Tagliatela College of Engineering

EAS112 – Methods of Engineering Analysis

TO: EAS 112 Students March 27, 2013

FROM: Dr. Collura, EAS112 Instructor

Subject: Project 2, Optimum Pipe Insulation

A long pipe is to be installed to transport steam from a boiler to another part of the plant. Insulation is

needed on the pipe for both safety and economic reasons. You are to develop a spreadsheet to

calculate the surface temperature of the insulated pipe and to model the heat loss to the surrounding

air as a function of the thickness of insulation on the outside of the pipe. Your model should allow for

variation in the key parameters to explore the effect of various changes. Using data generated by your

model, select the best insulation thickness to maximize the present value of net savings in comparison

to an un-insulated pipe. Provide appropriate plots and data tables to support your decision and to show

the financial penalty for using a different insulation thickness.

Heat Loss Calculation

The steam pipe is to be made from schedule 40 steel with a diameter in the range of 2 to 3.5 inches

(nominal pipe size). The pipe will be encased in fiberglass insulation with an aluminum sheet cladding to

protect from weather. Heat loss for this case can be modeled using a combination of convection and

conduction heat transfer rate equations. Heat from the steam is transferred to the inside wall of the

pipe by forced convection, then through each of three layers by conduction (pipe wall, insulation,

cladding) and finally from the outside of the cladding to the surrounding air by natural convection. The

governing equations are shown below to calculate heat transferred per unit length of pipe:

QS = rate of heat lost by

steam to inside pipe

wall

Q1, Q2, Q3 = rates of

heat transferred

through pipe wall,

insulation, aluminum

cladding, respectively

QA=rate of heat lost to air

A a A A

S s s s

Q h r T T C T T

C T T

r r

k T T Q

C T T

r r

k T T Q

C T T

r r

k T T Q

Q h r T T C T T

4 4 5 4

4 3 4

4 3

3 3 4

3

3 2 3

3 2

2 2 3

2

2 1 2

2 1

1 1 2

1

1 1 1 1

2

ln

2

ln

2

ln

2

2

– per meter length of steam pipe

Heat Transfer Rate for Each Layer

S

S

S

S

S

r1 r2

r3 r4

View of pipe looking along axis

Aluminum

Cladding

Fiberglass

Insulation

Steel Pipe

STEAM

The heat transfer rate equations include constants for the thermal conductivity of the materials and

heat transfer coefficients for the convective situations. Values for these will be fixed for the analysis.

The temperatures of the steam and the air will be fixed values, but the temperatures at each surface will

be dependent on the thickness of insulation and size of the pipe. The subscripts used for the

temperatures correspond to radial distances from the center of the pipe. The radii values will be fixed

for a particular case of pipe size and insulation thickness, but will be varied as part of the optimization

work. The intermediate temperatures, to be found by simultaneous solution of the equation set, are:

T1 = temperature of the inside wall of the pipe, at distance r1 from the pipe center axis

T2 = temperature of the outside pipe wall and the inside of the insulation, distance r2

T3=temperature of the outside of the insulation and inside of the aluminum cladding, distance r3

T4 = temperature of the outside surface of the cladding, exposed to the air, at distance r4

Average steady-state conditions will be used for the analysis of each case, thus the rate of heat lost from

the steam must equal the rate of heat transferred through each layer and ultimately the rate of heat lost

from the outside cladding to the air. Thus four linear equations can be obtained by setting QA = Q1,

Q1=Q2, etc. The resulting equations can be solved using matrix techniques to find the unknown

temperatures. Any one of the heat rate equations can then be used to find the heat loss rate. The

constants (h’s , k’s, ʌ and numbers) and the parameters (radii values) become the coefficients, and are

shown in the equations above as C1 through C4. For a given case, these will be easily calculated. Terms

containing the steam and air temperature are also constants (shift to the right side of equation). For

example, setting QS = Q1 and Q1 = Q2 results in the following:

Similar equations result from setting Q2 = Q3 and Q3 = QA.

Your spreadsheet should have a data section for setting the pipe diameter and insulation thickness along

with values for the constants, such as steam and air temperatures, thermal conductivity values, cost

information, etc. Develop the model such that entry of a pipe diameter and an insulation thickness

results in determination of the 4 temperatures and the rate of heat loss for the full pipe length.

Analysis of Insulation Thickness

Using your model, determine the optimum insulation thickness for different pipe diameters to achieve a

maximum net present value of savings. Savings here is defined as the dollar value of energy NOT lost as

a result of the insulation. To calculate this you must first determine the heat that would be lost if the

pipe was not insulated. Simply subtract the heat loss for a particular insulation thickness from the bare

pipe heat loss to determine the energy savings. The cost to insulate the pipe includes both the material

cost and the installation labor. A net installed cost is found by multiplying the material cost by an

0

C

Steady -State Heat Flow Rearranged for Matrix Solution :

1 2 2 1 2 3 2 3 2 1 2 3 2 3 3

1 1 1 2 1 2 1 2 1 2 2 1

Q Q C T T C T T C T C C T C T

QS Q C TS T C T T C T C T C TS

installation factor to account for labor and other installation expenses. Data is provided at the end of

this memo for physical properties, cost information etc.

Optimization work requires an objective to be maximized or minimized. In this project the “objective

function” is the present value of savings over a 5 year period using a specific interest rate with monthly

compounding. The installed cost of insulating the pipe occurs at time zero (present) and is negative, so

this is subtracted from the present value of 5 years of savings. Varying the insulation thickness will

affect this value, so you can determine if there is an optimum which maximizes the present value. You

should also be aware of safety concerns associated with a long run of steam pipe. In particular you

should assure that the outside surface temperature is no higher than 50o

C.

Report Requirements

At present, the diameter of the steam pipe has not determined, but it will be between 2 and 3 ½ inch

schedule 40 steel pipe. Dimensions for standard steel pipe are available in the literature and should be

used in this project. After creating the spreadsheet model, you should run simulations for cases in

which you vary the insulation thickness from 0.1 to 6.0 cm. Prepare plots showing surface temperature,

installed cost, annual savings and net present value as a function of insulation thickness. Create other

plots as you deem necessary to justify your design decisions regarding the insulation thickness. A full

analysis of this type should be performed for one pipe diameter. In addition, you should determine the

optimum thickness and required thickness to achieve an acceptable surface temperature for all pipe

sizes in the range given above. Note that nominal pipe sizes in this range are incremented in ½ inch

steps. For each pipe size, recommend an insulation thickness.

Your memo should give an overview of the project, discuss your approach, present results and discuss

methods used and assumptions made. Tables and plots should appear in the memo to with explanation

to make your points. Your concluding paragraph should include a discussion of what you learned in

doing the project. Your spreadsheet should, of course, be well-documented and well-organized to show

clearly how the work was done. The spreadsheet should include the following features:

x List of pipe diameters using the data validation methods

x Retrieval of dimensions for pipes from a table keyed to the selected pipe size (use Vlookup)

x Scroll bar to set the insulation thickness

x Use of Solver to vary thickness to maximize present value of net savings

x Check box to select either scroll bar or Solver for varying the insulation thickness

x Use of a button to run solver

x A Sub to copy key results to a table, attached to another button

x Any additional functional features you wish to include to make the simulation tool more useful

The project is due Wednesday, April 17, 2013, with a paper submission of the memo and attached

printout of the spreadsheet. The spreadsheet should also be submitted via Blackboard. Required data

is found on the next page.

Input data for use in project

Properties of pipe, insulation and outer cladding material Financial Analysis Parameters

Item Material k, thermal

conductivity density cost *Install

Factor

Energy

cost

Annual

Interest

Rate

Period

of

analysis

W/m-C kg/m3 $/kg $/$ $/kWh percent years

Pipe steel 43 7800 NA 5 $0.04 3.0% 5

insulation fiber glass 0.055 64.1 30 Per

month

months

Outer Layer aluminum 206 2700 40 0.25% 60

0.5 mm thick

* Installed cost = (total material cost) x installation factor

Heat Transfer Coefficients

W

m2

–

o

C

Other Parameters

From steam

to inside

pipe wall

From

outside pipe

cladding to

air

Steam

Temperature

Air

Temperature

Pipe

Length

hS hA C C Meter

50 5 150 10 50

Properties of Standard Steel Pipe

Schedule

40 Pipe Diameters Wall

thickness

Pipe OD, cm ID, cm cm

2 6.033 5.25 o.39

2.5 7.303 6.271 0.52

3 8.89 7.792 0.55

3.5 10.16 9.012 0.57