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This document was downloaded on September 02, 2013 at 04:20:14

Author(s) Knox, Robert Junior

TitleTemperature distribution in a metal hollow cylinder of finite radius under unsteady stateconditions

Publisher Annapolis, Maryland: Naval Postgraduate School

Issue Date 1948

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

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T ~ ~ E R A T U R E LISTRIBUTION IN A 1ffiTAL HOLLOW C Y L I ~ I D E R OFFINITE RADIUS UNDER TmSTEADY STATE CONDITIONS

byRobert Junior KnoxLieutenant C o r r ~ a n d e r , United States Navy

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This work is accepted as fulf i l l ing ,the thesis requirements for the degree of

MASTER OF MECHANICAL ENGINEERING

from theUnited States Naval Postgraduate School.

- Sr. Prof. P. J. KieferChairman

Department of Mechanical Engineering

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PREFACE

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Experiment Station and i t s Doctor William C. stewart forsuggesting the invest igation and cooperating to the ful les tin furnishing data. He wishes further to express apprecia-t ion for the counsel of Doctor Warren M. Rohsenow, Mass-achusetts Ins t i tute of Technology, Cambridge, Massachusetts.

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TABLE OF CONTENTS

TitleCertif ' icate of Approval.Preface.List of I l lust rat ions.Table of Symbols and Abbreviations.Chapter I Introduction.Chapter I I Origin of the Data.Chapter I I I Analytic Solution.

1 . The general solution.2 . Solutions to similar problems.3 . Further investigations.

Chapter IV Graphical Solution.

Pageii iv

vi1589

101419

1 . Available methods. 192 . A graphical solution of the EES problem. 20

BibliographyAppendix IAppendix I IAppendix I I I

23252830

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FigureLIST OF ILLUSTRATIONS

T i t l e Page1 Temperature v s . time graph; h eatin g medium,sa t ur a t e d steam.2

456

Temperature v s. time graph; h eatin g medium,s up er he at ed s te am .Tempera ture v s. time graph; heating medium,exhaust gas .Temperature v s . time graph; induction h e a t i n g .Temperature v s. time graph; h eatin g medium,exhaust g as.Temperature v s . time graph; h eatin g medium,sa t ur a t e d steam.

404142

7 D i r e c t graphical sol ut i on ( d e t a i l e d ) . 448 Temperature v s . time graph; h eatin g medium,s a t u r a t e d steam. 459 Basis ~ o r a g r a p h i c a l s o l u t i o n . 3010 Scale ~ o r the d i r e c t g r a p h i c a l method. 3311 Temperature v s . r a d i u s ; h e a t i n g medium,exhaust g a s . 4612 D ire ct g ra ph ic al sol ut i on (rapid) . _ 47

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TABLE OF SYMBOLS Al\1D JU3BREVIATIONS

B - Brit ish thermal unitc - specific heat - B/lb/FE - Modulus of elast ic i ty in tension and compressionF - degrees fahrenheitf - unit surface conductance - B/ft2 hr Fh - f /k f t - lI n - Bessel function f i r s t kind, order n.k - Thermal conductivity - B/hr/ft 2/ f t /FNn - Bessel function (Neumann) second kind, order n.n - any integerr - radiusr a - inside radius of a hollow cylinderrb - outside radius of a hollow cylindert - timeYn - Bessel function second kind, order n.0 ( - diffusivity - k/C/, in2/sec.

~ - coefficient of expansion

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er.dr - temperature at same time at r + Ll r and a t r-Ar.Qt - temperature a t time t .Qt ~ t - tempera ture a t time t + Ll t and t - I'! t at anyoneradius. .Q./\ - temperature a t sur.face .i\.Q - temperature at same time at A+dA and /\.-.o.J\.Atl'!AJ\. _.1n r / ra VI

- Poisson's rat io (unit la teral contraction/unitaxial elongation).- density - Ib/in3

6 - radial stress .~ - tangential s tress .~ - any .function.B&S -- Brown and Sharpe.Or - chromium.EES - Engineering Experiment Station .ft - . fee t .hr -- hour.I.D. - inside diameter.IPS - inside pipe s ize.in - inches.

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CHAPTER IINTRODUCTION

A metal pipe in i t ia l ly of uniform temperature, i s subjected to thermal shock by the sudden passage of a heatingmedium through the pipe. What the temperature distr ibutionis in the pipe wall before steady state conditions are a t-tained i s the subject of th is invest igat ion. The authorf i r s t became aware of th is problem in August 1947 duringan informal discussion with Doctor W. C. Stewart of the EES.

Thermal shock in the operation of a Navy steam propulsion plant is an abnormal condition. The possibi l i ty ofthermal shock conditions through a casualty is however alwayspresent. High temperature shock can be the resul t of fai lureto properly "warm up" a main steam l ine before admitting h ighpressure saturated steam from the boi ler . Cold shock can bethe resu l t of "carry-over" of a water slug from the boiler intothe l ines which have been carrying superheated steam. The

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greatest shock. Cooling had been done by water or com-pressed a i r .

Most of the tes ts indicated temperature readings onlyfor thermocouples located at the inside and outside walls.A_few tes t specimens were f i t ted with three thermocouples,the third thermocouple being located a t the mean radius ofthe pipe wall. The reading of the thermocouples were takenat time intervals of five'seconds.

Calculations of . thermal shock stresses had been madeby the EEB for representative tes ts . The author disagreeswith calculations made by the EES for the determination oftemperature distributions in the pipe wall in the unsteadystate condition. Their means of solving for the temperaturegradient is a graphical solution accomplished by a step-bystep method of averages for layers taken such that r 2! r l =

r l / r a ~ a constant. The method fai ls to f i t the data whenthe outside wall begins to increase in temperature. Inthe particular problem discussed in th is paper that condition arises a t time equal to six seconds. To allow for

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present in the EES problem nor in any other actual problem.The combination of three conditions is the cause for

the complexity of this part icular problem. These threeconditions are: I} steel pipe has a high thermal conduct ivi ty; 2) the heat source in the BES problem produces avariable surface temperature and; 3) the heat source pro-duces a variable rate of change of the surface temperature oThe analytic solutions pUblished in the l i terature are ofl i t t l e value as related to the EES problem.

Graphical solutions in the l i terature are developedfor the elementary boundary conditions only . The theoryof these, however, can be applied to the BES problem. Theauthor carries out this application for the case of heat-ing only and presents resul ts which he feels are more accuratethan those obtained to d ~ t e by the BES.

Attempts to find an analytic solution have proven unsuccessful. In the l ight of the authors investigations,there does not appear to be an analytic solution to this

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t o the thermocouples may be th e answer t o b e t t e r d a t a i nthe cr i t ica l time i n t e r v a l .

The numerical r e s u l t s i n t h i s paper are only a sr e l i a b l e a s the d a t a t a k e n . J ~ l values o f temperaturep r i o r to a time o f f i v e seconds have been obtained bye x t r a p o l a t i o n . A p l o t o f th e d ata taken when t h r e ethermQcouples were i n s t a l l e d i n the tes t i n d i c a t e s al a g i n th e temperature read in g s.* This may be an e r r o rintroduc e d by th e use of thermocouples. The l a g on th eo t h e r hand may be only the r e s u l t o f personnel e r r o r s i nt a k i n g th e temperature r e a d i n g s . I t i s ne c e ssa ry t h e r e -fore t o consider th e v a lu e o f t h i s paper p r i m a r i l y i nterms o f th e t he or y p re se nte d and i t s a p p l i c a t i o n t o as p e c i f i c problemo

* C . f . Appendix V.

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CHAPTER I I

ORIGIN OF THE DATA.

Tests were run on a length of 25-20 Cr-Ni, 3/4 inchIPS, schedule "double extra strong" pipe using saturatedsteam, superheated steam and exhaust gas as the heatingmedium. The resul tant curves of inside and outsidetemperatures vs. time are shown in figures 1, 2 and3 respectively.

One thermocouple was peened to the outside wall atmid length of the tubeo Another thermocouple was brazedto the bottom of a ~ inch diameter hole drilled to within0.01 inch of the inside wall, 1/8 inch distance from thepeened in thermocouple. In the t es t with exhaust g a s ~however, the la t ter- thermocouple was pressed against thebottom of the hole by means of an 18-8 Cr-Ni steel screwinsulated from the thermocouple. The thermocouple wireused was No. 29 B & S gage iron-constantan. The in le t

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temperatures ranging from 1040F to 1360F. Other runswere made by passing s u p e r h e a t ~ d steam at 500 psi pressureand 1050-l070F temperature through the sample. A thirdset of tes t runs was made by passing saturated steam a t500 psi pressure and 470F temperature through the tes tpipe.

In addition to these runs a series of tests wasmade on a similar sample of 18-8 Cr-Ni s teel pipe, 3 1/4inch O.D. and 1 3/4 inch I.D. with thermocouples located.04:t inches and .403 inches from the inside wall anda third peened to the outside wall. Thermal shockwas created by means of induction heating. A plot ofthe temperature vs. time curves for a l l three thermocouples is shown in figure 4.

Runs were also made on an 18-8 Or-Ni pipe, 3 1/2inches O.D. and 2 inches I.D. with thermocouples located.01 inches and .306 inches from the inner wall and a thirdpeened to the outer wall. Thermal shock was createdby means of exhaust gas a t temperatures ranging from

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749 inches in side rad iu s was set up for tes t in the samemanner as the in i t ia l tes t s . Thermocouples were located

. as for the previous tes t runs with saturated steam. Arepresentative plot of the resultant temperature curve isshown in figure 6.

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CHAPTER I I I

ANALYTIC SOLUTION

To calculate the thermal stresses in a lnite hollowcylinder the ollowing formulae given by Timoshenko ( l 6 ) ~ are used:

(1) C= BEt ( 1 -V ) r2 a f i ~ d r + a f e ~ r - e r ~a f e ~ d r a f i ~ d r )

I t is, not the author's intention to show thederivation of the above formulas since they follow as traight f o ~ a r d method and are presented clearly byTimoshenko'. The only purpose in introducing them ismerely to show their relation to the temperature dis-tr ibution in a th ick cylinde r.

To solve the integral ~ Q r d r i t is necessary toobtain the temperature distribution through the pipe

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( l4)( l3}(l l) can be adapted to a cylinder wall but largeerrors are introduced during the early stages of thesolution.

The maximum thermal stresses will occur in the earlystages of heat transfer . I t is therefore readily seen thatan analytic solution i s more desirable than the Schmidtgraphical solution.1. The general solution.

Assume the in i t i a l uniform temperature of the body aszero, the boundary conditions of the problem are:at t = 0 and r::: r aat t=-o and r > r a

e =0 .e =o.at t=t and r= r e = ( t)a aat t ::: t and r> r . Q= ( t )a , rThe differential equation for heat conduction in a

cylinder (Fourier-Poisson) when radial changes only areobserved is :

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(4) 1-

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(8) 9

Churchill (3) and Williamson and Adams (l?) presentpapers which give the ana ly tic solu tion for the transienttemperature distribution in a solid cylinder. The in i t ia ltemperature i s uniform. The surf'ace temperature r isesinstantaneously to a given temperature and 1s maintaineda t that value.,Williamson and Adams :f'orm:

(9)

where A is the nth root of Jo(p):::: O.Carslaw and Jaeger (2)* give a solution for the case

of the inf ini te hollow cylinder.Boundary conditions:a t t =0 andat t) 0 and

r:::r ;ar=r ;a

9=09= 91

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Dahl (4) gives a solution for the case of the f ini tehollow cylinder. The in i t ia l temperature is uniform. Theinner surface temperature is sUddenly raised to a valueand maintained at that value. The outer wall temperatureis maintained a t the in i t ia l temperature.Boundary conditions:at t::: 0 and r = r a ; 9 == 91a t t = 0 and r"> r a ; 9 = 0at t == t and r == r a r 9 =91a t t = t and r == rbi 9 ==0Solution:,

(11) 9- =: 91 log I'b!rlog r l l ra

n=coL 2J ( r) - ~ t- a [ J (11 r) _ ,- 0 CAn b N f 11 r ~ e ) In-n 0 Y-'n N (.u 'I"-L.) 0 YV-nn=l Oy n-o

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WherejU has the values of the positive roots of:

(lIe)

Jahnke und Emde (6) present excellent tables for theevaluation of bothp and the f i r s t two orders of N.

Carslaw and Jaeger (2) also present the case of af ini te hollow cylinder. The inside surface temperatureis raised instantaneously and maintained constant. Theouter wal l i s adjacent to an infini te fluid which maintainsi t s temperature constant at a point dis tant from theouter wall.Boundary conditions:at t = 0 and r a ~ r L. r b; e = 0a t t 0 and r = r a ; Q =Qla t t 0 and r = r b ; } Q / ~ r + f /k (e) =. 0where! is the unit surface conductance.Solution:

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where ~ n are the positive roots of:(12al Jo(,u.ra, [hYO (J.

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At t::: oo and r = r . Q:=Qa ' aa t t=oo and r ::: r b ; Q:= Qba t t : :00 and r a

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(2) cannot be assumed. The problem cannot be stretched asin the case of Perry and Berggren (12) who carefully choosethe cork lagging on a metal pipe and then work only withthe cork cylinder .

Another part icular solution is to le t J Q / ~ t =Cl-The solution of the differential equation is then:*(15)

Where Bl and Dl are cons tents of integration;" Q -= Qa- Qb.Another general solution therefore might be:. [0 ~ J ~ . 2(16) 9= ~ +B1ln r +D1 + 01J- J o ~ r ) + B N o ~ r ~ e-""jA tThis equation can b ~ used only for a limited time since thef i r s t term increases with t ime. The entire solution breaksdown on a substitution of in i t i a l conditions. At a ll valuesof r , time equal to zero, the f i r s t term and the temperaturedifference '(ea - eb ) are both zero. Again A equal to zerois the only choice for evaluating A.. Unless Q is considered

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the same value at infinite time. The Bessel term of thegeneral solution or the differential equation is alonenot enough since i t becomes zero at inf ini te time. Assumea general form:

(17)0 0

e =,ll(r,t) - L ~ J o ( P r ) .. B N o Y ' r ~,where ,0 ( r J t ) -= 02 when t =. 00and p( r , t ) = 0 when t =O.Boundary conditions again permit evaluation of B and jJ-,but again a l l terms go to zero when we attempt to evaluatethe constant A.

Assume the so lu tion o f the differential equation tobe 9 =R+ T., Differentiation and substitution results in :

(J.8)

On inspection i t is seen:

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through preference, an attempt wil l be made to find agraphical solutiono

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CHAPTER IV

GRAPHICAL SOLUTION

1. Available methods . Schmidt's graphical method (14) i s basically for a

slab. I t can be adapted to the cylinder, but as previouslystated, i t is very much in error during the early time in-te rvals . Williamson and Adams (17) point out, and Fishendenand Saunders (5) confirm the fact , that Lord Kelvin's (10)probabil i ty integral more nearly approximates the in i t i a lthermal wave penetration. This improvement over the Schmidtmethod was further advanced by Perry and Berggren (12).They determine a. graphical solution for the heat flowdifferent ia l equation for cylinders using small f ini te-differences.* This graphical solution is developed onlyfor the problem of a hollow cylinder with an instantaneoussurface heat source. The inside wall temperature is thenmaintained constant . Conduction to an inf ini te fluid is

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radius (12) (7). I t is merely a weighted mean value ofthe temperature of the two adjacent radi i before the timeinterval, namely:

For very small values of ~ r or as ~ r ~ O the equationagrees with Schmidt's equation for a slab (14):

The in i t i a l curve following the Lord Kelvin probabilityintegral or Dahl's equation* could be used over that f i r s tsmall time increment. Assume an instantaneous r ise oftemperature at the inner wall for the short time increment.The outer wall will remain a t the in i t i a l uniform temper-

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solution can be reduced to a minimum by a proper choiceof f). r . By progressing through the graphical analysisusing a tedious mathematical solution for each point , anaccurate curve can be drawn. In the EES problem the outersurface temperature increases with the inside temperature.I t is desirab le to prevent a dip in the curve below theouter wall temperature. Any temperature which graphicallyis determined to be below the oute r surface temperaturewil l automatically be given the same temperature as theouter surface. This condition wil l occur during earlytime intervals near the outer layer. An attempt to fa i rin a curve would introduce possible errors on the unsafeside of the actual temperature distribution curve. I tis considered better therefore to introduce a sl ightsafety factor by using the suggested procedure mentionedabove. A, similar procedure is used by Sherwood and Reed(15) for the .slab .'Jakob (7) and Perry and Berggren (12) give proofthat a scale of /J.r plotted as /)J\., where ..i\"'" ln r / ra '

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A careful evaluation of the problem graphicallyyields the resul ts as shown in figure 7 and figure B.I t appears, therefore, that the graphical solutionsuggested by Perry and B3rggren as applied to the EF3problGID is the only solution i f i t is desired to main-ta in the complex boundary conditionso I t is also asolution which can be calcu la ted with comparative easeby a direct graphical method.

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B 1 B L 1 0 G P ~ H Y

1 . ' Barker, L.R. The calculation of temperature stressesin tubes. Engineering. 124:443, October 1927.2. Carslaw, H.S., and Jaeger, J.C. Some two-dimensionalproblems in conduction of heat with circular symetry.

Proceedings of the London mathematical society.2:46:361-388, 1940. .3. Churchill, R.V. Modern operational mathematics inengineering. New York, McGraw-Hill, 1944.4. Dahl, O.G.C. Temperaturo and s tress d is tr ibu tion .in hollow cylinders. Transactions American societyof mechanical engineers. 46:161-208, 1924.50 Fishenden, M., and Saunders, O.A. The calculationof heat transmission. London, H.M.Stationeryoffice, 1932.6. Jahnke, E., und Erode,E. Funktionentafe1n mit formeln.und kurven. New York, Dover, 1943.7. Jakob,M. Fortschri t te der warmeforschung. Zeitschr i f t des vereines Deutscher ingenieure.75:969, 1931.8. Jakob, M., and HaWkins, G.A. Elements of heat t rans

fe r and insulation. New York, Wiley, 1942.-9. Lees, C.H. Thermal stresses in solid ~ n d in hollowcircular cylinders concentrically heated. Proceedings Royal society of London, A:10l:411-430,

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

14. Schmidt, E. Ueber die anwendung der differenzenrechnung auf technische anheiz- und abkuhlungsprobleme. Foeppls festschri f t . Berlin, Springer,1924.15. Sherwood, T.K., and Reed, a.E. Applied mathematicsin chemical engineering. New York, McGraw-Hill l1939.16. Timoshenko, S., Theory of elas t ic i ty . New York,McGraw-Hill, 1934.17.

18.

Williamson, F. D., and Adams, L. H. Temperaturedistribution in solids during heating and cooling. Physical review, 14:99-114, 1919.Woodruff, L. F. Principles o f electric powertransmission an d distr ibution. New York, Wiley,1938.

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APPENDIX ICarslaw and Jaeger (2) present the following solutions

in their paper "Some two-dimensional problems in conductionof heat with circular symetry".2t Cylinder of radius a, in i t ia l uniform temperature 0,surface t e m p e r a ~ u r e sUddenly brought to QI and maintainedconstant.3. Infinite hollow cylinder of radius a, in i t ia l uniformtemperature 0, inner surface temperature sUddenly broughtto el and maintained constant.5. Solid cylinder of radius a, in i t i a l uniform temperature91 and radiation a t the surface takes place into a mediuma t temperature o.6. Infinite hollow cylinder of radius a, in i t ia l uniformtemperature 91 and radiation takes place into a medium attemperature o.7. Hollow cylinder of in i t ia l uniform temperature o.

a) Outer surface maintained at 0 while inner surface

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ii} Inner surface suddenly raised. to G and maintainedconstant while outer su rface r a d i a ~ e s to a mediuma t temperature Qb.

i i i} Inner surface rad iates to a medium a t temperatureGa while outer su rface radiates to a medium a ttemperature Gb 9. Solid cy linder with an instantaneous heat source a tthe axiso

a} Outer surface temperature maintained a t a.b) Outer surface perfect ly insulated.c) Outer surface radiates to a fledium a t temperature O.

lao Solid cylinder with an instan taneous surface heatsource a t any radius r .

a) Outer surface temperature maintained at O.b) Outer surface perfect ly insulated.c) Outer surfa .ce rad iates to a medium a t temperature o.

11. Solid cylinder with an i n i t i a l temperature a (r ) .i ) Outer surface temperature maintained a t a.

i i ) Outer surface perfect ly insulated.i i i ) Outer surface rad iates to a medium a t temperature O.

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i ) In n er s urfa c e t em p er at ur e m a in ta in ed a t O.i i ) I n n e r su rface p e r f e c t l y i n s u l a t e d .

i i i ) Inne r su rface r a d i a t e s to a medium a t temper-a t u r e o.14. Hollow c y l i n d e r w i t h an i n s t a n t a n e o u s s u r f a c e h e a tsource a t any r a d i u s r .

a) I n s i d e and o u tsid e w a l l s maintained a t te mp e ratur.e O.i ) Outer w a l l mai nt ai ned a t temperature 0 w hile thei n n e r w a l l r a d i a t e s t o a medium a t t emper at ur e O.

i i ) I n n e r w a l l maintained a t t emper at ur e 0 w h ile th eo u t e r w a l l r a d i a t e s to a medium a t t emper at ur e o.i i i ) In n er and o ute r w alls both r a d i a t e t o a medium a ttemperature o.

1 5 . Hollow c y l i n d e r w ith an i ns ta nt an eo us h ea t source a tany s u r f a c e o f r a d i u s r and an i n i t i a l t emper at ur e a (r) .

i} I n s i d e and o u t s i d e w a l l s mai nt ai ned a t tempera t u r e O.i i } Outer w a l l maintained a t te mpe ra ture 0 w hi l e th ei n n e r w a l l r a d i a t e s t o a medium a t t emper at ur e O.

i i i} I n n e r w a l l maintained a t temperature 0 w hile th eo u t e r w a l l r a d ia t e s to a medium a t t emper at ur e O.

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APPENDIX I I

A par t icu lar solution to the di f feren t ia l equationfo r heat flow in a cylinder can be developed fo r ; ) e / ~ t = aconstant . The di f feren t ia l equation becomes:

d2e/dr2 + l / r de/dr =ClLet de/dr ='1, then

dy/dr + l/r '1 =ClLet v =: y / r , then

dy/dr = v + r dV/dror

2v 1" r dV/dr =C1f ~ T f C l ~ V 2 v(r2) (C -- 2v) ::; B1

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cl r 2 - 2r de/dr = Bfclr dr -fi dQ::o B fd;

C1r 2 - 4e = B t In r +- DQ = i 01r 2 + B f r In r 1- D'

A part icular solution to the pa rt ia l d if fe ren tia l heatt ransfer equation:

(3)

i s

when

a Q / ~ t ~ a oonstant 01

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J ~ . P P E N D I X I I I

A graphical interpretat ion of the di f feren t ia lequation for heat t ransfer in cylinders is presented byFerry and Berggren (12). The different ia l equationwhen axial temperature changes are zero i s :

(3 )

In terms of small ~ i n i t e differences the above equationbecomes:

(22)

e

IIIIIIIIII e-t-tl l t,n

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From f i g u r e 9 th e following eq u atio n s can be r e a d i l ydeterm ined:

L1 '" Qt - Qt= +L l t , r , rL lt L1t

.1 Q Qt - Qt._-:: ,r + L1 r _ r - b r2 Ll r

Q - Qt

Q Qn2Q t , r + ~ r , r t , r - t , r - D. r:: . . . ' : ; L \ ~ r " - - .=.A..... _.1 r2

Upon s U b s t i t u t i n g th e s e v a lu es into th e e q u a tio n o ff ini te d i f f e r e n c e s :

Qt , r t- A r - 2Qt , r+ Qt , r - A r +

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2( Q -Q ) "" Q -29'" Q +t+l\tJr t J r tJr+Ar t J r tJ r - Llr..iT-2r (Q - Qt Jr +Art , r - l\ r)

(20) e - 1. ~ l + - ~ } e + { l-t+l1t,r- 2 L' 2r t Jr+.t1r ~ ) Q Jr t Jr- lir~ h e resul tant equation is o ~ e which is a weighted meenfavoring the side of increased radius and the side ofgreater heat capacity.

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APPENDIX IV

A plot of temperature vs. In r / ra can be used for adirect graphical method of solution.

t I II I II I ,I I , II . I IIe I I II I II ~ , n - A " I II II ~ I II ~ t + . a t : . / l III '-.... I n-t-.I1J lI e I It.1l II I II I II I II I

. i \ - ~ ).., ./\ :A+L\A tFigure 10

Scale for the direct graphical methodFrom the geometry of the above figure i t is apparent

that :

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t.rr - -:=:'2--- - - - - - ~ - -r +

2

but r - 6.r/2 and r+ f:Jr/2 are the mid radi i of the twoadjacent ~ r layers of ro The scale division therefore

r -

must be inversely proportional to thei r correspondingrad i i .

This is sat is f ied by choosing a scale variable

so that di\ = dr / r or ,1 A= t . r /rThe mean radius of LL\.l i s r - Ar/2 and simila.rly forL1A 2 i s r+ t.r /2 .Then:

__ . . . ; ; ; i \ ; ; . ; ; r ~ _D.r---;r-

APPENDIX V

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A p l o t of stead y s t a t e temperature d i s t r i b u t i o n r a d i a l -l y in a hollow cylinder is a s t r a i g h t l i n e when pl ot t e d onc oo rd in ate s of temperature v s . In r / r a The d a t a as showni n f i g u r e 5 rep resen ts the readings o f thre e thermocouplesin a pipe wall. This d ata was r e p l o t t e d i n f i g u r e 11 astemperature v s . In r / ra

Two p e c u l i a r i t i e s are apparent upon examination of ther e s u l t a n t c urve s . Most a ppa re nt is th e change o f curvaturefrom concave up t o concave down a f t e r n i n e t y seconds. Thesecond p e c u l i a r i t y a t n i n e t y seconds is a change from i n c r e a s -ing slo p es with time to d ecreasing slo p es w ith time. Thela t te r is the e f f e c t o f the wave f r o n t a s the h eat penetratesth e pipe w a l l .

The change from concave up to concave down i n d i c a t e sa t once a descrepancy in the d a t a . Since stead y s t a t e i sthe u lti ma te c on diti on which can e x i s t under the co n d itio n sof the tes t , the concave down curves a r e an i m p o s s i b i l i t y .

The au th o r i s of the opinion t h a t the p l o t reveals a

APPENDIX VI

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The tedious graphical solution represented by figure7 immediately caused the author to inve stigate the possibi li ty of a more r ap id graph ical solution. A graphical solut ion , represented by figure 12, was then made for purposesof comparison.The f i r s t sixteen steps of both solutions are ident

ica l . Time increments in figure 12 are then rapidly in-.creased to speed up construction. The time to producefigure 12 was approximately one half the time required forfigure 7. A comparison of resul tant values of temperaturesfor r l , r 2 and r3 from both f igures was then made.A maximum dev iation of 6 degrees representing a 4%

error occured for r l , a t 1.512 seconds. This error rapidlydecreases and disappears af ter 6.048 seconds. For r2the maximum deviation was 4 degrees for a 2.8% error a t 2.1168seconds. The trend of a decreasing error a t an increasingtime for that maximum error with an increase in radius wasnoted fo r r3 . At r3 the maximum deviation was 3 degrees

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

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may wish to make oaloulations for a series of thermal shooktests for oomparative purposes.

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