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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.