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Setting NTUsum to one results in the following recommended node count: hA ⎛ 1 mf ,node ⎞ n = total ⎜ + ⎟ [6.3.35] rec m&⎜CmC⎟ f⎝f ss⎠ Figure 6.13 shows a progression of temperature profiles as the NTU sum passes through the recommended and critical node criteria. Figure 6.13(a) shows a temperature profile with the recommended node count, producing a stable and well-behaved temperature profile. Figure 6.13(b) shows that stable, but oscillatory, temperature profiles arise for node counts between the recommended and critical node counts. While stable, the profile in Figure 6.13(b) violates the second law as fluid temperature is shown dropping below the initial rock temperature. Figure 6.13(c) shows complete solution instability when there are fewer nodes than the ncrit. (a) (b) (c) Figure 6.13: The simplified explicit solution approaching instability as NTUsum approaches and exceeds 2. Implicit Numerical Model The stability issues experienced with the explicit model motivate the development of an analogous implicit model. Implicit models guarantee solution stability. In an implicit model, the values of fluid temperature and solid temperature are calculated based on their values at the end of a timestep or distance step rather than the beginning (as in an explicit model). The implicit finite-difference equations are formulated as follows based on equations [6.3.26] and [6.3.27]: Tf,(i,t+∆t) =Tf,(i−1,t) −hAnode(Tf,(i,t+∆t) −Ts,(i,t+∆t)) [6.3.36] m& f C f Ts,(i,t+∆t) =Ts,(i,t) +hAnode∆t(Tf,(i,t+∆t) −Ts,(i,t+∆t)) [6.3.37] msCs 111PDF Image | Organic Rankine Cycle Solar-Thermal Powerplants
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