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. 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 | \({\text{m}^{2}}\) |
| \({A_{\text{P}}}\) | Phase change material surface area | \({\text{m}^{2}}\) |
| \(\mathit{AR}\) | Aspect ratio | – |
| \({\mathit{AR}_{\text{max}}}\) | 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 | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
| \({C^{\text{S}}}\) | Specific heat capacity of a solid | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
| \({C^{\text{V}}}\) | Specific heat capacity of a vapour | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
| \({{C_{\text{P}}}^{\text{L}}}\) | Specific heat capacity of PCM as a liquid | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
| \({{C_{\text{P}}}^{\text{S}}}\) | Specific heat capacity of PCM as a solid | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
| \({C_{\text{tol}}}\) | Relative tolerance for conservation of energy | – |
| \({C_{\text{W}}}\) | 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 | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
| \({{{C_{\text{P}}}^{\text{L}}}_{\text{max}}}\) | Maximum specific heat capacity of PCM as a liquid | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
| \({{{C_{\text{P}}}^{\text{L}}}_{\text{min}}}\) | Minimum specific heat capacity of PCM as a liquid | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
| \({{{C_{\text{P}}}^{\text{S}}}_{\text{max}}}\) | Maximum specific heat capacity of PCM as a solid | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
| \({{{C_{\text{P}}}^{\text{S}}}_{\text{min}}}\) | Minimum specific heat capacity of PCM as a solid | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
| \(D\) | Diameter of tank | \({\text{m}}\) |
| \(E\) | Sensible heat | \({\text{J}}\) |
| \({E_{\text{P}}}\) | Change in heat energy in the PCM | \({\text{J}}\) |
| \({E_{\text{W}}}\) | Change in heat energy in the water | \({\text{J}}\) |
| \({{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}\) | Change in heat energy in the PCM at the instant when melting begins | \({\text{J}}\) |
| \(g\) | Volumetric heat generation per unit volume | \(\frac{\text{W}}{\text{m}^{3}}\) |
| \({H_{\text{f}}}\) | Specific latent heat of fusion | \(\frac{\text{J}}{\text{kg}}\) |
| \({{H_{\text{f}}}_{\text{max}}}\) | Maximum specific latent heat of fusion | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
| \({{H_{\text{f}}}_{\text{min}}}\) | Minimum specific latent heat of fusion | \(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\) |
| \(h\) | 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 | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
| \({h_{\text{min}}}\) | Minimum thickness of a sheet of PCM | \({\text{m}}\) |
| \({h_{\text{P}}}\) | Convective heat transfer coefficient between PCM and water | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
| \({{h_{\text{P}}}^{\text{max}}}\) | Maximum convective heat transfer coefficient between PCM and water | \(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\) |
| \({{h_{\text{P}}}^{\text{min}}}\) | Minimum convective heat transfer coefficient between PCM 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 | \({\text{kg}}\) |
| \({m_{\text{P}}}\) | Mass of phase change material | \({\text{kg}}\) |
| \({m_{\text{W}}}\) | Mass of water | \({\text{kg}}\) |
| \(\mathit{MINFRACT}\) | Minimum fraction of the tank volume taken up by the PCM | – |
| \(\boldsymbol{\hat{n}}\) | Unit outward normal vector for a surface | – |
| \(Q\) | Latent heat | \({\text{J}}\) |
| \({Q_{\text{P}}}\) | Latent heat energy added to PCM | \({\text{J}}\) |
| \(q\) | 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 | \(\frac{\text{W}}{\text{m}^{2}}\) |
| \({q_{\text{P}}}\) | Heat flux into the PCM from water | \(\frac{\text{W}}{\text{m}^{2}}\) |
| \(\boldsymbol{q}\) | Thermal flux vector | \(\frac{\text{W}}{\text{m}^{2}}\) |
| \(S\) | Surface | \({\text{m}^{2}}\) |
| \(T\) | Temperature | \({{}^{\circ}\text{C}}\) |
| \(ΔT\) | Change in temperature | \({{}^{\circ}\text{C}}\) |
| \({T_{\text{boil}}}\) | Boiling point 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 | \({{}^{\circ}\text{C}}\) |
| \({T_{\text{melt}}}\) | Melting point temperature | \({{}^{\circ}\text{C}}\) |
| \({{T_{\text{melt}}}^{\text{P}}}\) | Melting point temperature for PCM | \({{}^{\circ}\text{C}}\) |
| \({T_{\text{P}}}\) | Temperature of the phase change material | \({{}^{\circ}\text{C}}\) |
| \({T_{\text{W}}}\) | 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 | \({\text{s}}\) |
| \({{t_{\text{melt}}}^{\text{final}}}\) | Time at which melting of PCM ends | \({\text{s}}\) |
| \({{t_{\text{melt}}}^{\text{init}}}\) | Time at which melting of PCM begins | \({\text{s}}\) |
| \({t_{\text{step}}}\) | Time step for simulation | \({\text{s}}\) |
| \(V\) | Volume | \({\text{m}^{3}}\) |
| \({V_{\text{P}}}\) | Volume of PCM | \({\text{m}^{3}}\) |
| \({V_{\text{tank}}}\) | Volume of the cylindrical tank | \({\text{m}^{3}}\) |
| \({V_{\text{W}}}\) | Volume of water | \({\text{m}^{3}}\) |
| \(η\) | ODE parameter related to decay rate | – |
| \(π\) | Ratio of circumference to diameter for any circle | – |
| \(ρ\) | Density | \(\frac{\text{kg}}{\text{m}^{3}}\) |
| \({ρ_{\text{P}}}\) | Density of PCM | \(\frac{\text{kg}}{\text{m}^{3}}\) |
| \({{ρ_{\text{P}}}^{\text{max}}}\) | Maximum density of PCM | \(\frac{\text{kg}}{\text{m}^{3}}\) |
| \({{ρ_{\text{P}}}^{\text{min}}}\) | Minimum density of PCM | \(\frac{\text{kg}}{\text{m}^{3}}\) |
| \({ρ_{\text{W}}}\) | 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 | \(\frac{\text{kg}}{\text{m}^{3}}\) |
| \(τ\) | Dummy variable for integration over time | \({\text{s}}\) |
| \({{τ_{\text{P}}}^{\text{L}}}\) | ODE parameter for liquid PCM | \({\text{s}}\) |
| \({{τ_{\text{P}}}^{\text{S}}}\) | ODE parameter for solid PCM | \({\text{s}}\) |
| \({τ_{\text{W}}}\) | ODE parameter for water related to decay time | \({\text{s}}\) |
| \(ϕ\) | Melt fraction | – |
| \(∇\) | Gradient | – |
Table of Symbols