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Mathematics 2022, 10, 3167 10 of 27 T int TES TES,sec + n pcm pcm 5. 6. The specific enthalpy of layer j0 is updated accordingly at the end of the considered time period ∆t, as indicated in Equation (5): hpcm,j0 (t+∆t) = hpcm,j0 (t) + ∆Upcm,j0 . (5) mlay Having reached this point, two possibilities may arise: • Layer j remains in latent zone: hlat− ≤ h (t+∆t) ≤ hlat+. Therefore, 0 pcm pcm,j0 pcm the assumption made in Step 4 is confirmed and the specific enthalpy of the layers interior to j0 is not affected during the time period under study, as described in Equation (6): hpcm,j(t+∆t) = hpcm,j(t) ∀ j < j0. (6) However, layers exterior to j0 are shown to be already in the sensible zone. To compute their specific enthalpy, their temperature is first estimated as indi- cated in Equation (7), applying a steady-state thermal conduction model, where Rcond for every layer j is computed in a similar way to Rcond in Step 2: ∀ j > j . (7) 0 pcm,j pcm,j0 pcm,j h h pcm,j pcm,j (t+∆t) = hlat− − c Tlat pcm p,pcm pcm − T (t+∆t) pcm,j T pcm,j (t+∆t) = T pcm,j−1 (t+∆t) + Q ̇ pcm,j0 Rcond Then, the specific enthalpy of the layers exterior to j0 is computed at the end of the time period under study, depending on whether the layer is superheated or subcooled, as shown in Equation (8): (t+∆t) = hlat+ +c pcm T p,pcm pcm,j if j is subcooled layer, (t+∆t)−Tlat (8) The cooling power that the whole PCM cylinder transfers to the intermediate fluid, Q ̇ pcm(t), matches the cooling power transferred between the intermediate fluid and j0 layer, Q ̇ pcm,j0 (t). Eventually, the temperature of the intermediate fluid is updated using the discrete approximation given in Equation (9): (t + ∆t) = T (t) − ∆t Q ̇ int mintcp,int (t) + Q ̇ , (9) surr – Q ̇ TES: Cooling power transferred from the refrigerant to the intermediate fluid (typically positive). – Q ̇ TES,sec: Cooling power transferred from the secondary fluid to the inter- mediate fluid (typically negative). – Q ̇ pcm(t): Cooling power transferred from each PCM cylinder to the inter- mediate fluid (positive during discharging cycle, negative during charging cycle). – Q ̇ surr : Cooling power transferred from the intermediate fluid to the sur- roundings (typically negative). This cooling power can be easily estimated from the temperature gradient between the intermediate fluid and the sur- roundings and the thermal insulation features of the TES tank. Layer j0 leaves latent zone. In other words, the latent energy of layer j0 run out sometimebeforetheperiodunderstudyfinished,∆tj0 ≤∆t,andthenthelayer j0 got into sensible zone. The precise instant is estimated, considering a constant Q ̇ where the following sign criteria is considered: + Q ̇ pcm if j is superheated layer. •PDF Image | Refrigeration Systems with Thermal Energy Storage
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