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Natural convection heat transfer of cold water between concentric cylinders for high rayleigh numbers—a numerical study
Pergamoa
Inr. 1. Engag Sci. Vol. 32, No. 9, pp. 002&7225(93)EOfM&-S
1437-1450, 1994 Copyright @ 1994 Elaevier Science Ltd Printed in Great Britain. All rights reserved
0020-7225/94 $7.00+ 0.00
NATURAL CONVECTION HEAT TRANSFER OF COLD WATER BETWEEN CONCENTRIC CYLINDERS FOR HIGH RAYLEIGH NUMBERS-A NUMERICAL STUDY C. V. RAG~AVARAO Department
and Y. V. S. S. SANYASIRAJU
of Mathematics, Indian Institute of Technology, Madras-600 036, India (Communicated
by S.-I. PAI)
Abstract-A numerical study of the natural convection heat transfer of cold water, having a density inversion between two isothermal horizontal concentric cylinders is studied. The governing equations are solved by u wind finite difference method for different annulus radius ratios with Rayleigh number range lJ- ld and the inversion parameter y varies between -2 and 0. The results indicate that the flow patterns are greatly influenced by the inversion parameter, which determines the effect of the additional convection that arises from the inversion of density of water at 3.98”C within the annulus.
INTRODUCTION The steady laminar natural convection heat transfer in a horizontal concentric cylindrical annulus has been attracting many researchers in the past decade due to its association with numerous practical applications like cooling of electronic systems, underground electric transmission lines and many others. But most of the past investigators used the linear Boussinesq relation between fluid density and its temperature, which is acceptable for most of the fluids (hereafter called common fluids), but it cannot exhibit the non-linear nature of the water at its freezing point 3.98”C (the density of water increases with decrease in temperature up to its freezing point and later it decreases with decrease in temperature). Because of this special behavior of cold water and its various practical applications, both experimental and theoretical studies have been done in this particular field. Seki et al. [l] investigated experimentally the natural convection of cold water in a horizontal concentric cylindrical annulus. Recently Desai and Forbes [2], Watson [3] and Nguyen et al. [4] have investigated the same problem using perturbation and numerical methods. From these experimental and theoretical investigations it is found that the resulting flow is multicellular rather than uni~llul~ in common fluids. Also the heat transfer rate is greatly minimized when these convective cells are of the same size. In the present investigation we considered the steady laminar natural convection heat transfer between two concentric isothermal cylindrical annuli filled with cold water for moderate and high Rayleigh numbers. A second degree Boussinesq approximation is used to allow the density inversion of the cold water at its freezing point. An upwind finite difference method is adopted to solve the coupled momentum and energy equations in stream function and vorticity formulation. It is found from this numerical experiment that even though the upwind difference scheme reduces the order of approximation of non-linear terms, it helps the linear system to be stable even for high Rayleigh numbers. The fmal results are expressed in terms of stream lines, isotherms and heat transfer variation on the inner and outer cylinders. FORMULATION
OF THE PROBLEM
The schematic picture of the two-dimensional steady natural convection in cylindrical annulus is given in Fig. l(a). The gap between the two cylinders is filled with cold water. The inner and outer cylinders with radius ri and r, are maintained at constant temperatures T and ES%%#.a 1437
C. V. RAGHAVARAO
and Y. V. S. S. SANYASIRAJU
P :
ej)
Q : (ki* ej-1) R :
+
1)
Fig. 1. (a) Flow geometry and coordinate system; (b) finite difference grid.
1439
Natural convection heat transfer of cold water
T, respectively. All fluid properties except water density are taken to be constant evaluated at the mean temperature T = (F + T,)/2. The Boussinesq equation of state is 1- i
&(T
and are
(1)
- T;)“]
n=l
where p’ and T’ are the density and temperature of the fluid, p;‘s are expansion coefficients and the subscript r represents the reference state. By giving various values to N, i.e. by a Nth degree polynomial approximation the above equation of state can be applied to large variety of fluids. Here to exhibit the non-linear behavior of the cold water at its freezing point, we used a second degree polynomial by taking N = 2 in equation (l), i.e., p’ = p; 1 - i &(T [ PI=1
- T;)“]
(2)
with /I; = 2(T: - T;)P; p; = 8 x 10e6 CT2
and
T:, = 3.98”C. It is found from the investigation of Landolt-Bornstein [5] and Moore and Wiess [6], the above relation is accurate over the range 0-8°C. Using Boussinesq approximation [equation (2)], the momentum and energy equations in terms of stream function and vorticity in polar coordinate system can be written as Q2$ = -o Qzo=
’
la(+JW)+
‘OS8
Pr r a(r, 6)
after using the non-dimensional T=--”
---de+
a
(3) sin 6:
(Ra,T + Ra2T2)
(4)
parameters
T’ - T’ T; - T;’
rr
U=-
U’Ri
ff ’
r=Ri7
v=-
V’Ri
a
where (Yis the thermal diffusivity, r the radial coordinate and u, v are the velocity components which are related with stream function JI by the relations
u=laJI r ae'
The corresponding
v=--*
a+ ar
boundary conditions are ++
0 on both the cylinders
T= 1 on the inner cylinder and = 0 on the outer cylinder.
(6)
1440
C. V. RAGHAVARAO
and Y. V. S. S. SANYASIRAJU
The non-dimensional parameters which appear in the equations are Prandtl number Pr and two Rayleigh numbers Rat and Ra,, where
Ra
=
i!mw1-
1
Ra
CJ
vff =
&$2m1- cJ*
2
Vff
where v is the kinematic viscosity and g is the acceleration due to gravity. Since from the equation of the state, p, is a function of Tb while p2 is a characteristic constant of the fluid. It follows that Ra, depends on both TI and TL while Ra, depends only on the temperature difference (Tf - TL) like the Rayleigh number Ra in the case of common fluid. This suggests that one should choose Ra2 as the appropriate Rayleigh number and define a non-linear Rayleigh number Ra = Ra, together with the parameter -2(T& - T:)
PI
Ra, ‘=G=P2(T;-T:)=
(T;-T;)
’
(7)
This parameter is hereafter called the inversion parameter, which relates the temperature for maximum density Tk to the wall temperature. From equation (2), the value of y varies between -2 c y < 0 because in this range of y, Tk lies inside of the cylindrical annulus and there exists an inversion of density within the confined fluid. There will not be any density inversion for y > 0 and y < -2, i.e. the flow approaches to common fluid when Jyl<< 1. The same is observed by Nguyen et al. [4] by a perturbation method for the range of Rayleigh numbers ld to 8 X 103. Here our main interest is to correlate the heat transfer rates for moderate and high Rayleigh numbers ranging from lo* to 105, in the density inversion range -2 5 y s 0, with radius ratios 1.5 and 2.0.
NUMERICAL
METHOD
To get the advantage of easy discretization of the cylindrical annulus the governing equations are taken in a polar coordinate system. Then the equations are transformed into the (5, 0) plane with the transformation r = exp(5). The corresponding equations are (8) cost3-$+sinBd
2 V$= a+ a[*
at
Ra(yT + T*)
(9)
a iii?’
Only the upper semi-circular domain is considered for investigation because of the symmetry along 8 = 0 and 0 = x.
Natural convection heat transfer of cold water
1441
The boundary conditions are imposed on the two isothermal concentric cylindrical walls and two lines of symmetry, i.e. at 8 = 0 and 8 = zc of the form #=w=>i=O
$ = 0, * = 0,
one=Oand8=Ir
T = 1, Oi,j = - ““~~~~~ T = 0, wo,j= -
W~A, -
on the inner cylinder
r(l,,Af
on the outer cylinder
W02
where the subscripts i and o in vorticity boundary conditions represent the values at the inner and outer cylinders respectively. The above coupled equations along with boundary conditions are solved numerically by the upwind finite difference method. The second order derivatives in the equations (8)-(10) are approximated by central differences of order At2 and Ae2 (where At and A0 are the step lengths in 6 and 8 directions respectively). The non-linear terms of equations (9) and (10) are approximated with first order upwind differences of the form (fF), =O*5(f -
IflE+l,j+
Ifl~,j-0*5(f
+ Ifl)E-lj
where fis aJ//ag or a$/ae, is approximated by central differences at any point (ti, 0,) and F is o and T respectively in equations (9) and (10). The diagonal dominance is assured because of the upwind difference approximation for the non-linear terms even for high Rayleigh numbers. To minimize the oscillations of the solution in the convergence process it is necessary to use some initial solution in the iterative process. For Ra = 100 the starting values for I(I, w and T are taken as zeros and for the solutions of high Rayleigh numbers the earlier solution is taken as the starting solution at all the inner grid points. After experimenting with different grid sizes it is found that a uniform 41 X 31 grid is optimum for the present investigation for both the radius ratios 1.5 and 2.0. A part of the computational grid is given in Fig. l(b) obtained by taking intersection of 5 = constant (circles) and 8 = constant (radial lines). The Block SLOR method is used in the iteration process. The resulting algebraic equations are solved using a tri-diagonal solver along each line. The acceleration parameters for T, o and + are taken as 0.7, 0.5 and 0.7 respectively. The vorticity took a greater number of iterations compared with stream function and temperature because of the small acceleration parameter compared with other two. The iterations are stopped when IF(“+l) _ F’“‘J 5
lo-4
at all inner grid points, where n is the iteration number and F is one of the field variables T, w and +. The calculations are carried out on the Siemens Bs2000 mainframe of the Indian Institute of Technology, Madras, India and the graphs were drawn on a Wipro PC-AT using a software package developed by the authors. For a typical case Ra = 5 X lo4 with 41 X 31 mesh the present code required 40s of CPU time.
DISCUSSION
OF THE
RESULTS
The developed computer code is first tested with the cylindrical annulus filled with common fluid. The corresponding results are compared with Kuehn and Goldstein [7] for the annulus radius ratio 2.6, Prandtl number 0.7 and the Rayleigh number varies between 16 and 16. The flow pattern as well as the temperature distribution for these test computations are in excellent agreement with the results of Kuehn and Goldstein. For Rayleigh numbers Ra = lo4 and
C. V. RAGHAVARAO
0
(a)
u
and Y.
V. S. S. SANYASIRAJU
x
x
0
Fig. 2. Stream lines and isotherms for common fluid. (a) Present results; (b) Ref. [7].
5 X lo4 the stream lines and isotherm patterns are given in Fig. 2 along with the results of Kuehn and Goldstein. The results of the present problem are expressed in terms of stream lines and isotherm patterns in Figs 3-8 for different values of y and R. Since for Ra < 1000 the flow pattern is mostly dominated by conduction, the stream lines and isotherm patterns are given for 103-105. The isotherms are drawn for T = O-l with increments of 0.1 in all the graphs and the corresponding stream lines are drawn depending on the maximum and minimum values of II, for that particular case. The isotherms are given in upside down form (K - 0) in each figure and stream lines are on the left-hand side.
FLOW
PATTERNS
AND
ISOTHERM
DISTRIBUTIONS
The parameters which effect the flow pattern for cold water are inversion parameter y, Rayleigh number Ra and the radius ratio R. The effect of the flow pattern with respect to y at = 1.5 is given in Figs 6, 7 and 8 at various Rayleigh R=20isgiveninFigs3,4and5andatR numbers. For y = 0 and y = -2 a uni-cellular flow is formed with opposite Sow directions. As the Rayleigh number increases the center of the cell is moving up and down respectively for y = 0 and y = -2, The co~esponding isotherms shows as Rayleigh number increases the convective heat transfer is increasing. For y = -1, a b&cellular flow with the cells moving in
Natural convection heat transfer of cold water
1443
R.= Id
R
0
0
A
Ra =
1L
IO’
0
Fig. 3. Stream lines and isotherms at R = 2.0 and Y = -2.
C. V. RAGHAVARAO
1444
and Y. V. S. S. SANYASIRAJU
Rs=5
x 10~
Fig. 4. Stream lines and isotherms at R = 2.0 and y = -1.
opposite direction are formed. In this case even at Rayleigh number lo4 very little convective heat transfer is observed. Hence the dominant process for heat transfer is conduction. As the Rayleigh number increases the center of rotation of the two cells are moving in opposite directions. Finally the effect of the radius ratio R can be observed by comparing Figs 3,4 and 5 with Figs 6, 7 and 8. From these graphs it is clear that the flow pattern is not changed with R, but the heat transfer rates Nu and Nu defined later are increased with R.
HEAT
TRANSFER
RATES
The local heat transfer rates at the inner and outer surfaces of the cylinders for the different cases are compared in Figs 9(a) and (b) respectively. The local heat transfer rates are expressed in terms of corresponding Nusselt numbers Nui on inner cylinder and Nu, on outer cylinder defined as
The
curves of Nusselt number
. [ dr 1r=l,R
Nui,o = -In R r eT distribution
on the cylinder are drawn to 8 for the inner
Natural convection heat transfer of cold water
Ra = 10’
Ra = 10’
Ra = IO3
a
0
Fig. 5. Stream lines and isotherms at R = 2.0 and y = 0.
1445
1446
C. V. RAGHAVARAO
and
Y. V.
S. S. SANYASIRAJU
Ra=5n104
rc
0
Fig. 6. Stream lines and isotherms at
R = 1.5 and y = -2.
cylinder in Fig. 9(a) and for the outer surface in Fig. 9(b). In both the figures the local heat transfer rate is compared with the corresponding heat transfer rates of common fluid at Ra = 5 x 104. From Figs 9(a) and (b) it is clear that the heat transfer rates for the cold water are small compared with that of common fluid for all values of y between -2 and 0. For y = -1 (complete inversion) the heat transfer rates are almost equal to one, i.e. conduction is dominated. On the inner cylinder Ntti is increasing like common fluid between 8 = 0 and 180” for y = 0 and decreasing between 8 = 0 and 180” for y = -2. The pattern of heat transfer rates
1447
Natural convection heat transfer of cold water
Rx = 5 x
Ra =
10’
104
Fig. 7. Stream lines and isotherms at R = I.5 and y = -1.
on the outer cylinder Nu, are completely reversed to the above patterns. Here for y = 0, Nu, is decreasing between 8 = 0 and 180” and increasing for y = -2. The pattern of heat transfer rates do not change with respect to the radius ratio R, but the rate is increased with increase of R. To study the total heat Iiow across a surface, the overall Nusselt numbers are calculated for different cases of y and R using the definition
The present results giving a maximum difference of 1.9% in overall Nusselt number on inner and outer surfaces of the cylinders, which are to be equal because of the energy conservation. Finally, the overall heat transfer rates Nu for different cases are correlated independently in the form s=aRab using a least-squares regression analysis. The corresponding values of R and y are given in Table 1.
values of ii and & for different
CONCLUSIONS The steady laminar natural convection of cold water within a concentric cylindrical annulus has been studied numerically by an upwind finite difference method for high Rayleigh numbers. In this investigation the cases y = 0 (i.e. the maximum density is at the inner cylinder), y = -2
1448
C. V. RAGHAVARAO
and Y. V. S. S. SANYASIRAJU
Ra=Sx
IO4
Ra = 8 x IO3
x
0
Fig. 8. Stream lines and isotherms at R = 1.5 and y = 0.
(i.e. the maximum density is at the outer cylinder) and y = -1 (maximum density is at the center of the annulus) are discussed for the annulus radius ratios 1.5 and 2. From the present results it is observed that (1) The heat transfer rates are increasing with R for constant values of Ra and y. (2) The heat transfer rates are increasing with Ra for constant values of R and 3/. (3) Uni-cellular How for y = 0 and y = -2 for R I 2, in which for y = -2, the center of rotation is moving down with increasing Ra and for y = 0, the center of rotation is moving up with increasing Ra. (4) B&cellular flow for y = -1, for which the two cells moving in different directions.
Natural convectian heat transfer of cold water 6
1449
- (4
5
0 Common fluid A I.P. = 0, R = 2.0 0 I.P. = 2, R = 2.0 0 I.P. = 0, R = 1.5 + I.P.=2.R=1.5
4
0 I.P. = 1. R=
z”
1.5
3 . 21
I I/ 1 * 0
(b) 0 Common fluid A I.P. = 0. R = 2.0 0 I.P. = 2. R = 2.0 tI.P.=O,R=l.S +l.P.=2.R=1.5 0 I.P. = 1, R = 1.5
\
Fig. 9. Nusselt number distribution
at Ra = 5 X lo4 on (a) the inner cylinder and (b) the outer cylinder.
1450
C. V. RAGHAVARAO
and Y. V. S. S. SANYASIRAJU Table 1
Inner cylinder R
2.0
y
0 -2 1.5 0 -2
Ra Id-ld ld-105 Id-16 Id-ld
Outer cylinder
ll
b
a
b
0.24734 0.25815 0.35296 0.36198
0.21344 0.20913 0.12423 0.12199
0.25733 0.2506 0.3546 0.3566
0.2077 0.2133 0.1237 0.1237
REFERENCES 111N. SEKI, S. FUKKUSAKO and M. NAKAOKA, ASME. J. Heat Transfer 97,556 (1975). 21 V. S. DESAI and R. E. FORBES, ASME 41-47 (1971). :3] A. WATSON, Q. J. Mech. Appl. Math. 25,423 (1972). :4] T. H. NGUYE.?, P. VASSEUR and L. ROBILLARD, ht. J. Hear Mass Transfer Z&l559 (1982). :5] LANDOLT-BORNSTEIN, Zahlenweste und Funklionem, Vol. 2, pp. 36-37. Springer. Berlin (1971). 61 D. R. MOORE and N. 0. WIESS, JFM 61,553 (1973). ‘71 T. H. KUEHN and R. J. GOLDSTEIN, JFM 74,695 (1976). (Received 10 August 1993; accepted 5 October 1993)
Inr. 1. Engag Sci. Vol. 32, No. 9, pp. 002&7225(93)EOfM&-S
1437-1450, 1994 Copyright @ 1994 Elaevier Science Ltd Printed in Great Britain. All rights reserved
0020-7225/94 $7.00+ 0.00
NATURAL CONVECTION HEAT TRANSFER OF COLD WATER BETWEEN CONCENTRIC CYLINDERS FOR HIGH RAYLEIGH NUMBERS-A NUMERICAL STUDY C. V. RAG~AVARAO Department
and Y. V. S. S. SANYASIRAJU
of Mathematics, Indian Institute of Technology, Madras-600 036, India (Communicated
by S.-I. PAI)
Abstract-A numerical study of the natural convection heat transfer of cold water, having a density inversion between two isothermal horizontal concentric cylinders is studied. The governing equations are solved by u wind finite difference method for different annulus radius ratios with Rayleigh number range lJ- ld and the inversion parameter y varies between -2 and 0. The results indicate that the flow patterns are greatly influenced by the inversion parameter, which determines the effect of the additional convection that arises from the inversion of density of water at 3.98”C within the annulus.
INTRODUCTION The steady laminar natural convection heat transfer in a horizontal concentric cylindrical annulus has been attracting many researchers in the past decade due to its association with numerous practical applications like cooling of electronic systems, underground electric transmission lines and many others. But most of the past investigators used the linear Boussinesq relation between fluid density and its temperature, which is acceptable for most of the fluids (hereafter called common fluids), but it cannot exhibit the non-linear nature of the water at its freezing point 3.98”C (the density of water increases with decrease in temperature up to its freezing point and later it decreases with decrease in temperature). Because of this special behavior of cold water and its various practical applications, both experimental and theoretical studies have been done in this particular field. Seki et al. [l] investigated experimentally the natural convection of cold water in a horizontal concentric cylindrical annulus. Recently Desai and Forbes [2], Watson [3] and Nguyen et al. [4] have investigated the same problem using perturbation and numerical methods. From these experimental and theoretical investigations it is found that the resulting flow is multicellular rather than uni~llul~ in common fluids. Also the heat transfer rate is greatly minimized when these convective cells are of the same size. In the present investigation we considered the steady laminar natural convection heat transfer between two concentric isothermal cylindrical annuli filled with cold water for moderate and high Rayleigh numbers. A second degree Boussinesq approximation is used to allow the density inversion of the cold water at its freezing point. An upwind finite difference method is adopted to solve the coupled momentum and energy equations in stream function and vorticity formulation. It is found from this numerical experiment that even though the upwind difference scheme reduces the order of approximation of non-linear terms, it helps the linear system to be stable even for high Rayleigh numbers. The fmal results are expressed in terms of stream lines, isotherms and heat transfer variation on the inner and outer cylinders. FORMULATION
OF THE PROBLEM
The schematic picture of the two-dimensional steady natural convection in cylindrical annulus is given in Fig. l(a). The gap between the two cylinders is filled with cold water. The inner and outer cylinders with radius ri and r, are maintained at constant temperatures T and ES%%#.a 1437
C. V. RAGHAVARAO
and Y. V. S. S. SANYASIRAJU
P :
ej)
Q : (ki* ej-1) R :
+
1)
Fig. 1. (a) Flow geometry and coordinate system; (b) finite difference grid.
1439
Natural convection heat transfer of cold water
T, respectively. All fluid properties except water density are taken to be constant evaluated at the mean temperature T = (F + T,)/2. The Boussinesq equation of state is 1- i
&(T
and are
(1)
- T;)“]
n=l
where p’ and T’ are the density and temperature of the fluid, p;‘s are expansion coefficients and the subscript r represents the reference state. By giving various values to N, i.e. by a Nth degree polynomial approximation the above equation of state can be applied to large variety of fluids. Here to exhibit the non-linear behavior of the cold water at its freezing point, we used a second degree polynomial by taking N = 2 in equation (l), i.e., p’ = p; 1 - i &(T [ PI=1
- T;)“]
(2)
with /I; = 2(T: - T;)P; p; = 8 x 10e6 CT2
and
T:, = 3.98”C. It is found from the investigation of Landolt-Bornstein [5] and Moore and Wiess [6], the above relation is accurate over the range 0-8°C. Using Boussinesq approximation [equation (2)], the momentum and energy equations in terms of stream function and vorticity in polar coordinate system can be written as Q2$ = -o Qzo=
’
la(+JW)+
‘OS8
Pr r a(r, 6)
after using the non-dimensional T=--”
---de+
a
(3) sin 6:
(Ra,T + Ra2T2)
(4)
parameters
T’ - T’ T; - T;’
rr
U=-
U’Ri
ff ’
r=Ri7
v=-
V’Ri
a
where (Yis the thermal diffusivity, r the radial coordinate and u, v are the velocity components which are related with stream function JI by the relations
u=laJI r ae'
The corresponding
v=--*
a+ ar
boundary conditions are ++
0 on both the cylinders
T= 1 on the inner cylinder and = 0 on the outer cylinder.
(6)
1440
C. V. RAGHAVARAO
and Y. V. S. S. SANYASIRAJU
The non-dimensional parameters which appear in the equations are Prandtl number Pr and two Rayleigh numbers Rat and Ra,, where
Ra
=
i!mw1-
1
Ra
CJ
vff =
&$2m1- cJ*
2
Vff
where v is the kinematic viscosity and g is the acceleration due to gravity. Since from the equation of the state, p, is a function of Tb while p2 is a characteristic constant of the fluid. It follows that Ra, depends on both TI and TL while Ra, depends only on the temperature difference (Tf - TL) like the Rayleigh number Ra in the case of common fluid. This suggests that one should choose Ra2 as the appropriate Rayleigh number and define a non-linear Rayleigh number Ra = Ra, together with the parameter -2(T& - T:)
PI
Ra, ‘=G=P2(T;-T:)=
(T;-T;)
’
(7)
This parameter is hereafter called the inversion parameter, which relates the temperature for maximum density Tk to the wall temperature. From equation (2), the value of y varies between -2 c y < 0 because in this range of y, Tk lies inside of the cylindrical annulus and there exists an inversion of density within the confined fluid. There will not be any density inversion for y > 0 and y < -2, i.e. the flow approaches to common fluid when Jyl<< 1. The same is observed by Nguyen et al. [4] by a perturbation method for the range of Rayleigh numbers ld to 8 X 103. Here our main interest is to correlate the heat transfer rates for moderate and high Rayleigh numbers ranging from lo* to 105, in the density inversion range -2 5 y s 0, with radius ratios 1.5 and 2.0.
NUMERICAL
METHOD
To get the advantage of easy discretization of the cylindrical annulus the governing equations are taken in a polar coordinate system. Then the equations are transformed into the (5, 0) plane with the transformation r = exp(5). The corresponding equations are (8) cost3-$+sinBd
2 V$= a+ a[*
at
Ra(yT + T*)
(9)
a iii?’
Only the upper semi-circular domain is considered for investigation because of the symmetry along 8 = 0 and 0 = x.
Natural convection heat transfer of cold water
1441
The boundary conditions are imposed on the two isothermal concentric cylindrical walls and two lines of symmetry, i.e. at 8 = 0 and 8 = zc of the form #=w=>i=O
$ = 0, * = 0,
one=Oand8=Ir
T = 1, Oi,j = - ““~~~~~ T = 0, wo,j= -
W~A, -
on the inner cylinder
r(l,,Af
on the outer cylinder
W02
where the subscripts i and o in vorticity boundary conditions represent the values at the inner and outer cylinders respectively. The above coupled equations along with boundary conditions are solved numerically by the upwind finite difference method. The second order derivatives in the equations (8)-(10) are approximated by central differences of order At2 and Ae2 (where At and A0 are the step lengths in 6 and 8 directions respectively). The non-linear terms of equations (9) and (10) are approximated with first order upwind differences of the form (fF), =O*5(f -
IflE+l,j+
Ifl~,j-0*5(f
+ Ifl)E-lj
where fis aJ//ag or a$/ae, is approximated by central differences at any point (ti, 0,) and F is o and T respectively in equations (9) and (10). The diagonal dominance is assured because of the upwind difference approximation for the non-linear terms even for high Rayleigh numbers. To minimize the oscillations of the solution in the convergence process it is necessary to use some initial solution in the iterative process. For Ra = 100 the starting values for I(I, w and T are taken as zeros and for the solutions of high Rayleigh numbers the earlier solution is taken as the starting solution at all the inner grid points. After experimenting with different grid sizes it is found that a uniform 41 X 31 grid is optimum for the present investigation for both the radius ratios 1.5 and 2.0. A part of the computational grid is given in Fig. l(b) obtained by taking intersection of 5 = constant (circles) and 8 = constant (radial lines). The Block SLOR method is used in the iteration process. The resulting algebraic equations are solved using a tri-diagonal solver along each line. The acceleration parameters for T, o and + are taken as 0.7, 0.5 and 0.7 respectively. The vorticity took a greater number of iterations compared with stream function and temperature because of the small acceleration parameter compared with other two. The iterations are stopped when IF(“+l) _ F’“‘J 5
lo-4
at all inner grid points, where n is the iteration number and F is one of the field variables T, w and +. The calculations are carried out on the Siemens Bs2000 mainframe of the Indian Institute of Technology, Madras, India and the graphs were drawn on a Wipro PC-AT using a software package developed by the authors. For a typical case Ra = 5 X lo4 with 41 X 31 mesh the present code required 40s of CPU time.
DISCUSSION
OF THE
RESULTS
The developed computer code is first tested with the cylindrical annulus filled with common fluid. The corresponding results are compared with Kuehn and Goldstein [7] for the annulus radius ratio 2.6, Prandtl number 0.7 and the Rayleigh number varies between 16 and 16. The flow pattern as well as the temperature distribution for these test computations are in excellent agreement with the results of Kuehn and Goldstein. For Rayleigh numbers Ra = lo4 and
C. V. RAGHAVARAO
0
(a)
u
and Y.
V. S. S. SANYASIRAJU
x
x
0
Fig. 2. Stream lines and isotherms for common fluid. (a) Present results; (b) Ref. [7].
5 X lo4 the stream lines and isotherm patterns are given in Fig. 2 along with the results of Kuehn and Goldstein. The results of the present problem are expressed in terms of stream lines and isotherm patterns in Figs 3-8 for different values of y and R. Since for Ra < 1000 the flow pattern is mostly dominated by conduction, the stream lines and isotherm patterns are given for 103-105. The isotherms are drawn for T = O-l with increments of 0.1 in all the graphs and the corresponding stream lines are drawn depending on the maximum and minimum values of II, for that particular case. The isotherms are given in upside down form (K - 0) in each figure and stream lines are on the left-hand side.
FLOW
PATTERNS
AND
ISOTHERM
DISTRIBUTIONS
The parameters which effect the flow pattern for cold water are inversion parameter y, Rayleigh number Ra and the radius ratio R. The effect of the flow pattern with respect to y at = 1.5 is given in Figs 6, 7 and 8 at various Rayleigh R=20isgiveninFigs3,4and5andatR numbers. For y = 0 and y = -2 a uni-cellular flow is formed with opposite Sow directions. As the Rayleigh number increases the center of the cell is moving up and down respectively for y = 0 and y = -2, The co~esponding isotherms shows as Rayleigh number increases the convective heat transfer is increasing. For y = -1, a b&cellular flow with the cells moving in
Natural convection heat transfer of cold water
1443
R.= Id
R
0
0
A
Ra =
1L
IO’
0
Fig. 3. Stream lines and isotherms at R = 2.0 and Y = -2.
C. V. RAGHAVARAO
1444
and Y. V. S. S. SANYASIRAJU
Rs=5
x 10~
Fig. 4. Stream lines and isotherms at R = 2.0 and y = -1.
opposite direction are formed. In this case even at Rayleigh number lo4 very little convective heat transfer is observed. Hence the dominant process for heat transfer is conduction. As the Rayleigh number increases the center of rotation of the two cells are moving in opposite directions. Finally the effect of the radius ratio R can be observed by comparing Figs 3,4 and 5 with Figs 6, 7 and 8. From these graphs it is clear that the flow pattern is not changed with R, but the heat transfer rates Nu and Nu defined later are increased with R.
HEAT
TRANSFER
RATES
The local heat transfer rates at the inner and outer surfaces of the cylinders for the different cases are compared in Figs 9(a) and (b) respectively. The local heat transfer rates are expressed in terms of corresponding Nusselt numbers Nui on inner cylinder and Nu, on outer cylinder defined as
The
curves of Nusselt number
. [ dr 1r=l,R
Nui,o = -In R r eT distribution
on the cylinder are drawn to 8 for the inner
Natural convection heat transfer of cold water
Ra = 10’
Ra = 10’
Ra = IO3
a
0
Fig. 5. Stream lines and isotherms at R = 2.0 and y = 0.
1445
1446
C. V. RAGHAVARAO
and
Y. V.
S. S. SANYASIRAJU
Ra=5n104
rc
0
Fig. 6. Stream lines and isotherms at
R = 1.5 and y = -2.
cylinder in Fig. 9(a) and for the outer surface in Fig. 9(b). In both the figures the local heat transfer rate is compared with the corresponding heat transfer rates of common fluid at Ra = 5 x 104. From Figs 9(a) and (b) it is clear that the heat transfer rates for the cold water are small compared with that of common fluid for all values of y between -2 and 0. For y = -1 (complete inversion) the heat transfer rates are almost equal to one, i.e. conduction is dominated. On the inner cylinder Ntti is increasing like common fluid between 8 = 0 and 180” for y = 0 and decreasing between 8 = 0 and 180” for y = -2. The pattern of heat transfer rates
1447
Natural convection heat transfer of cold water
Rx = 5 x
Ra =
10’
104
Fig. 7. Stream lines and isotherms at R = I.5 and y = -1.
on the outer cylinder Nu, are completely reversed to the above patterns. Here for y = 0, Nu, is decreasing between 8 = 0 and 180” and increasing for y = -2. The pattern of heat transfer rates do not change with respect to the radius ratio R, but the rate is increased with increase of R. To study the total heat Iiow across a surface, the overall Nusselt numbers are calculated for different cases of y and R using the definition
The present results giving a maximum difference of 1.9% in overall Nusselt number on inner and outer surfaces of the cylinders, which are to be equal because of the energy conservation. Finally, the overall heat transfer rates Nu for different cases are correlated independently in the form s=aRab using a least-squares regression analysis. The corresponding values of R and y are given in Table 1.
values of ii and & for different
CONCLUSIONS The steady laminar natural convection of cold water within a concentric cylindrical annulus has been studied numerically by an upwind finite difference method for high Rayleigh numbers. In this investigation the cases y = 0 (i.e. the maximum density is at the inner cylinder), y = -2
1448
C. V. RAGHAVARAO
and Y. V. S. S. SANYASIRAJU
Ra=Sx
IO4
Ra = 8 x IO3
x
0
Fig. 8. Stream lines and isotherms at R = 1.5 and y = 0.
(i.e. the maximum density is at the outer cylinder) and y = -1 (maximum density is at the center of the annulus) are discussed for the annulus radius ratios 1.5 and 2. From the present results it is observed that (1) The heat transfer rates are increasing with R for constant values of Ra and y. (2) The heat transfer rates are increasing with Ra for constant values of R and 3/. (3) Uni-cellular How for y = 0 and y = -2 for R I 2, in which for y = -2, the center of rotation is moving down with increasing Ra and for y = 0, the center of rotation is moving up with increasing Ra. (4) B&cellular flow for y = -1, for which the two cells moving in different directions.
Natural convectian heat transfer of cold water 6
1449
- (4
5
0 Common fluid A I.P. = 0, R = 2.0 0 I.P. = 2, R = 2.0 0 I.P. = 0, R = 1.5 + I.P.=2.R=1.5
4
0 I.P. = 1. R=
z”
1.5
3 . 21
I I/ 1 * 0
(b) 0 Common fluid A I.P. = 0. R = 2.0 0 I.P. = 2. R = 2.0 tI.P.=O,R=l.S +l.P.=2.R=1.5 0 I.P. = 1, R = 1.5
\
Fig. 9. Nusselt number distribution
at Ra = 5 X lo4 on (a) the inner cylinder and (b) the outer cylinder.
1450
C. V. RAGHAVARAO
and Y. V. S. S. SANYASIRAJU Table 1
Inner cylinder R
2.0
y
0 -2 1.5 0 -2
Ra Id-ld ld-105 Id-16 Id-ld
Outer cylinder
ll
b
a
b
0.24734 0.25815 0.35296 0.36198
0.21344 0.20913 0.12423 0.12199
0.25733 0.2506 0.3546 0.3566
0.2077 0.2133 0.1237 0.1237
REFERENCES 111N. SEKI, S. FUKKUSAKO and M. NAKAOKA, ASME. J. Heat Transfer 97,556 (1975). 21 V. S. DESAI and R. E. FORBES, ASME 41-47 (1971). :3] A. WATSON, Q. J. Mech. Appl. Math. 25,423 (1972). :4] T. H. NGUYE.?, P. VASSEUR and L. ROBILLARD, ht. J. Hear Mass Transfer Z&l559 (1982). :5] LANDOLT-BORNSTEIN, Zahlenweste und Funklionem, Vol. 2, pp. 36-37. Springer. Berlin (1971). 61 D. R. MOORE and N. 0. WIESS, JFM 61,553 (1973). ‘71 T. H. KUEHN and R. J. GOLDSTEIN, JFM 74,695 (1976). (Received 10 August 1993; accepted 5 October 1993)