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112 This implicit model is derived from the simplified explicit model so it requires inclusion of the same timestep constraint (equation [6.3.23]). In the explicit model every calculation depends only on previously known values, so calculating forward in time and distance is trivial. In the case of the implicit model equations [6.3.36] and [6.3.37] result in a system of equations for each node and each timestep. Writing the system of equations for a single timestep and organizing them in matrix form yields the following tri-diagonal matrix equation of the standard form Ax=B: T+T+LLT+T+ f ,1 s,1 f ,n s,n node1,f ⎡1+A∆t −A∆t 0 0 0 0 ⎤ 11 ⎪+⎪⎢⎥ node1,s ⎢ −B∆t 1+B∆t 0 0 0 0 ⎥⎪Ts,1 ⎪ ⎢ Ts,1 ⎥ M⎢0 0OO0 0⎥⎪M⎪=⎢M⎥ M⎢00OO00⎥⎨M⎬M noden, f ⎢ noden,s ⎢ 1+ A ∆t n −A ∆t ⎥⎪T ⎪ 1+B∆t⎥ + 0 0 [6.3.37]. In Figure 6.14, An = hAnode and B = hAnode ⎣ ⎦⎪Ts,n⎪ ⎢⎣Ts,n ⎥⎦ ⎩⎭ 0 0 0 0 0 0 −B∆t Figure 6.14: The tridiagonal matrix that results from implementation of equations [6.3.36] and ⎧T+ ⎫ ⎡T ⎤ f,1 f,in ⎢T ⎥ n ⎪f,n⎪ ⎢f,n−1⎥ m& f C f , n ∆ t m s C s If the full explicit energy balance in equation [6.3.20] were used the matrix necessary to solve each timestep implicitly would be tridiagonal with each diagonal fully populated. That is, the system of equations would be fully coupled. In this case each node is represented by a discrete set of equations (n sets of two equations each with two unknowns) where equation [6.3.20] would yield a fully populated tri-diagonal matrix (2n equations with 2n unknowns). Figure 6.15 shows the implicit solution converging on the analytical solution reported in section 6.3.1 as the number of nodes increases (using equation [6.3.23] to constrain the solution timestep).PDF Image | Organic Rankine Cycle Solar-Thermal Powerplants
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