Modified Differential Approximation Research Articles

Insights into the hydrodynamic properties of slurry flow in a tubular photocatalytic reactor by PIV combined with LSIA

Although turbulent flow is generally needed for good suspension of photocatalysts, lower flow rates are preferred from an economic viewpoint for energy-efficient operation. However, no experimental work has been conducted to reveal how photocatalyst particles move and distribute in a tubular photocatalytic reactor under mild operating conditions, which is critical for reactor design and configuration of the light concentration. In our study, the photocatalyst itself was employed as a tracer particle for Particle Image Velocity (PIV) measurement. PIV combined with a new laser sheet image analysis (LSIA) technology was employed to investigate both the particle velocity and number distribution in the tubular reactor. It was found that, in the inlet, a higher velocity distribution of fluid generally occurred in the lower part of the tube. However, in the middle and outlet regions of the tube, a higher velocity distribution existed in the upper part of the tube. LSIA investigation showed that the transport capacity of the fluid and the initial particle size distribution are two essential factors influencing particle number distribution in the suspension. Regardless of the particle size, the middle part of the reactor holds the maximum number of particles while the outlet has the minimum number of particles. In the inlet, both small and large particles show similar number distribution trends against the flow rate. However, in the middle part of the tube, the number of small particles decreases with the flow rate while the number of the large particles shows the opposite trend. The difference in velocity distribution along the radial direction also significantly affects the particle distribution. One interesting finding is that, regardless of the particle size, the number of particles in the upper part of the tube is always higher than that in centre. Stokes’ drag law and the Saffman lift force were employed to explain this experimental finding. In the last section, the correlation between particle distribution and optical properties was numerically investigated by a modified differential approximation (MDA) method.

Advanced Differential Approximation Formulation of the PN Method for Radiative Transfer

The spherical harmonics (PN) method, especially its lowest order, i.e., the P1 or differential approximation, enjoys great popularity because of its relative simplicity and compatibility with standard models for the solution of the (overall) energy equation. Low-order PN approximations perform poorly in the presence of strongly nonisotropic intensity distributions, especially in optically thin situations within nonisothermal enclosures (due to variation in surface radiosities across the enclosure surface, causing rapid change of irradiation over incoming directions). A previous modification of the PN approximation, i.e., the modified differential approximation (MDA), separates wall emission from medium emission to reduce the nonisotropy of intensity. Although successful, the major drawback of this method is that the intensity at the walls is set to zero into outward directions, while incoming intensity is nonzero, resulting in a discontinuity at grazing angles. To alleviate this problem, a new approach, termed here the “advanced differential approximation (ADA),” is developed, in which the directional gradient of the intensity at the wall is minimized. This makes the intensity distribution continuous for the P1 method and mostly continuous for higher-order PN methods. The new method is tested for a 1D slab and concentric spheres and for a 2D medium. Results are compared with the exact analytical solutions for the 1D slab as well as the Monte Carlo-based simulations for 2D media.

Detailed radiation modeling of a partial-oxidation flame

A numerical study of a laminar methane partial oxidation flame (POX-flame) is presented to investigate different radiation modeling approaches. A laminar reference flame for large-scale gasification, which was previously investigated both experimentally and numerically by Stelzner et al. [1], is particularly suitable for such a study because of the distinct temperatures and radiating species concentrations. A second study of a well-known methane–air flame [2] is also included. The fully resolved and coupled solution of the reactive flows was achieved with an OpenFOAM® based solver using detailed approaches for diffusion and chemistry. Different combinations for solving the radiative transfer equation (RTE) and the evaluation of the radiative properties were implemented and investigated. The discrete ordinate method (DOM), the modified differential approximation (MDA), the P1 model and the optically thin model (OTM) were used for the solution of the RTE, whereas the spectral line-based weighted sum of gray gases model (SLW), the weighted sum of gray gases model (WSGG) and gray absorption (RADCAL) was used for the evaluation of the radiative properties.In both flame setups gaseous radiation was found to be important and needs to be considered. In the POX-flame radiative absorption has a significant impact on the radiative source terms and significant non-gray gas effects were found. Simple radiation modeling approaches, typical for combustion regimes, namely the OTM with RADCAL, were not able to capture these effects. More sophisticated radiation models were required. The MDA with both the SLW and the WSGG provided reasonable results at still acceptable computational costs. In contrast, in the methane–air flame, the simple radiation models achieved a reasonable agreement with the temperature measurements.Finally, particular combinations of RTE solution methods and property models are recommended for the different flame setups based on the differences in the temperature and the radiating species concentrations in the different flame regions.

Solution of the Radiative Transfer Equation in Three-Dimensional Participating Media Using a Hybrid Discrete Ordinates: Spherical Harmonics Method

In this article, a new hybrid solution to the radiative transfer equation (RTE) is proposed. Following the modified differential approximation (MDA), the radiation intensity is first split into two components: a “wall” component, and a “medium” component. Traditionally, the wall component is determined using a viewfactor-based surface-to-surface exchange formulation, while the medium component is determined by invoking the first-order spherical harmonics (P1) approximation. Recent studies have shown that although the MDA approach is accurate over a large range of optical thicknesses, it is prohibitive for complex three-dimensional geometry with obstructions, both from a computational efficiency as well as memory standpoint. The inefficiency stems from the use of the viewfactor-based approach for determination of the wall-emitted component. In this work, instead, the wall component is determined directly using the control angle discrete ordinates method (CADOM). The new hybrid method was validated for both two-dimensional (2D) and three-dimensional (3D) geometries against benchmark Monte Carlo results for gray media in which the optical thickness was varied over a large range. In all cases, the accuracy of the hybrid method was found to be within a few percent of Monte Carlo results, and comparable to the solutions of the RTE obtained directly using CADOM. Finally, the new hybrid method was explored for 3D nongray media in the presence of reflecting walls and various scattering albedos. As a noteworthy advantage, irrespective of the conditions used, it was always found to be computationally more efficient than standalone CADOM and up to 15 times more efficient than standalone CADOM for optically thick media with strong scattering.

Application of the modified differential approximation for radiative transfer to arbitrary geometry

The first-order spherical harmonics method (or P 1 approximation) has found prolific usage for approximate solution of the radiative transfer equation (RTE) in participating media. However, the accuracy of the P 1 approximation deteriorates as the optical thickness of the medium is decreased. The modified differential approximation (MDA) was originally proposed to remove the shortcomings of the P 1 approximation in optically thin situations. This article presents algorithms to apply the MDA to arbitrary geometry—in particular, geometry with obstructions, and inhomogeneous media. The wall-emitted component of the intensity was computed using a combined view-factor and ray-tracing approach. The Helmholtz equation, arising out of the medium-emitted component, was solved using an unstructured finite-volume procedure. The general procedure was validated for both two-dimensional (2D) and three-dimensional (3D) geometries against benchmark Monte Carlo results. The accuracy of MDA was found to be superior to the P 1 approximation for all optical thicknesses. Its accuracy, when compared with the discrete ordinates method (both S 6 and S 8), was found to be clearly superior in optically thin situations, but problem dependent in optically intermediate and thick situations. For 3D geometries, calculation and storage of the view-factor matrix was found to be a major shortcoming of the MDA. In addition, for inhomogeneous media, calculation of optical distances requires a ray-tracing procedure, which was found to be a bottleneck from a computational efficiency standpoint. Several strategies to reduce both memory and computational time are discussed and demonstrated.

Differential approximations for transient radiative transfer through a participating medium exposed to collimated irradiation

First, we apply the modified differential approximation (MDA) suggested by Chandrasekhar to transient radiative transfer in a scattering planar medium exposed to collimated pulse irradiation. Next, a hybrid method of the P 1/3 approximation suggested by Olson and the MDA is developed. The hybrid method may be referred to as the modified P 1/3 approximation (MP 1/3A) and is also applied to the same example. Comparisons of the results obtained by solving the MDA, the MP 1/3A and the exact integral equation are made. The comparisons show that the temporal distribution of the transmissivity obtained by the MDA contains a small protuberance or an abrupt slope change, which decreases with the decrease of the scattering albedo. The results obtained by the MP 1/3A are more accurate than those obtained by the MDA for most of the cases considered, because the MP 1/3A corrects the propagation speed of the transmitted radiation.

Three-dimensional radiation in absorbing-emitting-scattering media using the modified differential approximation

The modified differential approximation (MDA) first proposed by Olfe and recently extended by Modest is applied to a three-dimensional absorbing-emitting-scattering medium bounded by non-black walls. Numerical techniques are developed to solve for the radiation fields due to wall emission and medium emission. The techniques are evaluated in terms of the computer memory requirement, CPU time and numerical accuracy. For a model problem, the modified differential approximation is compared against the zone method, the P 1 approximation, the P, approximation, S 4 and S 6. MDA is shown to be superior to the P 1 and P 3 approximations and to compare favorably with S N solutions for all conditions of optical depth.

The Improved Differential Approximation for Radiative Transfer in Multidimensional Media

One particularly popular approximate method for radiative transfer in participating media is the so-called differential approximation or P-1 approximation. Unfortunately, the differential approximation is accurate only in optically thick media, and may become very inaccurate in optically thin situations, in particular in two- and three-dimensional geometries. To increase the accuracy, a number of improvements have been proposed. Olfe (1967, 1968) separated wall emission from medium emission in what he called the modified differential approximation (MDA). Modest (1974, 1975) combined the P-1 approximation with an optically thin correction term in what he called the improved differential approximation (IDA). While very accurate, both models were limited to relatively simple situations. Recently, Modest (1989) demonstrated that Olfe's MDA may be formulated for a general three-dimensional case with an absorbing/emitting, anisotropically scattering medium bounded by gray, diffusely emitting and reflecting walls. It is the purpose of the present note to show that the IDA may also be extended to the most general case, reducing to the correct optically thin and optically thick limits for all situations, resulting in accuracy similar to that of the MDA, but being computationally more efficient.