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1H hSs W Uc(1)R2 A method of characteristics given by Van Lew[7] is adopted in this study to solve the dimensionless energy balance equations for heat transfer between air and ceramic balls. Equation (11) and (12) can be reduced along the characteristic zz tt const and zz const respectively, and then be led to the solutions as Equation (16) and (17) via integral-discretization and trapezoidal rule: [1tt(11)]f2,2(1 tt)s2,2[1tt(11)]f1,1(tt)s1,1(tt) (16) 2XY2X2XY2XY (1 tt)s2,2 (tt)f 2,2 (1 tt)s2,1 (tt)f 2,1 (17) 2W 2W 2W 2W Equations (16) and (17) can be reposed as a group of algebraic equations for two unknowns of f 2,2 ands2,2 , whilef ands atgridpointf1,1ands2,1areknown. tttt tt tt ,1 s2,2 (1 ) ( ) (18) s2,1 2W2W 2W 2W Since Equation (18) is only for two equations, Cramer’s rule can be applied to obtain the solution, which contributes to the numerical computation with the minimum computing time. Fig.7 shows the solution matrix by using method of characteristics, where M is the spatial grid number, zz is the dimensionless spatial step length, N is the time grid number, and tt is the dimensionless time step length. FIGURE 7 Solution matrix by using method of characteristics In the solution matrix, shown as Fig.7, since the conditions at1,1 ,1,2 , and 2,1 are known for both phases based on the actual experiment conditions, the temperatures of ceramic balls and air at 2,2 will be easily calculated from Equation (18). Then extending the above sample calculation to all points in the grid of time and space for both phases, the entire matrix of solutions in time and space for both ceramic balls and air can be obtained eventually. (15) 1tt(1 1),tt [1tt(1 1)] 2 X Y 2Xf2,2 2 X Y f1,1 (tt) 2X s1,1 f2,1 (tt) Y ss 080027-6PDF Image | packed-bed thermal energy storage device
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