Calhoun: The NPS Institutional Archive

Theses and Dissertations Thesis Collection

1948

Temperature distribution in a metal hollow cylinder of

finite radius under unsteady state conditions

Knox, Robert Junior

Annapolis, Maryland: Naval Postgraduate School

http://hdl.handle.net/10945/31614

TEMPERATURE tISTRIBUTION IN A M.ETAL HOLLOW CYLINDER OF

FINITE RADIUS UNDER TmSTEADY STATE CONDITIONS

by

Robert Junior Knox

Lieutenant Commander, United States Navy

Submitted in partial fulfillment of the requirements for the degree of

MASTER OF MECHANICAL ENGINEERING

United States Naval Postgraduate School Annapolis, Maryland

1948

This work is accepted as fulfilling ,

the thesis requirements for the degree of

MASTER OF MECHANICAL ENGINEERING

from the

United States Naval Postgraduate School.

- Sr. Prof. P. J. Kiefer

Chairman

Department of Mechanical Engineering

Approved:

Academic Dean

1

PREFACE

The u. s. Naval Engineering Experiment Station,

Annapolis, Maryland has recently run tests to determine

the effects of thermal shock on steam piping for naval

use. After several preliminary tests, data were taken

for a section of pipe heated by the flow of saturated steam

through the pipe. Thea e da:ta cons is.tad of temperature

readings at the inside and outside walls of the pipe

at time intervals of five seconds. The calculations

of the stresses set up by thermal shock require a know­

ledge of the temperature distribution in the wall at all

radii.

With the hope of assisting the Engineering Experiment

Station in the evaluation of these test data, the author

chose the task of finding the means of determining temper­

ature distribution in a pipe wall at unsteady state con­

ditions.

The author wishes to express his deep appreciation

to those who ~ave aided in this investigation. He is

especially grateful to Associate Professor George H. Lee,

u. s. Naval Postgraduate School, without whose assistance

and continued guidance this investigation would not have

been carried out and for his careful review of the complet­

ed manuscript.

The author is also grateful to the Engineering

ii

Experiment Station and its Doctor William c. Stewart for

suggesting the investigation and cooperating to the fullest

in furnishing data. He wishes further to express apprecia­

tion for the counsel of Doctor Warren M. Rohsenow, Mass­

achusetts Institute of Technology, Cambridge, Massachusetts.

iii

TABLE OF CONTENTS

Title

Certificate of Approval.

Preface.

List of Illustrations.

Table of Symbols and Abbreviations.

Chapter I Introduction.

Chapter II Origin of the Data.

Chapter III Analytic Solution.

1. The general solution.

2. Solutions to similar problems.

3. Further investigations.

Chapter IV Graphical Solution.

Page

1

11

v

vi

l

5

8

9

10

14

19

1. Available methods. 19

2. A graphical solution of the EES problem. 20

Bibliography

Appendix I

Appendix II

Appendix III

Appendix IV

Appendix V

Appendix VI

iv

23

25

28

30

33

35

36

LIST OF ILLUSTRATIONS

Figure Title

1 Temperature vs. time graph; heating medium, saturated steam.

2

4

5

6

Temperature vs. time graph; heating medium, superheated steam.

Temperature VS• time graph; heating medium, exhaust gas •.

Temperature vs. time graph; induction heating.

Temperature vs. time graph; heating medium, exhaust gas.

Temperature vs. time graph; heating medium, saturated steam.

Page

40

41

42

7 Direct graphical solution (detailed). 44

8 Temperature vs. time graph; heating medium, saturated steam. 45

9 Basis for a graphical solution. 30

10 Scale for the direct graphical method. 33

11 Temperature vs. radius; heating medium, exhaust gas. 46

12 Direct graphical solution (rapid) •. _ 47

.. v

TABLE OF SYMBOLS AND JU3BREVIATIONS

B - British thermal unit

c - specific heat - B/lb/F

E - Modulus of elasticity in tension and compression

F - degrees fahrenheit

f - unit surface conductance - B/ft2 hr F

h - f /k ft-1

Jn - Bessel function first kind, order n.

k - Thermal conductivity - B/hr/ft2/ft/F

Nn - Bessel function (Neumann) second kind, order n.

n - any integer

r - radius

ra - inside radius of a hollow cylinder

rb - outside radius of a hollow cylinder

t - time

Yn - Bessel function second kind, order n.

o( - diffusivity - k/cf' in2/sec.

/5' - coefficient of expansion

~l - instantaneous surface temperature

Ae - small finite difference in temperature

Ar small finite difference in radius

4t - small finite difference in time

e - temperature

9a - temperature at inside radius

eb - temperature at outside radius

vi

er±.4r - temperature at same time at r + Ll r and at r-Ar.

Qt - temperature at time t.

Gt±P. t - temperature at time t + Ll t and t- flt at any one radius. ·

G/\ - temperature at sur£ace j\.

g - temperature at same time at A+dA and ~-~/\. .i\t.ftA.

./\. -- 1n r/ra•

- Poisson's ratio (unit lateral contraction/unit axial elongation).

- density - lb/in3.

<Jr - radial stress.

6'f" -- tangential stress.

~ - any £unction.

B&S -- Brown and Sharpe.

Cr - chromium.

EES - Engineering Experiment Station.

rt - £eet.

hr - hour.

I.D. - inside diameter.

IPS - inside pipe size.

in - inches.

ln - logarithm to Naperie.n base.

log - logarithm to base 10.

Ni - nickle.

O.D. -- outs:ide diameter.

psi - lbs/in2.

oO - in£inity.

vii

CHAPTER I

INTRODUCTION

A metal pipe initially of uniform temperature, is sub­

jected to thermal shock by the sudden passage of a heating

medium through the pipe. What the temperature distribution

is in the pipe wall before steady· state conditions are at­

tained is the subject of this investigation. The author

first became aware of this problem in August 1947 during

an informal discussion with Doctor W. C. Stewart of the EES.

Thermal shock in the operation of a Navy steam propul­

sion plant is an abnormal condition. The possibility of

thermal shock conditions through a casualty is however always

present. High temperature shock can be the result of failure

to properly "warm up" a main steam line before admitting high

pressure saturated steam from the boiler. Cold shock can be

the result of "carry-over" of a water slug from the boiler into

the lines which have been carrying superheated steam. The

probiem was assigned the EES for investigation.

By August 1947 the EES had obtained a sizeable amount of

data from cyclic thermal shock tests. These tests were run on

various sections of pipe to determine the physical effects of

cyclic thermal shocking. Heating ha.d been accomplished by

using exhaust gases, superheated steam, saturated steam and

induction coils. Saturated steam produced the greatest temp­

erature gradient through the pipe wall and therefore the

1

greatest shock. Cooling had been done by water or com­

pressed air.

Most of the tests indicated temperature readings only

for thermocouples located at the inside and outside walls.

A_few test specimens were fitted with three thermocouples,

the third thermocouple being located at the mean radius of

the pipe wall. The reading of the thermocouples were taken

at time intervals of five· seconds.

Calculations of .thermal shock stresses had been made

by the EES for representative tests. The author disagrees

with calculations made by the EES for the determination of

temperature distributions in the pipe wall in the unsteady

state condition. Their means of solving for the temperature

gradient is a graphical solution accomplished by a step-by­

step method of averages for layers taken such that r 2/r1 =

r1/ra~ a constant. The method fails to fit the data when

the outside wall begins to increase in temperature. In

the particular problem discussed in this paper that con­

dition arises at time equal to six seconds. To allow for

this inconsistency the EES progresses by adjusting the data

to agree with the boundary conditions.

Literature dealing with the analytic solution of

temperature distribution in a hollow cylinder is limited

to highly idealized problems. Peculiarly, all published

solutions have one connnon boundary condition, namely,

an instantaneous surface heat source which maintains the

surface at a constant temperature. This condition is not

2

present in the EES problem nor in any other actual problem.

The combination of three conditions is the cause for

the complexity of thi.s .particular problem. These three

conditions are: l} steel pipe has a high thermal conduc­

tivity; 2) the heat source in the EES problem produces a

variable surface temperature and; 3) the heat source pro-

duces a variable rate of change of the surface temperatureo

The analytic solutions published in the literature are of

little value as related to the EES problem.

Graphical solutions in the literature are developed

for the elementary boundary conditions only. The theory

of these, however, can be applied to the EES problem. The

author carries out this application for the case of heat-,_

ing only and presents results which he feels are more accurate

than those obtained to d~te by the EES.

Attempts to find an analytic solution have proven un­

successful. In the light of the authors investigations,

there does not appear to be an analytic solution to this

problem at present.

It has become quite obvious, as a result of this

investigation, that more detailed data should be taken in

future tests. The installation of several thermocouples

at proper radial points in the pipe walls would be desir­

able. The maximum temperature rate and consequently the

maxlmum thermal shock in the test occurs within five seconds

after heating begins. Automatic recording devices attached

3

to the thermocouples may be the answer to better data in

the critical time interval.

The numerical results in this paper are only as

reliable as the data taken. J..11 values of temperature

prior to a time of five seconds have been obtained by

extrapolation. A plot of the data taken when three

thermQcouples were installed in the test indicates a

lag in the temperature readings.* This may be an error

introduced by the use of thermocouples. The lag on the

other hand may be only the result of personnel errors in

taking the temperature readings. It is necessary there­

fore to consider the value of this paper primarily in

terms of the theory presented and its application to a

specific problemo

* C.f. Appendix V.

4

CHAPTER II

ORIGIN OF TEE DATA

Tests were run on a length of 25-20 Cr-Ni, 3/4 inch

IPS, schedule "double extre. strong" pipe using saturated

steam, superheated steam and exhaust gas as the heating

medium. The resultant curves of inside and outside

temperatures vs. time are shown in figures 1, 2 and

3 respectively.

One thermocouple was peened to the outside wall at

mid length of the tubeo Another thermocouple was brazed

to the bottom of a ~ inch diameter hole drilled to within

0.01 inch of the inside wall, 1/8 inch distance from the

peened in thermocouple. In the test with exhaust gas,

however, the latter· thermocouple was pressed against the

bottom of the hole by means of an 18-8 Cr-Ni steel screw

insulated from the thermocoupleo The thermocouple wire

used was No. 29 B & S gage iron-constantan. The inlet

and outlet pipes carrying the heating medium were drawn

up tightly against the counterbored ends of the test

sample by means of bolts secured to flanges welded to the

pipese Insulating gaskets were placed in the counter­

bored ends of the test sample to I!linimize heat transfer

due to conduction.

The specially bladed turbo supercharger wheel at the

EES was employed_to supply the combustion gas using gas

5

temperatures ranging from l040F to 1360F. Other runs

were made by passing superheat~d steam at 500 psi pressure

and 1050-1070F temperature through the sample. A third

set of test runs was made by passing saturated steam at

500 psi pressure and 470F temperature through the teat

pipe.

In addition to these runs a series of tests was

made on a similar sample of 18-8 Cr-Ni steel pipe, 3 1/4

inch O.D. and l 3/4 inch I.D. with thermocouples located

.o4r inches and .403 inches from the inside wall and

a third peened to the outside wall. Thermal shock

was created by means of induction heating. A plot of

the temperature vs. time curves for all three thermo­

couples is shown in :figure 4.

Runs were also made on an 18-8 Cr-Ni pipe, 3 1/2

inches o.D. and 2 inches I.D. with thermocouples located

.01 inches and .306 inches from the inner wall and a third

peened to the outer wall. Thermal shock was created

by means of exhaust gas at temperatures ranging :from

1260-1450F. A plot of the temperature vs. time curves

for all three thermocouples is shown in :figure 5.

The results of these tests clearly indicated that

saturated steam produced the greatest rate of temperature

rise. Thermal shock tests were then planned using sat­

urated steam as the heating medium. A section of two

inch 25-20 Cr-Ni pipe 1.189 inches outside radius,

6

.749 inches inside radius was set up for test in the same

manner as the initial tests. Thermocouples were located

· as for the previous test runs with saturated steam. A

representative plot of the resultant temperature curve is

shown in figure 6.

7

CHAPTER III

ANALYTIC SOLUTION

To calculate the thermal stresses in a finite hollow

cylinder the following formulae given by Timoshenko ( 16 )~·

are used:

(1) 0:= BE t c1- 7J >r2

It is_ not the author's intention to show the

derivation of the above formulas since they follow a

straight fot>ward method and are presented clearly by

Timoshanko'. The only purpose in introducing them is

merely to show their relation to the temperature dis­

tribution in a thick cylinder.

To solve the integral .J"Grdr it is necessary to

obtain the temperature distribution through the pipe

wall. It is therefore necessary to determine the

analytic solution or to use a graphical method of

solution._ The Schmidt graphical method for a slab

* Numbers in brackets refer to bibliography.

8

(14)(13}(11) can be adapted to a cylinder wall but large

errors are introduced during the early stages of the

solution.

The maximum thermal stresses will occur in the early

stages of heat transfer. It is therefore readily seen that

an analytic solution is more desirable than the Schmidt

graphical solution.

1. The general solution.

Assume the initial uniform temperature of the body as

zero, the boundary conditions of the problem are:

at t = O and r :: r a

at t = O and r > r a

at t=t and r=r a

at t :::: t and r>r a

e =O.

e = o. e = ¢ Ct)

a . Q=¢ (t) , r

The differential equation for heat conduction in a

cylinder (Fourier-Poisson) when radial changes only are

observed is:

( 3) 1 r

~ Q ) Jr

A general solution is obtained by assuming Q = R T where R

is a function of r only and T is a function of t only.

Differentiating and substituting:

9

{4) l o<.T

::: _l c-~~ +~ JR\_ R d r 2 r dr J -

where_/{ 2 is any arbitrary constant.

2

-/

A general solution therefore since T ::= e (Cl -.<;,}t)

and R :::A1J0 (,;< r) + B1N0 (,){r)

is~

oO

(5) Q = I ~J 0{/ r) +BN0 (__,,.u r)

I

where A and B are constants of integration.

A particular solution occurs at steady state when

JQ/ J t = 0 the solution having the form:

( 6) e = c log r + D

where C a.nd D 9.re constants of integration.

Under the conditions where the pipe wall eventually reaches

a steady state cpndition the general solution is;

oO

(7) Q =-C log r +D - I. ~ ... J0(~ r) + BN0

(/fr) e-o<.A2

t I

2. Solutions to similar problems.

Barker (1) and Lees (9) give the steady state

temperature distribution which is of interest but of little

value.

Barker's form:

10

(8) e

Churchill (3) and Williamson and Adams (l?) present

papers which give the analytic solution for the transient

temperature distribution in a solid cylinder. The initial

tempera.ture is uniform. The surf'ace temperature rises

instantaneously to a given temperature and is maintained

at that value.,

Williamson and Adams form:

(9) [

l - 2 ) ao Jo (.unr/l']) )

fJnJ1 <µ. n> /11=/ .

where A is the nth root of J 0 (J>·):::: o.

Carslaw and Jaeger (2)* give a solution for the case

of the infinite hollow cylinder.

Boundary conditions:

at t =O and

at t) 0 and

Solution:

r=:r ; a

r=r ; a

* c .r. Appendix I

e=o 9= 9 l

11

Dahl (4) gives a solution for the case of the finite

hollow cylinder. The initial temperature is uniform. The

inner surface temperature is suddenly raised to a value

and maintained at that value. The outer wall temperature

is maintained at the initial temperature.

Boundary conditions:

at t::: 0 and r=r8 ;

at t=O and r> ra;

at t=t and r= rar

at t=t and r= rb;

Solution:-

( 11 ) Q. = 91 log I'bf r log rb/'ra

9 ==91

9=0

Q =91

Q :::o

n=oo

L 2 J ( r ) -~ t - a[J ( 11 r} _ , __ o CAn b N r 11 r ~ e Pn -n o y-·n N ( .u ,...,,__ ) o v--n

n=l oy n-o

12

Where_µ has the values of the positive roots of:

(llc}

Jahnke und Emde (6) present excellent tables for the

evaluation of bothp and the first two orders of N.

Carslaw and Jaeger (2) also present the case of a

finite hollow cylinder. The inside surface temperature

is raised instantaneously and maintained constant. The

outer wall is adjacent to an infinite fluid which maintains

its temperature constant at a point distant from the

outer wall.

Boundary conditions:

at t = 0 and r a~ r L rb; e = O

at t > O and r = r a ; Q = G1

at t > o and r = rb ; }Q/c} r + f /k ( e) = o

where f is the unit surface conductance.

Solution:

13

where fln are the positive roots of:

( 12al J 0 (,,U. r al [hY0 ()-' rbl - µY1 (ft rbl] -

Y0 (,..Ur al [ hJ 0 ( }A rb) - )"-J 1 (prb] - 0

where h is the ratio f /k.

Jo Further investigations.

Dahl's equation appeared to be applicable at first.

Upon further investigation it was found that the actual bound-

ary conditions were not in agreement with Dahl's boundary

conditions. P..n attempt to apply temperature differences in a

step-by-step manner failed at the second step since Dahl's

equation no longer applied without an initial uniform temp­

erature distribution. Further consideration of Dahl's

equation was therefore abandoned.

For any one radius in the EES problem Q = ¢( t) is of

the form of Froelish's equation (18) for the saturation

curve.

(13) x t Q=----y + t

where x and y are arbitrary constants to be determined from

the shape of the curve.

14

At t ::: 00 and r=r . Q := Q a' a

at t =ex> and r = rb; Q:: Qb

at t ::oO and r a<. r < rb; Q =Barker's steady state distribu-

ti on. If the general solution were: 00

(14) Q=C+ I~J0(_;«r) + BN0(/rIT

I

then at t = o0 the second term disappears. Let C equal the

steady state condition and the final boundary condition

is satisfied. Although both inside and outside wall temp­

eratures are functions of time; they also satisfy this

boundary condition. They become constant at infinite time.

At time equal to zero and r= rb the Barker term becomes

zero (initial temperature) and the constant B of the Bessel

term can be evaluated. Also at time equal to zero and

r ~ra,/< must satisfy the roots of the Bessel term in the

above equation. It now remains only to evaluate the con­

stant A. This unfortunately is impossible. At zero time,

zero temperature, r = r, there exists only the value of zero

for A and the solution thus fails.

This practical problem fails to produce the very con-

venient boundary condition of an infinite surface heat

source. The EES is working ·with iron pipe of very high

thermal conductivity. The problem primarily is centered

around the initial two or three second period of time.

Boundary conditions used by Dahl (4) or Carslaw and Jaeger

15

(2) cannot be assumed. The problem cannot be stretched as

in the case of Perry and Berggren (12) who carefully choose

the cork lagging on a metal pipe and then work only with

the cork cylinder •.

Another particular solution is to let JQ/~t = c1 •

The solution of the differential equation is then:*

(15)

Where B1 and D1 are constants of integration;· e-= Q8

- eb.

Another general solution therefore might be:

(16) 9r.+B11n r +D1 +c1~ -r ~J0~r) + BN0 (/lr~ e-"'J'2t

This equation can b~ used only for a limited time since the

first term increases with time. The entire solution breaks

down on a substitution of initial conditions. At all values

of r, time equal to zero, the first term and the temperature

difference.Cea - eb) are both zero. Again A equal to zero

is the only choice for evaluating A.. Unless e is considered

as ea- 9b1 the solution is merely for a condition in which

all temperatures are rising at the same rate at all radii.

For a perfectly insulated outer wall the problem is

only slightly modified.. The outer and inner walls rise to

* c.£. Appendix II

16

the same value at infinite time. The Bessel term of the

general solution for the differential equation is alone

not enough since it becomes zero at infinite time. Assume

a general form:

00

(17) e =.!S(r,t) -L~Jo(}'r) +- BNolf'r~ I

where ,0 (r, t) -= c2 when t =. o0

and g5 ( r, t ) = O when t = o.

Boundary conditions again permit evaluation of B and j>-,

but again all terms go to zero when we attempt to evaluate

the constant A.

Assume the solution of the differential equation to

be Q =R+ T., Differentiation and substitution results in:

(J.8)

On inspection it is seen:

(19)

thus the result is the same as the solution of ae/~t-== cl. As a result of the above L.'1.vestigation there does not

appear, applying the above approach, to be an analytic

solution to the EES problem. Through necessity, rather than

17

through preference, an attempt will be made to find a

graphical solutiono

CHAPTER IV

GRAPHICAL SOLUTION

1. Available methods.

·Schmidt's graphical method (14) is basically for a

slab. It can be adapted to the cylinder, but as previously

stated, it is very much in error during the early time in­

tervals. Williamson and Adams (17) point out, and Fishenden

and Saunders (5) confirm the fact, that Lord Kelvin's (10)

probability integral more nearly approximates the initial

thermal wave penetration. This improvement over the Schmidt

method was further advanced by Perry and Berggren (12).

They determine a. graphical solution for the heat flow

differential equation for cylinders using small finite -

differences.* This graphical solution is developed only

for the problem of a hollow cylinder with an instantaneous

surface heat source. The inside wall temperature is then

maintained constant. Conduction to an infinite fluid is

assumed to take place at the outer wall. The results

are compared with values obtained by using Carslaw's

and Jaeger's formula for the same condition.**

The equation given by Perry and Berggren is for the

temperature rise during a time interval at any given

* C.f. Appendix III

**c.r. Chapter III, equation (12).

19

radius (12) (7). It is merely a weighted mean value of

the temperature of the two adjacent radii before the time

interval, namely:

For very sma11·values of ~r or as ~r~o the equation

agrees with Schmidt's equation for a slab (14):

The initial curve following the Lord Kelvin probability

integral or Dahl's equation* could be used over that first

small time increment. Assume an instantaneous rise of

temperature at the inner wall for the short time increment.

The outer wall will remain at the initial uniform temper­

ature. Where !). t is. very small Dahl's equation has little,

if any, advantage over Lord Kelvin's integral.

2. A graphical solution of the EES problem.

It is desirable to have small increments of ~r.

** Since Jt is proportional to Ar2 , tbe error of the graphical

* C.f. Chapter III, equation (11) ** C.f. Appendix III.

20

solution can be reduced to a minimum by a proper choice

of!). r. By progressing through the graphical analysis

using a tedious mathematical solution for each point, an

accurate curve can be drawn. In the EES problem the outer

surface temperature increases with the inside temperature.

It is desirable to prevent a dip in the curve below the

outer wall temperature. Any temperature which graphically

is determined to be below the outer surface temperature

will automatically be given the same temperature as the

outer surface. This condition will occur during early

time intervals near the outer layer. An attempt to fair

in a curve would introduce possible errors on the unsaf'e

side of the actual temperature distribution curve. It

is considered better therefore to introduce a slight

safety factor by using the suggested procedure mentioned

above. A, similar procedure is used by Sherwood and Reed

( 15) for the ·slab.,

Jakob (7) and Perry and Berggren (12) give proof

that a scale of Ar plotted as J')J\.., where . .i\""' ln r/r a'

can be used in a plot against temperature to produce

a direct graphical method of solution.* The temperature

rise in a cylindrical segment for a time interval ~t

can be obtained merely by drawing a straight line on

the graph between et, A -Ll.I\. and et, il +- A A • The

intersection Qt+ A t, ./\. is the desired temperature.

* c.r. Appendix IV

21

A careful evaluation of the problem graphically

yields the results as shown in figure 7 and figure B.

It appears, therefore, that the graphical solution

suggested by Perry and E3rggren as applied to the EF.3

problGm is the only solution if it is desired to main­

tain the complex boundary conditionso It is also a

solution which can be calculated with comparative ease

by a direct graphical method.

BIBLIOGRAPHY

1. , Barker, L.H. The calculation of temperature stresses in tubes. Engineering. 124:443, October 1927.

2. Carslaw, H.S., and Jaeger, J.C. Some two-dimensional problems in conduction of heat with circular symetry. Proceedings of the London mathematical society. 2:46:361-388, 1940. .

3o Churchill, R.V. Modern operational mathematics in engineering. New York, McGraw-Hill, 1944.

4. Dahl, o.G.c. Temperature,, and stress distribution . in hollow cylinders. Transactions American society of mechanical engineers. 46:161-208, 1924.

5o Fishenden, M., and Saunders, O.A. The calculation of heat transmission. London, H.M.Stationery office, 1932.

6. Jahnke, E.) und Emde, R. Funktionentafeln mit formeln. und kurven. New York, Dover, 1943.

7. Jakob,· M:. Fortschritte der warmeforsohung. Zeit­schrift des vereines Deutscher ingenieure. 75:969, 1931.

-

8. Jakob, M., and Hawkins, G.A. Elements of heat trans­fer and insulation. New York, Wiley, 1942.

9. Lees, C.H. Thermal stresses in solid ~nd in hollow circular cylinders concentrically heated. Pro­ceedings Royal society of London, A:l01:411-430, 19220

lOo Lord Kelvin (Sir vVilliam Thomson)• Mathematical and physical papers. III:435, Cambridge University press, 1882-19110

11. McAdams, W.H. Heat .transmission. New York, McGraw­Hill, 1942.

12. Perry, R.L., and Berggren, ';l.P. Transient heat conduction in hqllow cylinders after sudden change of inner-surface temperature. University of California publications in engineering, 4:3:59-88, 1944.

23

13. Schack, A. (Translated by H. Goldschmidt and E.P. Partridge). Industrial heat transfer. New York, Wiley, 1933.

14. Schmidt, E. Ueber die anwendung der differenzen­rechnung auf technische anheiz- und abkuhlungs­probleme. Foeppls festschrift. Berlin, Springer, 1924.

· 15. Sherwood, T.K., and Reed, c.E. Applied mathematics in chemical engineering. New York, McGraw-Hill 1

1939.

16. Timoshenko, s., Theory of elasticity. New York, McGraw-Hill, 1934.

17.

18.

Williamson, F. D., and Adams, L. H. Temperature distribution in solids during heating and cool­ing. Physical review, 14:99-114, 1919.

Woodruff, L. F. Principles of electric power transmission and distribution. New York, Wiley, 1938.

24

·APPENDIX I

Carslaw and Jaeger (2) present the following solutions

in their paper "Some two-dimensional problems in conduction

of heat with circular symetry".

2t Cylinder of radius a, initial uniform temperature o, surf ace tempera~ure suddenly brought to Ql and maintained

constant.

J. Infinite hollow cylinder of radius a, initial uniform

temperature o, inner surface temperature suddenly brought

to e1 and maintained constant.

5. Solid cylinder of radius a, initial uniform temperature

e1 and radiation at the surface takes place into a medium

at temperature o. 6. Infinite hollow cylinder of radius a, initial uniform

temperature e1 and radiation takes place into a medium -9.t

temperature o. 7. Hollow cylinder of initial uniform temperature o.

a) Outer surface maintained at O while inner surface is suddenly raised and maintained at e1 •

b) Inner surface maintained at O while outer surface is suddenly raised and maintained at e1 •

c) Inner surface raised suddenly to Qe. and maintained while outer surface is suddenly raised to 9b and maintained.

8. Rollow cylinder of initial uniform temperature o.

i) Outer surface suddenly raised to Qb and maintained constant while inner surface radiates to a medium of temperature ea•

* Numbers of _problems correspond to those used by Carslaw and Jaeger.

25

ii}

iii)

Inner nurface suddenly raised. to 9 and maintained constant while outer surface radiaies to a medium at temperature Qb.

Inner surface radiates to a medium at temperature 9a while outer surface radiates to a medium at temperature Gb.

9. Solid cylinder with an instantaneous heat source at

the axiso

a) Outer surface temperature maintained at Oo

b) Outer surface perfectly insulated.

c) Outer surface radiates to a medium at temperature

lOo Solid cylinder with an instantaneous surface heat

source at any radius r.

a) Outer surface temperature maintained e.t o.

b) Outer surf ace perfectly insulated.

c) Outer surface radiates to a medium at temperature

11. Solid cylinder with an initial temperature a ¢(r).

i) Outer surface temperature maintained at Oo

ii) Outer surface perfectly insulated.

o.

o.

iii) Outer surface radiates to a medium at temperature o. 12. Infinite hollow cylinder with an instantaneous surface

heat source at any radius r.

a) Inner surf ace temperature maintained at o. b) Inner surface perfectly insulated.

c) Inner surface radiates to a med.ium at temper-ature o.

13. Infinite hollow cylinder with an instantaneous surface

heat source at any radius rand initial temperature a ¢(r).

26

i) Inner surface temperature maintained at o.

ii) Inner surface perfectly insulated.

iii) Inner surf ace radiates to a medium at temper-ature o.

14. Hollow cylinder with an instantaneous surface heat

source at any radius r.

a) Inside and outside walls maintained at temper­atur.e o.

i) Outer wall maintained at temperature 0 while the inner wall radiates to a medium at temperature o.

ii) Inner wall maintained at temperature 0 while the outer wall radiates to a medium at temperature o.

iii) Inner and outer walls both radiate to a medium at temperature Oo

15. Hollow cylinder with an instantaneous heat source at

any surface of radius rand an initial temperature a ¢(r).

i} Inside and outside walls maintained at temper­ature o.

ii) Outer wall maintained at temperature 0 while the inner wall radiates to a medium at temperature o.

iii) Inner wall maintained at temperature 0 while the outer wall radiates to a medium e. t terrrperature o.

iv) Inner and outer walls both radiate to a medium at temperature o.

27

APPENDIX II

A particular solution to the differential equation

for heat flow in a cylinder can be developed f'or ;)e/~t = a

constant. The differential equation becomes:

d2e/dr2 + l/r • de/dr = c1

Let d9/dr = y, then

dy/dr + l/r • y =C1

Let v ,,,. y/r, then

or

Since v ~y/r

dy/dr = v + r dv/dr

2v -tr dv/dr =C1

J ~ r /c1~v2v

( r2) ( C -- 2v) => B l

Since y::::: d9/dr

28

c1r 2 - 2r d9/dr = B

jc1r dr -fa dQ ::o Bf d;

c 1 r2 - 49 = B' ln r +- D

Q = i C 1 r2 + B' ' ln r t- D'

A particular solution to the partial differential heat

transfer equation:

(3)

is

when

a9/~t ~ a constant c1

29

J~.PPENDIX III

A graphical interpretation of the differential

equation for heat transfer in cylinders is presented by

Ferry e.nd Berggren ( 12}. The differential equation

when axial temperature changes are zero is: ,

( 3) = o( f-) 29 + .1:._ k) \ ~ r 2 r ';)r

In terms of small ~inite differences the above equation

becomes:

(22) A!L. = o( (.?''\ + _l_ ~) t'.lt \Li r r l.l r

e

Figure 9

Basis for a graphical solution

30

From figure 9 the following equations can be readily

determined:

1"' Qt - Qt ~= +At,r ,r

.flt L1t

.1Q Q - Qt, t,r + L1 r r- fir --::

L\r 2 Ll r

Q - Qt Q Q f\29 t 1r + /J. r Ir t 2r - t 1r- D.r

= L\ A

.1r2 !1r

Upon substituting these values into the equation of

finite differences:

1

r

Qt 2r +- A r-2Qt ,r+ Qt ,r- Ar

( L\r)2

e - Q. t , r + Ll r t , r- /J. r

2 .1 r

Let ~t = ( 4 r) 2/2o<.. , and multiplying by ( L\ r) 2 :

.31

-::.

+

2( Q -Q ) "" g -2e + g + t+llt,r t,r t,r+Ar t,r t,r- Llr

..1r -2r

(g - g t , r + a r t , r- .1 r}

(20) 9 - l. ~l+-~) 9 + {1-t+llt,r- 2 L' 2r t,r+Llr ~)9 J ~r t ,r- llr

~he resultant equation is o~e which is a weighted mean

favoring the side of increased radius and the side of

greater heat capacity.

32

APPENDIX IV

A plot of temperature vs. ln r/ra can be used for a

direct graphical method of solution.

e

t

I ~,n-.o.1\

I I I I '

. I I

I I I I I e I

't,ll I

Figure 10

I I I

Scale for the direct graphical method

From the geometry of the above figure it is apparent

that:

But from

33

r + 2

but r - 6.r/2 and r+ llr/2 are the mid radii of the two

adjacent D.r layers of ro The scale division therefore

must be inversely proportional to their corresponding

radii.

This is satisfied by choosing a scale variable

so that d/\. = dr/r or Ll A.= llr/r

The mean radius of LL\.1 is r - Ar/2 and similarly for

L1A.2 is r + t:.r/2.

Then:

Lli'3_ = ~r L1A2 :::: Llr

r - nr r+ .llr ---;r- -r

and

These value coincide with the values required by figure 10.

This scale for plotting radius therefore provides a means

of making a direct graphical solution,

APPENDIX V

A plot of steady state temperature distribution radial­

ly in a hollow cylinder is a straight line when plotted on

coordinates of temperature vs. ln r/ra• The data as shown

in figure 5 represents the readings of three thermocouples

in a pipe wall. This data was replotted in figure 11 as

temperature vs. ln r/ra•

Two peculiarities are apparent upon examination of the

resultant curves. Most apparent is the change of curvature

from concave up to concave down after ninety seconds. The

second peculiarity at ninety seconds is a change from increas­

ing slopes with time to decreasing slopes with time. The

latter is the effect of the wave front a·s the heat penetrates

the pipe wall.

The change from concave up to concave down indicates

at once a descrepancy in the data. Since steady state is

the ultimate condition which can exist under the conditions

of the test, the concave down curves are an impossibility.

The author is of the opinion that the plot reveals a

lag in temperature readings. This may be an error introduced

by either the thermocouples, the indicating instrument, or

both. The lag, on the other hand, may be only the result of

personnel errors in taking the temperature readings. The

existence of this lag must, however, be acknowledged.

35

APPENDIX VI

The tedious graphical solution represented by figure

7 immediately caused the author to investigate the possibil­

ity of a more rapid graphical solution. A graphical solu­

tion, represented by figure 12, was then made for purposes

of comparison.

The first sixteen steps of both solutions are ident­

ical. Time increments in figure 12 are then rapidly in-.

creased to speed up construction. The time to produce

figure 12 was approximately one half the time required for

figure 7. A comparison of resultant values of temperatures

for r1, r2 and r3 from both figures was then made.

A maximum deviation of 6 degrees representing a 4%

error oc·cured for r1, at 1.512 seconds. This error rapidly

decreases and disappears after 6.048 seconds. For r 2

the maximum deviation was 4 degrees for a 2.8% error at 2.1168

seconds. The trend of a decreasing error at an increasing

time for that maximum error with an increase in radius was

noted for r3• At r 3 the maximum deviation was 3 degrees

for an error of 2.7%. The time was 3~628 seconds. The

deviation tends to disappear at r 2 and r 3 as time increases;

this is a trend characteristic of any graphical solution of

this type (15).

The maximum error introduced by the rapid graphical

solution is small and tends to decrease with time. Deter-

mination of the temperature distribution in a pipe wall

can be made by the rapid graphical method in two to three

36

hours. These factors should be of value to the EES who

may wish to make calculations for a series of thermal shock

tests for comparative purposes.

37

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