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[ Original Paper ] | |
Journal of the Korean Society of Marine Engineering - Vol. 40, No. 5, pp. 412-417 | |
Abbreviation: J. Korean Soc. of Marine Engineering (JKOSME) | |
ISSN: 2234-7925 (Print) 2234-8352 (Online) | |
Print publication date Jun 2016 | |
Received 14 Mar 2016 Revised 11 May 2016 Accepted 04 Jun 2016 | |
DOI: https://doi.org/10.5916/jkosme.2016.40.5.412 | |
Numerical analysis of natural convection heat transfer from a fin in parallel enclosure | |
1Engineering Research Institute, Department of Mechanical Engineering for Production, Gyeongsang National University, 501 jinjudae-ro, Jinju, Gyeongsangnam-do 52828, Korea, Tel: 055-772-1631 (mwbae@gnu.ac.kr) | |
2Professor Emeritus, Tokyo Institute of Technology, Tel: +81-429-24-5608 (ymochimaru-1947@cx.117.cx) | |
Correspondence to : †Engineering Research Institute, Department of Mechanical Engineering for Production, Gyeongsang National University, 501 jinjudae-ro, Jinju, Gyeongsangnam-do 52828, Korea, E-mail: mwbae@gnu.ac.kr, Tel: 055-772-1631 | |
Copyright © The Korean Society of Marine Engineering This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. | |
A fin of finite width with infinitely small thickness is assumed to be placed horizontally between two horizontal parallel plates of infinite extension in the exactly central position. The lower plate and the half of the upper plate are kept at a constant lower temperature, and the remaining upper plate is kept at a constant higher temperature. The fin is also kept at a constant temperature (variable). Steady-state two-dimensional laminar natural convection is analyzed as a problem of boundary value under a boundary-fitted conformal mapping system, using a spectral finite difference scheme, with a condition of doubly-connectedness. The steady-state solution is obtained as a limit of the transient solution.
Keywords: Natural convection heat transfer, Fin, Conformal mapping, Spectral finite difference scheme, Doubly-connectedness |
The numerical estimation of natural convection from a fin is an element of heat transfer problems. Basically, two-dimensional analysis is fundamental. Furthermore, if the introduction of a boundary-fitted coordinate system under the conformal mapping is possible, then this problem is reduced simply to a boundary-value problem, very similar to a Cartesian coordinate system, resulting in possibility of spectral decomposition [1]. Many theories and applications on conformal mapping have been presented in the past [2]-[5].
Among many numerical schemes for heat and fluid flow analysis, the spectral finite difference scheme using the conformal mapping [1] is very effective, although the special attention is required in a multiply-connection region [6], which is the case described in this paper. This spectral finite difference scheme has a good resolution in space and a high computation speed. Also, this scheme can cover various problems associated with infinite extension [7] and support the mixed types of boundary conditions [1][8]. From the viewpoint of experimental work, the heat transfer from fins in plane layers is treated by authors of references [9]-[13].
The spectral finite difference scheme developed by one of authors earlier [1][14] has the following features. For a coordinate system under analysis, the boundary-fitted-coordinate system using the conformal mapping is adopted, where all the boundaries in a two-dimensional problem, including infinity, if any, constitute a part or whole of one or two coordinate surfaces for the same component of one independent variable so that it can be reduced to a boundary-value problem (at least Dirichlet type, Neumann type, or mixed type [8]). If a part does not correspond to a physical boundary, it is necessary to introduce additional conditions such as continuity of scalars. Generally, such a conformal mapping is not necessarily unique, and a sufficient method to apply it for any configuration is not yet known.
Secondly, Fourier series is frequently used for the spectral decomposition of dependent variables, but is not restricted. In some cases, analytical solutions (at least first two terms [7] and asymptotic solution [15]) are obtained without any aid of finite difference approximation.
Thirdly, the condition of multiply-connectedness is exactly introduced in a mathematical point of view [1]/[14][15]; such a condition is not in the conventional FEM (Finite element method) and FDM (finite difference method). Decomposition of the governing equations exactly depends on mathematical theorems (usually depending on necessary and sufficient conditions), which does not introduce errors. Nevertheless, it is assumed that higher components decay faster, which means that the conditional convergence of series is not adequate for numerical solutions. The conventional FEM and FDM produce error(s) in formulation unless polynomial solutions are expected. Experimental values may be sensitive to conditions, not expressed explicitly, which can be covered by the present method in some cases [16].
Authors have been performing several works on natural convection heat transfer [17]-[19]. In this study, the numerical analysis of two-dimensional laminar natural convection heat transfer from a fin in the parallel enclosure is carried out by a spectral finite difference scheme.
Two-dimensional laminar natural convection in a vertical plane is considered in this study. Fluid is assumed to be enclosed between two horizontal parallel plates (CE and C’E’) of infinite extension equipped with a horizontal finite length fin(infinitesimally small thickness, upper surface OA, lower surface OA’) exactly in the middle of the plates as shown in Figure 1.
The boundary-fitted conformal mapping coordinate system (α, β) is introduced such that
(1) |
where (x, y) is a dimensionless Cartesian coordinate system with vertically upward y. In this system, the reference length, L, is given by the distance between two parallel plates, i.e., the upper and lower plates are given by y= 0.5 and y= −0.5 respectively. It is assumed that the point (α, β) = (0, 0) corresponds to point O, and (α, β) = (0, π/2) to point A. Consequently, the upper plate is designated by α = 0, 2 tan−1(1/
(2) |
(3) |
(4) |
Through Equation (1):
(5) |
The following fully Dirichlet type thermal boundary conditions (not producing stratification) are introduced: along the lower plate T = 0, along the left part of upper plate (DE) T = 0, and along the right part of upper plate (CD) T =1, and along the fin (both sides) T = TC (0 < TC < 1). As a dynamical boundary condition, no-slip flows on the plates and on the fin surfaces are assumed ; i.e., without loss of generality, ψ = 0 along the upper and lower plates, ψ = c (constant to be determined) along the fin surface, and
Under Equation (1), the continuity of scalar quantity ϕ(α, β) at the both sides of center line (AB, A’B’) and the continuity of gradient vector for the scalar quantity ϕ(α, β) are given by
(6) |
(7) |
which applies for ψ, ζ, and T.
At the single point given by r = 0, any scalar quantity and its gradient are independent of β, which is a necessary condition.
The condition for the region of doubly-connectedness is given by
(8) |
where p stands for pressure and f for integration around the fin. Equation (8) leads to
(9) |
Under the given thermal boundary condition, Equation (9) becomes
(10) |
Among many possibilities of complete sets, Fourier series expansion is applied; that is:
(11) |
The mathematical boundary conditions (6) and (7) for the arguments in Equation (11) become
(12) |
(13) |
if π/2 < |β| < 2 tan−1(1/
(14) |
(15) |
(16) |
The constant c (to be determined) can be included in Equation (10), noting that J = 0 if α = β = 0.
The system of Equation (2) - (4) can be split into each Fourier component, supplemented with the decomposed boundary conditions at r = 1 and r = 0. Spatial and time derivatives can be replaced by finite difference approximation (nonuniform grid spacing can be accepted), which can be semi-implicitly integrated with respect to time, using a suitably given initial thermal and fluid flow field to get a steady-state solution. Since the temperature on the fin surface is uniform, the total dimensionless force F (based on ρU2L, U≡
(17) |
The mean Nusselt number on the fin surface, Num, is given by
(18) |
In this problem, the Dirichlet thermal boundary condition is assumed. Thus if Num > 0 is positive, the heat is emitted outward from the fin as a heat source, and if Num < 0 is negative, the heat is absorbed inward to the fin, as a heat sink. Consequently, at −∞ < α ≤ 0, if
Figures 2 and 3 show streamlines and isotherms at
Undisturbed (without any fin) pure conduction temperature distribution is given by
(19) |
(20) |
which shows that Tc at O is 0.25. Thus, as long as
Natural convention heat transfer from a two-dimensional fin in the parallel enclosure was successfully analyzed, using a spectral finite difference scheme, supplemented with a condition of doubly-connectedness of the domain. The enclosure plates were placed horizontally and kept at constant temperatures. Problems were treated as a boundary-value problem, and variables were exactly decomposed into Fourier components. To get a steady-state solution, a time-marching scheme was applied. Depending on the uniform surface temperature of fin, the fin behaved as a heat source or sink.
Gr : | Grashof number based on the length L and temperature difference ΔT (> 0) ≡ temperature at the right half of the upper plate minus that at the left half. |
k : | Numerical parameter |
L : | :Reference length ≡ the distance between the upper and lower plates |
Num : | Mean Nusselt number |
Pr : | Prandtl number |
T : | Dimensionless temperature ≡ (local temperature minus the temperature at the left half of the upper plates) / ΔT |
x : | Cartesian coordinate (horizontal) |
y : | Cartesian coordinate (vertical) |
α : | Mapping coordinate |
β : | Conformal mapping coordinate |
ζ : | Dimensionless vorticity |
ρ : | Density |
ψ : | Dimensionless stream function |
f : | Fin surface |
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