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Radiation Heat Transfer
Chapter 3
Radiation Heat Transfer Chapter Outline 3.1 Heat Transfer by Thermal Radiation 3.2 Nonlinear Heat Transfer Problems with Thermal Radiation 3.2.1 Cooling of a Lumped System by Combined Convection and Radiation 3.3 Results and Discussion 3.3.1 ConvectiveRadiating Cooling of a Lumped System with Variable Specific Heat 3.3.2 Temperature Distribution in a Radiating Thick Rectangular Plate 3.3.3 Temperature Distribution in a Radiating Thick Rectangular Plate 3.4 Conclusion References Further Reading
106 109
110 116
118 133 136 149 150 151
ABSTRACT Heat transfer from a body with a high temperature to a body with lower temperature, when bodies are not in direct physical contact with each other or Nonlinear Systems in Heat Transfer. DOI: http://dx.doi.org/10.1016/B978-0-12-812024-8.00003-5 © 2018 Elsevier Inc. All rights reserved.
105
106
Nonlinear Systems in Heat Transfer
when they are separated in space, is called heat radiation. Unlike conduction and convection, heat transfer by thermal radiation does not necessarily need a material medium for the energy transfer. In the case of thermal radiation from a solid surface, the medium through which the radiation passes could be a vacuum, gas, or liquid. Within most practical engineering problems, usually all three heat transfer mechanisms, namely conduction, convection, and radiation, occur simultaneously. In this chapter, we will study some practical nonlinear heat transfer problems, in which radiation is of importance and acts concurrently with other heat transfer mechanisms. Differential equations, governing the physics of the problems will be introduced and finally solved via analytical approaches. Keywords: Heat transfer; thermal radiation; StefanBoltzmann constant; emissivity; perturbation method; homotopy perturbation method, variational iteration method
3.1 HEAT TRANSFER BY THERMAL RADIATION Heat transfer from a body with a high temperature to a body with a lower temperature, when bodies are not in direct physical contact with each other or when they are separated in space, is called heat radiation [1], as schematically shown in Fig. 3.1. All physical substances in solid, liquid, or
Radiation Heat Transfer Chapter | 3
107
FIGURE 3.1 Heat transfer by thermal radiation between two bodies.
gaseous states can emit energy via a process of electromagnetic radiation because of vibrational and rotational movement of their molecules and atoms [2]. The intensity of such energy flux depends upon the temperature of the body and the nature of its surface [3]. The radiation occurs at all temperatures, with the rate of emission increasing with the temperature. Unlike conduction and convection, heat transfer by thermal radiation does not necessarily need a material medium for the energy transfer. In the case of thermal radiation from a solid surface, the medium through which the radiation passes could be vacuum, gas, or liquid. Molecules and atoms of the medium can absorb, reflect, or transmit the radiation energy. If the medium is a vacuum, since there are no molecules or atoms, the radiation energy is not attenuated and, therefore, fully transmitted. Therefore, radiation heat transfer is more efficient in
108
Nonlinear Systems in Heat Transfer
a vacuum. In the case of a gas (e.g., air), energy can be slightly absorbed or reflected by air molecules and the balance is transmitted. For liquid medium, most of the radiation is absorbed is a thin layer close to the solid surface and nothing is transmitted [2]. In the context of heat radiation, a surface that absorbs all incident radiation and reflects none is called a black surface or black body. The StefanBoltzmann law of thermal radiation for a black body states that the rate of radiation energy from the surface per unit area is proportional to the fourth power of the temperature of the body [1]: q 5 σAT 4
ð3:1Þ
with q rate of energy emission from the surface, A surface area of the radiator and σ the SetfanBoltzmann constant. If we consider a black body with surface temperatures T1 which radiates to another black body with surface temperature T2 that completely surrounds it, the second black body completely absorbs the incident energy and emits radiant energy that is proportional to T24 . The net rate heat transfer by thermal radiation is then given by: ð3:2Þ q 5 σA T14 2 T24 A black body is a perfect radiator. Real bodies, however, do not act like a perfect radiator and emit at a lower rate. To take
Radiation Heat Transfer Chapter | 3
109
into account the real nature of the radiant bodies, a factor ε, called emissivity, is introduced. Emissivity is defined as the ratio of the emission from a real “gray” surface to the emission from a perfect “black” surface. Then, the rate of radiation heat transfer from a real body at temperature T1 which is surrounded by a black body at temperature T2 , is given by: ð3:3Þ q 5 σA1 ε1 T14 2 T24 It is worth pointing out that in most of the practical engineering problems, usually all three-heat transfer mechanisms, namely conduction, convection, and radiation, occur simultaneously.
3.2 NONLINEAR HEAT TRANSFER PROBLEMS WITH THERMAL RADIATION In this section, we will study some practical nonlinear heat transfer problems in which radiation is of importance and acts concurrently with other heat transfer mechanisms. Differential equations governing the physics of the problems will be introduced and finally solved via analytical approaches, as explained in Chapter 1, Introduction to Nonlinear Systems and Solution Methods.
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3.2.1 Cooling of a Lumped System by Combined Convection and Radiation Consider the problem of combined convective radiative cooling of a lumped system [4]. Let the system have volume V, surface area A, density ρ, specific heat c, emissivity E, and the initial temperature Ti . At t 5 0. The system is exposed to an environment with convective heat transfer with the coefficient of h and the temperature Ta . The system also loses heat through radiation, and the effective sink temperature is Ts . The cooling equation and the initial conditions are given as: ρVc
dT 1 hAðT 2 Ta Þ 1 EσAðT 4 2 Ts4 Þ 5 0 dt ð3:4Þ t50
T 5 Ti
ð3:5Þ
To solve the cooling equation, we introduce the following dimensionless parameters: θ5
T Ti
τ5
θa 5
Ta Ti
t ρVca =hA
Ts Ti EσTi3 ε5 h θs 5
ð3:6Þ
Radiation Heat Transfer Chapter | 3
111
After the parameter change, the heat transfer equation will transform into the following form: dθðτÞ 1 ½θðτÞ2 θa 1 εðθðτÞ4 2 θ4s Þ 5 0 ð3:7Þ dτ with the initial condition θ ð 0Þ 5 1
ð3:8Þ
For simplicity, let us assume the case θa 5 θs 5 0. Therefore, the nonlinear differential equation that governs the cooling process will reduce to: dθðτÞ 1 θðτÞ 1 εθðτÞ4 5 0 dτ
ð3:9Þ
with the initial condition θ ð 0Þ 5 1
ð3:10Þ
The exact solution of Eq. (3.9) is obtained [4] as: 1 1 1 εθ3 ln 5τ 3 ð1 1 εÞθ3
ð3:11Þ
3.2.1.1 Variational Iteration Method (VIM) In order to solve Eq. (3.9) using the VIM, we construct a correction functional, as follows: ðτ dθn ðtÞ 4 1θn ðtÞ1εθ~ n ðtÞ dt θn11 ðτÞ5θn ðτÞ1 λ dt 0 ð3:12Þ
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Its stationary conditions can be obtained as follows: λ0 ðtÞ 2 λðtÞ 5 0
ð3:13Þ
1 1 λðtÞjt5τ 5 0
ð3:14Þ
The Lagrangian multiplier can, therefore, be identified as: λ 5 2et2τ
ð3:15Þ
As a result, we obtain the following iteration formula: θn11 ðτÞ
dθ ðtÞ n 1 θn ðtÞ 1 εθ4n ðtÞ dt 5 θn ðτÞ 2 et2τ dt 0 ð3:16Þ ðτ
Now, we start with an arbitrary initial approximation that satisfies the initial condition: θ0 ðτÞ 5 e2τ
ð3:17Þ
Then, using the above variational formula (3.16), we have: ðτ 4 t2τ dθ0 ðtÞ 1θ0 ðtÞ1εθ0 ðtÞ dt θ1 ðτÞ5θ0 ðτÞ2 e dt 0 ð3:18Þ
Radiation Heat Transfer Chapter | 3
113
Substituting Eq. (3.17) into Eq. (3.18) and after simplifications, we obtain the firstorder iteration solution as: 1 1 θ1 ðτÞ 5 e2τ 2 εe2τ 1 εe24τ 3 3
ð3:19Þ
In the same way, θ2 ðτÞ can also be obtained as follows: 1 1 2 θ2 ðτÞ 5 e2τ 2 εe2τ 1 εe24τ 1 ε2 e2τ 3 3 9 1
1 4 2τ 2 3 2τ 1 5 2τ 4 2 24τ ε e 2 εe 2 εe 2 ε e 81 27 1215 9
1
2 4 27τ 2 3 27τ 4 4 24τ εe 2 εe 2 εe 27 9 81
2 2 2 1 ε3 e24τ 1 ε3 e210τ 1 ε2 e27τ 9 27 9 1
1 5 216τ 1 5 24τ εe εe 1 1215 243
2
1 5 213τ 1 4 213τ 2 5 210τ εe εe 1 εe 1 243 81 243
2
4 4 210τ 2 5 27τ 2 εe εe 81 243 ð3:20Þ
and so on. In the same manner the rest of the components of the iteration formula can
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Nonlinear Systems in Heat Transfer
be obtained to provide a more accurate approximate solution.
3.2.1.2 Homotopy Perturbation Method (HPM) After separating the linear and nonlinear parts of Eq. (3.9), we apply homotopy perturbation, as follows: Hðθ; pÞ 5 ð1 2 pÞ½LðθÞ 2 Lðu0 Þ 1 p½AðθÞ 2 f ðrÞ 5 0
ð3:21Þ
where LðθÞ is the linear part of the equation and Lðu0 Þ is the initial approximation [5]. We assume that the solution function can be written as a power series in p, as following: θ 5 θ0 1 pθ1 1 p2 θ2 1 ?
ð3:22Þ
Substituting Eq. (3.22) into Eq. (3.21) and rearranging based on powers of p-terms, we obtain: p0 :
dθ0 du0 1 θ0 2 2 u0 5 0 dτ dτ θ0 ð0Þ 5 1
p1 :
ð3:23Þ ð3:24Þ
dθ1 du0 1 θ1 1 1 u0 1 εθ40 5 0 ð3:25Þ dτ dτ θ1 ð0Þ 5 0
ð3:26Þ
Radiation Heat Transfer Chapter | 3
p2 :
dθ2 1 θ2 1 4εθ30 θ1 5 0 dτ θ2 ð0Þ 5 0
115
ð3:27Þ ð3:28Þ
One can easily solve Eqs. (3.23)(3.28) to find θi in Eq. (3.22). When p-1, the best approximate solution can be obtained as: 1 θðτÞ 5 e2τ 1 εðe24τ 2 e2τ Þ 3 ð3:29Þ 2 2 27τ 24τ 2τ 2 ε ð2 e 1 2e 2 e Þ 9
3.2.1.3 Conventional Perturbation Method For very small ε, we can assume a regular perturbation expansion [6] and calculate the first few terms in the form of θ 5 θ0 1 εθ1 1 ε2 θ2 1 . . .
ð3:30Þ
Substituting Eq. (3.30) into Eq. (3.9) and after expansion and rearranging based on coefficients of ε-terms, we have: ε0 : ε1 :
dθ0 1 θ0 5 0; dτ
dθ1 1 θ40 1 θ1 5 0; dτ
θ0 ð0Þ 5 1
ð3:31Þ
θ1 ð0Þ 5 0 ð3:32Þ
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ε2 :
dθ2 1 θ2 1 4θ30 θ1 5 0; θ2 ð0Þ 5 0 ð3:33Þ dτ
Solving Eqs. (3.31)(3.33) gives θ0 ðτ Þ 5 e2τ 1 1 θ1 ðτ Þ 5 e24τ 2 e2τ 3 3 θ2 ðτ Þ 5
2 27τ 4 24τ 2 2τ e 2 e 1 e 9 9 9
ð3:34Þ ð3:35Þ ð3:36Þ
and, therefore, the approximate solution based on the first three terms in the perturbation expansion becomes: 1 θðτ Þ 5 e2τ 1 ε e24τ 2 e2τ 3 ð3:37Þ 2 2 27τ 24τ 2τ 1 ε e 2 2e 1 e 9
3.3 RESULTS AND DISCUSSION The approximate solution obtained by the VIM and the HPM are compared to the exact solution of the nonlinear convective radiative cooling problem in Fig. 3.2, Fig. 3.3 and Fig. 3.4 [7]. As can be observed, the differences among VIM, HPM, and the exact solution are negligible when the dimensionless parameter ε is very small, e.g., less
Radiation Heat Transfer Chapter | 3
117
1
0.9
VIM Method HPM Method Exact Solution
θ(τ)
0.8
0.7
0.6
0.5
0.4 0
0.25
0.5
τ
0.75
1
FIGURE 3.2 The approximate solution of two analytical approaches, VIM and HPM, compared to the exact solution at ε 5 0:4. 1 0.9
VIM Method HPM Method Exact Solution
θ(τ)
0.8 0.7 0.6 0.5 0.4
0
0.25
0.5
τ
0.75
1
FIGURE 3.3 The approximate solution of two analytical approaches, VIM and HPM, compared to the exact solution at ε 5 0:65.
than 0.4. With an increase of the dimensionless parameter, the results of VIM and HPM slightly diverge from the exact solution.
Nonlinear Systems in Heat Transfer
1 0.9
VIM Method HPM Method Exact Solution
0.8 0.7
θ(τ)
118
0.6 0.5 0.4
0
0.25
0.5
0.75
1
τ
FIGURE 3.4 The approximate solution of two analytical approaches, VIM and HPM, compared to the exact solution at ε 5 0:85.
Overall, both methods provide a reasonably accurate approximate solution and it seems that VIM leads to slightly better results, even for large values of ε. This is illustrated in Fig. 3.5 and Table 3.1, which show the errors on the approximate analytical solutions as a function of ε.
3.3.1 ConvectiveRadiating Cooling of a Lumped System with Variable Specific Heat For another example, we consider the problem of combined convectiveradiative cooling of a lumped system with a temperaturedependent specific heat [4]. Let the system have volume V, surface area A, density ρ,
Radiation Heat Transfer Chapter | 3
119
0.07 0.065 0.06
VIM / Exact HPM / Exact
0.055 0.05
Error
0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0
0.25
0.5
ε
0.75
1
FIGURE 3.5 Errors of VIM and HPM solutions at τ 5 0:5 as a function of the dimensionless parameter ε.
specific heat c, emissivity E, and the initial temperature Ti . At t 5 0, the system is exposed to an environment with convective heat transfer coefficient of h and temperature Ta . The system also loses heat through radiation and the effective sink temperature is Ts . Assume that the specific heat c is a linear function of temperature in the form of: c 5 ca ½1 1 βðT 2 Ta Þ
ð3:38Þ
where ca is the specific heat at the temperature Ta and β is a constant. The convective radiating cooling equation and the initial conditions can be established as follows: ρVc
dT 1 hAðT 2 Ta Þ 1 EσAðT 4 2 Ts4 Þ 5 0 dt ð3:39Þ
TABLE 3.1 The Results of VIM and HPM Techniques and Their Errors at τ 5 0:5 ε
VIM
HPM
EXACT
Error of VIM
Error of HPM
0
0.606531
0.606531
0.606531
1.64872E-10
1.64872E-10
0.1
0.591617
0.591638
0.591591
4.28955E-05
7.80444E-05
0.2
0.578207
0.578371
0.578023
0.000318544
0.000602619
0.3
0.566185
0.566732
0.56562
0.000999783
0.00196688
0.4
0.55544
0.55672
0.554217
0.002207405
0.004516592
0.5
0.545868
0.548335
0.543681
0.004021512
0.008559256
0.6
0.537369
0.541576
0.533903
0.006490236
0.014371001
0.7
0.52985
0.536445
0.524793
0.009636553
0.022201898
0.8
0.523226
0.53294
0.516275
0.013463676
0.032280117
0.9
0.517412
0.531062
0.508284
0.017959388
0.04481523
1
0.512333
0.530812
0.500765
0.023099547
0.060000859
Radiation Heat Transfer Chapter | 3
t50
T 5 Ti
121
ð3:40Þ
To solve Eq. (3.39), let us do the following changes of parameters: θ5 τ5
T ; Ti
θa 5 t
ρVca =hA
;
Ta ; Ti
θs 5
ε1 5 βTi ;
Ts ; Ti EσTi3 ε2 5 h ð3:41Þ
which transforms Eq. (3.39) into the following form: ½1 1 ε1 ðθ 2 θa Þ
dθ 1 ðθ 2 θa Þ 1 ε2 θ4 2 θ4s 5 0 dτ
with the initial condition θð0Þ 5 1
ð3:42Þ
For the sake of simplicity, we assume the case of θa 5 θs 5 0. Therefore, the governing convectiveradiating cooling equation will be written as: ð1 1 ε1 θÞ
dθ 1 θ 1 ε2 θ4 5 0 dτ
ð3:43Þ
with the initial condition θ ð 0Þ 5 1
ð3:44Þ
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The exact solution of Eq. (3.43) is obtained [4] as: " 1=3 1 11ε2 θ3 1 ε1 1 ð11ε2 Þ3 ð11ε2 θ3 Þ ln ln 1 3 ð11ε2 Þθ3 3ε1=3 2 ð11ε2 Þð11ε1=3 θÞ3 2 2 !# 1=3 1=3 pffiffiffi 2ε 21 2ε θ21 5τ 1 3 arctan 2pffiffiffi 2arctan 2pffiffiffi 3 3 ð3:45Þ
3.3.1.1 Variational Iteration Method (VIM) First, we construct a correction functional that reads θn11 ðτÞ5θn ðτÞ ðτ dθ ðtÞ 4 n 1θn ðtÞ1ε2 θ~ n ðtÞ dt 1 λ ð11ε1 θ~ n ðtÞÞ dt 0 ð3:46Þ with the stationary conditions that can be obtained as follows: λ0 ðtÞ 2 λðtÞ 5 0
ð3:47Þ
1 1 λðtÞjt5τ 5 0
ð3:48Þ
Solving Eqs. (3.47) and (3.48) gives: λ 5 2et2τ
ð3:49Þ
Radiation Heat Transfer Chapter | 3
123
As a result, we obtain the following iteration formula: θn11 ðτÞ
ðτ
n dθn ðtÞ 5 θn ðτÞ 2 e ð1 1 ε1 θ~ n ðtÞÞ dt ð3:50Þ 0 o 4 1 θn ðtÞ 1 ε2 θ~ n ðtÞ dt t2τ
Now, let us begin with an arbitrary initial approximation, as follows: θ0 ðτÞ 5 e2τ
ð3:51Þ
Using the variational iteration formula (3.50), the first-order iteration can be obtained as: θ1 ðτÞ
ðτ
n dθ0 ðtÞ 5 θ0 ðτÞ 2 e ð1 1 ε1 θ~ 0 ðtÞÞ dt ð3:52Þ 0 o 4 ~ 1 θ0 ðtÞ 1 ε2 θ0 ðtÞ dt t2τ
Substituting Eq. (3.51) into Eq. (3.52) and after solving and some simplifications, we will obtain the first-order solution as: 1 θ1 ðτÞ 5 e2τ 1 ε1 e2τ 2 ε2 e2τ 2 ε1 e22τ 3 1 24τ 1 ε2 e 3 ð3:53Þ
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In the same way, θ2 ðτÞ can be obtained as follows: 3 2 2 2 17 θ2 ðτÞ 5 ε21 e23τ 1 ε1 ε2 e22τ 2 ε32 e2τ 1 ε22 e2τ 2 ε1 ε2 e25τ 2 3 27 9 12 2
4 3 2 210τ 1 3 2 29τ 12 3 26τ 1 2 23τ 4 4 28τ 1 ε1 ε2 e 1 ε1 ε2 e 2 ε2 ε1 e 2 ε2 ε1 e ε ε e 27 1 2 2 5 2 63
1
1 4 213τ 1 2 2 29τ 1 2 22τ 1 5 2τ 2 2 2 27τ 1 ε1 ε2 e 2 ε2 ε1 e 2 ε ε1 e ε e 1 ε1 ε2 e 81 2 2 9 1215 2 3
1
2 2 3 212τ 212 4 2τ 1 3 2τ 1 1 5 213τ 1 ε ε e ε ε1 e 1 ε1 e 1 ε42 e213τ 2 ε e 33 1 2 36855 2 6 81 243 2
4 4 4 2 1 2 ε22 e24τ 2 ε22 ε1 e24τ 2 ε32 ε1 e27τ 1 ε22 ε1 e27τ 1 ε42 e2τ 9 3 9 3 81 1 2 1 7 41 1 ε21 e2τ 1 ε42 e27τ 2 ε2 ε4 e2τ 2 ε1 ε2 e2τ 2 ε2 ε21 e25τ 2 27 105 1 12 12 1
2 2 3 210τ 4 4 2 2 ε ε e 1 ε32 ε1 e210τ 2 ε31 ε22 e26τ 2 ε32 ε1 e211τ 1 ε21 ε32 e24τ 27 1 2 27 5 15 9
2
40 2 28τ 8 2 2 ε ε1 e 2 ε2 ε41 e25τ 1 ε21 ε32 e28τ 1 ε21 ε32 e26τ 1 ε22 e27τ 63 2 21 15 9
2 7 65 2 3 2τ 8 2 ε31 e24τ 2 ε2 ε21 e2τ 2 ε ε e 2 ε2 ε41 e25τ 1 ε21 ε32 e28τ 3 20 4158 1 2 21 1
2 2 3 26τ 2 2 27τ 2 3 24τ 7 65 2 3 2τ ε ε e 1 ε2 e 2 ε1 e 2 ε2 ε21 e2τ 2 ε ε e 15 1 2 9 3 20 4158 1 2
1
37 3 2 2τ 1 3 2τ 2 4 27τ 2 5 27τ 2 2 22τ ε ε e 2 ε1 ε2 e 2 ε2 ε1 e 2 ε e 1 ε2 ε1 e 1890 1 2 15 3 243 2 3
6 59 3 2τ 4 2 2 26τ 4 3 2 24τ 1 4 28τ 1 ε2 ε41 e26τ 2 ε ε1 e 2 ε1 ε2 e 2 ε1 ε2 e 1 ε2 ε1 e 5 945 2 5 9 7 8 8 1 5 216τ 8 4 ε e 2 ε21 ε22 e28τ 1 ε2 ε21 e26τ 1 1 ε32 ε1 e28τ 1 ε32 ε1 e24τ 7 5 1215 2 21 9 4 2 1 2ε21 ε22 e25τ 2 3ε31 ε2 e25τ 2 ε21 ε22 e24τ 1 2ε2 ε21 e24τ 1 ε42 ε1 e27τ 3 27 1 2 1 4 4 214τ 4 2 ε21 ε32 e29τ 2 ε1 e22τ 2 ε32 e27τ 1 ε2 e24τ 2 1 ε1 ε2 e24τ ε ε1 e 6 9 3 351 2 3 2 41 4 1 1 2 ε21 ε32 e27τ 1 ε22 ε1 e25τ 2 ε42 ε1 e24τ 2 ε32 ε1 e25τ 1 ε2 ε41 e24τ 9 36 81 3 3 2
4 4 210τ 2 3 2 5 210τ 2 3 24τ 1 ε32 e210τ 1 ε31 e23τ 1 1 ε2 e ε ε1 e ε e 81 2 27 2 243 2 9
2
4 4 24τ 4 23 2 2 2τ 23 2 2τ ε e 2 2ε21 e22τ 2 ε31 e22τ 2 ε42 e210τ 1 ε ε e 1 ε2 ε1 e 81 2 81 210 1 2 84
1 ε31 ε22 e25τ 2
2 2 3 211τ 4 3 24τ 2 1 1 ε1 ε2 e 1 ε42 ε1 e211τ 1 ε22 ε1 e25τ ε ε e 15 1 2 3 45 27
1 4 2 4 1 5 24τ 2 ε21 ε32 e25τ 2 ε31 ε22 e28τ 2 ε31 ε2 e27τ 1 ε31 ε22 e27τ 1 ε e 3 7 3 9 243 2
ð3:54Þ
Radiation Heat Transfer Chapter | 3
125
and so on. In the same manner the rest of the components of the iteration formula can be obtained.
3.3.1.2 Homotopy Perturbation Method (HPM) Applying homotopy perturbation to Eq. (3.43) results in: dν 4 Lðν Þ2Lðθ0 Þ1pLðθ0 Þ1p ε1 ν 1ε2 ν 50 ð3:55Þ dτ with dν 1ν dτ dθ0 Lðθ0 Þ 5 1 θ0 dτ Expanding the parameter ν as: L ðν Þ 5
ν 5 ν 0 1 pν 1 1 p2 ν 2
ð3:56Þ ð3:57Þ ð3:58Þ
and substituting it into Eq. (3.55) will result in: dν 0 dν 1 dν 2 dθ0 1p 1 p2 1 ν 0 1 pν 1 1 p2 ν 2 2 dτ dτ dτ dτ dθ0 dν 0 dν 1 2 2 θ0 1 p 1 θ0 1ε1 ν 0 p 1 ε1 ν 0 p dτ dτ dτ dν 0 2 1 ε1 ν 1 p 1 ε2 ν 40 p 1 4ε2 ν 1 ν 30 p2 5 0 dτ
ð3:59Þ
After rearranging based on powers of p-terms, we will have: p0 :
dν 0 dθ0 1 ν0 2 2 θ0 5 0 dτ dτ
ð3:60Þ
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Nonlinear Systems in Heat Transfer
ν 0 ð0Þ 5 1 p1 :
dν 1 dθ0 dν 0 1 ν1 1 1 θ0 1 ε1 ν 0 1 ε2 ν 40 5 0 ð3:62Þ dτ dτ dτ ν 1 ð0Þ 5 0
p2 :
ð3:61Þ
ð3:63Þ
dν 2 dν 1 dν 0 1 ν 2 1 ε1 ν 0 1 ε1 ν 1 1 4ε2 ν 1 ν 30 5 0 ð3:64Þ dτ dτ dτ ν 2 ð0Þ 5 0
ð3:65Þ
To determine ν, Eqs. (3.60)(3.65) should be solved. Considering an initial approximation in the form of θ0 ðτ Þ 5 e2τ
ð3:66Þ
one can easily obtain: ð3:67Þ ν 0 5 e2τ 1 24τ 1 22τ ν 1 5 2ε1 e 1 ε2 e 1 ε1 2 ε2 e2τ ð3:68Þ 3 3 1 1 25τ ν 2 5ε21 e23τ 2 ε1 ε2 e 2ε1 ε1 2 ε2 e22τ 3 3 1 2 23τ 1 1 2 25τ 1 ε1 e 2 ε1 ε2 e 2ε1 ε1 2 ε2 e22τ 2ε1 ε2 e25τ 1 ε22 e27τ 2 12 3 9 4 1 1 1 24τ 2τ 2 1 ε2 ε1 2 ε2 e 1e 2ε1 1 ε1 ε2 12ε1 ε1 2 ε2 3 3 3 3 ! 1 1 2 4 1 1 e2τ 2 ε21 1 ε1 ε2 1ε1 ε2 2 ε22 2 ε2 ε1 2 ε2 2 12 9 3 3 ð3:69Þ
Radiation Heat Transfer Chapter | 3
127
As p-1, then ν-θ. Therefore, the approximate solution to the heat transfer equation can be obtained as: 1 1 2 17 θðτÞ5e2τ 12ε1 e2τ 1 ε2 e23τ 1ε1 2 ε2 1 ε1 ε2 e2τ 2 ε1 ε2 e24τ 3 3 3 12
!
2 4 4 7 1 2 22ε21 e2τ 1 ε22 e26τ 1 ε1 ε2 e23τ 2 ε22 e23τ 2 ε1 ε2 1 ε21 1 ε22 9 3 9 12 2 9 ð3:70Þ
3.3.1.3 Conventional Perturbation Method Since there are two different small parameters (ε1 and ε2 ), we assume θ as the following in solving Eq. (3.43) using the perturbation method [8]: θ 5 θ00 1 ε1 θ01 1 ε2 θ10 1 ε21 θ02 1 ε22 θ20
ð3:71Þ
Substituting Eq. (3.71) into Eq. (3.43), rearranging based on the coefficients of ε1 and ε2 and solving the resulting equations, the approximate solution by the perturbation method can be obtained as: 1 θ5e2τ 1ε1 e2τ 2e22τ 1 ε2 e24τ 2e2τ 3 2 1 1 ε21 e2τ 24e22τ 13e25τ 1 ε22 e2τ 22e24τ 1e27τ 2 9 1 1 ε1 ε2 27e2τ 18e22τ 116e24τ 117e25τ 12 ð3:72Þ
Nonlinear Systems in Heat Transfer
3.3.1.4
Results and Discussion
Fig. 3.6, Fig. 3.7, and Fig. 3.8 show the approximate solution to the heat transfer problem provided by HPM and VIM compared to the exact solution. The influence of small parameters, ε1 and ε2 , is shown. When both dimensionless parameters are relatively small, the analytical approaches provide accurate approximate solutions. By increasing the dimensionless parameter ε1 , the difference between the approximate solutions and the exact solution increases, while HPM seems to provide slightly more accurate results than VIM. The dimensionless parameter ε2 has a less significant influence of the results, and both analytical methods provide a reasonably accurate approximation. 1 Variational Method Homotopy Method Exact Solution
0.9
0.8
θ(τ)
128
0.7
0.6
0.5
0
0.25
0.5
0.75
1
τ
FIGURE 3.6 The approximate solution of HPM and VIM to the heat transfer problem, compared to the exact solution at ε1 5 0:4 and ε2 5 0:4.
Radiation Heat Transfer Chapter | 3
129
1 Variational Method Homotopy Method Exact Solution
0.9
θ(τ)
0.8
0.7
0.6
0.5 0
0.25
0.5
0.75
1
τ
FIGURE 3.7 The approximate solution of HPM and VIM to the heat transfer problem, compared to the exact solution at ε1 5 0:8 and ε2 5 0:4. 1 Variational Method Homotopy Method Exact Solution
0.9
θ(τ)
0.8
0.7
0.6
0.5
0.4 0
0.25
0.5
0.75
1
τ
FIGURE 3.8 The approximate solution of HPM and VIM to the heat transfer problem, compared to the exact solution at ε1 5 0:4 and ε2 5 0:8.
The errors of VIM and HPM at τ 5 0:5 and ε1 5 0:4 is shown in Fig. 3.9. It is obviously seen that the error of HPM is less than that
Nonlinear Systems in Heat Transfer
0.025 Variat./Exact Homot./Exact 0.02
Error
130
0.015
0.01
0.005
0.25
0.5
ε2
0.75
1
FIGURE 3.9 Errors of HPM and VIM as a function of the dimensionless parameter ε2 at τ 5 0:5 and ε1 5 0:4.
of VIM when ε2 , 0:5. If ε2 exceeds this limit, the error of HPM exceeds the error of VIM. A comparison between errors of VIM and HPM at τ 5 0:5 and ε1 5 0:8 is illustrated in Fig. 3.10, which shows that for a large magnitude of ε1 , the error of HPM is always less than the error of VIM. However, the overall error of VIM and HPM is reasonably low, which proves the capability and robustness of these analytical techniques for solving nonlinear heat transfer problems. A comparison between the homotopy perturbation method and the conventional perturbation method is provided in Fig. 3.11. It is worth pointing out that the simplest condition to solve Eq. (3.43) is when the dimensionless parameters ε1 5 ε2 5 0, where
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0.04 Variat./Exact Homoto./Exact
0.035
Error
0.03
0.025
0.02
0.015
0.01 0.25
0.5
ε2
0.75
1
FIGURE 3.10 Errors of HPM and VIM as a function of the dimensionless parameter ε2 at τ 5 0:5 and ε1 5 0:8.
FIGURE 3.11 Error (in %) of the conventional perturbation method, compared to that of the homotopy perturbation method at the initial time, when ε5ε1 5 ε2 .
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the equation changes from nonlinear to linear. By increasing the dimensionless parameters, the equation becomes more nonlinear. The calculated errors clearly illustrate that for very small dimensionless parameters, the conventional perturbation method is as accurate as the homotopy perturbation method. However, when the nonlinearity of the problem increases, the error of the conventional perturbation technique increases drastically. This proves the robustness and stability of HPM technique, even for highly nonlinear heat transfer problems. Table 3.2 shows the errors of the conventional perturbation method and the homotopy TABLE 3.2 Comparison of the Errors of the Conventional Perturbation Method and the Homotopy Perturbation Method at τ 5 0 and when ε1 6¼ ε2 ε1
ε2
Error/Perturb. (%)
Error/HPM (%)
0.1
0.2
5.66
4.5
0.2
0.1
5.66
8
0.8
0.3
68
8
0.3
0.8
68
10.5
1.0
0.04
11.33
0
0.04
1.0
11.33
1.92
0.03
0.07
0.59
1.45
0.07
0.03
0.59
3.25
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133
perturbation method when ε1 ¼ 6 ε2 . As can be seen, for different combination of these parameters, HPM always seems to be reasonably accurate, while the conventional perturbation method only provides accurate approximation solution when ε1 , ε2 or both are very small.
3.3.2 Temperature Distribution in a Radiating Thick Rectangular Plate Now we consider a nonlinear equation, governing the temperature distribution in a uniformly thick rectangular fin radiating to free space with nonlinearity of high order [9,10] d2θ 2 εθ4 ðxÞ 5 0 ð3:73Þ 2 dx subject to the boundary conditions of dθ ð3:74Þ θð1Þ 5 1 ð0Þ 5 0 dx
3.3.2.1 Variational Iteration Method (VIM) To solve Eqs. (3.73) and (3.74) using VIM, we will first construct the following correction functional as: ðx 2 d θn ðX Þ 4 θn11 ðxÞ 5 θn ðxÞ 1 λ 2 εθn ðX Þ dX dX 2 0 ð3:75Þ
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Making the above correction functional stationary, we obtain the following stationary conditions: λjX5x 5 0;
ð3:76Þ
0
ð3:77Þ
00
ð3:78Þ
1 2 λ jX5x 5 0; λ 50
The Lagrangian multiplier can therefore be identified as: λ5X2x
ð3:79Þ
Substituting Eq. (3.79) into the correction functional equation system (3.75) and considering that θn ðxÞ have to satisfy the boundary condition Eqs. (3.74), we define another coefficient as cn into Eq. (3.75), which has to be determined for each iteration subject to Eq. (3.74). Therefore, we will have the following iteration formula: θn11 ðxÞ 2
ðx d θn ðX Þ 4 2εθn ðX Þ dX 5cn θn ðxÞ1 ðX2xÞ dX 2 0 ð3:80Þ Using the iteration formula (3.80) and defining the initial approximation as θ0 ðxÞ 5 1, considering Eq. (3.74), the first two iterations can be calculated as: θ 1 ðx Þ 5
2 1 εx2 21ε
ð3:81Þ
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10;08015040 31x2 ε11680 4:51x4 ε2 1504 2:51x6 ε3 190x8 ε4 17x10 ε5 θ2 ðxÞ5 1080120;160ε19240ε2 11764ε3 190ε4 17ε5 ð3:82Þ
3.3.2.2
Results and Discussion
The exact solution cannot be easily obtained for the highly nonlinear heat transfer problem of Eq. (3.73). Therefore, to compare with the analytical approximate solutions, this equation is solved by the numerical method. Fig. 3.12 shows the comparison
FIGURE 3.12 The comparison of the results of the fourthorder VIM and the numerical solution.
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between the solution of fourth-order VIM and the numerical solution for various dimensionless parameters ε. As can be seen, when ε is very small, the difference between VIM and the numerical solution is negligible. By increasing ε, the difference increases. However, even for large ε, VIM provides an accurate solution.
3.3.3 Temperature Distribution in a Radiating Thick Rectangular Plate The example to be studied is the one-dimensional heat transfer in a straight fin with the length of L and the cross area of A and the perimeter of P (see Fig. 3.13). The fin surface transfers heat through both convection and radiation. Suppose the temperature of the surrounding air is T0 and the effective sink temperature for the radiative heat transfer is Ts. We assume that the base
FIGURE 3.13
Geometry of a straight fin.
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137
temperature of the fin is Tb and there is no heat transfer of the tip of the fin. It is also assumed that the convection heat transfer coefficient, h, and the emissivity coefficient of surface, Eg, are both constant while the conduction coefficient, k, can be variable. The energy equation and the boundary conditions for the fin are as follows [11]: d dT hp Eg σ 4 k 2 ðT 2Ta Þ2 T 2TS4 50 dx dx A A ð3:83Þ x50-
dT 50; x5L-T 5Tb dx
ð3:84Þ
Assuming k as a linear function of temperature, we have: k 5 ka ð1 1 βðT 2 Ta ÞÞ
ð3:85Þ
After making the equation dimensionless and changing parameters, we have: T Ta Ts ; θa 5 ; θs 5 ; Tb Tb Tb x 2 hpL2 Eg σTb3 pL3 X5 ;N 5 ; ε1 5 βTb ; ε2 5 L ka A ka A ð3:86Þ θ5
Substituting Eq. (3.86) into Eq. (3.83), we obtain:
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d dθ ½1 1 ε1 ðθ 2 θa Þ 2 N 2 ðθ 2 θ a Þ dx dX 2 ε2 θ4 2 θ4s 5 0 ð3:87Þ X 5 0-
dθ 5 0 X 5 1-θ 5 1 dX
ð3:88Þ
By assuming θa 5 θs 5 0, Eq. (3.87) transforms to 2 2 d2θ dθ d θ 2 2N θ1ε 1ε θ 2ε2 θ4 50 1 1 2 2 dX dX dX ð3:89Þ
3.3.3.1 Variational Iteration Method (VIM) According to the VIM, we can construct the correction functional as follows: ðx
00 2 θn11 ðxÞ5θn ðxÞ1 λ θn ðtÞ2N 2 θn t 1ε1 θ~ n ðtÞ 0
00 4 1ε1 θ~ n ðtÞθ n ðtÞ2ε2 θ~ n ðtÞ dt
ð3:90Þ where λ is general Langrange multiplier. Making the above correction functional stationary, we can obtain the following stationary conditions:
Radiation Heat Transfer Chapter | 3 00
139
0
λ ðtÞ2N 2 λðtÞ50; 12λ ðtÞjt5x 50; λðtÞjt5x 50 ð3:91Þ The Lagrange multiplier therefore can be identified as:
1 eNðx2tÞ 2 eNðt2xÞ λ52 ð3:92Þ 2 N Substituting Eq. (3.92) into Eq. (3.90) and considering that θn ðxÞ has to satisfy Eq. (3.88), another coefficient, Cn, can be introduced to Eq. (3.90), which has to be determined for each iteration subject to Eq. (3.88). We can then write down the following iteration formula: ðx θn11 ðxÞ5Cn θn ðxÞ2 (
1 eN ðx2tÞ 2eN ðt2xÞ 2 N
0
00 2 θ n ðtÞ2N 2 θn t 1ε1 θ~ n ðtÞ1ε1 θn ðtÞθ n ðtÞ ) ! 4 2ε2 θ~ ðtÞ dt ð3:93Þ 00
n
Now we are to start with an arbitrary initial approximation that satisfies the conditions: θ0 ðxÞ 5 sechðN Þ coshðNxÞ:
ð3:94Þ
Using the above variational formula (3.93), we have:
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Nonlinear Systems in Heat Transfer
( 1 eNðx2tÞ 2eNðt2xÞ 00 θ 0 ðtÞ2N 2 θ0 ðtÞ θ0 ðxÞ2 2 N 0 ) ! ðx
θ1 ðxÞ5C0
00
1ε1 θ20 ðtÞ1ε1 θ0 ðtÞθ 0 ðtÞ2ε2 θ40 ðtÞ dt ð3:95Þ Substituting Eq. (3.94) into Eq. (3.95) and after simplifications, we obtain: θ1 ðxÞ5 C0 sechðNÞcoshðNxÞ1 3
1 240ðN 2 cosh4 ðNÞÞ
40eð5NxÞ ε1 N 2 cosh2 ðNÞ224eð3NxÞ ε2 140eð3NxÞ
3ε1 N 2 cosh2 ðNÞ224eð5NxÞ ε2 2120ε1 N 2 cosh2 ðNÞeð4NxÞ 220ε2 eð2NxÞ 120ε1 N 2 eð6NxÞ cosh2 ðNÞ290ε2 eð4NxÞ 120ε1 N 2 eð2NxÞ cosh2 ðNÞ220ε2 eð6NxÞ ! 2ε2 eð8NxÞ 2ε2 eð24NxÞ ð3:96Þ where C0 5
and
1 A
ð3:97Þ
Radiation Heat Transfer Chapter | 3
A 5 sechðNÞcoshðNÞ 1
141
1 240ðN 2 cosh4 ðNÞÞ
3 ð40eð3NÞ ε1 N 2 cosh2 ðNÞ 2 24eð5NÞ ε2 1 40eð5NÞ ε1 N 2 cosh2 ðNÞ 2 24eð3NÞ ε2 2 120ε1 N 2 cosh2 ðNÞeð4NÞ 2 20ε2 eð6NÞ
ð3:98Þ
1 20ε1 N 2 eð2NÞ cosh2 ðNÞ 1 90ε2 eð4NÞ 1 20ε1 N 2 eð6NÞ cosh2 ðNÞ 2 20ε2 eð2NÞ 2 ε2 eð8NÞ 2 ε2 Þeð24NÞ and so in. In the same manner, the rest of the components of the iteration formula can be obtained.
3.3.3.2
Fin Efficiency
The heat transfer rate from the fin is found by using Newton’s law cooling [11] ðb Q5
pðT 2 Ta Þdx
ð3:99Þ
0
The ratio of the actual heat transfer from the surface to that would transfer if the whole fin surface were at the same temperature as the base is commonly called as the fin efficiency
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Nonlinear Systems in Heat Transfer
Q η5 5 Qideal
Ðb
0 pðT 2 Ta Þdx 5 pbðT 2 Ta Þ
ð1 θðxÞdx: x50
ð3:100Þ Integrating Eq. (3.100), the efficiency of straight fins is obtained as an analytical expression as follows: η5
E F
ð3:101Þ
where E52 2480N 2 eð5NÞ cosh3 ðNÞ140ε1 N 2 eð2NÞ 3cosh2 ðNÞ1480N 2 eð3NÞ cosh3 ðNÞ2160eð5NÞ 3ε1 N 2 cosh2 ðNÞ140ε2 eð6NÞ 1480ε1 N 3 eð4NÞ 3cosh2 ðNÞ296eð3NÞ ε2 2ε2 1160eð3NÞ ε1 N 2 3cosh2 ðNÞ240ε2 eð2NÞ 196eð5NÞ ε2 2360ε2 Neð4NÞ 1ε2 eð8NÞ 240ε1 N 2 eð6NÞ cosh2 ðNÞ
ð3:102Þ and F5 4N 240cosh4 ðNÞN 2 eð4NÞ 140eð3NÞ ε1 N 2 3cosh2 ðNÞ224eð5NÞ ε2140eð5NÞε1 N 2 cosh2 ðNÞ 224eð3NÞ ε2 2120ε1 N 2 eð4NÞ cosh2 ðNÞ 190ε2 eð4NÞ 120ε1 N 2 eð2NÞ cosh2 ðNÞ 220ε2 eð6NÞ 120ε1 N 2 eð6NÞ cosh2 ðNÞÞ
ð3:103Þ
Radiation Heat Transfer Chapter | 3
143
3.3.3.3 Homotopy Perturbation Method Let us construct the homotopy perturbation method in the following form 2 dν Lðν Þ 2 Lðθ0 Þ 1 pLðθ0 Þ 1 p ε1 dX ! 2 d ν 1 ε1 ν 2 ε2 ν 4 5 0 2 dX ð3:104Þ Assuming d 2 θ0 ν 5 ν 0 1 pν 1 1 p ν 2 ; θ0 5 5 0 ð3:105Þ dX 2 2
and substituting Eq. (3.105) into Eq. (3.104) and after organizing based on p-powers, we have: d2ν 0 p: 2 N 2 ν 0 5 0; 2 dX 0
X 5 0-
dν 0 5 0; dX
ð3:106Þ
X 5 1-ν 0 5 1 ð3:107Þ
2 2 d ν dν 0 d2ν 0 1 2 2 N ν 1 ε 1 ε ν p1 : 1 1 1 0 dX 2 dX dX 2 1 ε2 ν 40 5 0; ð3:108Þ
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Nonlinear Systems in Heat Transfer
X 5 0-
dν 1 5 0; dX
X 5 1-ν 1 5 1 ð3:109Þ
d2ν 2 dν 0 dν 1 d2ν 1 2 2 N ν2 1 2 p: 1 ε1 ν 0 dX 2 dX dX dX 2 d2ν 0 1 ε1 ν 1 1 4ε2 ν 1 ν 30 5 0; 2 dX ð3:110Þ 2
X 5 0-
dν 2 5 0; dX
X 5 1-ν 2 5 1 ð3:111Þ
Solving Eqs. (3.106)(3.111), we obtain: ν 0 5 sech N cosh NX
can
ð3:112Þ
3 sech4 N 4 ν 1 5sechN 2 2 ε2 sech N1ε2 8N 120N 2 !! 2 4 ε1 sech N ε2 sech N 3cosh4N1cosh2N 1 3 6N 2 sech4 N 3ε2 cosh4NX1 sech4 N 3coshNX2ε2 2 2 120N 8N ! ε1 sech2 N ε2 sech4 N cosh2NX 2 1 3 6N 2 ð3:113Þ
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145
ν 2 5sechNð2A3 cosh2N2A5 cosh3N 2A7 cosh4N2A9 cosh5N2A11 cosh7N2A14 Þ 3coshNX1A3 cosh2NX1A5 cosh3NX 1A7 cosh4NX1A9 cosh5NX1A11 cosh7NX1A14 ð3:114Þ where A3 5
2 C0 ε1 C0 sechN 2 sechN 3 6 ε2 C0 sech3 N ε2 C0 sech3 N 1 1 2 6N 2 2N 2
ð3:115Þ
ε2 sech5 N C00 sechN ε1 ε2 sech5 N 1 A5 5 1 240N 2 4 120N 2 ε1 ε1 C0 sechN ε1 ε2 1 1 C00 sechN 2 16 4 8 3 240N 2 ε1 C00 ε22 sech7 N 5 3 sech N 1 sechN 1 640N 4 16 3ε2 sech7 N 3ε2 C00 sech3 N 1 1 8N 2 160N 2 ð3:116Þ A7 5
ε22 C0 sech3 N 30N 2
ð3:117Þ
ε1 sech5 N ε1 ε2 1 sech5 N 1 A9 5 2 2 30 3 24N 15 3 24N ε1 ε2 ε22 sech7 N 5 sech N 2 24 3 80N 4 24 3 240N 2 ε2 C00 sech3 N 2 48N 2
ð3:118Þ
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Nonlinear Systems in Heat Transfer
2ε22 sech7 N A11 5 48 3 240N 4 A14 5 2C0 sechN 1
ε1 C0 sechN 2
3ε2 C0 sech3 N 2 2N 2
ð3:119Þ
ð3:120Þ
and 3 ε2 sech4 N 4 ε sech N 1 cosh4N 2 8N 2 120N 2 ! ! 2 4 ε1 sech N ε2 sech N cosh2N 1 1 3 6N 2
C0 5 sechN 2
ð3:121Þ C00 5
3.3.3.4
ε1 sech2 N ε2 sech4 N 1 3 6N 2
ð3:122Þ
Results and Discussion
Fig. 3.14 shows the temperature distribution in convectiveradiative conduction fins with variable thermal conductivity, obtained by VIM at N 5 1. Fig. 3.15 shows the variation of the fin efficiency with the thermo-geometric fin parameter for different values of the thermal conductivity parameter, obtained by VIM.
Radiation Heat Transfer Chapter | 3
147
FIGURE 3.14 Temperature distribution in convectiveradiative conduction fins with variable thermal conductivity at N 5 1.
FIGURE 3.15 Variation of the fin efficiency with the thermo-geometric fin parameter for different values of the thermal conductivity parameter.
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Nonlinear Systems in Heat Transfer
FIGURE 3.16 Three term approximated solution, obtained by HPM.
FIGURE 3.17 Approximate solution obtained via the conventional perturbation method.
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In Fig. 3.16, the three term approximated solution (ν 0 1 pν 1 1 p2 ν 2 , when p-1), obtained by HPM, is illustrated from X 5 0 to X 5 1, for a fixed ε2 5 0:2 and for different values of ε1 5 0; 0:2; 0:4; and 0:6. In Fig. 3.17, the approximate solution of Eq. (3.89), obtained via the conventional perturbation method [12], is depicted for the same ε1 and ε2 .
3.4 CONCLUSION In this chapter, a couple of nonlinear heat transfer problems were discussed, with a focus on heat radiation. In most of the practical engineering problems, usually all three heat transfer mechanisms, namely conduction, convection and radiation, occur simultaneously. Therefore, in the examples discussed, all of these heat transfer modes were present. The mathematical equations governing the physics of the problems were established and solved using the analytical approaches, as explained in Chapter 1, Introduction to Nonlinear Systems and Solution Methods. The exact solution or the numerical solution were used to assess the validity and accuracy of the analytical approaches. It was observed that analytical methods, providing approximate solutions, can be powerful tools to analyze the highly nonlinear heat transfer problems containing a combination
150
Nonlinear Systems in Heat Transfer
of conduction, convection, and radiation and provide accurate approximate solutions in problems that the exact solutions do not exit. Furthermore, they can be an easy-touse and yet accurate alternatives for numerical techniques, which could be difficult, time consuming, and sensitive to the initial solution and it is very difficult to converge to a solution in case of strong nonlinearity.
REFERENCES [1] Y. Rao, Heat Transfer, Universities Press (India) Limited, Hyderabad, India, 2001. [2] S. Sukhatme, A Textbook On Heat Transfer, Universities Press (India) Private Limited, Hyderabad, India, 2005. [3] J.H. Lienhard, A Heat Transfer Textbook, fourth ed., Dover Publications, Inc, Mineola, NY, 2011. [4] A. Aziz, T. Na, Perturbation Method in Heat Transfer, Hemisphere Publishing Corporation, 1984. [5] J. He, Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput. 135 (2003) 7379. [6] A. Nayfeh, Perturbation Methods, Wiley, New York, 1973. [7] H. Khaleghi, D. Ganji, A. Sadighi, Application of variational iteration and homotopy-perturbation methods to nonlinear heat transfer equations with variable coefficients, Numer. Heat Transf. Part A Appl. 52 (2007) 2542. [8] D. Ganji, A. Rajabi, Assessment of homotopy perturbation and perturbation methods in heat radiation equations, Int. Commun. Heat Mass Transf. vol. 33 (2006) 391400.
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151
[9] D. Ganji, The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer, Phys. Lett. A 355 (2006) 337341. [10] H. Tari, D. Ganji, H. Babazadeh, The application of He’s variational iteration method to nonlinear equations arising in heat transfer, Phys. Lett. A 363 (2007) 213217. [11] M.O. Miansari, D.D. Ganji, M.E. Miansari, Application of He’s variational iteration method to nonlinear heat transfer equations, Phys. Lett. A 372 (2008) 779785. [12] A. Aziz, E. Huq, Perturbation solution for convecting fin with variable thermal conductivity, J. Heat Transf. 97 (1995) 300310.
FURTHER READING A. Rajabi, D. Ganji, H. Taherian, Application of homotopy perturbation method in nonlinear heat conduction and convection equations, Phys. Lett. A 360 (2007) 570573.
Radiation Heat Transfer Chapter Outline 3.1 Heat Transfer by Thermal Radiation 3.2 Nonlinear Heat Transfer Problems with Thermal Radiation 3.2.1 Cooling of a Lumped System by Combined Convection and Radiation 3.3 Results and Discussion 3.3.1 ConvectiveRadiating Cooling of a Lumped System with Variable Specific Heat 3.3.2 Temperature Distribution in a Radiating Thick Rectangular Plate 3.3.3 Temperature Distribution in a Radiating Thick Rectangular Plate 3.4 Conclusion References Further Reading
106 109
110 116
118 133 136 149 150 151
ABSTRACT Heat transfer from a body with a high temperature to a body with lower temperature, when bodies are not in direct physical contact with each other or Nonlinear Systems in Heat Transfer. DOI: http://dx.doi.org/10.1016/B978-0-12-812024-8.00003-5 © 2018 Elsevier Inc. All rights reserved.
105
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when they are separated in space, is called heat radiation. Unlike conduction and convection, heat transfer by thermal radiation does not necessarily need a material medium for the energy transfer. In the case of thermal radiation from a solid surface, the medium through which the radiation passes could be a vacuum, gas, or liquid. Within most practical engineering problems, usually all three heat transfer mechanisms, namely conduction, convection, and radiation, occur simultaneously. In this chapter, we will study some practical nonlinear heat transfer problems, in which radiation is of importance and acts concurrently with other heat transfer mechanisms. Differential equations, governing the physics of the problems will be introduced and finally solved via analytical approaches. Keywords: Heat transfer; thermal radiation; StefanBoltzmann constant; emissivity; perturbation method; homotopy perturbation method, variational iteration method
3.1 HEAT TRANSFER BY THERMAL RADIATION Heat transfer from a body with a high temperature to a body with a lower temperature, when bodies are not in direct physical contact with each other or when they are separated in space, is called heat radiation [1], as schematically shown in Fig. 3.1. All physical substances in solid, liquid, or
Radiation Heat Transfer Chapter | 3
107
FIGURE 3.1 Heat transfer by thermal radiation between two bodies.
gaseous states can emit energy via a process of electromagnetic radiation because of vibrational and rotational movement of their molecules and atoms [2]. The intensity of such energy flux depends upon the temperature of the body and the nature of its surface [3]. The radiation occurs at all temperatures, with the rate of emission increasing with the temperature. Unlike conduction and convection, heat transfer by thermal radiation does not necessarily need a material medium for the energy transfer. In the case of thermal radiation from a solid surface, the medium through which the radiation passes could be vacuum, gas, or liquid. Molecules and atoms of the medium can absorb, reflect, or transmit the radiation energy. If the medium is a vacuum, since there are no molecules or atoms, the radiation energy is not attenuated and, therefore, fully transmitted. Therefore, radiation heat transfer is more efficient in
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a vacuum. In the case of a gas (e.g., air), energy can be slightly absorbed or reflected by air molecules and the balance is transmitted. For liquid medium, most of the radiation is absorbed is a thin layer close to the solid surface and nothing is transmitted [2]. In the context of heat radiation, a surface that absorbs all incident radiation and reflects none is called a black surface or black body. The StefanBoltzmann law of thermal radiation for a black body states that the rate of radiation energy from the surface per unit area is proportional to the fourth power of the temperature of the body [1]: q 5 σAT 4
ð3:1Þ
with q rate of energy emission from the surface, A surface area of the radiator and σ the SetfanBoltzmann constant. If we consider a black body with surface temperatures T1 which radiates to another black body with surface temperature T2 that completely surrounds it, the second black body completely absorbs the incident energy and emits radiant energy that is proportional to T24 . The net rate heat transfer by thermal radiation is then given by: ð3:2Þ q 5 σA T14 2 T24 A black body is a perfect radiator. Real bodies, however, do not act like a perfect radiator and emit at a lower rate. To take
Radiation Heat Transfer Chapter | 3
109
into account the real nature of the radiant bodies, a factor ε, called emissivity, is introduced. Emissivity is defined as the ratio of the emission from a real “gray” surface to the emission from a perfect “black” surface. Then, the rate of radiation heat transfer from a real body at temperature T1 which is surrounded by a black body at temperature T2 , is given by: ð3:3Þ q 5 σA1 ε1 T14 2 T24 It is worth pointing out that in most of the practical engineering problems, usually all three-heat transfer mechanisms, namely conduction, convection, and radiation, occur simultaneously.
3.2 NONLINEAR HEAT TRANSFER PROBLEMS WITH THERMAL RADIATION In this section, we will study some practical nonlinear heat transfer problems in which radiation is of importance and acts concurrently with other heat transfer mechanisms. Differential equations governing the physics of the problems will be introduced and finally solved via analytical approaches, as explained in Chapter 1, Introduction to Nonlinear Systems and Solution Methods.
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3.2.1 Cooling of a Lumped System by Combined Convection and Radiation Consider the problem of combined convective radiative cooling of a lumped system [4]. Let the system have volume V, surface area A, density ρ, specific heat c, emissivity E, and the initial temperature Ti . At t 5 0. The system is exposed to an environment with convective heat transfer with the coefficient of h and the temperature Ta . The system also loses heat through radiation, and the effective sink temperature is Ts . The cooling equation and the initial conditions are given as: ρVc
dT 1 hAðT 2 Ta Þ 1 EσAðT 4 2 Ts4 Þ 5 0 dt ð3:4Þ t50
T 5 Ti
ð3:5Þ
To solve the cooling equation, we introduce the following dimensionless parameters: θ5
T Ti
τ5
θa 5
Ta Ti
t ρVca =hA
Ts Ti EσTi3 ε5 h θs 5
ð3:6Þ
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111
After the parameter change, the heat transfer equation will transform into the following form: dθðτÞ 1 ½θðτÞ2 θa 1 εðθðτÞ4 2 θ4s Þ 5 0 ð3:7Þ dτ with the initial condition θ ð 0Þ 5 1
ð3:8Þ
For simplicity, let us assume the case θa 5 θs 5 0. Therefore, the nonlinear differential equation that governs the cooling process will reduce to: dθðτÞ 1 θðτÞ 1 εθðτÞ4 5 0 dτ
ð3:9Þ
with the initial condition θ ð 0Þ 5 1
ð3:10Þ
The exact solution of Eq. (3.9) is obtained [4] as: 1 1 1 εθ3 ln 5τ 3 ð1 1 εÞθ3
ð3:11Þ
3.2.1.1 Variational Iteration Method (VIM) In order to solve Eq. (3.9) using the VIM, we construct a correction functional, as follows: ðτ dθn ðtÞ 4 1θn ðtÞ1εθ~ n ðtÞ dt θn11 ðτÞ5θn ðτÞ1 λ dt 0 ð3:12Þ
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Its stationary conditions can be obtained as follows: λ0 ðtÞ 2 λðtÞ 5 0
ð3:13Þ
1 1 λðtÞjt5τ 5 0
ð3:14Þ
The Lagrangian multiplier can, therefore, be identified as: λ 5 2et2τ
ð3:15Þ
As a result, we obtain the following iteration formula: θn11 ðτÞ
dθ ðtÞ n 1 θn ðtÞ 1 εθ4n ðtÞ dt 5 θn ðτÞ 2 et2τ dt 0 ð3:16Þ ðτ
Now, we start with an arbitrary initial approximation that satisfies the initial condition: θ0 ðτÞ 5 e2τ
ð3:17Þ
Then, using the above variational formula (3.16), we have: ðτ 4 t2τ dθ0 ðtÞ 1θ0 ðtÞ1εθ0 ðtÞ dt θ1 ðτÞ5θ0 ðτÞ2 e dt 0 ð3:18Þ
Radiation Heat Transfer Chapter | 3
113
Substituting Eq. (3.17) into Eq. (3.18) and after simplifications, we obtain the firstorder iteration solution as: 1 1 θ1 ðτÞ 5 e2τ 2 εe2τ 1 εe24τ 3 3
ð3:19Þ
In the same way, θ2 ðτÞ can also be obtained as follows: 1 1 2 θ2 ðτÞ 5 e2τ 2 εe2τ 1 εe24τ 1 ε2 e2τ 3 3 9 1
1 4 2τ 2 3 2τ 1 5 2τ 4 2 24τ ε e 2 εe 2 εe 2 ε e 81 27 1215 9
1
2 4 27τ 2 3 27τ 4 4 24τ εe 2 εe 2 εe 27 9 81
2 2 2 1 ε3 e24τ 1 ε3 e210τ 1 ε2 e27τ 9 27 9 1
1 5 216τ 1 5 24τ εe εe 1 1215 243
2
1 5 213τ 1 4 213τ 2 5 210τ εe εe 1 εe 1 243 81 243
2
4 4 210τ 2 5 27τ 2 εe εe 81 243 ð3:20Þ
and so on. In the same manner the rest of the components of the iteration formula can
114
Nonlinear Systems in Heat Transfer
be obtained to provide a more accurate approximate solution.
3.2.1.2 Homotopy Perturbation Method (HPM) After separating the linear and nonlinear parts of Eq. (3.9), we apply homotopy perturbation, as follows: Hðθ; pÞ 5 ð1 2 pÞ½LðθÞ 2 Lðu0 Þ 1 p½AðθÞ 2 f ðrÞ 5 0
ð3:21Þ
where LðθÞ is the linear part of the equation and Lðu0 Þ is the initial approximation [5]. We assume that the solution function can be written as a power series in p, as following: θ 5 θ0 1 pθ1 1 p2 θ2 1 ?
ð3:22Þ
Substituting Eq. (3.22) into Eq. (3.21) and rearranging based on powers of p-terms, we obtain: p0 :
dθ0 du0 1 θ0 2 2 u0 5 0 dτ dτ θ0 ð0Þ 5 1
p1 :
ð3:23Þ ð3:24Þ
dθ1 du0 1 θ1 1 1 u0 1 εθ40 5 0 ð3:25Þ dτ dτ θ1 ð0Þ 5 0
ð3:26Þ
Radiation Heat Transfer Chapter | 3
p2 :
dθ2 1 θ2 1 4εθ30 θ1 5 0 dτ θ2 ð0Þ 5 0
115
ð3:27Þ ð3:28Þ
One can easily solve Eqs. (3.23)(3.28) to find θi in Eq. (3.22). When p-1, the best approximate solution can be obtained as: 1 θðτÞ 5 e2τ 1 εðe24τ 2 e2τ Þ 3 ð3:29Þ 2 2 27τ 24τ 2τ 2 ε ð2 e 1 2e 2 e Þ 9
3.2.1.3 Conventional Perturbation Method For very small ε, we can assume a regular perturbation expansion [6] and calculate the first few terms in the form of θ 5 θ0 1 εθ1 1 ε2 θ2 1 . . .
ð3:30Þ
Substituting Eq. (3.30) into Eq. (3.9) and after expansion and rearranging based on coefficients of ε-terms, we have: ε0 : ε1 :
dθ0 1 θ0 5 0; dτ
dθ1 1 θ40 1 θ1 5 0; dτ
θ0 ð0Þ 5 1
ð3:31Þ
θ1 ð0Þ 5 0 ð3:32Þ
116
Nonlinear Systems in Heat Transfer
ε2 :
dθ2 1 θ2 1 4θ30 θ1 5 0; θ2 ð0Þ 5 0 ð3:33Þ dτ
Solving Eqs. (3.31)(3.33) gives θ0 ðτ Þ 5 e2τ 1 1 θ1 ðτ Þ 5 e24τ 2 e2τ 3 3 θ2 ðτ Þ 5
2 27τ 4 24τ 2 2τ e 2 e 1 e 9 9 9
ð3:34Þ ð3:35Þ ð3:36Þ
and, therefore, the approximate solution based on the first three terms in the perturbation expansion becomes: 1 θðτ Þ 5 e2τ 1 ε e24τ 2 e2τ 3 ð3:37Þ 2 2 27τ 24τ 2τ 1 ε e 2 2e 1 e 9
3.3 RESULTS AND DISCUSSION The approximate solution obtained by the VIM and the HPM are compared to the exact solution of the nonlinear convective radiative cooling problem in Fig. 3.2, Fig. 3.3 and Fig. 3.4 [7]. As can be observed, the differences among VIM, HPM, and the exact solution are negligible when the dimensionless parameter ε is very small, e.g., less
Radiation Heat Transfer Chapter | 3
117
1
0.9
VIM Method HPM Method Exact Solution
θ(τ)
0.8
0.7
0.6
0.5
0.4 0
0.25
0.5
τ
0.75
1
FIGURE 3.2 The approximate solution of two analytical approaches, VIM and HPM, compared to the exact solution at ε 5 0:4. 1 0.9
VIM Method HPM Method Exact Solution
θ(τ)
0.8 0.7 0.6 0.5 0.4
0
0.25
0.5
τ
0.75
1
FIGURE 3.3 The approximate solution of two analytical approaches, VIM and HPM, compared to the exact solution at ε 5 0:65.
than 0.4. With an increase of the dimensionless parameter, the results of VIM and HPM slightly diverge from the exact solution.
Nonlinear Systems in Heat Transfer
1 0.9
VIM Method HPM Method Exact Solution
0.8 0.7
θ(τ)
118
0.6 0.5 0.4
0
0.25
0.5
0.75
1
τ
FIGURE 3.4 The approximate solution of two analytical approaches, VIM and HPM, compared to the exact solution at ε 5 0:85.
Overall, both methods provide a reasonably accurate approximate solution and it seems that VIM leads to slightly better results, even for large values of ε. This is illustrated in Fig. 3.5 and Table 3.1, which show the errors on the approximate analytical solutions as a function of ε.
3.3.1 ConvectiveRadiating Cooling of a Lumped System with Variable Specific Heat For another example, we consider the problem of combined convectiveradiative cooling of a lumped system with a temperaturedependent specific heat [4]. Let the system have volume V, surface area A, density ρ,
Radiation Heat Transfer Chapter | 3
119
0.07 0.065 0.06
VIM / Exact HPM / Exact
0.055 0.05
Error
0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0
0.25
0.5
ε
0.75
1
FIGURE 3.5 Errors of VIM and HPM solutions at τ 5 0:5 as a function of the dimensionless parameter ε.
specific heat c, emissivity E, and the initial temperature Ti . At t 5 0, the system is exposed to an environment with convective heat transfer coefficient of h and temperature Ta . The system also loses heat through radiation and the effective sink temperature is Ts . Assume that the specific heat c is a linear function of temperature in the form of: c 5 ca ½1 1 βðT 2 Ta Þ
ð3:38Þ
where ca is the specific heat at the temperature Ta and β is a constant. The convective radiating cooling equation and the initial conditions can be established as follows: ρVc
dT 1 hAðT 2 Ta Þ 1 EσAðT 4 2 Ts4 Þ 5 0 dt ð3:39Þ
TABLE 3.1 The Results of VIM and HPM Techniques and Their Errors at τ 5 0:5 ε
VIM
HPM
EXACT
Error of VIM
Error of HPM
0
0.606531
0.606531
0.606531
1.64872E-10
1.64872E-10
0.1
0.591617
0.591638
0.591591
4.28955E-05
7.80444E-05
0.2
0.578207
0.578371
0.578023
0.000318544
0.000602619
0.3
0.566185
0.566732
0.56562
0.000999783
0.00196688
0.4
0.55544
0.55672
0.554217
0.002207405
0.004516592
0.5
0.545868
0.548335
0.543681
0.004021512
0.008559256
0.6
0.537369
0.541576
0.533903
0.006490236
0.014371001
0.7
0.52985
0.536445
0.524793
0.009636553
0.022201898
0.8
0.523226
0.53294
0.516275
0.013463676
0.032280117
0.9
0.517412
0.531062
0.508284
0.017959388
0.04481523
1
0.512333
0.530812
0.500765
0.023099547
0.060000859
Radiation Heat Transfer Chapter | 3
t50
T 5 Ti
121
ð3:40Þ
To solve Eq. (3.39), let us do the following changes of parameters: θ5 τ5
T ; Ti
θa 5 t
ρVca =hA
;
Ta ; Ti
θs 5
ε1 5 βTi ;
Ts ; Ti EσTi3 ε2 5 h ð3:41Þ
which transforms Eq. (3.39) into the following form: ½1 1 ε1 ðθ 2 θa Þ
dθ 1 ðθ 2 θa Þ 1 ε2 θ4 2 θ4s 5 0 dτ
with the initial condition θð0Þ 5 1
ð3:42Þ
For the sake of simplicity, we assume the case of θa 5 θs 5 0. Therefore, the governing convectiveradiating cooling equation will be written as: ð1 1 ε1 θÞ
dθ 1 θ 1 ε2 θ4 5 0 dτ
ð3:43Þ
with the initial condition θ ð 0Þ 5 1
ð3:44Þ
122
Nonlinear Systems in Heat Transfer
The exact solution of Eq. (3.43) is obtained [4] as: " 1=3 1 11ε2 θ3 1 ε1 1 ð11ε2 Þ3 ð11ε2 θ3 Þ ln ln 1 3 ð11ε2 Þθ3 3ε1=3 2 ð11ε2 Þð11ε1=3 θÞ3 2 2 !# 1=3 1=3 pffiffiffi 2ε 21 2ε θ21 5τ 1 3 arctan 2pffiffiffi 2arctan 2pffiffiffi 3 3 ð3:45Þ
3.3.1.1 Variational Iteration Method (VIM) First, we construct a correction functional that reads θn11 ðτÞ5θn ðτÞ ðτ dθ ðtÞ 4 n 1θn ðtÞ1ε2 θ~ n ðtÞ dt 1 λ ð11ε1 θ~ n ðtÞÞ dt 0 ð3:46Þ with the stationary conditions that can be obtained as follows: λ0 ðtÞ 2 λðtÞ 5 0
ð3:47Þ
1 1 λðtÞjt5τ 5 0
ð3:48Þ
Solving Eqs. (3.47) and (3.48) gives: λ 5 2et2τ
ð3:49Þ
Radiation Heat Transfer Chapter | 3
123
As a result, we obtain the following iteration formula: θn11 ðτÞ
ðτ
n dθn ðtÞ 5 θn ðτÞ 2 e ð1 1 ε1 θ~ n ðtÞÞ dt ð3:50Þ 0 o 4 1 θn ðtÞ 1 ε2 θ~ n ðtÞ dt t2τ
Now, let us begin with an arbitrary initial approximation, as follows: θ0 ðτÞ 5 e2τ
ð3:51Þ
Using the variational iteration formula (3.50), the first-order iteration can be obtained as: θ1 ðτÞ
ðτ
n dθ0 ðtÞ 5 θ0 ðτÞ 2 e ð1 1 ε1 θ~ 0 ðtÞÞ dt ð3:52Þ 0 o 4 ~ 1 θ0 ðtÞ 1 ε2 θ0 ðtÞ dt t2τ
Substituting Eq. (3.51) into Eq. (3.52) and after solving and some simplifications, we will obtain the first-order solution as: 1 θ1 ðτÞ 5 e2τ 1 ε1 e2τ 2 ε2 e2τ 2 ε1 e22τ 3 1 24τ 1 ε2 e 3 ð3:53Þ
124
Nonlinear Systems in Heat Transfer
In the same way, θ2 ðτÞ can be obtained as follows: 3 2 2 2 17 θ2 ðτÞ 5 ε21 e23τ 1 ε1 ε2 e22τ 2 ε32 e2τ 1 ε22 e2τ 2 ε1 ε2 e25τ 2 3 27 9 12 2
4 3 2 210τ 1 3 2 29τ 12 3 26τ 1 2 23τ 4 4 28τ 1 ε1 ε2 e 1 ε1 ε2 e 2 ε2 ε1 e 2 ε2 ε1 e ε ε e 27 1 2 2 5 2 63
1
1 4 213τ 1 2 2 29τ 1 2 22τ 1 5 2τ 2 2 2 27τ 1 ε1 ε2 e 2 ε2 ε1 e 2 ε ε1 e ε e 1 ε1 ε2 e 81 2 2 9 1215 2 3
1
2 2 3 212τ 212 4 2τ 1 3 2τ 1 1 5 213τ 1 ε ε e ε ε1 e 1 ε1 e 1 ε42 e213τ 2 ε e 33 1 2 36855 2 6 81 243 2
4 4 4 2 1 2 ε22 e24τ 2 ε22 ε1 e24τ 2 ε32 ε1 e27τ 1 ε22 ε1 e27τ 1 ε42 e2τ 9 3 9 3 81 1 2 1 7 41 1 ε21 e2τ 1 ε42 e27τ 2 ε2 ε4 e2τ 2 ε1 ε2 e2τ 2 ε2 ε21 e25τ 2 27 105 1 12 12 1
2 2 3 210τ 4 4 2 2 ε ε e 1 ε32 ε1 e210τ 2 ε31 ε22 e26τ 2 ε32 ε1 e211τ 1 ε21 ε32 e24τ 27 1 2 27 5 15 9
2
40 2 28τ 8 2 2 ε ε1 e 2 ε2 ε41 e25τ 1 ε21 ε32 e28τ 1 ε21 ε32 e26τ 1 ε22 e27τ 63 2 21 15 9
2 7 65 2 3 2τ 8 2 ε31 e24τ 2 ε2 ε21 e2τ 2 ε ε e 2 ε2 ε41 e25τ 1 ε21 ε32 e28τ 3 20 4158 1 2 21 1
2 2 3 26τ 2 2 27τ 2 3 24τ 7 65 2 3 2τ ε ε e 1 ε2 e 2 ε1 e 2 ε2 ε21 e2τ 2 ε ε e 15 1 2 9 3 20 4158 1 2
1
37 3 2 2τ 1 3 2τ 2 4 27τ 2 5 27τ 2 2 22τ ε ε e 2 ε1 ε2 e 2 ε2 ε1 e 2 ε e 1 ε2 ε1 e 1890 1 2 15 3 243 2 3
6 59 3 2τ 4 2 2 26τ 4 3 2 24τ 1 4 28τ 1 ε2 ε41 e26τ 2 ε ε1 e 2 ε1 ε2 e 2 ε1 ε2 e 1 ε2 ε1 e 5 945 2 5 9 7 8 8 1 5 216τ 8 4 ε e 2 ε21 ε22 e28τ 1 ε2 ε21 e26τ 1 1 ε32 ε1 e28τ 1 ε32 ε1 e24τ 7 5 1215 2 21 9 4 2 1 2ε21 ε22 e25τ 2 3ε31 ε2 e25τ 2 ε21 ε22 e24τ 1 2ε2 ε21 e24τ 1 ε42 ε1 e27τ 3 27 1 2 1 4 4 214τ 4 2 ε21 ε32 e29τ 2 ε1 e22τ 2 ε32 e27τ 1 ε2 e24τ 2 1 ε1 ε2 e24τ ε ε1 e 6 9 3 351 2 3 2 41 4 1 1 2 ε21 ε32 e27τ 1 ε22 ε1 e25τ 2 ε42 ε1 e24τ 2 ε32 ε1 e25τ 1 ε2 ε41 e24τ 9 36 81 3 3 2
4 4 210τ 2 3 2 5 210τ 2 3 24τ 1 ε32 e210τ 1 ε31 e23τ 1 1 ε2 e ε ε1 e ε e 81 2 27 2 243 2 9
2
4 4 24τ 4 23 2 2 2τ 23 2 2τ ε e 2 2ε21 e22τ 2 ε31 e22τ 2 ε42 e210τ 1 ε ε e 1 ε2 ε1 e 81 2 81 210 1 2 84
1 ε31 ε22 e25τ 2
2 2 3 211τ 4 3 24τ 2 1 1 ε1 ε2 e 1 ε42 ε1 e211τ 1 ε22 ε1 e25τ ε ε e 15 1 2 3 45 27
1 4 2 4 1 5 24τ 2 ε21 ε32 e25τ 2 ε31 ε22 e28τ 2 ε31 ε2 e27τ 1 ε31 ε22 e27τ 1 ε e 3 7 3 9 243 2
ð3:54Þ
Radiation Heat Transfer Chapter | 3
125
and so on. In the same manner the rest of the components of the iteration formula can be obtained.
3.3.1.2 Homotopy Perturbation Method (HPM) Applying homotopy perturbation to Eq. (3.43) results in: dν 4 Lðν Þ2Lðθ0 Þ1pLðθ0 Þ1p ε1 ν 1ε2 ν 50 ð3:55Þ dτ with dν 1ν dτ dθ0 Lðθ0 Þ 5 1 θ0 dτ Expanding the parameter ν as: L ðν Þ 5
ν 5 ν 0 1 pν 1 1 p2 ν 2
ð3:56Þ ð3:57Þ ð3:58Þ
and substituting it into Eq. (3.55) will result in: dν 0 dν 1 dν 2 dθ0 1p 1 p2 1 ν 0 1 pν 1 1 p2 ν 2 2 dτ dτ dτ dτ dθ0 dν 0 dν 1 2 2 θ0 1 p 1 θ0 1ε1 ν 0 p 1 ε1 ν 0 p dτ dτ dτ dν 0 2 1 ε1 ν 1 p 1 ε2 ν 40 p 1 4ε2 ν 1 ν 30 p2 5 0 dτ
ð3:59Þ
After rearranging based on powers of p-terms, we will have: p0 :
dν 0 dθ0 1 ν0 2 2 θ0 5 0 dτ dτ
ð3:60Þ
126
Nonlinear Systems in Heat Transfer
ν 0 ð0Þ 5 1 p1 :
dν 1 dθ0 dν 0 1 ν1 1 1 θ0 1 ε1 ν 0 1 ε2 ν 40 5 0 ð3:62Þ dτ dτ dτ ν 1 ð0Þ 5 0
p2 :
ð3:61Þ
ð3:63Þ
dν 2 dν 1 dν 0 1 ν 2 1 ε1 ν 0 1 ε1 ν 1 1 4ε2 ν 1 ν 30 5 0 ð3:64Þ dτ dτ dτ ν 2 ð0Þ 5 0
ð3:65Þ
To determine ν, Eqs. (3.60)(3.65) should be solved. Considering an initial approximation in the form of θ0 ðτ Þ 5 e2τ
ð3:66Þ
one can easily obtain: ð3:67Þ ν 0 5 e2τ 1 24τ 1 22τ ν 1 5 2ε1 e 1 ε2 e 1 ε1 2 ε2 e2τ ð3:68Þ 3 3 1 1 25τ ν 2 5ε21 e23τ 2 ε1 ε2 e 2ε1 ε1 2 ε2 e22τ 3 3 1 2 23τ 1 1 2 25τ 1 ε1 e 2 ε1 ε2 e 2ε1 ε1 2 ε2 e22τ 2ε1 ε2 e25τ 1 ε22 e27τ 2 12 3 9 4 1 1 1 24τ 2τ 2 1 ε2 ε1 2 ε2 e 1e 2ε1 1 ε1 ε2 12ε1 ε1 2 ε2 3 3 3 3 ! 1 1 2 4 1 1 e2τ 2 ε21 1 ε1 ε2 1ε1 ε2 2 ε22 2 ε2 ε1 2 ε2 2 12 9 3 3 ð3:69Þ
Radiation Heat Transfer Chapter | 3
127
As p-1, then ν-θ. Therefore, the approximate solution to the heat transfer equation can be obtained as: 1 1 2 17 θðτÞ5e2τ 12ε1 e2τ 1 ε2 e23τ 1ε1 2 ε2 1 ε1 ε2 e2τ 2 ε1 ε2 e24τ 3 3 3 12
!
2 4 4 7 1 2 22ε21 e2τ 1 ε22 e26τ 1 ε1 ε2 e23τ 2 ε22 e23τ 2 ε1 ε2 1 ε21 1 ε22 9 3 9 12 2 9 ð3:70Þ
3.3.1.3 Conventional Perturbation Method Since there are two different small parameters (ε1 and ε2 ), we assume θ as the following in solving Eq. (3.43) using the perturbation method [8]: θ 5 θ00 1 ε1 θ01 1 ε2 θ10 1 ε21 θ02 1 ε22 θ20
ð3:71Þ
Substituting Eq. (3.71) into Eq. (3.43), rearranging based on the coefficients of ε1 and ε2 and solving the resulting equations, the approximate solution by the perturbation method can be obtained as: 1 θ5e2τ 1ε1 e2τ 2e22τ 1 ε2 e24τ 2e2τ 3 2 1 1 ε21 e2τ 24e22τ 13e25τ 1 ε22 e2τ 22e24τ 1e27τ 2 9 1 1 ε1 ε2 27e2τ 18e22τ 116e24τ 117e25τ 12 ð3:72Þ
Nonlinear Systems in Heat Transfer
3.3.1.4
Results and Discussion
Fig. 3.6, Fig. 3.7, and Fig. 3.8 show the approximate solution to the heat transfer problem provided by HPM and VIM compared to the exact solution. The influence of small parameters, ε1 and ε2 , is shown. When both dimensionless parameters are relatively small, the analytical approaches provide accurate approximate solutions. By increasing the dimensionless parameter ε1 , the difference between the approximate solutions and the exact solution increases, while HPM seems to provide slightly more accurate results than VIM. The dimensionless parameter ε2 has a less significant influence of the results, and both analytical methods provide a reasonably accurate approximation. 1 Variational Method Homotopy Method Exact Solution
0.9
0.8
θ(τ)
128
0.7
0.6
0.5
0
0.25
0.5
0.75
1
τ
FIGURE 3.6 The approximate solution of HPM and VIM to the heat transfer problem, compared to the exact solution at ε1 5 0:4 and ε2 5 0:4.
Radiation Heat Transfer Chapter | 3
129
1 Variational Method Homotopy Method Exact Solution
0.9
θ(τ)
0.8
0.7
0.6
0.5 0
0.25
0.5
0.75
1
τ
FIGURE 3.7 The approximate solution of HPM and VIM to the heat transfer problem, compared to the exact solution at ε1 5 0:8 and ε2 5 0:4. 1 Variational Method Homotopy Method Exact Solution
0.9
θ(τ)
0.8
0.7
0.6
0.5
0.4 0
0.25
0.5
0.75
1
τ
FIGURE 3.8 The approximate solution of HPM and VIM to the heat transfer problem, compared to the exact solution at ε1 5 0:4 and ε2 5 0:8.
The errors of VIM and HPM at τ 5 0:5 and ε1 5 0:4 is shown in Fig. 3.9. It is obviously seen that the error of HPM is less than that
Nonlinear Systems in Heat Transfer
0.025 Variat./Exact Homot./Exact 0.02
Error
130
0.015
0.01
0.005
0.25
0.5
ε2
0.75
1
FIGURE 3.9 Errors of HPM and VIM as a function of the dimensionless parameter ε2 at τ 5 0:5 and ε1 5 0:4.
of VIM when ε2 , 0:5. If ε2 exceeds this limit, the error of HPM exceeds the error of VIM. A comparison between errors of VIM and HPM at τ 5 0:5 and ε1 5 0:8 is illustrated in Fig. 3.10, which shows that for a large magnitude of ε1 , the error of HPM is always less than the error of VIM. However, the overall error of VIM and HPM is reasonably low, which proves the capability and robustness of these analytical techniques for solving nonlinear heat transfer problems. A comparison between the homotopy perturbation method and the conventional perturbation method is provided in Fig. 3.11. It is worth pointing out that the simplest condition to solve Eq. (3.43) is when the dimensionless parameters ε1 5 ε2 5 0, where
Radiation Heat Transfer Chapter | 3
131
0.04 Variat./Exact Homoto./Exact
0.035
Error
0.03
0.025
0.02
0.015
0.01 0.25
0.5
ε2
0.75
1
FIGURE 3.10 Errors of HPM and VIM as a function of the dimensionless parameter ε2 at τ 5 0:5 and ε1 5 0:8.
FIGURE 3.11 Error (in %) of the conventional perturbation method, compared to that of the homotopy perturbation method at the initial time, when ε5ε1 5 ε2 .
132
Nonlinear Systems in Heat Transfer
the equation changes from nonlinear to linear. By increasing the dimensionless parameters, the equation becomes more nonlinear. The calculated errors clearly illustrate that for very small dimensionless parameters, the conventional perturbation method is as accurate as the homotopy perturbation method. However, when the nonlinearity of the problem increases, the error of the conventional perturbation technique increases drastically. This proves the robustness and stability of HPM technique, even for highly nonlinear heat transfer problems. Table 3.2 shows the errors of the conventional perturbation method and the homotopy TABLE 3.2 Comparison of the Errors of the Conventional Perturbation Method and the Homotopy Perturbation Method at τ 5 0 and when ε1 6¼ ε2 ε1
ε2
Error/Perturb. (%)
Error/HPM (%)
0.1
0.2
5.66
4.5
0.2
0.1
5.66
8
0.8
0.3
68
8
0.3
0.8
68
10.5
1.0
0.04
11.33
0
0.04
1.0
11.33
1.92
0.03
0.07
0.59
1.45
0.07
0.03
0.59
3.25
Radiation Heat Transfer Chapter | 3
133
perturbation method when ε1 ¼ 6 ε2 . As can be seen, for different combination of these parameters, HPM always seems to be reasonably accurate, while the conventional perturbation method only provides accurate approximation solution when ε1 , ε2 or both are very small.
3.3.2 Temperature Distribution in a Radiating Thick Rectangular Plate Now we consider a nonlinear equation, governing the temperature distribution in a uniformly thick rectangular fin radiating to free space with nonlinearity of high order [9,10] d2θ 2 εθ4 ðxÞ 5 0 ð3:73Þ 2 dx subject to the boundary conditions of dθ ð3:74Þ θð1Þ 5 1 ð0Þ 5 0 dx
3.3.2.1 Variational Iteration Method (VIM) To solve Eqs. (3.73) and (3.74) using VIM, we will first construct the following correction functional as: ðx 2 d θn ðX Þ 4 θn11 ðxÞ 5 θn ðxÞ 1 λ 2 εθn ðX Þ dX dX 2 0 ð3:75Þ
134
Nonlinear Systems in Heat Transfer
Making the above correction functional stationary, we obtain the following stationary conditions: λjX5x 5 0;
ð3:76Þ
0
ð3:77Þ
00
ð3:78Þ
1 2 λ jX5x 5 0; λ 50
The Lagrangian multiplier can therefore be identified as: λ5X2x
ð3:79Þ
Substituting Eq. (3.79) into the correction functional equation system (3.75) and considering that θn ðxÞ have to satisfy the boundary condition Eqs. (3.74), we define another coefficient as cn into Eq. (3.75), which has to be determined for each iteration subject to Eq. (3.74). Therefore, we will have the following iteration formula: θn11 ðxÞ 2
ðx d θn ðX Þ 4 2εθn ðX Þ dX 5cn θn ðxÞ1 ðX2xÞ dX 2 0 ð3:80Þ Using the iteration formula (3.80) and defining the initial approximation as θ0 ðxÞ 5 1, considering Eq. (3.74), the first two iterations can be calculated as: θ 1 ðx Þ 5
2 1 εx2 21ε
ð3:81Þ
Radiation Heat Transfer Chapter | 3
135
10;08015040 31x2 ε11680 4:51x4 ε2 1504 2:51x6 ε3 190x8 ε4 17x10 ε5 θ2 ðxÞ5 1080120;160ε19240ε2 11764ε3 190ε4 17ε5 ð3:82Þ
3.3.2.2
Results and Discussion
The exact solution cannot be easily obtained for the highly nonlinear heat transfer problem of Eq. (3.73). Therefore, to compare with the analytical approximate solutions, this equation is solved by the numerical method. Fig. 3.12 shows the comparison
FIGURE 3.12 The comparison of the results of the fourthorder VIM and the numerical solution.
136
Nonlinear Systems in Heat Transfer
between the solution of fourth-order VIM and the numerical solution for various dimensionless parameters ε. As can be seen, when ε is very small, the difference between VIM and the numerical solution is negligible. By increasing ε, the difference increases. However, even for large ε, VIM provides an accurate solution.
3.3.3 Temperature Distribution in a Radiating Thick Rectangular Plate The example to be studied is the one-dimensional heat transfer in a straight fin with the length of L and the cross area of A and the perimeter of P (see Fig. 3.13). The fin surface transfers heat through both convection and radiation. Suppose the temperature of the surrounding air is T0 and the effective sink temperature for the radiative heat transfer is Ts. We assume that the base
FIGURE 3.13
Geometry of a straight fin.
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temperature of the fin is Tb and there is no heat transfer of the tip of the fin. It is also assumed that the convection heat transfer coefficient, h, and the emissivity coefficient of surface, Eg, are both constant while the conduction coefficient, k, can be variable. The energy equation and the boundary conditions for the fin are as follows [11]: d dT hp Eg σ 4 k 2 ðT 2Ta Þ2 T 2TS4 50 dx dx A A ð3:83Þ x50-
dT 50; x5L-T 5Tb dx
ð3:84Þ
Assuming k as a linear function of temperature, we have: k 5 ka ð1 1 βðT 2 Ta ÞÞ
ð3:85Þ
After making the equation dimensionless and changing parameters, we have: T Ta Ts ; θa 5 ; θs 5 ; Tb Tb Tb x 2 hpL2 Eg σTb3 pL3 X5 ;N 5 ; ε1 5 βTb ; ε2 5 L ka A ka A ð3:86Þ θ5
Substituting Eq. (3.86) into Eq. (3.83), we obtain:
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d dθ ½1 1 ε1 ðθ 2 θa Þ 2 N 2 ðθ 2 θ a Þ dx dX 2 ε2 θ4 2 θ4s 5 0 ð3:87Þ X 5 0-
dθ 5 0 X 5 1-θ 5 1 dX
ð3:88Þ
By assuming θa 5 θs 5 0, Eq. (3.87) transforms to 2 2 d2θ dθ d θ 2 2N θ1ε 1ε θ 2ε2 θ4 50 1 1 2 2 dX dX dX ð3:89Þ
3.3.3.1 Variational Iteration Method (VIM) According to the VIM, we can construct the correction functional as follows: ðx
00 2 θn11 ðxÞ5θn ðxÞ1 λ θn ðtÞ2N 2 θn t 1ε1 θ~ n ðtÞ 0
00 4 1ε1 θ~ n ðtÞθ n ðtÞ2ε2 θ~ n ðtÞ dt
ð3:90Þ where λ is general Langrange multiplier. Making the above correction functional stationary, we can obtain the following stationary conditions:
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0
λ ðtÞ2N 2 λðtÞ50; 12λ ðtÞjt5x 50; λðtÞjt5x 50 ð3:91Þ The Lagrange multiplier therefore can be identified as:
1 eNðx2tÞ 2 eNðt2xÞ λ52 ð3:92Þ 2 N Substituting Eq. (3.92) into Eq. (3.90) and considering that θn ðxÞ has to satisfy Eq. (3.88), another coefficient, Cn, can be introduced to Eq. (3.90), which has to be determined for each iteration subject to Eq. (3.88). We can then write down the following iteration formula: ðx θn11 ðxÞ5Cn θn ðxÞ2 (
1 eN ðx2tÞ 2eN ðt2xÞ 2 N
0
00 2 θ n ðtÞ2N 2 θn t 1ε1 θ~ n ðtÞ1ε1 θn ðtÞθ n ðtÞ ) ! 4 2ε2 θ~ ðtÞ dt ð3:93Þ 00
n
Now we are to start with an arbitrary initial approximation that satisfies the conditions: θ0 ðxÞ 5 sechðN Þ coshðNxÞ:
ð3:94Þ
Using the above variational formula (3.93), we have:
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( 1 eNðx2tÞ 2eNðt2xÞ 00 θ 0 ðtÞ2N 2 θ0 ðtÞ θ0 ðxÞ2 2 N 0 ) ! ðx
θ1 ðxÞ5C0
00
1ε1 θ20 ðtÞ1ε1 θ0 ðtÞθ 0 ðtÞ2ε2 θ40 ðtÞ dt ð3:95Þ Substituting Eq. (3.94) into Eq. (3.95) and after simplifications, we obtain: θ1 ðxÞ5 C0 sechðNÞcoshðNxÞ1 3
1 240ðN 2 cosh4 ðNÞÞ
40eð5NxÞ ε1 N 2 cosh2 ðNÞ224eð3NxÞ ε2 140eð3NxÞ
3ε1 N 2 cosh2 ðNÞ224eð5NxÞ ε2 2120ε1 N 2 cosh2 ðNÞeð4NxÞ 220ε2 eð2NxÞ 120ε1 N 2 eð6NxÞ cosh2 ðNÞ290ε2 eð4NxÞ 120ε1 N 2 eð2NxÞ cosh2 ðNÞ220ε2 eð6NxÞ ! 2ε2 eð8NxÞ 2ε2 eð24NxÞ ð3:96Þ where C0 5
and
1 A
ð3:97Þ
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A 5 sechðNÞcoshðNÞ 1
141
1 240ðN 2 cosh4 ðNÞÞ
3 ð40eð3NÞ ε1 N 2 cosh2 ðNÞ 2 24eð5NÞ ε2 1 40eð5NÞ ε1 N 2 cosh2 ðNÞ 2 24eð3NÞ ε2 2 120ε1 N 2 cosh2 ðNÞeð4NÞ 2 20ε2 eð6NÞ
ð3:98Þ
1 20ε1 N 2 eð2NÞ cosh2 ðNÞ 1 90ε2 eð4NÞ 1 20ε1 N 2 eð6NÞ cosh2 ðNÞ 2 20ε2 eð2NÞ 2 ε2 eð8NÞ 2 ε2 Þeð24NÞ and so in. In the same manner, the rest of the components of the iteration formula can be obtained.
3.3.3.2
Fin Efficiency
The heat transfer rate from the fin is found by using Newton’s law cooling [11] ðb Q5
pðT 2 Ta Þdx
ð3:99Þ
0
The ratio of the actual heat transfer from the surface to that would transfer if the whole fin surface were at the same temperature as the base is commonly called as the fin efficiency
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Q η5 5 Qideal
Ðb
0 pðT 2 Ta Þdx 5 pbðT 2 Ta Þ
ð1 θðxÞdx: x50
ð3:100Þ Integrating Eq. (3.100), the efficiency of straight fins is obtained as an analytical expression as follows: η5
E F
ð3:101Þ
where E52 2480N 2 eð5NÞ cosh3 ðNÞ140ε1 N 2 eð2NÞ 3cosh2 ðNÞ1480N 2 eð3NÞ cosh3 ðNÞ2160eð5NÞ 3ε1 N 2 cosh2 ðNÞ140ε2 eð6NÞ 1480ε1 N 3 eð4NÞ 3cosh2 ðNÞ296eð3NÞ ε2 2ε2 1160eð3NÞ ε1 N 2 3cosh2 ðNÞ240ε2 eð2NÞ 196eð5NÞ ε2 2360ε2 Neð4NÞ 1ε2 eð8NÞ 240ε1 N 2 eð6NÞ cosh2 ðNÞ
ð3:102Þ and F5 4N 240cosh4 ðNÞN 2 eð4NÞ 140eð3NÞ ε1 N 2 3cosh2 ðNÞ224eð5NÞ ε2140eð5NÞε1 N 2 cosh2 ðNÞ 224eð3NÞ ε2 2120ε1 N 2 eð4NÞ cosh2 ðNÞ 190ε2 eð4NÞ 120ε1 N 2 eð2NÞ cosh2 ðNÞ 220ε2 eð6NÞ 120ε1 N 2 eð6NÞ cosh2 ðNÞÞ
ð3:103Þ
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3.3.3.3 Homotopy Perturbation Method Let us construct the homotopy perturbation method in the following form 2 dν Lðν Þ 2 Lðθ0 Þ 1 pLðθ0 Þ 1 p ε1 dX ! 2 d ν 1 ε1 ν 2 ε2 ν 4 5 0 2 dX ð3:104Þ Assuming d 2 θ0 ν 5 ν 0 1 pν 1 1 p ν 2 ; θ0 5 5 0 ð3:105Þ dX 2 2
and substituting Eq. (3.105) into Eq. (3.104) and after organizing based on p-powers, we have: d2ν 0 p: 2 N 2 ν 0 5 0; 2 dX 0
X 5 0-
dν 0 5 0; dX
ð3:106Þ
X 5 1-ν 0 5 1 ð3:107Þ
2 2 d ν dν 0 d2ν 0 1 2 2 N ν 1 ε 1 ε ν p1 : 1 1 1 0 dX 2 dX dX 2 1 ε2 ν 40 5 0; ð3:108Þ
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X 5 0-
dν 1 5 0; dX
X 5 1-ν 1 5 1 ð3:109Þ
d2ν 2 dν 0 dν 1 d2ν 1 2 2 N ν2 1 2 p: 1 ε1 ν 0 dX 2 dX dX dX 2 d2ν 0 1 ε1 ν 1 1 4ε2 ν 1 ν 30 5 0; 2 dX ð3:110Þ 2
X 5 0-
dν 2 5 0; dX
X 5 1-ν 2 5 1 ð3:111Þ
Solving Eqs. (3.106)(3.111), we obtain: ν 0 5 sech N cosh NX
can
ð3:112Þ
3 sech4 N 4 ν 1 5sechN 2 2 ε2 sech N1ε2 8N 120N 2 !! 2 4 ε1 sech N ε2 sech N 3cosh4N1cosh2N 1 3 6N 2 sech4 N 3ε2 cosh4NX1 sech4 N 3coshNX2ε2 2 2 120N 8N ! ε1 sech2 N ε2 sech4 N cosh2NX 2 1 3 6N 2 ð3:113Þ
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ν 2 5sechNð2A3 cosh2N2A5 cosh3N 2A7 cosh4N2A9 cosh5N2A11 cosh7N2A14 Þ 3coshNX1A3 cosh2NX1A5 cosh3NX 1A7 cosh4NX1A9 cosh5NX1A11 cosh7NX1A14 ð3:114Þ where A3 5
2 C0 ε1 C0 sechN 2 sechN 3 6 ε2 C0 sech3 N ε2 C0 sech3 N 1 1 2 6N 2 2N 2
ð3:115Þ
ε2 sech5 N C00 sechN ε1 ε2 sech5 N 1 A5 5 1 240N 2 4 120N 2 ε1 ε1 C0 sechN ε1 ε2 1 1 C00 sechN 2 16 4 8 3 240N 2 ε1 C00 ε22 sech7 N 5 3 sech N 1 sechN 1 640N 4 16 3ε2 sech7 N 3ε2 C00 sech3 N 1 1 8N 2 160N 2 ð3:116Þ A7 5
ε22 C0 sech3 N 30N 2
ð3:117Þ
ε1 sech5 N ε1 ε2 1 sech5 N 1 A9 5 2 2 30 3 24N 15 3 24N ε1 ε2 ε22 sech7 N 5 sech N 2 24 3 80N 4 24 3 240N 2 ε2 C00 sech3 N 2 48N 2
ð3:118Þ
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2ε22 sech7 N A11 5 48 3 240N 4 A14 5 2C0 sechN 1
ε1 C0 sechN 2
3ε2 C0 sech3 N 2 2N 2
ð3:119Þ
ð3:120Þ
and 3 ε2 sech4 N 4 ε sech N 1 cosh4N 2 8N 2 120N 2 ! ! 2 4 ε1 sech N ε2 sech N cosh2N 1 1 3 6N 2
C0 5 sechN 2
ð3:121Þ C00 5
3.3.3.4
ε1 sech2 N ε2 sech4 N 1 3 6N 2
ð3:122Þ
Results and Discussion
Fig. 3.14 shows the temperature distribution in convectiveradiative conduction fins with variable thermal conductivity, obtained by VIM at N 5 1. Fig. 3.15 shows the variation of the fin efficiency with the thermo-geometric fin parameter for different values of the thermal conductivity parameter, obtained by VIM.
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FIGURE 3.14 Temperature distribution in convectiveradiative conduction fins with variable thermal conductivity at N 5 1.
FIGURE 3.15 Variation of the fin efficiency with the thermo-geometric fin parameter for different values of the thermal conductivity parameter.
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FIGURE 3.16 Three term approximated solution, obtained by HPM.
FIGURE 3.17 Approximate solution obtained via the conventional perturbation method.
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In Fig. 3.16, the three term approximated solution (ν 0 1 pν 1 1 p2 ν 2 , when p-1), obtained by HPM, is illustrated from X 5 0 to X 5 1, for a fixed ε2 5 0:2 and for different values of ε1 5 0; 0:2; 0:4; and 0:6. In Fig. 3.17, the approximate solution of Eq. (3.89), obtained via the conventional perturbation method [12], is depicted for the same ε1 and ε2 .
3.4 CONCLUSION In this chapter, a couple of nonlinear heat transfer problems were discussed, with a focus on heat radiation. In most of the practical engineering problems, usually all three heat transfer mechanisms, namely conduction, convection and radiation, occur simultaneously. Therefore, in the examples discussed, all of these heat transfer modes were present. The mathematical equations governing the physics of the problems were established and solved using the analytical approaches, as explained in Chapter 1, Introduction to Nonlinear Systems and Solution Methods. The exact solution or the numerical solution were used to assess the validity and accuracy of the analytical approaches. It was observed that analytical methods, providing approximate solutions, can be powerful tools to analyze the highly nonlinear heat transfer problems containing a combination
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of conduction, convection, and radiation and provide accurate approximate solutions in problems that the exact solutions do not exit. Furthermore, they can be an easy-touse and yet accurate alternatives for numerical techniques, which could be difficult, time consuming, and sensitive to the initial solution and it is very difficult to converge to a solution in case of strong nonlinearity.
REFERENCES [1] Y. Rao, Heat Transfer, Universities Press (India) Limited, Hyderabad, India, 2001. [2] S. Sukhatme, A Textbook On Heat Transfer, Universities Press (India) Private Limited, Hyderabad, India, 2005. [3] J.H. Lienhard, A Heat Transfer Textbook, fourth ed., Dover Publications, Inc, Mineola, NY, 2011. [4] A. Aziz, T. Na, Perturbation Method in Heat Transfer, Hemisphere Publishing Corporation, 1984. [5] J. He, Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput. 135 (2003) 7379. [6] A. Nayfeh, Perturbation Methods, Wiley, New York, 1973. [7] H. Khaleghi, D. Ganji, A. Sadighi, Application of variational iteration and homotopy-perturbation methods to nonlinear heat transfer equations with variable coefficients, Numer. Heat Transf. Part A Appl. 52 (2007) 2542. [8] D. Ganji, A. Rajabi, Assessment of homotopy perturbation and perturbation methods in heat radiation equations, Int. Commun. Heat Mass Transf. vol. 33 (2006) 391400.
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[9] D. Ganji, The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer, Phys. Lett. A 355 (2006) 337341. [10] H. Tari, D. Ganji, H. Babazadeh, The application of He’s variational iteration method to nonlinear equations arising in heat transfer, Phys. Lett. A 363 (2007) 213217. [11] M.O. Miansari, D.D. Ganji, M.E. Miansari, Application of He’s variational iteration method to nonlinear heat transfer equations, Phys. Lett. A 372 (2008) 779785. [12] A. Aziz, E. Huq, Perturbation solution for convecting fin with variable thermal conductivity, J. Heat Transf. 97 (1995) 300310.
FURTHER READING A. Rajabi, D. Ganji, H. Taherian, Application of homotopy perturbation method in nonlinear heat conduction and convection equations, Phys. Lett. A 360 (2007) 570573.