Detailed Description
This calculator estimates radiative heat loss through circular and rectangular openings in process heating equipment. It uses empirical polynomial approximations for the radiative view factor, based on the geometry of the opening and wall thickness. The calculator is suitable for modeling heat loss from doors, ports, and other openings, and is based on standard heat transfer references and ASHRAE research.
The calculation follows a top-down approach:
- Total Opening Heat Loss - Calculate radiative heat loss through opening based on Stefan-Boltzmann law
- View Factor - Determine geometric view factor using empirical polynomial correlations
- Polynomial Coefficients - Apply appropriate polynomial based on opening geometry and thickness ratio
Total Opening Heat Loss
\begin{equation}\label{eq:opening-total-heat-loss} Q_\text{open} = \varepsilon \sigma A (T_\text{in}^4 - T_\text{amb}^4) \cdot F \cdot \frac{t_\text{open}}{100}\end{equation}
- Symbols
\(Q_\text{open}\) Opening heat loss \([\unit{ \btu\per\hour}]\) \(\varepsilon\) Surface emissivity \([\unit{ \unitless}]\) \(\sigma\) Stefan-Boltzmann constant \([\unit{ \btu\per\hour\foot\squared\degreeRankine\tothe{4}}]\) \(A\) Opening area \([\unit{ \foot\squared}]\) \(T_\text{in}\) Inside temperature \([\unit{ \degreeRankine}]\) \(T_\text{amb}\) Ambient temperature \([\unit{ \degreeRankine}]\) \(F\) Radiative view factor \([\unit{ \unitless}]\) \(t_\text{open}\) Percent time open \([\unit{ \unitless}]\)
View Factor Calculation
The radiative view factor F is a function of the opening geometry and wall thickness.
The radiative view factor essentially relates the fraction of emitted radiation from surface A that strikes surface B. Typically, this is a straightforward calculation with some basic assumptions; however, it is impractical to assume no restrictions in radiative heat transfer from the interior of the furnace out to the radiation far field. This loss can be modelled by assuming the only opening between surface A and B is one through the depth of the furnace’s walls and insulation. This partially simplifies the process by allowing MEASUR to use empirical equations to determine the view factor based on physical opening geometry.
For circular openings, F is approximated by a polynomial in the thickness-to-diameter ratio. For rectangular openings, F is a function of thickness-to-length and lateral dimension ratios, using piecewise polynomial fits.
The empirical equations for rectangular openings that follow will be presented as cases which take this form of equation: Each polynomial is of the form where (TR) is the thickness-to-dimension ratio for the opening:
\begin{equation}\label{eq:opening-view-factor} F_\text{Case#} = \frac{C_0 + C_1 TR + C_2 TR^2 + C_3 TR^3 + C_4 TR^4 + C_5 TR^5}{100}\end{equation}
TR is taken to be the thickness ratio, which is defined by the height or length of the opening, whichever is shorter, divided by the furnace wall thickness. LDR, or lateral dimension ratio, is then defined by the larger of either length or height divided by shorter of the two. In calculation, TR is limited to no greater than 6.
- Note
- See Opening View Factor Polynomials for explicit polynomial forms.
Polynomial Approximations
Polynomial approximations for radiative view factor.
The following polynomials are used for different ranges of thickness-to-diameter or thickness-to-length ratios. Coefficients are fit to published view factor data for furnace openings.
- Polynomial Coefficients for View Factor Cases
- Each polynomial is of the form where (TR) is the thickness-to-dimension ratio for the opening:
\begin{equation}\label{eq:opening-view-factor-2} F_\text{Case#} = \frac{C_0 + C_1 TR + C_2 TR^2 + C_3 TR^3 + C_4 TR^4 + C_5 TR^5}{100}\end{equation}
The following table lists the coefficients (C₀–C₅) for each polynomial used to approximate the radiative view factor for various opening geometries in the code:
| Case | C₀ | C₁ | C₂ | C₃ | C₄ | C₅ |
|---|---|---|---|---|---|---|
| 1 | 1.1 | 92.8571 | -57.5893 | 15.625 | 0 | 0 |
| 2 | 29.5 | 26.8417 | -4.35417 | -8.33e-2 | .104167e-1 | -8.33e-3 |
| 3 | 3.5 | 89.5833 | -50.0 | 10.4167 | 0 | 0 |
| 4 | 24.0 | 39.3917 | -11.6042 | 1.85417 | -.145833 | 4.17e-3 |
| 5 | 2.7 | 112.679 | -70.9821 | 15.625 | 0 | 0 |
| 6 | 35.5 | 29.4583 | -4.52083 | -6.875e-1 | .270833 | -.208333e-2 |
| 7 | 13.0 | 123.75 | -100.0 | 31.25 | 0 | 0 |
| 8 | 27.0 | 64.5667 | -29.9167 | 7.14583 | -8.33e-1 | 3.75e-2 |
- Circular Openings
\begin{equation}\label{eq:opening-vf-case1} F_1(TR) = \frac{1.10 + 92.86TR - 57.59TR^2 + 15.62TR^3}{100}\end{equation}
\begin{equation}\label{eq:opening-vf-case2} F_2(TR) = \frac{29.50 + 26.84TR - 4.35TR^2 - 0.083TR^3 + 0.104TR^4 - 0.0083TR^5}{100}\end{equation}
- Circular Opening View Factor Case Table
- The following table shows which polynomial formula is used for each thickness ratio (TR) for circular openings:
| TR ≤ 0.1 | \(TR = 0.1\) \(F_1 * \frac{TR}{0.1}\) |
| 0.1 < TR ≤ 6 | \(F_1\) |
| 6 < TR | \(TR = 6\) \(F_2\) |
Where:
- TR = thickness ratio (diameter/thickness)
- For TR < 0.1, use TR = 0.1
- For 6 < TR, use TR = 6
- Rectangular Openings
\begin{equation}\label{eq:opening-vf-case3} F_3(TR) = \frac{3.50 + 89.58TR - 50.00TR^2 + 10.42TR^3}{100}\end{equation}
\begin{equation}\label{eq:opening-vf-case4} F_4(TR) = \frac{24.00 + 39.39TR - 11.60TR^2 + 1.85TR^3 - 0.146TR^4 + 0.0042TR^5}{100}\end{equation}
\begin{equation}\label{eq:opening-vf-case5} F_5(TR) = \frac{2.70 + 112.68TR - 70.98TR^2 + 15.62TR^3}{100}\end{equation}
\begin{equation}\label{eq:opening-vf-case6} F_6(TR) = \frac{35.50 + 29.46TR - 4.52TR^2 - 0.687TR^3 + 0.271TR^4 - 0.021TR^5}{100}\end{equation}
\begin{equation}\label{eq:opening-vf-case7} F_7(TR) = \frac{13.00 + 123.75TR - 100.00TR^2 + 31.25TR^3}{100}\end{equation}
\begin{equation}\label{eq:opening-vf-case8} F_8(TR) = \frac{27.00 + 64.57TR - 29.92TR^2 + 7.15TR^3 - 0.833TR^4 + 0.0375TR^5}{100}\end{equation}
- Rectangular Opening View Factor Case Table
- The following table shows which polynomial formula is used for each combination of thickness ratio (TR) and lateral dimension ratio (LDR) for rectangular openings:
| 1 ≤ LDR ≤ 2 | 2 < LDR ≤ 9 | 9 < LDR | |
|---|---|---|---|
| TR ≤ 0.1 | \(TR = 0.1\) \(F_3 + (F_5 - F_3) * (LDR - 1)\) | \(TR = 0.1\) \(F_5 + (F_7 - F_5) * \frac{(LDR - 2)}{8}\) | \(F_7(TR=0.1)*(\frac{TR}{0.1})\) |
| 0.1 < TR ≤ 0.9 | \(F_3 + (F_5 - F_3) * (LDR - 1)\) | \(F_5 + (F_7 - F_5) * \frac{(LDR - 2)}{8}\) | \(F_7\) |
| 0.9 < TR ≤ 6 | \(F_4 + (F_6 - F_4) * (LDR - 1)\) | \(F_6 + (F_8 - F_6) * \frac{(LDR - 2)}{8}\) | \(F_8\) |
| 6 < TR | \(TR = 6\) \(F_4 + (F_6 - F_4) * (LDR - 1)\) | \(TR = 6\) \(F_6 + (F_8 - F_6) * \frac{(LDR - 2)}{8}\) | \(TR = 6\) \(F_8\) |
Where:
- TR = thickness ratio (length/thickness or height/thickness)
- LDR = lateral dimension ratio (length/height or height/length)
- For TR < 0.1, use TR = 0.1
- For 6 < TR, use TR = 6
Symbols
Symbols used in opening heat loss formulas.
| \(Q_\text{open}\) | Opening heat loss \([\unit{ \btu\per\hour}]\) |
| \(\varepsilon\) | Surface emissivity \([\unit{ \unitless}]\) |
| \(\sigma\) | Stefan-Boltzmann constant \([\unit{ \btu\per\hour\foot\squared\degreeRankine\tothe{4}}]\) |
| \(A\) | Opening area \([\unit{ \foot\squared}]\) |
| \(T_\text{in}\) | Inside temperature \([\unit{ \degreeRankine}]\) |
| \(T_\text{amb}\) | Ambient temperature \([\unit{ \degreeRankine}]\) |
| \(F\) | Radiative view factor \([\unit{ \unitless}]\) |
| \(t_\text{open}\) | Percent time open \([\unit{ \unitless}]\) |
| \(d\) | Diameter of circular opening \([\unit{ \inch}]\) |
| \(t\) | Wall thickness \([\unit{ \inch}]\) |
| \(l\) | Length of rectangular opening \([\unit{ \inch}]\) |
| \(h\) | Height of rectangular opening \([\unit{ \inch}]\) |
- See also
- Kays & Crawford, "Convective Heat and Mass Transfer"; Incropera & DeWitt, "Fundamentals of Heat and Mass Transfer"; Glicksman (ASHRAE Transactions, 1972); Perry's Chemical Engineers' Handbook
- physics::us::kStefanBoltzmann for the Stefan-Boltzmann constant
Modules | |
| Opening Total Heat Loss Formula | |
| Opening View Factor Formula | |
| The radiative view factor F is a function of the opening geometry and wall thickness. | |
| Opening View Factor Polynomials | |
| Polynomial approximations for radiative view factor. | |
| Opening Heat Loss Symbols | |
| Symbols used in opening heat loss formulas. | |
Files | |
| file | opening_heat_loss.h |
Namespaces | |
| namespace | opening_heat_loss |
| Calculates heat losses from process heating equipment openings to the ambient. | |
Functions | |
| double | opening_heat_loss::totalHeatLoss (double area, double emissivity, double insideTemperature, double ambientTemperature, double viewFactor, double percentTimeOpen) |
| Calculates the total heat loss for a process heating equipment opening with a given area. | |
| double | opening_heat_loss::totalHeatLossQuad (double emissivity, double length, double width, double ambientTemperature, double insideTemperature, double percentTimeOpen, double viewFactor) |
| Calculates total heat loss for a rectangular opening. | |
| double | opening_heat_loss::totalHeatLossCircular (double emissivity, double diameter, double ambientTemperature, double insideTemperature, double percentTimeOpen, double viewFactor) |
| Calculates total heat loss for a circlular opening. | |
| double | opening_heat_loss::calculateViewFactorCircular (double thickness, double diameter) |
| Calculates the view factor for a circular opening. | |
| double | opening_heat_loss::calculateViewFactorQuad (double thickness, double length, double height) |
| Calculates the view factor for a rectangular opening. | |
Function Documentation
◆ calculateViewFactorCircular()
| double opening_heat_loss::calculateViewFactorCircular | ( | double | thickness, |
| double | diameter | ||
| ) |
- Parameters
-
[in] thickness Process heating equipment wall thickness \([\unit{\inch}]\) [in] diameter Diameter of opening \([\unit{\inch}]\)
- Returns
- View factor for radiative heat transfer \([\unit{\unitless}]\)
- See also
- opening_circular_view_factor_formula
◆ calculateViewFactorQuad()
| double opening_heat_loss::calculateViewFactorQuad | ( | double | thickness, |
| double | length, | ||
| double | height | ||
| ) |
- Parameters
-
[in] thickness Process heating equipment wall thickness \([\unit{\inch}]\) [in] length Length of opening \([\unit{\inch}]\) [in] height Height of opening \([\unit{\inch}]\)
- Returns
- View factor for radiative heat transfer \([\unit{\unitless}]\)
- See also
- opening_quad_view_factor_formula
◆ totalHeatLoss()
| double opening_heat_loss::totalHeatLoss | ( | double | area, |
| double | emissivity, | ||
| double | insideTemperature, | ||
| double | ambientTemperature, | ||
| double | viewFactor, | ||
| double | percentTimeOpen | ||
| ) |
This function computes the radiative heat loss from a process heating equipment opening to the surroundings, based on the opening area, surface emissivity, inside and ambient temperatures, view factor, and percent time open. The calculation assumes radiative losses only and does not account for convection or conduction.
- Parameters
-
[in] area Area of the opening \([\unit{\inch\squared}]\) [in] emissivity Emissivity of the opening surface \([\unit{\unitless}]\) [in] insideTemperature Inside temperature \([\unit{\degreeFahrenheit}]\) [in] ambientTemperature Ambient temperature \([\unit{\degreeFahrenheit}]\) [in] viewFactor View factor for radiative heat transfer \([\unit{\unitless}]\) [in] percentTimeOpen Percentage of time opening is open \([\unit{\percent}]\)
- Returns
- Total heat loss \([\unit{\btu\per\hour}]\)
- See also
- opening_heat_loss_formula
◆ totalHeatLossCircular()
| double opening_heat_loss::totalHeatLossCircular | ( | double | emissivity, |
| double | diameter, | ||
| double | ambientTemperature, | ||
| double | insideTemperature, | ||
| double | percentTimeOpen, | ||
| double | viewFactor | ||
| ) |
- Parameters
-
[in] emissivity Emissivity of the opening surface \([\unit{\unitless}]\) [in] diameter Diameter of opening \([\unit{\inch}]\) [in] ambientTemperature Ambient temperature \([\unit{\degreeFahrenheit}]\) [in] insideTemperature Inside temperature \([\unit{\degreeFahrenheit}]\) [in] percentTimeOpen Percentage of time opening is open \([\unit{\percent}]\) [in] viewFactor View factor for radiative heat transfer \([\unit{\unitless}]\)
- Returns
- Total heat loss \([\unit{\btu\per\hour}]\)
- See also
- opening_circular_heat_loss_formula
◆ totalHeatLossQuad()
| double opening_heat_loss::totalHeatLossQuad | ( | double | emissivity, |
| double | length, | ||
| double | width, | ||
| double | ambientTemperature, | ||
| double | insideTemperature, | ||
| double | percentTimeOpen, | ||
| double | viewFactor | ||
| ) |
- Parameters
-
[in] emissivity Emissivity of the opening surface \([\unit{\unitless}]\) [in] length Length of opening \([\unit{\inch}]\) [in] width Height of opening \([\unit{\inch}]\) [in] ambientTemperature Ambient temperature \([\unit{\degreeFahrenheit}]\) [in] insideTemperature Inside temperature \([\unit{\degreeFahrenheit}]\) [in] percentTimeOpen Percentage of time opening is open \([\unit{\percent}]\) [in] viewFactor View factor for radiative heat transfer \([\unit{\unitless}]\)
- Returns
- Total heat loss \([\unit{\btu\per\hour}]\)
- See also
- opening_quad_heat_loss_formula
