A FINITE DIFFERENCE STUDY OF TRANSIENT HEAT TRANSFER

INVOLVING HYDRODYNAMIC VARIATION IN THE THERMAL

ENTRANCE REGION OF A CIRCULAR TUBE

by

RAYMOND MILTON KLIEWER, B.S. in M.E., M.S. in M.E.

A DISSERTATION

IN

MECHANICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

August, 1970

\

ACKNOWLEDGMENTS

Appreciation is gratefully acknowledged to Professor M. E.

Davenport for his direction of this dissertation and to the

members of my committee. Professors T. A. Atchinson, D- P. Jordan,

and J. H. Lawrence.

ii

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS ii

LIST OF TABLES v

LIST OF FIGURES vi

NOMENCLATURE vii

I. INTRODUCTION 1

II. GOVERNING EQUATIONS 7

2.1. Assumptions 9

2.2. Governing Equations 9

2.3. Boundary Conditions 10

2.4. Variation of Fluid Properties 11

2.5. Non-dimensionalization of Variables 12

2.6. Dimensionless Governing Equations 13

2.7. Non-dimensionalized Boundary Conditions 15

2.8. Independent Parameters 16

2.9. Heat Transfer and Friction Parameters 16

III. FINITE DIFFERENCE METHOD 19

3.1. Finite Difference Approximations 19

3.2. Governing Equations in Finite Difference Form . . . 23

IV. SOLUTION 36

4.1. Method for Solving Governing Equations 36

4.2. Computer Program 41

iii

iv

Page

V. DISCUSSION OF RESULTS 46

5.1. Thermal Results 46

5.2. Hydrodynamic Results 52

LIST OF REFERENCES 59

APPENDIX 61

A. ERROR, CONVERGENCE, AND STABILITY CONSIDERATIONS

FOR FINITE DIFFERENCE SOLUTIONS 62

B. MESH SIZE AND WEIGHTING PARAMETERS 67

C. PROPERTY DATA FOR AIR 69

LIST OF TABLES

Table Page

B.l. Weighting Parameter Values and Mesh Size 68

'Vv/ftV

Figure

2.1

3.1

3.2

4.1

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

5.10

LIST OF FIGURES

Page

Problem Geometry 8

Designation of Mesh Points 20

Control Volume 33

Computer Flow Chart 43

Temperature Profile Near Entrance - Constant

Wall Temperature 47

Temperature Profile Down Tube - Constant Wall Temperature 47

Temperature Profile Near Entrance - Constant Wall Heat Flux 48

Temperature Profile Down Tube - Constant Wall

Heat Flux 48

Nusselt Number - Constant Wall Temperature 50

Nusselt Number - Constant Wall Heat Flux 51

Friction Factor - Constant Wall Temperature 53

Friction Factor - Constant Wall Heat Flux 54

Pressure - Constant Wall Temperature 56

Pressure - Constant Wall Heat Flux 57

vi

NOMENCLATURE

English Letter Symbols

c sonic velocity, —

. . , BTU C specific neat at constant pressure, • , op P lb R

m

/ BTU C specific heat at constant volume, , op v XD K

m

u2 Ec an Eckert number, TTTJTJ dimensionless

P 2T w

f friction factor, , dimensionless

"A g acceleration of gravity, -

2 sec^

,2^ 3. 2p'^wg Gr a Grashof number, , dimensionless

BTU h convective heat transfer coefficient.

K thermal conductivity,

hr ft^^R

BTU

hr ft*R

M Mach number, —, dimensionless

n total number of increments in the radial direction, dimensionless

hr Nu Nxjsselt number, 2 —r;—, dimensionless

P p r e s s u r e , f t ^

Pr Prandtl number, -~-, dimensionless

vii

Q heat flux.

viii BTU

hr ft2

r radial coordinate measured from the centerline of the tube, ft

lb ft R gas constant.

**R lb m

2pu r Re Reynolds number, - , dimensionless

t time, hr

T absolute temperature, °R

u axial velocity, —

V radial velocity, ;— hr

x axial coordinate, ft

Greek Letter Symbols C

Y specific heat ratio, p^, dimensionless v

0 weighting parameter with respect to axial position, dimensionless

lb y viscosity, ^^ ^^

lb p density,

ft3

^^f T shear stress,

ft2

^ weighting parameter with respect to time, dimensionless

ix Subscripts

B bulk value (with respect to the tube cross-section)

i grid mesh point in the axial direction

j grid mesh point in the radial direction

k grid mesh point in time

o conditions at thermal entrance of the tube

W evaluated at the wall

t evaluated at the centerline

Superscripts

a exponent governing viscosity variation with temperature, dimensionless

b exponent governing thermal conductivity variation with temperature, dimensionless

+ variable is non-dimensionalized

CHAPTER I

INTRODUCTION

The present investigation is a study of the transient behavior

of laminar gas flow in a circular tube with an imposed step increase

in tube wall temperature or heat flux. It is desired to determine

the thermal and hydrodynamic behavior during the transient. Variation

of the fluid properties is taken into account. An extensive search

of the available literature reveals no similar study of convection

transients where hydrodynamic variation associated with property

variation is included in the analysis.

Early investigations assumed that the convection heat transfer

corresponded to the instantaneous surface temperature or heat flux.

Thus steady state results were applied to a non-steady process. This

is the so called quasi-steady assumption. Also, these investigations

treated the problem using one-dimensional approximations meaning

that velocity and temperature variations over the flow passage

cross-section were neglected.

There is a difference between the actual heat transfer and the

quasi-steady heat transfer values. Sparrow and Gregg [11] computed

the first order deviations of the instantaneous heat transfer from

Numbers in brackets refer to similarly numbered references in List of References at the end of this report. Some passages used in this report are taken from the author's master's thesis.

2

the quasi-steady value in 1957. The analysis considers laminar flow

over a semi-infinite flat plate. A spatially uniform plate wall

temperature which may vary with time is considered. A series

solution is obtained with the terms after the first indicating the

departure of the instantaneous heat transfer from the quasi-steady

value.

In 1958, Sparrow and Siegel [12] published an analysis of

laminar forced convection heat transfer in the thermal entrance

region of a circular tube. Arbitrary time variations of either

surface temperature or heat flux are considered. Velocity and

temperature profiles are taken to be functions of radial position

Instead of being simply one-dimensional.

Using the energy differential equation for a fully developed

velocity profile with constant fluid properties, an exact solution

is obtained. The Poiseuille equation provides the fully developed

parabolic velocity profile. For the case of time dependent tube

wall temperatures a third order polynomial temperature profile

satisfying the boundary conditions is assumed; while in the case of

the time dependent wall heat flux, a parabolic temperature profile

is assumed. An artificial, clearly definable thermal boundary layer

is postulated. The two conditions associated with this thermal

boundary are that the fluid temperature at.the boundary layer is

equal to the fluid temperature before the initiation of the transient

and that the derivative of the temperature with respect to the

radial coordinate is equal to zero at the thermal boundary. Using

the parabolic velocity profile and the postulated temperature

profile, the energy equation is integrated across the thermal

boundary layer for step changes in tube wall temperature and heat

flux. A dimensionless differential equation is obtained in terms

of the thermal boundary layer thickness. A solution is found using

the method of characteristics. The solution consists of two parts,

one for the transient time period and the other for the steady

state time period.

Using the superposition technique of Duhamel's theorem, the

results are generalized to obtain a solution for arbitrary time

dependent variations of wall temperature or heat flux. By

appropriately shifting the time scale, the solution is extended to

cases with steady state heat transfer occurring before the initiation

of the thermal transient.

Transient laminar forced convection heat transfer in the

thermal entrance region of flat ducts is considered by Siegel and

Sparrow [10] in a similar analysis. Likewise, both the velocity

and temperature profiles are taken into account.

In 1960, Siegel [9] solved the same type problem for arbitrary

time variations in wall temperature using a different solution

method from that previously presented by Sparrow and Siegel. Laminar

forced convection heat transfer in both a circular tube and a

parallel plate channel is considered. The initiation of the thermal

transient can begin from an already established steady state

condition with heat transfer taking place, or the fluid and wall can

be initially at the same uniform temperature. The analysis assumes

a fully developed parabolic velocity distribution. The energy

equation, assuming constant fluid properties and neglecting viscous

dissipation and heat conduction in the axial direction, is employed.

Both steady state and transient solutions are presented. In

the case of the circular tube the steady state solution is simply

the Graetz problem, the solution being obtained by separation of

variables. A transient solution is obtained by a series expansion

about the steady state condition. The partial differential equation

made up of the integrated form of the energy equation is solved

using the method of characteristics. Eigenvalues for the solution

are obtained by numerical methods. The heat flux is evaluated

using Fourier's conduction law. The results are extended for an

arbitrary time dependent wall temperature by superposition techniques

In the case of a parallel plate channel a steady state

solution is assumed by expanding eigenfunctions of the solution

of the corresponding slug flow problem. The assumed solution is

then substituted into the energy equation. This equation is

multiplied separately by different cosine harmonics, each equation

being integrated. Thus a set of simultaneous equations is obtained.

By solving the equations simultaneously and applying the boundary

conditions, an infinite series solution for the temperature

distribution is found. A transient solution for the parallel plates

is assumed by noting similarities between the steady state solutions

for the circular tube and the parallel plate channel. The assumed

temperature distribution is substituted into the energy equation,

and the integrations are performed. It is noted that this equation

corresponds to the integrated energy equation obtained for the

circular tube. Then analogy is used to obtain the temperature

distribution. Fourier's conduction law is used to determine the

heat transfer rate.

The nature of the series solution makes it difficult to

calculate results very near to the thermal entrance because many terms

are required to obtain satisfactory convergence. The previous results

of Sparrow and Siegel did not extend far down the flow passage, the

solutions existing only until the thermal boundary layers met at

the centerline of the flow passage. It is claimed that by joining

the results of both analyses solutions can be obtained for all

positions down the flow passage.

As has been pointed out, all of the analytical solutions

described are limited in that some major simplifying assumptions are

made before the solution is obtained. The results of these

assumptions is always to either provide solutions of questionable

validity or to limit the range of validity of the solutions. The

problems solved by analytical means are only approximations to the

real problems. Therefore, they can never be called the "true"

solutions to the typical forced convection transient heat transfer

problems that they represent.

The analytical methods discussed here have required assumptions

which greatly simplify the governing equations. These simplified

equations have been solved using the analytical tools of mathematics.

An alternate method can be used to solve the governing differential

equations involving the use of numerical techniques where finite

differences are used to approximate the derivatives. Using numerical

techniques, more complex differential equations can be solved.

Thus it can be seen that at least two alternatives are available for

solving this type of problem. The exact method allowing the "exact"

solution of an approximate problem and the numerical method providing

an approximate solution to a more general problem.

The solution of differential equations by numerical methods is

by no means a simple one. The advent of the modern high speed

digital computer has made the solution of previously unmanageable

problems feasible. More modem finite difference methods have

likewise been developed resulting in improved control over stability

and convergence problems resulting from the use of these methods.

Several analyses have been made of the convection heat transfer > t

problem in circular tubes using finite difference methods [6] [13] 3

[14]. These analyses, however, have considered only the steady state 3 y

problem and neglected the transient problem. These analyses notably ^

include the variation of fluid properties which are proven to

appreciably affect the solution especially for large temperature

differences.

CHAPTER II

GOVERNING EQUATIONS

The problem to be solved here is for flow of a compressible

viscous fluid through a circular tube, the fluid suddenly being

heated either by a step increase in the tube wall temperature or heat

flux. The flow is through a vertical tube. Thus a cylindrical

coordinate system may be employed reducing the number of equations

that must be solved since axial symmetry exists.

It is desired to determine the temperature and velocity

variations with respect to time and axial position after initiation

of the thermal transient. Viscosity, thermal conductivity, and

density variations with temperature are considered. T and u o o

represent the bulk temperature and bulk velocity at the entrance to

the heated section, the entrance temperature being uniform. T, u, v,

X, and r denote the temperature, axial velocity, radial velocity,

axial coordinate, and radial coordinate, respectively. The geometry

of the problem is shown in Figure 2.1.

The general governing equations are well known and can be found

in Reference 2. These equations are derived from the fundamental

laws of mechanics and thermodynamics. Even using finite difference

methods, it is not feasible to consider solving the governing

equations in their most general form. In order to obtain a

manageable problem certain assumptions must be made.

7

8

01 iw or

\

T

To I U,

r Tube >

d. tl ft

I

S

Heated Section

1 Unheated Entrance Region

Direction of Fluid Flow

Problem Geometry

Figure 2.1

2.1. Assumptions

The following assumptions are made to simplify the problem:

1. The fluid flow is laminar.

2. The ideal gas equation of state is valid for the fluid

in the range of the solution.

3. The specific heats and gas constant remain constant in

the range of the solution.

4. The fluid is a Newtonian fluid.

5. Normal frictional stresses are negligible.

6. Axial conduction is negligible.

7. Viscous energy dissipation is negligible.

2.2. Governing Equations

The governing equations take the following simplified forms

consistent with the assumptions already listed, these being the

so-called boundary layer equations:

The energy differential equation

^-ririrK^) ^-" The axial momentum differential equation

X

(2-2)

1 ^

The continuity differential equation

The equation of state for an ideal gas

10

•i<(M+-ririevr)^^^o < - >

p= eRT (2-A)

2.3. Boundary Conditions

In order to solve the governing differential equations

certain boundary conditions must be incorporated consistent with

the specified problem. These boundary conditions are:

Entrance conditions

U(0^r,±)- UB,0 I I~ (^^/J (Poiseuille equation) (2-5)

T(0,r.t)=To (2-6)

V(Q/^i-)= 0 (2-8)

P(0,i)= Po (2-9)

Wall conditions

V (X, r..t) = 0 (2-11)

Centerline conditions ^^

V(X, 0,ir) = 0 (2-iA)

^CX,D,±) = 0 (2-15)

i!r(X,0.±) - 0 (2-16)

Initial condition

T(x,r.,o) = r (2-17) 2.4. Variation of Fluid Properties

For many gases it has been found that the specific heats vary

only slightly as functions of temperature for normal temperature values.

Viscosity and thermal conductivity, however, often increase as about

the 0.7 power of the absolute temperature while the density varies

inversely with the first power of the absolute temperature. It

might be noted that this variation corresponds to the ideal gas

equation of state which has already been introduced into this

analysis.

Deissler [3] allowed the viscosity and the thermal conductivity

to vary according to the power functions

and

U = JJO(TI) (2-18)

K~ KXTJ (2-19)

where a and b are constants. T is an aribtrary reference temperature,

and K and y are the reference thermal conductivity and viscosity o o

12 respectively which are evaluated at the reference temperature. The

values for the constants a and b are obtained from data given in

Reference 4.

2.5. Non-dimensionalization of Variables

The number of independent parameters in the problem can be

reduced to a minimum by the non-dimensionalization of variables.

This makes possible some generalization of the problem. Non-

dimensionalization also allows the use of variables of roughly the

same order of magnitude. This is important when using a digital

computer where numbers can be carried to a limited number of

significant figures.

The variables are non-dimensionalized according to the following

equations:

+ T T = — , non-dimensionalized temperature (2-20) o

p = • —, non-dimensionalized density (2-21) o

u = — , non-dimensionalized axial velocity (2-22) o

V = "r— Re Pr , non-dimensionalized radial velocity (2-23) 2u o o o

P -P P « , non-dimensionalized pressure (2-24)

2 p u^ o o

+ 2x x = —;:—=:—, non-dimensionalized axial coordinate (2-25)

r Re Pr w o o 4" r r * — , non-dimensionalized radial coordinate (2-26) r w

+ ^^o t = , non-dimensionalized time (2-27)

p C r^ o p w

+ W v Q^ " K~f~' non-dimensionalized wall heat flux (2-28)

o o

W = "j~- = iif-) t non-dimensionalized viscosity (2-29) o o

+ K T b ^ = = (zT") > non-dimensionalized thermal conductivity (2-30)

o o

The quantities used in the non-dimensionalization are listed below:

C , specific heat at constant pressure

K , thermal conductivity evaluated at T o " o

P , entrance pressure o* ^

C p^ Pr = —^—, Prandtl number evaluated at the entrance (2-31)

o

0 , wall heat flux

2p r u o w o

Re = , Reynolds number evaluated at the entrance (2-32)

o

T , entrance temperature

u , entrance bulk axial velocity

y , viscosity evaluated at T *o o

p , entrance density

2.6. Dimensionless Governing Equations

By substituting the non-dimensionalized variables into the

governing equations, the dimensionless forms of the governing equations

are obtained. These are:

The energy differential equation

\

where

Uc"

where

The equation of state for an ideal gas

where

14

(2-34)

The axial momentum differential equation

tars • " jJL^ -* (2-36)

/? . ^ ~ ^ (2-37)

7"^ C Co /y/y LAe

Aeo — yUe (2-38)

The continuity differential equation

ir^PW ^ -pi - fVr ; ^ir' = 0 (2-39)

e ' = T^ [ I - y/Ho' P^J (2-Ao)

A7o= c. =a/?7:)* (2-Ai)

15

2.7 Non-dimensionalized Boundary Conditions

The boundary conditions corresponding to the dimensionless

form of the governing equations can now be stated:

Entrance conditions

u*co,r:r)=^ 2.ci-r*') (2-^2) VCo> r:r) =• X = i (2-43) e^(0/'X)=- f'o'-i (2-44)

PY 0, r) = 0 (2-*6) Wall conditions

wa' I, e)=- 0 (2-47)

rcx\ I. r) = TJ />a i'^o (2-^9) or

Q"(X':IJ*') = Q^* A>Qi->(3 (2-50) Centerline conditions

^^Lr,DX)^0 (2-52)

#(X;0,t>0 (2-53)

Initial condition

r\X*r:o) = T:'^ / (2-5A)

16 2.8 Independent Parameters

The Eckert number found in the dimensionless energy

differential equation is not independent of dimensionless

parameters found in the ideal gas equation of state. That is for

an ideal gas

With the non-dimensionalization of the governing equations,

independent parameters appear for the problem. These parameters

can be divided into two groups [14], operational and property

parameters. The operational parameters are M , Gr /Re , and the

thermal boundary condition type and magnitude. The property

parameters are the Prandtl number, the specific heat ratio, and

the two exponents for the power law property variations. The

property parameters are independent in the mathematical sense;

however, they are not physically independent since they specify a

particular gas and a certain temperature range for which the

solution is valid. Similarly the operational parameters M and

Gr /Re are not completely independent since certain restrictions o o

are imposed on the relative magnitude of these parameters by

physical considerations.

2.9 Heat Transfer and Friction Parameters

Heat transfer results are generally presented in terms of

the Nusselt number which is defined as

A/a = 2 - ^ (2-56)

17 where h is the convective coefficient of heat transfer. This

parameter relates the wall heat flux, Q , to the difference between

the wall surface temperature, T , and a characteristic temperature

of the fluid at that point. The convective coefficient of heat

transfer is defined by

where the characteristic temperature is based on the bulk temperature.

The bulk fluid temperature is the temperature the fluid would

attain if it were perfectly mixed over the tube cross-section.

Mathematically the bulk temperature can be expressed as

-r ^ .\:yurrdr

or in non-dimensional form

7- _ .I'rarr^r (2-59) = Using Fourier's conduction equation evaluated at the wall, the

wall heat flux can also be expressed as

Using the preceding formulations and the variation of thermal

conductivity with temperature, the Nusselt number can be expressed

in terms of the dimensionless quantities as

A/ -.AKL^ -^n 'W (2-61)

1

Fluid flow results are generally presented in terms of the

friction factor. This parameter may be defined as

18

(2-62)

where x^ is the wall shear stress, p is the bulk fluid density, and

Ug is the bulk fluid velocity.

The bulk fluid density is defined as the density evaluated at

the bulk temperature which can be written mathematically in

functional notation as

P3 - ?CTe) (2-63)

The bulk fluid velocity is defined as

6(8 = £ rpg rdr

tsHv (2-64)

or in non-dimensional form

U = _ £rfUrWK

^xr (2-65)

For a Newtonian fluid the viscous wall shear stress can be

written as

Tvv- JJi^^^J^ (2-66)

Using the preceding formulations and the variation of viscosity

with temperature, the friction factor can be expressed in terms of

the dimensionless quantities as

i Veo (2-67)

,iS^ *•- «14-3P3'^??a^i'

CHAPTER III

FINITE DIFFERENCE METHOD

With the governing equations and boundary conditions

established in the preceding chapter, a method of solution can be

developed. The energy, axial momentum, and continuity differential

equations are solved numerically by replacing the derivatives with

finite difference quotients which are evaluated at discrete points

in the domain where the solution is sought. It might be noted here

that the difference quotients represent only approximations of the

derivatives in the governing equations. The equation of state is

solved algebraically at the mesh points in the domain. A numerical

solution is obtained utilizing a suitable solution scheme and a

digital computer.

3.1. Finite Difference Approximations

There are numerous finite difference schemes available for

solving differential equations numerically. Consider the function

F. . . with the subscripts i, j, and k associated with the x, r, and i»j »k

t variables respectively. The designation of the mesh points is

shown in Figure 3.1. Two weighting parameters, 0 and (j), lying

in the intervals 0 < 0 < 1 and 0 < <^ < 1 may be introduced. 0

represents the weighting parameter with respect to axial position,

X, and ( represents the weighting parameter with respect to time, t.

Using a Taylor series expansion in three variables

19

20

7 f

Designation of Mesh Points

Figure 3.1

•t-A^ityF(\nt) the following equations can be obtained where all derivatives are

evaluated at the point [(i+0)Ax,jAr, (k+cj)) At] :

21

(3-1)

_ JF . -L aiE

(3-2)

•h(l-e) L (i)iF^n. ht - a../. hi)^(h0XFj:,jti.k-B,^^,i)]

= j^+ie(i-e)§3rAx'-

(3-3)

'^wmm^

Jll-0MF^i»^,^?R^M,*F^^,)H[-SXf^^.,~?.Ey^F.^.,.)^ 22

^i^'-itif^^^^

+i:0(he)&^Ar

+ (t>ci-P)^rAt^ (3-4)

= -^^-ra-i0)ifAt

•^•tBCI-6)^AX^

•h-iU-3(j)+3^')^Ai%---Note that the value of F(xrfOAx,r,t+( At) can be appoxlmated

(3-5)

by

vhich represents linear interpolation.

(3-6)

When 6 and are zero, the difference approximations of eqtiaticns

(3-2) through (3-5) are explicit. For an explicit system of equations,

the unknowns are found directly in terms of known quantities. If 6

and are not zero, then a set of simultaneous equations must be solved

- 23

In order to determine the values of the unknown quantities. Such a

system of equations is Implicit. It might first appear that an explicit

scheme would be far simpler to use than an implicit scheme. Stability

and convergence considerations, however, reduce the desirability of

such a scheme. A brief discussion of stability and convergence is

presented in the Appendix.

3.2. Governing Equations in Finite Difference Form

Making derivative approximations by neglecting all but the

first term to the right of the equal sign in equations (3-2) through

(3-5), the governing differential equations can be written in finite

difference form. It is instructive to examine the energy and axial

momentum differential equations at the centerline of the tube. By

application of L'Hospital's rule, the symmetry condition

ipA=^ (3-7) -A=o and the centerline condition

v*a:o,f)=o (3-8) the energy differential equation reduced to

(3-9)

at the centerline. Similarly by application of L'Hospital's rule,

the symmetry condition

* ^ 1 = 0 (3-10) .irVfe

24

and the centerline condition

vxx:o.t')^o the axia l momentum d i f fe ren t ia l equation reduces to

. JOT o^M" d P \ O P / / ^ ^ , ProGj^'

(3-11)

(3-12)

The governing equations in finite difference form can be

expressed as:

The energy difference equation

where

(3-13)

(3-14)

(3-15)

(3-16)

(3-18)

miMi^-k'[-6Z:,,^).Hl-6XT^Mi-Xi,l)']

Hl-6X(t)C(t-*K*kj>,M-(t*K*ky,,i.,)

25 (3-19)

(3-20)

(3-21)

(3-22)

(3-23)

(3-24)

HI-eiH^,!., -i;.,>J +( l~4>XXi„A%,,S (3-25)

and

L^

-^^kf' Q^^iAhfk^l (3-26)

26

(3-27)

"aAf* ^(j^FiiMiMi (3-28)

"f f<?/ /V/, ; */ iEe.? -<v-/,/>*/ fif/H^,A*i FTIZJI-IA, A*i (3-29)

except a t the c e n t e r l i n e where

i- ^;-fi 6(J)EGI^^^j^^kH

G , /H/-,> / ^ ^r^^ Q^^^hi'H^, k^i

(3-30)

(3-31)

(3-32)

-^Dc.iH^Mi Wet,i*i^,A^i •^dEsi^4ii,i\k*iEee,dH^;k4-l (3-33)

and

^ ; = 0 (3-34)

The momentum difference equation

Amjxii,i,k*i Liji*;j'i,kH "^Om,L*t,Lkfi LLi^u^kn ^ Cm,C*u,k*iLLi*ixti.k*

'iMitlf,**! *En^*iJk,R{i,i 27

(3-35)

where

(3-36)

Amt,iHf-k.l = AX* l-(f>U\^,k^, HI-(ti)CU\^,A-(Ai.li ) ] (3-37)

(3-38)

(3-39)

(3-40)

(3-Al)

(3-42)

(3-43)

Hi-e)l(l)((.f'jjtk^u,hr C rfji'k^.,,k„)

H\-B)W.^.k.KH)ft.^.k']]

and

28

(3-44)

(3-45)

(3-46)

(3-47)

i 90

(3-48)

(3-49)

(3-50)

Pm,i*Ki,h\ "" /lryn,uu,i*i AMZ^L^^^bl Omt.i^^^k^x umi,i^k^\

'^^,i*isfMFni,(H,^M -&miji^\,^,k^l (3-51)

^,tV.,^> AX"- (3-52)

except at the centerline where

Am.i%h, = ~ZF^^P ^'"' ''•• •.*' (3-53)

Cm,;*^,k*\~~A^ Q^Emi.mi,kM (3_55)

fM,.V/,^>/ = AX"" <3-57)

and

i^O (3-58)

A somewhat different approximation for the partial derivative

with respect to the radial coordinate can be made with the continuity

differential equation since this equation contains no second order

derivatives with respect to the radial coordinate. The derivative

with respect to the radial coordinate is evaluated at j-1/2 instead

of at j. This approximation reduces the truncation error in the

%

'^DmiJ^iji^k^l '^EEm,i^ijLk*>Emz,Uuk*rGmi,i.M^^,k*l (3-56) ^

i

30 derivative approximation. Truncation error is discussed in the

Appendix. The finite difference equation used for the radial

derivative is

eld) (f?., j.hi - 5., J., tu )i-(l-rhX F,*,^.k -FM^-, t.)]

j m - ^^ '^WTP^' ^ • • • (3-">

4 ^ ' ^ e (/^>ii>"~^i/* ^|^(^^.^^/- '^" ~ ^ j ^

r where all derivatives are evaluated at the point [(i+0)Ax,(j-1/2)Ar, .J

(k+(^)At]. Again the derivative approximation is made by neglecting yj

all but the first term to the right of the equal sign in equation II-

(3-59).

Evaluating the partial derivatives with respect to the axial

coordinate and time, these approximations become

Cf A

f 0 ( />>/./i / - % *")^0- 0^^V-^ ~ /7/-1 (3-60)

4- ^(^ '{/ -"~ F*^i-' "^-^Cl- eX F/j-^ A*I - F^-, k) BAh

The value of F(x+0Ax,r-Ar/2,t+(J)At) can be approximated by

31

(3-61)

- ^ B/;0/r»j-.fa, Ki~6)FA,f,.k] Hi-&MF,i-, k.,-^ci-ii))Ej-,.)3

which represents linear interpolation.

The continuity difference equation is

where

(3-62)

(3-63)

(3-64)

5

^ 6(l/f(^-i, 1^ -d'Hj-i.k.)-^ChoXP^;/-i.^'-^l^y-/.A>) (3-66)

The equation of state is simply solved algebraically at the mesh

i' points in the domain. Jj

The equation of state is

p = - ^ r/-x/w:E>] (3-67) ^

An additional constraint can be added to the set of governing

equations, this being the integrated continuity equation. By adding

this equation, we add another unknown, pressure, to prevent over-

determination of the set of governing equations. The integrated

form of the continuity equation can be written for a control volume

as shown in Figure 3.2.

The equation takes the form

^-

Outflow

33

ri^t

Inflow

i

I

Control Volume

Figure 3.2

34 The indicated integrations can be carried out numerically with

standard numerical integration formulas. These are:

The trapezoidal rule

i^FW dx --t[r(xymx.)^mW^ • • •

+lnX,.,)+F(X.)]AX (3-69) which reduces to

CFOOdX-'ilF(X^ISX)+F(X)]AX o-m for the control volume shown in Figure 3.2.

Simpson's rule

where n is an even number and

Z\ r " n (3-72)

Performing some of the integrations, replacing p by p in

the time derivative, and approximating this derivative with a simple

finite difference form, the integrated continuity equation can be

written as

•/• f"ZlXY^''*"^^'''* + ^ ' • ' S'ti*j - Q (3-73)

i

I +l[F(r,)^ F(i;)+ • • • -hF(f7-z)]]Ar O-ID I

p^'

35

By substituting the equation of state into the first term of the

preceding equation, a convenient form is obtained.

The integrated continuity equation becomes

j ; U+i,A<-i

(3-74)

ll

%

CHAPTER IV

SOLUTION

With the governing equations established and expressed in

finite difference form, a solution scheme can be developed to solve

these equations. A scheme is developed for use with a digital

computer.

4.1. Method for Solving Governing Equations

Both the energy and momentum difference equations can be

represented as n+1 simultaneous equations which can be written in

matrix form for axial station i+1 and time step k+1. The axial

symmetry of the tube allows the solution to be made only from the

wall to the centerline instead of from wall to wall. These equations

can be expressed in matrix form as follows:

:0

%

36

37

+

K h^ M

I si-

+ I-

' O O O O ^ ' . . -

M r-V ^ ^

I Q Q. Q Q^

II

i

\

%

CSJ

O O OJ

. 4 .

-St

o o 5cfi' o o

o •H •P

cd

cr (U

§3 U 0) c: 0)

0)

A :r

05 O O

^ ^ O O Oi

wr

I

-«e ^ -5*J

? tf "

r:^ ^ r : ^

I o o o O

.4

I si-

38

i

•

•

i

«

•

•-»i*

d ^

5 •:t:

5

ft

0 1

—

5 - ^ 5 r^ .^ .+ • • •

iu5 u5 u5

-K ' * - -

' ^ < ^ ^ O^ -.- rvT

^ ^ » e" s'' 6"

II

4

u5

^ • -?^

cf

1

S

1

k

- ^

r

3 i

. 5 j i ^

5

o o O

o o CQ » • » o o

o •H

« cr

t i 0)

-- -- "J

d

I

'-<

^J CQ <q;

OQ -=^ ^

O

O

o

o

39

It should be noted that at the centerline

(4-3)

(4-4)

and by symmetry

(4-5)

(4-6)

In both cases the coefficient matrices are tridiagonal with the

exception of those coefficients associated with the difference

equations at the tube wall and centerline. That is, both coefficient

matrices have zeros for all elements except those on the main diagonal

and on the two neighboring diagonals. These sparse matrices allow

efficient use of the Gaussian elimination method of solution. First

by upper triangulation, the coefficient matrices are reduced to

bidiagonal form resulting in equations of the form

TJ^^^Afi ~ Cd,/V/,^>i /A/, *.,A*I " Ue,Ui,^,k^i (4-7)

and

(4-8) UCjl*U.k^\^ Cw>U.^/ LL^^I^hk*! ^Dy,^^i^l,^,k^\'^tmMfbihH,kM

where the primed coefficients represent the new values of these

quantities after the appropriate manipulations have been performed.

The difference equations for temperature can be solved by back

substitution from the wall to the centerline if the wall temperature

is known.

1

I

j ^ ^

40 Similarly once the value for P^., , ., is determined, the

i+1, k+1

difference equations for axial velocity can be solved from the tube

wall to the centerline since the wall boundary condition for velocity

is known.

+ ^i+1 k+1 ^ determined from the integrated continuity equation

using Simpson's rule to perform the integrations indicated in

equation (3-74) . Successive substitution of the momentum difference

equations (4-8) leads to a quadratic equation for P - , . of the form

X i-WoU.^') where A and B are constants resulting from successive

substitutions of the axial momentum difference equations (4-8) into

the left hand side of equation (3-74). This equation is solved for

P . 1 1 II hy iteration. i+1, k+1 " s

The equation of state is simply solved algebraically at the Sj

appropriate grid points.

The continuity difference equations can be solved for the radial

velocity by solving the equations from the centerline to the wall of

the tube. It might be noted that the coefficient matrix for these

equations is bidiagonal.

For the case when the tube wall heat flux is specified instead of

wall temperature an iterative procedure must be used to determine the

tube wall temperature from Fourier's conduction equation evaluated

at the wall.

The entire system of governing equations cannot be solved

pr

41 explicitly due to coupling and non-linearities found in these

equations. Note that the coefficient matrices associated with the

energy and axial momentum equations are not independent of temperature

and axial velocity.

To solve this complex set of simultaneous equations an iteration

scheme is employed. First, initial values are selected for the

unknown variables. These initial values may be those found at the

previous axial station at the same time step. Then new values of the

temperature at the grid points are calculated followed by calculation

of fluid property values which are functions of temperature including

the density. The bulk density is determined so that a new pressure

value can be found. Then the axial velocities are determined and ^

i finally the radial velocities are found. The whole process is iJ,

§1 iterated tmtil convergence is obtained. 2

.J

A higher order difference approximation gives better accuracy " 5

by reducing truncation error. Thus the derivatives evaluated at "

the tube wall can be expressed by [1]

^^jioEr Ul'^Fn-CODFn,•tCDOFn.^.

- iOO Fn-. F150 Fn-^ -Zi F:.S ) (*-10)

9T 3u This approximation is used to evaluate — — ) and ——) in the

3r ^ 8r ^ determination of the Nusselt number and the friction factor.

4.2. Computer Program

Actual computations are accomplished using a digital computer

A generalized flow chart showing the sequence of operation is

presented in Figure 4.1.

In Block 1 of the flow chart all necessary values for

computation are read as input data. The numerical values of the

input data are presented in the Appendix with the exception of the

operating conditions. The input parameters are:

1. The number of increments, n, in the radial direction and

the size of the axial increments and time increments

42

initially.

2. The weighting parameters, 0 and <j), used in the difference

equations.

3. Parameters governing the iteration convergence criteria.

4. The properties of the gas a, b, Y> SII< ^ •

5. The operating conditions M , Gr /Re , thermal boundary

condition (uniform wall temperature or uniform wall heat

flux), and the magnitude of the thermal boundary condition.

6. Criteria for changing the step sizes in the axial

direction, radial direction, and time.

In Block 2 the conditions before the transient are established

1 31 i >

I

and stored. Initially it is assumed that

T ^ = l (4-11)

and

v+ = 0 (4-12)

throughout the tube.

These conditions are later corrected by solving the governing

w

Calculate Entrance Conditions

Solve Governing Equations

No

43 Computer Flow Chart

Figure 4.1

10 Step in Axial Position

i t a

Step in Time

Calculate Heat Transfer and

Friction Parameters

I No

Print Output

44

equations for the case of no heating. The parabolic entrance velocity

profile is computed.

In Block 3 the time is set at zero.

Block 4 represents solving the governing equations. The

procedure is to compute the coefficient matrix for the energy equation

at the appropriate axial station and time. Upper triangulation and

back substitution are performed on the matrix to obtain the temperature

distribution. The density is determined at the grid points from the

equation of state. The value of the bulk fluid density is determined

followed by determination of the coefficient matrix for the axial

momentum equation. The axial momentum equation coefficient is then

reduced to a bidiagonal matrix by back substitution. This allows the

pressure to be determined from the integrated continuity equation.

Finally the axial velocity distribution across the tube is determined

by back substitution and the radial velocity distribution is

determined by solving the continuity equation from the centerline

to the tube wall.

In Block 5 a comparison is made of the values of the temperature

and axial velocity for the point next to the tube wall obtained

during the current iteration with the values obtained in the previous

iteration. When the temperature and axial velocity at the point

adjacent to the tube wall matches the previously calculated values

within a small tolerance, the execution of the program proceeds to

Block 6.

Block 6 represents the calculation of the heat transfer and

friction parameters.

3

J

45

In Block 8 a comparison is made of the values of the temperature

and axial velocity at the point next to the tube wall obtained at

the current time step with the values obtained at the previous time

step at the same axial station. If these values match within a

small tolerance, the program execution proceeds to Block 10. This

occurs when steady state is reached at the current axial station.

Otherwise, a new time step is made and the program execution

proceeds to Block 9.

Block 9 governs stepping with respect to time, and Block 10

governs stepping with respect to the axial coordinate.

0 i i i a

i

^Cti^

CHAPTER V

DISCUSSION OF RESULTS

The developed computer program is used to solve the transient

problem for both a step increase in tube wall temperature and heat

flux where

T ••• = 1.2 (5-1) w

and

C = -5 (5-2) A parabolic velocity profile is selected at the thermal entrance.

The gas selected is air. Air property data is given in the Appendix.

An entrance Mach number of .01 is chosen and the gravity term in

the momentum equation is set equal to zero. Thus only forced

convection is considered. The present analysis indicates that the

steady state Nusselt number is not strongly affected by the variation

of fluid properties. The transient Nusselt number values, however,

appear to be significantly affected. The friction factor deviates

significantly from the isothermal value.

3

5.1. Thermal Results

Figures 5.1, 5.2, 5.3, and 5.4 show the growth of the

temperature profiles with time at two axial stations for both of the

heating conditions. At early times the temperature profiles appear

to be those of pure radial conduction [12] and to be relatively

independent of axial position. This is due to the low gas velocities

46

kl Temperature Profile Near Entrance

Constant Wall Temperature

Figure 5.1

X

a

b

1(10"^)

t"*" = .0006

t = Steady State

Temperature Profile Down Tube Constant Wall Temperature

1.1 --

1.0

Figure 5.

+ X =

a

b

c

d

1(10" + t = + t = + t = + t =

2

h .0006

.0020

.0080

Steady State

p

t 9 «

i

\

48

1.10

T"*" 1.05

1.00

Temperature Profile Near Entrance Constant Wall Heat Flux

Figure 5.3

X

a

b

1(10"^)

t"*" = .0010

t"*" = Steady State

1.15 Temperature Profile Down Tube Constant Wall Heat Flux

Figure 5.4

1.10

1.05

+ X =

a

b

c

8(10"^)

t"*" = .0010

t"*" = .0080

t"*" = Steady State

{ 3

0»

PF'

near the wall and thermal boundary layer which has not yet

propagated in the radial direction to a steady state value.

The Nusselt number is a strong function of time, but it appears

to be a rather weak function of axial position until the steady

state condition is reached. Figures 5.5 and 5.6 show the Nusselt

number variation down the tube at various times for the constant

tube wall temperature and heat flux respectively. The damped

oscillations at the entrance are typical of those found near

boundaries for finite difference solutions. The constant property

results of Sparrow and Siegel [12] are shown for comparison. It

is worth noting that the characteristics method of the constant

properties solution assumes an abrupt arrival of the Nusselt number

at the steady state value. That is, steady state is achieved at a

definite value of time for each axial position. The present

solution, however, approaches the steady state value asymptotically.

It can be seen that during the transient the present solution

predicts somewhat larger Nusselt number values than those predicted

by the constant properties analysis. The steady state Nusselt

number results of Wors^e-Schmidt are shown on Figure 5.6 for the

constant heat flux case. It should be stated that Wors^e-Schmidt

includes the gravity term in the momentum equation.

At steady state the values obtained from the present analysis

and those obtained from the constant properties solution are very

close. Sparrow and Siegel claim very close correlation with the

steady state results of the Graetz solution of Lipkis (see

Discussion of Reference 5). The Graetz solution is a constant

49

\

.1

80 4-

Present Analysis

Constant Properties

Nu

Nuss Constant

elt Number Wall Temperature

Figure 5.5

a

b

c

d

e

t"*" = .0006

t**" = .0010

t"*" = .0020

t"*" = .0040

t = Steady State

8 10

x' do"^)

80 --

Nusselt Number Constant Wall Heat Flux

Figure 5.6

51

70 --

a

60 --

50 --

Nu

40

30

20

10 --

Present Analysis

Constant Properties

i.

8

x'^dO"^)

property eigenvalue solution. It should be noted that the Nusselt

number is not a function of temperature according to the transient

constant properties analysis. The present solution indicates that

it is a rather weak function of temperature as the solution

approaches steady state. This can be inferred by the close

correlation between the two solutions.

The relatively mild effect of property variation can be

associated with several opposing effects [14]. None of these effects

are dominating. These effects include increasing thermal

conductivity at the wall, which causes greater radial conduction;

larger axial velocities near the wall, which cause greater axial

convection; radial velocities near the wall causing radial

convection; and the decrease in gas density which tends to reduce

the heat transfer from the wall.

5.2 Hydrodynamic Results

Figures 5.7 and 5.8 show the friction factor variation down

the tube at various times for the constant tube wall temperature

and heat flux respectively. No known similar report of transient

friction factor values exists. The results indicate that the

friction factor is indeed affected by property variations due

primarily to warping of the axial velocity profile and the variation

of viscosity with temperature. It should be noted that the friction

factor variation takes opposite trends with respect to time

depending on v^ether the flow is excited by a step change in tube

wall temperature or heat flux. This is not surprising when one

52

i

- I

<: A

53

f XRe

54

Friction Factor

Constant Wall Heat Flux

Figure 5.8

f XRe

a

b

c d

e

t = .0006 ^+ t = .0010 + t = .0020 ^+ t «= .0040 .+ t = Steady State

5

x*(10"3)

considers the large viscosity changes instantaneously occurring at

the wall in the case of a step increase in wall temperature. It

might be argued that an instantaneous step change in wall temperature

is physically impossible. In the case of a step increase in wall

heat flux a rather gradual increase in wall temperature and

corresponding wall viscosity results. Thus a rather smooth variation

of the friction factor between the initial and final steady state

values is observed.

The steady state friction factor results of Wors^e-Schmidt

are shown on Figure 5.8 for the constant heat flux case. It should

again be noted that Wors^e-Schmidt included the gravity term in the

momentum equation. Also, WorsjJe-Schmidt used a somewhat different

definition of the friction factor and evaluated the Reynolds

number at the local value instead of the entrance value.

The friction factor shows an increase over the value found for

isothermal flow. For the flow of a gas the velocity profile flattens

with heating. The radial velocity allows for the rearrangement

necessary for the velocity profile to change shape. It is noted

that the larger values of radial velocity occur at small time values

and axial stations near the thermal entrance. This is where the

larger property variations occur and where the largest deviations

from the constant property values take place.

Figures 5.9 and 5.10 show the pressure distribution down the

tube at various times for the constant tube wall temperature and

heat flux respectively. The pressure variations can be attributed

to the warping of the axial velocity profile and the increase of

55

t

56

12 ~-Pressure

Constant Wall Temperature

Figure 5.9

10 --

.08 --

.06 --

.04 --

.02

.00

\

8 10

x" (10" )

.07 H-

.06 --

.05 --

.04 --

.03 --

.02

.01 --

57

Pressure Constant Wall Heat Flux

Figure 5.10

.00

l:

x' dO h

58

the viscosity with temperature. Both of these factors tend to

increase the fluid shear forces. In the case of a step increase in

tube wall temperature note that the pressure approaches the steady

state value quite rapidly. It might be stated tbat the pressure

drops in the type of flow considered are generally quite small.

^»ty*wii^i>»fi. n'."•^••^'•^mm ^K^^

LIST OF REFERENCES

1. Arbramowitz, Milton and Irene A. Stegun, Handbook of Mathematical Functions, Washington, D.C., U.S. Government Printing Office, 1964.

2. Bird, R. Byron, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena, New York, John Wiley & Sons, Inc., 1960.

3. Deissler, R. G., "Analytical Investigation of Fully Developed Laminar Flow in Tubes with Heat Transfer with Fluid Properties Variable Along the Radius," NACA Technical Note 2410, 1951.

4. Hilsenrath, Joseph and others. Tables of Thermodynamic and Transport Properties of Air, Argon, Carbon Dioxide, Carbon Monoxide, Hydrogen, Nitrogen, Oxygen, and Steam, New York, Pergamon Press, 1960.

5. Kays, W. M., "Numerical Solutions for Laminar-Flow Heat Transfer in Circular Tubes," Transactions ASME, Vol. 77, 1955, pp. 1265-1274.

6. Lee, William John, "A Theoretical Sutdy of Nonisothermal Flow and Heat Transfer in Vertical Tubes for Fluids with Variable Physical Properties," Ph.D. Thesis, Georgia Institute of Technology, 1962.

7. O'Brien, George G., Morton A. Hyman, and Sidney Kaplan, "A Study of the Numerical Solution of Partial Differential Equations," Journal of Mathematics and Physics, Vol. 29, 1950, pp. 223-251,

8. Richtmyer, Robert Davis and K. W. Morton, Difference Methods for Initial Value Problems, 2d ed.. New York, Interscience Publishers, Inc., 1967.

9. Siegel, Robert, "Heat Transfer for Laminar Flow in Ducts with Arbitrary Time Variations in Wall Temperature," Journal of Applied Mechanics. Transactions ASME, Vol. 27, 1960, pp. 241-249.

10. Siegel, R. and E. M. Sparrow, "Transient Heat Transfer for Laminar Forced Convection in the Thermal Entrance Region of Flat Ducts," Journal of Heat Transfer, Transactions ASME, Vol. 81, 1959, pp. 29-36.

59

60

11. Sparrow, E. M. and J. T. Gregg, "Prandtl Number Effects on Unsteady Forced Convection Heat Transfer," NACA Technical Note 4311, 1958.

12. Sparrow, E. M. and R. Siegel, "Thermal Entrance Region of a Circular Tube Under Transient Heating Conditions," Proceedings, Third U. S. National Congress of Applied Mechanics, Brown University, June, 1958, pp. 817-826.

13. Wilkins, Bert Jr., "Nonisothermal Laminar Flow and Heat Transfer with Temperature Dependent Physical Properties," Ph.D. Thesis, Georgia Institute of Technology, 1965.

14. Wors^e-Schmidt, Peder M., "Finite-Difference Solution for Laminar Flow of Gas in a Tube at a High Heating Rate," Stanford University, Stanford, California, November, 1964.

31

Wtfrm

APPENDIX

A. ERROR, CONVERGENCE, AND STABILITY CONSIDERATIONS FOR FINITE DIFFERENCE SOLUTIONS

B. MESH SIZE AND WEIGHTING PARAMETERS

C. PROPERTY DATA FOR AIR

61

62

APPENDIX A

ERROR, CONVERGENCE, AND STABILITY CONSIDERATIONS

FOR FINITE DIFFERENCE SOLUTIONS

Two major types of error are associated with finite difference

solutions of differential equations. One is truncation error which

results from the fact that there is a finite distance between the

mesh points of the solution grid. A second source of error is

round-off error due to the fact that calculations in practice can

be carried to only a limited number of significant figures.

For simplicity consider a partial differential equation which

is solved by finite difference methods where E is the exact solution

of the partial differential equation [7]. It should be remembered

that what are actually being solved, however, are the difference

equations associated with the partial differential equation. Let

the exact solution of the different equations be D. Then (E-D)

represents the truncation error. Note that as

AX.AFAi-^O (A-l)

in the equations used to approximate the derivatives, equations

(3-2) through (3-5), all terms but the first on the right hand side

of the equal sign approach zero. In fact it might be argued that

instead of solving the given partial differential equation, the

actual differential equation solved is that high order partial

differential equation resulting from adding the infinite number

63

of neglected terms in the equations used to approximate the

derivatives in the partial differential equation. Thus the solution

of the difference equations is said to converge to the exact

solution of the partial differential equation if

(A-2)

It should be noted that the numerical solution of the

difference equations, N, is not generally the exact solution of the

difference equations. Round-off errors are inherent in computing

machinery, and small errors are also introduced in the solution of

nonlinear simultaneous equations by iterative procedures. This

numerical error can be express as (D-N).

Stability is the condition under which the numerical error is

small throughout the domain of the solution. If the numerical

error increases as the solution progresses, the solution scheme is

said to be unstable. To make

(E-N) = (E-D) + (D-N)

small, the numerical scheme must be both convergent and stable.

Consider the partial differential equation

(A-3)

(A-4)

where a is a constant [8]. This can be written in finite difference

form as

4^%§^^ n-(^^vA>/-^f^>i..'^f^^/A.l)t(|-oi):/^,,;rZl^A^ (A-5)

64

where a is a weighting parameter such that 0 < a < 1. The function

Fj ^^ with the subscripts j,k represents the value of the function

F at the grid points, the subscripts being associated with the r

and w variables respectively. It is assumed that a solution to the

difference equation can be obtained in the form of the Fourier

series term

F,-Ar£'"'-i' where A and C are constants, m is an integer, and

i-FT This method of analysis is attributed to von Neumann. After

substituting this expression into the difference equation, an

equation for the growth factor, O , can be written as

^rm^- i-Chcy)LCl-COSmAr)

.(A-6)

(A-7)

i-\-cyLi\-CosmAr) (A-8)

where

I f

, ^ £o(AW

Max/EMl^ I (A-io)

then, according to the von Neumann stability criterion, the

difference equation is stable. Otherwise, some harmonic is

amplified without limit as k increases indicating an unstable

condition. For the finite difference formulation used here, it

can be seen that the growth factor never exceeds +1 for all real

m. The growth factor, however, can be less than -1. Consequently

the required stability condition is

no restriction

if 0 < 0 < 1/2

if 1/2 < o < 1

65

(A-11)

(A-12)

The constant coefficient linear partial differential equation

considered in the preceding stability considerations is, of course,

much simpler than the equations used in the present analysis. It is

seen that the implicit character of the difference equation implies

stability when the weighting parameter is greater than one-half.

Another property of the implicit formulation is that the solution is

obtained by solving a set of simultaneous equations, this being the

primary disadvantage of the method. Convergence considerations are

not made in the preceding analysis; however, for large classes of

linear equations it has been proven by Lax [6] that stability is

the necessary and sufficient condition for convergence. Generally

this is extended to variable coefficients and nonlinear equations

with only experimental and intuitive justification. For such cases

a mild strengthening of the stability criteria may be required [8].

It has been shown that lower order terms of a differential

equation have practically no effect on stability [8]. The constant

properties solution of Sparrow and Siegel using the method of

characteristics suggests that the solution can be divided into two

parts. The first part represents the solution during the transient

period, the solution being independent of axial position. The second

part represents the solution for steady state, the solution being

independent of time.

One might argue that the energy and momentum equations can be

approximated by

•kMr'Kifh

66

<A-13)

(A-14)

during the transient time period and

eu:£-TMrrf) (A-15)

^ ' o=3f fi .-iF<ry# ; (A-16)

at steady state. Further, the solutions indicate that the term

behaves very nearly as a constant term.

9P

3x

With some admittedly heuristic arguments we are able to see that

the present solution should tend to be stable; however, these

arguments are somewhat unsatisfactory. More satisfying is the

excellent agreement of the present finite difference solution with

other solutions. While complete transient solutions are not

available, it can be seen that good agreement is obtained where

solutions are available. Steady state results compare favorably

with those of other analyses. It is not possible to make a rigorous

stability analysis of the finite difference scheme due to the

complexity of the nonlinear variable coefficient governing equations

\^ich are not even independent of each other.

/ -r-

67

APPENDIX B

MESH SIZE AND WEIGHTING PARAMETERS

Considerable experimentation was done to determine suitable

+ + + values of 0,(|»,Ax ,Ar , and At to assure satisfactory convergence

and stability of the computation scheme. A procedure of guessing

trial values was used until acceptable values of these parameters

were found. Because of the large number of parameters involved

and their interdependence on each other with respect to convergence

and stability, a truly optimum set of values of these parameters

is very difficult, if not impossible, to obtain. Larger steps in

time are made as the transient progresses with satisfactory results.

Also larger steps in axial position are possible for stations farther

down the tube. Larger steps in the radial direction for

simultaneously large values for axial position and time also prove

to be satisfactory.

Some limitations are also imposed by limited storage

capabilities of the computer used to perform the computations.

Table B.l shows the weighting parameter values and mesh size

used in the computations.

68

0 = .75 (|) = .90

+ X

.000 - .001

.001 - .010

Ax-

.000250

.000500

0.0 < x" < .001, 0.0 < t"*" < .0040, n = 52

X > .001, t > .0040, n = 26

.00000 - .00195

.00195 - .00395

.00395 - .00795

.00795 - .01590

.01590 -

At

.000200

.000400

.000800

.001600

.003200

Weighting Parameter Values

and Mesh Size

Table B.l

69

APPENDIX C

PROPERTY DATA FOR AIR

Property data is evaluated at 610"R and one atmosphere pressure,

Pr « .7 [4]

Y - 1.4 [4]

a «= .67 [14]

b « .71 [14]

y.