Instance Models

This section transforms the problem defined in the problem description into one which is expressed in mathematical terms. It uses concrete symbols defined in the data definitions to replace the abstract symbols in the models identified in theoretical models and general definitions.

The goals GS:Predict-Water-Temperature, GS:Predict-PCM-Temperature, GS:Predict-Water-Energy, and GS:Predict-PCM-Energy are solved by IM:eBalanceOnWtr, IM:eBalanceOnPCM, IM:heatEInWtr, and IM:heatEInPCM. The solutions for IM:eBalanceOnWtr and IM:eBalanceOnPCM are coupled since the solutions for \({T_{\text{W}}}\) and \({T_{\text{P}}}\) depend on one another. IM:heatEInWtr can be solved once IM:eBalanceOnWtr has been solved. The solutions of IM:eBalanceOnPCM and IM:heatEInPCM are also coupled, since the temperature of the phase change material and the change in heat energy in the PCM depend on the phase change.

Detailed derivation of the energy balance on water:

To find the rate of change of \({T_{\text{W}}}\), we look at the energy balance on water. The volume being considered is the volume of water in the tank \({V_{\text{W}}}\), which has mass \({m_{\text{W}}}\) and specific heat capacity, \({C_{\text{W}}}\). Heat transfer occurs in the water from the heating coil as \({q_{\text{C}}}\) (GD:htFluxWaterFromCoil) and from the water into the PCM as \({q_{\text{P}}}\) (GD:htFluxPCMFromWater), over areas \({A_{\text{C}}}\) and \({A_{\text{P}}}\), respectively. The thermal flux is constant over \({A_{\text{C}}}\), since the temperature of the heating coil is assumed to not vary along its length (A:Temp-Heating-Coil-Constant-over-Length), and the thermal flux is constant over \({A_{\text{P}}}\), since the temperature of the PCM is the same throughout its volume (A:Temp-PCM-Constant-Across-Volume) and the water is fully mixed (A:Constant-Water-Temp-Across-Tank). No heat transfer occurs to the outside of the tank, since it has been assumed to be perfectly insulated (A:Perfect-Insulation-Tank). Since the assumption is made that no internal heat is generated (A:No-Internal-Heat-Generation-By-Water-PCM), \(g=0\). Therefore, the equation for GD:rocTempSimp can be written as:

\[{m_{\text{W}}}\,{C_{\text{W}}}\,\frac{\,d{T_{\text{W}}}}{\,dt}={q_{\text{C}}}\,{A_{\text{C}}}-{q_{\text{P}}}\,{A_{\text{P}}}\]

Using GD:htFluxWaterFromCoil for \({q_{\text{C}}}\) and GD:htFluxPCMFromWater for \({q_{\text{P}}}\), this can be written as:

\[{m_{\text{W}}}\,{C_{\text{W}}}\,\frac{\,d{T_{\text{W}}}}{\,dt}={h_{\text{C}}}\,{A_{\text{C}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)-{h_{\text{P}}}\,{A_{\text{P}}}\,\left({T_{\text{W}}}-{T_{\text{P}}}\right)\]

Dividing Equation (2) by \({m_{\text{W}}}\,{C_{\text{W}}}\), we obtain:

\[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}}\,{A_{\text{C}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)-\frac{{h_{\text{P}}}\,{A_{\text{P}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{W}}}-{T_{\text{P}}}\right)\]

Factoring the negative sign out of the second term of the right-hand side (RHS) of Equation (3) and multiplying it by \({h_{\text{C}}}\) \({A_{\text{C}}}\) / \({h_{\text{C}}}\) \({A_{\text{C}}}\) yields:

\[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}}\,{A_{\text{C}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)+\frac{{h_{\text{C}}}\,{A_{\text{C}}}}{{h_{\text{C}}}\,{A_{\text{C}}}}\,\frac{{h_{\text{P}}}\,{A_{\text{P}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{P}}}-{T_{\text{W}}}\right)\]

Rearranging this equation gives us:

\[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}}\,{A_{\text{C}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)+\frac{{h_{\text{P}}}\,{A_{\text{P}}}}{{h_{\text{C}}}\,{A_{\text{C}}}}\,\frac{{h_{\text{C}}}\,{A_{\text{C}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{P}}}-{T_{\text{W}}}\right)\]

By substituting \({τ_{\text{W}}}\) (from DD:balanceDecayRate) and \(η\) (from DD:balanceDecayTime), this can be written as:

\[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)+\frac{η}{{τ_{\text{W}}}}\,\left({T_{\text{P}}}-{T_{\text{W}}}\right)\]

Finally, factoring out \(\frac{1}{{τ_{\text{W}}}}\), we are left with the governing ODE for IM:eBalanceOnWtr:

\[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}+η\,\left({T_{\text{P}}}-{T_{\text{W}}}\right)\right)\]

Detailed derivation of the energy balance on the PCM during sensible heating phase:

To find the rate of change of \({T_{\text{P}}}\), we look at the energy balance on the PCM. The volume being considered is the volume of PCM (\({V_{\text{P}}}\)). The derivation that follows is initially for the solid PCM. The mass of phase change material is \({m_{\text{P}}}\) and the specific heat capacity of PCM as a solid is \({{C_{\text{P}}}^{\text{S}}}\). The heat flux into the PCM from water is \({q_{\text{P}}}\) (GD:htFluxPCMFromWater) over phase change material surface area \({A_{\text{P}}}\). The thermal flux is constant over \({A_{\text{P}}}\), since the temperature of the PCM is the same throughout its volume (A:Temp-PCM-Constant-Across-Volume) and the water is fully mixed (A:Constant-Water-Temp-Across-Tank). There is no heat flux output from the PCM. Assuming no volumetric heat generation per unit volume (A:No-Internal-Heat-Generation-By-Water-PCM), \(g=0\), the equation for GD:rocTempSimp can be written as:

\[{m_{\text{P}}}\,{{C_{\text{P}}}^{\text{S}}}\,\frac{\,d{T_{\text{P}}}}{\,dt}={q_{\text{P}}}\,{A_{\text{P}}}\]

Using GD:htFluxPCMFromWater for \({q_{\text{P}}}\), this equation can be written as:

\[{m_{\text{P}}}\,{{C_{\text{P}}}^{\text{S}}}\,\frac{\,d{T_{\text{P}}}}{\,dt}={h_{\text{P}}}\,{A_{\text{P}}}\,\left({T_{\text{W}}}-{T_{\text{P}}}\right)\]

Dividing by \({m_{\text{P}}}\) \({{C_{\text{P}}}^{\text{S}}}\) we obtain:

\[\frac{\,d{T_{\text{P}}}}{\,dt}=\frac{{h_{\text{P}}}\,{A_{\text{P}}}}{{m_{\text{P}}}\,{{C_{\text{P}}}^{\text{S}}}}\,\left({T_{\text{W}}}-{T_{\text{P}}}\right)\]

By substituting \({{τ_{\text{P}}}^{\text{S}}}\) (from DD:balanceSolidPCM), this can be written as:

\[\frac{\,d{T_{\text{P}}}}{\,dt}=\frac{1}{{{τ_{\text{P}}}^{\text{S}}}}\,\left({T_{\text{W}}}-{T_{\text{P}}}\right)\]

Equation (4) applies for the solid PCM. In the case where all of the PCM is melted, the same derivation applies, except that \({{C_{\text{P}}}^{\text{S}}}\) is replaced by \({{C_{\text{P}}}^{\text{L}}}\), and thus \({{τ_{\text{P}}}^{\text{S}}}\) is replaced by \({{τ_{\text{P}}}^{\text{L}}}\). Although a small change in surface area would be expected with melting, this is not included, since the volume change of the PCM with melting is assumed to be negligible (A:Volume-Change-Melting-PCM-Negligible).

In the case where \({T_{\text{P}}}={{T_{\text{melt}}}^{\text{P}}}\) and not all of the PCM is melted, the temperature of the phase change material does not change. Therefore, d \({T_{\text{P}}}\) / d \(t\) = 0.

This derivation does not consider the boiling of the PCM, as the PCM is assumed to either be in a solid state or a liquid state (A:No-Gaseous-State-PCM).