid spac- ing of 5 mm, determine the temperature distribution in the rod.
(b) With the boundary conditions unchanged, what
(a) A debate ensues concerning where the beam should be focused, location A or location B. Using a finite
heat generation rate will cause the midpoint tem- perature to reach 600 K?
4.54 A flue passing hot exhaust gases has a square cross section, 300 mm to a side. The walls are constructed of refractory brick 150 mm thick with a thermal conduc- tivity of 0.85 W/m · K. Calculate the heat loss from the flue per unit length when the interior and exterior sur- faces are maintained at 350 and 25 C, respectively. Use a grid spacing of 75 mm.
4.55 Steady-state temperatures (K) at three nodal points of a long rectangular rod are as shown. The rod experiences a uniform volumetric generation rate of 5 X 107 W/m3
Case Surface Boundary Condition
1 1 T = 100 C
T = 50 C
T = 50 C
T = 100 C
and has a thermal conductivity of 20 W/m · K. Two of its
sides are maintained at a constant temperature of 300 K,
while the others are insulated.
(a) Determine the temperatures at nodes 1, 2, and 3.
(b) Calculate the heat transfer rate per unit length (W/m) from the rod using the nodal temperatures. Compare this result with the heat rate calculated from knowledge of the volumetric generation rate and the rod dimensions.
(a) Determine the spatially averaged value of the ther-
mal conductivity k. Use this value to estimate the heat rate per unit length for cases 1 and 2.
(b) Using a grid spacing of 2 mm, determine the heat rate per unit depth for case 1. Compare your result to the estimated value calculated in part (a).
(c) Using a grid spacing of 2 mm, determine the heat rate per unit depth for case 2. Compare your result to the estimated value calculated in part (a).
4.57 Steady-state temperatures at selected nodal points of the symmetrical section of a flow channel are known to be T2 = 95.47°C, T3 = 117.3°C, T5 = 79.79°C, T6 = 77.29°C, T8 = 87.28°C, and T10 = 77.65°C. The wall experiences uniform volumetric heat generation of q = 106 W/m3 and has a thermal conductivity of k = 10 W/m · K. The inner and outer surfaces of the chan-
nel experience convection with fluid temperatures of Too,i = 50°C and Too,o = 25°C and convection coeffi- cients of hi = 500 W/m2 · K and ho = 250 W/m2 · K.
4.56 Functionally graded materials are intentionally fabri- cated to establish a spatial distribution of properties in the final product. Consider an L X L two-dimensional object with L = 20 mm. The thermal conductivity dis- tribution of the functionally graded material is k(x) =
20 W/m · K + (7070 W/m5/2 · K) x3/2. Two sets of boundary
4 5 6
8 9 10
Dx = Dy = 25 mm
conditions, denoted as cases 1 and 2, are applied.
(a) Determine the temperatures at nodes 1, 4, 7, and 9.
(b) Calculate the heat rate per unit length (W/m) from the outer surface A to the adjacent fluid.
(c) Calculate the heat rate per unit length from the inner fluid to surface B.
(d) Verify that your results are consistent with an over- all energy balance on the channel section.
4.58 Consider an aluminum heat sink (k = 240 W/m · K), such as that shown schematically in Problem 4.28. The
inner and outer widths of the square channel are w = 20 mm and W = 40 mm, respectively, and an outer surface temperature of Ts = 50°C is maintained by the array of electronic chips. In this case, it is not the inner surface temperature that is known, but conditions (Too, h) associated with coolant flow through the chan- nel, and we wish to determine the rate of heat transfer to the coolant per unit length of channel. For this pur- pose, consider a symmetrical section of the channel and a two-dimensional grid with l1x = l1y = 5 mm.
(a) For Too = 20°C and h = 5000 W/m2 · K, determine the unknown temperatures, T1, . . ., T7, and the rate of heat transfer per unit length of channel, q'.
Assess the effect of variations in h on the unknown temperatures and the heat rate.
Heat sink, k
Calculate the heat transfer per unit depth into the page, q', using l1x = l1y = l1r = 10 mm and l1 = r/8. The base of the rectangular subdomain is held at Th = 20 C, while the vertical surface of the cylindrical subdomain and the surface at outer radius ro are at Tc = 0 C. The remaining surfaces are adiabatic, and the thermal con- ductivity is k = 10 W/m · K.
4.60 Consider the two-dimensional tube of a noncircular cross section formed by rectangular and semicylindrical subdomains patched at the common dashed control sur- faces in a manner similar to that described in Problem
4.59. Note that, along the dashed control surfaces, tem- peratures in the two subdomains are identical and local conduction heat fluxes to the semicylindrical subdo- main are identical to local conduction heat fluxes from the rectangular subdomain. The bottom of the domain is held at Ts = 100 C by condensing steam, while the flowing fluid is characterized by the temperature and convection coefficient shown in the sketch. The
T1 T5 Ts
remaining surfaces are insulated, and the thermal con- ductivity is k = 15 W/m · K.
Coolant, T¥, h
k = 15 W/m⋅K
4.59 Conduction within relatively complex geometries can
sometimes be evaluated using the finite-difference methods of this text that are applied to subdomains and
T¥,i = 20°C
hi = 240 W/m2·K
patched together. Consider the two-dimensional
t = 10 mm y
Di = 40 mm
Ts = 100°C
domain formed by rectangular and cylindrical subdo- mains patched at the common, dashed control surface.
Note that, along the dashed control surface, tempera-
tures in the two subdomains are identical and local
L = Do = 80 mm
conduction heat fluxes to the cylindrical subdomain are identical to local conduction heat fluxes from the rectangular subdomain.
Find the heat transfer rate per unit length of tube, q',
using l1x = l1y = l1r = 10 mm and l1= r/8. Hint: Take advantage of the symmetry of the problem by considering only half of the entire domain.
4.61 The steady-state temperatures ( C) associated with selec- ted nodal points of a two-dimensional system having a thermal conductivity of 1.5 W/m · K are shown on the
ro = 50 mm
Tc = 0°C
129.4 T2 45.8
H = 30 mm
ri = 30 mm
W = 20 mm
x Th = 20°C
172.9 T1 132.8 67.0
T0 = 200°C
(a) Determine the temperatures at nodes 1, 2, and 3.
(b) Calculate the heat transfer rate per unit thickness normal to the page from the system to the fluid.
4.62 A steady-state, finite-difference analysis has been per- formed on a cylindrical fin with a diameter of 12 mm and a thermal conductivity of 15 W/m · K. The convec- tion process is characterized by a fluid temperature of 25 C and a heat transfer coefcient of 25 W/m 2 · K.
T0 = 100.0°C
D T1 = 93.4°C
T2 = 89.5°C
(a) The temperatures for the first three nodes, sepa- rated by a spatial increment of x = 10 mm, are given in the sketch. Determine the fin heat rate.
(b) Determine the temperature at node 3, T3.
4.63 Consider the two-dimensional domain shown. All sur- faces are insulated except for the isothermal surfaces at x = 0 and L.
(a) Determine the temperatures at nodes 1, 2, 3, and 4. Estimate the midpoint temperature.
Reducing the mesh size by a factor of 2, determine the corresponding nodal temperatures. Compare your results with those from the coarser grid.
From the results for the finer grid, plot the 75, 150,
and 250 C isotherms.
4.65 Consider a long bar of square cross section (0.8 m to the side) and of thermal conductivity 2 W/m · K. Three of these sides are maintained at a uniform temperature of 300 C. The fourth side is exposed to a uid at 100 C for which the convection heat transfer coefficient is 10 W/m2 · K.
(a) Using an appropriate numerical technique with a grid spacing of 0.2 m, determine the midpoint tem- perature and heat transfer rate between the bar and the fluid per unit length of the bar.
Reducing the grid spacing by a factor of 2, determine the midpoint temperature and heat transfer rate. Plot the corresponding temperature distribution across the surface exposed to the fluid. Also, plot the 200 and 250 C isotherms.
4.66 Consider a two-dimensional, straight triangular fin of length L = 50 mm and base thickness t = 20 mm. The thermal conductivity of the fin is k = 25 W/m · K. The base temperature is Tb = 50 C, and the n is exposed to
convection conditions characterized by h = 50 W/m2 · K,
Too = 20 C. Using a nite difference mesh with l1x = 10 mm and l1y = 2 mm, and taking advantage of symme- try, determine the fin efficiency, Yf. Compare your value of the fin efficiency with that reported in Figure 3.19.
L = 50 mm
(a) Use a one-dimensional analysis to estimate the
shape factor S.
(b) Estimate the shape factor using a finite difference analysis with l1x = l1y = 0.05L. Compare your
t = 20 mm
4 10 15
3 9 14 18
2 8 13 17 20
1 7 12 16 19 21
answer with that of part (a), and explain the differ- ence between the two solutions. y
4.64 Consider two-dimensional, steady-state conduction in a x
square cross section with prescribed surface temperatures.
y A common arrangement for heating a large surface area is to move warm air through rectangular ducts below the surface. The ducts are square and located midway between the top and bottom surfaces that are exposed to room air and insulated, respectively.
For the condition when the floor and duct temperatures are 30 and 80 C, respectively, and the thermal conductiv- ity of concrete is 1.4 W/m · K, calculate the heat rate from each duct, per unit length of duct. Use a grid spacing with l1x = 2 l1y, where l1y = 0.125L and L =150 mm.
Consider the gas turbine cooling scheme of Example 4.3. In Problem 3.23, advantages associated with applying a thermal barrier coating (TBC) to the exterior surface of a turbine blade are described. If a 0.5-mm-thick zirco- nia coating (k = 1.3 W/m · K, Rt”,c = 10-4 m2 · K/W) is applied to the outer surface of the air-cooled blade,
device to nonintrusively determine the surface tempera- ture distribution. Predict the temperature distribution of the painted surface, accounting for radiation heat trans- fer with large surroundings at Tsur = 25 C.
4.71 Consider using the experimental methodology of Prob- lem 4.70 to determine the convection coefficient distri- bution about an airfoil of complex shape.
determine the temperature field in the blade for the operating conditions of Example 4.3.
4.69 A long, solid cylinder of diameter D = 25 mm is formed
4 5 6 7
of an insulating core that is covered with a very thin (t = 50 µ,m), highly polished metal sheathing of thermal conductivity k = 25 W/m · K. Electric current flows through the stainless steel from one end of the cylinder to
30 28 27 26 25 24 23 22
19 18 17
the other, inducing uniform volumetric heating within the sheathing of . = 5 X 106 W/m3. As will become evident
in Chapter 6, values of the convection coefficient between the surface and air for this situation are spa- tially nonuniform, and for the airstream conditions of the experiment, the convection heat transfer coefficient varies with the angle e as h(e) = 26 + 0.637e – 8.92e2 for 0 e r/2 and h(e) = 5 for r/2 e r.
t = 50 µm
D = 25 mm
Accounting for conduction in the metal sheathing and
radiation losses to the large surroundings, determine the convection heat transfer coefficients at the locations shown. The surface locations at which the temperatures are measured are spaced 2 mm apart. The thickness of the metal sheathing is t = 20 µ,m, the volumetric gener-
ation rate is . = 20 X 106 W/m3, the sheathing’s ther-
mal conductivity is k = 25 W/m · K, and the emissivity of the painted surface is s = 0.98. Compare your results to cases where (i) both conduction along the sheathing and radiation are neglected, and (ii) when only radiation is neglected.
Temperature Temperature Temperature Location ( C) Location ( C) Location ( C)
q• = 5 x 106 W/m3
k = 25 W/m • K
1 27.77 11 34.29 21 31.13
Neglecting conduction in the e-direction within the
stainless steel, plot the temperature distribution
T(e) for 0 e r for Too = 25 C.
Accounting for e-direction conduction in the stain-
less steel, determine temperatures in the stainless
steel at increments of l1e = r/20 for 0 e r.
Compare the temperature distribution with that of part (a).
Hint: The temperature distribution is symmetrical about the horizontal centerline of the cylinder.
4.70 Consider Problem 4.69. An engineer desires to measure the surface temperature of the thin sheathing by painting it black (s= 0.98) and using an infrared measurement
A thin metallic foil of thickness 0.25 mm with a pattern
of extremely small holes serves as an acceleration grid to control the electrical potential of an ion beam. Such a grid is used in a chemical vapor deposition (CVD) process for the fabrication of semiconductors. The top surface of the grid is exposed to a uniform heat flux
caused by absorption of the ion beam, qs” = 600 W/m2. The edges of the foil are thermally coupled to water- cooled sinks maintained at 300 K. The upper and lower surfaces of the foil experience radiation exchange with the vacuum enclosure walls maintained at 300 K. The effective thermal conductivity of the foil material is 40 W/m · K, and its emissivity is 0.45.
Vacuum enclosure, Tsur
Ion beam, q”s
Grid hole pattern
Grid x L = 115 mm
Water-cooled electrode sink, Tsink
Assuming one-dimensional conduction and using a finite-difference method representing the grid by 10 nodes in the x-direction, estimate the temperature distri- bution for the grid. Hint: For each node requiring an energy balance, use the linearized form of the radiation rate equation, Equation 1.8, with the radiation coefficient hr, Equation 1.9, evaluated for each node.
A long bar of rectangular cross section, 0.4 m X 0.6 m on a side and having a thermal conductivity of 1.5 W/m · K, is subjected to the boundary conditions shown.
T = 200°C
(a) Using a finite-difference method with a mesh size of l1x = l1y = 40 mm, calculate the unknown nodal temperatures and the heat transfer rate per width of groove spacing (w) and per unit length normal to the page.
With a mesh size of l1x = l1y = 10 mm, repeat the foregoing calculations, determining the temperature field and the heat rate. Also, consider conditions for which the bottom surface is not at a uniform tempera- ture T2 but is exposed to a fluid at Too = 20 C. With l1x = l1y = 10 mm, determine the temperature field and heat rate for values of h = 5, 200, and 1000 W/m2 · K, as well as for h ® oo.
Refer to the two-dimensional rectangular plate of Prob- lem 4.2. Using an appropriate numerical method with l1x = l1y = 0.25 m, determine the temperature at the midpoint (1, 0.5).
The shape factor for conduction through the edge of adjoining walls for which D > L/5, where D and L are the wall depth and thickness, respectively, is shown in Table 4.1. The two-dimensional symmetrical element of the edge, which is represented by inset (a), is bounded by the diagonal symmetry adiabat and a sec- tion of the wall thickness over which the temperature distribution is assumed to be linear between T1 and T2.
Linear temperature distribution
T2 T2 T2
T = 200°C
Two of the sides are maintained at a uniform tempera- ture of 200 C. One of the sides is adiabatic, and the remaining side is subjected to a convection process with Too = 30 C and h = 50 W/m2 · K. Using an appro-
priate numerical technique with a grid spacing of 0.1 m, determine the temperature distribution in the bar and the heat transfer rate between the bar and the fluid per unit length of the bar.
4.74 The top surface of a plate, including its grooves, is main-
n • L
tained at a uniform temperature of T1 = 200 C. The lower surface is at T2 = 20 C, the thermal conductivity is 15 W/m · K, and the groove spacing is 0.16 m.
(a) Using the nodal network of inset (a) with L = 40 mm, determine the temperature distribution in the element for T1 = 100 C and T2 = 0 C. Evaluate the heat rate
per unit depth (D = 1 m) if k = 1 W/m · K. Determine the corresponding shape factor for the edge, and compare your result with that from Table 4.1.
(b) Choosing a value of n = 1 or n = 1.5, establish a nodal network for the trapezoid of inset (b) and determine the corresponding temperature field. Assess the validity of assuming linear temperature distributions across sections a–a and b–b.
The diagonal of a long triangular bar is well insulated, while sides of equivalent length are maintained at uni- form temperatures Ta and Tb.
Ta = 100°C
Tb = 0°C
(a) Establish a nodal network consisting of five nodes along each of the sides. For one of the nodes on the diagonal surface, define a suitable control volume and derive the corresponding finite-difference equation. Using this form for the diagonal nodes and appropri- ate equations for the interior nodes, find the temper- ature distribution for the bar. On a scale drawing of the shape, show the 25, 50, and 75 C isotherms.
(b) An alternate and simpler procedure to obtain the finite-difference equations for the diagonal nodes fol- lows from recognizing that the insulated diagonal sur- face is a symmetry plane. Consider a square 5 X 5 nodal network, and represent its diagonal as a symme- try line. Recognize which nodes on either side of the diagonal have identical temperatures. Show that you can treat the diagonal nodes as “interior” nodes and write the finite-difference equations by inspection.
A straight fin of uniform cross section is fabricated from a material of thermal conductivity 50 W/m · K, thickness w = 6 mm, and length L = 48 mm, and it is very long in the direction normal to the page. The convection heat transfer coefficient is 500 W/m2 · K with an ambient air temperature of Too = 30 C. The base of the n is main- tained at Tb = 100 C, while the n tip is well insulated.
(a) Using a finite-difference method with a space increment of 4 mm, estimate the temperature distri- bution within the fin. Is the assumption of one- dimensional heat transfer reasonable for this fin?
(b) Estimate the fin heat transfer rate per unit length normal to the page. Compare your result with the one-dimensional, analytical solution, Equation 3.81.
(c) Using the finite-difference mesh of part (a), compute and plot the fin temperature distribution for values of h = 10, 100, 500, and 1000 W/m2 · K. Determine
and plot the fin heat transfer rate as a function of h.
A rod of 10-mm diameter and 250-mm length has one end maintained at 100 C. The surface of the rod expe- riences free convection with the ambient air at 25 C and a convection coefficient that depends on the differ- ence between the temperature of the surface and the ambient air. Specifically, the coefficient is prescribed by a correlation of the form, hfc = 2.89[0.6 + 0.624
(T – Too)1/6]2, where the units are hfc (W/m2 · K) and T (K).
The surface of the rod has an emissivity s = 0.2 and
experiences radiation exchange with the surroundings at Tsur = 25 C. The n tip also experiences free convec- tion and radiation exchange.
Assuming one-dimensional conduction and using a finite-difference method representing the fin by five nodes, estimate the temperature distribution for the fin. Determine also the fin heat rate and the relative contri- butions of free convection and radiation exchange. Hint: For each node requiring an energy balance, use the linearized form of the radiation rate equation, Equa-
tion 1.8, with the radiation coefficient hr, Equation 1.9, evaluated for each node. Similarly, for the convection rate equation associated with each node, the free con- vection coefficient hfc must be evaluated for each node.
A simplified representation for cooling in very large-scale integration (VLSI) of microelectronics is shown in the
sketch. A silicon chip is mounted in a dielectric substrate, and one surface of the system is convectively cooled, while the remaining surfaces are well insulated from the surroundings. The problem is rendered two-dimensional
by assuming the system to be very long in the direction perpendicular to the paper. Under steady-state operation, electric power dissipation in the chip provides for uni-
form volumetric heating at a rate of . . However, the
heating rate is limited by restrictions on the maximum temperature that the chip is allowed to achieve.
(b) The grid spacing used in the foregoing finite-difference solution is coarse, resulting in poor precision for the temperature distribution and heat removal rate. Investigate the effect of grid spacing by consider- ing spatial increments of 50 and 25 µ,m.
(c) Consistent with the requirement that a + b = 400 µ,m, can the heat sink dimensions be altered in a manner that reduces the overall thermal resistance?
T¥ = 20°C
h = 500 W/m2•K
ks = 5 W/m•K
kc = 50 W/m•K
q• = 107 W/m3
4.82 A plate (k = 10 W/m · K) is stiffened by a series of lon- gitudinal ribs having a rectangular cross section with length L = 8 mm and width w = 4 mm. The base of the plate is maintained at a uniform temperature Tb = 45 C, while the rib surfaces are exposed to air at a tem- perature of Too = 25 C and a convection coefcient of h = 600 W/m2 · K.
L = 27 mm y
For the conditions shown on the sketch, will the maxi- mum temperature in the chip exceed 85 C, the maximum allowable operating temperature set by industry stan- dards? A grid spacing of 3 mm is suggested.
A heat sink for cooling computer chips is fabricated from copper (ks = 400 W/m · K), with machined microchan- nels passing a cooling fluid for which T = 25 C and
h = 30,000 W/m2 · K. The lower side of the sink experi- ences no heat removal, and a preliminary heat sink design
calls for dimensions of a = b = ws = wf = 200 µ,m. A
symmetrical element of the heat path from the chip to the fluid is shown in the inset.
(a) Using a finite-difference method with l1x = l1y =
2 mm and a total of 5 X 3 nodal points and regions, estimate the temperature distribution and the heat rate from the base. Compare these results with those obtained by assuming that heat transfer in the rib is one-dimensional, thereby approximating the behav- ior of a fin.
(b) The grid spacing used in the foregoing finite- difference solution is coarse, resulting in poor pre-
(a) Using the symmetrical element with a square nodal network of l1x = l1y = 100 µ,m, determine the corre- sponding temperature field and the heat rate q' to the coolant per unit channel length (W/m) for a maximum allowable chip temperature Tc, max = 75 C. Estimate the corresponding thermal resistance between the chip surface and the fluid, R't,c-ƒ(m · K/W). What is
the maximum allowable heat dissipation for a chip that measures 10 mm X 10 mm on a side?
cision for estimates of temperatures and the heat rate. Investigate the effect of grid refinement by reducing the nodal spacing to l1x = l1y = 1 mm (a 9 X 3 grid) considering symmetry of the center line.
(c) Investigate the nature of two-dimensional conduc- tion in the rib and determine a criterion for which the one-dimensional approximation is reasonable. Do so by extending your finite-difference analysis to deter- mine the heat rate from the base as a function of the length of the rib for the range 1.5 L/w 10, keep- ing the length L constant. Compare your results with those determined by approximating the rib as a fin.
4.83 The bottom half of an I-beam providing support for a furnace roof extends into the heating zone. The web is well insulated, while the flange surfaces experience
convection with hot gases at Too = 400 C and a convec- tion coefficient of h = 150 W/m2 · K. Consider the symmetrical element of the flange region (inset a), assuming that the temperature distribution across the web is uniform at Tw = 100 C. The beam thermal con- ductivity is 10 W/m · K, and its dimensions are wƒ = 80 mm, ww = 30 mm, and L = 30 mm.
(a) Using a grid spacing of 30 mm and the Gauss-Seidel iteration method, determine the nodal temperatures and the heat rate per unit length normal to the page into the bar from the air.
Determine the effect of grid spacing on the temper- ature field and heat rate. Specifically, consider a grid spacing of 15 mm. For this grid, explore the effect of changes in h on the temperature field and the isotherms.
4.85 A long trapezoidal bar is subjected to uniform tempera- tures on two surfaces, while the remaining surfaces are well insulated. If the thermal conductivity of the mate- rial is 20 W/m · K, estimate the heat transfer rate per unit
wo Uniform ?
length of the bar using a finite-difference method. Use the Gauss–Seidel method of solution with a space incre- ment of 10 mm.
T1 = 100°C 30 mm
T2 = 25°C
(a) Calculate the heat transfer rate per unit length to the beam using a 5 X 4 nodal network.
(b) Is it reasonable to assume that the temperature dis- tribution across the web–flange interface is uni- form? Consider the L-shaped domain of inset (b) and use a fine grid to obtain the temperature distri- bution across the web–flange interface. Make the distance wo > ww /2.
4.84 A long bar of rectangular cross section is 60 mm X 90 mm on a side and has a thermal conductivity of 1 W/m · K. One surface is exposed to a convection
4.86 Small-diameter electrical heating elements dissipating 50 W/m (length normal to the sketch) are used to heat a ceramic plate of thermal conductivity 2 W/m · K. The upper surface of the plate is exposed to ambient air at 30 C with a convection coefcient of 100 W/m 2 · K,
while the lower surface is well insulated.
Ceramic plate Heating element
process with air at 100 C and a convection coefcient of 100 W/m2 · K, while the remaining surfaces are main- tained at 50 C.
Ts = 50°C
(a) Using the Gauss–Seidel method with a grid spac- ing of l1x = 6 mm and l1y = 2 mm, obtain the tem- perature distribution within the plate.
(b) Using the calculated nodal temperatures, sketch four isotherms to illustrate the temperature distri- bution in the plate.
(c) Calculate the heat loss by convection from the plate to the fluid. Compare this value with the ele- ment dissipation rate.
(d) What advantage, if any, is there in not making
l1x = l1y for this situation?
With l1x = l1y = 2 mm, calculate the temperature field within the plate and the rate of heat transfer from the plate. Under no circumstances may the temperature at any location in the plate exceed 400 C. Would this limit be exceeded if the airow were terminated and heat transfer to the air were by natural convection with h = 10 W/m2 · K?
A straight fin of uniform cross section is fabricated from a material of thermal conductivity k = 5 W/m · K, thickness w = 20 mm, and length L = 200 mm. The fin is very long in the direction normal to the page. The base of the fin is maintained at Tb = 200 C, and the tip condition allows for convection (case A of Table 3.4), with h = 500 W/m2 · K and Too = 25 C.
T¥ = 100°C
h = 500 W/m2•K
Consider the long rectangular bar of Problem 4.84 with the prescribed boundary conditions.
(a) Using the finite-element method of FEHT, determine the temperature distribution. Use the View/ Tempera- ture Contours command to represent the isotherms. Identify significant features of the distribution.
(b) Using the View/Heat Flows command, calculate the heat rate per unit length (W/m) from the bar to the airstream.
(c) Explore the effect on the heat rate of increasing the convection coefficient by factors of two and three. Explain why the change in the heat rate is not pro- portional to the change in the convection coefficient.
Consider the long rectangular rod of Problem 4.53, which experiences uniform heat generation while its surfaces are maintained at a fixed temperature.
(a) Using the finite-element method of FEHT, determine the temperature distribution. Use the View/ Tempera- ture Contours command to represent the isotherms. Identify significant features of the distribution.
(b) With the boundary conditions unchanged, what
Tb = 200°C
x T¥, h
w = 20 mm
k = 5 W/m•K
L = 200 mm
heat generation rate will cause the midpoint tem- perature to reach 600 K?
Consider the symmetrical section of the flow channel of Problem 4.57, with the prescribed values of . , k, T ,
Too,o, hi, and ho. Use the finite-element method of FEHT
to obtain the following results.
(a) Determine the temperature distribution in the sym- metrical section, and use the View/Temperature
(a) Assuming one-dimensional heat transfer in the fin, calculate the fin heat rate, q'f (W/m), and the tip temperature TL. Calculate the Biot number for the
fin to determine whether the one-dimensional assumption is valid.
(b) Using the finite-element method of FEHT, perform a two-dimensional analysis on the fin to determine the fin heat rate and tip temperature. Compare your results with those from the one-dimensional, analyti- cal solution of part (a). Use the View/Temperature Contours option to display isotherms, and discuss key features of the corresponding temperature field and heat flow pattern. Hint: In drawing the outline of the fin, take advantage of symmetry. Use a fine mesh near the base and a coarser mesh near the tip. Why?
(c) Validate your FEHT model by comparing predic- tions with the analytical solution for a fin with thermal conductivities of k = 50 W/m · K and 500 W/m · K. Is the one-dimensional heat transfer assumption valid for these conditions?
Contours command to represent the isotherms. Identify significant features of the temperature dis- tribution, including the hottest and coolest regions and the region with the steepest gradients. Describe the heat flow field.
(b) Using the View/Heat Flows command, calculate the heat rate per unit length (W/m) from the outer surface A to the adjacent fluid.
(c) Calculate the heat rate per unit length from the inner fluid to surface B.
(d) Verify that your results are consistent with an over- all energy balance on the channel section.
The hot-film heat flux gage shown schematically may be used to determine the convection coefficient of an adjoining fluid stream by measuring the electric power dissipation per unit area, Pe” (W/m2), and the average surface temperature, Ts,f , of the film. The power dissi- pated in the film is transferred directly to the fluid by convection, as well as by conduction into the substrate.
If substrate conduction is negligible, the gage measure- ments can be used to determine the convection coeffi- cient without application of a correction factor. Your assignment is to perform a two-dimensional, steady- state conduction analysis to estimate the fraction of the power dissipation that is conducted into a 2-mm-thick quartz substrate of width W = 40 mm and thermal con- ductivity k = 1.4 W/m · K. The thin, hot-film gage has a width of w = 4 mm and operates at a uniform power
dissipation of 5000 W/m2. Consider cases for which the
uid temperature is 25 C and the convection coefcient
is 500, 1000, and 2000 W/m2 · K.
Determine the effect of grid spacing on the tem- perature field and heat loss per unit length to the air. Specifically, consider a grid spacing of 25 mm and plot appropriately spaced isotherms on a schematic of the system. Explore the effect of changes in the convection coefficients on the tem- perature field and heat loss.
4.93 Electronic devices dissipating electrical power can be cooled by conduction to a heat sink. The lower surface of the sink is cooled, and the spacing of the devices ws, the width of the device wd, and the thickness L and ther- mal conductivity k of the heat sink material each affect the thermal resistance between the device and the cooled surface. The function of the heat sink is to spread the heat dissipated in the device throughout the sink material.
ws = 48 mm Device, Td = 85°C
wd = 18 mm
L = 24 mm
Use the finite-element method of FEHT to analyze a symmetrical half-section of the gage and the quartz
k = 300 W/m•K
Cooled surface, Ts = 25°C
substrate. Assume that the lower and end surfaces of the substrate are perfectly insulated, while the upper surface experiences convection with the fluid.
(a) Determine the temperature distribution and the con- duction heat rate into the region below the hot film for the three values of h. Calculate the fractions of electric power dissipation represented by these rates. Hint: Use the View/Heat Flow command to find the heat rate across the boundary elements.
(b) Use the View/Temperature Contours command to view the isotherms and heat flow patterns. Describe the heat flow paths, and comment on features of the gage design that influence the paths. What limita- tions on applicability of the gage have been revealed by your analysis?
4.92 Consider the system of Problem 4.54. The interior sur- face is exposed to hot gases at 350 C with a convection coefficient of 100 W/m2 · K, while the exterior surface
experiences convection with air at 25 C and a convec-
(a) Beginning with the shaded symmetrical element, use a coarse (5 X 5) nodal network to estimate the thermal resistance per unit depth between the device
and lower surface of the sink, R't,d-s (m · K/W). How
does this value compare with thermal resistances based on the assumption of one-dimensional con- duction in rectangular domains of (i) width wd and length L and (ii) width ws and length L?
(b) Using nodal networks with grid spacings three and five times smaller than that in part (a), determine the effect of grid size on the precision of the ther- mal resistance calculation.
(c) Using the finer nodal network developed for part (b), determine the effect of device width on the thermal resistance. Specifically, keeping ws and L fixed, find the thermal resistance for values of wd /ws = 0.175, 0.275, 0.375, and 0.475.
4.94 Consider one-dimensional conduction in a plane composite wall. The exposed surfaces of materials A
tion coefficient of 5 W/m2 · K.
and B are maintained at T1
= 600 K and T2
= 300 K,
(a) Using a grid spacing of 75 mm, calculate the temper- ature field within the system and determine the heat loss per unit length by convection from the outer sur- face of the flue to the air. Compare this result with the heat gained by convection from the hot gases to the air.
respectively. Material A, of thickness La = 20 mm,
has a temperature-dependent thermal conductivity of ka = ko [1 + a(T – To)], where ko = 4.4 W/m · K, a = 0.008 K-1, To = 300 K, and T is in kelvins. Material B is of thickness Lb = 5 mm and has a ther- mal conductivity of kb = 1 W/m · K.
ka = ka(T) kb
that of IHT, or the finite-element method of FEHT to obtain the following results.
T1 = 600 K T2 = 300 K
x La La + Lb
(a) Calculate the heat flux through the composite wall by assuming material A to have a uniform thermal conductivity evaluated at the average temperature of the section.
(b) Using a space increment of 1 mm, obtain the finite-
T¥,o = 25°C
ho = 250 W/m2•K
Temperature uniformity of 5°C required
k = 15 W/m•K
T¥,i = 200°C
hi = 500 W/m2•K
difference equations for the internal nodes and calculate the heat flux considering the temperature- dependent thermal conductivity for Material A. If the IHT software is employed, call-up functions from Tools/Finite-Difference Equations may be used to obtain the nodal equations. Compare your result with that obtained in part (a).
(c) As an alternative to the finite-difference method of part (b), use the finite-element method of FEHT to calculate the heat flux, and compare the result with that from part (a). Hint: In the Specify/Material Properties box, properties may be entered as a func- tion of temperature (T ), the space coordinates (x, y), or time (t). See the Help section for more details.
4.95 A platen of thermal conductivity k = 15 W/m · K is heated by flow of a hot fluid through channels of width L = 20 mm, with Too,i = 200°C and hi = 500 W/m2 · K. The upper surface of the platen is used to heat a process fluid at Too,o = 25°C with a convection coefficient of
ho = 250 W/m2 · K. The lower surface of the platen is
insulated. To heat the process fluid uniformly, the tem- perature of the platen’s upper surface must be uniform to within 5°C. Use a finite-difference method, such as
(a) Determine the maximum allowable spacing W between the channel centerlines that will satisfy the specified temperature uniformity requirement.
(b) What is the corresponding heat rate per unit length from a flow channel?
4.96 Consider the cooling arrangement for the very large-scale integration (VLSI) chip of Problem 4.93. Use the finite- element method of FEHT to obtain the following results.
(a) Determine the temperature distribution in the chip- substrate system. Will the maximum temperature exceed 85 C?
(b) Using the FEHT model developed for part (a), determine the volumetric heating rate that yields a maximum temperature of 85 C.
(c) What effect would reducing the substrate thickness have on the maximum operating temperature? For a
volumetric generation rate of . = 107 W/m3, reduce
the thickness of the substrate from 12 to 6 mm, keep- ing all other dimensions unchanged. What is the max- imum system temperature for these conditions? What fraction of the chip power generation is removed by convection directly from the chip surface?
A thermocouple junction, which may be approximated as a sphere, is to be used for tempera- ture measurement in a gas stream. The convection coefficient between the junction surface and the gas is h = 400 W/m2 · K, and the junction thermophysical properties are k = 20 W/m · K, c = 400 J/kg · K, and p = 8500 kg/m3. Determine the junction diameter needed for the thermocouple to have a time constant of 1 s. If the junction is at 25°C and is placed in a gas stream that is at 200°C, how long will it take for the junction to reach 199°C?
Consider the thermocouple and convection conditions of Example 5.1, but now allow for radiation exchange with the walls of a duct that encloses the gas stream. If the duct walls are at 400°C and the emissivity of the thermocouple bead is 0.9, calculate the steady-state tem- perature of the junction. Also, determine the time for the junction temperature to increase from an initial condition of 25°C to a temperature that is within 1°C of its steady-state value.
A 3-mm-thick panel of aluminum alloy (k = 177 W/m · K, c = 875 J/kg · K, and p = 2770 kg/m3) is finished on both sides with an epoxy coating that must be cured at or above Tc = 150°C for at least 5 min. The production line for the curing operation involves two steps: (1) heating in a large oven with air at Too,o = 175°C and a convection coefficient
of ho = 40 W/m2 · K, and (2) cooling in a large chamber with air at Too,c = 25°C and a con- vection coefficient of hc = 10 W/m2 · K. The heating portion of the process is conducted over a time interval te, which exceeds the time tc required to reach 150°C by 5 min (te = tc + 300 s). The coating has an emissivity of e= 0.8, and the temperatures of the oven and chamber walls are 175 and 25°C, respectively. If the panel is placed in the oven at
an initial temperature of 25°C and removed from the chamber at a safe-to-touch tempera- ture of 37°C, what is the total elapsed time for the two-step curing operation?
Air to be supplied to a hospital operating room is first purified by forcing it through a single- stage compressor. As it travels through the compressor, the air temperature initially increases due to compression, then decreases as it is returned to atmospheric pressure. Harmful pathogen particles in the air will also be heated and subsequently cooled, and they will be
destroyed if their maximum temperature exceeds a lethal temperature Td. Consider spherical pathogen particles (D = 10 ,um, p = 900 kg/m3, c = 1100 J/kg · K, and k = 0.2 W/m · K) that are dispersed in unpurified air. During the process, the air temperature may be described by
an expression of the form Too(t) = 125°C – 100°C · cos(27Tt/tp), where tp is the process time
associated with flow through the compressor. If tp = 0.004 s, and the initial and lethal pathogen temperatures are Ti = 25°C and Td = 220°C, respectively, will the pathogens be destroyed? The value of the convection heat transfer coefficient associated with the pathogen particles is h = 4600 W/m2 · K.
Consider a steel pipeline (AISI 1010) that is 1 m in diameter and has a wall thickness of 40 mm. The pipe is heavily insulated on the outside, and, before the initiation of flow, the walls of the pipe are at a uniform temperature of -20°C. With the initiation of flow, hot oil at 60°C is pumped through the pipe, creating a convective condition corresponding to h = 500 W/m2 · K at the inner surface of the pipe.
1. What are the appropriate Biot and Fourier numbers 8 min after the initiation of flow?
2. At t = 8 min, what is the temperature of the exterior pipe surface covered by the insulation?
3. What is the heat flux q”(W/m2) to the pipe from the oil at t = 8 min?
4. How much energy per meter of pipe length has been transferred from the oil to the pipe at t = 8 min?
A new process for treatment of a special material is to be evaluated. The material, a sphere of radius ro = 5 mm, is initially in equilibrium at 400°C in a furnace. It is suddenly removed from the furnace and subjected to a two-step cooling process.
On a hot and sunny day, the concrete deck surrounding a swimming pool is at a tempera- ture of Td = 55°C. A swimmer walks across the dry deck to the pool. The soles of the swimmer’s dry feet are characterized by an Lsf = 3-mm-thick skin/fat layer of thermal con-
ductivity ksf = 0.3 W/m · K. Consider two types of concrete decking; (i) a dense stone mix
and (ii) a lightweight aggregate characterized by density, specific heat, and thermal conduc- tivity of plw = 1495 kg/m3, cp,lw = 880 J/kg · K, and klw = 0.28 W/m · K, respectively. The density and specific heat of the skin/fat layer may be approximated to be those of liquid water, and the skin/fat layer is at an initial temperature of Tsf,i = 37°C. What is the tempera- ture of the bottom of the swimmer’s feet after an elapsed time of t = 1 s?
Derive an expression for the ratio of the total energy transferred from the isothermal surfaces of a plane wall to the interior of the plane wall, Q/Qo, that is valid for Fo 0.2. Express your results in terms of the Fourier number Fo.
A proposed cancer treatment utilizes small, composite nanoshells whose size and composi- tion are carefully specified so that the particles efficiently absorb laser irradiation at particu- lar wavelengths . Prior to treatment, antibodies are attached to the nanoscale particles. The doped particles are then injected into the patient’s bloodstream and are distributed throughout the body. The antibodies are attracted to malignant sites, and therefore carry and adhere the nanoshells only to cancerous tissue. After the particles have come to rest within the tumor, a laser beam penetrates through the tissue between the skin and the can- cer, is absorbed by the nanoshells, and, in turn, heats and destroys the cancerous tissues.
Consider an approximately spherical tumor of diameter Dt = 3 mm that is uniformly infiltrated with nanoshells that are highly absorptive of incident radiation from a laser located outside the patient’s body.
1. Estimate the heat transfer rate from the tumor to the surrounding healthy tissue for a steady-state treatment temperature of Tt,ss = 55°C at the surface of the tumor. The ther- mal conductivity of healthy tissue is approximately k = 0.5 W/m · K, and the body temperature is Tb = 37°C.
2. Find the laser power necessary to sustain the tumor surface temperature at Tt,ss = 55°C if the tumor is located d = 20 mm beneath the surface of the skin, and the laser heat flux decays exponentially, ql”(x) = ql”,o(1 – p) e-Kx, between the surface of the body and the tumor. In the preceding expression, ql”,o is the laser heat flux outside the body, p = 0.05 is the reflectivity of the skin surface, and K = 0.02 mm-1 is the extinction coefcient of the tissue between the tumor and the surface of the skin. The laser beam has a diameter of Dl = 5 mm.
3. Neglecting heat transfer to the surrounding tissue, estimate the time at which the tumor temperature is within 3°C of Tt,ss = 55°C for the laser power found in part 2. Assume the tissue’s density and specific heat are that of water.
4. Neglecting the thermal mass of the tumor but accounting for heat transfer to the sur- rounding tissue, estimate the time needed for the surface temperature of the tumor to reach Tt = 52°C.
A nanostructured dielectric material has been fabricated, and the following method is used to measure its thermal conductivity. A long metal strip 3000 angstroms thick, w = 100 ,um wide, and L = 3.5 mm long is deposited by a photolithography technique on the top surface of a d = 300-,um-thick sample of the new material. The strip is heated periodically by an electric current supplied through two connector pads. The heating rate is qs(t) = Llqs + Llqs sin(wt), where Llqs is 3.5 mW. The instantaneous, spatially averaged temperature of the metal strip is found experimentally by measuring the time variation of its electrical resis- tance, R(t) = E(t)/I(t), and by knowing how the electrical resistance of the metal varies with temperature. The measured temperature of the metal strip is periodic; it has an ampli- tude of LlT = 1.37 K at a relatively low heating frequency of w = 27T rad/s and 0.71 K at a frequency of 2007T rad/s. Determine the thermal conductivity of the nanostructured dielec-
tric material. The density and specific heats of the conventional version of the material are 3100 kg/m3 and 820 J/kg · K, respectively.
A fuel element of a nuclear reactor is in the shape of a plane wall of thickness 2L = 20 mm and is convectively cooled at both surfaces, with h = 1100 W/m2 · K and Too = 250°C. At normal operating power, heat is generated uniformly within the element at a volumetric rate of q1 = 107 W/m3. A departure from the steady-state conditions associated with normal operation will occur if there is a change in the generation rate. Consider a sudden change to q2 = 2 X 107 W/m3, and use the explicit finite-difference method to determine the fuel element temperature distribution after 1.5 s. The fuel element thermal properties are k = 30 W/m · K and a = 5 X 10-6 m2/s.
A thick slab of copper initially at a uniform temperature of 20°C is suddenly exposed to radiation at one surface such that the net heat flux is maintained at a constant value of 3 X 105 W/m2. Using the explicit and implicit finite-difference techniques with a space increment of Llx = 75 mm, determine the temperature at the irradiated surface and at an interior point that is 150 mm from the surface after 2 min have elapsed. Compare the results with those obtained from an appropriate analytical solution.
5.1 Consider a thin electrical heater attached to a plate and backed by insulation. Initially, the heater and plate are at the temperature of the ambient air, Too. Suddenly, the power to the heater is activated, yielding a constant heat flux q”o (W/m2) at the inner surface of the plate.
is at a uniform temperature corresponding to that of the airstream. Suddenly, a radiation heat source is switched on, applying a uniform flux q”o to the outer surface.
q”o for t > 0
x = L
(a) Sketch and label, on T x coordinates, the tempera- ture distributions: initial, steady-state, and at two intermediate times.
(b) Sketch the heat flux at the outer surface q”x (L, t) as a function of time.
(a) Sketch and label, on T x coordinates, the tempera- ture distributions: initial, steady-state, and at two intermediate times.
(b) Sketch the heat flux at the outer surface q”x(L, t) as a function of time.
5.2 The inner surface of a plane wall is insulated while the outer surface is exposed to an airstream at Too. The wall
5.3 A microwave oven operates on the principle that appli- cation of a high-frequency field causes electrically polarized molecules in food to oscillate. The net effect is a nearly uniform generation of thermal energy within the food. Consider the process of cooking a slab of beef of thickness 2L in a microwave oven and compare it with cooking in a conventional oven, where each side of
the slab is heated by radiation. In each case the meat is to be heated from 0°C to a minimum temperature of 90°C. Base your comparison on a sketch of the tempera- ture distribution at selected times for each of the cook- ing processes. In particular, consider the time t0 at which heating is initiated, a time t1 during the heating process, the time t2 corresponding to the conclusion of heating, and a time t3 well into the subsequent cooling process.
5.4 A plate of thickness 2L, surface area As, mass M, and specific heat cp, initially at a uniform temperature Ti, is suddenly heated on both surfaces by a convection process (Too, h) for a period of time to, following which the plate is insulated. Assume that the midplane tem- perature does not reach Too within this period of time.
(a) Assuming Bi � 1 for the heating process, sketch and label, on T x coordinates, the following temperature distributions: initial, steady-state (t l oo), T(x, to), and at two intermediate times between t = to and t l oo.
(b) Sketch and label, on T t coordinates, the midplane and exposed surface temperature distributions.
(c) Repeat parts (a) and (b) assuming Bi � 1 for the plate.
(d) Derive an expression for the steady-state tempera- ture T(x, oo) = Tf , leaving your result in terms of plate parameters (M, cp), thermal conditions (Ti, Too, h), the surface temperature T(L, t), and the heating time to.
5.5 For each of the following cases, determine an appropri- ate characteristic length Lc and the corresponding Biot number Bi that is associated with the transient thermal response of the solid object. State whether the lumped capacitance approximation is valid. If temperature infor- mation is not provided, evaluate properties at T = 300 K.
(a) A toroidal shape of diameter D = 50 mm and cross-sectional area Ac = 5 mm2 is of thermal conductivity k = 2.3 W/m · K. The surface of the torus is exposed to a coolant corresponding to a convection coefficient of h = 50 W/m2 · K.
(b) A long, hot AISI 304 stainless steel bar of rectan- gular cross section has dimensions w = 3 mm, W = 5 mm, and L = 100 mm. The bar is subjected to a coolant that provides a heat transfer coefficient of h = 15 W/m2 · K at all exposed surfaces.
(c) A long extruded aluminum (Alloy 2024) tube of inner and outer dimensions w = 20 mm and W = 24 mm, respectively, is suddenly submerged in water, result- ing in a convection coefficient of h = 37 W/m2 · K at the four exterior tube surfaces. The tube is plugged at both ends, trapping stagnant air inside the tube.
(d) An L = 300-mm-long solid stainless steel rod of diam- eter D = 13 mm and mass M = 0.328 kg is exposed to a convection coefficient of h = 30 W/m2 · K.
(e) A solid sphere of diameter D = 12 mm and thermal conductivity k = 120 W/m · K is sus- pended in a large vacuum oven with internal wall temperatures of Tsur = 20°C. The initial sphere temperature is Ti = 100°C, and its emissivity is s = 0.73.
(f) A long cylindrical rod of diameter D = 20 mm, den- sity p = 2300 kg/m3, specific heat cp = 1750 J/kg · K, and thermal conductivity k = 16 W/m · K is suddenly exposed to convective conditions with Too = 20°C. The rod is initially at a uniform temperature of Ti = 200°C and reaches a spatially averaged tempera- ture of T = 100°C at t = 225 s.
(g) Repeat part (f) but now consider a rod diameter of
D = 200 mm.
5.6 Steel balls 12 mm in diameter are annealed by heating to
h = 55 W/m2·K
1150 K and then slowly cooling to 400 K in an air envi- ronment for which Too = 325 K and h = 20 W/m2 · K. Assuming the properties of the steel to be k = 40 W/m · K, p = 7800 kg/m3, and c = 600 J/kg · K, estimate the time
required for the cooling process.
5.7 Consider the steel balls of Problem 5.6, except now the air temperature increases with time as Too(t) = 325 K + at, where a = 0.1875 K/s.
2L = 1.2 mm
T¥ = 300°C
(a) Sketch the ball temperature versus time for 0
t 1 h. Also show the ambient temperature, Too, in
your graph. Explain special features of the ball
(b) Find an expression for the ball temperature as a function of time T(t), and plot the ball temperature for 0 t 1 h. Was your sketch correct?
5.8 The heat transfer coefficient for air flowing over a sphere is to be determined by observing the temperature–time history of a sphere fabricated from pure copper. The sphere, which is 12.7 mm in diameter, is at 66°C before it is inserted into an airstream having a temperature of 27°C. A thermocouple on the outer surface of the sphere indicates 55°C 69 s after the sphere is inserted into the airstream. Assume and then justify that the sphere behaves as a spacewise isothermal object and calculate the heat transfer coefficient.
5.9 A solid steel sphere (AISI 1010), 300 mm in diameter, is coated with a dielectric material layer of thickness 2 mm and thermal conductivity 0.04 W/m · K. The coated sphere is initially at a uniform temperature of 500°C and is suddenly quenched in a large oil bath for
The base plate of an iron has a thickness of L = 7 mm and is made from an aluminum alloy (p = 2800 kg/m3, c = 900 J/kg · K, k = 180 W/m · K, s = 0.80). An electric resistance heater is attached to the inner surface of the plate, while the outer surface is exposed to ambient air and large surroundings at Too = Tsur = 25°C. The areas of both the inner and outer surfaces are As = 0.040 m2.
If an approximately uniform heat flux of q”h = 1.25 X
which Too = 100°C and h = 3300 W/m2 · K. Estimate
is applied to the inner surface of the base
the time required for the coated sphere temperature to reach 140°C. Hint: Neglect the effect of energy storage in the dielectric material, since its thermal capacitance (pcV ) is small compared to that of the steel sphere.
5.10 A flaked cereal is of thickness 2L = 1.2 mm. The density, specific heat, and thermal conductivity of the flake are p = 700 kg/m3, cp = 2400 J/kg · K, and k = 0.34 W/m · K, respectively. The product is to be baked by increasing its temperature from Ti = 20°C to Tf = 220°C in a convec- tion oven, through which the product is carried on a con- veyor. If the oven is Lo = 3 m long and the convection heat transfer coefficient at the product surface and oven air temperature are h = 55 W/m2 · K and Too = 300°C, respectively, determine the required conveyor velocity,
V. An engineer suggests that if the flake thickness is reduced to 2L = 1.0 mm the conveyor velocity can be increased, resulting in higher productivity. Determine the required conveyor velocity for the thinner flake.
plate and the convection coefficient at the outer surface is h = 10 W/m2 · K, estimate the time required for the
plate to reach a temperature of 135°C. Hint: Numerical integration is suggested in order to solve the problem.
5.12 Thermal energy storage systems commonly involve a packed bed of solid spheres, through which a hot gas flows if the system is being charged, or a cold gas if it is being discharged. In a charging process, heat transfer from the hot gas increases thermal energy stored within the colder spheres; during discharge, the stored energy decreases as heat is transferred from the warmer spheres to the cooler gas.
(r, c, k, Ti) D
Consider a packed bed of 75-mm-diameter alumi- num spheres (p = 2700 kg/m3, c = 950 J/kg · K, k = 240 W/m · K) and a charging process for which gas enters the storage unit at a temperature of Tg,i = 300°C.
for the sheet to reach a temperature of T = 102°C. Plot the copper temperature versus time for 0 t 0.5 s. On the same graph, plot the copper temperature history assuming the heat transfer coefficient is constant, eval-
If the initial temperature of the spheres is Ti = 25°C
and the convection coefficient is h = 75 W/m2 · K, how long does it take a sphere near the inlet of the system to
uated at the average copper temperature Assume lumped capacitance behavior.
T = 100°C.
accumulate 90% of the maximum possible thermal energy? What is the corresponding temperature at the center of the sphere? Is there any advantage to using copper instead of aluminum?
5.13 A tool used for fabricating semiconductor devices consists of a chuck (thick metallic, cylindrical disk) onto which a very thin silicon wafer (p = 2700 kg/m3, c = 875 J/kg · K, k = 177 W/m · K) is placed by a robotic arm. Once in position, an electric field in the chuck is energized, creating an electrostatic force that
holds the wafer firmly to the chuck. To ensure a repro- ducible thermal contact resistance between the chuck and the wafer from cycle to cycle, pressurized helium gas is introduced at the center of the chuck and flows (very slowly) radially outward between the asperities of the interface region.
5.15 Carbon steel (AISI 1010) shafts of 0.1-m diameter are
heat treated in a gas-fired furnace whose gases are at 1200 K and provide a convection coefficient of 100 W/m2 · K. If the shafts enter the furnace at 300 K, how long must they remain in the furnace to achieve a centerline temperature of 800 K?
5.16 A thermal energy storage unit consists of a large rectan- gular channel, which is well insulated on its outer surface and encloses alternating layers of the storage material and the flow passage.
Tw(0) = Tw,i = 100°C
w = 0.758 mm
Interface region, greatly exaggerated
Helium gas purge
Chuck, Tc = 23°C
Each layer of the storage material is an aluminum slab of width W = 0.05 m, which is at an initial temperature
An experiment has been performed under conditions for which the wafer, initially at a uniform temperature Tw,i = 100°C, is suddenly placed on the chuck, which is at a uniform and constant temperature Tc = 23°C. With the wafer in place, the electrostatic force and the helium gas flow are applied. After 15 s, the temperature of the wafer is determined to be 33°C. What is the thermal contact
resistance R”t,c (m2 · K/W) between the wafer and chuck? Will the value of R”t,c increase, decrease, or remain the same if air, instead of helium, is used as the purge gas?
5.14 A copper sheet of thickness 2L = 2 mm has an initial temperature of Ti = 118°C. It is suddenly quenched in liquid water, resulting in boiling at its two surfaces. For boiling, Newton’s law of cooling is expressed as q'' = h(Ts – Tsat), where Ts is the solid surface tem- perature and Tsat is the saturation temperature of the fluid (in this case Tsat = 100°C). The convection heat transfer coefficient may be expressed as h = 1010 W/m2 · K 3(T – Tsat)2. Determine the time needed
of 25°C. Consider conditions for which the storage unit is charged by passing a hot gas through the passages, with the gas temperature and the convection coefficient assumed to have constant values of Too = 600°C and
h = 100 W/m2 · K throughout the channel. How long
will it take to achieve 75% of the maximum possible energy storage? What is the temperature of the alu- minum at this time?
5.17 Small spherical particles of diameter D = 50 ,um contain a fluorescent material that, when irradiated with white light, emits at a wavelength corresponding to the mater- ial’s temperature. Hence the color of the particle varies with its temperature. Because the small particles are neu- trally buoyant in liquid water, a researcher wishes to use them to measure instantaneous local water temperatures in a turbulent flow by observing their emitted color. If the particles are characterized by a density, specific heat, and thermal conductivity of p = 999 kg/m3,
k = 1.2 W/m · K, and cp = 1200 J/kg · K, respectively,
determine the time constant of the particles. Hint: Since the particles travel with the flow, heat transfer between the particle and the fluid occurs by conduction. Assume lumped capacitance behavior.
5.18 A spherical vessel used as a reactor for producing pharmaceuticals has a 5-mm-thick stainless steel wall (k = 17 W/m · K) and an inner diameter of Di = 1.0 m. During production, the vessel is filled with reactants for which p = 1100 kg/m3 and c = 2400 J/kg · K, while exothermic reactions release energy at a volumetric rate
of q = 104 W/m3. As first approximations, the reactants may be assumed to be well stirred and the thermal capacitance of the vessel may be neglected.
(a) The exterior surface of the vessel is exposed to ambient air (Too = 25°C) for which a convection
If the chemical is to be heated from 300 to 450 K in 60 min, what is the required length L of the submerged tubing?
5.20 An electronic device, such as a power transistor mounted on a finned heat sink, can be modeled as a spatially isothermal object with internal heat generation and an external convection resistance.
(a) Consider such a system of mass M, specific heat c, and surface area As, which is initially in equilib- rium with the environment at Too. Suddenly, the electronic device is energized such that a constant heat generation Eg (W) occurs. Show that the tem- perature response of the device is
() = exp (- t
coefficient of h = 6 W/m2 · K may be assumed. If
the initial temperature of the reactants is 25°C, what is the temperature of the reactants after 5 h of process time? What is the corresponding tempera- ture at the outer surface of the vessel?
Explore the effect of varying the convection coeffi- cient on transient thermal conditions within the reactor.
5.19 Batch processes are often used in chemical and pharma- ceutical operations to achieve a desired chemical composition for the final product and typically involve a transient heating operation to take the product from room temperature to the desired process temperature. Consider a situation for which a chemical of density p = l200 kg/m3 and specific heat c = 2200 J/kg · K occu- pies a volume of V = 2.25 m3 in an insulated vessel. The chemical is to be heated from room temperature, Ti = 300 K, to a process temperature of T = 450 K by passing saturated steam at Th = 500 K through a coiled, thin-walled, 20-mm-diameter tube in the vessel. Steam condensation within the tube maintains an interior con- vection coefficient of hi = 10,000 W/m2 · K, while the highly agitated liquid in the stirred vessel maintains an outside convection coefficient of ho = 2000 W/m2 · K.
where () = T – T(oo) and T(oo) is the steady-state temperature corresponding to t l oo; ()i = Ti – T(oo); Ti = initial temperature of device; R = thermal resistance 1/hAs; and C = thermal capacitance Mc.
(b) An electronic device, which generates 60 W of heat, is mounted on an aluminum heat sink weigh- ing 0.31 kg and reaches a temperature of 100°C in ambient air at 20°C under steady-state conditions. If the device is initially at 20°C, what temperature will it reach 5 min after the power is switched on?
5.21 Molecular electronics is an emerging field associated with computing and data storage utilizing energy transfer at the molecular scale. At this scale, thermal energy is associ- ated exclusively with the vibration of molecular chains. The primary resistance to energy transfer in these pro- posed devices is the contact resistance at metal-molecule interfaces. To measure the contact resistance, individual molecules are self-assembled in a regular pattern onto a very thin gold substrate. The substrate is suddenly heated by a short pulse of laser irradiation, simultaneously trans- ferring thermal energy to the molecules. The molecules vibrate rapidly in their “hot” state, and their vibrational intensity can be measured by detecting the randomness of the electric field produced by the molecule tips, as indi- cated by the dashed, circular lines in the schematic.
Slowly vibrating molecules
Rapidly vibrating molecules
Randomly vibrating electric field
Initial, cool state Hot state
Molecules that are of density p = 180 kg/m3 and spe- cific heat cp = 3000 J/kg · K have an initial, relaxed length of L = 2 nm. The intensity of the molecular vibration increases exponentially from an initial value of Ii to a steady-state value of Iss > Ii with the time constant associated with the exponential response being TI = 5 ps. Assuming the intensity of the molecular vibration represents temperature on the molecular scale and that each molecule can be viewed as a cylinder of initial length L and cross-sectional area Ac, determine the thermal contact resistance, Rt”,c, at the metal–molecule interface.
5.22 A plane wall of a furnace is fabricated from plain carbon steel (k = 60 W/m · K, p = 7850 kg/m3, c = 430 J/kg · K) and is of thickness L = 10 mm. To protect it from the
corrosive effects of the furnace combustion gases, one surface of the wall is coated with a thin ceramic film that, for a unit surface area, has a thermal resistance of
R”t,f = 0.01 m2 · K/W. The opposite surface is well insu-
lated from the surroundings.
R”t, f Carbon steel,
r, c, k, Ti
At furnace start-up the wall is at an initial temperature of Ti = 300 K, and combustion gases at Too = 1300 K enter the furnace, providing a convection coefficient of h = 25 W/m2 · K at the ceramic film. Assuming the film to have negligible thermal capacitance, how long will it take for the inner surface of the steel to achieve a temperature of Ts,i = 1200 K? What is the tempera- ture Ts,o of the exposed surface of the ceramic film at this time?
A steel strip of thickness o = 12 mm is annealed by passing it through a large furnace whose walls are maintained at a temperature Tw corresponding to that of combustion gases flowing through the furnace (Tw = Too). The strip, whose density, specific heat, ther- mal conductivity, and emissivity are p = 7900 kg/m3, cp = 640 J/kg · K, k = 30 W/m · K, and s = 0.7, respec- tively, is to be heated from 300°C to 600°C.
(a) For a uniform convection coefficient of h = 100 W/m2 · K and Tw = Too = 700°C, determine the time required to heat the strip. If the strip is moving at 0.5 m/s, how long must the furnace be?
(b) The annealing process may be accelerated (the strip speed increased) by increasing the environmental temperatures. For the furnace length obtained in part (a), determine the strip speed for Tw = Too = 850°C and Tw = Too = 1000°C. For each set of envi- ronmental temperatures (700, 850, and 1000°C), plot the strip temperature as a function of time over the range 25°C T 600°C. Over this range, also
plot the radiation heat transfer coefficient, hr, as a function of time.
5.24 In a material processing experiment conducted aboard the space shuttle, a coated niobium sphere of 10-mm diameter is removed from a furnace at 900°C and cooled to a temperature of 300°C. Although properties of the niobium vary over this temperature range, con- stant values may be assumed to a reasonable approxi-
mation, with p = 8600 kg/m3, c = 290 J/kg · K, and k =
63 W/m · K.
(a) If cooling is implemented in a large evacuated chamber whose walls are at 25°C, determine the time required to reach the final temperature if the coating is polished and has an emissivity of e= 0.1. How long would it take if the coating is oxidized and s = 0.6?
(b) To reduce the time required for cooling, considera- tion is given to immersion of the sphere in an inert gas stream for which Too = 25°C and h = 200 W/m2 · K. Neglecting radiation, what is the time required for cooling?
(c) Considering the effect of both radiation and con- vection, what is the time required for cooling if h = 200 W/m2 · K and s = 0.6? Explore the effect on the cooling time of independently varying h and s.
5.25 Plasma spray-coating processes are often used to pro- vide surface protection for materials exposed to hostile environments, which induce degradation through fac- tors such as wear, corrosion, or outright thermal failure. Ceramic coatings are commonly used for this purpose. By injecting ceramic powder through the nozzle (anode) of a plasma torch, the particles are entrained by the plasma jet, within which they are then accelerated and heated.
5.26 The plasma spray-coating process of Problem 5.25 can be used to produce nanostructured ceramic coatings. Such coatings are characterized by low thermal conduc- tivity, which is desirable in applications where the coat- ing serves to protect the substrate from hot gases such as in a gas turbine engine. One method to produce a nanostructured coating involves spraying spherical particles, each of which is composed of agglomerated Al2O3 nanoscale granules. To form the coating, parti-
cles of diameter Dp = 50 ,um must be partially molten
when they strike the surface, with the liquid Al2O3 providing a means to adhere the ceramic material to
the surface, and the unmelted Al2O3 providing the
many grain boundaries that give the coating its low thermal conductivity. The boundaries between individ- ual granules scatter phonons and reduce the thermal
conductivity of the ceramic particle to kp
= 5 W/m · K.
Plasma jet with entrained ceramic particles (T¥, h)
Ceramic coating Substrate
The density of the porous particle is reduced to
p = 3800 kg/m3. All other properties and conditions are as specified in Problem 5.25.
(a) Determine the time-in-ight corresponding to 30% of the particle mass being melted.
(b) Determine the time-in-ight corresponding to the particle being 70% melted.
(c) If the particle is traveling at a velocity V = 35 m/s, determine the standoff distances between the noz- zle and the substrate associated with your answers in parts (a) and (b).
During their time-in-ight, the ceramic particles must be heated to their melting point and experience com- plete conversion to the liquid state. The coating is formed as the molten droplets impinge (splat) on the substrate material and experience rapid solidification. Consider conditions for which spherical alumina (Al2O3) particles of diameter Dp = 50 ,um, density pp =
3970 kg/m3, thermal conductivity kp = 10.5 W/m · K,
and specific heat cp = 1560 J/kg · K are injected into an arc plasma, which is at Too = 10,000 K and provides a coefficient of h = 30,000 W/m2 · K for convective heat- ing of the particles. The melting point and latent heat of fusion of alumina are Tmp = 2318 K and hsf = 3577 kJ/kg, respectively.
(a) Neglecting radiation, obtain an expression for the time-in-flight, ti-f, required to heat a particle from its initial temperature Ti to its melting point Tmp, and, once at the melting point, for the particle to experience complete melting. Evaluate ti-f for Ti = 300 K and the prescribed heating conditions.
(b) Assuming alumina to have an emissivity of sp = 0.4 and the particles to exchange radiation with large surroundings at Tsur = 300 K, assess the validity of neglecting radiation.
5.27 A chip that is of length L = 5 mm on a side and thick- ness t = 1 mm is encased in a ceramic substrate, and its exposed surface is convectively cooled by a dielectric liquid for which h = 150 W/m2 · K and Too = 20°C.
In the off-mode the chip is in thermal equilibrium with the coolant (Ti = Too). When the chip is energized, how- ever, its temperature increases until a new steady state is established. For purposes of analysis, the energized chip is characterized by uniform volumetric heating
with q = 9 X 106 W/m3. Assuming an infinite contact resistance between the chip and substrate and negligible conduction resistance within the chip, determine the steady-state chip temperature Tf. Following activation
of the chip, how long does it take to come within 1°C of
this temperature? The chip density and specific heat are
p = 2000 kg/m3 and c = 700 J/kg · K, respectively.
5.28 Consider the conditions of Problem 5.27. In addition to treating heat transfer by convection directly from the chip to the coolant, a more realistic analysis would account for indirect transfer from the chip to the sub- strate and then from the substrate to the coolant. The total thermal resistance associated with this indirect route includes contributions due to the chip–substrate interface (a contact resistance), multidimensional conduction in the substrate, and convection from the surface of the substrate to the coolant. If this total ther- mal resistance is Rt = 200 K/W, what is the steady- state chip temperature Tf? Following activation of the
chip, how long does it take to come within 1°C of this
5.29 A long wire of diameter D = 1 mm is submerged in an oil bath of temperature Too = 25°C. The wire has an electrical resistance per unit length of Re' = 0.01 D/m. If a current of I = 100 A flows through the wire and the convection coefficient is h = 500 W/m2 · K, what is the steady-state temperature of the wire? From the time the current is applied, how long does it take for the wire to reach a temperature that is within 1°C of the steady- state value? The properties of the wire are p = 8000 kg/m3, c = 500 J/kg · K, and k = 20 W/m · K.
5.30 Consider the system of Problem 5.1 where the tempera- ture of the plate is spacewise isothermal during the transient process.
(a) Obtain an expression for the temperature of the plate as a function of time T(t) in terms of qo”, Too, h, L, and the plate properties p and c.
(b) Determine the thermal time constant and the steady-state temperature for a 12-mm-thick plate of pure copper when Too = 27°C, h = 50 W/m2 · K,
and qo” = 5000 W/m2. Estimate the time required to reach steady-state conditions.
For the conditions of part (b), as well as for h = 100 and 200 W/m2 · K, compute and plot the corresponding temperature histories of the plate for 0 t 2500 s.
Shape memory alloys (SMAs) are metals that undergo a change in crystalline structure within a relatively nar- row temperature range. A phase transformation from martensite to austenite can induce relatively large changes in the overall dimensions of the SMA. Hence, SMAs can be employed as mechanical actuators. Con- sider an SMA rod that is initially Di = 2 mm in diame-
ter, Li = 40 mm long, and at a uniform temperature of
Ti = 320 K. The specific heat of the SMA varies signif- icantly with changes in the crystalline phase, hence c
varies with the temperature of the material. For a particular SMA, this relationship is well described by c = 500 J/kg · K + 3630 J/kg · K X e(-0.808 K X|T-336K|). The density and thermal conductivity of the SMA material are p = 8900 kg/m3 and k = 23 W/m · K, respectively.
The SMA rod is exposed to a hot gas character- ized by Too = 350 K, h = 250 W/m2 · K. Plot the rod temperature versus time for 0 t 60 s for the cases of variable and constant (c = 500 J/kg · K) specific heats. Determine the time needed for the rod temperature to experience 95% of its maximum temperature change. Hint: Neglect the change in the dimensions of the SMA rod when calculating the thermal response of the rod.
5.32 Before being injected into a furnace, pulverized coal is preheated by passing it through a cylindrical tube whose surface is maintained at Tsur = 1000°C. The coal pellets are suspended in an airflow and are known to move with a speed of 3 m/s. If the pellets may be approximated as spheres of 1-mm diameter and it may be assumed that they are heated by radiation transfer from the tube surface, how long must the tube be to heat coal entering at 25°C to a temperature of 600°C? Is the use of the lumped capacitance method justified?
5.33 As noted in Problem 5.3, microwave ovens operate by rapidly aligning and reversing water molecules within the food, resulting in volumetric energy generation and, in turn, cooking of the food. When the food is initially frozen, however, the water molecules do not readily oscillate in response to the microwaves, and the volu- metric generation rates are between one and two orders of magnitude lower than if the water were in liquid form. (Microwave power that is not absorbed in the food is reflected back to the microwave generator, where it must be dissipated in the form of heat to pre- vent damage to the generator.)
(a) Consider a frozen, 1-kg spherical piece of ground beef at an initial temperature of Ti = -20°C placed in a microwave oven with Too = 30°C and h =
15 W/m2 · K. Determine how long it will take the
beef to reach a uniform temperature of T = 0°C, with all the water in the form of ice. Assume the properties of the beef are the same as ice, and assume 3% of the oven power (P = 1 kW total) is absorbed in the food.
(b) After all the ice is converted to liquid, determine how long it will take to heat the beef to Tf = 80°C if 95% of the oven power is absorbed in the food. Assume the properties of the beef are the same as liquid water.
(c) When thawing food in microwave ovens, one may observe that some of the food may still be frozen while other parts of the food are overcooked.
Explain why this occurs. Explain why most microwave ovens have thaw cycles that are associ- ated with very low oven powers.
5.34 A metal sphere of diameter D, which is at a uniform temperature Ti, is suddenly removed from a furnace and suspended from a fine wire in a large room with air at a uniform temperature Too and the surrounding walls at a temperature Tsur.
(a) Neglecting heat transfer by radiation, obtain an expression for the time required to cool the sphere
purpose is termed a Liquid Droplet Radiator (LDR). The heat is first transferred to a high vacuum oil, which is then injected into outer space as a stream of small droplets. The stream is allowed to traverse a distance L, over which it cools by radiating energy to outer space at absolute zero temperature. The droplets are then col- lected and routed back to the space station.
Tsur = 0 K
to some temperature T.
(b) Neglecting heat transfer by convection, obtain an expression for the time required to cool the sphere to the temperature T.
(c) How would you go about determining the time
required for the sphere to cool to the temperature T L
if both convection and radiation are of the same
order of magnitude?
Consider an anodized aluminum sphere (s = 0.75)
Cold oil return
50 mm in diameter, which is at an initial tempera- ture of Ti = 800 K. Both the air and surroundings are at 300 K, and the convection coefficient is 10 W/m2 · K. For the conditions of parts (a), (b),
and (c), determine the time required for the sphere to cool to 400 K. Plot the corresponding tempera- ture histories. Repeat the calculations for a polished aluminum sphere (e= 0.1).
A horizontal structure consists of an LA = 10-mm-thick layer of copper and an LB = 10-mm-thick layer of alu- minum. The bottom surface of the composite structure receives a heat flux of q'' = 100 kW/m2, while the top surface is exposed to convective conditions character- ized by h = 40 W/m2 · K, Too = 25°C. The initial tem- perature of both materials is Ti,A = Ti,B = 25°C, and a contact resistance of Rt”,c = 400 X 10-6 m2 · K/W exists at the interface between the two materials.
(a) Determine the times at which the copper and alu- minum each reach a temperature of Tf = 90°C. The copper layer is on the bottom.
(b) Repeat part (a) with the copper layer on the top.
Hint: Modify Equation 5.15 to include a term associ- ated with heat transfer across the contact resistance. Apply the modified form of Equation 5.15 to each of the two slabs. See Comment 3 of Example 5.2.
5.36 As permanent space stations increase in size, there is an attendant increase in the amount of electrical power they dissipate. To keep station compartment tempera- tures from exceeding prescribed limits, it is necessary to transfer the dissipated heat to space. A novel heat rejection scheme that has been proposed for this
Consider conditions for which droplets of emissivity s = 0.95 and diameter D = 0.5 mm are injected at a tem- perature of Ti = 500 K and a velocity of V = 0.1 m/s.
Properties of the oil are p = 885 kg/m3, c = 1900 J/kg · K,
and k = 0.145 W/m · K. Assuming each drop to radiate to deep space at Tsur = 0 K, determine the distance L required for the droplets to impact the collector at a final temperature of Tf = 300 K. What is the amount of thermal energy rejected by each droplet?
5.37 Thin film coatings characterized by high resistance to abrasion and fracture may be formed by using microscale composite particles in a plasma spraying process. A spherical particle typically consists of a ceramic core, such as tungsten carbide (WC), and a metallic shell, such as cobalt (Co). The ceramic provides the thin film coat- ing with its desired hardness at elevated temperatures, while the metal serves to coalesce the particles on the coated surface and to inhibit crack formation. In the plasma spraying process, the particles are injected into a plasma gas jet that heats them to a temperature above the melting point of the metallic casing and melts the casing before the particles impact the surface.
Consider spherical particles comprised of a WC core of diameter Di = 16 ,um, which is encased in a Co shell of outer diameter Do = 20 ,um. If the particles flow in a plasma gas at Too = 10,000 K and the coeffi- cient associated with convection from the gas to the particles is h = 20,000 W/m2 · K, how long does it take to heat the particles from an initial temperature of Ti = 300 K to the melting point of cobalt, Tmp = 1770 K? The density and specific heat of WC (the core of the particle) are pc = 16,000 kg/m3 and
cc = 300 J/kg · K, while the corresponding values for Co (the outer shell) are ps = 8900 kg/m3 and cs = 750 J/kg · K. Once having reached the melting point, how much additional time is required to com- pletely melt the cobalt if its latent heat of fusion is hsf = 2.59 X 105 J/kg? You may use the lumped capac- itance method of analysis and neglect radiation exchange between the particle and its surroundings.
A long, highly polished aluminum rod of diameter D = 35 mm is hung horizontally in a large room. The initial rod temperature is Ti = 90°C, and the room air is Too = 20°C. At time t1 = 1250 s, the rod temperature is T1 = 65°C, and, at time t2 = 6700 s, the rod tempera- ture is T2 = 30°C. Determine the values of the con- stants C and n that appear in Equation 5.26. Plot the rod temperature versus time for 0 t 10,000 s. On the same graph, plot the rod temperature versus time for a constant value of the convection heat transfer coeffi- cient, evaluated at a rod temperature of T = (Ti + Too)/2. For all cases, evaluate properties at T = (Ti + Too)/2.
5.39 Thermal stress testing is a common procedure used to assess the reliability of an electronic package. Typi- cally, thermal stresses are induced in soldered or wired connections to reveal mechanisms that could cause failure and must therefore be corrected before the prod- uct is released. As an example of the procedure, consider an array of silicon chips (pch = 2300 kg/m3,
cch = 710 J/kg · K) joined to an alumina substrate
(psb = 4000 kg/m3, csb = 770 J/kg · K) by solder balls (psd = 11,000 kg/m3, csd = 130 J/kg · K). Each chip of width Lch and thickness tch is joined to a unit substrate section of width Lsb and thickness tsb by solder balls of diameter D.
involving the same convection coefficient h. Assuming no reduction in surface area due to con- tact between a solder ball and the chip or substrate, obtain expressions for the thermal time constant of each component. Heat transfer is to all surfaces of a chip, but to only the top surface of the substrate. Evaluate the three time constants for Lch = 15 mm,
tch = 2 mm, Lsb = 25 mm, tsb = 10 mm, D = 2 mm, and a value of h = 50 W/m2 · K, which is character-
istic of an airstream. Compute and plot the temper- ature histories of the three components for the heating portion of a cycle, with Ti = 20°C and Too = 80°C. At what time does each component expe- rience 99% of its maximum possible temperature rise, that is, (T – Ti)/(Too – Ti) = 0.99? If the maximum stress on a solder ball corresponds to the maximum difference between its temperature and that of the chip or substrate, when will this maximum occur?
(b) To reduce the time required to complete a stress test, a dielectric liquid could be used in lieu of air to provide a larger convection coefficient of h = 200 W/m2 · K. What is the corresponding sav- ings in time for each component to achieve 99% of its maximum possible temperature rise?
5.40 The objective of this problem is to develop thermal models for estimating the steady-state temperature and the transient temperature history of the electrical trans- former shown.
Solder ball (D, rsd, csd)
Substrate (rsb, csb)
The external transformer geometry is approximately cubical, with a length of 32 mm to a side. The com-
A thermal stress test begins by subjecting the multichip module, which is initially at room temperature, to a hot fluid stream and subsequently cooling the module by exposing it to a cold fluid stream. The process is repeated for a prescribed number of cycles to assess the integrity of the soldered connections.
(a) As a first approximation, assume that there is negligible heat transfer between the components (chip/solder/substrate) of the module and that the thermal response of each component may be determined from a lumped capacitance analysis
bined mass of the iron and copper in the transformer is
0.28 kg, and its weighted-average specific heat is 400 J/kg · K. The transformer dissipates 4.0 W and is operating in ambient air at Too = 20°C, with a convec-
tion coefficient of 10 W/m2 · K. List and justify the
assumptions made in your analysis, and discuss limita- tions of the models.
(a) Beginning with a properly defined control volume, develop a model for estimating the steady-state temperature of the transformer, T(oo). Evaluate T(oo) for the prescribed operating conditions.
(b) Develop a model for estimating the thermal response (temperature history) of the transformer if it is initially at a temperature of Ti = Too and power is suddenly applied. Determine the time required for the transformer to come within 5°C of its steady-state operating temperature.
5.41 In thermomechanical data storage, a processing head,
undergo a cooling process can remain in a supercooled liquid state well below their equilibrium freezing tem- perature, Tf, particularly when the liquid is not in con- tact with any solid material. Droplets of liquid water in the atmosphere have a supercooled freezing tempera- ture, Tf,sc, that can be well correlated to the droplet diameter by the expression Tf,sc = -28 + 0.87 ln(Dp)
consisting of M heated cantilevers, is used to write data
in the diameter range 10-7
m, where Tf,sc
onto an underlying polymer storage medium. Electrical resistance heaters are microfabricated onto each can- tilever, which continually travel over the surface of the medium. The resistance heaters are turned on and off by controlling electrical current to each cantilever. As a cantilever goes through a complete heating and cooling cycle, the underlying polymer is softened, and one bit of data is written in the form of a surface pit in the polymer. A track of individual data bits (pits), each sep- arated by approximately 50 nm, can be fabricated. Mul- tiple tracks of bits, also separated by approximately 50 nm, are subsequently fabricated into the surface of the storage medium. Consider a single cantilever that is fabricated primarily of silicon with a mass of 50 X 10-18 kg and a surface area of 600 X 10-15 m2. The cantilever is initially at Ti = Too = 300 K, and the heat
transfer coefficient between the cantilever and the
ambient is 200 X 103 W/m2 · K.
(a) Determine the ohmic heating required to raise the cantilever temperature to T = 1000 K within a
has units of degrees Celsius and Dp is expressed in units of meters. For a droplet of diameter D = 50 ,um and
initial temperature Ti = 10°C subject to ambient condi- tions of Too = -40°C and h = 900 W/m2 · K, compare the time needed to completely solidify the droplet for case A, when the droplet solidifies at Tf = 0°C, and case B, when the droplet starts to freeze at Tf,sc. Sketch the temperature histories from the initial time to the time when the droplets are completely solid. Hint: When the droplet reaches Tf,sc in case B, rapid solidifi- cation occurs during which the latent energy released by the freezing water is absorbed by the remaining liq- uid in the drop. As soon as any ice is formed within the droplet, the remaining liquid is in contact with a solid (the ice) and the freezing temperature immediately shifts from Tf,sc to Tf = 0°C.
5.43 Consider the series solution, Equation 5.42, for the plane wall with convection. Calculate midplane (x* = 0) and surface (x* = 1) temperatures ()* for Fo = 0.1 and 1, using Bi = 0.1, 1, and 10. Consider only the first four eigenvalues. Based on these results, discuss the validity of the approximate solutions, Equa- tions 5.43 and 5.44.
5.44 Consider the one-dimensional wall shown in the sketch, which is initially at a uniform temperature Ti and is sud- denly subjected to the convection boundary condition with a fluid at Too.
heating time of th = 1 ,us. Hint: See Problem 5.20.
(b) Find the time required to cool the cantilever from 1000 K to 400 K (tc) and the thermal processing time required for one complete heating and cooling cycle, tp = th + tc.
(c) Determine how many bits (N) can be written onto a 1 mm X 1 mm polymer storage medium. If M = 100 cantilevers are ganged onto a single processing head, determine the total thermal processing time
Wall, T(x, 0) = Ti, k, a
needed to write the data.
5.42 The melting of water initially at the fusion temperature, Tf = 0°C, was considered in Example 1.6. Freezing of water often occurs at 0°C. However, pure liquids that
For a particular wall, case 1, the temperature at x = L1 after t1 = 100 s is T1(L1, t1) = 315°C. Another wall, case 2, has different thickness and thermal conditions as shown.
L a k Ti Too h
to a uniform temperature of 190°C, calculate the
Case (m) (m2/s) (W/m · K) (C) (C) (W/m 2 · K)
elapsed time te
required for the midplane of the
0.10 15 X 10-6
0.40 25 X 10-6
How long will it take for the second wall to reach 28.5°C at the position x = L2? Use as the basis for analysis, the dimensionless functional dependence for the transient temperature distribution expressed in Equation 5.41.
5.45 Copper-coated, epoxy-filled fiberglass circuit boards are treated by heating a stack of them under high pressure, as shown in the sketch. The purpose of the pressing–heating operation is to cure the epoxy that bonds the fiberglass sheets, imparting stiffness to the boards. The stack, referred to as a book, is comprised of 10 boards and 11 pressing plates, which prevent epoxy from flowing between the boards and impart a smooth finish to the cured boards. In order to perform simpli- fied thermal analyses, it is reasonable to approximate the book as having an effective thermal conductivity (k) and an effective thermal capacitance (pcp). Calculate
the effective properties if each of the boards and
plates has a thickness of 2.36 mm and the following thermophysical properties: board (b) pb = 1000 kg/m3, cp,b = 1500 J/kg · K, kb = 0.30 W/m · K; plate (p) pp =
book to reach the cure temperature of 170°C.
(b) If, at this instant of time, t = te, the platen tempera- ture were reduced suddenly to 15°C, how much energy would have to be removed from the book by the coolant circulating in the platen, in order to return the stack to its initial uniform temperature?
5.47 A constant-property, one-dimensional plane slab of width 2L, initially at a uniform temperature, is heated convectively with Bi = 1.
(a) At a dimensionless time of Fo1, heating is suddenly stopped, and the slab of material is quickly covered with insulation. Sketch the dimensionless surface and midplane temperatures of the slab as a function of dimensionless time over the range 0 Fo oo. By changing the duration of heating to Fo2, the steady-state midplane temperature can be set equal to the midplane temperature at Fo1. Is the value of Fo2 equal to, greater than, or less than Fo1? Hint: Assume both Fo1 and Fo2 are greater than 0.2.
(b) Letting Fo2 = Fo1 + LlFo, derive an analytical expression for LlFo, and evaluate LlFo for the con- ditions of part (a).
(c) Evaluate LlFo for Bi = 0.01, 0.1, 10, 100, and oo
8000 kg/m3, cp,p = 480 J/kg · K, kp = 12 W/m · K.
are both greater than 0.2.
Platens with circulating fluid
Metal pressing plate
5.48 Referring to the semiconductor processing tool of Problem 5.13, it is desired at some point in the manu- facturing cycle to cool the chuck, which is made of aluminum alloy 2024. The proposed cooling scheme passes air at 20°C between the air-supply head and the chuck surface.
Air supply, 20°C
5.46 Circuit boards are treated by heating a stack of them under high pressure, as illustrated in Problem 5.45. The platens at the top and bottom of the stack are main- tained at a uniform temperature by a circulating fluid. The purpose of the pressing–heating operation is to cure the epoxy, which bonds the fiberglass sheets, and impart stiffness to the boards. The cure condition is
L = 25 mm
Exit air Chuck
Heater coil (deactivated)
achieved when the epoxy has been maintained at or above 170°C for at least 5 min. The effective thermo- physical properties of the stack or book (boards
and metal pressing plates) are k = 0.613 W/m · K and
pcp = 2.73 X 106 J/m3 · K.
(a) If the book is initially at 15°C and, following appli- cation of pressure, the platens are suddenly brought
(a) If the chuck is initially at a uniform temperature of 100°C, calculate the time required for its lower surface to reach 25°C, assuming a uniform convec- tion coefficient of 50 W/m2 · K at the head–chuck
Generate a plot of the time-to-cool as a function of the convection coefficient for the range
10 h 2000 W/m2 · K. If the lower limit repre- sents a free convection condition without any head present, comment on the effectiveness of the head design as a method for cooling the chuck.
5.49 Annealing is a process by which steel is reheated and then cooled to make it less brittle. Consider the reheat stage for a 100-mm-thick steel plate (p = 7830 kg/m3, c = 550 J/kg · K, k = 48 W/m · K), which is initially at a
thickness 2L required to transfer a total amount of energy such that Q/Qo = 0.90 over a t = 8-h period. The initial concrete temperature is Ti = 40°C.
uniform temperature of Ti = 200°C and is to be heated
to a minimum temperature of 550°C. Heating is effected in a gas-fired furnace, where products of com- bustion at Too = 800°C maintain a convection coeffi-
cient of h = 250 W/m2 · K on both surfaces of the plate.
How long should the plate be left in the furnace?
5.50 Consider an acrylic sheet of thickness L = 5 mm that is used to coat a hot, isothermal metal substrate at Th = 300°C. The properties of the acrylic are p = 1990 kg/m3, c = 1470 J/kg · K, and k = 0.21 W/m · K.
Neglecting the thermal contact resistance between the acrylic and the metal substrate, determine how long it will take for the insulated back side of the acrylic to reach its softening temperature, Tsoft = 90°C. The initial acrylic temperature is Ti = 20°C.
5.51 The 150-mm-thick wall of a gas-fired furnace is con-
5.54 A plate of thickness 2L = 25 mm at a temperature of 600°C is removed from a hot pressing operation and must be cooled rapidly to achieve the required physical properties. The process engineer plans to use air jets to control the rate of cooling, but she is uncertain whether it is necessary to cool both sides (case 1) or only one side (case 2) of the plate. The concern is not just for the time-to-cool, but also for the maximum temperature difference within the plate. If this temperature differ- ence is too large, the plate can experience significant warping.
structed of fireclay brick (k = 1.5 W/m · K, p = 2600 kg/m3, cp = 1000 J/kg · K) and is well insulated at its outer surface. The wall is at a uniform initial tempera-
ture of 20°C, when the burners are fired and the inner surface is exposed to products of combustion for which
Case 1: cooling, both sides
T(x, 0) = Ti
Case 2: cooling, one side only
T(x, 0) = Ti
Too = 950°C and h = 100 W/m2 · K.
(a) How long does it take for the outer surface of the wall to reach a temperature of 750°C?
Plot the temperature distribution in the wall at the
k T¥, h
k T¥, h
foregoing time, as well as at several intermediate times.
5.52 Steel is sequentially heated and cooled (annealed) to relieve stresses and to make it less brittle. Consider a 100-mm-thick plate (k = 45 W/m · K, p = 7800 kg/m3, cp = 500 J/kg · K) that is initially at a uniform tempera- ture of 300°C and is heated (on both sides) in a gas-fired furnace for which Too = 700°C and h = 500 W/m2 · K. How long will it take for a minimum temperature of 550°C to be reached in the plate?
5.53 Stone mix concrete slabs are used to absorb thermal energy from flowing air that is carried from a large con- centrating solar collector. The slabs are heated during the day and release their heat to cooler air at night. If the daytime airflow is characterized by a temperature and convection heat transfer coefficient of Too = 200°C and h = 35 W/m2 · K, respectively, determine the slab
The air supply is at 25°C, and the convection coefficient on the surface is 400 W/m2 · K. The thermophysical prop- erties of the plate are p = 3000 kg/m3, c = 750 J/kg · K, and k = 15 W/m · K.
(a) Using the IHT software, calculate and plot on one graph the temperature histories for cases 1 and 2 for a 500-s cooling period. Compare the times required for the maximum temperature in the plate to reach 100°C. Assume no heat loss from the unexposed surface of case 2.
(b) For both cases, calculate and plot on one graph the variation with time of the maximum temperature difference in the plate. Comment on the relative magnitudes of the temperature gradients within the plate as a function of time.
5.55 During transient operation, the steel nozzle of a rocket engine must not exceed a maximum allowable operating
temperature of 1500 K when exposed to combustion gases characterized by a temperature of 2300 K and a convection coefficient of 5000 W/m2 · K. To extend the duration of engine operation, it is proposed that a ceramic thermal barrier coating (k = 10 W/m · K, a = 6 X 10-6 m2/s) be applied to the interior surface of the nozzle.
(a) If the ceramic coating is 10 mm thick and at an ini- tial temperature of 300 K, obtain a conservative estimate of the maximum allowable duration of engine operation. The nozzle radius is much larger than the combined wall and coating thickness.
Compute and plot the inner and outer surface tem- peratures of the coating as a function of time for 0 t 150 s. Repeat the calculations for a coating thickness of 40 mm.
5.56 Two plates of the same material and thickness L are at different initial temperatures Ti,1 and Ti,2, where Ti,2 > Ti,1. Their faces are suddenly brought into contact. The external surfaces of the two plates are insulated.
(a) Let a dimensionless temperature be defined as T* (Fo) = (T – Ti,1)/( Ti,2 – Ti,1). Neglecting the ther- mal contact resistance at the interface between the plates, what are the steady-state dimensionless tem- peratures of each of the two plates, T*ss,1 and Ts*s,2? What is the dimensionless interface temperature Ti*nt at any time?
(b) An effective overall heat transfer coefficient between the two plates can be defined based on the instantaneous, spatially averaged dimensionless
rubber wall (assumed to be untreaded) is taken from an initial temperature of 25°C to a midplane temperature of 150°C.
(a) If steam flow over the tire surfaces maintains a convection coefficient of h = 200 W/m2 · K, how long will it take to achieve the desired midplane temperature?
To accelerate the heating process, it is recom- mended that the steam flow be made sufficiently vigorous to maintain the tire surfaces at 200°C throughout the process. Compute and plot the mid- plane and surface temperatures for this case, as well as for the conditions of part (a).
5.59 A plastic coating is applied to wood panels by first depositing molten polymer on a panel and then cool- ing the surface of the polymer by subjecting it to airflow at 25°C. As first approximations, the heat of reaction associated with solidification of the polymer may be neglected and the polymer/wood interface may be assumed to be adiabatic.
Plastic coating (k, a, Ti)
U*eff = q*/(T*2 – T*1 ). Noting
If the thickness of the coating is L = 2 mm and it has an
that a dimensionless heat transfer rate to or from
either of the two plates may be expressed as q* = d(Q/Qo)/dFo, determine an expression for Ue*ff for Fo > 0.2.
5.57 In a tempering process, glass plate, which is initially at a uniform temperature Ti, is cooled by suddenly reduc-
initial uniform temperature of Ti = 200°C, how long will it take for the surface to achieve a safe-to-touch temperature of 42°C if the convection coefficient is h = 200 W/m2 · K? What is the corresponding value of the interface temperature? The thermal conductivity and diffusivity of the plastic are k = 0.25 W/m · K and
ing the temperature of both surfaces to Ts. The plate is 20 mm thick, and the glass has a thermal diffusivity of 6 X 10-7 m2/s.
a = 1.20 X 10-7
(a) How long will it take for the midplane temperature to achieve 50% of its maximum possible tempera- ture reduction?
(b) If (Ti – Ts) = 300°C, what is the maximum tem- perature gradient in the glass at the time calculated in part (a)?
5.58 The strength and stability of tires may be enhanced by heating both sides of the rubber (k = 0.14 W/m · K, a = 6.35 X 10-8 m2/s) in a steam chamber for which Too = 200°C. In the heating process, a 20-mm-thick
5.60 A long rod of 60-mm diameter and thermophysical properties p = 8000 kg/m3, c = 500 J/kg · K, and k = 50 W/m · K is initially at a uniform temperature and is heated in a forced convection furnace maintained at 750 K. The convection coefficient is estimated to be 1000 W/m2 · K.
(a) What is the centerline temperature of the rod when the surface temperature is 550 K?
In a heat-treating process, the centerline tempera- ture of the rod must be increased from Ti = 300 K to T = 500 K. Compute and plot the centerline temperature histories for h = 100, 500, and 1000 W/m2 · K. In each case the calculation may be terminated when T = 500 K.
5.61 A long cylinder of 30-mm diameter, initially at a uni- form temperature of 1000 K, is suddenly quenched in a large, constant-temperature oil bath at 350 K. The cylin- der properties are k = 1.7 W/m · K, c = 1600 J/kg · K, and p = 400 kg/m3, while the convection coefficient is 50 W/m2 · K.
(a) Calculate the time required for the surface of the cylinder to reach 500 K.
Compute and plot the surface temperature history for 0 t 300 s. If the oil were agitated, provid- ing a convection coefficient of 250 W/m2 · K, how would the temperature history change?
5.62 Work Problem 5.47 for a cylinder of radius ro and length L = 20 ro.
5.63 A long pyroceram rod of diameter 20 mm is clad with a very thin metallic tube for mechanical protection. The bonding between the rod and the tube has a thermal
contact resistance of R't,c = 0.12 m · K/W.
Thin metallic tube
Ceramic rod Bonding interface
D = 20 mm
(a) If the rod is initially at a uniform temperature of 900 K and is suddenly cooled by exposure to an airstream for which Too = 300 K and h =
100 W/m2 · K, at what time will the centerline reach
Cooling may be accelerated by increasing the air- speed and hence the convection coefficient. For values of h = 100, 500, and 1000 W/m2 · K, com- pute and plot the centerline and surface tempera- tures of the pyroceram as a function of time for 0 t 300 s. Comment on the implications of achieving enhanced cooling solely by increasing h.
5.64 A long rod 40 mm in diameter, fabricated from sapphire (aluminum oxide) and initially at a uniform temperature of 800 K, is suddenly cooled by a fluid at 300 K having a heat transfer coefficient of
1600 W/m2 · K. After 35 s, the rod is wrapped in insula-
tion and experiences no heat losses. What will be the temperature of the rod after a long period of time?
5.65 A cylindrical stone mix concrete beam of diameter D =
0.5 m initially at Ti = 20°C is exposed to hot gases at
Too = 500°C. The convection coefficient is h = 10 W/m2 · K.
(a) Determine the centerline temperature of the beam after an exposure time of t = 6 h.
(b) Determine the centerline temperature of a second beam that is of the same size and exposed to the same conditions as in part (a) but fabricated of lightweight aggregate concrete with density p = 1495 kg/m3,
thermal conductivity k = 0.789 W/m · K, and specific
heat cp = 880 J/kg · K.
5.66 A long plastic rod of 30-mm diameter (k = 0.3 W/m · K and pcp = 1040 kJ/m3 · K) is uniformly heated in an oven as preparation for a pressing operation. For best results, the temperature in the rod should not be less than 200°C. To what uniform temperature should the rod be heated in the oven if, for the worst case, the rod sits on a conveyor for 3 min while exposed to convec- tion cooling with ambient air at 25°C and with a con-
vection coefficient of 8 W/m2 · K? A further condition for good results is a maximum–minimum temperature difference of less than 10°C. Is this condition satisfied? If not, what could you do to satisfy it?
5.67 As part of a heat treatment process, cylindrical, 304 stainless steel rods of 100-mm diameter are cooled from an initial temperature of 500°C by suspending them in an oil bath at 30°C. If a convection coefficient of
500 W/m2 · K is maintained by circulation of the oil,
how long does it take for the centerline of a rod to reach a temperature of 50°C, at which point it is withdrawn from the bath? If 10 rods of length L = 1 m are processed per hour, what is the nominal rate at which energy must be extracted from the bath (the cooling load)?
5.68 In a manufacturing process, long rods of different diameters are at a uniform temperature of 400°C in a curing oven, from which they are removed and cooled by forced convection in air at 25°C. One of the line operators has observed that it takes 280 s for a 40-mm- diameter rod to cool to a safe-to-handle temperature of 60°C. For an equivalent convection coefficient, how long will it take for an 80-mm-diameter rod to cool to the same temperature? The thermophysical properties
of the rod are p = 2500 kg/m3, c = 900 J/kg · K, and
k = 15 W/m · K. Comment on your result. Did you anticipate this outcome?
5.69 The density and specific heat of a particular material are known (p = l200 kg/m3, cp = 1250 J/kg · K), but its thermal conductivity is unknown. To determine the thermal conductivity, a long cylindrical specimen of diameter D = 40 mm is machined, and a thermocouple
is inserted through a small hole drilled along the centerline.
T¥, h, V
is desirable to increase the surface temperature for a short time without significantly warming the interior of the ball. This type of heating can be accomplished by sudden immersion of the ball in a molten salt bath with Too = 1300 K and h = 5000 W/m2 · K. Assume that any location within the ball whose temperature exceeds 1000 K will be hardened. Estimate the time required to harden the outer millimeter of a ball of diameter 20 mm, if its initial temperature is 300 K.
Material of unknown
thermal conductivity ( r, cp, Ti)
5.72 A cold air chamber is proposed for quenching steel ball bearings of diameter D = 0.2 m and initial temperature
The thermal conductivity is determined by performing an experiment in which the specimen is heated to a uni- form temperature of Ti = 100°C and then cooled by passing air at Too = 25°C in cross flow over the cylin- der. For the prescribed air velocity, the convection coefficient is h = 55 W/m2 · K.
(a) If a centerline temperature of T(0, t) = 40°C is recorded after t = 1136 s of cooling, verify that the material has a thermal conductivity of k = 0.30 W/m · K.
For air in cross flow over the cylinder, the pre- scribed value of h = 55 W/m2 · K corresponds to a velocity of V = 6.8 m/s. If h = CV0.618, where the
Ti = 400°C. Air in the chamber is maintained at -15°C by a refrigeration system, and the steel balls pass through the chamber on a conveyor belt. Optimum bearing production requires that 70% of the initial ther- mal energy content of the ball above -15°C be removed. Radiation effects may be neglected, and the convection heat transfer coefficient within the chamber
is 1000 W/m2 · K. Estimate the residence time of the
balls within the chamber, and recommend a drive veloc- ity of the conveyor. The following properties may be used for the steel: k = 50 W/m · K, a = 2 X 10-5 m2/s, and c = 450 J/kg · K.
constant C has units of W · s0.618/m2.618 · K, how does the centerline temperature at t = 1136 s vary with velocity for 3 V 20 m/s? Determine the center- line temperature histories for 0 t 1500 s and velocities of 3, 10, and 20 m/s.
In Section 5.2 we noted that the value of the Biot number significantly influences the nature of the temper- ature distribution in a solid during a transient conduction process. Reinforce your understanding of this important concept by using the IHT model for one-dimensional transient conduction to determine radial temperature distributions in a 30-mm-diameter, stainless steel rod
(k = 15 W/m · K, p = 8000 kg/m3, cp = 475 J/kg · K), as
it is cooled from an initial uniform temperature of 325°C by a fluid at 25°C. For the following values of the con- vection coefficient and the designated times, determine
the radial temperature distribution: h = 100 W/m2 · K
(t = 0, 100, 500 s); h = 1000 W/m2 · K (t = 0, 10, 50 s);
h = 5000 W/m2 · K (t = 0, 1, 5, 25 s). Prepare a separate graph for each convection coefficient, on which tempera- ture is plotted as a function of dimensionless radius at the designated times.
5.71 In heat treating to harden steel ball bearings (c = 500 J/kg · K, p = 7800 kg/m3, k = 50 W/m · K), it
5.73 A soda lime glass sphere of diameter D1 = 25 mm is encased in a bakelite spherical shell of thickness L = 10 mm. The composite sphere is initially at a uniform temperature, Ti = 40°C, and is exposed to a fluid at
Too = 10°C with h = 30 W/m2 · K. Determine the center
temperature of the glass at t = 200 s. Neglect the ther-
mal contact resistance at the interface between the two materials.
5.74 Stainless steel (AISI 304) ball bearings, which have uni- formly been heated to 850°C, are hardened by quench- ing them in an oil bath that is maintained at 40°C. The ball diameter is 20 mm, and the convection coefficient
associated with the oil bath is 1000 W/m2 · K.
(a) If quenching is to occur until the surface tempera- ture of the balls reaches 100°C, how long must the balls be kept in the oil? What is the center tempera- ture at the conclusion of the cooling period?
(b) If 10,000 balls are to be quenched per hour, what is the rate at which energy must be removed by the oil bath cooling system in order to maintain its temper- ature at 40°C?
5.75 A sphere 30 mm in diameter initially at 800 K is quenched in a large bath having a constant temperature of 320 K with a convection heat transfer coefficient of 75 W/m2 · K. The thermophysical properties of the sphere material are: p = 400 kg/m3, c = 1600 J/kg · K, and k = 1.7 W/m · K.
(a) Show, in a qualitative manner on T t coordinates, the temperatures at the center and at the surface of the sphere as a function of time.
(b) Calculate the time required for the surface of the sphere to reach 415 K.
(c) Determine the heat flux (W/m2) at the outer surface of the sphere at the time determined in part (b).
(d) Determine the energy (J) that has been lost by the sphere during the process of cooling to the surface temperature of 415 K.
(e) At the time determined by part (b), the sphere is quickly removed from the bath and covered with perfect insulation, such that there is no heat loss from the surface of the sphere. What will be the temperature of the sphere after a long period of time has elapsed?
Compute and plot the center and surface tempera- ture histories over the period 0 t 150 s. What effect does an increase in the convection coefficient to h = 200 W/m2 · K have on the foregoing temper- ature histories? For h = 75 and 200 W/m2 · K, com- pute and plot the surface heat flux as a function of time for 0 t 150 s.
5.76 Work Problem 5.47 for the case of a sphere of radius ro.
5.77 Spheres A and B are initially at 800 K, and they are simultaneously quenched in large constant temperature baths, each having a temperature of 320 K. The follow- ing parameters are associated with each of the spheres and their cooling processes.
Sphere A Sphere B
Diameter (mm) 300 30
Density (kg/m3) 1600 400
Specific heat (kJ/kg · K) 0.400 1.60
Thermal conductivity (W/m · K) 170 1.70
Convection coefficient (W/m2 · K) 5 50
(a) Show in a qualitative manner, on T – t coordinates, the temperatures at the center and at the surface for each sphere as a function of time. Briefly explain the reasoning by which you determine the relative positions of the curves.
(b) Calculate the time required for the surface of each sphere to reach 415 K.
(c) Determine the energy that has been gained by each of the baths during the process of the spheres cool- ing to 415 K.
5.78 Spheres of 40-mm diameter heated to a uniform temperature of 400°C are suddenly removed from the oven and placed in a forced-air bath operating at 25°C with a convection coefficient of 300 W/m2 · K on the sphere surfaces. The thermophysical properties of the sphere material are p = 3000 kg/m3, c = 850 J/kg · K, and k = 15 W/m · K.
(a) How long must the spheres remain in the air bath for 80% of the thermal energy to be removed?
(b) The spheres are then placed in a packing carton that prevents further heat transfer to the environment. What uniform temperature will the spheres eventu- ally reach?
5.79 To determine which parts of a spider’s brain are trig- gered into neural activity in response to various optical stimuli, researchers at the University of Massachusetts Amherst desire to examine the brain as it is shown images that might evoke emotions such as fear or hunger. Consider a spider at Ti = 20°C that is shown a frightful scene and is then immediately immersed in liquid nitrogen at Too = 77 K. The brain is subsequently dissected in its frozen state and analyzed to determine which parts of the brain reacted to the stimulus. Using your knowledge of heat transfer, determine how much time elapses before the spider’s brain begins to freeze. Assume the brain is a sphere of diameter Db = 1 mm,
centrally located in the spider’s cephalothorax, which may be approximated as a spherical shell of diameter Dc = 3 mm. The brain and cephalothorax properties correspond to those of liquid water. Neglect the effects of the latent heat of fusion and assume the heat transfer coefficient is h = 100 W/m2 · K.
5.80 Consider the packed bed operating conditions of Problem 5.12, but with Pyrex (p = 2225 kg/m3, c = 835 J/kg · K, k = 1.4 W/m · K) used instead of alu- minum. How long does it take a sphere near the inlet of the system to accumulate 90% of the maximum possi- ble thermal energy? What is the corresponding temper- ature at the center of the sphere?
5.81 The convection coefficient for flow over a solid sphere may be determined by submerging the sphere, which is initially at 25°C, into the flow, which is at 75°C, and measuring its surface temperature at some time during the transient heating process.
(a) If the sphere has a diameter of 0.1 m, a thermal con- ductivity of 15 W/m · K, and a thermal diffusivity of 10-5 m2/s, at what time will a surface temperature
of 60°C be recorded if the convection coefficient is 300 W/m2 · K?
Assess the effect of thermal diffusivity on the ther- mal response of the material by computing center and surface temperature histories for a = 10-6, 10-5, and 10-4 m2/s. Plot your results for the period 0 t 300 s. In a similar manner, assess the effect of thermal conductivity by considering val- ues of k = 1.5, 15, and 150 W/m · K.
Consider the sphere of Example 5.6, which is initially at a uniform temperature when it is suddenly removed from the furnace and subjected to a two-step cooling process. Use the Transient Conduction, Sphere model of IHT to obtain the following solutions.
(a) For step 1, calculate the time required for the center temperature to reach T(0, t) = 335°C, while cool- ing in air at 20°C with a convection coefficient of 10 W/m2 · K. What is the Biot number for this cool-
ing process? Do you expect radial temperature gra- dients to be appreciable? Compare your results to those of the example.
(b) For step 2, calculate the time required for the center temperature to reach T(0, t) = 50°C, while cooling in a water bath at 20°C with a convection coeffi- cient of 6000 W/m2 · K.
(c) For the step 2 cooling process, calculate and plot the temperature histories, T(r, t), for the center and surface of the sphere. Identify and explain key fea- tures of the histories. When do you expect the tem- perature gradients in the sphere to be the largest?
5.83 Two large blocks of different materials, such as copper and concrete, have been sitting in a room (23°C) for a very long time. Which of the two blocks, if either, will feel colder to the touch? Assume the blocks to be semi-infinite solids and your hand to be at a tempera- ture of 37°C.
5.84 A plane wall of thickness 0.6 m (L = 0.3 m) is made of steel (k = 30 W/m · K, p = 7900 kg/m3, c = 640 J/kg · K). It is initially at a uniform temperature and is then exposed to air on both surfaces. Consider two different convection conditions: natural convection, characterized by h = 10 W/m2 · K, and forced convection, with h = 100 W/m2 · K. You are to calculate the surface tem-
perature at three different times—t = 2.5 min, 25 min, and 250 min—for a total of six different cases.
(a) For each of these six cases, calculate the nondimen- sional surface temperature, ()*s = (Ts – Too)/(Ti – Too),
using four different methods: exact solution, first- term-of-the-series solution, lumped capacitance, and semi-infinite solid. Present your results in a table.
(b) Briefly explain the conditions for which (i) the first-term solution is a good approximation to the exact solution, (ii) the lumped capacitance solu- tion is a good approximation, (iii) the semi-infinite solid solution is a good approximation.
5.85 Asphalt pavement may achieve temperatures as high as 50°C on a hot summer day. Assume that such a temper- ature exists throughout the pavement, when suddenly a rainstorm reduces the surface temperature to 20°C. Cal- culate the total amount of energy (J/m2) that will be transferred from the asphalt over a 30-min period in which the surface is maintained at 20°C.
5.86 A thick steel slab (p = 7800 kg/m3, c = 480 J/kg · K, k = 50 W/m · K) is initially at 300°C and is cooled by water jets impinging on one of its surfaces. The temper- ature of the water is 25°C, and the jets maintain an extremely large, approximately uniform convection coefficient at the surface. Assuming that the surface is maintained at the temperature of the water throughout the cooling, how long will it take for the temperature to reach 50°C at a distance of 25 mm from the surface?
5.87 A tile-iron consists of a massive plate maintained at 150°C by an embedded electrical heater. The iron is placed in contact with a tile to soften the adhesive, allowing the tile to be easily lifted from the subflooring. The adhesive will soften sufficiently if heated above 50°C for at least 2 min, but its temperature should not exceed 120°C to avoid deterioration of the adhesive. Assume the tile and subfloor to have an initial tempera- ture of 25°C and to have equivalent thermophysical
properties of k = 0.15 W/m · K and pcp = 1.5 X 106 J/m3 · K.
Tile, 4–mm thickness Subflooring
(a) How long will it take a worker using the tile-iron to lift a tile? Will the adhesive temperature exceed 120°C?
(b) If the tile-iron has a square surface area 254 mm to the side, how much energy has been removed from it during the time it has taken to lift the tile?
5.88 A simple procedure for measuring surface convection heat transfer coefficients involves coating the surface with a thin layer of material having a precise melting point temperature. The surface is then heated and, by determining the time required for melting to occur, the
convection coefficient is determined. The following experimental arrangement uses the procedure to deter- mine the convection coefficient for gas flow normal to a surface. Specifically, a long copper rod is encased in a super insulator of very low thermal conductivity, and a very thin coating is applied to its exposed surface.
Surface coating Copper rod,
k = 400 W/m•K, a = 10–4m2/s
If the rod is initially at 25°C and gas flow for which h = 200 W/m2 · K and Too = 300°C is initiated, what is the melting point temperature of the coating if melting is observed to occur at t = 400 s?
An insurance company has hired you as a consultant to improve their understanding of burn injuries. They are especially interested in injuries induced when a portion of a worker’s body comes into contact with machinery that is at elevated temperatures in the range of 50 to 100°C. Their medical consultant informs them that irre- versible thermal injury (cell death) will occur in any living tissue that is maintained at T > 48°C for a dura- tion Llt > 10 s. They want information concerning the extent of irreversible tissue damage (as measured by distance from the skin surface) as a function of the machinery temperature and the time during which con- tact is made between the skin and the machinery. Assume that living tissue has a normal temperature of 37°C, is isotropic, and has constant properties equiva- lent to those of liquid water.
(a) To assess the seriousness of the problem, compute locations in the tissue at which the temperature will reach 48°C after 10 s of exposure to machinery at 50°C and 100°C.
(b) For a machinery temperature of 100°C and 0 t 30 s, compute and plot temperature histories at tissue locations of 0.5, 1, and 2 mm from the skin.
5.90 A procedure for determining the thermal conductivity of a solid material involves embedding a thermocouple in a thick slab of the solid and measuring the response to a prescribed change in temperature at one surface. Consider an arrangement for which the thermocouple is embedded 10 mm from a surface that is suddenly
brought to a temperature of 100°C by exposure to boil- ing water. If the initial temperature of the slab was 30°C and the thermocouple measures a temperature of 65°C, 2 min after the surface is brought to 100°C, what is its thermal conductivity? The density and specific heat of the solid are known to be 2200 kg/m3 and
700 J/kg · K.
5.91 A very thick slab with thermal diffusivity 5.6 X 10-6 m2/s and thermal conductivity 20 W/m · K is ini- tially at a uniform temperature of 325°C. Suddenly, the surface is exposed to a coolant at 15°C for which the convection heat transfer coefficient is 100 W/m2 · K.
(a) Determine temperatures at the surface and at a depth of 45 mm after 3 min have elapsed.
Compute and plot temperature histories (0 t 300 s) at x = 0 and x = 45 mm for the following parametric variations: (i) a = 5.6 X 10-7, 5.6 X 10-6, and 5.6 X 10-5 m2/s; and (ii) k = 2, 20, and 200 W/m · K.
5.92 A thick oak wall, initially at 25°C, is suddenly exposed to combustion products for which Too = 800°C and h = 20 W/m2 · K.
(a) Determine the time of exposure required for the surface to reach the ignition temperature of 400°C.
Plot the temperature distribution T(x) in the medium at t = 325 s. The distribution should extend to a location for which T = 25°C.
5.93 Standards for firewalls may be based on their thermal response to a prescribed radiant heat flux. Consider a 0.25-m-thick concrete wall (p = 2300 kg/m3,
c = 880 J/kg · K, k = 1.4 W/m · K), which is at an initial
temperature of Ti = 25°C and irradiated at one surface by lamps that provide a uniform heat flux of qs” = 104 W/m2. The absorptivity of the surface to the irradiation is as = 1.0. If building code requirements dictate that the temperatures of the irradiated and back surfaces must not exceed 325°C and 25°C, respec- tively, after 30 min of heating, will the requirements be met?
5.94 It is well known that, although two materials are at the same temperature, one may feel cooler to the touch than the other. Consider thick plates of copper and glass, each at an initial temperature of 300 K. Assuming your finger to be at an initial temperature of 310 K and to have thermophysical properties of p = 1000 kg/m3,
c = 4180 J/kg · K, and k = 0.625 W/m · K, determine
whether the copper or the glass will feel cooler to the touch.
5.95 Two stainless steel plates (p = 8000 kg/m3, c = 500 J/kg · K, k = 15 W/m · K), each 20 mm thick and
insulated on one surface, are initially at 400 and 300 K when they are pressed together at their uninsulated sur- faces. What is the temperature of the insulated surface of the hot plate after 1 min has elapsed?
5.96 Special coatings are often formed by depositing thin layers of a molten material on a solid substrate. Solidifi- cation begins at the substrate surface and proceeds until the thickness S of the solid layer becomes equal to the thickness o of the deposit.
r , hsf
kw, a w
r , hsf
Just before the start of solidification (S = 0), the mold wall is everywhere at an initial uniform temperature Ti and the molten metal is everywhere at its fusion (melt- ing point) temperature of Tf. Following the start of solidification, there is conduction heat transfer into the mold wall and the thickness of the solidified metal S increases with time t.
(a) Consider conditions for which molten material at its fusion temperature Tf is deposited on a large substrate that is at an initial uniform temperature Ti. With S = 0 at t = 0, develop an expression for estimating the time td required to completely solidify the deposit if it remains at Tf throughout the solidification process. Express your result in terms of the substrate thermal conductivity and thermal diffusivity (ks, as), the density and latent heat of fusion of the deposit (p, hsf), the deposit thickness o, and the relevant temperatures (Tf, Ti).
(b) The plasma spray deposition process of Problem
5.25 is used to apply a thin (o = 2 mm) alumina coating on a thick tungsten substrate. The substrate has a uniform initial temperature of Ti = 300 K, and its thermal conductivity and thermal diffusivity may be approximated as ks = 120 W/m · K and as = 4.0 X 10-5 m2/s, respectively. The density and latent heat of fusion of the alumina are p = 3970 kg/m3 and hsf = 3577 kJ/kg, respectively, and the alumina solidifies at its fusion temperature (Tf = 2318 K). Assuming that the molten layer is instantaneously deposited on the substrate, estimate the time required for the deposit to solidify.
5.97 When a molten metal is cast in a mold that is a poor conductor, the dominant resistance to heat flow
(a) Sketch the one-dimensional temperature distribu- tion, T(x), in the mold wall and the metal at t = 0 and at two subsequent times during the solidification. Clearly indicate any underlying assumptions.
(b) Obtain a relation for the variation of the solid layer thickness S with time t, expressing your result in terms of appropriate parameters of the system.
5.98 Joints of high quality can be formed by friction welding. Consider the friction welding of two 40-mm-diameter Inconel rods. The bottom rod is stationary, while the top rod is forced into a back-and-forth linear motion characterized by an instantaneous horizontal dis- placement, d(t) = a cos(wt) where a = 2 mm and w = 1000 rad/s. The coefficient of sliding friction between the two pieces is ,u = 0.3. Determine the compressive force that must be applied to heat the joint to the Inconel melting point within t = 3 s, starting from an initial temperature of 20°C. Hint: The fre- quency of the motion and resulting heat rate are very high. The temperature response can be approximated as if the heating rate were constant in time, equal to its average value.
Top moving cylindrical rod
is within the mold wall. Consider conditions for
which a liquid metal is solidifying in a thick-walled mold of thermal conductivity kw and thermal diffusiv- ity aw. The density and latent heat of fusion of the metal are designated as p and hsf, respectively, and in both its molten and solid states, the thermal conduc- tivity of the metal is very much larger than that of the mold.
Bottom stationary cylindrical rod
5.99 A rewritable optical disc (DVD) is formed by sand- wiching a 15-nm-thick binary compound storage mater- ial between two 1-mm-thick polycarbonate sheets. Data are written to the opaque storage medium by irradiating it from below with a relatively high-powered laser beam of diameter 0.4 ,um and power 1 mW, resulting in rapid heating of the compound material (the polycar- bonate is transparent to the laser irradiation). If the temperature of the storage medium exceeds 900 K, a noncrystalline, amorphous material forms at the heated spot when the laser irradiation is curtailed and the spot is allowed to cool rapidly. The resulting spots of amor- phous material have a different reflectivity from the sur- rounding crystalline material, so they can subsequently be read by irradiating them with a second, low-power laser and detecting the changes in laser radiation trans- mitted through the entire DVD thickness. Determine the irradiation (write) time needed to raise the storage medium temperature from an initial value of 300 K to 1000 K. The absorptivity of the storage medium is 0.8. The polycarbonate properties are p = 1200 kg/m3,
k = 0.21 W/m · K, and cp = 1260 J/kg · K.
Heat pump Buried tubing Top view
To what depth should the tubing be buried so that the soil can be viewed as an infinite medium at constant temperature over a 12-month period? Account for the periodic cooling (heating) of the soil due to both annual changes in ambient conditions and variations in heat pump operation from the winter heating to the summer cooling mode.
5.101 To enable cooking a wider range of foods in microwave ovens, thin, metallic packaging materials have been developed that will readily absorb microwave energy. As the packaging material is heated by the microwaves, conduction simultaneously occurs from the hot packag-
Read laser (on)
Write laser (on/off)
ing material to the cold food. Consider the spherical piece of frozen ground beef of Problem 5.33 that is now wrapped in the thin microwave-absorbing packaging material. Determine the time needed for the beef that is immediately adjacent to the packaging material to reach T = 0°C if 50% of the oven power (P = 1 kW total) is absorbed in the packaging material.
5.102 Derive an expression for the ratio of the total energy transferred from the isothermal surface of an infinite cylinder to the interior of the cylinder, Q/Qo, that is valid for Fo 0.2. Express your results in terms of
the Fourier number Fo.
5.100 Ground source heat pumps operate by using the soil,
rather than ambient air, as the heat source (or sink) for heating (or cooling) a building. A liquid transfers energy from (to) the soil by way of buried plastic tub- ing. The tubing is at a depth for which annual varia- tions in the temperature of the soil are much less than those of the ambient air. For example, at a location such as South Bend, Indiana, deep-ground tempera- tures may remain at approximately 11°C, while annual excursions in the ambient air temperature may range from –25°C to +37°C. Consider the tubing to be laid out in a closely spaced serpentine arrangement.
5.103 The structural components of modern aircraft are com- monly fabricated of high-performance composite mate- rials. These materials are fabricated by impregnating mats of extremely strong fibers that are held within a form with an epoxy or thermoplastic liquid. After the liquid cures or cools, the resulting component is of extremely high strength and low weight. Periodically, these components must be inspected to ensure that the fiber mats and bonding material do not become delami- nated and, in turn, the component loses its airworthiness. One inspection method involves application of a
uniform, constant radiation heat flux to the surface being inspected. The thermal response of the surface is mea- sured with an infrared imaging system, which captures the emission from the surface and converts it to a color- coded map of the surface temperature distribution. Con- sider the case where a uniform flux of 5 kW/m2 is applied to the top skin of an airplane wing initially at 20°C. The opposite side of the 15-mm-thick skin is adja- cent to stagnant air and can be treated as well insulated. The density and specific heat of the skin material are 1200 kg/m3 and 1200 J/kg · K, respectively. The effec- tive thermal conductivity of the intact skin material is k1 = 1.6 W/m · K. Contact resistances develop internal to the structure as a result of delamination between the fiber mats and the bonding material, leading to a reduced effective thermal conductivity of k2 = 1.1 W/m · K. Determine the surface temperature of the component after 10 and 100 s of irradiation for (i) an area where the material is structurally intact and (ii) an adjacent area where delamination has occurred within the wing.
the lumped capacitance approximation is accurate to within 10%.
5.105 Problem 4.9 addressed radioactive wastes stored under- ground in a spherical container. Because of uncertainty in the thermal properties of the soil, it is desired to mea- sure the steady-state temperature using a test container (identical to the real container) that is equipped with internal electrical heaters. Estimate how long it will take the test container to come within 10°C of its steady-state value, assuming it is buried very far underground. Use the soil properties from Table A.3 in your analysis.
5.106 Derive an expression for the ratio of the total energy transferred from the isothermal surface of a sphere to the interior of the sphere Q/Qo that is valid for Fo 0.2. Express your result in terms of the Fourier number, Fo.
5.107 Consider the experimental measurement of Example
5.10. It is desired to measure the thermal conductivity of an extremely thin sample of the same nanostruc- tured material having the same length and width. To minimize experimental uncertainty, the experimenter wishes to keep the amplitude of the temperature response, LlT, above a value of 0.1°C. What is the min- imum sample thickness that can be measured? Assume the properties of the thin sample and the magnitude of the applied heating rate are the same as those mea- sured and used in Example 5.10.
The stability criterion for the explicit method requires that the coefficient of the Tp term of the one-dimen- sional, finite-difference equation be zero or positive. Consider the situation for which the temperatures at the
two neighboring nodes (Tm -1, Tm +1) are 100°C while
the center node (T p ) is at 50°C. Show that for values of Fo > 1 the finite-difference equation will predict a value of T p+1 that violates the second law of thermodynamics.
5.104 Consider the plane wall of thickness 2L, the infinite cylinder of radius ro, and the sphere of radius ro. Each configuration is subjected to a constant surface heat flux q”s . Using the approximate solutions of Table 5.2b for Fo > 0.2, derive expressions for each of the three geometries for the quantity (Ts,act – Ti)/(Ts,lc – Ti). In this expression, Ts,act is the actual surface temperature as determined by the relations of Table 5.2b, and
5.109 A thin rod of diameter D is initially in equilibrium
with its surroundings, a large vacuum enclosure at temperature Tsur. Suddenly an electrical current I (A) is passed through the rod having an electrical resistivity pe and emissivity s. Other pertinent thermophysical properties are identified in the sketch. Derive the tran- sient, finite-difference equation for node m.
D x D x e
Ts,lc is the temperature associated with lumped capaci- I
tance behavior. Determine criteria associated with (Ts,act – Ti)/ (Ts,lc – Ti) 1.1, that is, determine when
m – 1
m m + 1
re, r, c, k
5.110 A one-dimensional slab of thickness 2L is initially at a uniform temperature Ti. Suddenly, electric current is passed through the slab causing uniform volumetric
heating . (W/m3). At the same time, both outer sur-
faces (x = ±L) are subjected to a convection process at Too with a heat transfer coefficient h.
T1 = 0°C
0, q• = 2 ´ 107 W/m3
0 1 2
D x –L +L
Write the finite-difference equation expressing conser- vation of energy for node 0 located on the outer sur- face at x = -L. Rearrange your equation and identify any important dimensionless coefficients.
5.111 Consider Problem 5.9 except now the combined vol- ume of the oil bath and the sphere is Vtot = 1 m3. The oil bath is well mixed and well insulated.
(a) Assuming the quenching liquid’s properties are that of engine oil at 380 K, determine the steady- state temperature of the sphere.
(b) Derive explicit finite difference expressions for the sphere and oil bath temperatures as a func- tion of time using a single node each for the sphere and oil bath. Determine any stability requirements that might limit the size of the time step Llt.
(c) Evaluate the sphere and oil bath temperatures after one time step using the explicit expressions of part (b) and time steps of 1000, 10,000, and 20,000 s.
(d) Using an implicit formulation with Llt = 100 s, determine the time needed for the coated sphere to reach 140°C. Compare your answer to the time associated with a large, well-insulated oil bath. Plot the sphere and oil temperatures as a function of time over the interval 0 h t 15 h. Hint: See Comment 3 of Example 5.2.
5.112 A plane wall (p = 4000 kg/m3, cp = 500 J/kg · K, k = 10 W/m · K) of thickness L = 20 mm initially has a linear, steady-state temperature distribution with boundaries maintained at T1 = 0°C and T2 = 100°C. Suddenly, an electric current is passed through the wall, causing uniform energy generation at a rate q = 2 X 107 W/m3. The boundary conditions T1 and T2 remain fixed.
(a) On T x coordinates, sketch temperature distrib- utions for the following cases: (i) initial condi- tion (t 0); (ii) steady-state conditions (t l oo), assuming that the maximum temperature in the wall exceeds T2; and (iii) for two intermedi- ate times. Label all important features of the distributions.
(b) For the system of three nodal points shown schematically (1, m, 2), define an appropriate control volume for node m and, identifying all rel- evant processes, derive the corresponding finite- difference equation using either the explicit or implicit method.
With a time increment of Llt = 5 s, use the finite- difference method to obtain values of Tm for the first 45 s of elapsed time. Determine the corre- sponding heat fluxes at the boundaries, that is, q”x (0, 45 s) and qx” (20 mm, 45 s).
(d) To determine the effect of mesh size, repeat your analysis using grids of 5 and 11 nodal points (Llx = 5.0 and 2.0 mm, respectively).
5.113 A round solid cylinder made of a plastic material (a = 6 X 10-7 m2/s) is initially at a uniform tempera- ture of 20°C and is well insulated along its lateral sur- face and at one end. At time t = 0, heat is applied to the left boundary causing T0 to increase linearly with time at a rate of 1°C/s.
Using the explicit method with Fo = 1, derive the finite-difference equations for nodes 1, 2, 3, and 4.
(b) Format a table with headings of p, t(s), and the nodal temperatures T0 to T4. Determine the surface temperature T0 when T4 = 35°C.
5.114 Derive the explicit finite-difference equation for an interior node for three-dimensional transient
conduction. Also determine the stability criterion. Assume constant properties and equal grid spacing in all three directions.
5.115 Derive the transient, two-dimensional finite-difference equation for the temperature at nodal point 0 located on the boundary between two different materials.
A wall 0.12 m thick having a thermal diffusivity of
1.5 X 10-6 m2/s is initially at a uniform temperature of 85°C. Suddenly one face is lowered to a temperature of 20°C, while the other face is perfectly insulated.
(a) Using the explicit finite-difference technique with space and time increments of 30 mm and 300 s, respectively, determine the temperature distribu- tion at t = 45 min.
(b) With Llx = 30 mm and Llt = 300 s, compute T(x, t) for 0 t tss, where tss is the time required for the temperature at each nodal point to reach a value that is within 1°C of the steady-state temper- ature. Repeat the foregoing calculations for Llt = 75 s. For each value of Llt, plot temperature histories for each face and the midplane.
5.117 A molded plastic product (p = 1200 kg/m3, c = 1500 J/kg · K, k = 0.30 W/m · K) is cooled by exposing one surface to an array of air jets, while the opposite surface is well insulated. The product may be approxi- mated as a slab of thickness L = 60 mm, which is ini- tially at a uniform temperature of Ti = 80°C. The air jets are at a temperature of Too = 20°C and provide a uniform convection coefficient of h = 100 W/m2 · K at the cooled surface.
Consider a one-dimensional plane wall at a uniform initial temperature Ti. The wall is 10 mm thick, and has a thermal diffusivity of C= 6 X 10-7 m2/s. The left face is insulated, and suddenly the right face is lowered to a temperature Ts,r.
(a) Using the implicit finite-difference technique with Llx = 2 mm and Llt = 2 s, determine how long it will take for the temperature at the left face Ts,l to achieve 50% of its maximum possible temperature reduction.
(b) At the time determined in part (a), the right face is suddenly returned to the initial temperature. Deter- mine how long it will take for the temperature at the left face to recover to a 20% temperature reduction, that is, Ti – Ts,l = 0.2(Ti – Ts,r).
The plane wall of Problem 2.60 (k = 50 W/m · K, a = 1.5 X 10-6 m2/s) has a thickness of L = 40 mm and an initial uniform temperature of To = 25°C. Sud- denly, the boundary at x = L experiences heating by a fluid for which Too = 50°C and h = 1000 W/m2 · K, while heat is uniformly generated within the wall at q = 1 X 107 W/m3. The boundary at x = 0 remains at To.
(a) With Llx = 4 mm and Llt = 1 s, plot temperature distributions in the wall for (i) the initial condi- tion, (ii) the steady-state condition, and (iii) two intermediate times.
(b) On qx” – t coordinates, plot the heat flux at x = 0 and x = L. At what elapsed time is there zero heat flux at x = L?
5.120 Consider the fuel element of Example 5.11. Initially, the element is at a uniform temperature of 250°C with no heat generation. Suddenly, the element is inserted into the reactor core, causing a uniform volumetric heat gen-
eration rate of . = 108 W/m3. The surfaces are convec-
tively cooled with Too = 250°C and h = 1100 W/m2 · K. Using the explicit method with a space increment of 2 mm, determine the temperature distribution 1.5 s after the element is inserted into the core.
Consider two plates, A and B, that are each initially isothermal and each of thickness L = 5 mm. The faces of the plates are suddenly brought into contact in a joining process. Material A is acrylic, initially at Ti,A =
T¥, h L
20°C with pA
= 1990 kg/m3, cA
= 1470 J/kg · K, and
Plastic (Ti, r, c, k)
kA = 0.21 W/m · K. Material B is steel initially at Ti,B =
300°C with pB = 7800 kg/m3, cB = 500 J/kg · K, and kB = 45 W/m · K. The external (back) surfaces of the acrylic and steel are insulated. Neglecting the thermal
Using a finite-difference solution with a space incre-
ment of Llx = 6 mm, determine temperatures at the cooled and insulated surfaces after 1 h of exposure to the gas jets.
contact resistance between the plates, determine how long it will take for the external surface of the acrylic to reach its softening temperature, Tsoft = 90°C. Plot the acrylic’s external surface temperature as well as the
average temperatures of both materials over the time span 0 t 300 s. Use 20 equally spaced nodal points.
Consider the fuel element of Example 5.11, which oper- ates at a uniform volumetric generation rate of q = 107 W/m3, until the generation rate suddenly
changes to . = 2 X 107 W/m3. Use the Finite-Difference
Equations, One-Dimensional, Transient conduction model builder of IHT to obtain the implicit form of the finite-difference equations for the 6 nodes, with Llx = 2 mm, as shown in the example.
(a) Calculate the temperature distribution 1.5 s after the change in operating power, and compare your results with those tabulated in the example.
(b) Use the Explore and Graph options of IHT to calcu- late and plot temperature histories at the midplane
(00) and surface (05) nodes for 0 t 400 s. What are the steady-state temperatures, and approxi- mately how long does it take to reach the new equi- librium condition after the step change in operating power?
In a thin-slab, continuous casting process, molten steel leaves a mold with a thin solid shell, and the molten material solidifies as the slab is quenched by water jets en route to a section of rollers. Once fully solidified, the slab continues to cool as it is brought to an accept- able handling temperature. It is this portion of the process that is of interest.
slab is cooled at its top and bottom surfaces by water jets (Too = 50°C), which maintain an approximately uniform convection coefficient of h = 5000 W/m2 · K at both sur- faces. Using a finite-difference solution with a space increment of Llx = 1 mm, determine the time required to cool the surface of the slab to 200°C. What is the corre- sponding temperature at the midplane of the slab? If the slab moves at a speed of V = 15 mm/s, what is the required length of the cooling section?
Determine the temperature distribution at t = 30 min for the conditions of Problem 5.116.
(a) Use an explicit finite-difference technique with a time increment of 600 s and a space increment of 30 mm.
(b) Use the implicit method of the IHT Finite-Difference Equation Tool Pad for One-Dimensional Tran- sient Conduction.
A very thick plate with thermal diffusivity
5.6 X 10-6 m2/s and thermal conductivity 20 W/m · K is initially at a uniform temperature of 325°C. Sud- denly, the surface is exposed to a coolant at 15°C for which the convection heat transfer coefficient is 100 W/m2 · K. Using the finite-difference method with a space increment of Llx = 15 mm and a time incre- ment of 18 s, determine temperatures at the surface and at a depth of 45 mm after 3 min have elapsed.
5.126 Referring to Example 5.12, Comment 4, consider a sudden exposure of the surface to large surroundings at an elevated temperature (Tsur) and to convection (T , h).
(Ti = 1400°C)
2L = 200 mm
(a) Derive the explicit, finite-difference equation for the surface node in terms of Fo, Bi, and Bir.
(b) Obtain the stability criterion for the surface node. Does this criterion change with time? Is the crite- rion more restrictive than that for an interior node? A thick slab of material (k = 1.5 W/m · K, a = 7 X 10-7 m2/s, s = 0.9), initially at a uniform
temperature of 27°C, is suddenly exposed to large surroundings at 1000 K. Neglecting convection and using a space increment of 10 mm, determine temperatures at the surface and 30 mm from the surface after an elapsed time of 1 min.
A constant-property, one-dimensional plane wall of width 2L, at an initial uniform temperature Ti, is heated convectively (both surfaces) with an ambient fluid at Too = Too,1, h = h1. At a later instant in time,
Consider a 200-mm-thick solid slab of steel
(p = 7800 kg/m3, c = 700 J/kg · K, k = 30 W/m · K), initially at a uniform temperature of Ti = 1400°C. The
t = t1, heating is curtailed, and convective cooling is
initiated. Cooling conditions are characterized by
Too = Too,2 = Ti, h = h2.
(a) Write the heat equation as well as the initial and boundary conditions in their dimensionless form for the heating phase (Phase 1). Express the equations in terms of the dimensionless quanti- ties ()*, x*, Bi1, and Fo, where Bi1 is expressed in terms of h1.
(b) Write the heat equation as well as the initial and boundary conditions in their dimensionless form for the cooling phase (Phase 2). Express the equations in terms of the dimensionless quantities ()*, x*, Bi2, Fo1, and Fo where Fo1 is the dimensionless time associated with t1, and Bi2 is expressed in terms of h2. To be consistent with part (a), express the dimensionless temperature in terms of Too = Too,1.
(c) Consider a case for which Bi1 = 10, Bi2 = 1, and
Fo1 = 0.1. Using a finite-difference method with
wall can be approximated as isothermal and repre- sented as a lumped capacitance (Equation 5.7). For the conditions shown schematically, we wish to compare predictions based on the one-term approxi- mation, the lumped capacitance method, and a finite- difference solution.
T(x, t), T(x, 0) = Ti = 250°
r = 7800 kg/m3
c = 440 J/kg•K
k = 15 W/m•K
T¥ = 25°C
h = 500 W/m2•K
x L = 20 mm
Llx* = 0.1 and LlFo = 0.001, determine the tran- sient thermal response of the surface (x* = 1), midplane (x* = 0), and quarter-plane (x* = 0.5) of the slab. Plot these three dimensionless temper- atures as a function of dimensionless time over the range 0 Fo 0.5.
(d) Determine the minimum dimensionless tempera-
1 2 3 4 5
ture at the midplane of the wall, and the dimen-
sionless time at which this minimum temperature x
(Dt = 1s)
Consider the thick slab of copper in Example 5.12, which is initially at a uniform temperature of 20°C and is suddenly exposed to a net radiant flux of 3 X 105 W/m2. Use the Finite-Difference Equations/ One-Dimensional/Transient conduction model builder of IHT to obtain the implicit form of the finite-difference equations for the interior nodes. In your analysis, use a space increment of Llx = 37.5 mm with a total of 17 nodes (00–16), and a time increment of Llt = 1.2 s. For the surface node 00, use the finite-difference equa- tion derived in Section 2 of the Example.
(a) Calculate the 00 and 04 nodal temperatures at t = 120 s, that is, T(0, 120 s) and T(0.15 m, 120 s), and compare the results with those given in Com- ment 1 for the exact solution. Will a time increment of 0.12 s provide more accurate results?
(b) Plot temperature histories for x = 0, 150, and 600 mm, and explain key features of your results.
5.129 In Section 5.5, the one-term approximation to the series solution for the temperature distribution was developed for a plane wall of thickness 2L that is ini- tially at a uniform temperature and suddenly sub- jected to convection heat transfer. If Bi 0.1, the
(a) Determine the midplane, T(0, t), and surface, T(L, t), temperatures at t = 100, 200, and 500 s using the one-term approximation to the series solu- tion, Equation 5.43, What is the Biot number for the system?
(b) Treating the wall as a lumped capacitance, calcu- late the temperatures at t = 50, 100, 200, and 500 s. Did you expect these results to compare favorably with those from part (a)? Why are the temperatures considerably higher?
Consider the 2- and 5-node networks shown schematically. Write the implicit form of the finite- difference equations for each network, and deter- mine the temperature distributions for t = 50, 100, 200, and 500 s using a time increment of Llt = 1 s. You may use IHT to solve the finite-difference equations by representing the rate of change of the nodal temperatures by the intrinsic function, Der(T, t). Prepare a table summarizing the results of parts (a), (b), and (c). Comment on the relative differences of the predicted temperatures. Hint: See the Solver/Intrinsic Functions section of IHT/Help or the IHT Examples menu (Example 5.2) for guidance on using the Der(T, t) function.
Steel-reinforced concrete pillars are used in the con- struction of large buildings. Structural failure can occur at high temperatures due to a fire because of softening of the metal core. Consider a 200-mm-thick composite pillar consisting of a central steel core (50 mm thick) sandwiched between two 75-mm-thick concrete walls. The pillar is at a uniform initial temperature of Ti =
27°C and is suddenly exposed to combustion products
at Too = 900°C, h = 40 W/m2 · K on both exposed sur- faces. The surroundings temperature is also 900°C.
(a) Using an implicit finite difference method with Llx = 10 mm and Llt = 100 s, determine the tem- perature of the exposed concrete surface and the center of the steel plate at t = 10,000 s. Steel
properties are: ks = 55 W/m · K, ps = 7850 kg/m3, and cs = 450 J/kg · K. Concrete properties are: kc = 1.4 W/m · K, pc = 2300 kg/m3, cc = 880 J/kg · K, and e= 0.90. Plot the maximum and minimum
concrete temperatures along with the maximum and minimum steel temperatures over the duration 0 t 10,000 s.
(b) Repeat part (a) but account for a thermal contact resistance of Rt”,c = 0.20 m2 · K/W at the concrete- steel interface.
(c) At t = 10,000 s, the fire is extinguished, and the surroundings and ambient temperatures return to Too = Tsur = 27°C. Using the same convection heat transfer coefficient and emissivity as in parts (a) and (b), determine the maximum steel temperature and the critical time at which the maximum steel temperature occurs for cases with and without the contact resistance. Plot the concrete surface tem- perature, the concrete temperature adjacent to the steel, and the steel temperatures over the duration 10,000 t 20,000 s.
Consider the bonding operation described in Problem 3.115, which was analyzed under steady-state condi- tions. In this case, however, the laser will be used to heat the film for a prescribed period of time, creating the transient heating situation shown in the sketch.
The strip is initially at 25°C and the laser provides a uniform flux of 85,000 W/m2 over a time interval of Llton = 10 s. The system dimensions and thermo- physical properties remain the same, but the convec- tion coefficient to the ambient air at 25°C is now 100 W/m2 · K and w1 = 44 mm.
Using an implicit finite-difference method with Llx = 4 mm and Llt = 1 s, obtain temperature histories for 0 t 30 s at the center and film edge, T(0, t) and T(w1/2, t), respectively, to determine if the adhesive is satisfactorily cured above 90°C for 10 s and if its degradation temperature of 200°C is exceeded.
5.132 One end of a stainless steel (AISI 316) rod of diameter 10 mm and length 0.16 m is inserted into a fixture maintained at 200°C. The rod, covered with an insulat- ing sleeve, reaches a uniform temperature throughout its length. When the sleeve is removed, the rod is sub- jected to ambient air at 25°C such that the convection
heat transfer coefficient is 30 W/m2 · K.
(a) Using the explicit finite-difference technique with a space increment of Llx = 0.016 m, estimate the time required for the midlength of the rod to reach 100°C.
(b) With Llx = 0.016 m and Llt = 10 s, compute T(x, t) for 0 t t1, where t1 is the time required for the midlength of the rod to reach 50°C. Plot the temperature distribution for t = 0, 200 s, 400 s, and t1.
A tantalum rod of diameter 3 mm and length 120 mm is supported by two electrodes within a large vacuum enclosure. Initially the rod is in equilibrium with the electrodes and its surroundings, which are main- tained at 300 K. Suddenly, an electrical current, I = 80 A, is passed through the rod. Assume the emissivity of the rod is 0.1 and the electrical resistiv-
ity is 95 X 10-8 D · m. Use Table A.1 to obtain the
other thermophysical properties required in your solution. Use a finite-difference method with a space increment of 10 mm.
Laser source, q”o
t 300 K
Electrode, 300 K
(a) Estimate the time required for the midlength of the rod to reach 1000 K.
(b) Determine the steady-state temperature distribu- tion and estimate approximately how long it will take to reach this condition.
A support rod (k = 15 W/m · K, a = 4.0 X 10-6 m2/s) of diameter D = 15 mm and length L = 100 mm spans a channel whose walls are maintained at a temperature of Tb = 300 K. Suddenly, the rod is exposed to a cross flow of hot gases for which Too = 600 K and h = 75 W/m2 · K. The channel walls are cooled and remain at 300 K.
Tb = 300 K
(b) The foil is operating under steady-state conditions when, suddenly, the ion beam is deactivated. Obtain a plot of the subsequent midspan temperature–time history. How long does it take for the hottest point on the foil to cool to 315 K, a safe-to-touch condition?
Circuit boards are treated by heating a stack of them under high pressure as illustrated in Problem 5.45 and described further in Problem 5.46. A finite-difference method of solution is sought with two additional con- siderations. First, the book is to be treated as having distributed, rather than lumped, characteristics, by using a grid spacing of Llx = 2.36 mm with nodes at the center of the individual circuit board or plate. Sec- ond, rather than bringing the platens to 190°C in one sudden change, the heating schedule Tp(t) shown in the sketch is to be used to minimize excessive thermal stresses induced by rapidly changing thermal gradients in the vicinity of the platens.
(a) Using an appropriate numerical technique, deter- mine the thermal response of the rod to the con- vective heating. Plot the midspan temperature as a function of elapsed time. Using an appropriate analytical model of the rod, determine the steady- state temperature distribution, and compare the result with that obtained numerically for very long elapsed times.
(b) After the rod has reached steady-state conditions,
Text Box: Tp (°C)160