Table of Symbols
The symbols used in this document are summarized in the Table of Symbols along with their units. The choice of symbols was made to be consistent with the heat transfer literature and with existing documentation for solar water heating systems with no phase change material. The symbols are listed in alphabetical order. For vector quantities, the units shown are for each component of the vector.
| Symbol | Description | Units |
|---|---|---|
| \({A_{\text{C}}}\) | Heating coil surface area | \({\text{m}^{2}}\) |
| \({{A_{\text{C}}}^{\text{max}}}\) | Maximum surface area of coil | \({\text{m}^{2}}\) |
| \({A_{\text{in}}}\) | Surface area over which heat is transferred in | \({\text{m}^{2}}\) |
| \({A_{\text{out}}}\) | Surface area over which heat is transferred out | \({\text{m}^{2}}\) |
| \({A_{\text{tol}}}\) | Absolute tolerance | – |
| \({\mathit{AR}_{\text{max}}}\) | Maximum aspect ratio | – |
| \({\mathit{AR}_{\text{min}}}\) | Minimum aspect ratio | – |
| \(C\) | Specific heat capacity | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
| \({C^{\text{L}}}\) | Specific heat capacity of a liquid | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
| \({C_{\text{W}}}\) | Specific heat capacity of water | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
| \({{C_{\text{W}}}^{\text{max}}}\) | Maximum specific heat capacity of water | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
| \({{C_{\text{W}}}^{\text{min}}}\) | Minimum specific heat capacity of water | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
| \(D\) | Diameter of tank | \({\text{m}}\) |
| \(E\) | Sensible heat | \({\text{J}}\) |
| \({E_{\text{W}}}\) | Change in heat energy in the water | \({\text{J}}\) |
| \(g\) | Volumetric heat generation per unit volume | \(\frac{\text{W}}{\text{m}^{3}}\) |
| \(h\) | Convective heat transfer coefficient | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
| \({h_{\text{C}}}\) | Convective heat transfer coefficient between coil and water | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
| \({{h_{\text{C}}}^{\text{max}}}\) | Maximum convective heat transfer coefficient between coil and water | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
| \({{h_{\text{C}}}^{\text{min}}}\) | Minimum convective heat transfer coefficient between coil and water | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
| \(L\) | Length of tank | \({\text{m}}\) |
| \({L_{\text{max}}}\) | Maximum length of tank | \({\text{m}}\) |
| \({L_{\text{min}}}\) | Minimum length of tank | \({\text{m}}\) |
| \(m\) | Mass | \({\text{kg}}\) |
| \({m_{\text{W}}}\) | Mass of water | \({\text{kg}}\) |
| \(\boldsymbol{\hat{n}}\) | Normal vector | – |
| \(q\) | Heat flux | \(\frac{\text{W}}{\text{m}^{2}}\) |
| \({q_{\text{C}}}\) | Heat flux into the water from the coil | \(\frac{\text{W}}{\text{m}^{2}}\) |
| \({q_{\text{in}}}\) | Heat flux input | \(\frac{\text{W}}{\text{m}^{2}}\) |
| \({q_{\text{out}}}\) | Heat flux output | \(\frac{\text{W}}{\text{m}^{2}}\) |
| \(\boldsymbol{q}\) | Thermal flux vector | \(\frac{\text{W}}{\text{m}^{2}}\) |
| \({R_{\text{tol}}}\) | Relative tolerance | – |
| \(S\) | Surface | \({\text{m}^{2}}\) |
| \(T\) | Temperature | \({{}^{\circ}\text{C}}\) |
| \(ΔT\) | Change in temperature | \({{}^{\circ}\text{C}}\) |
| \({T_{\text{C}}}\) | Temperature of the heating coil | \({{}^{\circ}\text{C}}\) |
| \({T_{\text{env}}}\) | Temperature of the environment | \({{}^{\circ}\text{C}}\) |
| \({T_{\text{init}}}\) | Initial temperature | \({{}^{\circ}\text{C}}\) |
| \({T_{\text{W}}}\) | Temperature of the water | \({{}^{\circ}\text{C}}\) |
| \(t\) | Time | \({\text{s}}\) |
| \({t_{\text{final}}}\) | Final time | \({\text{s}}\) |
| \({{t_{\text{final}}}^{\text{max}}}\) | Maximum final time | \({\text{s}}\) |
| \({t_{\text{step}}}\) | Time step for simulation | \({\text{s}}\) |
| \(V\) | Volume | \({\text{m}^{3}}\) |
| \({V_{\text{tank}}}\) | Volume of the cylindrical tank | \({\text{m}^{3}}\) |
| \({V_{\text{W}}}\) | Volume of water | \({\text{m}^{3}}\) |
| \(π\) | Ratio of circumference to diameter for any circle | – |
| \(ρ\) | Density | \(\frac{\text{kg}}{\text{m}^{3}}\) |
| \({ρ_{\text{W}}}\) | Density of water | \(\frac{\text{kg}}{\text{m}^{3}}\) |
| \({{ρ_{\text{W}}}^{\text{max}}}\) | Maximum density of water | \(\frac{\text{kg}}{\text{m}^{3}}\) |
| \({{ρ_{\text{W}}}^{\text{min}}}\) | Minimum density of water | \(\frac{\text{kg}}{\text{m}^{3}}\) |
| \({τ_{\text{W}}}\) | ODE parameter for water related to decay time | \({\text{s}}\) |
| \(∇\) | Gradient | – |
Table of Symbols