Theoretical Models

This section focuses on the general equations and laws that SWHS is based on.

RefnameTM:consThermE
LabelConservation of thermal energy
Equation\[-∇\cdot{}\boldsymbol{q}+g=ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t}\]
Description
  • \(∇\) is the gradient (Unitless)
  • \(\boldsymbol{q}\) is the thermal flux vector (\(\frac{\text{W}}{\text{m}^{2}}\))
  • \(g\) is the volumetric heat generation per unit volume (\(\frac{\text{W}}{\text{m}^{3}}\))
  • \(ρ\) is the density (\(\frac{\text{kg}}{\text{m}^{3}}\))
  • \(C\) is the specific heat capacity (\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\))
  • \(t\) is the time (\({\text{s}}\))
  • \(T\) is the temperature (\({{}^{\circ}\text{C}}\))
Notes
  • The above equation gives the law of conservation of energy for transient heat transfer in a given material.
  • For this equation to apply, other forms of energy, such as mechanical energy, are assumed to be negligible in the system (A:Thermal-Energy-Only).
SourceFourier Law of Heat Conduction and Heat Equation
RefByGD:rocTempSimp
RefnameTM:sensHtE
LabelSensible heat energy
Equation\[E=\begin{cases}{C^{\text{S}}}\,m\,ΔT, & T\lt{}{T_{\text{melt}}}\\{C^{\text{L}}}\,m\,ΔT, & {T_{\text{melt}}}\lt{}T\lt{}{T_{\text{boil}}}\\{C^{\text{V}}}\,m\,ΔT, & {T_{\text{boil}}}\lt{}T\end{cases}\]
Description
  • \(E\) is the sensible heat (\({\text{J}}\))
  • \({C^{\text{S}}}\) is the specific heat capacity of a solid (\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\))
  • \(m\) is the mass (\({\text{kg}}\))
  • \(ΔT\) is the change in temperature (\({{}^{\circ}\text{C}}\))
  • \({C^{\text{L}}}\) is the specific heat capacity of a liquid (\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\))
  • \({C^{\text{V}}}\) is the specific heat capacity of a vapour (\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\))
  • \(T\) is the temperature (\({{}^{\circ}\text{C}}\))
  • \({T_{\text{melt}}}\) is the melting point temperature (\({{}^{\circ}\text{C}}\))
  • \({T_{\text{boil}}}\) is the boiling point temperature (\({{}^{\circ}\text{C}}\))
Notes
  • Sensible heating occurs as long as the material does not reach a temperature where a phase change occurs. A phase change occurs if \(T={T_{\text{boil}}}\) or \(T={T_{\text{melt}}}\). If this is the case, refer to TM:latentHtE.
SourceDefinition of Sensible Heat
RefByIM:heatEInPCM and IM:heatEInWtr
RefnameTM:latentHtE
LabelLatent heat energy
Equation\[Q\left(t\right)=\int_{0}^{t}{\frac{\,dQ\left(τ\right)}{\,dτ}}\,dτ\]
Description
  • \(Q\) is the latent heat (\({\text{J}}\))
  • \(t\) is the time (\({\text{s}}\))
  • \(τ\) is the dummy variable for integration over time (\({\text{s}}\))
Notes
  • \(Q\) is the change in thermal energy (latent heat energy).
  • \(Q\left(t\right)=\int_{0}^{t}{\frac{\,dQ\left(τ\right)}{\,dτ}}\,dτ\) is the rate of change of \(Q\) with respect to time \(τ\).
  • \(t\) is the time elapsed, as long as the phase change is not complete.
  • The status of the phase change depends on the melt fraction (from DD:meltFrac).
  • Latent heating stops when all material has changed to the new phase.
SourceDefinition of Latent Heat
RefByTM:sensHtE and IM:heatEInPCM
RefnameTM:nwtnCooling
LabelNewton’s law of cooling
Equation\[q\left(t\right)=h\,ΔT\left(t\right)\]
Description
  • \(q\) is the heat flux (\(\frac{\text{W}}{\text{m}^{2}}\))
  • \(t\) is the time (\({\text{s}}\))
  • \(h\) is the convective heat transfer coefficient (\(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\))
  • \(ΔT\) is the change in temperature (\({{}^{\circ}\text{C}}\))
Notes
  • Newton’s law of cooling describes convective cooling from a surface. The law is stated as: the rate of heat loss from a body is proportional to the difference in temperatures between the body and its surroundings.
  • \(h\) is assumed to be independent of \(T\) (from A:Heat-Transfer-Coeffs-Constant).
  • \(ΔT\left(t\right)=T\left(t\right)-{T_{\text{env}}}\left(t\right)\) is the time-dependant thermal gradient between the environment and the object.
SourceincroperaEtAl2007 (pg. 8)
RefByGD:htFluxPCMFromWater and GD:htFluxWaterFromCoil