Energy Input Exhaust Gas Heat Loss Calculation

Detailed Description

The following formulas describe the calculation of available heat, heat delivered, exhaust gas losses, and all intermediate steps. Each step is documented with its own formula and symbol table.

The calculation follows a top-down approach:

1. Exhaust Gas Losses - Calculate heat lost through exhaust gases based on available heat
2. Heat Delivered - Calculate useful heat delivered based on input and available heat percentage
3. Available Heat - Calculate percentage of input heat available after exhaust losses
4. Air Corrections - Calculate corrections for excess air and combustion air preheat
5. Specific Heat of Air - Temperature-dependent specific heat calculation

Exhaust Gas Losses

---

Total heat loss through exhaust gases.

The exhaust gas losses represent heat carried away by hot exhaust gases. It is calculated from the delivered heat and the fraction of unavailable heat.

\begin{equation}\label{eq:energy-input-exhaust-gas-total-heat-loss-cpp} Q_{loss} = Q_{delivered} \cdot \frac{100 - AH}{AH}\end{equation}

Symbols

\(Q_{loss}\)	Exhaust gas losses \([\unit{ \btu\per\hour}]\)
\(Q_{delivered}\)	Heat delivered \([\unit{ \btu\per\hour}]\)
\(AH\)	Available heat percent \([\unit{ \percent}]\)
\(100\)	Total heat percentage \([\unit{ \percent}]\)

Heat Delivered

---

Useful heat delivered to the furnace.

The heat delivered is the fraction of total input heat that remains after exhaust gas losses, based on the available heat percentage.

\begin{equation}\label{eq:energy-input-exhaust-gas-heat-delivered-cpp} Q_{delivered} = Q_{input} \cdot \frac{AH}{100}\end{equation}

Symbols

\(Q_{delivered}\)	Heat delivered \([\unit{ \btu\per\hour}]\)
\(Q_{input}\)	Total heat input \([\unit{ \btu\per\hour}]\)
\(AH\)	Available heat percent \([\unit{ \percent}]\)
\(100\)	Percentage conversion factor \([\unit{ \percent}]\)

Available Heat

---

Percentage of input heat available after exhaust losses.

The available heat represents the fraction of input energy that remains available for useful work after accounting for exhaust gas temperature, excess air, and combustion air preheat.

\begin{equation}\label{eq:energy-input-exhaust-gas-available-heat-cpp} AH = BH + AC + CAC\end{equation}

Symbols

\(AH\)	Available heat percent \([\unit{ \percent}]\)
\(BH\)	Base heat percentage \([\unit{ \percent}]\)
\(AC\)	Air correction \([\unit{ \percent}]\)
\(CAC\)	Combustion air correction \([\unit{ \percent}]\)

Base Heat

---

Base available heat before corrections.

The base heat formula calculates the starting percentage of input heat available for useful work, before accounting for losses due to excess air and combustion air preheat. It is determined by subtracting the exhaust temperature loss from a model-specific base value (95%). This step isolates the effect of exhaust gas temperature on available heat, prior to applying air corrections.

\begin{equation}\label{eq:energy-input-exhaust-gas-base-heat-cpp} BH = 95 - 0.025 \cdot T_{exh}\end{equation}

Symbols

\(BH\)	Base heat percentage \([\unit{ \percent}]\)
\(95\)	Base available heat at zero exhaust temperature \([\unit{ \percent}]\)
\(0.025\)	Exhaust temperature loss coefficient \([\unit{ \percent\per\degreeFahrenheit}]\)
\(T_{exh}\)	Exhaust gas temperature \([\unit{ \degreeFahrenheit}]\)

Air Correction

---

Correction for excess air in exhaust.

The air correction accounts for heat lost due to excess air being heated to exhaust temperature. Excess air reduces available heat by carrying away sensible heat.

\begin{equation}\label{eq:energy-input-exhaust-gas-air-correction-cpp} AC = -[(-1.078913827 + SH_{air} \cdot T_{exh})] \cdot EA/100\end{equation}

Symbols

\(AC\)	Air correction \([\unit{ \percent}]\)
\(-1.078913827\)	Air correction base - see physics::us::kAirCorrectionBase \([\unit{ \btu\per\pound}]\)
\(SH_{air}\)	Specific heat of air \([\unit{ \btu\per\pound\degreeFahrenheit}]\)
\(T_{exh}\)	Exhaust gas temperature \([\unit{ \degreeFahrenheit}]\)
\(EA\)	Excess air as percent \([\unit{ \percent}]\)
\(100\)	Percentage conversion factor \([\unit{ \percent}]\)

Combustion Air Correction

---

Correction for preheated combustion air.

The combustion air correction accounts for sensible heat added to the system when combustion air is preheated. Preheating combustion air increases available heat.

\begin{equation}\label{eq:energy-input-exhaust-gas-combustion-air-correction-cpp} CAC = (-1.078913827 + SH_{air} \cdot T_{air}) \cdot (1 + EA/100)\end{equation}

Symbols

\(CAC\)	Combustion air correction \([\unit{ \percent}]\)
\(-1.078913827\)	Air correction base - see physics::us::kAirCorrectionBase \([\unit{ \btu\per\pound}]\)
\(SH_{air}\)	Specific heat of air \([\unit{ \btu\per\pound\degreeFahrenheit}]\)
\(T_{air}\)	Combustion air temperature \([\unit{ \degreeFahrenheit}]\)
\(EA\)	Excess air as percent \([\unit{ \percent}]\)
\(100\)	Percentage conversion factor \([\unit{ \percent}]\)

Specific Heat of Air

---

Temperature-dependent specific heat of air.

The specific heat of air increases with temperature. This linear correlation provides the specific heat as a function of temperature for use in air correction calculations.

\begin{equation}\label{eq:energy-input-exhaust-gas-specific-heat-air-cpp} SH_{air} = 0.017828518 + 0.000002556 \cdot T_{air}\end{equation}

Symbols

\(SH_{air}\)	Specific heat of air \([\unit{ \btu\per\pound\degreeFahrenheit}]\)
\(0.017828518\)	Base specific heat of air - see physics::us::kSpecificHeatAirBase \([\unit{ \btu\per\pound\degreeFahrenheit}]\)
\(0.000002556\)	Specific heat temperature coefficient - see physics::us::kSpecificHeatAirCoeff \([\unit{ \btu\per\pound\degreeFahrenheit\per\degreeFahrenheit}]\)
\(T_{air}\)	Combustion air temperature \([\unit{ \degreeFahrenheit}]\)

See also

physics::us::kSpecificHeatAirBase for base specific heat of air

physics::us::kSpecificHeatAirCoeff for specific heat temperature coefficient

physics::us::kAirCorrectionBase for air correction base constant

Modules
	Exhaust Gas Losses Formula
	Total heat loss through exhaust gases.
	Heat Delivered Formula
	Useful heat delivered to the furnace.
	Available Heat Formula
	Percentage of input heat available after exhaust losses.
	Base Heat Formula
	Base available heat before corrections.
	Air Correction Formula
	Correction for excess air in exhaust.
	Combustion Air Correction Formula
	Correction for preheated combustion air.
	Specific Heat of Air Formula
	Temperature-dependent specific heat of air.

Files
file	energy_input_exhaust_gas_heat_loss.h

Namespaces
namespace	energy_input_exhaust_gas_heat_loss
	Calculates available heat, heat delivered, and exhaust gas losses for non-EAF electric furnaces.

Classes
struct	energy_input_exhaust_gas_heat_loss::EnergyInputExhaustGasResult
	Result struct for energy input exhaust gas heat loss calculations. More...

Functions
double	energy_input_exhaust_gas_heat_loss::availableHeat (double excess_air, double combustion_air_temp, double exhaust_gas_temp, double total_heat_input=1)
	Calculates available heat for a non-EAF electric furnace.
double	energy_input_exhaust_gas_heat_loss::heatDelivered (double available_heat, double total_heat_input)
	Calculates heat delivered for a non-EAF electric furnace.
double	energy_input_exhaust_gas_heat_loss::totalHeatLoss (double heat_delivered, double available_heat)
	Calculates exhaust gas losses for a non-EAF electric furnace.
EnergyInputExhaustGasResult	energy_input_exhaust_gas_heat_loss::calculate (double excess_air, double combustion_air_temp, double exhaust_gas_temp, double total_heat_input=1)
	Calculates available heat, heat delivered, and exhaust gas losses for a non-EAF electric furnace.

Function Documentation

availableHeat()

double energy_input_exhaust_gas_heat_loss::availableHeat	(	double	excess_air,
		double	combustion_air_temp,
		double	exhaust_gas_temp,
		double	total_heat_input = 1
	)

Computes the available heat percentage based on excess air, combustion air temperature, exhaust gas temperature, and total heat input. If total heat input is zero, available heat is set to 100%.

Parameters

[in]	excess_air	Excess air as percent \([\unit{\percent}]\)
[in]	combustion_air_temp	Temperature of combustion air \([\unit{\degreeFahrenheit}]\)
[in]	exhaust_gas_temp	Temperature of exhaust gas \([\unit{\degreeFahrenheit}]\)
[in]	total_heat_input	Total heat input from all sources \([\unit{\btu\per\hour}]\). If zero, available heat is 100%.

Returns

Available heat percent \([\unit{\percent}]\)

See also

energy_input_exhaust_gas_heat_loss_formula

calculate()

EnergyInputExhaustGasResult energy_input_exhaust_gas_heat_loss::calculate	(	double	excess_air,
		double	combustion_air_temp,
		double	exhaust_gas_temp,
		double	total_heat_input = 1
	)

Computes all three quantities and returns them as a tuple.

Parameters

[in]	excess_air	Excess air as percent \([\unit{\percent}]\)
[in]	combustion_air_temp	Temperature of combustion air \([\unit{\degreeFahrenheit}]\)
[in]	exhaust_gas_temp	Temperature of exhaust gas \([\unit{\degreeFahrenheit}]\)
[in]	total_heat_input	Total heat input from all sources \([\unit{\btu\per\hour}]\). If zero, available heat is 100%.

Returns

EnergyInputExhaustGasResult struct containing: available_heat \([\unit{\percent}]\), heat_delivered \([\unit{\btu\per\hour}]\), and exhaust_gas_losses \([\unit{\btu\per\hour}]\)

See also

energy_input_exhaust_gas_heat_loss_formula

heatDelivered()

double energy_input_exhaust_gas_heat_loss::heatDelivered	(	double	available_heat,
		double	total_heat_input
	)

Computes the heat delivered to the furnace based on available heat percentage and total heat input.

Parameters

[in]	available_heat	Available heat percent \([\unit{\percent}]\)
[in]	total_heat_input	Total heat input from all sources \([\unit{\btu\per\hour}]\)

Returns

Heat delivered \([\unit{\btu\per\hour}]\)

See also

energy_input_exhaust_gas_heat_loss_formula

totalHeatLoss()

double energy_input_exhaust_gas_heat_loss::totalHeatLoss	(	double	heat_delivered,
		double	available_heat
	)

Computes the exhaust gas losses based on heat delivered and available heat percentage.

Parameters

[in]	heat_delivered	Heat delivered \([\unit{\btu\per\hour}]\)
[in]	available_heat	Available heat percent \([\unit{\percent}]\)

Returns

Exhaust gas losses \([\unit{\btu\per\hour}]\)

See also

energy_input_exhaust_gas_heat_loss_formula