General Definitions

This section collects the laws and equations that will be used to build the instance models.

RefnameGD:rocTempSimp
LabelSimplified rate of change of temperature
Equation\[m\,C\,\frac{\,dT}{\,dt}={q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V\]
Description
  • \(m\) is the mass (\({\text{kg}}\))
  • \(C\) is the specific heat capacity (\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\))
  • \(t\) is the time (\({\text{s}}\))
  • \(T\) is the temperature (\({{}^{\circ}\text{C}}\))
  • \({q_{\text{in}}}\) is the heat flux input (\(\frac{\text{W}}{\text{m}^{2}}\))
  • \({A_{\text{in}}}\) is the surface area over which heat is transferred in (\({\text{m}^{2}}\))
  • \({q_{\text{out}}}\) is the heat flux output (\(\frac{\text{W}}{\text{m}^{2}}\))
  • \({A_{\text{out}}}\) is the surface area over which heat is transferred out (\({\text{m}^{2}}\))
  • \(g\) is the volumetric heat generation per unit volume (\(\frac{\text{W}}{\text{m}^{3}}\))
  • \(V\) is the volume (\({\text{m}^{3}}\))
Source–
RefByGD:rocTempSimp, IM:eBalanceOnWtr, and IM:eBalanceOnPCM

Detailed derivation of simplified rate of change of temperature:

Integrating TM:consThermE over a volume (\(V\)), we have:

\[-\int_{V}{∇\cdot{}\boldsymbol{q}}\,dV+\int_{V}{g}\,dV=\int_{V}{ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t}}\,dV\]

Applying Gauss’s Divergence Theorem to the first term over the surface \(S\) of the volume, with \(\boldsymbol{q}\) as the thermal flux vector for the surface and \(\boldsymbol{\hat{n}}\) as a unit outward normal vector for a surface:

\[-\int_{S}{\boldsymbol{q}\cdot{}\boldsymbol{\hat{n}}}\,dS+\int_{V}{g}\,dV=\int_{V}{ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t}}\,dV\]

We consider an arbitrary volume. The volumetric heat generation per unit volume is assumed constant. Then Equation (1) can be written as:

\[{q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V=\int_{V}{ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t}}\,dV\]

Where \({q_{\text{in}}}\), \({q_{\text{out}}}\), \({A_{\text{in}}}\), and \({A_{\text{out}}}\) are explained in GD:rocTempSimp. The integral over the surface could be simplified because the thermal flux is assumed constant over \({A_{\text{in}}}\) and \({A_{\text{out}}}\) and \(0\) on all other surfaces. Outward flux is considered positive. Assuming \(ρ\), \(C\), and \(T\) are constant over the volume, which is true in our case by A:Constant-Water-Temp-Across-Tank, A:Temp-PCM-Constant-Across-Volume, A:Density-Water-PCM-Constant-over-Volume, and A:Specific-Heat-Energy-Constant-over-Volume, we have:

\[ρ\,C\,V\,\frac{\,dT}{\,dt}={q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V\]

Using the fact that \(ρ\)=\(m\)/\(V\), Equation (2) can be written as:

\[m\,C\,\frac{\,dT}{\,dt}={q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V\]

RefnameGD:htFluxWaterFromCoil
LabelHeat flux into the water from the coil
Units\(\frac{\text{W}}{\text{m}^{2}}\)
Equation\[{q_{\text{C}}}={h_{\text{C}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)\right)\]
Description
  • \({q_{\text{C}}}\) is the heat flux into the water from the coil (\(\frac{\text{W}}{\text{m}^{2}}\))
  • \({h_{\text{C}}}\) is the convective heat transfer coefficient between coil and water (\(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\))
  • \({T_{\text{C}}}\) is the temperature of the heating coil (\({{}^{\circ}\text{C}}\))
  • \({T_{\text{W}}}\) is the temperature of the water (\({{}^{\circ}\text{C}}\))
  • \(t\) is the time (\({\text{s}}\))
Notes
Sourcekoothoor2013
RefByIM:eBalanceOnWtr
RefnameGD:htFluxPCMFromWater
LabelHeat flux into the PCM from water
Units\(\frac{\text{W}}{\text{m}^{2}}\)
Equation\[{q_{\text{P}}}={h_{\text{P}}}\,\left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right)\]
Description
  • \({q_{\text{P}}}\) is the heat flux into the PCM from water (\(\frac{\text{W}}{\text{m}^{2}}\))
  • \({h_{\text{P}}}\) is the convective heat transfer coefficient between PCM and water (\(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\))
  • \({T_{\text{W}}}\) is the temperature of the water (\({{}^{\circ}\text{C}}\))
  • \(t\) is the time (\({\text{s}}\))
  • \({T_{\text{P}}}\) is the temperature of the phase change material (\({{}^{\circ}\text{C}}\))
Notes
Sourcekoothoor2013
RefByIM:eBalanceOnWtr and IM:eBalanceOnPCM