General Definitions

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

Simplified rate of change of temperature

Refname	GD:rocTempSimp
Label	Simplified 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	–
RefBy	GD:rocTempSimp and IM:eBalanceOnWtr

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. Assuming $ρ$ , $C$ , and $T$ are constant over the volume, which is true in our case by A:Constant-Water-Temp-Across-Tank, A:Density-Water-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:

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

Heat flux into the water from the coil

Refname	GD:htFluxWaterFromCoil
Label	Heat 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	${q_{\text{C}}}$ is found by assuming that Newton’s law of cooling applies (A:Newton-Law-Convective-Cooling-Coil-Water). This law (defined in TM:nwtnCooling) is used on the surface of the heating coil. A:Temp-Heating-Coil-Constant-over-Time
Source	koothoor2013
RefBy	IM:eBalanceOnWtr

Software Requirements Specification for Solar Water Heating Systems

General Definitions

Simplified rate of change of temperature

Detailed derivation of simplified rate of change of temperature:

Heat flux into the water from the coil