The evaluation of direct exchange areas (DEA) in zonal method is the most important task due to the heavy computer cost of multi-integrals together with the existing of singularities. A technique of variable transformation to reduce the fold of integrals, which was developed originally by Erkku (1959) to calculate the DEAs of uniformly zonal dividing cylindrical system, was extended by Tian and Chiu (2003) for nonuniformly zonal dividing cylindrical system with large thermal gradients. In this paper, we further extend the reduced integration scheme (RIS) to calculate the DEAs in three-dimensional rectangular system. The detail deductions of six-, five-, and fourfold integrals to threefold ones are presented; the DEAs in a rectangular system with assumption of gray medium are computed by the Gaussian quadrature integration (GQI) and the RIS comparatively. The comparisons reveal that the RIS can provide remarkable higher accuracy and efficiency than GQI. More interestingly and practicably, the singularities of DEAs can be decomposed and weakened obviously by RIS.
In recent decades, the zonal method [
Larsen and Howell [
The applications of zonal method can be found in many thermal processes, whether for offline analysis of parameter design or for online control of operating parameter optimization. Chapman et al. [
Recently, we have made the mathematical modeling of thermal processes both in conventional reheating furnace and regenerative reheating furnace [
In fact, the definitions of DEAs of all kinds of zone pair, say, surface-surface, gas-gas (or volume-volume), and gas-surface (or volume-surface), can be found in [
Radiative exchange between surface zone
Similarly, as shown in Figure
Radiative exchange between gas zone
The formula of DEA between one gas zone
Radiative exchange between gas zone
Due to the feature of radiative heat transfer, there exists the reciprocity for DEAs and it implies
The energy conservation of radiative heat transfer leads to the following relations for surface zone of area
According to the reciprocity, only half of the DEAs need to be evaluated. The energy conservation can be used to check the accuracy of computation.
As described in [
There is a twofold integral; for example,
Set
After the variable substitution, the integral can be executed in three regions as shown in Figure
Regions of integration after the transformation.
Changing the order of integration
Since
Please note that this variable transformation is available for all computation of DEAs in rectangular systems. Now the twofold integral (
Considering a rectangular enclosure as shown in Figure
Zone arrangement in a rectangular enclosure. (a) Gas zones
Set
Set
If the zone dividing is uniform in all directions, the above integrals can be further simplified as
There exist six cases for the surface zone the surface zones are perpendicular to the the surface zones are perpendicular to the the surface zones are perpendicular to the the surface zones are perpendicular to the the surface zones are perpendicular to the the surface zones are perpendicular to the
Consider the case (A); now the cosine function in (
Set
Using the RIS, formula (
If the gas zone can be projected onto the surface zone completely along one direction, or its projection can overlay the surface zone completely after parallel moving along one direction, formula (
As to cases (B)~(F), the cosine functions and
one surface zone is perpendicular to the other; one surface zone is parallel to the other.
For case (A), taking one surface zone on the
Set
Now
Using the RIS, formula (
For other situations, one surface zone on the
For case (B), taking two surface zones, which are perpendicular to the
Set
Using the RIS, formula (
If the lengths of both surface zones are equal in
The formulae of other situations for case (B) can be deduced in the same way. We state that the situations with shielding effect do not belong to the cases studied here.
Up to now we know that, if the zone grids along three directions are uniform, the integrals of gas-gas zones can be reduced from one sixfold integral to eight threefold integrals and those of gas-surface zones from one fivefold to four threefold integrals. For the case of parallel surface zones, one fourfold integral is reduced to four twofold integrals; for the case of perpendicular surface zones, one fourfold integral is reduced to two threefold integrals. If the quadrature point number
Returning to Figure
Same as in [
Taking the number of integral steps
DEAs of gas zone
DEAs |
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GQI | 3.55586 | 0.64400 | 0.64400 | 0.24433 | 0.64400 | 0.24433 | 0.24433 | 0.13122 |
RIS | 3.79281 | 0.64415 | 0.64415 | 0.24433 | 0.64415 | 0.24433 | 0.24433 | 0.13123 |
DEAs of gas zone
DEAs |
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GQI | 3.05222 | 0.43781 | 0.43781 | 0.14722 | 0.30129 | 0.16836 | 0.16836 | 0.10348 |
RIS | 2.70782 | 0.43819 | 0.43819 | 0.14723 | 0.30129 | 0.16836 | 0.16836 | 0.10348 |
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DEAs |
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GQI | 3.05222 | 0.43781 | 0.43781 | 0.14722 | 0.30129 | 0.16836 | 0.16836 | 0.10348 |
RIS | 2.70782 | 0.43819 | 0.43819 | 0.14723 | 0.30129 | 0.16836 | 0.16836 | 0.10348 |
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DEAs |
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GQI | 3.05222 | 0.43781 | 0.43781 | 0.14722 | 0.30129 | 0.16836 | 0.16836 | 0.10348 |
RIS | 2.70782 | 0.43819 | 0.43819 | 0.14723 | 0.30129 | 0.16836 | 0.16836 | 0.10348 |
DEAs of surface zone
DEAs |
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GQI | 0 | 0 | 0 | 0 | 0.24342 | 0.15657 | 0.15657 | 0.10507 |
RIS | 0 | 0 | 0 | 0 | 0.24342 | 0.15657 | 0.15657 | 0.10507 |
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DEAs |
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GQI | 1.65273 | 0.15096 | 0.25190 | 0.07788 | 0.15096 | 0.11658 | 0.07788 | 0.07402 |
RIS | 1.57526 | 0.15096 | 0.25234 | 0.07788 | 0.15096 | 0.11658 | 0.07788 | 0.07402 |
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DEAs |
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GQI | 1.65273 | 0.15096 | 0.25190 | 0.07788 | 0.15096 | 0.11658 | 0.07788 | 0.07402 |
RIS | 1.57526 | 0.15096 | 0.25234 | 0.07788 | 0.15096 | 0.11658 | 0.07788 | 0.07402 |
It is a common knowledge that the accuracy of integrals will increase with the number of integral step
DEAs against the number of integral steps
Figure
Differences of DEAs against the number of integral steps
As mentioned in Section
Verifications and relative errors of conservation for
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GQI | Relative errors of GQI (%) | RIS | Relative errors of RIS (%) |
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2 | 2.31407 | 15.70 | 1.99713 | 0.14 |
4 | 2.02711 | 1.36 | 1.99936 | 0.03 |
8 | 2.04049 | 2.03 | 2.00097 | 0.05 |
As to the singularities that appeared in the computation of DEAs, there exit three cases. The first case is the DEA of one gas zone to itself, the second case is the DEA of one gas zone to its adjacent surface, and the third case is the DEA between two surfaces with an intersecting line. In the computations of all DEAs by RIS, the folds are always reduced. That means the possibilities of singularity can be reduced by RIS, just as said in [
The best way to evaluate the efficiency of one approach is the comparison of CPU time. Again we first take the single DEA of
Comparison of single DEA CPU time in seconds for different
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GQI | RIS | GQI | RIS | GQI | RIS | |
2 | 0.0781 | 0.0000 | 0.0780 | 0.0000 | 0.0000 | 0.0000 |
4 | 1.3122 | 0.0000 | 0.0781 | 0.0000 | 0.0156 | 0.0000 |
8 | 84.5153 | 0.0156 | 2.5781 | 0.0156 | 0.1094 | 0.0000 |
16 | 5897.4692 | 0.1564 | 63.4372 | 0.0625 | 2.3281 | 0.0781 |
32 | — | 1.5621 | 2569.1563 | 0.6402 | 43.1250 | 0.7500 |
64 | — | 10.4374 | — | 8.6440 | 633.6250 | 5.9844 |
Then taking the sum of DEAs of
Comparison of sum DEAs CPU time in seconds for different
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GQI | RIS | GQI | RIS | |
2 | 1.9066 | 0.0469 | 1.0165 | 0.0781 |
4 | 421.7854 | 0.2548 | 26.7033 | 0.6092 |
8 | 24649.4421 | 1.1723 | 9799.5624 | 12.8447 |
The reduced integration scheme is extended to calculate the direct exchange areas in three-dimensional rectangular system. The detailed reductions of multifold (six-, five-, and fourfold) integrals to threefold integrals are presented for nonuniform zonal dividing. A uniform zonal dividing case with gray medium is adopted to comparatively check the accuracy and efficiency of the reduced integration scheme. The standard Gaussian quadrature integration is also applied for the purpose of comparisons. The comparisons reveal that only
Area,
Integrand function
Direct exchange area between volume zones
Direct exchange area between volume zone
Direct exchange area between surface zones
Number of integral step
Distance between two elements, m
Volume,
Volume of gas element,
Coordinates in Cartesian system, m
Distances along corresponding coordinates, m.
Extinction coefficient,
Angle between normal and
Absorption coefficient,
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was financially supported by the Natural Science Foundation of China with granted contract nos. 51176026 and 51206043.