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Mathematics 2022, 10, 3167 9 of 27 elling approach proposed in [29] is applied, due to the high speed requirements imposed by the predictive scheduler, since the model is expected to run countless times over a certain horizon within the optimization procedure. In the modelling approach presented in [29], the PCM elements (in this case, cylinders) are divided into a certain number of layers nlay, each one of them featuring the same mass mlay. The number of considered layers represents a trade-off between accuracy and computational load. Some assumptions have been made to develop the time-efficient discrete model. On the one hand, the intrinsic dynamics of the layers in the sensible zone are assumed to be negligible, thus their temperatures are computed according to a steady-state thermal conduction model. On the other hand, cold energy is considered to be transferred at constant rate, provided that the set of layers in latent and sensible zone does not vary. As mentioned above, the TES tank state vector xTES is made up of the temperature of the intermediate fluid Tint and an enthalpy description of the cold energy stored inside every PCM cylinder, which may be defined as indicated in Equation (1): hpcm,1 hpcm,2 ∈Rnlay+1, (1) where the specific enthalpies hpcm,j ∀j = 1, ..., nlay of all the cylindrical layers are included in the state vector. The description of the recursive algorithm implementing the simplified dynamic model of the TES tank, given the TES tank charging Q ̇ TES and discharging Q ̇ TES,sec cooling powers as virtual manipulated inputs and the surroundings temperature Tsurr as a measurable disturbance, is given in a step-by-step sketch, similarly to the original time- efficient discrete model described in [29]: 1. From a given state vector xTES(t), an inward scanning sequence of the PCM layers is performed, looking for the first layer j0 in latent zone, as indicated in Equation (2): j = maxj∈{1,...n } | hlat− ≤h (t)≤hlat+. (2) 0 lay pcm pcm,j pcm 2. The thermal resistance Rcond , corresponding to the spherical shell including all PCM pcm,j0 layers exterior to layer j0: j ∈ {j0 +1, . . . , nlay}, is computed as if it was a single, clustered thermal-conduction layer. 3. The cooling power transferred between the intermediate fluid and j0 layer is computed as shown in Equation (3): Q ̇ pcm,j (t) = Tint(t) − Tpcm,j0 (t) , (3) 0 Rcond + Rcond,wall + Rconv,ext pcm,j0 pcm pcm where Rcond,wall refers to the thermal conduction resistance due to the PCM cylinder pcm coating, and Rconv,ext represents the thermal resistance caused by external natural pcm convection with the intermediate fluid. 4. Layer j0 is initially considered to remain in latent zone during the complete time period ∆t, in such a way that cold energy is assumed to be transferred at the constant rate Q ̇ pcm,j0 (t), as shown in Equation (4): xTES = ··· h p c m , n Tint lay ∆Upcm,j0 = t+∆t ̇ ̇ Qpcm,j0 dt ≈ Qpcm,j0 (t) ∆t. (4) tPDF Image | Refrigeration Systems with Thermal Energy Storage
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