Abstract
We consider the one-dimensional steady fin problem with the Dirichlet boundary condition at one end and the Neumann boundary condition at the other. Both the thermal conductivity and the heat transfer coefficient are given as arbitrary functions of temperature. We perform preliminary group classification to determine forms of the arbitrary functions appearing in the considered equation for which the principal Lie algebra is extended. Some invariant solutions are constructed. The effects of thermogeometric fin parameter and the exponent on temperature are studied. Also, the fin efficiency is analyzed.
1. Introduction
A search for exact and numerical solutions for models arising in heat flow through extended surfaces continues to be of scientific interest. The literature in this area is sizeable (see, e.g., [1] and references cited therein). Perhaps such interest has been instilled by frequent encounters of fin problems in many engineering applications to enhance heat transfer. Fins play an important role in enhancing heat dissipation from a hot surface. They are used in air conditioning, air-cooled craft engines, refrigeration, cooling of computer processors, cooling of oil carrying pipe line, and so on.
In recent years, many authors have been interested in the steady-state problems [2–5] describing heat flow in one-dimensional longitudinal rectangular fins. The symmetry analysis, in particular, group classification of the unsteady fin problem has attracted some interest (see, e.g., [6–10]).
Few exact solutions exist for one-dimensional problems. Perhaps this is due to highly nonlinearity of the fin models. In fact, existing solutions are constructed only when both thermal conductivity and heat transfer coefficient are given as constant [2]. Recently, in [5], exact solutions of the one-dimensional fin problem given nonlinear thermal conductivity and heat transfer coefficient have been constructed.
In this paper, we determine the cases of thermal conductivity and heat transfer coefficient terms for which extra symmetries are admitted. We then select the realistic cases and analyze the problem. In Section 2, we provide the mathematical formulation of the problem. Symmetry analysis is performed in Section 3. We determine the principal Lie algebra, equivalence transformations, and list the cases for which the principal Lie algebra is extended. In Section 4, we employ symmetry techniques to determine wherever possible, the invariant solutions.
2. Mathematical Models
Consider longitudinal rectangular one-dimensional fin with a cross-sectional area as shown in Figure 1. The perimeter and length are given by and , respectively. The fin is attached to a fixed base surface of temperature and extends into a fluid of temperature . The fin is insulated at the tip. The steady energy balance equation is given by [4] where and are temperature-dependent thermal conductivity and heat transfer coefficient, respectively (see, e.g., [2, 3]). The spatial variable is .
The relevant boundary conditions are given by
Introducing the dimensionless variables reduces (2.1) to the relevant dimensionless energy equation and the boundary conditions become
Setting leads to the equivalent definition of thermogeometric fin parameter , where is the Biot number, and is the extension factor with being the fin thickness. Since is an arbitrary function of temperature, we equate the product to . Note that the thermogeometric fin parameter is specified. The parameters and are the heat transfer coefficient at the fin base and the fluid thermal conductivity. The analysis of (2.4) was conducted in [5], wherein the heat transfer coefficient was assumed to be given by the power law function of temperature. In this paper, we allow both the heat transfer coefficient and thermal conductivity to be arbitrary functions of temperature and employ preliminary group classification techniques to determine the forms which lead to exact solutions. We consider the governing equation
We note that (2.6) is linearizable provided that is a differential consequence of . The proof of this statement follows from chain rule [11]. This implies that (2.6) may be linearizable for any such that its derivative is . Also, the linearization of (2.6) was performed in [12] wherein approximate techniques were employed to solve the problem. In this paper, we apply Lie point symmetry techniques to analyze the problem.
3. Symmetry Analysis
The theory and applications of symmetry analysis may be found in excellent text such as those of [13–18]. In the next subsections, we construct the equivalence algebra and hence equivalence group of transformations admitted by (2.6). Furthermore we determine the Lie point symmetries admitted by (2.6) with arbitrary functions and ; that is, we seek the principal Lie algebra. Symmetry technique are algorithmic and tedious. Here we utilize the interactive computer software algebra REDUCE [19] to facilitate the calculations.
3.1. Equivalence Transformations
In brief, an equivalence transformation of a differential equation is an invertible transformation of dependent and independent variables which leave the form of the equation in question unchanged [20]. However the form of the arbitrary functions appearing in the transformed equation may be distinct from those of the original equation. To determine the equivalence transformation, one may seek the equivalence algebra generated by the vector field The second prolongation is given by where with and being the total derivative operator defined by respectively. The prime implies differentiation with respect to . The invariance surface condition is given by This system of equations yields the infinite dimensional equivalence algebra spanned by the base vectors admitted by (2.6). Here and are arbitrary functions of and , respectively.
3.2. Principal Lie Algebra
In this subsection, we seek classical Lie point symmetries generated by the vector field admitted by the governing equation for any arbitrary functions and . We seek invariance in the form
Here is the second prolongation defined by where the prolongation formulae are given above. The principal Lie algebra is one dimensional and spanned by space translation. For nontrivial function and , we obtain the determining equations(1), (2), (3), (4).
The determining equation (2.1) implies that and , where and are arbitrary functions of and , respectively. The determining equations (2.2), (2.3), and (2.4) become(2*), (3*), (4*).
It appears that full group classification of (2.6) may be difficult to achieve. Hence, we resort to the preliminary group classification techniques.
3.3. Preliminary Group Classification
We follow the sketch of the preliminary group classification technique as outlined in [20]. We note that the (2.6) admits an infinite equivalence algebra as given in Section 3.1. So we are free to take any finite dimensional subalgebra as large as we desire and use it for preliminary group classification. We choose a five-dimensional equivalence algebra spanned by the vectors Recall that and are dependent. Thus, we consider the projections of (3.10) on the space of . The nonzero projections of operators (3.10) are
Proposition 3.1 (see, e.g., [20]). Let be an -dimensional subalgebra of the algebra . Denote by , a basis of and by the elements of the algebra such that is the projections of on . If equations are invariant with respect to the algebra then the equation admits the operator
Proposition 3.2 (see, e.g., [20]). Let (3.13) and equation be constructed according to Proposition 3.1 via subalgebras and , respectively. If and , are similar subalgebras in then (3.13) and (3.15) are equivalent with respect to the equivalence group generated by . These propositions imply that the problem of preliminary group classification of (2.6) is reduced to the algebraic problem of constructing nonsimilar subalgebras of or optimal system of subalgebras [20]. We explore methods in [13] to construct the one-dimensional optimal systems. The set of nonsimilar one-dimensional subalgebras is Here and are arbitrary constants.
As an example, we apply 1 to one of the element of the optimal system. Since this involves routine calculations of invariants, we list the rest of cases in Table 1, wherein , and are arbitrary constants. Note that the power law was obtained in [5], therefore we omit this case in this manuscript.
Consider the subalgebra
where, without loss of generality, we have assumed to be unity. A basis of invariants is obtained from the equation
and the forms of and are For simplicity, we have allowed both integration constants to vanish. Further cases are listed in Table 1. By applying Proposition 3.1, we obtain the symmetry generator . We shall show in Section 4.2 that, for these forms of and , one may obtain seven more Lie point symmetry generators.
4. Symmetry Reductions and Invariant Solutions
The main use of symmetries is to reduce the number of independent variables of the given equation by one. If a partial differential equation (PDE) is reduced to an ordinary differential equation (ODE), one may or may not solve the resulting ODE exactly. If a second-order ODE admits a two-dimensional Lie algebra, then one can use Lie’s method of canonical coordinates to completely integrate the equation (see, e.g., [21]).
4.1. Example 1
As an illustrative example, we consider the case and where . In this case (2.4) admits a non-Abelian two-dimensional Lie algebra spanned by the base vectors listed in Table 1. This noncommuting pair of symmetries leads to the canonical variables where is an arbitrary constant. We have two cases, the “particular” canonical variables when and the “general” canonical variables given a nonzero , say .
4.1.1. Particular Canonical Form
The corresponding canonical forms of and are Writing transforms (2.6) to Here prime is the total derivative with respect to . Three cases arise.
Case 1. For , we obtain the constant solution which is not related to the original problem. Thus, we ignore it.
Case 2. If the term in the square bracket vanishes, then we obtain in terms of original variables the exact “particular” solution Note that this exact solution satisfies the boundary only at one end. The Neumann’s boundary condition leads to a contradiction since the thermogeometric fin parameter is a nonzero constant.
Case 3. Solving the entire equation (4.3) we, obtain the solution in complicated quadratures, and therefore we omit it.
4.1.2. General Canonical Form
In this case, the transformed equations are given by Clearly reduces (4.5) to (4.3). We herein omit further analysis.
4.2. Example 2
We consider as an example (2.6) with thermal conductivity given as exponential function of temperature; that is, and heat transfer coefficient is given as the quotient . Given , then (2.6) admits a maximal eight-dimensional symmetry algebra spanned by the base vectors
Equation (2.6) is linearizable or equivalent to (see, e.g., [21]). In fact, we note that the point transformation , linearizes (2.6) given . Following a simple manipulation, we obtain the invariant solutions satisfying the prescribed boundary conditions, namely, Solution (4.7) is depicted in Figures 2 and 3. Note that, for and , we obtain solutions which have no physical significance for heat transfer in fins. Therefore, we herein omit such solutions.
The fin efficiency is defined as the ratio of actual heat transfer from the fin surface to the surrounding fluid while the whole fin surface is kept at the same temperature (see, e.g., [1]). Given (4.7) fin efficiency () is given by
We use MAPLE package to evaluate this integral. The plot is depicted in Figure 4.
5. Some Discussions and Concluding Remarks
We considered a one-dimensional fin model describing steady-state heat transfer in longitudinal rectangular fins. Here, the thermal conductivity and heat transfer coefficient are temperature dependent. As such the considered problem is highly nonlinear. This is a significant improvement to the results presented in the literature (see, e.g., [2, 3]). Preliminary group classification led to a number of cases of thermal conductivity and heat transfer coefficient for which extra symmetries are obtained. Exact solutions are constructed when thermal conductivity and heat transfer coefficient increase exponential with temperature. We observed, in Figure 2, that temperature inversely proportional to the values of the thermogeometric fin parameter. Furthermore, we observe that for certain values of , the solution is not physically sound (see also, [22]). One may recall that the thermogeometric fin parameter depends also on heat transfer coefficient at the base of the fin. We notice that the exponential temperature-dependent heat transfer coefficient in this paper leads to lower values of for which the solutions are realistic. That is, the maximum values of , say for which the solutions are physically sound, is around 2. We observe, in Figure 5, that as values of increase beyond 2, the temperature profile becomes negative. This contradicts the rescaling of temperature (the dimensionless temperature). Unlike [5, 23] whereby heat transfer is given by a power law, this value is much higher. The reasons behind this observation is studied elsewhere. In Figure 3, temperature increases with increased values of the exponent . Furthermore, fin efficiency decreases with increased values of the thermogeometric fin parameter. We observed, in Figure 4, that the maximum value of the thermogeometric fin parameter for which the fin efficiency is realistic is again around 2.
Acknowledgment
R. J. Moitsheki wishes to thank the National Research Foundation of South Africa under Thuthuka program, for the continued generous financial support.