Instance Models

This section transforms the problem defined in the problem description into one which is expressed in mathematical terms. It uses concrete symbols defined in the data definitions to replace the abstract symbols in the models identified in theoretical models and general definitions.

The goal GS:Predict-Water-Temperature is met by IM:eBalanceOnWtr and the goal GS:Predict-Water-Energy is met by IM:heatEInWtr.

Energy balance on water to find the temperature of the water

Refname	IM:eBalanceOnWtr
Label	Energy balance on water to find the temperature of the water
Input	${T_{\text{C}}}$ , ${T_{\text{init}}}$ , ${t_{\text{final}}}$ , ${A_{\text{C}}}$ , ${h_{\text{C}}}$ , ${C_{\text{W}}}$ , ${m_{\text{W}}}$
Output	${T_{\text{W}}}$
Input Constraints	${T_{\text{C}}}\geq{}{T_{\text{init}}}$
Output Constraints
Equation	$\frac{\,d{T_{\text{W}}}}{\,dt}+\frac{1}{{τ_{\text{W}}}}\,{{T_{\text{W}}}}=\frac{1}{{τ_{\text{W}}}}\,{T_{\text{C}}}$
Description	$t$ is the time ( ${\text{s}}$ ) ${T_{\text{W}}}$ is the temperature of the water ( ${{}^{\circ}\text{C}}$ ) ${τ_{\text{W}}}$ is the ODE parameter for water related to decay time ( ${\text{s}}$ ) ${T_{\text{C}}}$ is the temperature of the heating coil ( ${{}^{\circ}\text{C}}$ )
Notes	${τ_{\text{W}}}$ is calculated from DD:balanceDecayRate. The above equation applies as long as the water is in liquid form, $0\lt{}{T_{\text{W}}}\lt{}100$ ( ${{}^{\circ}\text{C}}$ ) where $0$ ( ${{}^{\circ}\text{C}}$ ) and $100$ ( ${{}^{\circ}\text{C}}$ ) are the melting and boiling point temperatures of water, respectively (A:Water-Always-Liquid).
Source	koothoor2013 (with PCM removed)
RefBy	UC:No-Internal-Heat-Generation, FR:Output-Values, FR:Find-Mass, and FR:Calculate-Values

Detailed derivation of the energy balance on water:

To find the rate of change of ${T_{\text{W}}}$ , we look at the energy balance on water. The volume being considered is the volume of water in the tank ${V_{\text{W}}}$ , which has mass ${m_{\text{W}}}$ and specific heat capacity, ${C_{\text{W}}}$ . Heat transfer occurs in the water from the heating coil as ${q_{\text{C}}}$ (GD:htFluxWaterFromCoil), over area ${A_{\text{C}}}$ . No heat transfer occurs to the outside of the tank, since it has been assumed to be perfectly insulated (A:Perfect-Insulation-Tank). Since the assumption is made that no internal heat is generated (A:No-Internal-Heat-Generation-By-Water), $g=0$ . Therefore, the equation for GD:rocTempSimp can be written as:

${m_{\text{W}}}\,{C_{\text{W}}}\,\frac{\,d{T_{\text{W}}}}{\,dt}={q_{\text{C}}}\,{A_{\text{C}}}$

Using GD:htFluxWaterFromCoil for ${q_{\text{C}}}$ , this can be written as:

${m_{\text{W}}}\,{C_{\text{W}}}\,\frac{\,d{T_{\text{W}}}}{\,dt}={h_{\text{C}}}\,{A_{\text{C}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)$

Dividing Equation (2) by ${m_{\text{W}}}\,{C_{\text{W}}}$ , we obtain:

$\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}}\,{A_{\text{C}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)$

By substituting ${τ_{\text{W}}}$ (from DD:balanceDecayRate), this can be written as:

$\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)$

Heat energy in the water

Refname	IM:heatEInWtr
Label	Heat energy in the water
Input	${T_{\text{init}}}$ , ${m_{\text{W}}}$ , ${C_{\text{W}}}$ , ${m_{\text{W}}}$
Output	${E_{\text{W}}}$
Input Constraints
Output Constraints
Equation	${E_{\text{W}}}\left(t\right)={C_{\text{W}}}\,{m_{\text{W}}}\,\left({T_{\text{W}}}\left(t\right)-{T_{\text{init}}}\right)$
Description	${E_{\text{W}}}$ is the change in heat energy in the water ( ${\text{J}}$ ) $t$ is the time ( ${\text{s}}$ ) ${C_{\text{W}}}$ is the specific heat capacity of water ( $\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}$ ) ${m_{\text{W}}}$ is the mass of water ( ${\text{kg}}$ ) ${T_{\text{W}}}$ is the temperature of the water ( ${{}^{\circ}\text{C}}$ ) ${T_{\text{init}}}$ is the initial temperature ( ${{}^{\circ}\text{C}}$ )
Notes	The above equation is derived using TM:sensHtE. The change in temperature is the difference between the temperature at time $t$ ( ${\text{s}}$ ), ${T_{\text{W}}}$ and the initial temperature, ${T_{\text{init}}}$ ( ${{}^{\circ}\text{C}}$ ). This equation applies as long as $0\lt{}{T_{\text{W}}}\lt{}100$ ${{}^{\circ}\text{C}}$ (A:Water-Always-Liquid, A:Atmospheric-Pressure-Tank).
Source	koothoor2013
RefBy	FR:Output-Values and FR:Calculate-Values

Software Requirements Specification for Solar Water Heating Systems

Instance Models

Energy balance on water to find the temperature of the water

Detailed derivation of the energy balance on water:

Heat energy in the water