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110 Tf ,(i,t+∆t) −Tf ,(i−1,t) =−NTUfluid,node(Tf ,(i−1,t) −Ts,(i,t)) [6.3.30] Ts,(i,t+∆t) −Ts,(i,t) = NTUsolid,node (Tf ,(i−1,t) −Ts,(i,t) ) [6.3.31] Solution instability arises when the calculated temperature difference between solid and fluid increases. Increasing temperature differences result in the uncontrolled, geometric change in predicted fluid and solid temperatures. Based on this understanding a very simple stability criterion can be considered: if the temperature difference between solid and fluid is decreasing from timestep to timestep then the temperature profile that develops will be stable (although not necessarily accurate). This criterion is satisfied for any initial fluid and solid temperatures if the sum of the two nodal NTU values (NTUsum) is less than or equal to two: NTU sum = hAnode ∆t + hAnode ≤ 2 [6.3.32] m s C s m& f C f If NTUsum is greater than two, equations [6.3.30] and [6.3.31] dictate that the nodal temperature difference at the end of a timestep will be larger than the temperature difference at the beginning of the timestep, causing the solution to become unstable. Equation [6.3.32] can be used to derive a critical node count for achieving solution stability by recognizing: total A =A [6.3.33] n Then substituting equations [6.3.23] and [6.3.33] into [6.3.32] and solving for the critical number of nodes to ensure solution stability: node hA ⎛ 1 mf ,node ⎞ n = total ⎜ + ⎟ [6.3.34] crit 2m&⎜C mC⎟ f⎝f ss⎠ Criterion [6.3.34] ensures only solution stability; it does not guarantee agreement with the Second Law. If ncrit nodes are used the predicted solid and fluid temperatures can exceed what would be their equilibrium temperature given infinite time for heat transfer. To ensure both solution stability and the agreement with the second law NTUsum must not exceed one, implying that the sum of the solid and fluid temperature changes during the timestep does not exceed the temperature difference at the beginning of the timestep.PDF Image | Organic Rankine Cycle Solar-Thermal Powerplants
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