Solid Load Charge Material Heat Required Calculator

Detailed Description

The following formulas describe the calculation of total heat required for a solid load/charge material, including sensible heat, latent heat, vaporization, reaction heat, and additional heat. Each step is documented with its own formula and symbol table.

The calculation follows a top-down approach:

1. Total Heat Required - Sum of moisture removal, solid heating, reaction heat, and additional heat
2. Moisture Removal - Heat to evaporate water content (varies based on discharge temperature)
   • Below boiling: Sensible heat only
   • Above boiling: Sensible heat to boiling + latent heat + sensible heat of vapor
3. Solid Heating - Heat to raise temperature of dry solid (varies based on melting)
   • Below melting: Sensible heat only
   • Above melting: Sensible heat to melting + latent heat + sensible heat of liquid/remaining solid
4. Heat of Reaction - Endothermic reaction heat if applicable

Total Heat Required

---

\begin{equation}\label{eq:solid-load-charge-material-total-heat-required-cpp} Q_{total} = Q_{moisture} + Q_{solid} + Q_{reaction} + Q_{additional}\end{equation}

Symbols

\(Q_{total}\)	Total heat required \([\unit{ \btu\per\hour}]\)
\(Q_{moisture}\)	Heat required for moisture removal \([\unit{ \btu\per\hour}]\)
\(Q_{solid}\)	Heat required for solid heating/melting \([\unit{ \btu\per\hour}]\)
\(Q_{reaction}\)	Heat of reaction \([\unit{ \btu\per\hour}]\)
\(Q_{additional}\)	Additional heat required \([\unit{ \btu\per\hour}]\)

Moisture Removal (Below boiling point)

---

The total heat required when water vapor discharge temperature is below boiling point.

\begin{equation}\label{eq:solid-load-charge-material-moisture-removal-cpp} Q_{moisture} = m_{feed} \cdot f_{charged} \cdot (T_{vapor,discharge} - T_{initial})\end{equation}

Symbols

\(Q_{moisture}\)	Heat required for moisture removal \([\unit{ \btu\per\hour}]\)
\(m_{feed}\)	Charge feed rate \([\unit{ \pound\per\hour}]\)
\(f_{charged}\)	Fraction of water charged \([\unit{ \unitless}]\)
\(T_{vapor,discharge}\)	Water vapor discharge temperature \([\unit{ \degreeFahrenheit}]\)
\(T_{initial}\)	Initial temperature \([\unit{ \degreeFahrenheit}]\)

Moisture Removal (Above boiling point)

---

The total heat required when water vapor discharge temperature is above boiling point.

\begin{equation}\label{eq:solid-load-charge-material-moisture-removal-above-boiling-cpp} Q_{moisture} = Q_{sens,boil} + Q_{vap} + Q_{sens,vapor}\end{equation}

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

\begin{equation}\label{eq:solid-load-charge-material-vaporization-heat-cpp} Q_{vap} = m_{feed} \cdot (f_{charged} - f_{discharged}) \cdot H_{evap}\end{equation}

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

Symbols

\(Q_{moisture}\)	Heat required for moisture removal \([\unit{ \btu\per\hour}]\)
\(Q_{sens,boil}\)	Sensible heat to boiling point \([\unit{ \btu\per\hour}]\)
\(Q_{vap}\)	Vaporization heat \([\unit{ \btu\per\hour}]\)
\(Q_{sens,vapor}\)	Sensible heat for vapor above boiling \([\unit{ \btu\per\hour}]\)
\(m_{feed}\)	Charge feed rate \([\unit{ \pound\per\hour}]\)
\(f_{charged}\)	Fraction of water charged \([\unit{ \unitless}]\)
\(f_{discharged}\)	Fraction of water discharged \([\unit{ \unitless}]\)
\(H_{evap}\)	Latent heat of evaporation \([\unit{ \btu\per\pound}]\)
\(C_{p,vapor}\)	Specific heat of water vapor \([\unit{ \btu\per\pound\degreeFahrenheit}]\)
\(T_{vapor,discharge}\)	Water vapor discharge temperature \([\unit{ \degreeFahrenheit}]\)
\(T_{boil}\)	Boiling point of water \([\unit{ \degreeFahrenheit}]\)
\(T_{initial}\)	Initial temperature \([\unit{ \degreeFahrenheit}]\)

Heat for Remaining Moisture

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Heat required for moisture remaining in the solid after discharge.

\begin{equation}\label{eq:solid-load-charge-material-remaining-moisture-cpp} Q_{moisture,rem} = m_{feed} \cdot f_{discharged} \cdot (T_{vapor,discharge} - T_{boil})\end{equation}

Symbols

\(Q_{moisture,rem}\)	Heat for remaining moisture \([\unit{ \btu\per\hour}]\)
\(m_{feed}\)	Charge feed rate \([\unit{ \pound\per\hour}]\)
\(f_{discharged}\)	Fraction of water discharged \([\unit{ \unitless}]\)
\(T_{vapor,discharge}\)	Water vapor discharge temperature \([\unit{ \degreeFahrenheit}]\)
\(T_{boil}\)	Boiling point \([\unit{ \degreeFahrenheit}]\)

Solid Heating (Below melting point)

---

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

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

Symbols

\(Q_{solid}\)	Heat required for solid heating \([\unit{ \btu\per\hour}]\)
\(m_{feed}\)	Charge feed rate \([\unit{ \pound\per\hour}]\)
\(f_{charged}\)	Fraction of water charged \([\unit{ \unitless}]\)
\(C_{p,solid}\)	Specific heat of solid \([\unit{ \btu\per\pound\degreeFahrenheit}]\)
\(T_{discharge}\)	Discharge temperature \([\unit{ \degreeFahrenheit}]\)
\(T_{initial}\)	Initial temperature \([\unit{ \degreeFahrenheit}]\)

Solid Heating (Above melting point)

---

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 \([\unit{ \degreeFahrenheit}]\)

Heat of Reaction

---

Heat required for endothermic reaction.

\begin{equation}\label{eq:solid-load-charge-material-heat-of-reaction-cpp} Q_{reaction} = m_{feed} \cdot (1 - f_{charged}) \cdot f_{reacted} \cdot H_{reaction}\end{equation}

Symbols

\(Q_{reaction}\)	Heat of reaction \([\unit{ \btu\per\hour}]\)
\(m_{feed}\)	Charge feed rate \([\unit{ \pound\per\hour}]\)
\(f_{charged}\)	Fraction of water charged \([\unit{ \unitless}]\)
\((1 - f_{charged})\)	Fraction of dry charge \([\unit{ \unitless}]\)
\(f_{reacted}\)	Fraction of dry charge reacted \([\unit{ \unitless}]\)
\(H_{reaction}\)	Heat of reaction \([\unit{ \btu\per\pound}]\)

See also

Solid Load Charge Material Database

physics::us::kWaterBoilingPointF for water boiling point

physics::us::kWaterEvaporation for latent heat of evaporation

physics::us::kSpecificHeatWaterVapor for water vapor specific heat

Modules
	Total Heat Required Formula
	Moisture Removal (Below boiling point) Formula
	Moisture Removal (Above boiling point) Formula
	Heat for Remaining Moisture Formula
	Solid Heating (Below melting point) Formula
	Solid Heating (Above melting point) Formula
	Heat of Reaction Formula

Files
file	solid_load_charge_material_heat_required.h
	Calculates total heat required for a solid load/charge material.

Namespaces
namespace	solid_load_charge_material_heat_required
	Namespace for solid load/charge material heat required calculation.

Functions
double	solid_load_charge_material_heat_required::totalHeatRequired (LoadChargeMaterial::ThermicReactionType thermic_reaction_type, double specific_heat_solid, double latent_heat, double specific_heat_liquid, double melting_point, double charge_feed_rate, double water_content_charged, double water_content_discharged, double initial_temperature, double discharge_temperature, double water_vapor_discharge_temperature, double charge_melted, double charge_reacted, double reaction_heat, double additional_heat)
	Calculates the total heat required for a solid load/charge material.

Function Documentation

totalHeatRequired()

double solid_load_charge_material_heat_required::totalHeatRequired	(	LoadChargeMaterial::ThermicReactionType	thermic_reaction_type,
		double	specific_heat_solid,
		double	latent_heat,
		double	specific_heat_liquid,
		double	melting_point,
		double	charge_feed_rate,
		double	water_content_charged,
		double	water_content_discharged,
		double	initial_temperature,
		double	discharge_temperature,
		double	water_vapor_discharge_temperature,
		double	charge_melted,
		double	charge_reacted,
		double	reaction_heat,
		double	additional_heat
	)

Parameters

[in]	thermic_reaction_type	Thermic reaction type (endothermic, exothermic, or none)
[in]	specific_heat_solid	Average specific heat of solid material (dry) \([\unit{\btu\per\pound\degreeFahrenheit}]\)
[in]	latent_heat	Latent heat of fusion \([\unit{\btu\per\pound}]\)
[in]	specific_heat_liquid	Specific heat of liquid from molten material \([\unit{\btu\per\pound\degreeFahrenheit}]\)
[in]	melting_point	Melting point \([\unit{\degreeFahrenheit}]\)
[in]	charge_feed_rate	Charge (wet) feed rate \([\unit{\pound\per\hour}]\)
[in]	water_content_charged	Water content as charged (percent, e.g., 10 for 10%)
[in]	water_content_discharged	Water content as discharged (percent, e.g., 5 for 5%)
[in]	initial_temperature	Initial temperature \([\unit{\degreeFahrenheit}]\)
[in]	discharge_temperature	Charge material discharge temperature \([\unit{\degreeFahrenheit}]\)
[in]	water_vapor_discharge_temperature	Water vapor discharge temperature \([\unit{\degreeFahrenheit}]\)
[in]	charge_melted	Charge melted (percent of dry charge, e.g., 10 for 10%)
[in]	charge_reacted	Charge reacted (percent of dry charge, e.g., 5 for 5%)
[in]	reaction_heat	Heat of reaction \([\unit{\btu\per\pound}]\)
[in]	additional_heat	Additional heat required \([\unit{\btu\per\hour}]\)

Returns

Total heat required \([\unit{\btu\per\hour}]\).