Entropy generation minimization in transient heat conduction processes PART II Transient heat conduction in solids

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1 BULLEIN OF HE POLISH ACADEMY OF SCIENCES ECHNICAL SCIENCES, Vol. 6, No., 1 DOI: 1.78/bpasts-1-97 Entropy generation minimization in transient heat conduction processes PAR II ransient heat conduction in solids Z.S. KOLENDA J.S. SZMYD AGH University of Science echnology, Department of Fundamental Research in Energy Engineering, 3 Mickiewicza Ave., 3-59 Kraków, Pol Abstract. Formulation solution of the initial boundary-value problem of heat conduction in solids have been presented when an entropy generation minimization principle is imposed as the arbitrary constraint. Using an entropy balance equation the Euler-Lagrange variational approach a new form of the heat conduction equation (non-linear partial difference equation) is derived. Key words: transient heat conduction, initial boundary value problem, entropy generation minimization. Nomenclature c constants, c p specific heat capacity [kj/kg K], i internal, k heat conduction coefficient [W/m K], L Wiedemann-Franz constant [W/A K], ṁ mass flow rate [kg/h], q heat flux [W/m ], q v intensity of internal heat source [W/m 3 ], ṡ entropy flux [W/m K], Ṡ gen entropy generation due to the process irreversibility [W/m 3 K], absolute temperature [K], x, y Cartesian coordinates, Θ transformed temperature, Ω domain, operator nabla, - time, ρ density, λ Lagrange multiplier, δ difference. Indexes He helium, in entering, o environment, out leaving, t total, x,y partial derivatives with respects to x y. 1. Introduction he transient heat conduction equation assuming minimization of the entropy generation rate, Ṡ gen,min, can be derived in two ways from the Euler-Lagrange variational principle or direct minimization of the expression describing entropy generation rate. Euler-Lagrange transient heat conduction equation. he function to be minimized is Ṡ gen = Ω = k d, where k is thermal conductivity coefficient (assumed constant), = (x, y, z, ) temperature field, denotes time Ω is the domain model consideration, with additional condition S dωd = C, (1) Ω = where S = S(Ω, ) is entropy of the system C is constant. Condition (1) represents entropy change of the system from initial to final equilibrium state. he problem can be dealt with by means of Lagrange s method of undetermined multipliers as follows Find function (x, y, z, ) which satisfying required initial boundary conditions minimizes simultaneously integral Ṡ gen = Ω = k d minimum over the whole domain Ω provided that S dωd = C. Ω = kolenda@agh.edu.pl 883

2 Z.S. Kolenda J.S. Szmyd o solve the problem a new function [1] K = k + λ S is introduced where λ is the Lagrange multiplier. he extremals must satisfy the Euler-Lagrange equation K 3 ( ) d K =, dx i xi i=1 where xi denotes gradient components / x i, (x i = x, y, z). Introducing from thermodynamics ds d = ρc p function K is K = k + λρc p () Varational calculus requires the function () must satisfy the Euler-Lagrange equation K ( ) K ( ) K ( ) K =. x x y y z z After calculations k k + λ ρc p =. (3) From the fact that the third component of Eq. (3) does not depend on the path of the process, value of λ can be chosen arbitrary. Assuming λ =, Eq. (3) becomes in the final form a a =, () where a = k/ρc p is diffusion coefficient. Additional external heat source q v,ad is described by the second term of Eq. () is. Numerical example q v,a = a. Consider 1D classical initial-boundary problem given by: (in dimensionless form) governing equations x =, = (x, ), x ( 1, 1), (5) (1, ) = (1) = 1, ( 1, ) = ( 1) = 1, (x, ) = (x) = e. A general scheme is presented in Fig. 1. Analytical solution can be easily obtained with separation of variables method is (x, ) = 1 + ( 1) n [ π n + 1 exp (n + 1) π ] n= cos(n + 1) π (6) x. Fig. 1. Initial boundary value problem When the entropy generation minimization approach is applied, the initial-boundary value problem takes the form: (dimensionless form) governing equations x 1 with boundary conditions initial conditions Introducing new variable ( ) = x, = (x, ), x ( 1, 1), ( 1, ) = 1, (1, ) = 1, (x, ) = (x) = e. θ (x, ) = ln(x, ) initial-boundary value problem becomes θ x = θ, θ = θ(x, ) (7) θ (, ) =, θ (1, ) =, θ (x, ) = 1. Its analytical solution is [] θ(x, ) = ( 1) n [ π n + 1 exp (n + 1) π ] n= cos(n + 1) π (8) x (x, ) = exp (θ(x, )). Solutions (6) (8) are presented in Figs. 3 respectively. Difference δ = (x, ) clasical (x, )Ṡgen (9) 88 Bull. Pol. Ac.: ech. 6() 1

3 Entropy generation minimization in transient heat conduction processes. Part II. Fig.. Solution of classical problem Fig. 3. Solution with entropy generation minimization condition Fig.. emperature difference δ(x,) according to Eq. (9) Bull. Pol. Ac.: ech. 6() 1 885

4 Z.S. Kolenda J.S. Szmyd is shown in Fig. where (x, ) clasical (x, )Ṡgen are solutions of classical entropy generation minimization initial-boundary value problems (5) (7). Solutions (6) (8) allow to calculate entropy generation rates Ṡ gen,t = 1 Ṡ gen,t = 1 1 Numerical calculations gives ( ) (x, ) dxd x ( ) θ(x, ) dxd. x classical initial-boundary value problem Ṡ gen,cl =.33 (J/m 3 K), entropy generation minimization Ṡ gen,cl =.67 (J/m 3 K). 3. Example of general solution 3.1. Initial-boundary value problem when boundary conditions depends on time. governing equations (dimensionless form) After transformation the problem is x 1 ( ) = x, = (x, ), x (, 1), (, ) = φ 1 (), (1, ) = φ (), (x, ) = f(x). θ (x, ) = ln(x, ) θ x = θ,, θ (, ) = lnφ 1 (), θ (1, ) = lnφ (), θ (x, ) = lnf(x) its solution is given with the use of Duhamel s theorem []: θ(x, ) = exp ( n π ) sin(nπx) + nπ 1 n= lnf(x ) π sin(nπx )dx + exp ( n π λ ) {lnφ 1 (x) ( 1) n lnφ (x)} (x, ) = exp (θ(x, )). In such cases entropy generation rate calculation requires numerical calculation. 3.. Consider D initial-boundary value problem. governing equations (dimensionless form) [ ( x + y 1 ) ( ) ] + x y After introducing the problem takes the form = (x, ), x, y ( 1, 1), (x, 1, ) = (x, 1, ) = ( 1, y, ) = (1, y, ) = 1, (x, y, ) = (x, y). θ (x, y, ) = ln(x, y, ) θ x + θ y = θ =, θ (x, 1, ) = θ (x, 1, ) = θ ( 1, y, ) = θ (1, y, ) =, θ (x, y, ) = θ (x, y) = ln (x, y). he solution for θ is [3]: x θ (x, y, ) = ψ(x, 1) ψ(y, 1)erf, where finally ψ(x, 1) = π n= ( 1) n [ n + 1 exp (n + 1) π ] cos(n + 1) π x (x, y, ) = exp (θ(x, y, )). 886 Bull. Pol. Ac.: ech. 6() 1

5 Entropy generation minimization in transient heat conduction processes. Part II. Solutions of many initial boundary-value problems can be found in heat transfer literature, [] in monographs on analytical solutions of non-linear partial differential equations [3].. Conclusions he results illustrate a practical aspect of the Principle of Entropy Generation Minimization [] that: he entropy of a system can be reduced only if it is made to interact with one or more auxiliary systems in a process which impacts to these at least a compensating amount of entropy. In the case of heat conduction processes interaction with outside systems are realized by additional internal heat source which becomes as the component of Euler-Lagrange equation. In this way it is possible to explain formation processes of dissipative structures [5]. Additional heat source decreases irreversibility ratio. Analysis of the data presented in Fig. (δ(x, )) points out that the temperature changes with time are more intensive (faster) when the source is included. It leads directly to more effective consumption of driving energy minimization of natural resources. he method is directly connected with exergy optimization. Acknowledgements. Authors are grateful for the support of the Ministry of Science Higher Education (grant AGH no ). REFERENCES [1] A. Bejan, Entropy Generation Minimizations, CRC Press, Boca Raton, [] H.S. Carslaw J.C. Jaeger, Conduction of Heat in Solids, Clarendon Press, Oxford, [3] A.P. Polyanin V.F. Zaitsev, Hbook of Nonlinear Partial Differential Equations, Chatman & Hall/CRC, Boca Raton, 3. [] F. Reif, Statistical physics, in Berkley Physics Course, vol. 5, p. 31, MacGraw-Hill, New York, [5] D. Kondepudi I. Prigogine, Modern hermodynamics, John Wiley&Sons, Chichester, Bull. Pol. Ac.: ech. 6() 1 887

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