Analytical Solutions for Steady Heat Transfer in Longitudinal Fins with Temperature-Dependent Properties

Ndlovu, Partner L.; Moitsheki, Raseelo Joel

doi:10.1155/2013/273052

Cited by 25 publications

(31 citation statements)

References 31 publications

(45 reference statements)

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“…From Figure , we note the temperature θ increases when increasing x (in other words, the temperature increases when approaching from heat source). Remark When β = n , Equations and become

\begin{matrix} θ & = θ_{0} {Cosh}^{\frac{1}{1 + n}} (M \sqrt{1 + n} x) \\ θ_{0} & = {Cosh}^{- \frac{1}{1 + n}} (M \sqrt{1 + n}) . \end{matrix}

From , we obtain

θ = {((), \frac{Cosh ((), M \sqrt{1 + n} x)}{Cosh ((), M \sqrt{1 + n})})}^{\frac{1}{1 + n}} .

Equation is obtained in , used in , and obtained as a special case when

n = - \frac{4}{3}

in .…”

Section: Numerical Examplementioning

confidence: 89%

“…The steady rate of heat transfer from the entire fin Q f i n can be determined from Fourier's law of heat conduction

Q_{fin} = A_{c} K (T) \frac{dT}{dX} |_{X = L} .

The dimensionless heat transfer rate Q is defined by

Q = \frac{Q_{fin}}{A_{c} K_{b} (T_{b} - T_{a}) / L} = \frac{A_{c} K (T) \frac{dT}{dX} |_{X = L}}{A_{c} K_{b} (T_{b} - T_{a}) / L} .

Using , Equation becomes

Q = θ^{β} \frac{dθ}{dx} |_{x = 1} .

From and , when x = 1, we obtain

Q = θ^{'} (1) .

Substituting Equation into Equation , we obtain

θ^{'} = \frac{\sqrt{2} M}{\sqrt{2 + n + β}} θ^{- β} \sqrt{θ^{2 + n + β} - θ_{0}^{2 + n + β}} .

…”

Section: Heat Fluxmentioning

confidence: 99%

“…The efficiency of the fin can be defined by the ratio of the actual heat transfer to the ideal heat transfer . So, the efficiency of the fin is given by

η = \frac{Q_{actual}}{Q_{ideal}} = \frac{\int_{0}^{L} ph (T) ((), T - T_{a}) dX}{pL h_{b} ((), T_{b} - T_{a})} = \int_{0}^{1} θ^{n + 1} dx .

Using Equation , Equation becomes

η = \frac{1}{M^{2}} \int_{0}^{1} \frac{d}{dx} ((), θ^{β} \frac{dθ}{dx}) dx = \frac{1}{M^{2}} ((), θ {(1)}^{β} θ^{'} (1) - θ {(0)}^{β} θ^{'} (0)) .

Using Equation , Equation becomes

η = \frac{1}{M^{2}} θ^{'} (1) .

From and , we obtain relation between the fin efficiency η and the heat transfer rate Q

Q = M^{2} η .

From and , we obtain

1 = \frac{i}{M \sqrt{2 α (β + 1)}} {(1 - ρ)}^{α - \frac{1}{2}} ((), {(1 - ρ)}^{- α}_{2} F)

…”

Section: Fin Efficiencymentioning

confidence: 99%

“…Approximate solutions of Equation are investigated using differential transform method in . In the case of steady state, exact analytical solution of Equation when

n = β = - \frac{4}{3}

is obtained in , but when β ≠ n , numerical solution only of Equation is obtained.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

General exact solution of the fin problem with the power law temperature-dependent thermal conductivity

Kader

Latif

Nour

2015

Math. Meth. Appl. Sci.

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In this paper, the general exact implicit solution of the second-order nonlinear ordinary differential equation governing heat transfer in rectangular fin is obtained using Lie point symmetry method. General relationship among the fin efficiency, the rate of heat transfer from the entire fin, the fin effectiveness, and the thermo-geometric fin parameter is obtained for any value of the mode of heat transfer n and the constantˇ.

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“…From Figure , we note the temperature θ increases when increasing x (in other words, the temperature increases when approaching from heat source). Remark When β = n , Equations and become

\begin{matrix} θ & = θ_{0} {Cosh}^{\frac{1}{1 + n}} (M \sqrt{1 + n} x) \\ θ_{0} & = {Cosh}^{- \frac{1}{1 + n}} (M \sqrt{1 + n}) . \end{matrix}

From , we obtain

θ = {((), \frac{Cosh ((), M \sqrt{1 + n} x)}{Cosh ((), M \sqrt{1 + n})})}^{\frac{1}{1 + n}} .

Equation is obtained in , used in , and obtained as a special case when

n = - \frac{4}{3}

in .…”

Section: Numerical Examplementioning

confidence: 89%

“…The steady rate of heat transfer from the entire fin Q f i n can be determined from Fourier's law of heat conduction

Q_{fin} = A_{c} K (T) \frac{dT}{dX} |_{X = L} .

The dimensionless heat transfer rate Q is defined by

Q = \frac{Q_{fin}}{A_{c} K_{b} (T_{b} - T_{a}) / L} = \frac{A_{c} K (T) \frac{dT}{dX} |_{X = L}}{A_{c} K_{b} (T_{b} - T_{a}) / L} .

Using , Equation becomes

Q = θ^{β} \frac{dθ}{dx} |_{x = 1} .

From and , when x = 1, we obtain

Q = θ^{'} (1) .

Substituting Equation into Equation , we obtain

θ^{'} = \frac{\sqrt{2} M}{\sqrt{2 + n + β}} θ^{- β} \sqrt{θ^{2 + n + β} - θ_{0}^{2 + n + β}} .

…”

Section: Heat Fluxmentioning

confidence: 99%

“…The efficiency of the fin can be defined by the ratio of the actual heat transfer to the ideal heat transfer . So, the efficiency of the fin is given by

η = \frac{Q_{actual}}{Q_{ideal}} = \frac{\int_{0}^{L} ph (T) ((), T - T_{a}) dX}{pL h_{b} ((), T_{b} - T_{a})} = \int_{0}^{1} θ^{n + 1} dx .

Using Equation , Equation becomes

η = \frac{1}{M^{2}} \int_{0}^{1} \frac{d}{dx} ((), θ^{β} \frac{dθ}{dx}) dx = \frac{1}{M^{2}} ((), θ {(1)}^{β} θ^{'} (1) - θ {(0)}^{β} θ^{'} (0)) .

Using Equation , Equation becomes

η = \frac{1}{M^{2}} θ^{'} (1) .

From and , we obtain relation between the fin efficiency η and the heat transfer rate Q

Q = M^{2} η .

From and , we obtain

1 = \frac{i}{M \sqrt{2 α (β + 1)}} {(1 - ρ)}^{α - \frac{1}{2}} ((), {(1 - ρ)}^{- α}_{2} F)

…”

Section: Fin Efficiencymentioning

confidence: 99%

“…Approximate solutions of Equation are investigated using differential transform method in . In the case of steady state, exact analytical solution of Equation when

n = β = - \frac{4}{3}

is obtained in , but when β ≠ n , numerical solution only of Equation is obtained.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

General exact solution of the fin problem with the power law temperature-dependent thermal conductivity

Kader

Latif

Nour

2015

Math. Meth. Appl. Sci.

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show abstract

“…A brief review of the relevant literature published before 2011 can be found in our previous paper [20]. Research achievements in recent years include nonlinear steady heat conduction analysis for a convective n [21], convective-radiative ns of porous material [22], convective-radiative moving ns (or plates) [23][24][25], a Tshaped n [26] and variable thickness ns [27][28][29], and nonlinear transient heat conduction analysis for variable thickness ns [30]. Recently, the range of application of the DTM has been extended to the analysis of heat conduction in nonhomogeneous bodies [31,32] and phase-change problems [33,34].…”

Section: Introductionmentioning

confidence: 99%

Series solution to coupled nonlinear heat and moisture transfer in slabs with temperature-dependent diffusivities

Chiba

2014

Nonlinear Engineering

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The two-dimensional di erential transform method (DTM) is applied to solve the one-dimensional coupled heat and moisture di usion problem for a slab with temperature-dependent thermal and moisture diffusivities, which are expressed by a linear function and an exponential function of temperature, respectively. One surface of the slab is subjected to convective hygrothermal loading and the other has constant prescribed temperature and moisture. Approximate analytical (series) solutions for the temperature and moisture pro les in the slab are derived. The transformed functions included in the solutions are obtained through a simple recursive procedure. Numerical results for a slab subjected to a sudden change in surface temperature illustrate the e ects of temperature-dependent di usivities on the transient temperature and moisture pro les of the slab. The results indicate that the nonlinear e ect originating from the varying moisture di usivity is not negligible for resin composites. The DTM is a useful new analytical method for solving nonlinear coupled transient problems.

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Analysis of unsteady heat transfer of specific longitudinal fins with Temperature-dependent thermal coefficients by DTM

Pasha

Jalili

Ganji

2018

Alexandria Engineering Journal

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Analytical Solutions for Steady Heat Transfer in Longitudinal Fins with Temperature-Dependent Properties

Cited by 25 publications

References 31 publications

General exact solution of the fin problem with the power law temperature-dependent thermal conductivity

General exact solution of the fin problem with the power law temperature-dependent thermal conductivity

Series solution to coupled nonlinear heat and moisture transfer in slabs with temperature-dependent diffusivities

Analysis of unsteady heat transfer of specific longitudinal fins with Temperature-dependent thermal coefficients by DTM

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