The total heat required when discharge temperature is above melting point.

\begin{equation}\label{eq:solid-load-charge-material-full-heat-cpp} Q_{solid} = Q_{sens,solid} + Q_{melt} + Q_{sens,liquid} + Q_{sens,remaining}\end{equation}

\begin{equation}\label{eq:solid-load-charge-material-sensible-heat-to-melting-cpp} Q_{sens,solid} = m_{feed} \cdot (1 - f_{charged}) \cdot C_{p,solid} \cdot (T_{melt} - T_{initial})\end{equation}

\begin{equation}\label{eq:solid-load-charge-material-melting-heat-cpp} Q_{melt} = m_{feed} \cdot (1 - f_{charged}) \cdot f_{melted} \cdot H_{latent}\end{equation}

\begin{equation}\label{eq:solid-load-charge-material-sensible-heat-liquid-cpp} Q_{sens,liquid} = m_{feed} \cdot (1 - f_{charged}) \cdot f_{melted} \cdot C_{p,liquid} \cdot (T_{discharge} - T_{melt})\end{equation}

\begin{equation}\label{eq:solid-load-charge-material-sensible-heat-remaining-cpp} Q_{sens,remaining} = m_{feed} \cdot (1 - f_{charged}) \cdot (1 - f_{melted}) \cdot C_{p,solid} \cdot (T_{discharge} - T_{melt})\end{equation}

Symbols

\(Q_{solid}\)	Heat required for solid heating/melting \([\unit{ \btu\per\hour}]\)
\(Q_{sens,solid}\)	Sensible heat to melting point \([\unit{ \btu\per\hour}]\)
\(Q_{melt}\)	Melting heat \([\unit{ \btu\per\hour}]\)
\(Q_{sens,liquid}\)	Sensible heat for melted fraction \([\unit{ \btu\per\hour}]\)
\(m_{feed}\)	Charge feed rate \([\unit{ \pound\per\hour}]\)
\(f_{melted}\)	Fraction of charge melted \([\unit{ \unitless}]\)
\(C_{p,solid}\)	Specific heat of solid \([\unit{ \btu\per\pound\degreeFahrenheit}]\)
\(C_{p,liquid}\)	Specific heat of liquid \([\unit{ \btu\per\pound\degreeFahrenheit}]\)
\(H_{latent}\)	Latent heat of melting \([\unit{ \btu\per\pound}]\)
\(T_{discharge}\)	Discharge temperature \([\unit{ \degreeFahrenheit}]\)
\(T_{melt}\)	Melting point \([\unit{ \degreeFahrenheit}]\)
\(T_{initial}\)	Initial temperature \([\unit{ \degreeFahrenheit}]\)