_{1}

^{*}

Numerical solution of a radiative radial fin with temperature-dependent thermal conductivity is presented. Calculations are implemented along the lines of a boundary integral technique coupled with domain discretization. Localized solutions of the nonlinear governing differential equation are sought on each element of the problem domain after enforcing inter-nodal connectivity as well as the boundary conditions for the dependent variables. A finite element-type assembly of the element equations and matrix solution yield the scalar profile. Comparison of the numerical results with those found in literature validates the formulation. The effects of such problem parame ters as radiation-sink temperature, thermal conductivity, radiation-conduction fin parameter, volumetric heat generation, on the scalar profile were found to be in conformity with the physics of the problem. We also observed from this study that the volumetric heat generation plays a significant role in the overall heat transfer activity for a fin. For relatively high values of internal heat generation, a situation arises where a greater percentage of this energy can not escape to the environment and the fin ends up gaining energy instead of losing it. And the overall fin performance deteriorates. The same can also be said for the radiation-conduction parameter , whose increases can only give physically realistic results below a certain threshold value.

Fins are vastly used in different heat transfer applications such as air-conditioning systems, heat exchangers, power plants, refrigeration, chemical processing equipment, computers, fluid-conveyance structures etc. Their chief purpose is the transfer of heat from a surface to the surrounding. This process is usually enhanced by attaching highly conductive materials to a surface in order to facilitate heat flow between a source and a sink. Though linear differential equations can be used to model rectangular fins, however thermal conductivity is known to depend on temperature because of the high temperature gradient that exists in fin operation. The heat transfer coefficient should therefore reflect the temperature variation along the fin or the local temperature difference between the fin and the surrounding fluid. Since this is a highly significant consideration, we should end up considering a nonlinear differential equation for modelling fin energy transfer performance and thermal optimization [

The other broad area of fin literature focuses on the fin’s thermophysical properties for any chosen geometry. Some of the work treated in this area, involved the use of analytic and semi analytic techniques in predicting the fin’s scalar profiles especially for idealized cases. Atay and Coskun [

Considerable work has also been devoted to the application of domain-based numerical techniques to fin study and analysis since most of the analytic and semi-analytic methods contain complex terms which may not be convenient for fin design. For example the finite difference technique presents an accurate and a straight forward way to resolve nonlinear fin equations (Jain et al. [

It comes as no surprise that earlier work in fin calculations were phrased in terms of finite difference schemes. It was only after the numerical advantages of integral formulations became apparent that attempts to use boundary integral and finite element became widespread. This trend has caught up so fast in the field of solid mechanics to the extent that approximations based solely on finite-difference approximations are hardly ever used. This is however not the case in fluid dynamical computations where the oftentimes the nonlinear inertia terms renders the governing equations non-self adjoint and as a result differential approximations come as a natural choice.

In the work reported herein, we adopt a domain-decomposed singular integral formulation to solve the one-dimensional nonlinear fin equation whose complimentary equation comprises a one-dimensional Poisson equation with a dirac-delta forcing function. Its solution is known as the fundamental solution and together with the Green’s second identity yield the basis for the integral representation of the governing equation in a generic element of the discretized problem domain. The boundary integral representation of the governing equation on each element of the discretized domain is similar to a finite element system of equations and come with all the advantages and properties of the finite element method (FEM) ability of dealing with nonlinearity and non-homogeneity. The classic nonlinear fin equation with a temperature dependent thermal conductivity is solved and the results are validated through the use of benchmark solutions. To the author’s best knowledge, the version of the integral method of solution of the fin problem adopted herein is novel or barely existent in fin literature.

Symbols

b: Fin tip length, m

D: Problem domain

H: Constant

K: Temperature-dependent thermal conductivity Wm^{−1}K^{−1}

K 0 : Thermal conductivity at base temperature, Wm^{−1}K^{−1}

q: Volumetric heat generation Wm^{−3}

Q: Dimensionless heat generation

Q f : Heat transfer rate from the surface of a fin

T: Temperature, K

T b : Fin’s base temperature

T s : Radiation sink temperature

χ ( k ) : Transformed analytical function

x ( k ) : Original analytical function

w: Semi thickness of fin, m.

Greek Symbols

β : Thermal conductivity parameter

ε : Emissivity

η : Fin efficiency

λ : Slope of the thermal conductivity temperature curve, K^{−1}

σ : Stefan-Boltzman constant, Wm^{−2}K^{−4}

θ : Dimensionless temperature

θ s : Dimensionless radiation sink temperature

ψ : Radiation-conduction fin parameter

Problem specification and non-dimensionalization described herein follow that of Torabi et al. [

The energy balance equation for a differential element of the fin [

2 w d d x [ k ( T ) d T d x ] − 2 ε σ ( T 4 − T s 4 ) + q = 0 (1)

where k ( T ) , σ are thermal conductivity and the Stefan Boltzman constant, respectively. The thermal conductivity of the fin material is assumed to vary linearly with temperature and is specified as

k ( T ) = k 0 [ 1 + λ ( T − T a ) ] (2)

where k 0 , T a are the thermal conductivity at the T a temperature of the fin and λ is the measure of variation of the thermal conductivity with temperature. In order to ease the process of dicretization, Equation (1) is non-dimensionalized in the following manner.

θ = T T b , θ a = T a T b , θ s = T s T b , ξ = x b , β = λ T b , ψ = ε σ b 2 T b 3 k 0 w , Q = b 2 q T b k o (3)

The non-dimensional version of the fin problem becomes

d d ξ [ ( 1 + β ( θ − θ a ) ) d θ d ξ ] − ψ ( θ 4 − θ s 4 ) + Q = 0 (4a)

Equation (4a) is a one-dimensional, nonlinear, boundary-value problem with the following boundary conditions

d θ d ξ ( 0 ) = 0 , θ ( 1 ) = 1.0 (4b)

A very useful parameter for fin design is the efficiency defined by the ratio of actual and ideal heat transfer rate of the fin.

η = Q f Q f , i d e a l = 1 b ∫ 0 b T 4 T b 4 d x = ∫ 0 1 θ 4 d ξ (4c)

A major challenge here is to create an integral analog of Equation (4a). From a theoretical viewpoint, we need a transformation that will convert our governing differential equation to an integral form that can then be resolved within a problem domain. A key feature of this approach, involves the use of the Laplacian 1-D operator as an auxiliary equation: d 2 G / d x 2 = δ ( x − x i ) , x ∈ ( − ∞ , ∞ ) with a fundamental solution G ( x , x i ) = ( | x − x i | + k a ) / 2 , where k a represents an arbitrary constant often taken as the length of the longest element in the problem domain, and the distance between the source point and any other point is given by x − x i . Applying the Green’s identity together with the auxiliary equation to Equation (4a) and relating the whole procedure to a generic element defined by the span [ x 1 , x 2 ] inside the problem domain yields

− 2 ζ θ i + [ H ( ξ 2 − ξ i ) − H ( ξ i − ξ 2 ) ] θ 2 − [ H ( ξ 1 − ξ i ) − H ( ξ i − ξ 1 ) ] θ 1 − ( | ξ 2 − ξ i | + l ˜ ) φ 2 + ( | ξ 1 − ξ i | + l ˜ ) φ 1 + ∫ ξ 1 ξ 2 ( | ξ − ξ i | + l ˜ ) [ − ∂ ln D ( θ ) ∂ ξ + 1 D ( θ ) ( f ( ξ , t ) − Q ) ] d ξ = 0 (5)

where φ = d θ / d ξ , f ( ξ , t ) = ψ ( θ 4 − θ s 4 ) , D ( θ ) = 1 + β ( θ − θ a ) H is the Heaviside function, and λ = 0.5 , when ζ i = 0.5 , for ξ i = ξ 1 or ξ i = ξ 2 .

We deal with the line integral in Equation (5) by approximating the dependent variable and its function with linear interpolating functions in space, i.e. f ( ξ , t ) ≈ Ω ( ς ) f j ( t ) where Ω j is the interpolating function and ς = ( ξ − ξ 1 ) / l is a local coordinate for an element length l = ξ 2 − ξ 1 . Equation (5) is the element integral analog the governing differential equation. And is solved at each node of the elementized problem domain to give the following equations at node 1 and then at node 2

− θ 1 + θ 2 + l m φ 1 − ( l m + l ) φ 2 + ∫ ξ 1 ξ 2 ( | ξ − ξ 1 | + l m ) Ω j φ j ( − 1 l d Ω n d ς Θ n ) d ξ + ∫ ξ 1 ξ 2 ( | ξ − ξ 1 | + l m ) Ω n χ n Ω j ( f j + Q ) d ξ = 0 (6a)

θ 1 − θ 2 − ( l m + l ) φ 1 − l m φ 2 + ∫ ξ 1 ξ 2 ( | ξ − ξ 1 | + l m ) Ω j φ j ( − 1 l d Ω n d ς Θ n ) d ξ + ∫ ξ 1 ξ 2 ( | ξ − ξ 1 | + l m ) Ω n χ n Ω j ( f j + Q ) d ξ = 0 (6b)

A compact matrix representation of Equations (6a) and (6b) results in

R i j θ j + ( L i j − B i n j Θ n ) ψ j + E i n j χ n ( f j + Q n ) (6c)

where the element coefficient matrices are defined as

R i j = [ − 1 1 1 − 1 ] , L i j = [ l m − ( l m + l ) ( l m + l ) − l m ] , B i n j = ∫ ς 0 ς 1 G ( ς , ς i ) d Ω n d ς Ω j d ξ , E i n j = ∫ ς 0 ς 1 G ( ς , ς i ) Ω n Ω j d ς (7a)

For the moment, it is significant to note that by discretizing the problem domain and formulating element equations, BEM integral formulation is now painlessly incorporated into an algorithm usually associated with finite element method (FEM). We now have an integral form of equation which possesses better numerical and accurate properties than its differential equivalent.

At this point, there are only two major concerns left for this hybrid formulation:

1) How to deal with nonlinearity

2) How to implement the calculations for the dependent variables with an efficient computational procedure.

Domain discretization, despite its spatial and temporal locality, has always been considered a huge disadvantage in BEM circles. Its major feature ensures that local updates of the dependent variable can be processed in contiguous elements. This allows efficient handling of the coefficient matrix as well as provide a viable numerical technique for dealing with issues related to nonlinearity, inhomogeneity, transience, and body force terms. We exploit this characteristic further by assuming that the nonlinear diffusivity is uniform within a generic element and can therefore be weight-averaged as shown.

D ( θ ¯ ) = D [ α θ ¯ ( κ + 1 ) + ω θ ¯ ( κ ) ] , 0 ≤ α ≤ 1 , ω = 1 − α (7b)

where θ ¯ ( κ + 1 ) = ( θ 1 ( κ + 1 ) + θ 2 ( κ + 1 ) ) / 2 , θ ¯ ( κ ) = ( θ 1 ( κ ) + θ 2 ( κ ) ) / 2 and κ is the iteration counter. As a consequence of this approximation, ∂ D / ∂ θ = 0 and Equation (6c) now becomes

R i j θ j + ( L i j ) ψ j + E i n j χ n ( f j + Q n ) (8)

Equation (8) is still nonlinear and is handled by the Picard algorithm. θ j ( κ + 1 ) = [ A i j ( k ) ] − 1 π i ( θ j ( κ ) ) where A i j ( κ ) is made up of elements that depend on the solution at the previous time step. The nonlinear system of equations is solved iteratively until ‖ θ ( κ + 1 ) − θ ( κ ) ‖ ≤ ε n . The main drawback of the Picard iteration is its linear convergence rate, otherwise unlike the Newton’s technique it does not require Jacobian information.

The validity of the formulation developed herein was tested by comparing the numerical results with those of Torabi et al. [

ξ coordinate | Current results | DTM Results [ |
---|---|---|

0.00 | 0.82939e+00 | 0.82940e+00 |

0.04 | 0.82963e+00 | 0.82964e+00 |

0.08 | 0.83034e+00 | 0.83035e+00 |

0.10 | 0.83088e+00 | 0.83089e+00 |

o.14 | 0.83231e+00 | 0.83232e+00 |

0.16 | 0.83321e+00 | 0.83321e+00 |

0.20 | 0.83536e+00 | 0.83537e+00 |

0.24 | 0.83801e+00 | 0.83801e+00 |

0.30 | 0.84291e+00 | 0.84292e+00 |

0.34 | 0.84681e+00 | 0.84682e+00 |

0.40 | 0.85364e+00 | 0.85365e+00 |

0.44 | 0.85886e+00 | 0.85887e+00 |

0.50 | 0.86774e+00 | 0.86774e+00 |

0.54 | 0.87437e+00 | 0.87437e+00 |

0.60 | 0.88543e+00 | 0.88543e+00 |

0.64 | 0.89358e+00 | 0.89358e+00 |

0.70 | 0.90704e+00 | 0.90702e+00 |

0.74 | 0.91686e+00 | 0.91683e+00 |

0.80 | 0.93296e+00 | 0.93290e+00 |

0.84 | 0.94466e+00 | 0.94455e+00 |

0.90 | 0.96373e+00 | 0.96354e+00 |

0.94 | 0.97753e+00 | 0.9772e+00 |

0.96 | 0.98478e+00 | 0.98446e+00 |

0.98 | 0.99227e+00 | 0.99188e+00 |

Next we consider the net effect of θ a variation on the dimensionless temperature profile along the fin length. The governing differential equation (Equation (4a)) gives us a clue. The more θ a is increased, the less the thermal conductivity parameter β plays a role in the fin’s heat transfer thermal operation. Energy conservation requires that the radiation-conduction fin parameter ψ becomes more dominant. It can also be noted that the fin dimensionless temperature θ raised to the power 4 is multiplied by ψ . This results in an overall increase in radiative transport and a consequential decrease in temperature along the fin. This is the reason why

In the work reported herein a simplified, hybrid integral formulation [

ψ | Q = 0.8 | Q = 0.8 | Q = 0.1 | Q = 0.1 |
---|---|---|---|---|

β = − 0.6 | β = − 0.2 | β = − 0.2 | β = − 0.6 | |

1.0 | 0.436610 | 0.4553182 | 0.2388557 | 0.2015405 |

1.2 | 0.400292 | 0.4138134 | 0.2245216 | 0.1891190 |

1.3 | 0.381490 | 0.3965850 | 0.2183046 | 0.1837692 |

1.4 | 0.364861 | 0.3811712 | 0.2126007 | 0.1788790 |

1.5 | 0.350031 | 0.3672752 | 0.2073447 | 0.1743871 |

conductivity, radiation-conduction fin parameter, dimensionless radiation sink parameter, volumetric heat generation all play a significant role in the fin’s energy heat transfer analysis [

The author declares no conflicts of interest regarding the publication of this paper.

Onyejekwe, O.O. (2019) Simplified Integral Calculations for Radial Fin with Temperature-Dependent Thermal Conductivity. Journal of Applied Mathematics and Physics, 7, 513-526. https://doi.org/10.4236/jamp.2019.73037