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Mathematics 2022, 10, 3167 11 of 27 rate, Q ̇ pcm,j0 (t), and the remaining latent energy of layer j0 at instant t, as shown in Equation (10) for a charging process or Equation (11) for a discharging process: m (h (t) − hlat−) ∆tj = lay pcm,j0 0 Q ̇ p c m , j 0 pcm , (10) m (hlat+ − h ∆tj = lay pcm pcm,j0 (t)) . (11) 0 Q ̇ p c m , j 0 Then, a shorter time interval, ∆tj0−1, can be computed as shown in Equation (12): ∆tj0−1 = ∆t−∆tj0. (12) This time interval ∆tj0−1 is used for successive estimation, in which the next inner layer, j0−1, is considered as the new first layer in the latent zone, and the algorithm restarts from Step 2, but using the corresponding time period ∆tj0−1, instead of ∆t. Then, the cooling power transferred between each PCM cylinder and the inter- mediate fluid, Q ̇ pcm eventually has several successive contributions along the original time period ∆t, as shown in Equation (13): Q ̇pcm =Q ̇pcm,j0,Q ̇pcm,j0−1,...Q ̇pcm,j0−i, (13) j0−i being the most inner layer found in the latent state during the time period under study ∆t. Obviously, the addition of the partial time intervals matches the complete interval, as shown in Equation (14): l=i ∑∆tj0−l = ∆t. (14) l=0 For each one of the partial time intervals, Tint is updated using Equation (9), giving rise to the succession shown in Equation (15): T(t+∆t)=T(t)−∆tk0 Q ̇+Q ̇ +nQ ̇ +Q ̇ int j0 int mint cp,int TES TES,sec pcm pcm,j0 surr l=1 Tint(t+ ∑∆tj0−l) = Tint(t+∆tj0)− m c QTES +QTES,sec +npcmQpcm,j0−1 +Qsurr l=0 ∆tj0−1 ̇ ̇ ̇ ̇ int p,int l=i−1 l=0 int p,int l=i Tint(t+ ∑∆tj0−l) = Tint(t+ ∑ ∆tj0−i)− m c QTES +QTES,sec +npcmQpcm,j0−i +Qsurr ∆tj0−i ̇ ̇ ̇ ̇ ··· (15) l=0 l=0 The proposed model can be expressed as shown in Equation (16), where the function g stands for the complete discrete model, also shown in Figure 4: xTES(t+∆t)=g(xTES(t),Q ̇ref (t),Q ̇ref (t),Tsurr(t)). (16) As stated before, since the dynamic model presented in [25] involves a small sampling time to properly describe the fastest system dynamics, it does not turn out to be suitable or even computationally affordable to be used as the prediction model within a predictive scheduling strategy. This is where the simplified dynamic model proposed in this section, l=i Tint(t+∑∆tj0−l) ≡ Tint(t+∆t). TES TES,secPDF Image | Refrigeration Systems with Thermal Energy Storage
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