Leakage Heat Loss Calculator

Detailed Description

This calculator estimates the heat loss due to hot gases escaping from a furnace through openings other than the flue, under positive pressure conditions. The calculation uses furnace draft pressure, opening area, gas and ambient temperatures, a coefficient of discharge, specific gravity, and a correction factor. The method is based on standard industrial practice and is suitable for furnaces operating at positive pressure.

The calculation follows a top-down approach:

1. Total Heat Loss - Calculate heat carried away by leaking gases
2. Standard Cubic Feet per Hour (SCFH) - Convert actual flow to standard conditions
3. Cubic Feet per Hour (CFH) - Calculate leakage flow rate from pressure and opening area
4. Specific Heat - Compute temperature-dependent specific heat of gas

Relevant formulas and symbol definitions are documented below.

Total Leakage Heat Loss

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Total heat loss due to hot gas leakage.

The heat loss is calculated by multiplying the specific heat at average temperature, the leakage flow rate at standard conditions, and the temperature difference between leaking gas and ambient.

\begin{equation}\label{eq:leakage-heat-loss} Q = c_{psv} \cdot \text{SCFH} \cdot (T_{\text{leak}} - T_{\text{amb}})\end{equation}

Symbols

\(Q\)	Heat loss \([\unit{ \btu\per\hour}]\)
\(c_{psv}\)	Specific heat of gas at average temperature \([\unit{ \btu\per\cubicFoot\degreeFahrenheit}]\)
\(\text{SCFH}\)	Standard cubic feet per hour of leakage \([\unit{ \cubicFoot\per\hour}]\)
\(T_{\text{leak}}\)	Temperature of leaking gases \([\unit{ \degreeFahrenheit}]\)
\(T_{\text{amb}}\)	Ambient temperature \([\unit{ \degreeFahrenheit}]\)

Standard Cubic Feet per Hour

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Conversion of actual flow rate to standard conditions.

The standard cubic feet per hour is calculated by adjusting the actual volumetric flow rate for temperature to obtain flow at standard conditions (60°F, 1 atm).

\begin{equation}\label{eq:leakage-scfh} \text{SCFH} = \text{CFH} \cdot \sqrt{\frac{520}{460 + T_{\text{leak}}}}\end{equation}

Symbols

\(\text{SCFH}\)	Standard cubic feet per hour of leakage \([\unit{ \cubicFoot\per\hour}]\)
\(\text{CFH}\)	Volumetric flow rate of leakage \([\unit{ \cubicFoot\per\hour}]\)
\(520\)	Standard gas temperature - see physics::us::kStandardGasTemperatureR \([\unit{ \degreeRankine}]\)
\(460\)	Fahrenheit to Rankine offset - see physics::conversions::kFahrenheitToRankineOffset \([\unit{ \degreeRankine}]\)
\(T_{\text{leak}}\)	Temperature of leaking gases \([\unit{ \degreeFahrenheit}]\)

Cubic Feet per Hour

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Leakage flow rate from opening area and pressure.

The volumetric flow rate is calculated from the opening area, furnace draft pressure, specific gravity of gas, coefficient of discharge, and correction factor using standard orifice flow equations.

\begin{equation}\label{eq:leakage-cfh} \text{CFH} = 1655 \cdot C_{d} \cdot f_{\text{corr}} \cdot (A \cdot 144) \cdot \sqrt{\frac{P}{SG}}\end{equation}

Symbols

\(\text{CFH}\)	Volumetric flow rate of leakage \([\unit{ \cubicFoot\per\hour}]\)
\(1655\)	Flow constant for orifice equation \([\unit{ \unitless}]\)
\(C_{d}\)	Coefficient of discharge or flow coefficient \([\unit{ \unitless}]\)
\(f_{\text{corr}}\)	Correction factor \([\unit{ \unitless}]\)
\(A\)	Opening area \([\unit{ \foot\squared}]\)
\(144\)	Conversion factor (square feet to square inches) \([\unit{ \inch\squared\per\foot\squared}]\)
\(P\)	Furnace draft pressure \([\unit{ \inchWaterColumn}]\)
\(SG\)	Specific gravity of gas \([\unit{ \unitless}]\)

Note

The coefficient of discharge depends on the angle of convergence in degrees. (Source: Eclipse Combustion Engineering Guide, 1986, Chapter 1, Page #6) PDF link

Specific Heat Calculations

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Temperature-dependent specific heat formulas.

The specific heat of the gas varies with temperature and is calculated using polynomial correlations. The specific heat per standard volume is derived from the mass-based specific heat.

Specific Heat per Standard Volume

\begin{equation}\label{eq:leakage-cpsv} c_{psv} = 0.0793 \cdot c_{pm}\end{equation}

Mass-Based Specific Heat

\begin{equation}\label{eq:leakage-cpm} c_{pm} = 0.2371 - 9 \times 10^{-6} \cdot T_{\text{midpoint}} + 7 \times 10^{-8} \cdot T_{\text{midpoint}}^2 - 3 \times 10^{-11} \cdot T_{\text{midpoint}}^3\end{equation}

Midpoint Temperature

\begin{equation}\label{eq:leakage-midpoint} T_{\text{midpoint}} = \frac{T_{\text{leak}} + T_{\text{amb}}}{2}\end{equation}

Symbols

\(c_{psv}\)	Specific heat of gas at average temperature \([\unit{ \btu\per\cubicFoot\degreeFahrenheit}]\)
\(c_{pm}\)	Specific heat of gas at midpoint temperature \([\unit{ \btu\per\pound\degreeFahrenheit}]\)
\(T_{\text{midpoint}}\)	Midpoint temperature \([\unit{ \degreeFahrenheit}]\)
\(T_{\text{leak}}\)	Temperature of leaking gases \([\unit{ \degreeFahrenheit}]\)
\(T_{\text{amb}}\)	Ambient temperature \([\unit{ \degreeFahrenheit}]\)
\(0.0793\)	Conversion factor from mass-based to volume-based specific heat \([\unit{ \unitless}]\)

Modules
	Leakage Heat Loss Formula
	Total heat loss due to hot gas leakage.
	Standard Cubic Feet per Hour Formula
	Conversion of actual flow rate to standard conditions.
	Cubic Feet per Hour Formula
	Leakage flow rate from opening area and pressure.
	Specific Heat Calculations
	Temperature-dependent specific heat formulas.

Files
file	leakage_heat_loss.h

Namespaces
namespace	leakage_heat_loss
	Calculator for total energy loss for gas leakages.

Functions
double	leakage_heat_loss::totalHeatLoss (double draft_pressure, double opening_area, double leakage_gas_temperature, double ambient_temperature, double coefficient_discharge, double specific_gravity, double correction_factor)
	Calculates the total heat loss from hot gas leakage.

Function Documentation

totalHeatLoss()

double leakage_heat_loss::totalHeatLoss	(	double	draft_pressure,
		double	opening_area,
		double	leakage_gas_temperature,
		double	ambient_temperature,
		double	coefficient_discharge,
		double	specific_gravity,
		double	correction_factor
	)

This function computes the heat loss caused by gases leaking from the furnace. The calculation uses:

Parameters

[in]	draft_pressure	Furnace draft pressure \([\unit{\inchWaterColumn}]\)
[in]	opening_area	Opening area \([\unit{\foot\squared}]\)
[in]	leakage_gas_temperature	Temperature of leaking gases \([\unit{\degreeFahrenheit}]\)
[in]	ambient_temperature	Ambient temperature \([\unit{\degreeFahrenheit}]\)
[in]	coefficient_discharge	Coefficient of discharge or flow coefficient \([\unit{\unitless}]\)
[in]	specific_gravity	Specific gravity \([\unit{\unitless}]\)
[in]	correction_factor	Correction factor \([\unit{\unitless}]\)

Returns

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