A nongray-wall emissivity model for the Wide-Band Correlated K-distribution method

Highlights

• A nongray-wall emissivity model of WBCK is proposed for nongray-wall radiative problems.

• An absorption-coefficient-based optimized band interval for WBCK is proposed.

• The model provides good accuracy for fly-ash deposit, GH536, and soot deposit wall.

• The nongray-wall emissivity model can improve the computational efficiency.

Abstract

The walls of combustion systems are usually assumed to be black or gray in radiative calculations, which may cause large errors. The Planck-function-weighted emissivity is usually used as the gray-wall emissivity when using the Wide-Band Correlated K-distribution method. This approach can demonstrate good accuracy when coupled with the emissivity-based optimized band interval approach proposed by Solovjov et al., 2013. To improve computational efficiency without losing accuracy, a nongray-wall emissivity model and an absorption-coefficient-based method for determining the band interval are proposed. The accuracy of nongray-wall and gray-wall emissivity models along with different approaches to determine the band intervals is evaluated in three 1D isothermal and homogeneous cases bounded by fly-ash deposit, a high-temperature alloy, and soot deposit and a 3D fuel-air flame bounded by fly-ash deposit. The results show that the nongray-wall emissivity model is much more accurate than the gray-wall one when the number of bands is greater than 1. Coupled with the absorption-coefficient-based band interval approach, the nongray-wall emissivity model becomes more accurate when the number of bands is larger than 2, especially for low-temperature walls. It is sufficient to divide the entire spectrum into 4 bands for the cases tested here.

Introduction

Radiative heat transfer plays an important role in high-temperature industrial combustion systems, such as coal-fired boilers and gas turbine combustors [1]. Most modeling studies conducted by far have assumed that the walls of combustors are black or gray in radiative heat transfer calculations for simplicity in spite of the fact that the wall materials have different spectral emissivity [2,3]. This simplification may introduce large errors in prediction of the radiative source term and the heat flux [4], which may strongly affect the predicted performance of the combustion devices in terms of thermal protection [5] and pollutant emissions [6]. Therefore, comprehensive and accurate radiative models are necessary for accurately predicting radiative heat transfer in practical problems involving nongray walls and for optimal design of combustion devices.

Radiative property models with different spectral resolutions, such as narrow-band models [7], [8], [9], [10], wide-band models [11,12] and full-spectrum models [10,[13], [14], [15]], provide different accuracy and computational efficiency [16]. Narrow-band models, such as the Statistical Narrow-Band (SNB) method [7,8], the Statistical Narrow-Band Correlated-K (SNBCK) method [9] and the Narrow-Band Correlated K-distribution (NBCK) method [10], demonstrate the best accuracy in comparison with the Line-By-Line (LBL) method because of small changes in wall emissivity and radiative property of participating media within a narrow spectral range. Although the number of radiative transfer equations (RTEs) to be solved in narrow-band models is much less than that of LBL, the computing resources required by narrow-band models are still unacceptable in practical applications. Therefore, various computationally more efficient wide-band models and full-spectrum models have been developed. Full-spectrum models, such as the Full-Spectrum Correlated K-distribution (FSCK) method [10], the Spectral-Line-based Weighted-Sum-of-Gray-Gases (SLW) method [13] and the Weighted-Sum-of-Gray-Gases (WSGG) method [15], provide the highest computational efficiency at the cost of the loss of accuracy for practical problems involving nongray walls [17]. In contrast, wide-band models have a great potential in achieving the best compromise between efficiency and accuracy for practical problems with nongray walls. For example, Solovjov et al. [17] proposed a method to optimize the band interval according to the wall emissivity for the extension of SLW named the cumulative wavenumber model, in which only a few RTEs need to be solved. The banded method was employed by Bordbar et al. [18,19] to solve such problems as well. It was shown that these two methods provide a good compromise between accuracy and efficiency for nongray-wall problems.

The Planck-function-weighted emissivity at the wall temperature is usually employed as the gray-wall emissivity to ensure the conservation of radiative energy at the wall [16]. This gray-wall emissivity model can achieve good accuracy when coupled with the emissivity-based optimized band interval approach [17] mentioned above for nongray-wall problems. To further improve the computational efficiency without significant loss of accuracy, several other methods have been proposed recently. For instance, Fonseca et al. [20] discretized the entire spectrum into several spectral bands over which the nongray-wall emissivity can be considered constant and introduced the fraction of blackbody emission over each spectral band to use the WSGG model. Fonseca et al. [21] and Silva et al. [22] used a reference medium temperature, rather than the wall temperature, as the Planck temperature to obtain the gray-wall absorptivity, and solved the RTEs of WSGG and SLW, respectively. Liu et al. [23] proposed a grouping strategy to improve the correlation between the absorption spectrum of the radiating medium and the emission spectrum of the nongray wall and then solved the RTE of the Multi-Scale Multi-Group FSK (MSMGFSK) method. It was shown that these methods can demonstrate good accuracy and computational efficiency, especially the methods of Fonseca [21] and Silva [22], which can effectively improve the accuracy without reducing efficiency.

The objective of this work is to propose a nongray-wall emissivity model for the Wide-Band Correlated K-distribution (WBCK) method and compare this nongray-wall emissivity model with, on one hand, the benchmark LBL solution and, on the other hand, the gray-wall emissivity model, i.e., the Planck-function-weighted emissivity, in three 1D isothermal and homogeneous cases and a 3D fuel-air flame. All the 1D test cases are bounded by nongray walls composed of one of three typical materials found in industrial combustion devices, namely fly-ash deposit, a high-temperature alloy, and soot deposit, while the 3D case is bounded by walls coated with fly-ash deposit. Similar to the treatments of Solovjov et al. [17], the present WBCK method is formulated by generating the Planck-function-weighted wide-band k-distribution. This article is organized as follows. The second section presents the theoretical background. The third and fourth sections are devoted to the comparison of different models in three 1D cases and one 3D fuel-air flame, respectively. Section 5 summarizes the conclusions drawn from the present work.