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This article was downloaded by: [Southern Illinois University] On: 23 December 2014, At: 21:47 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Heat Transfer Engineering Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/uhte20 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 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. 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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; Downloaded by [Southern Illinois University] at 21:47 23 December 2014 > 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 Downloaded by [Southern Illinois University] at 21:47 23 December 2014 = 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 Downloaded by [Southern Illinois University] at 21:47 23 December 2014 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