General Definitions

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

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:

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