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Heat Transfer Engineering
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Some New Solutions for Extended Surface Heat
Transfer Using Symbolic Algebra
a
Abdul Aziz & Greg McFadden
a
a
Department of Mechanical Engineering , Gonzaga University , Spokane, WA
Published online: 15 Aug 2006.
To cite this article: Abdul Aziz & Greg McFadden (2005) Some New Solutions for Extended Surface Heat Transfer Using
Symbolic Algebra, Heat Transfer Engineering, 26:9, 30-40, DOI: 10.1080/01457630500205679
To link to this article: http://dx.doi.org/10.1080/01457630500205679
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Heat Transfer Engineering, 26(9):30–40, 2005
C Taylor & Francis Inc.
Copyright ISSN: 0145-7632 print / 1521-0537 online
DOI: 10.1080/01457630500205679
Some New Solutions for Extended
Surface Heat Transfer Using
Symbolic Algebra
Downloaded by [Southern Illinois University] at 21:47 23 December 2014
ABDUL AZIZ and GREG MCFADDEN
Department of Mechanical Engineering, Gonzaga University, Spokane, WA
The paper reports some new solutions for heat transfer through extended surfaces or fins using the symbolic algebra package
Maple 8, which is widely available. The four specific problems chosen for the present study are: (a) a rectangular convection
fin with the heat transfer coefficient varying either linearly or exponentially with the distance from the base, (b) a truncated
conical spine with convection at both ends, (c) a heat-generating annular fin with a constant base heat flux and an adiabatic
tip, and (d) a convection fin array made of a rectangular fin and two triangular fins. Each problem is formulated in a manner
that makes its solution novel and distinct from what is available in the literature. Solutions are provided in symbolic forms.
Using the numerical and graphical capabilities of Maple, the results are presented in the form of numerical data as well as
graphical displays. The paper demonstrates that Maple provides an effective and convenient tool for the analysis of extended
surface heat transfer problems that otherwise demand tedious algebraic manipulations.
INTRODUCTION
not exist in open literature. Besides its symbolic prowess, Maple
also has powerful numerical and graphical capabilities that facilitate parametric studies of the solutions and their graphical
display [2].
The four specific problems chosen for this study deal with
the thermal performance of the following fin designs:
Extended surfaces or fins find use in numerous applications
where the heat transfer between a hot surface and a cold adjacent
fluid needs to be enhanced. Applications range from the thermal
management of electronic components to power and process
heat exchangers. A comprehensive book on extended surface
heat transfer by Kraus et al. was published in 2001 [1].
The steady-state heat transfer analysis of an extended surface
involves solving second-order ordinary differential equations
under a variety of boundary conditions. For complex geometries
and/or boundary conditions, the solution procedure, though conceptually simple, involves very tedious algebraic manipulations.
Despite this difficulty, the number of analytical solutions for fin
heat transfer problems is vast. Current researchers on the subject
steadily continue to report new solutions.
Symbolic algebra packages, such as Maple and Mathematica, provide an alternative to hand analysis and thus alleviate
the drudgery of tedious algebraic efforts. This paper exploits the
power of the current version of Maple (i.e., Maple 8) to solve
some new fin heat transfer problems for which the solutions do
1. a rectangular convecting fin with variable heat transfer coefficient,
2. a truncated conical spine with convection at both ends,
3. a heat-generating annular fin with a constant base heat flux
and an adiabatic tip,
4. a convecting fin array consisting of a rectangular fin and two
triangular fins.
Each problem is formulated in a manner that makes the solution novel and distinct from what is available in the literature.
In particular, cases 2 and 3 have received no treatment in the
extended surface literature. However, other variations of cases
1 and 4 have been reported in the literature, and these would be
discussed in the paper.
Because the symbolic solutions are rather lengthy, their actual
displays would be omitted to conserve space. The reader can see
the results by a simple change from a colon to a semicolon at
the end of the Maple statement.
The authors are grateful to Kashif Aziz for preparing the drawings for this
paper.
Address correspondence to Dr. Abdul Aziz, Department of Mechanical Engineering, Gonzaga University, Spokane, WA 99258. E-mail: aziz@gonzaga.edu
30
A. AZIZ AND G. McFADDEN
31
where in Eq. (1a), h0 is the average heat transfer coefficient, and
in Eq. (1b), h0 is the heat transfer coefficient at x = 0. The tip
convects to the environment through a heat transfer coefficient
ht . The thermal conductivity of the fin is denoted by k.
The differential equation governing the temperature distribution in the fin may be written as
d 2θ
− N Xθ = 0
d X2
(2)
d 2θ
− N eX θ = 0
d X2
(3)
for case (i), and
Downloaded by [Southern Illinois University] at 21:47 23 December 2014
for case (ii), where
θ=
T − T∞
,
Tb − T∞
X=
x
,
b
N=
4h 0 b2
kδ
(4)
The boundary conditions to be satisfied are
X = 0,
θ=1
(5a)
X = 1,
dθ
+ Bit θ = 0
dX
(5b)
where Bit is the fin tip Biot number and is given by
Bit =
ht b
k
(6)
The heat transfer rate, q, from the fin may be expressed in dimensionless form as
qb
dθ Q=
(7)
=−
kδL(Tb − T∞ )
d X X =0
Figure 1 (a) Rectangular fin with variable heat transfer coefficient; (b) Linear
variation of heat transfer coefficient with distance; (c) Exponential variation of
heat transfer coefficient with distance.
A RECTANGULAR CONVECTING FIN WITH VARIABLE
HEAT TRANSFER COEFFICIENT
Mathematical Model
Figure 1a shows a rectangular fin of thickness δ, height b, and
length L (normal to the paper) attached to a primary surface at
a temperature Tb . The fin operates in a convective environment
at temperature T∞ . The convective heat transfer coefficient h is
assumed to be a function of x: h = h(x). Two forms of variation
of h(x) with x are considered:
x
(i)
linear, h(x) = 2h 0 (Fig. 1b) (1a)
b
(ii)
exponential,
h(x) = h 0 e x/b (Fig. 1c)
(1b)
heat transfer engineering
The solution of this problem for case (i) and with Bit = 0 has
been provided by Kraus and Bar-Cohen [3]. Han and Lefkowitz
[4] also provide a solution for Bit = 0 and h(x) = (γ + 1) h0 (x/b)
where γ is a constant. A heat transfer coefficient variation of the
form
h(x) = h 0 1 − ae−c(x/b) [1 − (a/c)(1 − e−c )]
was considered by Chen and Zyskowski [5]. Here, a and c are
constants. However, the specific problem considered here remains unexplored.
Maple Solution
Maple 8 is used to solve Eq. (2) in a symbolic form. An
excellent learning resource for Maple 8 is the learning guide
published by Waterloo Maple Inc. [6]. The company’s website
also provides a list of Maple books categorized by discipline.
Books dealing specifically with the solution of ordinary differential equations are those of Abell and Braselton [7], Coombes
et al. [8], and Betounes [9].
vol. 26 no. 9 2005
A. AZIZ AND G. McFADDEN
32
Temperature Distribution and Heat Transfer Rate
We first create Eq. (2) in Maple and call it Eq. (1). Note the
use of the command assume to tell Maple that N is greater than
zero. This precludes the generation of a complex solution for
θ(X ).
> restart; assume(N>0): Eq1
:=diff(theta(X),X,X)-N∗ X∗ theta(X)=0;
2
d
Eq1 :=
θ(X ) − N ∼ X θ(X ) = 0
d X2
Downloaded by [Southern Illinois University] at 21:47 23 December 2014
Next, Eq. (1) is solved using the dsolve command, which generates a general solution in terms of Airy functions Ai and Bi.
> Eq2 : = dsolve(Eq1,theta(X));
Eq2 := θ(X ) = C1 AiryAi N ∼ (1/3) X
+ C2 AiryBi N ∼(1/3) X
An expression for the derivative of θ(X ) is obtained, and boundary conditions (5b) and (5a) are created. The two linear algebraic
equations created by the application of the two boundary conditions are solved to obtain the constants of integration. These
constants are substituted (the assign command serves that purpose) into the general solution. Finally, Maple is asked to output
the temperature distribution, θ(X ); use Eq. (7) to generate the
dimensionless heat transfer rate, Q; and substitute X = 1 in
the expression for θ(X ) to provide the expression for the tip
temperature, θ(1).
> Eq3:=diff(Eq2,X);
d
Eq3 :=
θ(X ) = C1 AiryAi 1, N ∼ (1/3)X
dX
× N ∼ (1/3) + C2 AiryBi 1, N ∼(1/3) X N ∼(1/3)
> bc1:=subs(X=1,rhs(Eq3))+Bi[t]∗ subs(X=1,rhs(Eq2));
bc1 := C1AiryAi 1, N ∼(1/3) N ∼(1/3) + C2AiryBi
× 1, N ∼(1/3) N ∼(1/3) +Bit C1 AiryAi
× N ∼(1/3) + C2 AiryBi N ∼(1/3)
> bc2:=subs (X=0, rhs (Eq2));
bc2 := C1 AiryAi(0) + C2 AiryBi(0)
> consts:=simplify(solve({bc1 = 0,bc2 = 1}, { C1, C2})):
> assign(consts):
> Temp distribution:=simplify(rhs(Eq2)):
> Heat transfer rate:=−subs(X=0,rhs(Eq3)):
> Tip temperature:=subs(X=1,rhs(Eq2)):
Although the boundary conditions can be specified with the
equation in the dsolve command, they are specified separately
for clarity and for making changes in the boundary conditions
easy.
heat transfer engineering
Figure 2 Dimensionless temperature distribution: case (i).
As a numerical example, we consider a fin for which N = 1
and Bit = 2 and use Maple to calculate θ(X ), Q, and θ(1). A
plot of θ(X ) versus X is also created, as shown in Figure 2.
> Bi[t] :=2;
Bit := 2
> Eq4:=evalf(subs(N=1,Temp distribution));
Eq4 := 2.989036371 AiryAi(X )
− 0.09951067879 AiryBi(X )
> evalf(subs(N=1, Heat transfer rate));
0.8182300902
> evalf(subs(N=1, Tip temperature));
> plot(Eq4,X=0..1, labels=[“X”, “Theta”]):
For case (ii), the procedure for solving Eq. (3) subject to the
boundary conditions (5a, 5b) is the same as case (i). The following is the Maple worksheet for case (ii). Figure 3 plots the
resultant temperature distribution.
> restart; assume(N>0):
Eq1 :=diff(theta(X),X,X)-N∗ exp(X)∗ theta(X) = 0;
2
d
Eq1 :=
θ(X ) − N ∼ e X θ(X ) = 0
d X2
> Eq2:=dsolve(Eq1,theta(X));
√
X
Eq2 := θ(X ) = C1 BesselI 0, 2 N ∼ e( 2 )
√
X
+ C2 BesselK 0, 2 N ∼ e( 2 )
vol. 26 no. 9 2005
A. AZIZ AND G. McFADDEN
33
TRUNCATED CONICAL SPINE WITH CONVECTION
AT BOTH ENDS
Downloaded by [Southern Illinois University] at 21:47 23 December 2014
Mathematical Model
Figure 3 Dimensionless temperature distribution: case (ii).
A truncated conical spine of base radius rb , tip radius rt , and
height l is shown in Figure 4. The origin of the x coordinate is
located at a point remote from the fin tip. The spine material has a
thermal conductivity k. The fin base is in contact with a hot fluid
at temperature T f , which provides a heat transfer coefficient h f .
The lateral surface of the spine loses heat by convection to an environment at temperature T∞ through a heat transfer coefficient
h. The convection process at the fin tip is characterized by a heat
transfer coefficient h t and an environment temperature T∞ .
Beginning with the general differential equation appearing
in [1], it can be shown that for a truncated conical spine, the
equations governing the temperature distribution are
> Eq3:=diff(Eq2,X);
√
√
d
X
X
Eq3 :=
θ(X ) = C1 BesselI 1, 2 N ∼ e( 2 ) N ∼ e( 2 )
dX
√
√
X
X
− C2 BesselK 1, 2 N ∼ e( 2 ) N ∼ e( 2 )
> bc1:=subs(X=1,rhs(Eq3));
√
√
bc1 := C1 BesselI 1, 2 N ∼ e(1/2) N ∼ e(1/2)
√
√
− C2 BesselK 1, 2 N ∼ e(1/2) N ∼ e(1/2)
X2
(8)
X = 1,
dθ
= Bi f (1 − θ)
dX
(9a)
X = R,
dθ
= Bit θ
dX
(9b)
where
θ=
> bc2:=subs(X=0,rhs(Eq2));
√
bc2 := C1 BesselI(0, 2 N ∼ e0 )
√
+ C2 BesselK(0, 2 N ∼ e0 )
d 2θ
dθ
+ 2X
− m2 X θ = 0
2
dX
dX
T − T∞
,
T f − T∞
X=
x
,
b
R=
rt
,
rb
b=
l
1− R
(10)
Bi f =
> consts:=simplify(solve({bc1=0, bc2=1},{ C1, C2})):
> assign(consts):
> Temp distribution:=simplify(rhs(Eq2)):
> Heat transfer rate:=-subs(X=0,rhs(Eq3)):
> Tip temperature:=subs(X=1,rhs(Eq2)):
hfb
,
k
Bit =
ht b
,
k
m2 = b
4h 2 b2 + rb2
1/2
rb2 k 2
The heat transfer rate, q, from the spine may be expressed in
dimensionless form as
qb
dθ Q= 2
(11)
=
d X X =1
πrb k(T f − T∞ )
> Eq4:=evalf(subs(N=1,Temp distribution));
Eq4 := 0.04318749671 BesselI 0., 2. e(0.5000000000 X )
+ 7.915706052 BesselK 0., 2. e(0.5000000000 X )
> evalf(subs(N=1,Heat transfer rate));
1.038441583
> evalf(subs(N=1,Tip temperature));
0.4644125941
> plot(Eq4,X=0.1,labels = [“X”, “Theta”]):
heat transfer engineering
Figure 4 Truncated conical spine with convection at both ends.
vol. 26 no. 9 2005
A. AZIZ AND G. McFADDEN
34
Maple Solution
The solution procedure follows the steps of the first example
and need not be repeated here. Because the base temperature,
θ (1), is not known, it is found with the results for the temperature distribution, θ(X ), the heat transfer rate Q, and the tip
temperature, θ(R). The Maple worksheet now follows.
> restart;Eq1:=Xˆ2∗ diff(theta(X),X,X)
+ 2∗ X∗ diff(theta(X),X)-m∧ 2∗ X∗ theta(X)=0;
2
d
d
Eq1 := X 2
θ(X
)
+
2X
θ(X
)
d X2
dX
Downloaded by [Southern Illinois University] at 21:47 23 December 2014
− m 2 X θ(X ) = 0
> Eq2:=dsolve(Eq1,theta(X));
√
C1 BesselI(1, 2m X )
Eq2:=θ(X ) =
√
X
√
C2 BesselK(1, 2m X )
+
√
X
> Eq3:=simplify(diff(Eq2, X));
√
d
Eq3 :=
θ(X ) = (− C1 BesselI(1, 2m X )
dX
√
√
+ C1 X BesselI(0, 2 m X )m
√
− C2 BesselK(1, 2 m X )
√
√
− C2 X BesselK(0, 2 m X )m)/ X (3/2)
> bc1:=subs(X=R,rhs(Eq3))-Bi[t]∗ (subs(X=R,rhs(Eq2));
√
bc1 := (− C1 BesselI(1, 2 m R)
√
√
+ C1 R BesselI(0, 2 m R)m
√
− C2 BesselK(1, 2, m R)
√
√
− C2 R BesselK(0, 2m R)m)/R (3/2)
√
C1 BesselI(1, 2, m R)
− Bit
√
R
√ C1 BesselK(1, 2, m R)
+
√
R
Figure 5
Dimensionless temperature distribution in a conical truncated spine.
> Temp distribution:=simplify(rhs(Eq2)):
> Heat transfer rate:=simplify(subs(X=1,rhs(Eq3))):
> Base temperature:=simplify(subs(X=1,rhs(Eq2))):
> Tip temperature:=simplify(subs(X=R,rhs(Eq2))):
As a numerical example, a truncated conical spine is considered
with rb = 0.02 m, rt = 0.01 m, l = 0.05 m, h t = 80 W/m2 K, h f = 150 W/m2 -K, k = 25 W/m-K, and h = 25 W/m2 -K,
and θ(X ), Q, θ(1), and θ(R) are calculated. Figure 5 shows the
temperature distribution.
> rt := 0.01;rb:=0.02;l:=0.05;ht:=80;hf:=150;k
:=25;R:= rt/rb;b:=l/(1-R);h:=25;m
:=2*b*h*sqrt(b∧ 2+(rb)∧ 2)/(rb*k);Bi[t]
:=ht*b/k;Bi[f]:=hf*b/k;
r t := 0.01
r b := 0.02
l := 0.05
ht := 80
h f := 150
∗
> bc2:=subs(X=1,rhs(Eq3))−Bi[f] (1-subs(X=1,rhs(Eq2)));
bc2 := − C1 BesselI(1, 2 m) + C1 BesselI(0, 2 m)m
k := 25
R := 0.5000000000
− C2 BesselK(1, 2 m) − C2 BesselK(0, 2 m)m
b := 0.1000000000
− Bi f (1 − C1 BesselI(1, 2 m)
h := 25
− C2 BesselK(1, 2 m))
m := 1.019803903
> consts:=simplify(solve({bc1=0, bc2=0},{ Cl, C2})):
> assign(consts):
heat transfer engineering
Bit := 0.3200000000
Bi f := 0.6000000000
vol. 26 no. 9 2005
A. AZIZ AND G. McFADDEN
35
where
> Eq4:=Temp distribution;
θ = T − T∞
Eq4 := 0.6812965608(0.5160684716
√
BesselI(1, 2.039607806 X )
m 2 = 2h/kδ.
+ 0.074082603 BesselK
√
√
(1, 2.039607806 X ))/ X
The boundary conditions of specified heat flow out of the fin
base may be expressed as
> Heat transfer rate;
r = rb ,
0.2478113205
> Base temperature;
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> Tip temperature;
r = ra ,
0.4817494179
> plot(Eq4,X=0.5..1,0..1,labels=[“X”,“Theta”]):
A HEAT-GENERATING ANNULAR FIN WITH
CONSTANT BASE HEAT FLUX AND AN ADIABATIC TIP
The heat transfer analysis for a convecting annular or circular fin is usually based on the boundary conditions of constant
base temperature and an adiabatic tip. Yovanovich et al. [10]
modified the standard analysis by imposing convective boundary conditions at both ends of the fin. We make our problem
different by assuming a uniform heat generation rate in the fin
and imposing the boundary conditions of constant base heat flux
and an adiabatic tip.
Mathematical Model
dr
2
(13)
dθ
=0
dr
(14)
Because the base heat flow qb is specified, the quantities of
interest are the temperature excess distribution θ(r ), the base
temperature excess θ(rb ), and the tip temperature excess, θ(ra ).
Maple Solution
The formulation of the problem and its solution using Maple
can be easily understood from the worksheet below.
> restart;Eq1:=diff(theta(r),r,r)+1/r∗ diff(theta(r),r)
−m∧ 2∗ theta(r)+g/k=0;
d
2
θ(r )
d
g
dr
Eq1 :=
− m 2 θ(r ) + = 0
θ(r ) +
2
r
k
dr
> Eq2:=dsolve(Eq1,theta(r));
Figure 6 shows an annular fin of base radius rb , tip radius ra ,
and a uniform thickness δ. The fin loses heat by convection from
its top and bottom surfaces to an environment that is characterized by a heat transfer coefficient h and a temperature T∞ . The
fin is experiencing a volumetric heat generation at the rate of g
(W/m3 ). The heat flow rate out of the fin base is qb (W). There is
no heat loss from the tip of the fin. For a constant thermal conductivity, k, the standard annular fin equation appearing in [1] may
be modified to include the heat generation term g and written as
d θ
dθ
− qb = 0
dr
For no heat loss from the tip, the condition to be met is
0.5869811324
2
−2πrb k
+
1 dθ
g
− m2θ + = 0
r dr
k
(12)
Eq2:=θ(r ) = BesselI(0, m r ) C2
+ BesselK(0, m r ) C1 +
g
m2 k
> Eq3:=diff(Eq2,r);
d
Eq3: =
θ(r ) = BesselI(1, m r ) m C2 − BesselK
dr
×(1, m r ) m C1
> bc1:=-2∗ Pi∗ k∗ rb∗ delta∗ subs(r=rb,rhs(Eq3))-q[b];
bc1 : = −2πkr bδ(BesselI(1, m r b)m C2 − BesselK
(1, m r b)m C1) − qb
> bc2:=subs(r=ra,rhs(Eq3));
bc2 := BesselI(1, m ra)m C2 − BesselK(1, m r a)m C1
Figure 6 Annular fin with internal heat generation.
heat transfer engineering
> consts:=solve({bc1=0,bc2=0}, { C1, C2}:
> assign(consts):
> Temp distribution in the fin:=Eq2:
> Base temperature:=subs(r=rb,rhs(Eq2)):
> Tip temperature:=subs(r=ra,rhs(Eq2)):
vol. 26 no. 9 2005
A. AZIZ AND G. McFADDEN
36
As an example, consider an annular fin with a base radius of
0.06 m, tip radius of 0.12 m, and thickness of 0.002 m. The
base heat flow rate is to be 30 W. The volumetric heat generation rate of 105 W/m3 is to be accommodated. The fin has a
thermal conductivity of 30W/m-K and a convection heat transfer coefficient of 50 W/m2 -K. We need to find the temperature
distribution in the fin, the base temperature excess, and the tip
temperature excess.
Note that the heat generated in the fin is
qgen = gπ ra2 − rb2 δ
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= 105 (π)(0.122 − 0.062 )(0.002)
= 6.8W
and therefore the total heat dissipated by the fin is 30 + 6.8 =
36.8 W. This number can be verified by evaluating the convective
heat loss.
qconv = 4πh
ra
r θ(r )dr
(15)
rb
>q[b]:=30;h:=50;k:=30;delta:=0.002;rb:=0.06;ra:=0.12;g
:=100000;m:=sqrt(2∗ h/(k∗ delta));
qb := 30
h := 50
k := 30
δ := 0.002
r b := 0.06
ra := 0.12
Figure 7 Temperature excess distribution in an annular fin.
THREE-FIN CONVECTING ARRAY
In a finned array such as the one shown in Figure 8, the
analysis requires the solution of a set of ordinary differential
equations with common temperature boundary conditions at the
junction. If the differential equations and the boundary conditions are linear, the solution procedure is conceptually simple
but algebraically tedious. Kraus et al. [11] and Mikhailov and
Ozisik [12] have proposed solution strategies that are based on
a matrix formulation of the problem. Because the tedium of
algebraic manipulations can be delegated to Maple, it offers a
viable alternative to the techniques proposed in [11, 12]. Furthermore, with its built in numerical and graphical capabilities,
Maple can be used to carry out parametric studies and tabulate
and/or graphically display the results.
g := 100000
m := 40.82482905
> temp:=evalf(Temp distribution in the fin);
temp := θ(r ) = 0.08478239988 BesselI(0., 40.82482905 r )
+ 415.4262986 BesselK(0., 40.82482905 r )
+ 2.000000000
> q conv:=evalf(4∗ Pi∗ h∗ int(r∗ rhs(temp),r=0.06..0.12));
q conv:= 36.78584012
> evalf(Base temperature);
29.76948837
> evalf(Tip temperature);
5.823447207
> plot(rhs(temp),r=0.06..0.12,0..30,labels=[“r”, “Theta”]):
Figure 7 shows the temperature distribution in the fin.
heat transfer engineering
Figure 8 A three-fin convecting array.
vol. 26 no. 9 2005
A. AZIZ AND G. McFADDEN
Mathematical Model
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To demonstrate the effectiveness of Maple, a three-fin convecting array of Figure 4 is analyzed. The array consists of
two triangular fins and a rectangular fin. The fin heights are
b1 , b2 , and b3 . The thickness of the rectangular fin is δ, which
is also the base thickness of the two triangular fins. The length
of the fins (normal to the paper) is 1 m. The entire assembly is
made of a material with thermal conductivity k and operates in a
convective environment characterized by a heat transfer coefficient h and a temperature T∞ .
With the temperature excess θ = T − T∞ , the differential
equations for the individual fins may be written as
Fin 1:
x
d 2 θ1 (x) dθ1 (x)
+
− m 21 θ1 (x) = 0
dx2
dx
(16)
Fin 2:
x
d 2 θ2 (x) dθ2 (x)
+
− m 22 θ2 (x) = 0
dx2
dx
(17)
2
Fin 3:
d θ3 (x)
− m 23 θ3 (x) = 0
dx2
(18)
The boundary conditions to be satisfied by Eqs. (16–19) are
x = 0,
θ1 = a finite value
The value of q obtained from Eq. (22) should match qconv , which
is given by
b1
b2
b2
qconv = 2h
θ1 (x)d x +
θ2 (x)d x +
θ2 (x)d x
0
0
0
(23)
Maple Solution
The Maple worksheet for formulating and solving the problem is provided below.
> restart;Eq1:=x∗ diff(theta[1](x),x,x)+diff(theta[1](x),x)-m
[1]∧ 2∗ theta[1](x)=0;
Eq1 := x
d2
d
θ
(x)
+
(x)
− m 21 θ1 (x) = 0
θ
1
1
dx2
dx
> Eq2:=dsolve(Eq1,theta[1](x));
√
Eq2 := θ1 (x) = C1 BesselI(0, 2 m 1 x)
√
+ C2 BesselK(0, 2 m 1 x)
> Eq3:=subs( C2=0,Eq2);
√
Eq3 := θ1 (x) = C1 BesselI(0, 2 m 1 x)
> Eq4:=diff(rhs(Eq3),x);
θ2 = a finite value
(19)
Eq4 :=
θ3 = θb = Tb − T∞
x = b1 , θ 1 = θ j
(20a)
x = b2 , θ2 = θ j
(20b)
x = b3 , θ3 = θ j
(20c)
where θ j is the temperature excess at the junction,
= 2hb1 /
kδ, m 22 = 2hb2 /kδ, and m 23 = 2hb3 /kδ.
To determine the junction temperature excess, θ j , the condition of continuity of heat flux at the junction may be invoked to
give
m 21
dθ1 (x) dθ2 (x) dθ3 (x) +
+
=0
d x x=b1
d x x=b2
d x x=b3
(21)
The rate of heat dissipation by the array is equal to the heat flow
rate through the base of the rectangular fin (fin 3) and is given
by
dθ3 q = −kδL
d x x=0
37
√
C1 BesselI(1m 1 , 2 m 1 x)
√
x
> Eq5:=x∗ diff(theta[2](x),x,x)+diff(theta[2](x),x)-m[2]∧ 2∗
theta[2](x)=0;
2
d
d
Eq5:=x
θ2 (x) +
θ2 (x) − m 22 θ2 (x) = 0
dx2
dx
> Eq6:=dsolve(Eq5,theta[2](x));
√
Eq6:=θ2 (x) = C1 BesselI(0, 2 m 2 x)
√
+ C2 BesselK(0, 2 m 2 x)
> Eq7:=subs( C1= C3, C2= C4,Eq6);
√
Eq7 := θ2 (x) = C3 BesselI(0, 2 m 2 x)
√
+ C4 BesselK(0, 2 m 2 x)
> Eq8 := subs( C4 = 0,Eq7);
√
Eq8 := θ2 (x) = C3BesselI(0, 2 m 2 x)
> Eq9:=diff(rhs(Eq8),x);
(22)
heat transfer engineering
Eq9 :=
√
C3 BesselI(1m 1 , 2 m 2 x)
√
x
vol. 26 no. 9 2005
A. AZIZ AND G. McFADDEN
38
> Eq10:=diff(theta[3](x),x,x)-m[3]∧ 2∗ theta[3](x)=0;
2
d
Eq10 :=
θ3 (x) − m 23 θ3 (x) = 0
dx2
> Eq11:=dsolve(Eq10,theta[3](x));
Eq11 := θ3 (x) = C1e
(m 3 x)
+ C2 e
(−m 3 x)
> Eq12:=subs( C1= C5, C2= C6,Eq11);
Eq12 := θ3 (x) = C5e
(m 3 x)
+ C6 e
(−m 3 x)
Downloaded by [Southern Illinois University] at 21:47 23 December 2014
> Eq13:=diff(rhs(Eq12),x);
Eq13 := C5m 3 e
(m 3 x)
− C6m 3 e
(−m 3 x)
> bc3:=simplify(subs(x=0,rhs(Eq12)))-theta[b];
bc3 := C5 + C6 − θb
Figure 9 Temperature excess distribution in fin 1.
> bc4:=subs(x=b[3],rhs(Eq12))-theta[j];
bc4 := C5 e
(m 3 b3 )
+ C6 e
(−m 3 b3 )
>b[1]:=0.02;b[2]:=0.03;b[3]:=0.04;delta:=0.0025;h:=50;
k:=100;theta[b]:=30;m[1]:=sqrt(2∗ h∗ b[1]/(k∗ delta));m[2]
:=sqrt(2∗ h∗ b[2]/(k∗ delta));m[3]:=sqrt(2∗ h/(k∗ delta));
− θj
> bc5:=subs(x=b[1],rhs(Eq3))-theta[j];
bc5 := C1 BesselI(0, 2 m 1 b1 ) − θ j
b1 := 0.02
> bc6:=subs(x=b[2],rhs(Eq8))-theta[j];
bc6 := C3 BesselI(0, 2 m 2 b2 ) − θ j
b2 := 0.03
b3 := 0.04
> consts:=solve({bc3=0,bc4=0,bc5=0,bc6=0},{ C1, C3, C5,
C6}):
> assign(consts):
> Eq14:=simplify(Eq3):
> Eq15:=simplify(Eq8):
> Eq16:=simplify(convert(Eq12,trig)):
> Eq17:=subs(x=b[1],Eq4)+subs(x=b[2],Eq9)+subs(x=b[3],
Eq13):
> theta[j]:=simplify(convert((solve(Eq17,theta[j])),trig)):
> Temp distribution in fin 1:=simplify(Eq14):
> Temp distribution in fin 2:=simplify(Eq15):
> Temp distribution in fin 3:=simplify(Eq16):
As an example, consider the array with the following data
b1 = 0.02m,
δ = 0.0025m,
b2 = 0.03m,
b3 = 0.04m
h = 50W/m 2 · K ,
k = 100W/m · K
θb = Tb − T∞ = 30◦ C
We calculate the temperature distributions in the fins, the junction temperature excess, the heat transfer rate, the tip temperature
excess, and the convection heat loss. The temperature profiles for
fins 1, 2, and 3 are shown in Figures 9, 10, and 11, respectively.
heat transfer engineering
Figure 10 Temperature excess distribution in fin 2.
vol. 26 no. 9 2005
A. AZIZ AND G. McFADDEN
39
> Heat transfer rate:=evalf(-k∗ delta∗ subs(x=0,Eq13));
Heat transfer rate := 146.3117504
> Tip temperature fin 1:=evalf(subs(x=0,Eq3));
Tip temperature fin 1 := θ1 (0) = 12.11727339
> Tip temperature fin 2:=evalf(subs(x=0,Eq8));
Tip temperature fin 2 := θ2 (0) = 10.14186752
> q conv:=2∗ h∗ (int(rhs(Temp1),x=0..0.02)+int(rhs(Temp2),
x=0..0.03)+int(rhs(Temp3),x=0..0.04));
Downloaded by [Southern Illinois University] at 21:47 23 December 2014
q conv:= 146.3117503
CONCLUDING REMARKS
Figure 11 Temperature excess distribution in fin 3.
Maple 8 has proved to be an effective and convenient tool
for analyzing a wide variety of extended surface heat transfer
problems. The drudgery of lengthy algebraic manipulation is
avoided by delegating that task to Maple. Maple also provides
a convenient platform for solving a specific problem or for conducting parametric studies. Results can be output in the form of
tables and/or graphs.
δ := 0.0025
h := 50
k := 100
θb := 30
NOMENCLATURE
m 1 := 2.828427125
a
AiryAi
AiryBi
b
m 2 := 3.464101615
m 3 := 20.00000000
> Temp1:=Temp distribution in fin 1;
Temp1: = θ1 (x)
= 12.11727338 BesselI(0, 5.656854250
√
x)
> plot(rhs(Temp1),x=0..b[1],10..16,labels=[“x”,“Theta”]):
> Temp2:=Temp distribution in fin 2;
Temp2 := θ2 (x)
√
= 10.14186752 BesselI(0, 6.928203230 x)
> plot(rhs(Temp2),x=0..b[2],10..16,labels=[“x”, “Theta”]):
> Temp3:=Temp distribution in fin 3;
Temp3 := θ3 (x) = −29.26235008 sinh(20.00000000 x)
+29.99999996 cosh(20.00000000 x)
> plot(rhs(Temp3),x=0..b[3],10..30,labels=[“x”, “Theta”]):
> theta[j];
14.13498023
heat transfer engineering
distance from the origin to the spine tip
Airy function of the first kind
Airy function of the second kind
fin height or dimensionless height or distance from the origin to the spine base
BesselI
Bessel function of the first kind
BesselK
Bessel function of the second kind
Bi
Biot number
C1, C2, . . . , C6 constants of integration
g
volumetric rate of heat generation
h
convection heat transfer coefficient
k
thermal conductivity
l
spine height
L
fin length
m
fin or spine parameter
N
dimensionless fin parameter
q
heat flux or heat transfer rate
Q
dimensionless heat transfer rate
r
radius or radial coordinate
R
ratio of radii
T
temperature
x
axial coordinate
X
dimensionless axial coordinate
Greek Symbols
δ
θ
fin thickness
temperature excess or dimensionless temperature
vol. 26 no. 9 2005
A. AZIZ AND G. McFADDEN
40
Subscripts
Downloaded by [Southern Illinois University] at 21:47 23 December 2014
a
b
f
j
o
t
∞
1
2
3
[10] Yovanovich, M. M., Culham, J. R., and Lemczyk, T. F., Simplified
Solutions to Circular Annular Fins with Contact Resistance and
End Cooling, AIAA J. Thermophys. Heat Transfer, vol. 2, pp. 152–
157, 1988.
[11] Kraus, A. D., Snider, A. D., and Dotty, L. F., An Efficient Algorithm for Evaluating Arrays of Extended Surfaces, Journal of
Heat Transfer, vol. 100, pp. 288–293, 1988.
[12] Mikhailov, M. D., and Ozisik, M. N., On the Solution of Heat
Transfer through an Array of Extended Surfaces, International
Journal of Heat and Mass Transfer, vol. 27, no. 6, pp. 893–899,
1984.
fin tip
fin base
base fluid
junction of fins
average
fin tip
ambient
fin 1
fin 2
fin 3
REFERENCES
[1] Kraus, A. D., Aziz, A., and Welty, J., Extended Surface Heat
Transfer, John Wiley, New York, 2001.
[2] Aziz, A., Performance Analysis of a Cascaded RectangularTriangular Fin Using Maple. Proceedings ASME 2005 International Design Engineering Technical Conferences & Computers
and Information in Engineering Conference, September 24–28,
2005, Long Beach, CA, Paper No. DETC 2005-84103.
[3] Kraus, A. D., and Bar-Cohen, A., Design and Analysis of Heat
Sinks, John Wiley, New York, 1995.
[4] Han, L. S., and Lefkowitz, S. G., Constant Cross Section Fin
Efficiencies for Non-Uniform Surface Heat Transfer Coefficients,
ASME Paper 60-WA-41, ASME, New York, 1960.
[5] Chen, S. Y., and Zyskowski, G. L., Steady State Heat Conduction in a Straight Fin with Variable Heat Transfer Coefficient, 6th
National Heat Transfer Conference, Boston, MA, ASME Paper
63-HT-1, ASME, New York, 1963.
[6] Maple Soft, Maple 8 Learning Guide, Waterloo, Canada, 2002.
[7] Abell, M. L., and Braselton, J. P., Differential Equations with
Maple V, Academic Press, Inc., New York, 1994.
[8] Coombes, K. R., Hunt, B. R., Lipsman, R. L., Osborn, J. E.,
and Stuck, G. J., Differential Equations with Maple, John Wiley,
New York, 1996.
[9] Betounes, D., Differential Equations: Theory and Applications
with Maple, Springer-Verlag, New York, 2001.
heat transfer engineering
Abdul Aziz is currently a professor of mechanical engineering at Gonzaga University, Spokane,
Washington. He received his B.E. (mechanical
and electrical) degree from the NED Engineering College, University of Karachi, Pakistan, in
1963, and his Ph.D. in mechanical engineering
from the University of Leeds, England, in 1967.
Previously, he has worked at the University of
Michigan; Imperial College; University of Riyadh,
Saudi Arabia; Babcock & Wilcox Co, Renfrew, Scotland; and Karachi Electric
Supply Corporation, Karachi, Pakistan. He is the author or coauthor of 85 papers and two books, Perturbation Methods in Heat Transfer (Taylor and Francis,
1984) and Extended Surface Heat Transfer (John Wiley, 2001). He has contributed chapters to the Handbook of Numerical Heat Transfer (John Wiley,
1988) and Heat Transfer Handbook (John Wiley, 2002). Aziz is a Fellow of the
ASME. He has won a gold medal for his undergraduate performance (1963),
Gonzaga University’s Distinguished Scholar award (1992), and Alpha Sigma
Nu National Jesuit book award (2001).
Greg McFadden earned a B.S. in mechanical
engineering from Gonzaga University in 2003.
While at Gonzaga, he developed a keen interest
in thermal sciences as a result of advanced course
work and research collaboration with Dr. Aziz.
He is currently a graduate student in the Department of Mechanical Engineering at the University
of Minnesota in Minneapolis, pursuing a MS degree program. His research work is focused on the
study of heat and fluid flow in random-wired regenerators.
vol. 26 no. 9 2005