- Email: [email protected]
Fin performance in an oscillating temperature environment
Appliedl:nergt9 ( 1981113 21
FIN P E R F O R M A N C E IN A N O S C I L L A T I N G TEMPERATURE ENVIRONMENT A. A z I z and H. SOFRATA
Department of Mechanical Engineering, Co//ege t~f Engineering, PO Box 800, Riyadh (Saudi Arabia)
SUMMARY
An anah'tical solution is presented to describe the perjormance of a straight convecting.fin operating in an oscillating temperature environment. The solution involves the amplitude parameter, A, and the frequency parameter, B, in addition to the steady-state convection conduction parameter, N. The effects of these parameters on the amplitude and phase angle of spatial temperature and base heat flux arc discussed. For most practical .finned st,~wes, i.e. N < 2, the time-average jin ~[ficiene.v isjound to be practically independent of B in the range 0"01 10 hut increases as A increases. This is in contrast with the previousO" reported study qf periodic variation of base temperature wherein,Jbr N < 1, the time-average fin efficiency wa.~ found to reduce sign!ficant O' with the increase of parameters A and B in the same range.
NOMENCLATURE
Dimensionless amplitude parameter, defined by eqn. (1). Fin thickness. Dimensionless frequency parameter (~oL2/~). B Temperature amplitude, eqn. (18). C D Heat transfer rate amplitude, eqn. (22). .fl, J2 Functions of u, v and X, eqns (15) and (16). F1, F2 Functions of u and t,, eqns (24) and (25). Convective heat transfer coefficient. h Thermal conductivity of fin. k Fin length. L Convection~zonduction parameter ](2h/bk ) 1,2 L. N q Heat transfer rate. 13 A h
Applied Energy 0306-2619/81/0009-0013/$02-50 ,~'~Applied Science Publishers Ltd, England, 1981 Printed in Great Britain
14
Q t la
tb tm
T TI
T2 hi, U X
X O~
0
(o 7; O9
A. AZIZ, H. SOFRATA
Dimensionless heat transfer rate, q L / k b ( t b - t,,). Temperature. Environment temperature. Fin base temperature. Mean environment temperature. Dimensionless temperature (t - t m ) / ( t b -- tin). Steady component of T. Oscillatory component of T. Parameters defined by eqn. (14). Distance from fin tip. Dimensionless distance, x / L . Thermal diffusivity. Instantaneous fin efficiency, eqn. (26). Time-average fin efficiency, eqn. (27). Time. Temperature phase angle. Heat transfer rate phase angle. Function of X, eqn. (6). Dimensionless time, ~ O / Z 2. Frequency.
I.
INTRODUCTION
Fins attached to electronic components, internal combustion engines, solar collectors, etc. often operate under unsteady thermal conditions. Because the existing fin literature is largely devoted to steady-state performance, it has recently been felt necessary to analyse time-dependent fin problems. However, the number of contributions in this area has been rather limited. For convecting fins these include papers by Chapman, 1 Donaldson and Shouman, z Yang, 3 Aziz, 4 Suryanarayana5'6 and Kim. v These analyses stipulate either a step or a periodic change in the base temperature or base heat flux and employ classical techniques such as separation of variables, complex combination and Laplace transformations. The more complex problem of the transient response of a purely radiating fin has been treated by Russel and Chapman s who used the free-parameter method to determine the base temperature time variations which permitted similarity solutions for an infinitely long fin. Most recently, fins with simultaneous surface convection and radiation have been analysed by Compo 9'1° and by Eslinger and Chung 11 using combined variational and numerical techniques. Two assumptions which restrict the applicability of the studies mentioned above are uniform fin thickness and constant environment temperature. Only Compo 9 has relaxed the latter assumption by examining the behaviour of a
FIN PERFORMANCE IN AN OSCILLATING TEMPERATURE ENVIRONMENT
15
convectin~radiating fin with the environment temperature decreasing exponentially with time. It appears that there are many, practically interesting, fin situations for which no analytical or experimental information is available. One such situation, studied here, is that of a convecting fin operating in an oscillating temperature environment. This problem, which arises frequently in practice, can be solved using the method of complex combination. The closed form solution derived contains the amplitude parameter, A, and the frequency parameter, B, in addition to the steadystate fin parameter, N. Results illustrating the effects of these parameters on spatial temperature, base heat flux and time-average fin efficiency are discussed. A qualitative comparison is made with the previously reported study dealing with periodic variations of the base temperature. 3'~ A separate paper will address the more general problem of periodic heat transfer in convecting fins of arbitrary profiles with periodic variation of either base temperature or environment temperature. In that paper a numerical scheme is used in conjunction with the method of complex combination and results are presented tot triangular, convex parabolic and concave parabolic geometries.
2.
ANALYSIS
We consider a straight fin of length, L, and uniform thickness, b, attached to a primary surface at temperature, tb, as shown in Fig. 1. The fin convects on both faces (to an environment at temperature, t,) with a heat transfer coefficient, h. The fin tip is assumed to be insulated. The environment temperature is allowed to oscillate around the mean temperature, tin, with a frequency, ~o, as follows (see Fig. 1): ta
--
t m
tb -
t,,
-AcosoJO
A< 1
(l/
where A is the non-dimensional amplitude parameter. Assuming one-dimensional conduction and constant thermal properties, the transient energy equation for the fin can be written as:
~2t
2h
~x 2
b k (t - to) = ~ , ~
1 ~'~t
(2)
for which: x = 0
~t 0x
0;
x = L, t =
tb
(3)
Substituting eqn. (1) into eqn. (2) and non-dimensionalising, one obtains: •2T OX z
~T N2(T
-
A
cos B z ) = ~ -
(4)
16
k. AZIZ, H. SOFRATA
for which: 8T 8X
X=0
0;
X=
1, T = 1
(5)
where: T = (t - t,,)/(t b - t,,), X = x / L , N 2 = 2 h L Z / b k , B = o~LZ/~, r = ~ O / L 2. The symbols are defined in the N o m e n c l a t u r e . • fa
I
it~ x
I •
tb
tb
,
I 11"
,
I
21"1"
,
I
,
311
I
/,]I"
Fig. 1. Fin geometry and environment temperature oscillation. F o r the regular periodic solution, the temperature, T, is assumed to be the sum of a steady c o m p o n e n t , T 1(X), and an oscillatory c o m p o n e n t , T 2 ( X , z). In accordance with the m e t h o d of complex c o m b i n a t i o n , T 2 ( X , r) is assumed to be of the form: T z ( X , z) = R[A~9(X) e x p ( i B z ) ]
(6)
where R stands for the real part of the quantity in square brackets. Accordingly, eqns (4) and (5) become equivalent to the following ordinary differential equations: d 2T 1 dX 2
N 2T 1 = 0
(7)
for which: dT1 dX
X=0
-0;
X=
1, T 1 =
1
d21// --- (N 2 + iB)qs + N 2 = 0 dX 2
(8)
(9)
for which: X=O
d~ dX -
-
=
O;
X
=
1, t~
=
0
(10)
FIN P E R F O R M A N C E IN A N O S C I L L A T I N G T E M P E R A T U R E E N V I R O N M E N T
17
The solutions for T 1 and ~, can be obtained as: cosh N X T1 = c o s h N N2
(ll)
cosh ( N 2 + iB)l 2X~ 1 - ~ 0 7 h ~ 2 +~BF T !
(
- N 2 +iB
(12)
Substituting eqn. (12) into eqn. (6) and taking the real part, the solution for T2(X, z), after some algebraic manipulation, finally becomes:
T2
-
AN 2 (u2 +t,2)2((N2j]_Bf2)cosB.c
+ (N
2.12 "
+
BJ;)sinBr)
( 13)
where:
U = ( N2+(N4+B2)12 2 ) 1 2
,11 = 1 -
.t2 =
Z" = ( N 2 - ( N 4 + B 2 ) 1 2
2)12
(14)
cosh u cos c cosh uX cos vX + sinh u sin c sinh uX sin cX (cosh u c o s v) 2 + (sinh u sin c) 2
15)
cosh u cos v sinh uXsin t'X - sinh u sin t' cosh u X c o s vX (cosh u cos t,)2 + (sinh u sin t,)2
16)
In c o m p a c t form, eqn. (13) can be expressed as: T 2 = C c o s ( B ~ - qS)
17)
where the amplitude, C, and the phase angle, qb, are: A N2 c =
+
~b = tan
t J7
+.17) '2
lsl
\N2fl - B12)
19)
The instantaneous heat transfer rate, q, from the fin can now be evaluated in terms of the temperature gradient at the base. In dimensionless form it is:
qL ?T xdTl ?T2] Q = bh'(t~St,,) = ~X 1 = d X ix= ~ + ? X x--~
(20)
Using eqns (I 1) and (13) to evaluate the gradients, Q is finally obtained as: Q = N t a n h N + D c o s ( B r + ~)
(21)
where: D .....
AN 2
U 2 ~-
i 5 ( g 2 + F22)1 2
(22)
18
A. AZIZ, H. SOFRATA
I{N2F2 + BFlX~ = tan-
F~
=
\ N - - ~ --
vsinvcosv
-
(23)
BF2/]
usinhucoshu
(24)
cosh 2 u - sin 2 v vsinhucoshu + usinvcosv
F2 =
cosh 2 u -
(25)
sin 2 v
The efficiency of the fin, defined as the ratio of actual heat transfer to ideal heat transfer for a fin of infinite thermal conductivity, can be expressed as: q q -
2(t b -
ta)Lh -
tanh N N(1 - A cos B z )
+
Dcos (Bz + ~) N2(1 - A cos B z )
(26)
The time-average fin efficiency over a cycle is defined as: = ~
q d(Bz)
(27)
Using eqn. (26) in eqn. (27), 0 follows as: tanhN
D
q -- N ( i ~ x 1 2 ) 1 2 q- A N 2
3.
(
1 (1 -- A 2 ) l 2 -
) 1 cos~
(28)
RESULTS AND DISCUSSION
The analytical solution presented in section 2 involves the amplitude parameter, A, the frequency parameter, B, and the convection-conduction parameter, N. Equations (18) and (19) show that the amplitude, C, of the oscillatory component of the temperature is proportional to A whilst the phase angle, 95, is independent of A. Figures 2 and 3 show the effect of the remaining parameters, N and B, on the amplitude and phase angle of the temperature oscillation. For a given N, the effect of B is slight in the range 0.01-1. As B increases further, the amplitude decreases whilst the phase angle increases. This effect becomes more pronounced at lower values of N. On the other hand, for a given B, the amplitude increases and the phase angle decreases as N increases. For any given N and B, the amplitude and phase angle increases from zero at the base (which has a constant imposed temperature) to the highest value at the tip (which is insulated). A qualitative comparison with the case of periodic variation of the base temperature 3 shows that the effect of parameter B is similar in both cases. However, as compared with the significant influence of N in the present study, Yang 3 found that N had very little effect on temperature oscillation. Furthermore, because of different imposed boundary conditions at the fin base in the two cases, the amplitude
FIN PERFORMANCE IN AN OSCILLATING TEMPERATURE ENVIRONMENT
I"°F- . . . . .
s
[-"
~---_..~-\,
---
.
= s
0.~,~
~ ' - . .
oo 0.0
0.2
0.4
\
0.6
~
0.8
1.0
X
Fig. 2. Amplitude of temperature oscillation.
2.0 1.8
1.6 1.4
N
=
I
.... N --.--N
=
2
=
5
B
¢,1.2
1.0
06 0.6
\ \
0.4 0.2
¢.
:
0.0 0.0
02
0.6
0.¢
0.8
X
Fig. 3.
Phase angle of temperature oscillation.
1.0
19
20
A. AZ1Z, H. SOFRATA
N
1. Of
,,X3
z
o.a[
3
0.6t
2
3
2
02
5
!
0.01 Fig. 4.
0.1
B
1
10
Amplitude of base heat flux oscillation.
O.OI 0 01
0.1
1
10
B
Fig. 5.
Phase angle of base heat flux oscillation.
of temperature oscillation in Yang's case decreased from the base to the tip whereas, in the present case, it increases from zero at the base to the highest value at the tip. The heat transfer rate is also composed of a steady component and an oscillatory component. According to eqns (22) and (23) the amplitude, D, of the oscillatory component is directly proportional to A but the phase angle, ~, is independent of A. The influence of B and N is shown in Figs 4 and 5. In the range 0.01 1 the parameter B has little effect on the amplitude which is essentially governed by N and increases as N increases. As B increases beyond unity, the amplitude decreases gradually at high N but quite rapidly at lower values of N. The phase, ~, is negligibly small in the range B = 0.01 0.1 but increases sharply as B increases further. In contrast to the amplitude, the phase angle, ~, for a given B decreases with increasing N. Comparison with Yang's results 3 reveals that the influence of parameters N and B on the heat flux amplitude is seen to be reversed. In his case the amplitude was essentially governed by B whilst N had no significant effect. However, the effects of N on the phase angle in the two cases are similar. Calculations for the time-average fin efficiency given by eqn. (28) were made for the range of variables: B = 0.01 to 10, N = 0.5 to 5 and A = 0 to 0.5. It was found that at low values of N the contribution of the second term in eqn. (28) is negligible compared with that of the first term for all values of B. As N increases the second
FIN PERFORMANCE IN AN OSCILLATING TEMPERATURE ENVIRONMENT
21
term becomes gradually significant, particularly at the higher values o f B. But even at N = 5 and B = 10, the contribution of the second term is only about 10 per cent o f that of the first term. Because, in practice, N rarely exceeds 2, the second term in eqn. (28) may be ignored without any significant loss o f accuracy. Considering the first term ofeqn. (28), the time-average is seen to increase with A. For example, it increases by 0-5 per cent at A = 0.1 and by about 15 per cent at ,4 = 0.5. It is interesting to note the contrast between the present results and those of Yang 3 and Aziz 4 for the periodic variation of base temperature. In the latter studies. the time-average efficiency for N < 1 was found to be reduced significantly with increases of parameters A and B.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
A. J. CHAPMAN,Chem. Eng. Symp. Series, 55 (1959), p-.195. A. B. DONALDSONand A. R. SHOUMAN, Appl. Sei. Res., 26 (1972). p. 75. J. W. YANG, J. Heat TransJer, 94 (1972), p. 310. A. Aztz, J. Heat Tran3Jer, 97 (1975), p. 302. N. V. SUI~YANARAVANA,J. Heat TransJer, 97 (1975), p. 417. N. V. SURVANARAYANA,J. Heat Transfer, 98 (1976), p. 324. R. H. KJM, Letters in Heat and Mass Tran.~ler, 3 (1976), p. 73. L. D. RUSSEL and A. J. CHAPMAN, A I A A J., 6 (1968), p. 90. A. COMPO, Wiirme-und Stoffh'bertragung, 9 (1976), p. 139. A. COMPO, Wiirme-und Stoffffbertragung, 10 (1977), p. 203. R. ESLINGER and B. T. F. CHUNG. Proc. 2nd A I A A / A S M E Thermophysics and Heat "Fran,~/~,r Con/~'renee, Palo Alto, ('alVin, 1978. Paper 78-893.
FIN P E R F O R M A N C E IN A N O S C I L L A T I N G TEMPERATURE ENVIRONMENT A. A z I z and H. SOFRATA
Department of Mechanical Engineering, Co//ege t~f Engineering, PO Box 800, Riyadh (Saudi Arabia)
SUMMARY
An anah'tical solution is presented to describe the perjormance of a straight convecting.fin operating in an oscillating temperature environment. The solution involves the amplitude parameter, A, and the frequency parameter, B, in addition to the steady-state convection conduction parameter, N. The effects of these parameters on the amplitude and phase angle of spatial temperature and base heat flux arc discussed. For most practical .finned st,~wes, i.e. N < 2, the time-average jin ~[ficiene.v isjound to be practically independent of B in the range 0"01 10 hut increases as A increases. This is in contrast with the previousO" reported study qf periodic variation of base temperature wherein,Jbr N < 1, the time-average fin efficiency wa.~ found to reduce sign!ficant O' with the increase of parameters A and B in the same range.
NOMENCLATURE
Dimensionless amplitude parameter, defined by eqn. (1). Fin thickness. Dimensionless frequency parameter (~oL2/~). B Temperature amplitude, eqn. (18). C D Heat transfer rate amplitude, eqn. (22). .fl, J2 Functions of u, v and X, eqns (15) and (16). F1, F2 Functions of u and t,, eqns (24) and (25). Convective heat transfer coefficient. h Thermal conductivity of fin. k Fin length. L Convection~zonduction parameter ](2h/bk ) 1,2 L. N q Heat transfer rate. 13 A h
Applied Energy 0306-2619/81/0009-0013/$02-50 ,~'~Applied Science Publishers Ltd, England, 1981 Printed in Great Britain
14
Q t la
tb tm
T TI
T2 hi, U X
X O~
0
(o 7; O9
A. AZIZ, H. SOFRATA
Dimensionless heat transfer rate, q L / k b ( t b - t,,). Temperature. Environment temperature. Fin base temperature. Mean environment temperature. Dimensionless temperature (t - t m ) / ( t b -- tin). Steady component of T. Oscillatory component of T. Parameters defined by eqn. (14). Distance from fin tip. Dimensionless distance, x / L . Thermal diffusivity. Instantaneous fin efficiency, eqn. (26). Time-average fin efficiency, eqn. (27). Time. Temperature phase angle. Heat transfer rate phase angle. Function of X, eqn. (6). Dimensionless time, ~ O / Z 2. Frequency.
I.
INTRODUCTION
Fins attached to electronic components, internal combustion engines, solar collectors, etc. often operate under unsteady thermal conditions. Because the existing fin literature is largely devoted to steady-state performance, it has recently been felt necessary to analyse time-dependent fin problems. However, the number of contributions in this area has been rather limited. For convecting fins these include papers by Chapman, 1 Donaldson and Shouman, z Yang, 3 Aziz, 4 Suryanarayana5'6 and Kim. v These analyses stipulate either a step or a periodic change in the base temperature or base heat flux and employ classical techniques such as separation of variables, complex combination and Laplace transformations. The more complex problem of the transient response of a purely radiating fin has been treated by Russel and Chapman s who used the free-parameter method to determine the base temperature time variations which permitted similarity solutions for an infinitely long fin. Most recently, fins with simultaneous surface convection and radiation have been analysed by Compo 9'1° and by Eslinger and Chung 11 using combined variational and numerical techniques. Two assumptions which restrict the applicability of the studies mentioned above are uniform fin thickness and constant environment temperature. Only Compo 9 has relaxed the latter assumption by examining the behaviour of a
FIN PERFORMANCE IN AN OSCILLATING TEMPERATURE ENVIRONMENT
15
convectin~radiating fin with the environment temperature decreasing exponentially with time. It appears that there are many, practically interesting, fin situations for which no analytical or experimental information is available. One such situation, studied here, is that of a convecting fin operating in an oscillating temperature environment. This problem, which arises frequently in practice, can be solved using the method of complex combination. The closed form solution derived contains the amplitude parameter, A, and the frequency parameter, B, in addition to the steadystate fin parameter, N. Results illustrating the effects of these parameters on spatial temperature, base heat flux and time-average fin efficiency are discussed. A qualitative comparison is made with the previously reported study dealing with periodic variations of the base temperature. 3'~ A separate paper will address the more general problem of periodic heat transfer in convecting fins of arbitrary profiles with periodic variation of either base temperature or environment temperature. In that paper a numerical scheme is used in conjunction with the method of complex combination and results are presented tot triangular, convex parabolic and concave parabolic geometries.
2.
ANALYSIS
We consider a straight fin of length, L, and uniform thickness, b, attached to a primary surface at temperature, tb, as shown in Fig. 1. The fin convects on both faces (to an environment at temperature, t,) with a heat transfer coefficient, h. The fin tip is assumed to be insulated. The environment temperature is allowed to oscillate around the mean temperature, tin, with a frequency, ~o, as follows (see Fig. 1): ta
--
t m
tb -
t,,
-AcosoJO
A< 1
(l/
where A is the non-dimensional amplitude parameter. Assuming one-dimensional conduction and constant thermal properties, the transient energy equation for the fin can be written as:
~2t
2h
~x 2
b k (t - to) = ~ , ~
1 ~'~t
(2)
for which: x = 0
~t 0x
0;
x = L, t =
tb
(3)
Substituting eqn. (1) into eqn. (2) and non-dimensionalising, one obtains: •2T OX z
~T N2(T
-
A
cos B z ) = ~ -
(4)
16
k. AZIZ, H. SOFRATA
for which: 8T 8X
X=0
0;
X=
1, T = 1
(5)
where: T = (t - t,,)/(t b - t,,), X = x / L , N 2 = 2 h L Z / b k , B = o~LZ/~, r = ~ O / L 2. The symbols are defined in the N o m e n c l a t u r e . • fa
I
it~ x
I •
tb
tb
,
I 11"
,
I
21"1"
,
I
,
311
I
/,]I"
Fig. 1. Fin geometry and environment temperature oscillation. F o r the regular periodic solution, the temperature, T, is assumed to be the sum of a steady c o m p o n e n t , T 1(X), and an oscillatory c o m p o n e n t , T 2 ( X , z). In accordance with the m e t h o d of complex c o m b i n a t i o n , T 2 ( X , r) is assumed to be of the form: T z ( X , z) = R[A~9(X) e x p ( i B z ) ]
(6)
where R stands for the real part of the quantity in square brackets. Accordingly, eqns (4) and (5) become equivalent to the following ordinary differential equations: d 2T 1 dX 2
N 2T 1 = 0
(7)
for which: dT1 dX
X=0
-0;
X=
1, T 1 =
1
d21// --- (N 2 + iB)qs + N 2 = 0 dX 2
(8)
(9)
for which: X=O
d~ dX -
-
=
O;
X
=
1, t~
=
0
(10)
FIN P E R F O R M A N C E IN A N O S C I L L A T I N G T E M P E R A T U R E E N V I R O N M E N T
17
The solutions for T 1 and ~, can be obtained as: cosh N X T1 = c o s h N N2
(ll)
cosh ( N 2 + iB)l 2X~ 1 - ~ 0 7 h ~ 2 +~BF T !
(
- N 2 +iB
(12)
Substituting eqn. (12) into eqn. (6) and taking the real part, the solution for T2(X, z), after some algebraic manipulation, finally becomes:
T2
-
AN 2 (u2 +t,2)2((N2j]_Bf2)cosB.c
+ (N
2.12 "
+
BJ;)sinBr)
( 13)
where:
U = ( N2+(N4+B2)12 2 ) 1 2
,11 = 1 -
.t2 =
Z" = ( N 2 - ( N 4 + B 2 ) 1 2
2)12
(14)
cosh u cos c cosh uX cos vX + sinh u sin c sinh uX sin cX (cosh u c o s v) 2 + (sinh u sin c) 2
15)
cosh u cos v sinh uXsin t'X - sinh u sin t' cosh u X c o s vX (cosh u cos t,)2 + (sinh u sin t,)2
16)
In c o m p a c t form, eqn. (13) can be expressed as: T 2 = C c o s ( B ~ - qS)
17)
where the amplitude, C, and the phase angle, qb, are: A N2 c =
+
~b = tan
t J7
+.17) '2
lsl
\N2fl - B12)
19)
The instantaneous heat transfer rate, q, from the fin can now be evaluated in terms of the temperature gradient at the base. In dimensionless form it is:
qL ?T xdTl ?T2] Q = bh'(t~St,,) = ~X 1 = d X ix= ~ + ? X x--~
(20)
Using eqns (I 1) and (13) to evaluate the gradients, Q is finally obtained as: Q = N t a n h N + D c o s ( B r + ~)
(21)
where: D .....
AN 2
U 2 ~-
i 5 ( g 2 + F22)1 2
(22)
18
A. AZIZ, H. SOFRATA
I{N2F2 + BFlX~ = tan-
F~
=
\ N - - ~ --
vsinvcosv
-
(23)
BF2/]
usinhucoshu
(24)
cosh 2 u - sin 2 v vsinhucoshu + usinvcosv
F2 =
cosh 2 u -
(25)
sin 2 v
The efficiency of the fin, defined as the ratio of actual heat transfer to ideal heat transfer for a fin of infinite thermal conductivity, can be expressed as: q q -
2(t b -
ta)Lh -
tanh N N(1 - A cos B z )
+
Dcos (Bz + ~) N2(1 - A cos B z )
(26)
The time-average fin efficiency over a cycle is defined as: = ~
q d(Bz)
(27)
Using eqn. (26) in eqn. (27), 0 follows as: tanhN
D
q -- N ( i ~ x 1 2 ) 1 2 q- A N 2
3.
(
1 (1 -- A 2 ) l 2 -
) 1 cos~
(28)
RESULTS AND DISCUSSION
The analytical solution presented in section 2 involves the amplitude parameter, A, the frequency parameter, B, and the convection-conduction parameter, N. Equations (18) and (19) show that the amplitude, C, of the oscillatory component of the temperature is proportional to A whilst the phase angle, 95, is independent of A. Figures 2 and 3 show the effect of the remaining parameters, N and B, on the amplitude and phase angle of the temperature oscillation. For a given N, the effect of B is slight in the range 0.01-1. As B increases further, the amplitude decreases whilst the phase angle increases. This effect becomes more pronounced at lower values of N. On the other hand, for a given B, the amplitude increases and the phase angle decreases as N increases. For any given N and B, the amplitude and phase angle increases from zero at the base (which has a constant imposed temperature) to the highest value at the tip (which is insulated). A qualitative comparison with the case of periodic variation of the base temperature 3 shows that the effect of parameter B is similar in both cases. However, as compared with the significant influence of N in the present study, Yang 3 found that N had very little effect on temperature oscillation. Furthermore, because of different imposed boundary conditions at the fin base in the two cases, the amplitude
FIN PERFORMANCE IN AN OSCILLATING TEMPERATURE ENVIRONMENT
I"°F- . . . . .
s
[-"
~---_..~-\,
---
.
= s
0.~,~
~ ' - . .
oo 0.0
0.2
0.4
\
0.6
~
0.8
1.0
X
Fig. 2. Amplitude of temperature oscillation.
2.0 1.8
1.6 1.4
N
=
I
.... N --.--N
=
2
=
5
B
¢,1.2
1.0
06 0.6
\ \
0.4 0.2
¢.
:
0.0 0.0
02
0.6
0.¢
0.8
X
Fig. 3.
Phase angle of temperature oscillation.
1.0
19
20
A. AZ1Z, H. SOFRATA
N
1. Of
,,X3
z
o.a[
3
0.6t
2
3
2
02
5
!
0.01 Fig. 4.
0.1
B
1
10
Amplitude of base heat flux oscillation.
O.OI 0 01
0.1
1
10
B
Fig. 5.
Phase angle of base heat flux oscillation.
of temperature oscillation in Yang's case decreased from the base to the tip whereas, in the present case, it increases from zero at the base to the highest value at the tip. The heat transfer rate is also composed of a steady component and an oscillatory component. According to eqns (22) and (23) the amplitude, D, of the oscillatory component is directly proportional to A but the phase angle, ~, is independent of A. The influence of B and N is shown in Figs 4 and 5. In the range 0.01 1 the parameter B has little effect on the amplitude which is essentially governed by N and increases as N increases. As B increases beyond unity, the amplitude decreases gradually at high N but quite rapidly at lower values of N. The phase, ~, is negligibly small in the range B = 0.01 0.1 but increases sharply as B increases further. In contrast to the amplitude, the phase angle, ~, for a given B decreases with increasing N. Comparison with Yang's results 3 reveals that the influence of parameters N and B on the heat flux amplitude is seen to be reversed. In his case the amplitude was essentially governed by B whilst N had no significant effect. However, the effects of N on the phase angle in the two cases are similar. Calculations for the time-average fin efficiency given by eqn. (28) were made for the range of variables: B = 0.01 to 10, N = 0.5 to 5 and A = 0 to 0.5. It was found that at low values of N the contribution of the second term in eqn. (28) is negligible compared with that of the first term for all values of B. As N increases the second
FIN PERFORMANCE IN AN OSCILLATING TEMPERATURE ENVIRONMENT
21
term becomes gradually significant, particularly at the higher values o f B. But even at N = 5 and B = 10, the contribution of the second term is only about 10 per cent o f that of the first term. Because, in practice, N rarely exceeds 2, the second term in eqn. (28) may be ignored without any significant loss o f accuracy. Considering the first term ofeqn. (28), the time-average is seen to increase with A. For example, it increases by 0-5 per cent at A = 0.1 and by about 15 per cent at ,4 = 0.5. It is interesting to note the contrast between the present results and those of Yang 3 and Aziz 4 for the periodic variation of base temperature. In the latter studies. the time-average efficiency for N < 1 was found to be reduced significantly with increases of parameters A and B.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
A. J. CHAPMAN,Chem. Eng. Symp. Series, 55 (1959), p-.195. A. B. DONALDSONand A. R. SHOUMAN, Appl. Sei. Res., 26 (1972). p. 75. J. W. YANG, J. Heat TransJer, 94 (1972), p. 310. A. Aztz, J. Heat Tran3Jer, 97 (1975), p. 302. N. V. SUI~YANARAVANA,J. Heat TransJer, 97 (1975), p. 417. N. V. SURVANARAYANA,J. Heat Transfer, 98 (1976), p. 324. R. H. KJM, Letters in Heat and Mass Tran.~ler, 3 (1976), p. 73. L. D. RUSSEL and A. J. CHAPMAN, A I A A J., 6 (1968), p. 90. A. COMPO, Wiirme-und Stoffh'bertragung, 9 (1976), p. 139. A. COMPO, Wiirme-und Stoffffbertragung, 10 (1977), p. 203. R. ESLINGER and B. T. F. CHUNG. Proc. 2nd A I A A / A S M E Thermophysics and Heat "Fran,~/~,r Con/~'renee, Palo Alto, ('alVin, 1978. Paper 78-893.