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.

Energy balance on water to find the temperature of the water

Refname	IM:eBalanceOnWtr
Label	Energy balance on water to find the temperature of the water
Input	${m_{\text{W}}}$ , ${C_{\text{W}}}$ , ${h_{\text{C}}}$ , ${A_{\text{P}}}$ , ${h_{\text{P}}}$ , ${A_{\text{C}}}$ , ${T_{\text{P}}}$ , ${t_{\text{final}}}$ , ${T_{\text{C}}}$ , ${T_{\text{init}}}$
Output	${T_{\text{W}}}$
Input Constraints	${T_{\text{C}}}\gt{}{T_{\text{init}}}$
Output Constraints
Equation	$\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)+η\,\left({T_{\text{P}}}\left(t\right)-{T_{\text{W}}}\left(t\right)\right)\right)$
Description	$t$ is the time ( ${\text{s}}$ ) ${T_{\text{W}}}$ is the temperature of the water ( ${{}^{\circ}\text{C}}$ ) ${τ_{\text{W}}}$ is the ODE parameter for water related to decay time ( ${\text{s}}$ ) ${T_{\text{C}}}$ is the temperature of the heating coil ( ${{}^{\circ}\text{C}}$ ) $η$ is the ODE parameter related to decay rate (Unitless) ${T_{\text{P}}}$ is the temperature of the phase change material ( ${{}^{\circ}\text{C}}$ )
Notes	${T_{\text{P}}}$ is defined by IM:eBalanceOnPCM. The input constraint ${T_{\text{init}}}\leq{}{T_{\text{C}}}$ comes from A:Charging-Tank-No-Temp-Discharge. ${τ_{\text{W}}}$ is calculated from DD:balanceDecayRate. $η$ is calculated from DD:balanceDecayTime. The initial conditions for the ODE are ${T_{\text{W}}}\left(0\right)={T_{\text{P}}}\left(0\right)={T_{\text{init}}}$ following A:Same-Initial-Temp-Water-PCM. The ODE applies as long as the water is in liquid form, $0\lt{}{T_{\text{W}}}\lt{}100$ ( ${{}^{\circ}\text{C}}$ ) where $0$ ( ${{}^{\circ}\text{C}}$ ) and $100$ ( ${{}^{\circ}\text{C}}$ ) are the melting and boiling point temperatures of water, respectively (from A:Water-Always-Liquid and A:Atmospheric-Pressure-Tank).
Source	koothoor2013
RefBy	IM:eBalanceOnWtr, IM:eBalanceOnPCM, UC:No-Internal-Heat-Generation, FR:Output-Values, FR:Find-Mass, and FR:Calculate-Values

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:

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

Energy Balance on PCM to find temperature of PCM

Refname	IM:eBalanceOnPCM
Label	Energy Balance on PCM to find temperature of PCM
Input	${{T_{\text{melt}}}^{\text{P}}}$ , ${t_{\text{final}}}$ , ${T_{\text{init}}}$ , ${A_{\text{P}}}$ , ${h_{\text{P}}}$ , ${m_{\text{P}}}$ , ${{C_{\text{P}}}^{\text{S}}}$ , ${{C_{\text{P}}}^{\text{L}}}$
Output	${T_{\text{P}}}$
Input Constraints	${{T_{\text{melt}}}^{\text{P}}}\gt{}{T_{\text{init}}}$
Output Constraints
Equation	$\frac{\,d{T_{\text{P}}}}{\,dt}=\begin{cases}\frac{1}{{{τ_{\text{P}}}^{\text{S}}}}\,\left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right), & {T_{\text{P}}}\lt{}{{T_{\text{melt}}}^{\text{P}}}\\\frac{1}{{{τ_{\text{P}}}^{\text{L}}}}\,\left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right), & {T_{\text{P}}}\gt{}{{T_{\text{melt}}}^{\text{P}}}\\0, & {T_{\text{P}}}={{T_{\text{melt}}}^{\text{P}}}\land{}0\lt{}ϕ\lt{}1\end{cases}$
Description	$t$ is the time ( ${\text{s}}$ ) ${T_{\text{P}}}$ is the temperature of the phase change material ( ${{}^{\circ}\text{C}}$ ) ${{τ_{\text{P}}}^{\text{S}}}$ is the ODE parameter for solid PCM ( ${\text{s}}$ ) ${T_{\text{W}}}$ is the temperature of the water ( ${{}^{\circ}\text{C}}$ ) ${{τ_{\text{P}}}^{\text{L}}}$ is the ODE parameter for liquid PCM ( ${\text{s}}$ ) ${{T_{\text{melt}}}^{\text{P}}}$ is the melting point temperature for PCM ( ${{}^{\circ}\text{C}}$ ) $ϕ$ is the melt fraction (Unitless)
Notes	${T_{\text{W}}}$ is defined by IM:eBalanceOnWtr. The input constraint ${T_{\text{init}}}\leq{}{{T_{\text{melt}}}^{\text{P}}}$ comes from A:PCM-Initially-Solid. The temperature remains constant at ${{T_{\text{melt}}}^{\text{P}}}$ , even with the heating (or cooling), until the phase change has occurred for all of the material; that is as long as $0\lt{}ϕ\lt{}1$ . $ϕ$ (from DD:meltFrac) is determined as part of the heat energy in the PCM, as given in (IM:heatEInPCM). ${{τ_{\text{P}}}^{\text{S}}}$ is calculated in DD:balanceSolidPCM. ${{τ_{\text{P}}}^{\text{L}}}$ is calculated in DD:balanceLiquidPCM. The initial conditions for the ODE are ${T_{\text{W}}}\left(0\right)={T_{\text{P}}}\left(0\right)={T_{\text{init}}}$ following A:Same-Initial-Temp-Water-PCM.
Source	koothoor2013
RefBy	IM:eBalanceOnWtr, UC:No-Internal-Heat-Generation, UC:No-Gaseous-State, FR:Output-Values, FR:Find-Mass, FR:Calculate-Values, FR:Calculate-PCM-Melt-End-Time, and FR:Calculate-PCM-Melt-Begin-Time

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).

Heat energy in the water

Refname	IM:heatEInWtr
Label	Heat energy in the water
Input	${T_{\text{init}}}$ , ${m_{\text{W}}}$ , ${C_{\text{W}}}$ , ${m_{\text{W}}}$
Output	${E_{\text{W}}}$
Input Constraints
Output Constraints
Equation	${E_{\text{W}}}\left(t\right)={C_{\text{W}}}\,{m_{\text{W}}}\,\left({T_{\text{W}}}\left(t\right)-{T_{\text{init}}}\right)$
Description	${E_{\text{W}}}$ is the change in heat energy in the water ( ${\text{J}}$ ) $t$ is the time ( ${\text{s}}$ ) ${C_{\text{W}}}$ is the specific heat capacity of water ( $\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}$ ) ${m_{\text{W}}}$ is the mass of water ( ${\text{kg}}$ ) ${T_{\text{W}}}$ is the temperature of the water ( ${{}^{\circ}\text{C}}$ ) ${T_{\text{init}}}$ is the initial temperature ( ${{}^{\circ}\text{C}}$ )
Notes	The above equation is derived using TM:sensHtE. The change in temperature is the difference between the temperature at time $t$ ( ${\text{s}}$ ), ${T_{\text{W}}}$ and the initial temperature, ${T_{\text{init}}}$ ( ${{}^{\circ}\text{C}}$ ). This equation applies as long as $0\lt{}{T_{\text{W}}}\lt{}100$ ${{}^{\circ}\text{C}}$ (A:Water-Always-Liquid, A:Atmospheric-Pressure-Tank).
Source	koothoor2013
RefBy	FR:Output-Values, FR:Find-Mass, and FR:Calculate-Values

Heat energy in the PCM

Refname	IM:heatEInPCM
Label	Heat energy in the PCM
Input	${{T_{\text{melt}}}^{\text{P}}}$ , ${t_{\text{final}}}$ , ${T_{\text{init}}}$ , ${A_{\text{P}}}$ , ${h_{\text{P}}}$ , ${m_{\text{P}}}$ , ${{C_{\text{P}}}^{\text{S}}}$ , ${{C_{\text{P}}}^{\text{L}}}$ , ${T_{\text{P}}}$ , ${H_{\text{f}}}$ , ${{t_{\text{melt}}}^{\text{init}}}$
Output	${E_{\text{P}}}$
Input Constraints	${{T_{\text{melt}}}^{\text{P}}}\gt{}{T_{\text{init}}}$
Output Constraints
Equation	${E_{\text{P}}}=\begin{cases}{{C_{\text{P}}}^{\text{S}}}\,{m_{\text{P}}}\,\left({T_{\text{P}}}\left(t\right)-{T_{\text{init}}}\right), & {T_{\text{P}}}\lt{}{{T_{\text{melt}}}^{\text{P}}}\\{{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}+{H_{\text{f}}}\,{m_{\text{P}}}+{{C_{\text{P}}}^{\text{L}}}\,{m_{\text{P}}}\,\left({T_{\text{P}}}\left(t\right)-{{T_{\text{melt}}}^{\text{P}}}\right), & {T_{\text{P}}}\gt{}{{T_{\text{melt}}}^{\text{P}}}\\{{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}+{Q_{\text{P}}}\left(t\right), & {T_{\text{P}}}={{T_{\text{melt}}}^{\text{P}}}\land{}0\lt{}ϕ\lt{}1\end{cases}$
Description	${E_{\text{P}}}$ is the change in heat energy in the PCM ( ${\text{J}}$ ) ${{C_{\text{P}}}^{\text{S}}}$ is the specific heat capacity of PCM as a solid ( $\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}$ ) ${m_{\text{P}}}$ is the mass of phase change material ( ${\text{kg}}$ ) ${T_{\text{P}}}$ is the temperature of the phase change material ( ${{}^{\circ}\text{C}}$ ) $t$ is the time ( ${\text{s}}$ ) ${T_{\text{init}}}$ is the initial temperature ( ${{}^{\circ}\text{C}}$ ) ${{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}$ is the change in heat energy in the PCM at the instant when melting begins ( ${\text{J}}$ ) ${H_{\text{f}}}$ is the specific latent heat of fusion ( $\frac{\text{J}}{\text{kg}}$ ) ${{C_{\text{P}}}^{\text{L}}}$ is the specific heat capacity of PCM as a liquid ( $\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}$ ) ${{T_{\text{melt}}}^{\text{P}}}$ is the melting point temperature for PCM ( ${{}^{\circ}\text{C}}$ ) ${Q_{\text{P}}}$ is the latent heat energy added to PCM ( ${\text{J}}$ ) $ϕ$ is the melt fraction (Unitless)
Notes	The above equation is derived using TM:sensHtE and TM:latentHtE. ${E_{\text{P}}}$ for the solid PCM is found using TM:sensHtE for sensible heating, with the specific heat capacity of the solid PCM, ${{C_{\text{P}}}^{\text{S}}}$ ( $\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}$ ) and the change in the PCM temperature from the initial temperature ( ${{}^{\circ}\text{C}}$ ). ${E_{\text{P}}}$ for the melted PCM ( ${T_{\text{P}}}\gt{}{{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}$ ) is found using TM:sensHtE for sensible heat of the liquid PCM plus the energy when melting starts, plus the energy required to melt all of the PCM. The energy required to melt all of the PCM is ${H_{\text{f}}}\,{m_{\text{P}}}$ ( ${\text{J}}$ ) (from DD:htFusion). The change in temperature is ${T_{\text{P}}}-{{T_{\text{melt}}}^{\text{P}}}$ ( ${{}^{\circ}\text{C}}$ ). ${E_{\text{P}}}$ during melting of the PCM is found using the energy required at the instant melting of the PCM begins, ${{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}$ plus the latent heat energy added to the PCM, ${Q_{\text{P}}}$ ( ${\text{J}}$ ) since the time when melting began ${{t_{\text{melt}}}^{\text{init}}}$ ( ${\text{s}}$ ). The heat energy for boiling of the PCM is not detailed, since the PCM is assumed to either be in a solid or liquid state (A:No-Gaseous-State-PCM) (A:PCM-Initially-Solid).
Source	koothoor2013
RefBy	IM:eBalanceOnPCM, UC:No-Gaseous-State, FR:Output-Values, FR:Find-Mass, and FR:Calculate-Values

Software Requirements Specification for Solar Water Heating Systems Incorporating PCM