Thermal Resistance and Capacity Formulation

The central point is that use of the concepts of thermal resistance and capacitance enables us to write the forward-difference equation for all nodes and boundary conditions in the single compact form of Eq. (4-41). The setup for a numerical solution then becomes a much more organized process which can be adapted quickly to the computational methods at hand. 

Equation (4-41) is developed by using the forward-difference concept to produce an explicit relation for each Tf+l. As in our previous discussion, we could also write the energy balance using backward differences, with the heat transfers into each ith node calculated in terms of the temperatures at the p + 1 time increment. Thus, 

as before, the set of equations produces an implicit set which must be solved simultaneously for the Tf+1, etc. The solution can be carried out by a number of methods, as discussed in Chap. 3. If the solution is to be performed 

It is interesting to note that in the steady-state limit of At - this equation becomes identical with Eq. (3-32), the formulation we employed for the iterative solution in Chap. 3. 

The stability requirement in the explicit formulation may be examined by solving Eq. (4-41) for 7 f+ 1  

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We should remark that the resistance-capacity formulation is easily adapted to take into account thermal-property variations with temperature. One need only calculate the proper values of p, c, and k for inclusion in the C, and R . Depending on the nature of the problem and accuracy required, it may be necessary to calculate new values of C, and R0 for each iteration. Example 4-16 illustrates the effects of variable conductivity. 

The principal criteria for the formulations were mechanical and thermal stability, homogeneity, radiation resistance, high containment capacity, low volume, corrosion resistance, low leachability, easy fabrication (mastered technology), and flexibility with regard to the composition of waste to be conditioned. 

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