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Optimum shapes of convective pin fins with variable heat transfer coefficient
Optimum Shapes of Convective Pin Fins with Variable Heat Transfer Coejkient by
UPENDRA
Department
NATARAJAN
and
UDAY
v.
of Chemical Engineering,
SHENOY
Indian Institute of Technology,
Powai,
Bombay 400 076, India
ABSTRACT : Principles of variational calculus are used to determine the shapes of convective pin fins that maximize heat dissipation, given the amount of fin material. The analysis considers the convective heat transfer coeficient h to depend on thefin diameter D according to the relationship h CT l/D”, where n takes on values from 0.2 to 0.5 depending on the type of’pin,fin configuration andflow~ condition (1: AIChE .I, Vol. 29, p. 1043, !983). The Euler equations, which are nonlinear and coupled, are ,formulated and solved,for the cases of both length and weight constraints as well as only weight constraint. The resulting quadrature formulae are represented in the form of a convenient design plot, from which the optimum design parameters may be obtained and used to determine the$n and temperature profiles as well as the cooling performance. The solutions under both constraints yield considerably simple results for the case of only weight constraint, which corresponds to the diameter and excess temperature of the fin tip being zero. An important result is that the Schmidt criterion (2 : Z. Verein. Deutsch lng., Vol. 70, pp. 885, 947, 1926) of a linear temperature profile also holds for pinJins of specified weight with a variable heat transfer coeflcient. Finally, by using Pontryagin’s minimum principle (3 : The Mathematical theory of Optimal Processes, Wiley,, New York, 1962), it is demonstrated that the problems of maximizing cooling ($or a given$n weight) and minimizing weight (for a given& cooling) are identical as both are governed by the same optimum design equations.
Nomenclature b c D e E F h h, H k K n
fin length constant defined in (15) diameter of pin fin of circular cross-section error term (in the length-of-arc assumption discussed in Appendix functional given in (9), equal to the cooling effect, H functional defined in (10) convective heat transfer coefficient defined in (2) cooling effect defined in (6) thermal conductivity of fin fin volume (or, equivalently, fin weight) exponent defined in (2) (describing the variation of h)
N, Nu
removal number, Nusselt number
&th
~;The Franklm lnstitute001&0032/90 $3.00+000
B)
fin/qnofin
965
U. Natatzjan 4 Re x i
r? /I
e
Subscripts 0 b k nf P
and U. V. Shenoy heat transfer rate Reynolds number distance from the fin base along the length of the fin dimensionless excess temperature, e/O, variation fin efficiency constant Lagrange multiplier temperature difference between a point on the fin and its surroundings (temperature excess) Hamiltonian function defined in (A3) adjoint (costatc variables) system
fin base (x = 0) fin tip (X = b) division by k no fin condition parameter (used in plots and defined in Section VI)
Superscript * division by base diameter
D,, (to make fin diameter
dimensionless)
I. Introduction surfaces are often used in heat exchange devices to increase the heat transfer between the primary surface and the surrounding medium. In addition to longitudinal and radial fins, a common fin geometry is the pin fin or spine (4). usually of circular cross-section (Fig. 1). The fin optimization problem generally involves determining the shape and dimensions of the fin that (a) dissipates the maximum heat for a given mass or (b) utilizes the minimum mas? to dissipate a given heat. Furthermore, a constraint may or may not be imposed on the length of the fin (5). Hrymak et al. (6) have presented a detailed review (not included here) of the existing literature on the optimization of extended surfaces. Extended
b
FIG.
I.
Spine (pin fin) of circular cross-section. Journalof Ihe Franklm
966
lnst,tute Pergamon Pres plc
Optimum
Shapes ?f Cont’ectirre Pin Fins
Previous studies (1,7) on optimum pin fins have focused attention on determining the optimum dimensions of cylindrical spines of constant diameter by differential calculus. On the other hand, this work attempts to determine the optimum fin shape (in terms of the variation of the fin diameter D with fin distance x) as well as the optimum dimensions by variational calculus. It may be noted that Sonn and Bar-Cohen (7) found the optimum diameter for a cylindrical pin fin with a constant convective heat transfer coefficient h, while Li (1) analysed the same problem assuming a variable h. Though Razelos and Imre (8) determined the optimum shapes of longitudinal and radial fins for a power-law spatial variation of the heat transfer coefficient, they obtained solutions for pin fins only in the special case of constant h. As pointed out by Razelos and Imre (8) there is considerable theoretical as well as experimental evidence that the convective heat transfer coefficient h varies markedly along the fin surface. Large errors may be introduced (9) if optimum fins are designed under the assumption of constant h. Hence, the more realistic assumption (1) of h K l/D” is used throughout the analysis presented here. Letting n assume values from 0.2 to 0.5 allows considering pin fins in isolation and in arrays (both in-line and staggered) under various flow conditions. Further, the results for the case of constant h can be easily obtained from the general expressions derived below by setting n to be zero. While examining optimum shapes for cooling fins, an early study by Schmidt (2) suggested that the temperature must vary linearly along the length of the fin. The linear temperature profile, referred to as the Schmidt criterion in the literature, was formally shown by Duffin (10) to be optimum for longitudinal and radial fins with a weight constraint. In the present work, it is demonstrated that the Schmidt criterion holds even for optimum pin fins of a given weight with a variable heat transfer coefficient.
II. The Dzrerential
Formulation
The one-dimensional problem of a cooling pin fin of circular cross-section (Fig. 1) transferring heat to the surroundings by convection is considered here. If the temperature difference between a point on the fin surface and its surroundings is 8, then a heat balance at steady state over a differential element of length dx (assuming the curvature of the convective surface is small) yields the following governing differential equation :
The fin diameter D is taken to be a function of the distance from the fin base x. Though the thermal conductivity k is assumed constant, the convective heat transfer coefficient h is assumed to vary with the fin diameter as (1) h cc l/D” where hD is a constant. Vol 327. No. 6. pp. 965-982. 1990 Prmted ,n Great Britain
or
The above dependence
hD” = h,
(2)
of h on the diameter D directly follows 967
U. Nutarujan
and U. V. Sheno_v TARLEI Dinxwsio&ss
groups jbr design of’optimum
pit1 fin jiir wrious n
~ Rt>
Physical situation
Configurationi
IO’-10’
S, ILA, SA S, SA
II D
t Pin fin configuration array.
: S denotes
ILA
S
ILA, SA Constant
0.4
0.37
0.3
0.2
0.0
1.824
1.835
I.839
1.849
I.865
1.904
1.5
I.4
I.37
I.3
1.2
I.0
1.767
1.539
1.474
1.327
I.131
0.785
0.625
0.650
0.658
0.675
0.700
0.750
=
‘7 = (3 -n)i4
_____
2 x IO’-1 x IO”
0.5
I + ,I 2,1 ,I
(l6h,,K’/j$k)”
IO’-2 x 105
sin&
h
fin, ILA denotes in-line array. SA denotes staggered
from the analysis of experimental work (11, 12), which shows that the Nusselt number NU for heat transfer to tubes in crossflow depends on the Reynolds number Re according to NU cc Re’-“. The values of y1 typically vary from 0.2 to 0.5 (1) depending on the flow conditions and the types of tube arrays (refer to Table I). Clearly, n = 0 corresponds to the case where the convective coefficient is a constant. If Q. is the temperature excess at the base (.Y = 0) and u is defined as the dimensionless excess temperature g/O,, then (1) on combining with (2) gives
= h,,D’
no.
(3)
If the heat transferred through the tip of the fin of length b is negligible compared to that leaving through its lateral surface, then the appropriate boundary conditions are 14 =
1
at
x=0
(4)
and D2g=0
at
x=6.
Equation (5) implies that at the fin tip, the temperature gradient is zero (which corresponds to the case of both length and weight constraints as shown later) or the diameter is zero (which corresponds to the case of only weight constraint as demonstrated later).
Optimum Shapes cf Concectice
Pin Fins
The goal is to obtain non-negative, differentiable, continuous functions for u and D (in the interval 0 6 x d 6) that respectively define the temperature and fin profiles which maximize the cooling effect H given by
s s b
Hr
~,ZQ-“udx=
_
0
H is directly related to the heat transfer rate at the fin base q. as H = qo/eo. A constraint on the fin volume (or, equivalently, the weight) may be imposed as
’
0
nD2 ~ dx = K, a given positive constant.
4
(7)
III. The Variational Formulation To obtain the variational formulation of the steady one-dimensional physical problem discussed above, the formal procedure outlined by Arpaci (13) is followed. Taking the governing linear second-order ordinary differential equation (3) as the Euler equation associated with the desired variational problem and considering the variation in u only, we obtain ,,=l
Integrating
(&(k$$)-,h,,‘-.,),,dx=O.
the first term by parts,
Since the boundary conditions (4) and (5) are natural, remaining term in the above equation may be rewritten
the first term is zero. The as
~~(f+!!(!!!):+%!!!!)dx=O~ Thus, the variational
and the corresponding
formulation
functional
of the problem
is
is (9)
Both Liu (5) and Duffin (10) determined optimum shapes for longitudinal fins with and without heat generation respectively, by starting with. integral expressions similar to (9) ; however, they did not attempt to establish the functional E through formal techniques as done above. Vol. 327. ho. 6. pp. Yhj-9x2. Printed in Great Bntam
1990
969
U. Nutarqjan
and U. V. Sheno!,
Having derived the variational formulation, attention is focused on attaching a physical meaning to E by relating E to H. Integrating the first term in (9) by parts, and utilizing the physics of the problem (3), (4) and (5), comparison with (6) directly shows that E = H. Importantly, if the functional E is maximized, then the cooling effect H of the fin is maximized. Using the Lagrange multiplier ,J (a constant), the weight constraint is readily incorporated at this stage by defining the functional F to be equivalent to (7) and (9). Thus,
As shown earlier, the Euler equation ation of II is the original differential has no dependence on 14.
resulting equation
from (10) for any admissible vari(3) since the weight constraint (7)
IV. Analysis for both Length and Weight Constraints The case where the length is a constraining factor has been typically ignored by previous investigators with the exception of Liu (S), who studied longitudinal and radial fins. In this section, the case of a pin fin with a length constraint (fixed h) and a weight constraint (non-zero 1.) is considered. The variation in Ffor a variation in D is
For F to be an extremum, 6F must be zero for any admissible the following equation must be valid :
variation
SD and
The fin shape optimization problem requires solving a system ofcoupled differential equations of the second order (3) and second degree (12) for u and D. The solution procedure adopted below is along the lines originally suggested by Pitchumani and Shenoy (14). Both sides of (3) may be multiplied by (du/dx) and the equation rewritten as (13) Utilizing pulations
(12) to eliminate that
(du/dx)
from (13), it may be shown after some mani-
d&(3-n)u’D’ The left-hand side is an exact differential is easily written as 970
m’-/zD2)
= 0.
(14)
; thus, a useful relation
between
u and D
Journal of the Franklin Inst~tulr Pergamon Prtx plc
Optimum h,(3-n)u2D’-“-AD’ where of the in (5) value
Shapes of’ Conoectiw
Pin Fins
= C,
(15)
C is the constant of integration. If the subscript h is used to denote evaluation quantities at x = b, then (12) and (15) on imposing the boundary condition yield Dd +’ = 2h,(l -n)u,‘/A and C = A(1 +n)D?/[2(1 -n)]. Substituting this for C, (15) may be written in the following convenient form : u2
=
j+IJ+n)
5
2(1-n) Imposing the boundary by D,, (16) yields
condition
0D
in (4) and denoting
2
(16)
.
the base diameter
of the fin
(17) Then A is easily eliminated the temperature profile :
from (16) and (17) to give the following
J
U=
expression
2(1-n)D*2+(l+n)D,*2
FiP”[2(1
for
(18)
-nj+(l+Gj5*2)’
The asterisk in the superscript denotes that the fin diameter has been made dimensionless on dividing by the diameter at the base. Now, the quadrature formula for D* in terms of x is obtained by eliminating u and (du/dx) using (12), (16) and the differentiated form of (18). Noting that (du/dx) is negative, we get after some manipulation
~~ ~~~~~ s~~ I
(1 -n)
J(Gj-j
D*2~1-n)+(l+n)Db**]
(2D** - D;2)D*+
‘)I2dD* -fix
[2(l--n)D*2+(1+n)D~2][D*2-D~2] (19)
where Ak = A/k. Equation (19) provides an expression for the fin length h, when x = h and the lower limit of integration is set to D$ Substituting from (17) and (19) into (7), the expression for the fin weight is 42h,k(l K=
-n)(3
-n>l2r(‘+n)(l -n>J(l
+n)(3-n) x (2D*2 - Dt2)D *(3+n)‘2d?
S’oz 8[1,(2(1 -n)
+ (1 +n)Df2)](5f”)/(21
2n) __
___
m-n)D*2+-n)Df2] where hnk = h,,/k. Using (12) and (17) along with the definition given by H = nk[2h,,(l
The utility
of the above equations
[D*2 - Dz2]
of H in (6), the cooling
-n)12”‘+“)[(3 -r~)/&](~~~)l(~+*~) &l 4[2(1-+
(20)
(1 + ~I)D*~] 2!(‘fn) b will be demonstrated
+n) (1 -Dt2) ~~
in a subsequent
effect is
.
(21)
section
U. Nutarajan
and U. V. Shenoy
through a design plot as well as figures of the optimum fin and temperature profiles. As the above expressions can be considerably simplified if the length constraint is relaxed, the next section is devoted to this special case.
V. Analysis for only Weight Constraint When there is no constraint on the length, in F for a variation in b results in
then b is a variable
and a variation
For the optimum fin under conditions of variable b, the integrated term must vanish at both limits. At x = 0, the term is zero as 6x = 0. However, at x = b, 6x is not zero and a boundary condition is obtained for the fin to be optimum by using (12). Thus, +h,Dl-nu2
_
402
=
hmD’
4
-“cl
+‘)f
2
The above equation along with (5) and (12) indicates from (15), it is clear that C = 0 and we obtain
= 0
(22) now directly
s = h.
that Dh must be zero. Then,
D ‘+’ = h,(3 -n)u2/L Equation
at
(22)
implies that Dh = 0
and
u,, = 0.
(23)
Thus, the temperature at the fin tip is equal to the ambient fluid temperature and the tip diameter is zero. Then (12) may be solved for the optimum temperature profile after eliminating D using (22). The results obtained, using the boundary conditions (4) and (23), are du dx=
(24)
-
and u = l-x/b.
(25)
Equation (25) indicates that the temperature profile is linear; thus, the classical Schmidt criterion (2), which was formally shown by Duffin (10) to hold for longitudinal and radial fins under a weight constraint, is valid for optimum pin fins of specified weight, which maximize heat dissipation. Substitution of (24) and (25) into (22) gives the optimum fin profile as D = [hna(l +n)(b-x)‘]““+“). According to (26), the classical constant heat transfer coefficient
parabolic profile is optimum (i.e. n = 0). Since the concave
(26) for a fin with a parabolic pin fin
Optimum
Shupes
ef Convective
Pin Fins
analysed by Kraus (15) assumes the base diameter is given, it is not optimum. However, if the base diameter is chosen to be hkb2 as per (26), the spine is optimum for constant h. Substituting (26) into (7) it is seen that K=
71/[4(5fn)](hDk)2/(l+n)(l+n)(3+n)1(1+n)b(5+n):(l+n).
(27)
It may be emphasized that for the case of only weight constraint, the length of the fin is obviously not specified and must be determined from (27). The cooling effect thus obtained by substituting (25) and (26) into (6) is H = (&/4b)
[h&l
+n)b2]2’(‘+n).
(28)
As expected, (16)-(21) on setting Db = 0 and & = (3-n)/[(l+n)b2] reduce to (22)-(28). It is of interest to note that (22)-(28) also specify the optimum pin fin of minimum weight for a given heat dissipation and no length constraint (refer to Appendix A). This has been shown by Focke (16) to be true for the case when n = 0. Again, for n = 0 and no length constraint, the minimum weight convective pin fin problem has been formally solved by Bhargava and Duffin (17) (using a special transformation of variables along with Holder’s inequality) as well as by Razelos and Imre (8) (using Pontryagin’s principle). Also, it may be pointed out that the minimum weight radiation pin fin problem has been analysed by Wilkins (18). VI. Results and Discussion In designing optimum pin fins, the equations conveniently utilized in the form of various plots.
derived
thus far may be most
Generating the Plots Plots are included in this paper only for the specific case of n = 0.5, as the plots for other values of n (between 0.2 and 0.5) will exhibit very similar features. The case of n = 0.5 (see Table I) corresponds to a single pin fin as well as a pin fin array (both in-line and staggered), when the Reynolds number Re is between 100 and 1000 (1). The various plots relevant to the optimum design of pin fins, and the procedures adopted in generating them are briefly explained below. (i) A design plot of fin weight parameter Kp vs fin length parameter b, is made (Fig. 2) by choosing different values of & (in the range 0.1 - 10) and D: (in the range C-0.99). b and K/(nhif’+“)/8) are computed by numerical integration from (19) and (20) respectively ; subsequently, they are transformed to K,, = 1 -exp [-K/(xhDk 2’(‘+n)/ S)] and 6, s 1 -exp [-b]. The transformation is such that Kp and b, vary from 0 to 1, as K and b vary from 0 to co. In Fig. 2, the dotted lines represent constant ilk curves, while the solid lines denote constant DC curves. The design curve for Dz = 0 (which is merely (27) plotted in terms of Kp and b,) provides an upper bound on the fin length, as it corresponds to the case of only weight constraint. (ii) A plot of the fin profiles in terms of D* vs X, is made (Fig. 3) by choosing different values of 02 (in the range 04.9). D* may be obtained as a function of 973
U. Nutarujun
and W. V. Shenoy
- _ LENGTH
PARAMETER
3
bp
FIG.2. Design plot for n = 0.5.
fix by performing the necessary numerical integration in (19). ,,&x may be transformed to a dimensionless distance X, (which varies from 0 to 1) on dividing by its maximum value. This maximum is given by &h for the case when D? = 0 ; thus, (24) directly suggests that X, be defined by J&( 1 +rz)x/& n). It may be observed on Fig. 3 that under only a weight constraint (D,f = 0), the optimum fin profile is given by D* = (1 -XJzl(‘+“) as per (26). (iii) A temperature profiles plot in terms of u vs X, is next made (Fig. 4) by selecting the same values of Dtas in (ii). The D* data may be directly converted to dimensionless excess temperature u by use of (1X). It may be noticed on Fig. 4 that when only the weight is specified (D$= 0), then the optimum temperature profile is linear (U = 1 -X,) as per (25). (iv) A dimensionless performance curve relating the cooling effect parameter H,, to Dt is finally made (Fig. 5) by first calculating the quantity H;1&3m n)!(2+2n)/ [mkhAf’+“)/4] from (21). This quantity is next made dimensionless to obtain H,, on dividing by its maximum value which is attained at Dt= 0; (28) indicates that this maximum value is (1 fn)‘;’ (3-n)c’p”);(2+2”). Thus, HP is defined as H[;lk/(3_n)](3~“)i(2+2”)/[7ck(l+n)1:2h~~l+”) / 4 1. Clearly, from Fig. 5, the most cooling for a specified amount of material is obtained when there is no constraint on the fin length, i.e. when DC= 0. It may be noted that in the case of both length and weight constraints, expressions for the fin profile (19) and fin volume (20) are in terms of quadrature formulae ; so, the numerical integration was performed using the IMSL routine QDAGS (19). Journalofthe Frankhn 974
In,utute Pergdmon Press plc
Optimum
DIMENSIONLESS
FIG. 3. Optimum
0.0
DISTANCE-
DIMENSIONLESS
FIG. 4. Optimum
Xp
fin profiles for n = 0.5.
0.4
0.2
Shapes qf Conzrectire Pin Fins
temperature
0.6
DISTANCE
0.8
Xp
profiles for n = 0.5.
975
U. Natarajun
and U. V. Shenoy
FIG. 5. Performance plots for n = 0.5.
Utilizing the Plots The plots generated may be effectively used to design the optimum pin fin, when the weight and length of the required fin along with the thermal properties (namely, h, and k) are given. The value of n is also specified as per the fin configuration and the flow condition (see Table I). (i) From the given values of K and h, the values of Kp and h, are calculated. These are used to read off the values of 0: and 1, from the design plot in Fig. 2. If the design point lies to the right of the Df = 0 curve, then the length has been overspecified; in this case, the curve for only weight constraint (namely, Dt= 0 curve) should be used to obtain I+. (ii) With a knowledge of the optimum Df and &, (17) may be used to calculate the base diameter Do. (iii) The appropriate curve in terms of D* vs X, is chosen from the fin profiles plot depending on the value of D,*obtained in (i). Interpolation may be required to get an approximate curve if D,*is not very close to the values of 0; plotted in Fig. 3. The D* data are easily converted to the actual fin diameter D by multiplying by the Do value calculated in (ii). The X, data are transformed to the actual fin distance x by substituting the known value of I+. This procedure yields the actual optimum fin profile in terms of diameter D vs distance X. (iv) The optimum temperature profile is obtained in a straightforward manner from Fig. 4, if the procedure for converting A’,,to x as outlined in (iii) is followed. 976
Journal
of the FrankIm Inst~tutc Pergamon Pre*s plc
Optimum
Shapes qf‘ Concectiue Pin Fins
, ‘(4 F rom the known value of D$, the value of HP can be read off from Fig. 5. The optimum cooling effect H can then be calculated from the definition of H,, on substituting the value of Ak. Calculuting the Fin .E#cienq The fin efficiency r] is conventionally used in engineering practice to indicate the effectiveness of a fin in transferring a given quantity of heat. It is defined as the ratio of the heat actually dissipated by the fin to that dissipated by a hypothetical isothermal (at the base temperature 0,) fin of identical shape and size. The expression for the optimum efficiency is obtained from (17), (19) and (21). Thus,
s b
7chgD’-” dx
1
_
r
0
_______
_
H
’
2 1 -n Jol”“c-~cr
+Fn)m(2D*2 (
- Dz2) dD*
s @ J[1-D~2][2(1-n)D*2+(1+n)D,*2][D*2-D~2~~ (29)
For D$ = 0, the above equation
simplifies
to
u] = (3-n)/4. The efficiency is computed from (29) by numerical of Dz and the result is plotted in Fig. 5.
(30) integration
for different
values
Determining the l$ect c?f’n 011the Fin Design,ftir only Weight Constraint The variation of the heat transfer coefficient h depends on the value of the exponent n (2). The value of n in turn depends on the flow condition of the medium surrounding the fin, and on whether the fin is solitary or in an array (see Table I). Li (1) has given the values of n for various possible conditions normally encountered, and tabulated the corresponding values of the optimum design parameters for a cylindrical pin fin of specified weight. Here, attention is focused on our optimum pin fin with only the weight constrained (discussed in Section V) ; then, the same dimensionless groups used by Li (1) are calculated using our (26))(28) and (30). The results, which are given in Table I, can be directly used to design optimum fins with the profile defined by (26), when n, K and the thermal properties are specified. Thus, from the dimensionless groups in the first column in Table I, the optimum values of Do, b, H and q are easily calculated in that order.
VIZ, Concluding Remarks In most practical situations of fin design, the assumption causing the greatest errors is that of a uniform convection coefficient over the entire fin surface. In this paper, solutions for the optimum design of pin fins are developed under the more realistic conditions of a varying h. For the case where the length is constrained (Section IV), (19))(21) constitute a system of three equations in five variables (K, b, H, D$ and &). If the designer specifies two quantities (the weight K and the length b), then the remaining three quantities (H, 0: and iWr)may be determined ; thus, the design of a maximum
U. Nutarajan
and U. V. Shenc?);
cooling tin (where K and h are given) may be done. It must be emphasized that the design of a minimum mass fin (for a specified heat dissipation) may be done using the very same equations (refer to Appendix A). The desired cooling effect H and the length h are given in this case, and (19))(21) may be utilized to determine K, 0: and I,,\. If there is no length constraint (Section V), then (27) and (28) constitute a system of two equations in three variables (K, h and H). For the maximum cooling fin (given the weight K), h and H are determined from (27) and (28). On the other hand, for the minimum mass fin (given the required cooling H), b and Kare found from the same two equations. For this case where the length is not constrained, it may be noted that 0: = 0 and &. is directly related to h by (24). The analysis developed for the optimum fin in this paper is classical in the sense that it considers the fin to be one-dimensional and neglects the profile curvature in the arc-length calculation. A brief discussion on these assumptions is presented in Appendix B. Both these assumptions have been recently shown to be valid (20) for pin fins with constant thickness, triangular and parabolic profiles. In particular, neglecting the arc-length factor that multiplies the periphery has been shown to be a reasonable assumption, and is a “mild restriction” in useful fins (8). The formulation incorporating the curvature is relatively straightforward as it merely involves multiplying the perimeter ZD (in Eq. I) by the factor ,/l +0.25(dD/dx)2; however, the solution procedure will be relatively complicated and numerical in nature. The optimization, without making the length-of-arc assumption, will involve a nonlinear two-point boundary-value problem and will be similar to the minimum-weight one-dimensional fin optimization done by Maday (21). Considering the errors involved (reduction in weight by 1.6%) under the length-of-arc assumption for the specific case solved by Maday (21) and the waviness in the resulting fin profile, the simpler analysis presented here should be of more practical utility.
Acknowledgement The authors thank M. S. N. Mm-thy (Department of Chemical Engineering, Institute of Technology, Bombay) for discussions that aided in the preparation various plots.
Indian of the
References (1) C-H. Li, “Optimum cylindrical pin fin”, AlChE J., Vol. 29, p. 1043, 1983. (2) E. Schmidt, “Die Wlrmeubertragung durch Rippen”, Z. Verein. Deutsch Ing., Vol. 70, pp. 885 and 947, 1926. (3) L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko, “The Mathematical Theory of Optimal Processes”, Wiley, New York, 1962. (4) D. Q. Kern and A. D. Kraus, “Extended Surface Heat Transfer”, McGraw-Hill, New York, 1972. (5) C. Y. Liu, “A variational problem relating to cooling fins with heat generation”, Q. Appl. Math., Vol. 19, p. 245, 1961.
978
Journaloflhc
FrankIln ,nat,lutc Pergamon Press plc
Optimum
Shapes qf’ Contrectire Pin Fins
. A. N. Hrymak, G. J. McRae, and A. W. Westerberg, “Combined analysis and optimization of extended heat transfer surfaces”, ASME J. Heat Trarufer, Vol. 107, p. 527, 1985. A. Sonn and A. Bar-Cohen, “Optimum cylindrical pin fin”, ASME J. Heat TranJfer, (7) Vol. 103, p. 814, 1981. (8) P. Razelos and K. Imre, “Minimum mass convective fins with variable heat transfer coefficients”, J. Franklin Inst., Vol. 315, p. 269, 1983. (9) P. Razelos and K. Imre, “The optimum dimensions of circular fins with variable thermal parameters”, ASME J. Heat Transfer, Vol. 102, p. 420, 1980. (10) R. J. Duffin, “A variational problem relating to cooling fins”, J. Math. Mech., Vol. 8, p. 47, 1959. (11) A. A. Zukauskas, “Heat transfer from tubes in crossflow”, Adz?. Heat Transfer, Vol. 8, Academic Press, New York, 1972. E. R. G. Eckert and R. M. Drake, “Analysis of Heat and Mass Transfer”, McGraw(12) Hill, New York, 1972. Heat Transfer”, Addison-Wesley, Reading, MA, 1966. (13) V. S. Arpaci, “Conduction R. Pitchumani and U. V. Shenoy, “A unified approach to determining optimum shapes (14) for cooling fins of various geometries”, Private communication, 1988. of Extended Surface Thermal Systems”, (15) A. D. Kraus, “Analysis and Evaluation Hemisphere, Washington D.C., 1982. Forschung auf dem Gdiete Lies Ingenieur(16) R. Focke, “Die Nadel als Kuhlelement”, wesens, Vol. 13, p. 34, 1942. (17) S. Bhargava and R. J. Duffin, “On the non-linear method of Wilkins for cooling fin optimization”, SfAM J. Appl. Math.. Vol. 24, p. 441, 1973. (18) .I. E. Wilkins, Jr., “Minimum mass thin fins which transfer heat only by radiation to surroundings at absolute zero”, J. Sot. Indust. Appl. Math., Vol. 8, p. 630, 1960. Mathematical and Statistical Libraries, Problem (19) IMSL, Version 1.O, International Solving Software Systems, Houston, TX, 1987. of convective pin fins with internal heat (20) P. Razelos, “The optimum dimensions generation”. J. Franklin Inst., Vol. 321, p. 1, 1986. straight fin”, ASME J. Engng (21) C. J. Maday, “The minimum weight one-dimensional Znd., Vol. 96, p. 161, 1974. Theory with Applications”. Wiley, New York, 1969. (22) D. A. Pierre, “Optimization Blaisdell Pub. Co., (23) A. E. Bryson, Jr., and Y. C. Ho, “Applied Optimal Control”, Massachusetts, 1969. fin solution”, ASME J. Heat Transfer, (24) R. K. Irey, “Errors in the one-dimensional Vol. 90, p. 175, 1968. (25) S. Whitaker, “Elementary Heat Transfer Analysis”, Pergamon Press, New York, 1976.
Appendix A : Minimum Mass Convective Pin Fin The objective here is to formulate the optimization problem which minimizes the fin volume K for a specified amount of cooling H. The problem is similar to the optimum control problem (22) of extremizing an integral (7) subject to an isoparametric constraint (6). The procedure adopted is analogous to that of Maday (21) and Razclos and Imre (8). It utilizes Pontryagin’s principle (3), which provides an alternative approach to the classical variational calculus approach and may be considered a generalized Weierstrass excess-function test. Both approaches yield the same optimum solutions for problems involving continuous control. Bryson and Ho (23) favor the Pontryagin principle approach for it may often be the more direct method providing some insight into the actual processes
U. Nutamjun
and U. V. Shenoy
involved in the optimization problem. As the variational calculus approach has been adopid in the main paper, the methodology based on Pontgryagin’s principle will be applied in this Appendix. Equation (I) may be re-written in terms of state variables (dimensionless excess temperature u and heat transfer rate 4) as du d.u
--Y k(7-cD2/4)8, ’
and
dq ~ = -hnDO,,u d.u
C-41)
Equations (Al) provide a description of the basic heat transfer mechanisms in the fin: conduction through the cross-sectional area according to Fourier’s law, and convection through the circumferential area according to Newton’s law of cooling. The boundary conditions in (4) and (5) then become u=l The optimum
minimum
at.x = 0,
and
mass pin fin is obtained
q=O
at.v=h.
on minimizing
(A2) the system Hamiltonian (A3)
where &, and I.,, make up the adjoint system (costate Euler-Lagrange differential equations : dA,
ax
du
c)y
variables).
They satisfy the following
A,,
(A4)
k(nD’/4)0,,
and (A5) Here, D is the control namely,
variable
which must satisfy
Pontryagin’s
The state variables q, u are not fully specified at the boundaries L,, R,, must satisfy the following transversality condition.
minimum
principle
; hence. the costate variables
1,6q + /1,,6u = 0. The boundary
conditions
l/(kO,,~D*/4)
(A7)
(A2) are used to get I,, = 0
Eliminating
(3),
at x = 0,
and
,I,, = 0
atu = b.
and hnDQ,, from (A4) and (A5) respectively
(A8) by using (Al) yields (A9)
(Al01 Adding
(A9) and (AlO), we get Journal
980
of the Franklin Pergamon
lnst~tutc Press plc
Optimum
Shapes
ef’ Concectit’e
(d,+qi,)
= Cl = 0.
Pin Fins
.
.
&(&+qi,)=0
The constant of integration Combining (AlO) and (All)
or
C, is zero from gives
the boundary
(Al 1)
conditions
(A2) and (A8).
qd!Ln.d4=0. dx The above equation
may be easily solved to obtain i,,, = c,q.
(A12)
The constant CZ is to be determined from the optimization used along with (A12) gives the expression for i, as
process.
Equation
/$ = -c,u. Inserting
the expressions
tA13)
for 1, and 1, into (A6), we get XD y+-
Substituting
(A 11) when
C&(1
-n)nQ,uz D”
Czq2 + k(nD3/8)0,
= ”
(A14)
for q from (Al), we obtain after rearrangement
Comparison of the above result with (12) clearly shows that the differential equations are identical with C,O, = - l/l?. Thus, the optimization problem which minimizes the fin volume K for a given H is equivalent to the optimization problem which maximizes the cooling effect H for a given K; consequently, the design equations developed in the paper for a maximum cooling fin hold for a minimum mass fin.
Appendix B: Validity of the Classical Fin Assumptions The profile curvature
may be included
by modifying
(1) as follows :
hlzD,,/l + e0
(Bl)
where e = (dD/dx)*/4. The curvature effect is negligible if e K 1. We wish to estimate the magnitude of e for the specific case of only weight constraint, and thus quantify the errors involved in the length-of-arc assumption. From (26), it is seen that emax =
&
[hDk(l +n)b’-“]2”‘+n)
= (I:,,
It may be noted that the “usefulness” of a fin can be expressed in terms of the “removal number” N, (8) which is the ratio of the heat transferred with the fin to that which would be transferred without a fin. Thus, using (26), Vol. 727. No 6. pp. Y65SY82. IYYO Prmcd in Great Bntam
981
U. Natarujun
and U. V. Shenoy - . N _
I
H
+4~212~“+‘;_ k -hh,,,. h,,,W~
= (7$/4~)[Ml
H,r,
‘(B-3)
At this stage, the convection coefficient h,, in the absence of the fin must be specified. Let h,,, be approximated by the convection coefkient h, at the base in the presence of the fin; then. /I,,, = h,, = h,,jD% as per (2). On using (26). WCget N, = (I +n)b/D,,. Combining
(B4)
(B2), (B4) and (26), we get I 1 h,,D, em.!X- -i = N; (1+/z) k
(B5)
If we now choose a value of N, greater than or equal to 5 for economic justification as done by Razelos and Imre (8) and Razclos (ZO), then the P term will typically lead to an error less than 4%. Also, the assumption of a one-dimensional fin with substantial temperature gradients occurring only in one direction holds good, if the fin is thermally thin, i.e. the Biot number lz,,DJk is sufficiently small (24. 25). This implies a large value of N, and a small value of omilx; thus, both the assumptions of a one-dimensional fin as well as of negligible curvature will bc reasonably justified for a “useful” fin with a small Biot number.
Journal
982
of the Frankhn Pcrgamon
Institute Press plc
UPENDRA
Department
NATARAJAN
and
UDAY
v.
of Chemical Engineering,
SHENOY
Indian Institute of Technology,
Powai,
Bombay 400 076, India
ABSTRACT : Principles of variational calculus are used to determine the shapes of convective pin fins that maximize heat dissipation, given the amount of fin material. The analysis considers the convective heat transfer coeficient h to depend on thefin diameter D according to the relationship h CT l/D”, where n takes on values from 0.2 to 0.5 depending on the type of’pin,fin configuration andflow~ condition (1: AIChE .I, Vol. 29, p. 1043, !983). The Euler equations, which are nonlinear and coupled, are ,formulated and solved,for the cases of both length and weight constraints as well as only weight constraint. The resulting quadrature formulae are represented in the form of a convenient design plot, from which the optimum design parameters may be obtained and used to determine the$n and temperature profiles as well as the cooling performance. The solutions under both constraints yield considerably simple results for the case of only weight constraint, which corresponds to the diameter and excess temperature of the fin tip being zero. An important result is that the Schmidt criterion (2 : Z. Verein. Deutsch lng., Vol. 70, pp. 885, 947, 1926) of a linear temperature profile also holds for pinJins of specified weight with a variable heat transfer coeflcient. Finally, by using Pontryagin’s minimum principle (3 : The Mathematical theory of Optimal Processes, Wiley,, New York, 1962), it is demonstrated that the problems of maximizing cooling ($or a given$n weight) and minimizing weight (for a given& cooling) are identical as both are governed by the same optimum design equations.
Nomenclature b c D e E F h h, H k K n
fin length constant defined in (15) diameter of pin fin of circular cross-section error term (in the length-of-arc assumption discussed in Appendix functional given in (9), equal to the cooling effect, H functional defined in (10) convective heat transfer coefficient defined in (2) cooling effect defined in (6) thermal conductivity of fin fin volume (or, equivalently, fin weight) exponent defined in (2) (describing the variation of h)
N, Nu
removal number, Nusselt number
&th
~;The Franklm lnstitute001&0032/90 $3.00+000
B)
fin/qnofin
965
U. Natatzjan 4 Re x i
r? /I
e
Subscripts 0 b k nf P
and U. V. Shenoy heat transfer rate Reynolds number distance from the fin base along the length of the fin dimensionless excess temperature, e/O, variation fin efficiency constant Lagrange multiplier temperature difference between a point on the fin and its surroundings (temperature excess) Hamiltonian function defined in (A3) adjoint (costatc variables) system
fin base (x = 0) fin tip (X = b) division by k no fin condition parameter (used in plots and defined in Section VI)
Superscript * division by base diameter
D,, (to make fin diameter
dimensionless)
I. Introduction surfaces are often used in heat exchange devices to increase the heat transfer between the primary surface and the surrounding medium. In addition to longitudinal and radial fins, a common fin geometry is the pin fin or spine (4). usually of circular cross-section (Fig. 1). The fin optimization problem generally involves determining the shape and dimensions of the fin that (a) dissipates the maximum heat for a given mass or (b) utilizes the minimum mas? to dissipate a given heat. Furthermore, a constraint may or may not be imposed on the length of the fin (5). Hrymak et al. (6) have presented a detailed review (not included here) of the existing literature on the optimization of extended surfaces. Extended
b
FIG.
I.
Spine (pin fin) of circular cross-section. Journalof Ihe Franklm
966
lnst,tute Pergamon Pres plc
Optimum
Shapes ?f Cont’ectirre Pin Fins
Previous studies (1,7) on optimum pin fins have focused attention on determining the optimum dimensions of cylindrical spines of constant diameter by differential calculus. On the other hand, this work attempts to determine the optimum fin shape (in terms of the variation of the fin diameter D with fin distance x) as well as the optimum dimensions by variational calculus. It may be noted that Sonn and Bar-Cohen (7) found the optimum diameter for a cylindrical pin fin with a constant convective heat transfer coefficient h, while Li (1) analysed the same problem assuming a variable h. Though Razelos and Imre (8) determined the optimum shapes of longitudinal and radial fins for a power-law spatial variation of the heat transfer coefficient, they obtained solutions for pin fins only in the special case of constant h. As pointed out by Razelos and Imre (8) there is considerable theoretical as well as experimental evidence that the convective heat transfer coefficient h varies markedly along the fin surface. Large errors may be introduced (9) if optimum fins are designed under the assumption of constant h. Hence, the more realistic assumption (1) of h K l/D” is used throughout the analysis presented here. Letting n assume values from 0.2 to 0.5 allows considering pin fins in isolation and in arrays (both in-line and staggered) under various flow conditions. Further, the results for the case of constant h can be easily obtained from the general expressions derived below by setting n to be zero. While examining optimum shapes for cooling fins, an early study by Schmidt (2) suggested that the temperature must vary linearly along the length of the fin. The linear temperature profile, referred to as the Schmidt criterion in the literature, was formally shown by Duffin (10) to be optimum for longitudinal and radial fins with a weight constraint. In the present work, it is demonstrated that the Schmidt criterion holds even for optimum pin fins of a given weight with a variable heat transfer coefficient.
II. The Dzrerential
Formulation
The one-dimensional problem of a cooling pin fin of circular cross-section (Fig. 1) transferring heat to the surroundings by convection is considered here. If the temperature difference between a point on the fin surface and its surroundings is 8, then a heat balance at steady state over a differential element of length dx (assuming the curvature of the convective surface is small) yields the following governing differential equation :
The fin diameter D is taken to be a function of the distance from the fin base x. Though the thermal conductivity k is assumed constant, the convective heat transfer coefficient h is assumed to vary with the fin diameter as (1) h cc l/D” where hD is a constant. Vol 327. No. 6. pp. 965-982. 1990 Prmted ,n Great Britain
or
The above dependence
hD” = h,
(2)
of h on the diameter D directly follows 967
U. Nutarujan
and U. V. Sheno_v TARLEI Dinxwsio&ss
groups jbr design of’optimum
pit1 fin jiir wrious n
~ Rt>
Physical situation
Configurationi
IO’-10’
S, ILA, SA S, SA
II D
t Pin fin configuration array.
: S denotes
ILA
S
ILA, SA Constant
0.4
0.37
0.3
0.2
0.0
1.824
1.835
I.839
1.849
I.865
1.904
1.5
I.4
I.37
I.3
1.2
I.0
1.767
1.539
1.474
1.327
I.131
0.785
0.625
0.650
0.658
0.675
0.700
0.750
=
‘7 = (3 -n)i4
_____
2 x IO’-1 x IO”
0.5
I + ,I 2,1 ,I
(l6h,,K’/j$k)”
IO’-2 x 105
sin&
h
fin, ILA denotes in-line array. SA denotes staggered
from the analysis of experimental work (11, 12), which shows that the Nusselt number NU for heat transfer to tubes in crossflow depends on the Reynolds number Re according to NU cc Re’-“. The values of y1 typically vary from 0.2 to 0.5 (1) depending on the flow conditions and the types of tube arrays (refer to Table I). Clearly, n = 0 corresponds to the case where the convective coefficient is a constant. If Q. is the temperature excess at the base (.Y = 0) and u is defined as the dimensionless excess temperature g/O,, then (1) on combining with (2) gives
= h,,D’
no.
(3)
If the heat transferred through the tip of the fin of length b is negligible compared to that leaving through its lateral surface, then the appropriate boundary conditions are 14 =
1
at
x=0
(4)
and D2g=0
at
x=6.
Equation (5) implies that at the fin tip, the temperature gradient is zero (which corresponds to the case of both length and weight constraints as shown later) or the diameter is zero (which corresponds to the case of only weight constraint as demonstrated later).
Optimum Shapes cf Concectice
Pin Fins
The goal is to obtain non-negative, differentiable, continuous functions for u and D (in the interval 0 6 x d 6) that respectively define the temperature and fin profiles which maximize the cooling effect H given by
s s b
Hr
~,ZQ-“udx=
_
0
H is directly related to the heat transfer rate at the fin base q. as H = qo/eo. A constraint on the fin volume (or, equivalently, the weight) may be imposed as
’
0
nD2 ~ dx = K, a given positive constant.
4
(7)
III. The Variational Formulation To obtain the variational formulation of the steady one-dimensional physical problem discussed above, the formal procedure outlined by Arpaci (13) is followed. Taking the governing linear second-order ordinary differential equation (3) as the Euler equation associated with the desired variational problem and considering the variation in u only, we obtain ,,=l
Integrating
(&(k$$)-,h,,‘-.,),,dx=O.
the first term by parts,
Since the boundary conditions (4) and (5) are natural, remaining term in the above equation may be rewritten
the first term is zero. The as
~~(f+!!(!!!):+%!!!!)dx=O~ Thus, the variational
and the corresponding
formulation
functional
of the problem
is
is (9)
Both Liu (5) and Duffin (10) determined optimum shapes for longitudinal fins with and without heat generation respectively, by starting with. integral expressions similar to (9) ; however, they did not attempt to establish the functional E through formal techniques as done above. Vol. 327. ho. 6. pp. Yhj-9x2. Printed in Great Bntam
1990
969
U. Nutarqjan
and U. V. Sheno!,
Having derived the variational formulation, attention is focused on attaching a physical meaning to E by relating E to H. Integrating the first term in (9) by parts, and utilizing the physics of the problem (3), (4) and (5), comparison with (6) directly shows that E = H. Importantly, if the functional E is maximized, then the cooling effect H of the fin is maximized. Using the Lagrange multiplier ,J (a constant), the weight constraint is readily incorporated at this stage by defining the functional F to be equivalent to (7) and (9). Thus,
As shown earlier, the Euler equation ation of II is the original differential has no dependence on 14.
resulting equation
from (10) for any admissible vari(3) since the weight constraint (7)
IV. Analysis for both Length and Weight Constraints The case where the length is a constraining factor has been typically ignored by previous investigators with the exception of Liu (S), who studied longitudinal and radial fins. In this section, the case of a pin fin with a length constraint (fixed h) and a weight constraint (non-zero 1.) is considered. The variation in Ffor a variation in D is
For F to be an extremum, 6F must be zero for any admissible the following equation must be valid :
variation
SD and
The fin shape optimization problem requires solving a system ofcoupled differential equations of the second order (3) and second degree (12) for u and D. The solution procedure adopted below is along the lines originally suggested by Pitchumani and Shenoy (14). Both sides of (3) may be multiplied by (du/dx) and the equation rewritten as (13) Utilizing pulations
(12) to eliminate that
(du/dx)
from (13), it may be shown after some mani-
d&(3-n)u’D’ The left-hand side is an exact differential is easily written as 970
m’-/zD2)
= 0.
(14)
; thus, a useful relation
between
u and D
Journal of the Franklin Inst~tulr Pergamon Prtx plc
Optimum h,(3-n)u2D’-“-AD’ where of the in (5) value
Shapes of’ Conoectiw
Pin Fins
= C,
(15)
C is the constant of integration. If the subscript h is used to denote evaluation quantities at x = b, then (12) and (15) on imposing the boundary condition yield Dd +’ = 2h,(l -n)u,‘/A and C = A(1 +n)D?/[2(1 -n)]. Substituting this for C, (15) may be written in the following convenient form : u2
=
j+IJ+n)
5
2(1-n) Imposing the boundary by D,, (16) yields
condition
0D
in (4) and denoting
2
(16)
.
the base diameter
of the fin
(17) Then A is easily eliminated the temperature profile :
from (16) and (17) to give the following
J
U=
expression
2(1-n)D*2+(l+n)D,*2
FiP”[2(1
for
(18)
-nj+(l+Gj5*2)’
The asterisk in the superscript denotes that the fin diameter has been made dimensionless on dividing by the diameter at the base. Now, the quadrature formula for D* in terms of x is obtained by eliminating u and (du/dx) using (12), (16) and the differentiated form of (18). Noting that (du/dx) is negative, we get after some manipulation
~~ ~~~~~ s~~ I
(1 -n)
J(Gj-j
D*2~1-n)+(l+n)Db**]
(2D** - D;2)D*+
‘)I2dD* -fix
[2(l--n)D*2+(1+n)D~2][D*2-D~2] (19)
where Ak = A/k. Equation (19) provides an expression for the fin length h, when x = h and the lower limit of integration is set to D$ Substituting from (17) and (19) into (7), the expression for the fin weight is 42h,k(l K=
-n)(3
-n>l2r(‘+n)(l -n>J(l
+n)(3-n) x (2D*2 - Dt2)D *(3+n)‘2d?
S’oz 8[1,(2(1 -n)
+ (1 +n)Df2)](5f”)/(21
2n) __
___
m-n)D*2+-n)Df2] where hnk = h,,/k. Using (12) and (17) along with the definition given by H = nk[2h,,(l
The utility
of the above equations
[D*2 - Dz2]
of H in (6), the cooling
-n)12”‘+“)[(3 -r~)/&](~~~)l(~+*~) &l 4[2(1-+
(20)
(1 + ~I)D*~] 2!(‘fn) b will be demonstrated
+n) (1 -Dt2) ~~
in a subsequent
effect is
.
(21)
section
U. Nutarajan
and U. V. Shenoy
through a design plot as well as figures of the optimum fin and temperature profiles. As the above expressions can be considerably simplified if the length constraint is relaxed, the next section is devoted to this special case.
V. Analysis for only Weight Constraint When there is no constraint on the length, in F for a variation in b results in
then b is a variable
and a variation
For the optimum fin under conditions of variable b, the integrated term must vanish at both limits. At x = 0, the term is zero as 6x = 0. However, at x = b, 6x is not zero and a boundary condition is obtained for the fin to be optimum by using (12). Thus, +h,Dl-nu2
_
402
=
hmD’
4
-“cl
+‘)f
2
The above equation along with (5) and (12) indicates from (15), it is clear that C = 0 and we obtain
= 0
(22) now directly
s = h.
that Dh must be zero. Then,
D ‘+’ = h,(3 -n)u2/L Equation
at
(22)
implies that Dh = 0
and
u,, = 0.
(23)
Thus, the temperature at the fin tip is equal to the ambient fluid temperature and the tip diameter is zero. Then (12) may be solved for the optimum temperature profile after eliminating D using (22). The results obtained, using the boundary conditions (4) and (23), are du dx=
(24)
-
and u = l-x/b.
(25)
Equation (25) indicates that the temperature profile is linear; thus, the classical Schmidt criterion (2), which was formally shown by Duffin (10) to hold for longitudinal and radial fins under a weight constraint, is valid for optimum pin fins of specified weight, which maximize heat dissipation. Substitution of (24) and (25) into (22) gives the optimum fin profile as D = [hna(l +n)(b-x)‘]““+“). According to (26), the classical constant heat transfer coefficient
parabolic profile is optimum (i.e. n = 0). Since the concave
(26) for a fin with a parabolic pin fin
Optimum
Shupes
ef Convective
Pin Fins
analysed by Kraus (15) assumes the base diameter is given, it is not optimum. However, if the base diameter is chosen to be hkb2 as per (26), the spine is optimum for constant h. Substituting (26) into (7) it is seen that K=
71/[4(5fn)](hDk)2/(l+n)(l+n)(3+n)1(1+n)b(5+n):(l+n).
(27)
It may be emphasized that for the case of only weight constraint, the length of the fin is obviously not specified and must be determined from (27). The cooling effect thus obtained by substituting (25) and (26) into (6) is H = (&/4b)
[h&l
+n)b2]2’(‘+n).
(28)
As expected, (16)-(21) on setting Db = 0 and & = (3-n)/[(l+n)b2] reduce to (22)-(28). It is of interest to note that (22)-(28) also specify the optimum pin fin of minimum weight for a given heat dissipation and no length constraint (refer to Appendix A). This has been shown by Focke (16) to be true for the case when n = 0. Again, for n = 0 and no length constraint, the minimum weight convective pin fin problem has been formally solved by Bhargava and Duffin (17) (using a special transformation of variables along with Holder’s inequality) as well as by Razelos and Imre (8) (using Pontryagin’s principle). Also, it may be pointed out that the minimum weight radiation pin fin problem has been analysed by Wilkins (18). VI. Results and Discussion In designing optimum pin fins, the equations conveniently utilized in the form of various plots.
derived
thus far may be most
Generating the Plots Plots are included in this paper only for the specific case of n = 0.5, as the plots for other values of n (between 0.2 and 0.5) will exhibit very similar features. The case of n = 0.5 (see Table I) corresponds to a single pin fin as well as a pin fin array (both in-line and staggered), when the Reynolds number Re is between 100 and 1000 (1). The various plots relevant to the optimum design of pin fins, and the procedures adopted in generating them are briefly explained below. (i) A design plot of fin weight parameter Kp vs fin length parameter b, is made (Fig. 2) by choosing different values of & (in the range 0.1 - 10) and D: (in the range C-0.99). b and K/(nhif’+“)/8) are computed by numerical integration from (19) and (20) respectively ; subsequently, they are transformed to K,, = 1 -exp [-K/(xhDk 2’(‘+n)/ S)] and 6, s 1 -exp [-b]. The transformation is such that Kp and b, vary from 0 to 1, as K and b vary from 0 to co. In Fig. 2, the dotted lines represent constant ilk curves, while the solid lines denote constant DC curves. The design curve for Dz = 0 (which is merely (27) plotted in terms of Kp and b,) provides an upper bound on the fin length, as it corresponds to the case of only weight constraint. (ii) A plot of the fin profiles in terms of D* vs X, is made (Fig. 3) by choosing different values of 02 (in the range 04.9). D* may be obtained as a function of 973
U. Nutarujun
and W. V. Shenoy
- _ LENGTH
PARAMETER
3
bp
FIG.2. Design plot for n = 0.5.
fix by performing the necessary numerical integration in (19). ,,&x may be transformed to a dimensionless distance X, (which varies from 0 to 1) on dividing by its maximum value. This maximum is given by &h for the case when D? = 0 ; thus, (24) directly suggests that X, be defined by J&( 1 +rz)x/& n). It may be observed on Fig. 3 that under only a weight constraint (D,f = 0), the optimum fin profile is given by D* = (1 -XJzl(‘+“) as per (26). (iii) A temperature profiles plot in terms of u vs X, is next made (Fig. 4) by selecting the same values of Dtas in (ii). The D* data may be directly converted to dimensionless excess temperature u by use of (1X). It may be noticed on Fig. 4 that when only the weight is specified (D$= 0), then the optimum temperature profile is linear (U = 1 -X,) as per (25). (iv) A dimensionless performance curve relating the cooling effect parameter H,, to Dt is finally made (Fig. 5) by first calculating the quantity H;1&3m n)!(2+2n)/ [mkhAf’+“)/4] from (21). This quantity is next made dimensionless to obtain H,, on dividing by its maximum value which is attained at Dt= 0; (28) indicates that this maximum value is (1 fn)‘;’ (3-n)c’p”);(2+2”). Thus, HP is defined as H[;lk/(3_n)](3~“)i(2+2”)/[7ck(l+n)1:2h~~l+”) / 4 1. Clearly, from Fig. 5, the most cooling for a specified amount of material is obtained when there is no constraint on the fin length, i.e. when DC= 0. It may be noted that in the case of both length and weight constraints, expressions for the fin profile (19) and fin volume (20) are in terms of quadrature formulae ; so, the numerical integration was performed using the IMSL routine QDAGS (19). Journalofthe Frankhn 974
In,utute Pergdmon Press plc
Optimum
DIMENSIONLESS
FIG. 3. Optimum
0.0
DISTANCE-
DIMENSIONLESS
FIG. 4. Optimum
Xp
fin profiles for n = 0.5.
0.4
0.2
Shapes qf Conzrectire Pin Fins
temperature
0.6
DISTANCE
0.8
Xp
profiles for n = 0.5.
975
U. Natarajun
and U. V. Shenoy
FIG. 5. Performance plots for n = 0.5.
Utilizing the Plots The plots generated may be effectively used to design the optimum pin fin, when the weight and length of the required fin along with the thermal properties (namely, h, and k) are given. The value of n is also specified as per the fin configuration and the flow condition (see Table I). (i) From the given values of K and h, the values of Kp and h, are calculated. These are used to read off the values of 0: and 1, from the design plot in Fig. 2. If the design point lies to the right of the Df = 0 curve, then the length has been overspecified; in this case, the curve for only weight constraint (namely, Dt= 0 curve) should be used to obtain I+. (ii) With a knowledge of the optimum Df and &, (17) may be used to calculate the base diameter Do. (iii) The appropriate curve in terms of D* vs X, is chosen from the fin profiles plot depending on the value of D,*obtained in (i). Interpolation may be required to get an approximate curve if D,*is not very close to the values of 0; plotted in Fig. 3. The D* data are easily converted to the actual fin diameter D by multiplying by the Do value calculated in (ii). The X, data are transformed to the actual fin distance x by substituting the known value of I+. This procedure yields the actual optimum fin profile in terms of diameter D vs distance X. (iv) The optimum temperature profile is obtained in a straightforward manner from Fig. 4, if the procedure for converting A’,,to x as outlined in (iii) is followed. 976
Journal
of the FrankIm Inst~tutc Pergamon Pre*s plc
Optimum
Shapes qf‘ Concectiue Pin Fins
, ‘(4 F rom the known value of D$, the value of HP can be read off from Fig. 5. The optimum cooling effect H can then be calculated from the definition of H,, on substituting the value of Ak. Calculuting the Fin .E#cienq The fin efficiency r] is conventionally used in engineering practice to indicate the effectiveness of a fin in transferring a given quantity of heat. It is defined as the ratio of the heat actually dissipated by the fin to that dissipated by a hypothetical isothermal (at the base temperature 0,) fin of identical shape and size. The expression for the optimum efficiency is obtained from (17), (19) and (21). Thus,
s b
7chgD’-” dx
1
_
r
0
_______
_
H
’
2 1 -n Jol”“c-~cr
+Fn)m(2D*2 (
- Dz2) dD*
s @ J[1-D~2][2(1-n)D*2+(1+n)D,*2][D*2-D~2~~ (29)
For D$ = 0, the above equation
simplifies
to
u] = (3-n)/4. The efficiency is computed from (29) by numerical of Dz and the result is plotted in Fig. 5.
(30) integration
for different
values
Determining the l$ect c?f’n 011the Fin Design,ftir only Weight Constraint The variation of the heat transfer coefficient h depends on the value of the exponent n (2). The value of n in turn depends on the flow condition of the medium surrounding the fin, and on whether the fin is solitary or in an array (see Table I). Li (1) has given the values of n for various possible conditions normally encountered, and tabulated the corresponding values of the optimum design parameters for a cylindrical pin fin of specified weight. Here, attention is focused on our optimum pin fin with only the weight constrained (discussed in Section V) ; then, the same dimensionless groups used by Li (1) are calculated using our (26))(28) and (30). The results, which are given in Table I, can be directly used to design optimum fins with the profile defined by (26), when n, K and the thermal properties are specified. Thus, from the dimensionless groups in the first column in Table I, the optimum values of Do, b, H and q are easily calculated in that order.
VIZ, Concluding Remarks In most practical situations of fin design, the assumption causing the greatest errors is that of a uniform convection coefficient over the entire fin surface. In this paper, solutions for the optimum design of pin fins are developed under the more realistic conditions of a varying h. For the case where the length is constrained (Section IV), (19))(21) constitute a system of three equations in five variables (K, b, H, D$ and &). If the designer specifies two quantities (the weight K and the length b), then the remaining three quantities (H, 0: and iWr)may be determined ; thus, the design of a maximum
U. Nutarajan
and U. V. Shenc?);
cooling tin (where K and h are given) may be done. It must be emphasized that the design of a minimum mass fin (for a specified heat dissipation) may be done using the very same equations (refer to Appendix A). The desired cooling effect H and the length h are given in this case, and (19))(21) may be utilized to determine K, 0: and I,,\. If there is no length constraint (Section V), then (27) and (28) constitute a system of two equations in three variables (K, h and H). For the maximum cooling fin (given the weight K), h and H are determined from (27) and (28). On the other hand, for the minimum mass fin (given the required cooling H), b and Kare found from the same two equations. For this case where the length is not constrained, it may be noted that 0: = 0 and &. is directly related to h by (24). The analysis developed for the optimum fin in this paper is classical in the sense that it considers the fin to be one-dimensional and neglects the profile curvature in the arc-length calculation. A brief discussion on these assumptions is presented in Appendix B. Both these assumptions have been recently shown to be valid (20) for pin fins with constant thickness, triangular and parabolic profiles. In particular, neglecting the arc-length factor that multiplies the periphery has been shown to be a reasonable assumption, and is a “mild restriction” in useful fins (8). The formulation incorporating the curvature is relatively straightforward as it merely involves multiplying the perimeter ZD (in Eq. I) by the factor ,/l +0.25(dD/dx)2; however, the solution procedure will be relatively complicated and numerical in nature. The optimization, without making the length-of-arc assumption, will involve a nonlinear two-point boundary-value problem and will be similar to the minimum-weight one-dimensional fin optimization done by Maday (21). Considering the errors involved (reduction in weight by 1.6%) under the length-of-arc assumption for the specific case solved by Maday (21) and the waviness in the resulting fin profile, the simpler analysis presented here should be of more practical utility.
Acknowledgement The authors thank M. S. N. Mm-thy (Department of Chemical Engineering, Institute of Technology, Bombay) for discussions that aided in the preparation various plots.
Indian of the
References (1) C-H. Li, “Optimum cylindrical pin fin”, AlChE J., Vol. 29, p. 1043, 1983. (2) E. Schmidt, “Die Wlrmeubertragung durch Rippen”, Z. Verein. Deutsch Ing., Vol. 70, pp. 885 and 947, 1926. (3) L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko, “The Mathematical Theory of Optimal Processes”, Wiley, New York, 1962. (4) D. Q. Kern and A. D. Kraus, “Extended Surface Heat Transfer”, McGraw-Hill, New York, 1972. (5) C. Y. Liu, “A variational problem relating to cooling fins with heat generation”, Q. Appl. Math., Vol. 19, p. 245, 1961.
978
Journaloflhc
FrankIln ,nat,lutc Pergamon Press plc
Optimum
Shapes qf’ Contrectire Pin Fins
. A. N. Hrymak, G. J. McRae, and A. W. Westerberg, “Combined analysis and optimization of extended heat transfer surfaces”, ASME J. Heat Trarufer, Vol. 107, p. 527, 1985. A. Sonn and A. Bar-Cohen, “Optimum cylindrical pin fin”, ASME J. Heat TranJfer, (7) Vol. 103, p. 814, 1981. (8) P. Razelos and K. Imre, “Minimum mass convective fins with variable heat transfer coefficients”, J. Franklin Inst., Vol. 315, p. 269, 1983. (9) P. Razelos and K. Imre, “The optimum dimensions of circular fins with variable thermal parameters”, ASME J. Heat Transfer, Vol. 102, p. 420, 1980. (10) R. J. Duffin, “A variational problem relating to cooling fins”, J. Math. Mech., Vol. 8, p. 47, 1959. (11) A. A. Zukauskas, “Heat transfer from tubes in crossflow”, Adz?. Heat Transfer, Vol. 8, Academic Press, New York, 1972. E. R. G. Eckert and R. M. Drake, “Analysis of Heat and Mass Transfer”, McGraw(12) Hill, New York, 1972. Heat Transfer”, Addison-Wesley, Reading, MA, 1966. (13) V. S. Arpaci, “Conduction R. Pitchumani and U. V. Shenoy, “A unified approach to determining optimum shapes (14) for cooling fins of various geometries”, Private communication, 1988. of Extended Surface Thermal Systems”, (15) A. D. Kraus, “Analysis and Evaluation Hemisphere, Washington D.C., 1982. Forschung auf dem Gdiete Lies Ingenieur(16) R. Focke, “Die Nadel als Kuhlelement”, wesens, Vol. 13, p. 34, 1942. (17) S. Bhargava and R. J. Duffin, “On the non-linear method of Wilkins for cooling fin optimization”, SfAM J. Appl. Math.. Vol. 24, p. 441, 1973. (18) .I. E. Wilkins, Jr., “Minimum mass thin fins which transfer heat only by radiation to surroundings at absolute zero”, J. Sot. Indust. Appl. Math., Vol. 8, p. 630, 1960. Mathematical and Statistical Libraries, Problem (19) IMSL, Version 1.O, International Solving Software Systems, Houston, TX, 1987. of convective pin fins with internal heat (20) P. Razelos, “The optimum dimensions generation”. J. Franklin Inst., Vol. 321, p. 1, 1986. straight fin”, ASME J. Engng (21) C. J. Maday, “The minimum weight one-dimensional Znd., Vol. 96, p. 161, 1974. Theory with Applications”. Wiley, New York, 1969. (22) D. A. Pierre, “Optimization Blaisdell Pub. Co., (23) A. E. Bryson, Jr., and Y. C. Ho, “Applied Optimal Control”, Massachusetts, 1969. fin solution”, ASME J. Heat Transfer, (24) R. K. Irey, “Errors in the one-dimensional Vol. 90, p. 175, 1968. (25) S. Whitaker, “Elementary Heat Transfer Analysis”, Pergamon Press, New York, 1976.
Appendix A : Minimum Mass Convective Pin Fin The objective here is to formulate the optimization problem which minimizes the fin volume K for a specified amount of cooling H. The problem is similar to the optimum control problem (22) of extremizing an integral (7) subject to an isoparametric constraint (6). The procedure adopted is analogous to that of Maday (21) and Razclos and Imre (8). It utilizes Pontryagin’s principle (3), which provides an alternative approach to the classical variational calculus approach and may be considered a generalized Weierstrass excess-function test. Both approaches yield the same optimum solutions for problems involving continuous control. Bryson and Ho (23) favor the Pontryagin principle approach for it may often be the more direct method providing some insight into the actual processes
U. Nutamjun
and U. V. Shenoy
involved in the optimization problem. As the variational calculus approach has been adopid in the main paper, the methodology based on Pontgryagin’s principle will be applied in this Appendix. Equation (I) may be re-written in terms of state variables (dimensionless excess temperature u and heat transfer rate 4) as du d.u
--Y k(7-cD2/4)8, ’
and
dq ~ = -hnDO,,u d.u
C-41)
Equations (Al) provide a description of the basic heat transfer mechanisms in the fin: conduction through the cross-sectional area according to Fourier’s law, and convection through the circumferential area according to Newton’s law of cooling. The boundary conditions in (4) and (5) then become u=l The optimum
minimum
at.x = 0,
and
mass pin fin is obtained
q=O
at.v=h.
on minimizing
(A2) the system Hamiltonian (A3)
where &, and I.,, make up the adjoint system (costate Euler-Lagrange differential equations : dA,
ax
du
c)y
variables).
They satisfy the following
A,,
(A4)
k(nD’/4)0,,
and (A5) Here, D is the control namely,
variable
which must satisfy
Pontryagin’s
The state variables q, u are not fully specified at the boundaries L,, R,, must satisfy the following transversality condition.
minimum
principle
; hence. the costate variables
1,6q + /1,,6u = 0. The boundary
conditions
l/(kO,,~D*/4)
(A7)
(A2) are used to get I,, = 0
Eliminating
(3),
at x = 0,
and
,I,, = 0
atu = b.
and hnDQ,, from (A4) and (A5) respectively
(A8) by using (Al) yields (A9)
(Al01 Adding
(A9) and (AlO), we get Journal
980
of the Franklin Pergamon
lnst~tutc Press plc
Optimum
Shapes
ef’ Concectit’e
(d,+qi,)
= Cl = 0.
Pin Fins
.
.
&(&+qi,)=0
The constant of integration Combining (AlO) and (All)
or
C, is zero from gives
the boundary
(Al 1)
conditions
(A2) and (A8).
qd!Ln.d4=0. dx The above equation
may be easily solved to obtain i,,, = c,q.
(A12)
The constant CZ is to be determined from the optimization used along with (A12) gives the expression for i, as
process.
Equation
/$ = -c,u. Inserting
the expressions
tA13)
for 1, and 1, into (A6), we get XD y+-
Substituting
(A 11) when
C&(1
-n)nQ,uz D”
Czq2 + k(nD3/8)0,
= ”
(A14)
for q from (Al), we obtain after rearrangement
Comparison of the above result with (12) clearly shows that the differential equations are identical with C,O, = - l/l?. Thus, the optimization problem which minimizes the fin volume K for a given H is equivalent to the optimization problem which maximizes the cooling effect H for a given K; consequently, the design equations developed in the paper for a maximum cooling fin hold for a minimum mass fin.
Appendix B: Validity of the Classical Fin Assumptions The profile curvature
may be included
by modifying
(1) as follows :
hlzD,,/l + e0
(Bl)
where e = (dD/dx)*/4. The curvature effect is negligible if e K 1. We wish to estimate the magnitude of e for the specific case of only weight constraint, and thus quantify the errors involved in the length-of-arc assumption. From (26), it is seen that emax =
&
[hDk(l +n)b’-“]2”‘+n)
= (I:,,
It may be noted that the “usefulness” of a fin can be expressed in terms of the “removal number” N, (8) which is the ratio of the heat transferred with the fin to that which would be transferred without a fin. Thus, using (26), Vol. 727. No 6. pp. Y65SY82. IYYO Prmcd in Great Bntam
981
U. Natarujun
and U. V. Shenoy - . N _
I
H
+4~212~“+‘;_ k -hh,,,. h,,,W~
= (7$/4~)[Ml
H,r,
‘(B-3)
At this stage, the convection coefficient h,, in the absence of the fin must be specified. Let h,,, be approximated by the convection coefkient h, at the base in the presence of the fin; then. /I,,, = h,, = h,,jD% as per (2). On using (26). WCget N, = (I +n)b/D,,. Combining
(B4)
(B2), (B4) and (26), we get I 1 h,,D, em.!X- -i = N; (1+/z) k
(B5)
If we now choose a value of N, greater than or equal to 5 for economic justification as done by Razelos and Imre (8) and Razclos (ZO), then the P term will typically lead to an error less than 4%. Also, the assumption of a one-dimensional fin with substantial temperature gradients occurring only in one direction holds good, if the fin is thermally thin, i.e. the Biot number lz,,DJk is sufficiently small (24. 25). This implies a large value of N, and a small value of omilx; thus, both the assumptions of a one-dimensional fin as well as of negligible curvature will bc reasonably justified for a “useful” fin with a small Biot number.
Journal
982
of the Frankhn Pcrgamon
Institute Press plc