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Theoretical Models

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

Conservation of thermal energy

Refname	TM:consThermE
Label	Conservation 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).
Source	Fourier Law of Heat Conduction and Heat Equation
RefBy	GD:rocTempSimp

Sensible heat energy

Refname	TM:sensHtE
Label	Sensible 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.
Source	Definition of Sensible Heat
RefBy	IM:heatEInWtr and IM:heatEInPCM

Latent heat energy

Refname	TM:latentHtE
Label	Latent 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.
Source	Definition of Latent Heat
RefBy	TM:sensHtE and IM:heatEInPCM

Newton’s law of cooling

Refname	TM:nwtnCooling
Label	Newton’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.
Source	incroperaEtAl2007 (pg. 8)
RefBy	GD:htFluxPCMFromWater and GD:htFluxWaterFromCoil