Exact and approximate solutions of convective-radiative fins with temperature-dependent thermal conductivity using integral equation method

International Journal of Heat and Mass Transfer 150 (2020) 119303

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Exact and approximate solutions of convective-radiative fins with temperature-dependent thermal conductivity using integral equation method Yong Huang a, Xian-Fang Li b,∗ a b

School of Mathematics and Big Data, Foshan University, Foshan, Guangdong 528000, China School of Civil Engineering, Central South University, Changsha 410075, China

a r t i c l e

i n f o

Article history: Received 19 October 2019 Revised 20 December 2019 Accepted 31 December 2019

Keywords: Convective-radiative fin Exact solution Explicit temperature distribution Temperature-dependent thermal conductivity Fin efficiency Nonlinear fin problem

a b s t r a c t The nonlinear thermal performance of convective and radiative longitudinal cooling fins with temperature-dependent thermal conductivity is studied. For linearly temperature-dependent thermal conductivity, the exact temperature distribution is obtained analytically in an implicit integral form. By converting the resulting integral equation to an algebraic equation, a simpler explicit expression of quadratic polynomials for the temperature excess in the whole convective-radiative fins is derived and the numerical results are compared with the exact one and the previous ones. Additionally, analytical expressions for the temperature change at the fin tip and for the fin efficiency are respectively given in terms of the thermal and geometric parameters of extended surfaces. The accuracy of approximate solution is examined. The influences of nonlinearity and hybrid Biot number on the temperature distribution, the fin-tip temperature change, and the fin efficiency are analyzed.

1. Introduction There are two basic approaches to enhance the efficiency of heat transfer. One is to select preferable materials or change the material properties like heat transfer coefficients, and the other is to optimize the structures’ geometric shapes. As a convenient and economic mean, fins or extended surfaces are frequently used to design various shapes to enhance the requirements of highperformance heat transfer equipment [1,2], especially in thermal engineering applications, and increasing the heat transfer coefficient is not easy. These heat transfer equipments have wide applications in cooling systems for computer hardware, air conditioning, refrigeration, chemical processes, energy systems equipments, and heat exchangers. In a variety of fin profiles, longitudinal fins have been widely applied due to simplicity of their design and their easy manufacturing process. A great number of theoretical analyses on thermal performance of longitudinal fins have been made. For large temperature change in fins, the thermal conductivity does no longer remain a constant, but depends on the local temperature, which leads to a nonlinear fin problem. Another source of nonlinearity

∗

Corresponding author. E-mail address: xfl[email protected] (X.-F. Li).

https://doi.org/10.1016/j.ijheatmasstransfer.2019.119303 0017-9310/© 2020 Elsevier Ltd. All rights reserved.

© 2020 Elsevier Ltd. All rights reserved.

arises from radiation. For example, the measured data in an experiment show that from the temperature distribution along a fin cooled by natural convection and radiation, the heat loss due to radiation arrives at typically 15–20 percent of the total [3]. So, radiation heat transfer also strongly affects the performance of heat exchangers, especially at high temperatures. Thermal radiation and natural convection exhibit strong effects on nanofluid in annulus [4,5]. By considering Cattaneo–Christov heat flux model, Dogonchi et al. analyzed radiative nanofluid and heat transfer between parallel disks with penetrable and stretchable walls [6]. Thus, similar to convection and conduction, radiation has a significant effect on the temperature distribution and plays a critical role in improving the thermal performance of fins, in particular for the devices with low convection heat transfer coefficient. In addition to conduction, convection and radiation must be taken into account simultaneously in assessing the thermal performances of conductive, convective and radiative fins. To date, great progress on the heat transfer characteristics of conductive, convective, and radiative fins has been made. For longitudinal fins with uniform cross-section including circular and rectangular cross-section, the fins can be understood as sufficiently long so that the temperature is assumed to be only dependent on the axial location. Thus, the heat transfer is related to one-dimensional nonlinear problems, where the thermal conductivity and heat transfer coefficient are possibly dependent

2

Y. Huang and X.-F. Li / International Journal of Heat and Mass Transfer 150 (2020) 119303

Nomenclature Fin cross-sectional area (m2 ) Convection heat transfer coefficient (at the base) (W/m2 K) k Temperature-dependent thermal conductivity (W/mK) ka Thermal conductivity at temperature of a surrounding fluid (W/mK) L Axial length (m) M Dimensionless thermogeometric parameter m Power exponent of variable thermal conductivity n Power exponent of variable heat transfer coefficient P Perimeter of cross-section (m) Qactual (Qideal ) Actual (ideal) heat losses (W) Qc Convection heat losses (W) Qr Radiation heat losses from (W)i Ta Temperature of a surrounding fluid (K) Tb Temperature at fin’s base (K) Ts Sink temperature (K) X (x) (Dimensionless) Axial distance measured from the fin’s tip A h (hb )

Greek symbols α A measure of dimensionless thermal conductivity δ Dimensionless constant of thermal conductivity ε surface emissivity λ A measure of thermal conductivity variation with temperature η Fin efficiency σ Stefan–Boltzmann constant (W/m2 K4 ) θ (θ 0 ) Dimensionless temperature excess (at the fin tip) on the temperature. A large number of researchers have investigated power-law temperature-dependent heat transfer coefficients. Various analytic methods have been put forward to determine the temperature distribution in the longitudinal fins with powerlaw heat transfer coefficients, and they include the hypergeometric functions or elliptic integrals [7–9], the perturbation techniques [10–12], the decomposition method [13,14], the series method [15,16], the differential transformation method [17–20], et al. Although the exact solution can be achieved by the above-mentioned various methods, most of the solutions are expressed in terms of an infinite series. As a consequence, the practical use of the above approaches is still inconvenient. In addition, Bouaziz and Aziz [21] used a novel double optimal linearization to obtain a simple and accurate solution for the temperature distribution in a straight rectangular convective–radiative fin with temperaturedependent thermal conductivity. For a straight fin with both the thermal conductivity and the convection heat transfer coefficient being power-law temperature-dependent, Sun and Li derived an exact solution in an implicit integral form [22]. Abbasbandy and Shivanian gave the exact solution of a natural convection porous fin with temperature-dependent thermal conductivity and internal heat generation [23]. Although a large number of papers have been reported to seek the exact solution for conductive, convective, and radiative fins with temperature-dependent thermal conductivity, most of the obtained results are in implicit form, not explicit form, except some extreme cases. There is little information on an explicit expression for estimating the temperature change in the whole fin together with the fin efficiency. This paper investigates heat transfer characteristics of a straight conductive, convective, and radiative fin with temperature-

Fig. 1. Schematic of a longitudinal convective-radiative fin with an insulated tip.

dependent thermal conductivity. It has two-fold aims. One is to present an exact solution of a nonlinear problem for 1D longitudinal convective-radiative fins, and the other is to derive an explicit expression for the approximate temperature distribution and the fin efficiency in simple form with high accuracy. The validity and effectiveness of the approximate solution are examined by comparing them with the exact one and those from other approaches. The influences of the temperature-dependent thermal conduction, convection, and radiation on the temperature distribution and the fin efficiency are displayed graphically. 2. Statement of the problem Consider a heat transfer problem of a 1D convective-radiative straight fin of rectangular profile with cross-sectional area A, length L, and perimeter P, as shown in Fig. 1. The fin is bonded to a base surface at fixed temperature Tb and extends into a surrounding fluid at temperature Ta . At the fin surface, heat loss occurs due to convection and radiation, while at the fin end, it is assumed to be insulated since heat transfer through the tip end is relatively small and is negligible. If denoting the sink temperature for convention and radiation as Ta and Ts , respectively, for the problem mentioned above we can write the one-dimensional governing ordinary differential equation as follows:

d dX



kA

dT dX







− hP (T − Ta ) − εσ P T 4 − Ts4 = 0, 0 < X < L

(1)

where k denotes thermal conductivity, h heat convection transfer coefficient, ε surface emissivity, and σ Stefan–Boltzmann constant, and X is the distance measured from the fin tip. In the above, the following assumptions have been adopted: the fin is homogeneous and isotropic solid material, 1D steady heat conduction model in the fin is used, there are no heat sources or sinks in the fin, the temperature at the fin base and in the convection environment is uniform, heat transfer from the fin tip is neglected, heat dissipated from the fin surface radiation obeys the Stefan–Boltzmann law, the fin is stationary in the environment fluid, convection heat transfer coefficient between the fin and the environment is constant and uniform over the entire surface of the fin. For simplicity, here the

Y. Huang and X.-F. Li / International Journal of Heat and Mass Transfer 150 (2020) 119303

thermal conductivity k is linearly dependent on the local temperature change at any position, namely

k = ka [1 + λ(T − Ta )],

(2)

ka and λ being the thermal conductivity when the temperature takes Ta and a parameter, respectively, and the other material properties such as h and ε are assumed to be constants and independent of the local temperature change. At the base and the fin tip, appropriate boundary conditions can be stated below

dT = 0, at X = 0, dX

(3)

T = Tb , at X = L.

(4)

For practical problems, the thermal conductivity is positive, which implies that the parameter λ must meet necessary conditions. For convenience of later analysis, let us introduce the following dimensionless parameters

θ=

T − Ta X k (T ) , x = , k (θ ) = , Tb − Ta L ka



α = λ(Tb − Ta ), M = L

hP , ka A

εr =

(5)

εσ PL2 Tb3 ka A

.

(6)

Here θ is dimensionless temperature excess, M2 is called the hybrid Biot number, meaning the ratio of internal resistance L/ka A in the x-direction to gross external resistance 1/hPL [24]. In this study, the surroundings are assumed to be completely absorbing, i.e., black and at the same temperature as the ambient fluid, i.e., Ts = Ta . Moreover, similar to the treatment in [21], Ts = Ta = 0 is adopted. Using the above-introduced these dimensionless parameters, the 1D nonlinear heat transfer balance Eq. (1) may be rewritten as



d dθ (1 + αθ ) dx dx



− M 2 θ − εr θ 4 = 0, 0 < x < 1

(7)

The boundary conditions (3) and (4) can be also rewritten in the dimensionless form:

θ  ( 0 ) = 0,

θ ( 1 ) = 1,

(8)

where the prime denotes differentiation with respect to the argument,

3

minimum at the fin tip x = 0 and its maximum at the base x = 1. Thus the temperature θ monotonically increases from 0 to 1, or dθ /dx > 0. Consequently, we rewrite Eq. (9) as follows:



1 + αθ M2

θ + α 2 3

2

M2

θ 3 + 25 εr θ 5 + 13 αεr θ 6 + C

dθ = dx.

(11)

Performing integration, one easily acquires

θ θ0



M2 θ 2 +

2 3

1 + αθ dθ = x. α M2 θ 3 + 25 εr θ 5 + 13 αεr θ 6 + C

(12)

Therefore, the constant C can be determined by setting x = 1 in the above result, i.e.,



1

θ0



M2 θ 2 +

2 3

1 + αθ d θ = 1. α M2 θ 3 + 25 εr θ 5 + 13 αεr θ 6 + C

(13)

Note that in Eq. (13) θ is, in fact, a dummy variable, and substituting (10) into (13) yields an equation, which contains a unique unknown constant θ 0 , only if other parameters M, α , ε r are given beforehand. Once the constant θ 0 and then C are determined, we plug C into Eq. (12) and obtain an integral equation for θ , which can be used to exactly determine the desired temperature distribution. As a check, by setting α = εr = 0, from (13) one has



1

θ0



1

θ 2 − θ02

dθ = M,

(14)

which gives

ln

1+



1 − θ02

θ0

= M.

(15)

By solving the above equation, the unknown constant C is obtained to be θ0 = 1/ cosh M. An exact solution

cosh Mx cosh M

θ=

(16)

is then recovered from the above. For other cases, an explicit expression for the temperature change seems to be difficult. Instead, numerical results of the temperature change can be calculated by solving Eq. (12) under the condition (13).

3. Exact solution The determination of exact solutions is still much desired since they are useful not only to design engineers but also to researchers as benchmark solutions for checking the convergence, validity, and accuracy of numerical methods. Here, we make an effort to derive an exact solution of Eq. (7) subject to the boundary conditions in (8). To this end, after multiplying both sides of the above differential Eq. (7) by 2(1 + αθ )dθ /dx, from (7) we then integrate both sides to get



dθ dx

2

2 3

2 5

1 3

(1 + αθ )2 = M2 θ 2 + α M2 θ 3 + εr θ 5 + αεr θ 6 + C, (9)

where C is an integration constant, which can be determined by taking x = 0 in (9). Thus, using the first condition in (8), one finds



C = − M2 θ02 +

2 2 1 α M2 θ03 + εr θ05 + αεr θ06 3 5 3

(10)

where θ 0 specifies θ (0). Next, for a cooling fin problem of under consideration, with reference to the coordinate system, the temperature θ arrives at its

4. Approximate solution In the previous section, although we have derived an integral equation for exactly determining the temperature change, its solution is not provided explicitly. It is still inconvenient in practice due to lack of an explicit dependence of the temperature on the model parameters. Therefore, it is much desired to obtain an approximate simple expression for the temperature change which provides good insights into the significance of various system parameters and saves the computation costs and time. In the following, here we still utilize an integral equation approach to derive an approximate solution in an explicit form. In order to solve the above-stated problem more conveniently, we rewrite Eq. (7) as

d2 dx2



θ+

α 2

θ 2 − M 2 θ − εr θ 4 = 0.

(17)

Then we integrate both sides of the above-resulting ordinary differential equation twice and have

d dx



θ+

α 2

x x

θ 2 − M2 θ (s )ds − εr θ 4 (s )ds = C1 0

0

(18)

4

Y. Huang and X.-F. Li / International Journal of Heat and Mass Transfer 150 (2020) 119303

and

α

θ (x ) + −εr

2



x 0

θ (x ) − M 2



x

2 0



(x − s )θ 4 (s )ds = C1 x + C2

×

(19)

C1 = 0.

(20)

1+

2

− M2

1 0

(1 − s )θ (s )ds − εr



1 0

(1 − s )θ 4 (s )ds = C2 .

(21)

Next we insert C2 in (21) into the integral Eq. (19) to lead to a nonlinear Fredholm integral equation as follows:

θ (x ) + +εr

α 2



1 0

where

R(x, s ) =

θ (x ) + M 2

2

1

0

R(x, s )θ (s )ds

R(x, s )θ 4 (s )ds = 1 +

1 − x, 1 − s,

α 2

.

(22)

(23)

θ ( x ) = c1 + c2 x ,

(24)



1 + α + 0.4 M 2 + 4εr

M 2 + εr

 ×



2 1 + α + 0.4 M 2 + 4εr



2 1 c1 c2 + c22 + M2 3 5 6 2 2 4 c c + c1 c23 + 35 1 2 63

1

c1 +

1 c2 15



3

1 4 α c2 = 1 + . 99 2

c1 + c2 = 1 .

(25)

(26)

Eliminating another unknown constant c1 from (25) and (26) one gets



M2 + εr − 2 1 + α + 0.4M2 + 1.6εr c2 + 0.8(α + 5. 14εr )c22 −2. 44εr c23 + 0.55εr c24 = 0.

(29)





1+





1 + α + 0.4 M 2 + 4εr

 

2



1 − x2 .

(30)

Thus, we have obtained a simple expression for estimating the temperature distribution and such a simple approximation seems not to be obtained before. 5. Fin efficiency The fin efficiency is the ratio of the actual heat transferred from the fin surface to the surrounding fluid to the amount of heat to be transferred if the entire fin area is at the base temperature. In fact, the actual heat losses arise from convection and radiation, i.e.,

(31)

where

Qc =

L 0

hP (T − Ta )dX, Qr =



L 0

  εσ P T 4 − Ts4 dX.

(32)

The ideal heat loss is that when the entire fin surface is kept at the base temperature, i.e.,





Qideal = hP L(Tb − Ta ) + εσ P L Tb4 − Ts4 .

η

Q = actual = Qideal

η=

L 0

  εσ P T 4 − Ts4 dX   . hP L(Tb − Ta ) + εσ P L Tb4 − Ts4

hP (T − Ta )dX +

(33)

L 0

(34)

(27)

(28)

Although the root of Eq. (27) can be exactly obtained either by commercial software or by the analytic formula for seeking the roots of a quartic equation. It leads to the calculated results is accurate, but quite complicated. To avoid relatively cumbersome form of the root of Eq. (27), we may approximately obtain a suitable root of Eq. (27) in a relatively simple form

M2

1 0

θ (x )dx + εr M 2 + εr

1 0

θ 4 (x )dx

=

1+α  θ (1 ) M 2 + εr

(35)

Therefore, it is seen that η is always less than unity for θ (x) > 0 or 1 + α > 0 [25]. If using the exact solution, θ  (1) can be evaluated by (9) together with (10), namely

(1 + α )θ  (1 )   =

This is a quartic equation in c2 . Although a formula for the exact solution to (27) is available, we do not go ahead along this approach. Instead, we straightforward apply commercial software to solve a suitable root. Once the desired root is determined, the temperature distribution is given by

  θ ( x ) = 1 − c2 1 − x2 .

2 .

In dimensionless form, with the assumption Ta = Ts = 0, the fin efficiency is expressed as

On the other hand, the boundary condition θ (1 ) = 1 gives





0.2(α + 5. 14εr ) M2 + εr

The fin efficiency is evaluated by

where c1 and c2 are unknown constants to be determined. Upon substitution of (24) into Eq. (22), we then integrate both sides over the interval [0, 1], yielding





θ (x ) = 1 − 



2

1 α 2 c2 + c + 3 2 1 1 4 3 +εr c4 + c c2 + 3 1 15 1



0.2(α + 5. 14εr ) M2 + εr

Qactual = Qc + Qr ,

sx

In order to seek an approximate solution of the resulting nonlinear Fredholm integral Eq. (22), for simplicity we express the unknown function as a polynomial. Since θ  (0 ) = 0 must be satisfied, the simplest polynomial is as follows:

c1 +



As a result, one gets the following approximation of the temperature distribution in the whole fin

Furthermore, if setting x = 1 in Eq. (19) and remembering the second condition in (8), we have

α



2 1 + α + 0.4 M 2 + 4εr

(x − s )θ (s )ds

where C1 and C2 are integration constants to be determined. Setting x = 0 in Eq. (18) and recalling the first condition in (8) one gets

1+

M 2 + εr



c2 =



M2 1 − θ02 +

  2   1   2 α M2 1 − θ03 + εr 1 − θ05 + αεr 1 − θ06 , (36) 3 5 3

which gives the fin efficiency as

η=

1 M 2 + εr



×





M2 1 − θ02 +

  2   1   2 α M2 1 − θ03 + εr 1 − θ05 + αεr 1 − θ06 . 3 5 3 (37)

On the other hand, according to the approximate solution (28), we have



1

0

0

1

2 3

θ (x )dx = 1 − c2 8 3

θ 4 (x )dx = 1 − c2 +

(38) 16 2 64 3 128 4 c − c + c 5 2 35 2 315 2

(39)

Y. Huang and X.-F. Li / International Journal of Heat and Mass Transfer 150 (2020) 119303

5

Table 1 Comparison of the present and previous results for θ (x)with α = 0.2, εr = 0.8. M=0

M = 0.5

M=1

M = 1.5

x

Eq. (12)

Eq. (30)

Eq. (12)

Eq. (30)

[16]

Eq. (12)

Eq. (30)

[16]

Eq. (12)

Eq. (30)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.8221 0.8236 0.8284 0.8363 0.8476 0.8625 0.8810 0.9036 0.9306 0.9625 1.0000

0.8206 0.8224 0.8278 0.8368 0.8493 0.8655 0.8852 0.9085 0.9354 0.9659 1.0000 0.55%

0.7692 0.7713 0.7774 0.7878 0.8025 0.8218 0.8459 0.8752 0.9102 0.9515 1.0000

0.7688 0.7711 0.7781 0.7896 0.8058 0.8266 0.8521 0.8821 0.9168 0.9561 1.0000 0.79%

0.7692 0.7712 0.7774 0.7878 0.8025 0.8218 0.8459 0.8752 0.9102 0.9515 1.0000

0.6256 0.6289 0.6389 0.6558 0.6797 0.7109 0.7500 0.7975 0.8542 0.9212 1.0000

0.6290 0.6327 0.6439 0.6624 0.6884 0.7218 0.7626 0.8108 0.8664 0.9295 1.0000 1.68%

0.6256 0.6289 0.6389 0.6558 0.6797 0.7110 0.7500 0.7975 0.8542 0.9211 1.0000

0.4449 0.4497 0.4640 0.4882 0.5226 0.5680 0.6251 0.6952 0.7796 0.8803 1.0000

0.4449 0.4505 0.4671 0.4949 0.5337 0.5837 0.6448 0.7169 0.8002 0.8945 1.0000 3.14%

δ

Table 2 Temperature distribution θ (x) with α = 0.5, M = 1.

εr = 0

εr = 0.5

εr = 1

Eq. (12)

Eq. (30)

Eq. (12)

Eq. (30)

Eq. (12)

Eq. (30)

Eq. (12)

Eq. (30)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.7297 0.7324 0.7404 0.7538 0.7725 0.7967 0.8263 0.8614 0.9020 0.9482 1.0000

0.7296 0.7323 0.7404 0.7539 0.7728 0.7972 0.8269 0.8621 0.9026 0.9486 1.0000 0.08%

0.6866 0.6895 0.6985 0.7134 0.7344 0.7617 0.7954 0.8357 0.8829 0.9375 1.0000

0.6871 0.6903 0.6996 0.7153 0.7372 0.7653 0.7998 0.8404 0.8874 0.9406 1.0000 0.57%

0.6547 0.6578 0.6674 0.6833 0.7058 0.7352 0.7717 0.8157 0.8679 0.9289 1.0000

0.6617 0.6651 0.6752 0.6921 0.7158 0.7463 0.7835 0.8275 0.8782 0.9357 1.0000 1.53%

0.6294 0.6327 0.6426 0.6593 0.6829 0.7137 0.7523 0.7992 0.8552 0.9216 1.0000

0.6448 0.6483 0.6590 0.6767 0.7016 0.7336 0.7726 0.8188 0.8721 0.9325 1.0000 2.78%

δ

which are substituted into Eq. (35) to lead to the fin efficiency in the following form



η





2 2 2 2 c2 M + 4 εr 16 c2 εr 63 − 36c2 + 8c2 =1− + 3 315 M 2 + εr M 2 + εr



(40)

where c2 is given by (29). 6. Results and discussion In the foregoing two sections, we have obtained an exact solution and an approximate solution. To confirm the effectiveness of the method, let us carry out numerical computations for a straight convective-radiative fin and compare our numerical results with the previous ones. First, for the case of constant thermal conductivity, Sobamowo [26] applied the Galerkin method to give an approximate solution of a convective fin without radiation as

θ (x ) = 1 − 

5M 2

2 5+

2M 2

εr = 1.5

x

   1 − x2 .

(41)

For comparison and suitability, we do not look for the exact solution to the quartic Eq. (27). Instead, we employ the simple expression (30) for the temperature to avoid seeking an exact root of the quartic Eq. (27). Although a more accurate c can be determined by taking it as the exact root of (27) in (28), in what follows we prefer to select the simple expression (30) for the temperature. Note that the result (30) does not contain other parameters, and all the parameters involved are prescribed. Clearly, in the case of α = εr = 0, our approximation result (30) collapses to the one (41). Furthermore, in order to validate the derived solution and examine their accuracy, we give a comparison of our numerical results with those obtained previously through other approaches such as the differential transformation method in Table 1, where

we have assumed the parameters α = 0.2, εr = 0.8. For convenience, we denote the dimensionless temperature distribution as θ ex (x) and θ ap (x), respectively, when they are evaluated through solving Eqs. (12) and (30). From Table 1, we find that our results are in satisfactory agreement with those in [16]. In particular, we also present partial results for the hybrid Biot number M2 = 2.25. No matter whether the parameter is larger or less than unity, Eq. (30) provides a relatively accurate approximation of the temperature distribution over the whole fin length. For other various cases, we also calculated the temperature distribution in Tables 2 and 3. By comparing the results, we find that for all the cases under consideration, the maximum relative errors of the temperature change occur at the fin tip, where the relative error is defined by

δ = max

x∈[0,1]

|θap (x ) − θex (x )| % θex (x )

(42)

Moreover, the maximum relative errors are less than 1.5% for all the parameters in a range between 0 and 1, which indicates that the approximate solution (30) gives very high accuracy. With the parameters M and ε r increasing, their errors progressively become somewhat large. However, higher accuracy is achieved for larger α values. So for those parameters M and ε r greater than unity, it is better to employ Eq. (12) to determine the exact results, rather than using the approximate Eq. (30). It is emphasized that Eq. (30) provides a quite satisfactory approximation for lower values of M and ε r and is helpful for engineering applications since the maximum relative error does not excess 1.7%. Although the temperature distribution in a longitudinal convective-radiative fin can be determined as above, of much interest are some particular temperature change and temperature gradient such as the temperature change at the fin tip, θ 0 , and the temperature gradient at the fin base, θ  (1). The former reflects the

6

Y. Huang and X.-F. Li / International Journal of Heat and Mass Transfer 150 (2020) 119303 Table 3 Temperature distribution θ (x) with M = 0.5, εr = 0.8.

α=0

α = 0.5

α=1

α = 1.5

x

Eq. (12)

Eq. (30)

Eq. (12)

Eq. (30)

Eq. (12)

Eq. (30)

Eq. (12)

Eq. (30)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.7465 0.7487 0.7553 0.7664 0.7821 0.8028 0.8289 0.8608 0.8992 0.9452 1.0000

0.7458 0.7483 0.7560 0.7687 0.7865 0.8093 0.8373 0.8703 0.9085 0.9517 1.0000 1.11%

0.7966 0.7984 0.8040 0.8135 0.8268 0.8441 0.8656 0.8917 0.9224 0.9584 1.0000

0.7964 0.7985 0.8046 0.8147 0.8290 0.8473 0.8697 0.8962 0.9267 0.9613 1.0000 0.51%

0.8301 0.8317 0.8365 0.8446 0.8560 0.8708 0.8890 0.9108 0.9365 0.9661 1.0000

0.8282 0.8318 0.8369 0.8454 0.8573 0.8726 0.8913 0.9133 0.9388 0.9677 1.0000 0.27%

0.8541 0.8555 0.8597 0.8668 0.8767 0.8895 0.9052 0.9241 0.9460 0.9713 1.0000

0.8529 0.8556 0.8599 0.8672 0.8774 0.8906 0.9066 0.9256 0.9475 0.9723 1.0000 0.17%

δ

Fig. 2. Variation of the fin tip temperature change θ 0 against M for a) εr = 0, b) εr = 0.8.

Fig. 3. Variation of the fin tip temperature change θ 0 against ε r for a) M = 0, b) M = 0.5.

temperature excess at the fin tip, and the heat dissipated through the cooling fin is more if its value at the fin tip becomes less. So for cooling fins, it is natural to choose a fin having a lower value θ 0 . On the other hand, the latter θ  (1) describes the slope of the temperature change or the temperature gradient and is proportional to the fin efficiency η as seen from Eq. (35). Thus, opposite to the trend of θ 0 , in this case an increase of the fin ef-

ficiency η is achieved by requiring θ  (1) to become larger. Based on the above basic observation, in the following, we demonstrate the variation of the fin tip temperature change θ 0 and the fin efficiency η as a function of α , M, and ε r . Similar to the above, we give two approaches to obtain the temperature change at the fin tip, one corresponding to the exact, denoted by solid lines in the following figures, by solving Eq. (13) after substitution of (10) for C

Y. Huang and X.-F. Li / International Journal of Heat and Mass Transfer 150 (2020) 119303

7

Fig. 4. Variation of the fin efficiency η against M for a) εr = 0, b) εr = 0.8.

Fig. 5. Variation of the fin efficiency η against ε r for a) M = 0, b) M = 0.5.

and the other to the approximate, denoted by scattered plus symbols in the following figures, which can be simply expressed in an explicit form as

M 2 + εr

θ0 = 1 −   ×



2 1 + α + 0.4 M 2 + 4εr

1+





0.2(α + 5. 14εr ) M2 + εr





1 + α + 0.4 M 2 + 4εr



2

(43)

Figs. 2 and 3 show the variation of the fin tip temperature change θ 0 against M and ε r for different values of α , respectively. From Figs. 2 and 3, we find that the fin tip temperature change θ 0 decreases with the parameters M and ε r rising. Moreover, θ 0 increases as α is raised. These trends are in agreement with the previous results. The expression (43) for the fin tip temperature change θ 0 provides a quite satisfactory approximation, in particular for M and ε r lower than unity. With M or ε r rising to greater than unity, the error between the exact and approximate results progressively becomes a little larger. Figs. 4 and 5 display the variation of the fin efficiency η against M and ε r for different values of α , respectively. In Figs. 4 and 5, the exact fin efficiency η is evaluated by (37) where θ 0 is determined

by solving Eq. (13) and the corresponding results are demonstrated by solid lines, while the approximate fin efficiency η is evaluated by (40) with substitution of (29) and the obtained results by scattered plus symbols. Since the latter provides a simple expression, the dependence of η on all the parameters is very clear. From Figs. 4 and 5, the expression (40) can achieve a good approximation of the exact ones for smaller M and ε r . As observed before, with M or ε r rising, the effectiveness of the approximation is deteriorated. 7. Conclusions In this paper, the temperature distribution and the fin efficiency for a nonlinear longitudinal convective-radiative fin problem were analyzed. The nonlinearity contains temperature-dependent thermal conductivity and heat radiation. In the present study, we have given two approaches for obtaining a solution. One can be used to determine the exact solutions in an integral implicit form, and the other is to provide an explicit polynomial of degree two for the temperature distribution as an approximate solution, which has high accuracy. Especially, the latter gives a simple analytical expression in terms of the parameters involved. The influences of all

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Y. Huang and X.-F. Li / International Journal of Heat and Mass Transfer 150 (2020) 119303

the parameters on the temperature distribution, the fin tip temperature change, and the fin efficiency were presented graphically and discussed. Declaration of Competing Interest The authors declare that they have no conflict of interest. Acknowledgment This work was supported by the National Natural Science Foundation of China (Grant No. 11872379). Supplementary material Supplementary material associated with this article can be found, in the online version, at 10.1016/j.ijheatmasstransfer.2019. 119303 References [1] A. Aziz, Optimum dimensions of extended surfaces operating in a convective evironment, Appl. Mech. Rev. 45 (5) (1992) 155–173. [2] A.D. Kraus, A. Aziz, J. Welty, Extended Surface Heat Transfer, John Wiley, New York, 2002. [3] J. Mueller, D. W., H.I. Abu-Mulaweh, Prediction of the temperature in a fin cooled by natural convection and radiation, Appl. Therm. Eng. 26 (14–15) (2006) 1662–1668. [4] A.S. Dogonchi, Hashim, Heat transfer by natural convection of Fe3 O4 -water nanofluid in an annulus between a wavy circular cylinder and a rhombus, Int. J. Heat Mass Transf. 130 (2019) 320–332. [5] A.S. Dogonchi, M. Waqas, S.M. Seyyedi, M. Hashemi-Tilehnoee, D.D. Ganji, CVFEM analysis for Fe3 O4 -H2 O nanofluid in an annulus subject to thermal radiation, Int. J. Heat Mass Transf. 132 (2019) 473–483. [6] A.S. Dogonchi, A.J. Chamkha, S.M. Seyyedi, D.D. Ganji, Radiative nanofluid flow and heat transfer between parallel disks with penetrable and stretchable walls considering cattaneo-christov heat flux model, Heat Transf. Asian Res. 47 (5) (2018) 735–753. [7] A.K. Sen, S. Trinh, An exact solution for the rate of heat transfer from a rectangular fin governed by power law-type temperature dependence, J. Heat Transf. 108 (2) (1986) 457–459. [8] M. Anbarloei, E. Shivanian, Exact closed-form solution of the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient, J. Heat Transf. 138 (11) (2016) 114501–1145016. [9] S. Abbasbandy, E. Shivanian, Exact closed form solutions to nonlinear model of heat transfer in a straight fin, Int. J. Therm. Sci. 116 (2017) 45–51.

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