Convective-radiative fin with temperature dependent thermal conductivity, heat transfer coefficient and wavelength dependent surface emissivity

Propulsion and Power Research 2014;3(4):207–221

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http://ppr.buaa.edu.cn/

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ORGINAL ARTICLE

Convective-radiative fin with temperature dependent thermal conductivity, heat transfer coefficient and wavelength dependent surface emissivity Surjan Singha,n, Dinesh Kumara, K.N. Raib a

Research Scholar DST-Centre for Interdisciplinary Mathematical Sciences, Banaras Hindu University, Varanasi 221005, Utter Pradesh, India b Department of Mathematical Sciences, IIT, Banaras Hindu University, Varanasi, India Received 26 April 2014; accepted 3 November 2014 Available online 31 December 2014

KEYWORDS Fin; Wavelet; Convective; Emissivity; Wavelength

Abstract In this paper, we have studied heat transfer process in a continuously moving fin whose thermal conductivity, heat transfer coefficient varies with temperature and surface emissivity varies with temperature and wavelength. Heat transfer coefficient is assumed to be a power law type form where exponent represent different types of convection, nucleate boiling, condensation, radiation etc. The thermal conductivity is assumed to be a linear and quadratic function of temperature. Exact solution obtained in case of temperature independent thermal conductivity and in absence of radiation conduction parameter is compared with those obtained by present method and is same up to ten decimal places. The whole analysis is presented in dimensionless form and the effect of variability of several parameters namely convection-conduction, radiation-conduction, thermal conductivity, emissivity, convection sink temperature, radiation sink temperature and exponent on the temperature distribution in fin and surface heat loss are studied and discussed in detail. & 2014 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. All rights reserved.

1. Introduction n

Corresponding author. Tel.: þ91 8726598030.

E-mail addresses: [email protected] (Surjan Singh), [email protected] (Dinesh Kumar), [email protected] (K.N. Rai). Peer review under responsibility of National Laboratory for Aeronautics and Astronautics, China.

In mechanical process heat is generated in machines. The significant question arises here is that how to release this heat in environment. Fins or extended surfaces are used to release heat in environment. In many industrial applications such as hot rolling, optical fiber and casting, exchange of heat with the

2212-540X & 2014 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jppr.2014.11.003

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Nomenclature Ac Cp C0 h hb hp k(T) ka L P q T Ta Tb Ts U x

cross section area of fin (unit: m2) specific heat of the material (unit: J/(kg  K)) speed of light (unit: m/s) convection heat transfer coefficient (unit: W/(m2  K)) convection heat transfer coefficient at the base (unit: W/(m2  K)) plank constant (unit: J  s) temperature-dependent thermal conductivity (unit: W/(m  K)) thermal conductivity at the convection sink temperature Ta (unit: W/(m  K)) fin length (unit: m) fin perimeter (unit: m) surface heat loss by combined convection and radiation (unit: W) local fin temperature (unit: K) convection sink temperature (unit: K) fin base temperature (unit: K) sink temperature for radiation (unit: K) speed of moving fin (unit: m/s) axial distance measured from the base of fin (unit: m)

exp Nc Nr Pe Q Qa Qc X

exponential function convection-conduction parameter radiation-conduction parameter Peclet number surface heat loss, dimensionless advection component of heat loss base heat conduction axial distance measured from the base of fin

Greek symbols β ε εs λ θ θa θs ρ σ

measure of surface emissivity variation with temperature (unit: K-1) fin surface emissivity dimensionless surface emissivity at the radiation sink temperature Ts wavelength (unit: m) dimensionless temperature dimensionless convection sink temperature dimensionless radiation sink temperature density of material (unit: kg/m3) Stefan-Boltzmann constant (unit: W/(m3  K4))

Dimensionless parameters A B

thermal conductivity parameter surface emissivity parameter, when surface emissivity depend on temperature

ambient while it is in continuous motion. The book written by A.D. Kraus et al. [1] provides knowledge about different type extended surfaces and used power law type heat transfer coefficient in chapter 18. Heaslet and Lomax [2] first analyzed radiating fins with variable thermal conductivity and variable emissivity on the fin faces. They considered longitudinal fins of rectangular profile. Stockman and Kramer [3] studied one dimensional heat flow in fin and tube configuration, and considered the variation of thermal conductivity and emissivity as linear functions of temperature. Campo and Wolko [4] studied rectangular fin with power law type variations in thermal conductivity and emissivity with temperature as k¼ k0Tn, ε¼ ε0Tm. They considered fin with constant base temperature and insulated tip. Jaluria and Singh [5] modeled such a problem and solved it numerically. They studied the effect of Biot number and Peclet number on temperature distribution in the material and the surface heat loss. Karwe and Jaluriya [6] used Crank-Nicolson finite difference method to compute the temperature fields in the fluid and in the moving material. Choudhury and Jaluria [7] found a double series solution for the two dimensional, transient temperature distributions in a moving rod or a plate moving with a constant speed and losing heat by convection to the ambient fluid through a constant heat transfer coefficient. Aziz and Lopez [8] studied the heat transfer process in a continuously moving

sheet or rod of variable thermal conductivity that releases heat by simultaneous convection and radiation. They solved this problem using Runge-Kutta-Fehlberg method and effect of several parameters was studied in detail. Aziz and Khani [9] studied convection-radiation of a continuously moving fin of variable thermal conductivity. They solved this problem using homotopy analysis method and analysed the effect of several parameters. Torabi et al. [10] studied heat transfer in a moving fin with variable thermal conductivity, which releases heat by simultaneous convection and radiation to its surroundings. Differential Transform Method was used in solution and the effects of several parameters were also studied. Aziz and Torabi [11] studied convective-radiative fin with thermal conductivity and surface emissivity as a linear function of temperature, and also studied heat transfer coefficient as power law type form of temperature. The Runge-Kutta-Fehlberg method of fourth and fifth order was used in solution [11]. Shukla [12] studied the temperature distribution in a sublimation-cooled coated cylinder in convective and radiative environments. Torabi and Yaghoobi [13], studied heat transfer in straight fin with a step thickness and thermal conductivity considered as temperature dependent. Differential transform method (DTM) and variational iteration method (VIM) have been used in the solution. They conclude that DTM results are more accurate in comparison to VIM and HPM.

Convective-radiative fin with temperature dependent thermal conductivity

In power law type form exponent ‘n’ has specific meaning in heat transfer process and depends on the mode of heat transfer. For example exponent n is 1/4 for laminar natural convection and 1/3 for turbulent natural convection and many other values of n studied by Holman [14] in chapter 7. Typical values of ‘n’ are -1/4 for laminar film boiling or condensation, 2 for nucleate boiling, 3 for radiation and 0 for constant heat transfer coefficient. Several researchers studied straight rectangular fin problems with power law type heat transfer coefficient and using different method. Chang [15], Atay and Coskum [16] and Khaini et al. [17] used the adomian decomposition method, variation iteration method and homotopy analysis method respectively. Efficiency of fin with temperature dependent thermal conductivity was obtained as a function of thermo-geometric fin parameter. Cengel [18], and Bergman et al. [19] studied fundamental of radiation and emissivity. The emissivity of a real surface is not a constant. Rather, it varies with temperature as well as the wavelength and the direction of emitted radiation. No solution is available when thermal conductivity, heat transfer coefficient and surface emissivity varies with temperature and surface emissivity varies with temperature and wavelength. In this study an attempt has been made to solve the nonlinear boundary value problem describing the process of heat transfer through continuously moving fin with simultaneous variation of thermal conductivity, power law type heat transfer coefficient. We consider a real surface of fin material whose emissivity varies (i) linearly with temperature and (ii) temperature and wavelength. Four particular cases discussed in detail (i) when thermal conductivity and emissivity is a linear function of temperature (ii) when thermal conductivity is quadratic function of temperature and emissivity is a linear function of temperature (iii) when thermal conductivity is linear function of temperature and emissivity depends on temperature and wavelength (iv) when thermal conductivity is quadratic function of temperature and emissivity depends on temperature and wavelength.

2. Formulation of the problem We consider the thermal processing of a plate or a rod of cross-sectional area ‘Ac’ and perimeter ‘P’, while it moves horizontally with a constant speed ‘U’. The hot plate or rod emerges from a die or furnace at a constant temperature ‘Tb’. The motion of the plate or rod may induce a flow field otherwise quiescent surrounding medium or alternatively, the plate or rod makes us feel an externally driven flow over its surface. We consider hot plate or rod release heat in surrounding medium by convection and radiation. If natural or forced convection is weak then radiation plays an important role in heat transfer. ‘Tb’ is the base temperature, the sink temperatures for convection and radiation are denoted by ‘Ta’ and ‘Ts’ respectively. Fin tip is assumed to be adiabatic. The thermal conductivity of fin material k (T), the convective heat transfer coefficient h(T) are

209

Figure 1 Geometry of moving fin.

assumed to be function of temperature of the form
 T ; kðTÞ ¼ ka f Tb 

T Ta hðTÞ ¼ ha f Tb Ta

n

ð1Þ

ð2Þ

We consider a real surface of fin material whose emissivity varies (i) Linearly with temperature [11] i.e. εðTÞ ¼ εs ½1 þ βðT  T s Þ;

ð3Þ

for real material εs oo 1, and (ii) With temperature and wavelength [16] i.e. Eðemisive power of real bodyÞ εðλ; TÞ ¼ Eb ðemisive hpower  of black  ibodyÞ ¼

2πhp C 20 =λ5 exp

1

σT 4 1

0

¼ εs @

hp C 0 λK b T

 exp

1

hp C 0 λK b T



1

A

ð4Þ

2πh C 2 =λ5

p 0 for real material εs ¼ σT oo1, 4 where ka is the thermal conductivity of fin corresponding to the base temperature Tb, hb is the convection heat transfer corresponding to the temperature difference ‘Tb - Ta’ and ‘εs’ is the surface emissivity at the radiation sink temperature Ts, β is the measure of variation of surface emissivity with temperature. hp ¼ 6.626  10  34 J  s, Plank constant, Kb ¼ 1.381  10  23 J/K, Boltzmann constant, C0 ¼ 2.988  108 m/s, speed of light in vacuum ‘λ’ wavelength (m). In Figure 1, geometry of moving fin is presented which is given in [8]. The steady state energy balance equation for the material moving with a constant speed and loosing heat by simultaneous convection and radiation may be written as:
 d dT hðTÞP εðTÞσP  4 kðTÞ ðT  T a Þ T  T 4s  dx dx Ac Ac



ka dT ¼ 0; U αAc dx

ð5Þ

where α¼ ka/(ρCp) is the thermal diffusivity of the material, ρ is the density and Cp is the specific heat. The last term on the left in Eq. (5) is the advection term.

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Surjan Singh et al.

The system subjected to boundary conditions as follows x ¼ 0;

T ¼ Tb dT ¼ 0: dx

x ¼ L;

ð6Þ

where 2k  1 m  1

CT ψðXÞ ¼ ∑ ∑ C n;m ψ n;m ðXÞ n¼1m¼0

ð7Þ and Z

1

Introducing the dimensionless variables and similarity criteria

C n;m ¼

T Ta Ts x hb PL2 ; θa ¼ ; θs ¼ ; X ¼ ; Nc ¼ Tb L Tb Tb k a Ac εs σPL2 T 3b UL hp C 0 Nr ¼ ; Pe ¼ ; H¼ : αAc k a Ac λK b T b

C and ψ(X) are M  1 matrices given by " #T c1;0 ; c1;1 ; :::; c1;M  1 ; c2;0 ; c2;1 ; :::; c2;M  1 ; C¼ c2k  1 ;0 ; c2k  1 ;1 ; :::; c2k  1 ;M  1;

θ¼

ð8Þ

0

f ðXÞψ n;m ðXÞ:

ð14Þ

ð15Þ

and The Eqs. (1) to (3) can be written in dimensionless form k(T) ¼ ka f(θ), where f(θ) is function of temperature.
 θ  θa n hðTÞ ¼ hb ; 1  θa εðTÞ ¼ εs ½1 þ Bðθ  θs Þ; 2 3   1 1  5; εðT; λÞ ¼ εs ¼ εs 4  1 expðH=θÞ 1 1 exp ð1=HÞθ where 1/H is surface emissivity parameter, when surface emissivity depend on wavelength and temperature, 0r1/Hr1.

2

3 ψ 1;0 ðXÞ; ψ 1;1 ðXÞ; :::; ψ 1;M  1 ðXÞ; ψ 2;0 ðXÞ; T 6 7 ψðXÞ ¼ 4 ψ 2;1 ðXÞ; :::; ψ 2;M  1 ðXÞ; ψ 2k  1 ;0 ðXÞ; 5 ψ 2k  1 ;1 ðXÞ; :::; ψ 2k  1 ;M  1 ðXÞ

ð16Þ The Legendre wavelet ψ n;m ðXÞ ¼ ψðk; n̂; m; XÞ; n̂ ¼ 2n  1; n ¼ 1, 2, …, 2k  1, k is any positive integer, m is the order of Legendre polynomials and ‘X’ is defined on the interval [0, 1] by ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm þ 1=2Þ 2k=2 Pm ð2k X n̂ Þ; ψ n;m ðXÞ ¼ 0;

n̂  1 2k

r X r n2þ1 k ; otherwise ̂

ð17Þ

The system of Eqs. (5) to (7) reduce in the following form (i) Surface emissivity varies linearly with temperature f ðθÞ


2
 d2 θ dθ θ  θa n þ f 'ðθÞ  Nc ðθ  θ Þ a dX 1 θa dX 2  4 dθ ¼ 0; 0r X r 1;  Nr½1 þ Bðθ  θs Þ θ  θ4s  Pe dX

ð9Þ

X ¼ 0;

θ¼1

X ¼ 1;

dθ ¼ 0: dX

P0 ðXÞ ¼ 1; P1 ðXÞ ¼ X; Pmþ1 ðXÞ ¼ 

(ii) Surface emissivity varies with temperature and wavelength
2
 d2 θ dθ θ  θa n f ðθÞ 2 þ f 'ðθÞ  Nc ðθ  θa Þ dX 1  θa dX  4 1 dθ 4  ¼ 0; 0 r X r1;  Nr θ θs  Pe dX exp Hθ 1

where m ¼ 0, 1, …, M 1 and n¼ 1, 2,…, 2k-1. Here Pm(X) is the well known Legendre polynomials of order m.

m Pm  1 ðXÞ; m ¼ 1; 2; 3; :::; M  1; mþ1

ð11Þ ð12Þ

ð18Þ

Integrating Eq. (13) with respect to X from 0 to X, we have θ0 ðXÞ ¼ θ0 ð0Þ þ CT PψðXÞ

ð19Þ

where P is 2 M  2 M, k¼ 1, operational matrix of integration given by Razzaghi and Yosefi [20]. k-1

ð10Þ

2m þ 1 X Pm ðXÞ mþ1

2

1

p1ffiffi 3

k-1

0

6 p1ffiffi p1ffiffiffiffi 6 3 0 15 6 6 ffi 0 60  p1ffiffiffi 15 16 P¼ 6 ffi 0 0  p1ffiffiffi 26 35 6 6 6⋮ 6 4 0 0 ⋯

0 0 p1ffiffiffiffi 35

⋱

⋯

⋯

0

⋯

0

3

7 7 7 7 7 ⋯ 0 7 7 7 ⋮ 7 pffiffiffiffiffiffiffiffiffiffi 7 2Mpffiffiffiffiffiffiffiffiffiffi 3 7 0 ð2M  3Þ 2M  1 7 5 pffiffiffiffiffiffiffiffiffiffi  2M 1 pffiffiffiffiffiffiffiffiffiffi 0 ð2M  1Þ 2M  3

ð20Þ Put X ¼ 1 in Eq. (19), we have

3. Legendre wavelet collocation method

θ0 ð0Þ ¼ θ0 ð1Þ  CT Pψð1Þ ) θ0 ð0Þ ¼  CT Pψð1Þ

Let

Eq. (19) becomes

θ″ðXÞ ¼ C ψðXÞ; T

ð13Þ

θ0 ðXÞ ¼  CT Pψð1Þ þ CT PψðXÞ:

ð21Þ

Convective-radiative fin with temperature dependent thermal conductivity

Again Integrating Eq. (21) with respect to X from 0 to X, we have θðXÞ ¼ 1  CT Pψð1ÞdT PψðXÞ þ CT P2 ψðXÞ:

f ðθÞCT ψðXÞ þ f 0 ðθÞf  CT Pψð1Þ þ CT PψðXÞg2  Ncf1 CT Pψð1ÞdT PψðXÞ þ CT P2 ψðXÞ θa g  n 1 CT Pψð1ÞdT PψðXÞ þ CT P2 ψðXÞ  θa  1  θa

  Nr 1 þ Bf1 CT Pψð1ÞdT PψðXÞ þ CT P2 ψðXÞ  θs g

  f1  CT Pψð1ÞdT PψðXÞ þ CT P2 ψðXÞg4  θ4s  Pef  CT Pψð1Þ þ CT PψðXÞg ¼ R1ðX i ; CÞ ð23Þ

and

ð24Þ

As θ(X) is an approximate solution of Eq. (9) and Eq. (10). Choosing n collocation points Xi, i¼ 1,2,3,…,n in the interval [0, 1], at which residual R(X, c1, c2,…, cn) equal to zero. The number of such points must be equal to the number of coefficients " #T c1;0 ; c1;1 ; :::; c1;M  1 ; c2;0 ; c2;1 ; :::; c2;M  1 ; C¼ c2k  1 ;0 ; c2k  1 ;1; ; :::; c2k  1 ;M  1 Thus, we get R1ðX i ; ; CÞ ¼ 0; i ¼ 1; 2; 3; …; n

ð27Þ

ð22Þ

Substituting θ″ðXÞ; θ0 ðXÞand θðXÞ in Eq. (9) and Eq. (10), we get

f ðθÞCT ψðXÞ þ f 0 ðθÞf  CT Pψð1Þ þ CT PψðXÞg2  Ncf1 CT Pψð1ÞdT PψðXÞ þ CT P2 ψðXÞ θa g  n 1 CT Pψð1ÞdT PψðXÞ þ CT P2 ψðXÞ  θa  1  θa 1 n o  Nr H exp 1  CT Pψð1ÞdT PψðXÞþCT P2 ψðXÞ  1

  f1  CT Pψð1ÞdT PψðXÞ þ CT P2 ψðXÞg4  θ4s  Pef  CT Pψð1Þ þ CT PψðXÞg ¼ R2ðX i ; CÞ:

211

  f1 CT Pψð1ÞdT PψðXÞ þ CT P2 ψðXÞg4 θ4s PefCT Pψð1Þ þ CT PψðXÞg ¼ 0

ð25Þ

Case II: when emissivity is a linear function of temperature and thermal conductivity is a quadratic function of temperature i.e. f ðθÞ ¼ 1 þ Aðθ  θa Þ2 ; Eq. (25) reduces to

1 þ Af1 CT Pψð1ÞdT PψðXÞ  þCT P2 ψðXÞ  θa g2 CT ψðXÞ  þ2A 1 CT Pψð1ÞdT PψðXÞ þCT P2 ψðXÞ  θa gf  CT Pψð1Þþ 2 þCT PψðXÞ  Ncf1 CT Pψð1ÞdT PψðXÞ þCT P2 ψðXÞ  θa g  n 1 CT Pψð1ÞdT PψðXÞ þ CT P2 ψðXÞ  θa  1  θa

T  Nr 1 þ Bf1 C Pψð1ÞdT PψðXÞ  þCT P2 ψðXÞ  θs g f1 CT Pψð1ÞdT PψðXÞ  þCT P2 ψðXÞg4  θ4s  Pef  CT Pψð1Þ ð28Þ þCT PψðXÞg ¼ 0

Case III: when emissivity varies with temperature and wavelength and thermal conductivity is a linear function of temperature, Eq. (26) reduces to

1 þ Af1 CT Pψð1ÞdT PψðXÞ  þCT P2 ψðXÞ  θa g CT ψðXÞ þAf  CT Pψð1Þ þ CT PψðXÞg2  Ncf1  CT Pψð1ÞdT PψðXÞ þ CT P2 ψðXÞ θa g  n 1 CT Pψð1ÞdT PψðXÞ þ CT P2 ψðXÞ  θa  1  θa 1 n o  Nr H exp 1  CT Pψð1ÞdT PψðXÞþCT P2 ψðXÞ  1

  f1  CT Pψð1ÞdT PψðXÞ þ CT P2 ψðXÞg4  θ4s  Pef  CT Pψð1Þ þ CT PψðXÞg ¼ 0: ð29Þ

And R2ðX i ; ; CÞ ¼ 0; i ¼ 1; 2; 3; …; n

ð26Þ

Particular Cases: Case I: when emissivity and thermal conductivity are linear function of temperature i.e. f ðθÞ ¼ 1 þ Aðθ  θa Þ ; Eq. (25) reduces to

 1 þ Af1  CT Pψð1ÞdT PψðXÞ þ CT P2 ψðXÞ  θa g CT ψðXÞ

þAf CT Pψð1Þ þ CT PψðXÞg2  Ncf1 CT Pψð1ÞdT PψðXÞ þ CT P2 ψðXÞ θa g  n 1  CT Pψð1ÞdT PψðXÞ þ CT P2 ψðXÞ  θa  1  θa

  Nr 1 þ Bf1 CT Pψð1ÞdT PψðXÞ þ CT P2 ψðXÞ  θs g

Case IV: When emissivity varies with temperature and wavelength and thermal conductivity is a quadratic function of temperature, Eq. (26) reduces to

1 þ Af1 CT Pψð1ÞdT PψðXÞ  þCT P2 ψðXÞ  θa g2 CT ψðXÞ þ2Af1  CT Pψð1ÞdT PψðXÞ þ CT P2 ψðXÞ θa g f  CT Pψð1Þ þ CT PψðXÞg2  Ncf1  CT Pψð1ÞdT PψðXÞ þCT P2 ψðXÞ  θa g  n 1 CT Pψð1ÞdT PψðXÞ þ CT P2 ψðXÞ  θa  1  θa

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Surjan Singh et al.

Table 1

Comparison of exact and WCM result for temperature distribution in fin.

Temperature (θ) A¼ 0, B ¼ 0, Nc¼0.5, Nr¼ 0, θa ¼ 0.5, θs ¼ 0, n¼ 0 X

Pe¼0 (Exact)

Pe¼0 (WCM)

Pe¼0.5 (Exact)

Pe¼ 0.5 (WCM)

Pe¼ 1 (Exact)

Pe¼ 1 (WCM)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1.0000000000 0.9797061485 0.9618118272 0.9462275274 0.9328752949 0.9216883410 0.9126107074 0.9055969871 0.9006120968 0.8976311018 0.8966390909

1.0000000000 0.9797061485 0.9618118272 0.9462275274 0.9328752949 0.9216883410 0.9126107074 0.9055969871 0.9006120968 0.8976311018 0.8966390909

1.0000000000 0.9833400787 0.9683051551 0.9549014271 0.9431437765 0.9330562919 0.9246728633 0.9180378587 0.9132068893 0.9102476706 0.9092409909

1.0000000000 0.9833400787 0.9683051551 0.9549014271 0.9431437766 0.9330562919 0.9246728633 0.9180378587 0.9132068893 0.9102476706 0.9092409909

1.0000000000 0.9861553908 0.9734122669 0.9618194344 0.9514368837 0.9423372812 0.9346076853 0.9283515174 0.9236908273 0.9207688941 0.9197532129

1.0000000000 0.9861553908 0.9734122669 0.9618194344 0.9514368837 0.9423372812 0.9346076853 0.9283515175 0.9236908273 0.9207688941 0.9197532129

 Nr

n

1

o

1

  f1  C Pψð1Þd PψðXÞ þ C P ψðXÞg4  θ4s  Pef  CT Pψð1Þ þ CT PψðXÞg ¼ 0: ð30Þ H exp 1  CT Pψð1ÞdT PψðXÞþC T 2 P ψðXÞ T T T 2

Solving system of Eqs. (27) to (30) separately by NewtonRaphson Method for Case I to IV, we obtain value of C. Substituting these values of C in Eq. (22), we obtain required temperature.

4. Surface heat loss A quantity of fundamental interest in thermal processing application is the amount of surface heat loss from the moving material to its surrounding. The energy entering at x¼ 0 consists of conduction heat transfer plus the energy advection. A portion of this energy is lost to the surrounding fluid and the remainder is advected at x¼ L. The energy conducted at x¼ L is zero. The energy balance equation can be expressed as follows  KðTð0ÞÞAc

dTð0Þ þ ρUAc Cp T b  q ¼ ρUAc Cp TðLÞ; dx

Figure 2 WCM solution for Case I & II effect of ‘A’ on temperature distribution.

ð31Þ

where T(0) and T(L) are the temperature at x ¼ 0 and x ¼ L respectively and q is the surface heat loss. Introducing dimensionless variables Q¼

qL ; K a T b Ac

Qc ¼  ½1 þ Afθð0Þ θa g

Qa ¼

dθð0Þ ; dX

ρUC p L f1 θð1Þg ¼ Pef1  θð1Þg Ka

ð32Þ Figure 3 WCM solution for Case I & II effect of ‘B’ on temperature. distribution.

ð33Þ

and using dimensionless parameters ‘Q’ becomes Q ¼ Qc þ Qa

ð34Þ

where θ(0) and θ(1) are dimensionless temperatures at X ¼ 0 and X ¼ 1 respectively. Eq. (32) represents conduction energy at X ¼ 0 and Eq. (33) represents

Convective-radiative fin with temperature dependent thermal conductivity

213

Figure 7 WCM solution for Case I & II effect of ‘θa’ on temperature distribution. Figure 4 WCM solution for Case I & II effect of ‘Pe’ on temperature distribution.

Figure 8 WCM solution for Case I & II effect of ‘θs’ on temperature distribution. Figure 5 WCM solution for Case I & II effect of ‘Nc’ on temperature distribution.

Figure 6 WCM solution for Case I & II effect of ‘Nr’ on temperature distribution.

Figure 9 WCM solution for Case I & II effect of ‘n’ on temperature distribution.

214

Figure 10 Effect of convection-conduction parameter ‘Nc’ and radiation-conduction parameter ‘Nr’ on surface heat loss (in linear case).

Surjan Singh et al.

Figure 13 Effect of radiation-conduction parameter ‘Nr’, on and sink temperature ‘θa’ on surface heat loss (in linear case).

Figure 11 Effect of thermal conductivity parameter ‘A’ and sink temperature ‘θa’ on surface heat loss (in linear case). Figure 14 Effect of variable thermal conductivity and ‘Pe’ on surface heat loss (in linear case).

Figure 12 Effect of convection-conduction parameter ‘Nc’, and sink temperature ‘θa’ on surface heat loss (in linear case).

Figure 15 Effect of sink temperature and ‘Pe’ on surface heat loss (in linear case).

Convective-radiative fin with temperature dependent thermal conductivity

Figure 16 Effect of radiation-sink temperature and ‘Pe’ on surface heat loss (in linear case). Figure 19

215

Effect of ‘B’ and ‘n’ on surface heat loss (in linear case).

Figure 17 Effect of radiation-conduction parameter and ‘Pe’ on surface heat loss (in linear case). Figure 20 Effect of convection-conduction parameter ‘Nc’ and exponent ‘n’ on surface heat loss.

Figure 18 Effect of thermal conductivity and ‘n’ on surface heat loss (in linear case).

Figure 21 Effect of radiation-conduction parameter ‘Nr’ and exponent ‘n’ on surface heat loss.

216

Surjan Singh et al.

Figure 22 Effect of ‘Pe’ and exponent ‘n’ on surface heat loss. Figure 25 Effect of variable thermal conductivity on advection, base conduction and surface heat loss (in linear case).

Figure 23 Effect of ‘θa’ and exponent ‘n’ on surface heat loss.

Figure 26 Effect of sink temperature on advection, base conduction and surface heat loss (in linear case)

Effect of ‘θs’ and exponent ‘n’ on surface heat loss.

Figure 27 Effect of radiation-sink temperature ‘θs’ on advection, base conduction and surface heat loss (in linear case).

Figure 24

Convective-radiative fin with temperature dependent thermal conductivity

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advection energy at X ¼ 1 that is lost to the surrounding.

5. Comparison with exact result If f ðθÞ ¼ 1ðA ¼ 0Þ; B ¼ 0; n ¼ 0; Nr ¼ 0; then Eq. (9) becomes d2 θ dθ  Nc ðθ  θa Þ ¼ 0; 0 r X r 1:  Pe ð35Þ dX dX 2 The exact solution of above equation can be put in the form θðXÞ ¼ C 1 expðm1 XÞ þ C2 expðm2 XÞ þ θa ; where

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Pe þ Pe2 þ 4Nc 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Pe  Pe2 þ 4Nc m2 ¼ 2 ð1  θa Þm1 C1 ¼ ; m1  m2 expðm2  m1 Þ ð1 θa Þm1 C 2 ¼ 1 θa  ; m1  m2 exp ðm2  m1 Þ m1 ¼

for validation of the present method we compare WCM with exact result. For, Nc ¼ 0.5, θa ¼ 0.5, θs ¼ 0 and Pe ¼ 0, 0.5, 1 Exact and WCM results are presented in Table 1. We observe that as value of Pe increases, temperature in fin increases. WCM and exact both results are same and correct up to ten decimal places. Thus method provides a very good result for this problem. In numerical computation MATLAB 2012a software has been used.

6. Result and discussion In Case I and II, the computation has been made and results are presented in 29 figures (Figures 1 to 29). On the

Figure 28 Effect of ‘Pe’ and exponent ‘n’ on advection, base conduction and surface heat loss (in linear case).

Figure 29 Effect of radiation-conduction parameter on advection, base conduction and surface heat loss (in linear case).

figures presented in this study, only the parameters whose values different from the reference valued are indicated. The selected reference values include B ¼ 0.5, Nc ¼ 0.5, Nr ¼ 0.5, Pe ¼ 0.5, θa ¼ 0.5, θs ¼ 0.5, n¼ 2. The dimensionless temperature θ is a function of space coordinate X for different thermal conductivity parameter A is shown in Figure 2. We observe that the temperature in moving fin material increases as thermal conductivity parameter of material increases from 0 to 1. Here the most important point which we found that temperature in fin increases slowly. In Case I temperature in fin is higher than Case II. So Case II is more effective for cooling process. Effect of variable surface emissivity parameter ‘B’ on temperature distribution in moving fin is presented in Figure 3. In this figure, B ¼ 0 represents surface emissivity of fin that is constant and its value is &s. As we increase value of B, temperature in fin decreases. In Case I temperature in fin is higher than Case II for all values of B. Thus, for cooling process the surface emissivity of the fin material is more effective in Case II than Case I. The effect of Peclet number Pe (dimensionless speed of moving material) on the temperature distribution in fin is presented in Figure 4. In this figure we observe that as we increase value of Pe, temperature in moving fin increases. At Pe¼ 0 (no speed of fin), in Case II temperature in fin is lower than Case I. Here we conclude that high speed of moving fin takes small time to release heat in environment, so that temperature in fin increases. In Case I temperature in fin is higher than in Case II. In Figure 5, effect of convectionconduction parameter Nc on temperature distribution in fin is presented for Nc¼ 0, 1, 2. We observe that as value of Nc increases, temperature in fin decreases. Nc¼ 0 means no convection-conduction is there. We know that heat releases fast in its surroundings due to high convection-conduction process, this confirms from Figure 5. Thus the temperature in fin decreases for higher values of Nc, consequently cooling is more effective in this process. In Figure 6 radiationconduction parameter Nr varies from 0 to 2 for Case I and II.

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Top two curves represent temperature distribution in fin for Case I and II when Nr¼ 0 that there is no radiation; consequently temperature in fin is very high. In this figure, we observe that temperature in fin decreases as value of radiation-conduction parameter increases. The temperature is lower in Case II in comparison to Case I. In any physical problem, if heat is released through radiation, then temperature must decrease, which confirms from Figure 6. As heat radiation becomes higher, the radiative cooling becomes more effective, consequently temperature in moving fin decreases. Effect of convection sink temperature ‘θa’ on temperature distribution in moving fin is presented in Figure 7. Due to low convective heat loss, temperature in moving material increases. In this figure, we observe that as we increase the value of ‘θa’ temperature in fin increases. In Case I temperature is higher than Case II. Effect of the radiation sink temperature ‘θs’ on temperature distribution in moving fin is presented in Figure 8. As we increase ‘θs’ temperature in fin increases due to radiative heat loss. For higher value of ‘θs’ temperature is higher. Effect of exponent ‘n’ presented in Figure 9, shows that as we increase value of ‘n’ temperature distribution in fin increases. We observe that for laminar film boiling or condensation, cooling is more effective. In Case I temperature is higher than Case II. In Case I effect of different parameters on surface heat loss is presented in Figures 10–17. Effect of Nr and Nc on surface heat loss is shown in Figure 10. In our problem if there is no radiation and convection (Nc ¼ Nr ¼ 0) surface heat loss must be zero; this confirms from Figure 10. In this figure top curve represent high surface heat loss for high convection. We observe that as we increase convection and radiation, either individually or together, surface heat loss increases as required. In Figure 11 we observe that as we increase sink temperature and thermal conductivity parameter ‘A’; surface heat loss decreases rapidly. When thermal conductivity of fin material is zero then surface heat loss should be minimum. An effect of convection parameter and sink temperature on surface heat loss is presented in Figure 12. Surface heat loss decreases as value of sink temperature increases. When convection is zero, surface heat loss is represented by lower line. Effect of sink

Table 2 Nr

0.25 0.25 0.50 0.25 0.50 0.25 0.50 0.50

temperature and ‘Nr’ on surface heat loss is shown in Figure 13 which is similar as Figure 12. Effect of Pe and ‘A’ on surface heat loss is presented in Figure 14. In this figure bottom line represent surface heat loss when thermal conductivity is zero. For higher thermal conductivity, surface heat loss is higher. We observe that surface heat loss increases as thermal conductivity and Peclet number increases. Effect of Peclet number and convective sink temperature on surface heat loss is presented in Figure 15. We observe that surface heat loss is higher for low sink temperature. As we increase the value of sink temperature, surface heat loss decreases. As Pe increases, surface heat loss increases. In Figure 16 effect of radiation sink temperature and Pe on surface heat loss is presented; observations are similar as Figure 15. Effect of Peclet number and radiation on surface heat loss is shown in Figure 17. For Nr ¼ 0 i.e. no radiation, we observe that from bottom curve, surface heat loss increases very slowly. Surface heat loss increases as value of Pe and Nr increases. Top curve represents high surface heat loss due to high radiation. Effect of exponent ‘n’ and other combinations on surface heat loss are shown in Figures 18 to 24. In Figure 18 we observe that as we increase exponent ‘n’, surface heat loss decreases. As the thermal conductivity parameter increases, surface heat loss increases. In Figure 19 we observe that as we increase emissivity parameter ‘B’ surface heat loss increases but as we increase value of exponent ‘n’ surface heat loss decreases. In Figures 20, 21 and 22 effect of Nc, Nr and ‘Pe’ on surface heat loss with increasing ‘n’ are studied. In these three figures lower line represent surface heat loss for n ¼ 3 (radiation); i.e. for higher value of ‘n’ surface heat loss is lower. As we increase convection, radiation and advection then surface heat loss increases as required in physical problem. Effect of θa and θs on surface heat loss with increasing value of ‘n’ is presented in Figures 23 and 24. As value of convection sink temperature and radiation sink temperature increases, surface heat loss decreases with increasing exponent ‘n’. In Figures 25 to 29 we have studied separate effect of different parameters on surface heat loss Q, advection heat loss Qa and the base heat loss Qc. In Figure 25, the effect of

Comparison of WCM and DTM results for temperature distribution in fin when X¼θa ¼ θs ¼ n ¼0. Nc

0.25 0.50 0.25 0.25 0.25 0.50 0.50 0.50

Pe

0.25 0.25 0.25 0.50 0.50 0.50 0.25 0.50

WCM

DTM

A¼0

A ¼0.5

A ¼ 1.0

A ¼0

A¼ 0.5

A ¼1.0

0.8358 0.7652 0.7944 0.8447 0.8039 0.7774 0.7340 0.7461

0.8803 0.8190 0.8384 0.8803 0.8441 0.8255 0.7895 0.7961

0.9029 0.8531 0.8672 0.9030 0.8706 0.8573 0.8256 0.8298

0.8358 0.7652 0.7944 0.8447 0.8040 0.7774 0.7340 0.7461

0.8756 0.8192 0.8384 0.8808 0.8443 0.8266 0.7890 0.7967

0.9001 0.8536 0.8670 0.9034 0.8710 0.8585 0.8255 0.8308

Convective-radiative fin with temperature dependent thermal conductivity

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Table 3 Case III: Effect of ‘n’ and ‘λ’ on temperature distribution in fin when thermal conductivity is the linear function of temperature and emissivity depends on wavelength. Tb ¼800 K, A ¼0.5, B¼ 0.5, Nc¼ 0.5, Nr ¼0.5, Pe¼0.5, θa ¼0.5, θs ¼0.5 n¼  1/4

n¼1/4

n¼ 3

X

λ ¼1 μm H¼ 17.9924

λ ¼10 μm H ¼1.7992

λ ¼1 μm H ¼17.9924

λ ¼10 μm H¼1.7992

λ ¼1 μm H¼17.9924

λ ¼10 μm H ¼1.7992

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1.0000000000 0.9854693884 0.9722713290 0.9604414843 0.9500194803 0.9410491424 0.9335787151 0.9276610562 0.9233537976 0.9207194623 0.9198255282

0.9999999999 0.9819081712 0.9656316357 0.9511603821 0.9384978512 0.9276600717 0.9186750403 0.9115823001 0.9064326798 0.9032881647 0.9022218784

1.0000000000 0.9861379638 0.9736174226 0.9624506342 0.9526556512 0.9442565695 0.9372837792 0.9317742171 0.9277716159 0.9253267470 0.9244976519

0.9999999999 0.9826452304 0.9671167736 0.9533781692 0.9414086886 0.9312023940 0.9227674878 0.9161259048 0.9113131135 0.9083780997 0.9073835210

0.9999999996 0.9885548926 0.9784646600 0.9696508780 0.9620554757 0.9556381756 0.9503746107 0.9462549965 0.9432832661 0.9414766130 0.9408654160

0.9999999986 0.9852942166 0.9724272428 0.9612589623 0.9516847523 0.9436298556 0.9370452895 0.9319049507 0.9282036696 0.9259560558 0.9251960658

Table 4 Case IV: Effect of ‘n’ and ‘λ’ on temperature distribution in fin when thermal conductivity is the quadratic function of temperature and emissivity depends on wavelength. Tb ¼800 K, A¼ 0.5, B ¼ 0.5, Nc¼ 0.5, Nr ¼0.5, Pe¼0.5, θa ¼0.5, θs ¼ 0.5 n¼  1/4

n¼1/4

n¼ 3

X

λ ¼1 μm H¼ 17.9924

λ ¼10 μm H ¼1.7992

λ ¼1 μm H¼17.9924

λ ¼10 μm H¼1.7992

λ ¼1 μm H¼17.9924

λ ¼10 μm H¼1.7992

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.9999999991 0.9895882227 0.9768108402 0.9631095514 0.9495111640 0.9367728831 0.9254609910 0.9160091487 0.9087591525 0.9039889061 0.9019301083

0.9999999994 0.9842579366 0.9678391562 0.9516758472 0.9364443608 0.9226507768 0.9106831823 0.9008491270 0.8934022846 0.8885614150 0.8865231468

0.9999999992 0.9900323680 0.9779376657 0.9650648157 0.9523554721 0.9404894620 0.9299676534 0.9211713805 0.9144031166 0.9099137686 0.9079195751

0.9999999997 0.9849469404 0.9693372156 0.9540478698 0.9396969873 0.9267349410 0.9155012545 0.9062643233 0.8992487891 0.8946542306 0.8926671700

0.9999999998 0.9891492667 0.9788734432 0.9693437669 0.9607007397 0.9530642893 0.9465406262 0.9412269097 0.9372143251 0.9345900980 0.9334388575

0.9999999992 0.9849498684 0.9715505742 0.9596907593 0.9493120675 0.9403922192 0.9329349750 0.9269640542 0.9225194875 0.9196556577 0.9184407798

‘Pe’ and ‘A’ on Q, Qc and Qa is presented. When thermal conductivity A ¼ 0, base heat loss Qc decreases; surface heat loss Q and advection heat loss Qa increases as Pe increases. Top three lines represents value of Q, middle three lines represents Qc and lower three lines represents Qa. Effect of dimensionless sink temperature and Peclet number is presented in Figure 26. In this figure top three curves represent surface heat loss Q. We observe that surface heat loss increases as value of Pe increases, Pe increases only when ratio of UL/αAc increases i.e. speed of fin U increases or value of αAc decreases. For higher sink temperature, surface heat loss is lower. Middle three curves represent base heat conduction Qc. As value of Peclet number and sink temperature increases, base heat conduction Qc decreases. As Peclet number ‘Pe’ increases, the

advection heat loss (three bottom curves) also increases. Similar effect has been observed of Pe and dimensionless radiation sink temperature θs and ‘n’ on Q, Qc and Qa as shown in Figures 27 and 28 respectively. Effect of radiationconduction parameter and Peclet number on surface heat loss, advection heat loss and base heat loss are presented in Figure 29. For clarity and explanation we have used colored figures. Three magenta color lines represent the Case of no radiation i.e. Nr ¼ 0. As ‘Pe’ increases advection heat loss increases but the base heat loss decreases. Surface heat loss increases as value of ‘Pe’ and ‘Nr’ increases. For Case I, we compare Exact and WCM result in Table 1 and observe that both results are exactly the same. For n¼ 0, and B ¼ 0, results obtained by wavelet collocation

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method and DTM by Torabi and Yaghoobi [10], are nearly the same upto four decimal places as shown in Table 2. The error in DTM increases with a variation in thermal conductivity parameter ‘A’. In Case I and II, effect of different parameters studied in detail and presented in 29 figures. For Case III and IV results presented in Tables 3 and 4. Effect of exponent and wavelength on temperature distribution in fin is studied in Table 3, for Tb ¼ 800 K, A ¼ 0.5, B ¼ 0.5, Nc ¼ 0.5, Nr ¼ 0.5, Pe ¼ 0.5, θa ¼ 0.5, θs ¼ 0.5, n¼  1/4, n¼ 1/4, n¼ 3, λ¼ 1 μm and λ¼ 10 μm, when thermal conductivity is the linear function of temperature. We observe that as we increase value of wavelength, temperature in fin decreases, as value of ‘n’ increases temperature in fin increases. Due to high emission, energy release fast in environment, consequently cooling becomes more effective. It has been observed that when heat transfer through radiation (i.e. n¼ 3), the temperature in fin is highest. In Table 4, for Tb ¼ 800 K, A ¼ 0.5, B ¼ 0.5, Nc ¼ 0.5, Nr¼ 0.5, Pe ¼ 0.5, θa ¼ 0.5, θs ¼ 0.5, n ¼  1/4, n¼ 1/4, n¼ 3, λ ¼ 1 μm and λ ¼ 10 μm, we obtained temperature in fin when thermal conductivity is the quadratic function of temperature and emissivity depends on wavelength, which is presented in Eq. (4). When we compare Tables 3 and 4 we observe that cooling is more effective when thermal conductivity is the quadratic function of temperature.

6. Conclusion A mathematical model describing heat transfer in a continuously moving fin with temperature dependent thermal conductivity and heat transfer coefficient whose surface emissivity varies with temperature and wavelength is studied. The WCM has been briefly described and then used to obtain approximate numerical solution of this problem. The obtained results are compared with exact solution of a particular Case and are same up to ten decimal places. Four particular Cases of technical importance are dealt in detail. It has been observed that cooling is more effective when fin materials having quadratic type thermal conductivity comparison to linear type thermal conductivity. For higher values of exponent, convection and radiation sink temperature, heat releases fast from fin. Cooling process is fast in nucleate boiling. As value of ‘n’ increases, surface heat loss in fin decreases. Due to high convectionconduction and radiation-Conduction parameters heat releases fast in environment consequently cooling becomes more effective. However, as the fin moves faster, i.e. Peclet number ‘Pe’ increases the temperature in fin increases i.e. the fin experiences the slower cooling. For High wavelength, temperature in fin decreases consequently cooling becomes more effective. We observe that temperature in fin is low when thermal conductivity is quadratic. Temperature in fin decreases as value of exponent and wavelength increases. It has been observed that as dependency of

Surjan Singh et al.

emissivity increases with wavelength, the temperature of fin decreases.

Acknowledgements 1. Authors are grateful to Professor Umesh Singh Coordinator DST-Centre for interdisciplinary Mathematical Sciences Banaras Hindu University Varanasi, India for providing necessary facilities. 2. We are thankful to Dr. Satish kumar and Ms. Suman Mehla Dept. of English, Gurgaon Institute of Technology and Management Gurgaon (Haryana), India for language editing.

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