ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING Asia-Pac. J. Chem. Eng. 2012; 7: 182–205 Published online 8 February 2012 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/apj.654
Research article
Dynamic computer modeling of environmental systems for decision making, risk assessment and design Nikos C. Markatos* School of Chemical Engineering, National Technical University of Athens, Greece Received 7 June 2011; Revised 9 September 2011; Accepted 11 November 2011
ABSTRACT: This paper describes the mathematical modeling and associated computer simulations of environmental problems related to flow and heat/mass transfer. Many key ‘issues’ in designing environmental protection systems and in performing environmental risk assessment and control are related to the behavior of fluids in turbulent flow, often involving more than one phase, with chemical reaction or heat transfer. Computational-fluid-dynamics (CFD) techniques have shown great potential for analyzing these processes and are very valuable to the environmental engineer and scientist, by reducing the need to resort to ‘cut and try’ approaches to the design of complex environmental protection systems and to any relevant decision-making process. Multidimensional, multiphase dynamic models for the dispersion of air, water, and soil pollutants and for the prediction of environmental risks are presented. Up-to-date procedures and methods for solving the relevant equations are included, together with indications of the computational resources necessary to obtain realistic representations. Results using model simulations are presented for several cases of atmospheric and marine pollution, as well as for the environmental risks of fires and of petrol-tank explosions. It is concluded that the results are physically plausible and can be used with confidence for environmental policy decisions and for the design of environmentally friendly processes and equipment. Air, water, and soil management systems can be improved by the application of these computational prediction techniques. Copyright © 2012 Curtin University of Technology and John Wiley & Sons, Ltd. KEYWORDS: environmental; dynamic modeling; turbulence; pollutants dispersion; CFD; fires; explosions; risk assessment; consequence modeling; atmospheric pollution; marine pollution
INTRODUCTION General considerations Computers and computational methods have been used since the early 1970’s to approach engineering problems in design, control, and operations.[1–3] Progress has been slower in the environmental field of applications, as the scale of the environmental problems is orders of magnitude larger than that of equipment design and, therefore, awaited the appearance of largecapacity computers and/or sophisticated solvers and techniques, as for example, parallel processing.[4,5] The modern computer’s ability to handle complex mathematics and to permit the solution of detailed models allows present-day scientists, engineers, and policy makers to model environmental process physics and chemistry at all scales, to construct models that incorporate all relevant phenomena (emissions, diffusion,
*Correspondence to: Nikos C. Markatos, School of Chemical Engineering, National Technical University of Athens, Greece. E-mail:
[email protected] © 2012 Curtin University of Technology and John Wiley & Sons, Ltd. Curtin University is a trademark of Curtin University of Technology
convection, radiation, chemical reaction, etc.), and optimize more on the basis of computed theoretical predictions and less on empirical. This development allows us today to conduct, for example, the prediction of volatile organic compounds dispersion emitted from flooring materials in buildings,[6] the prediction and control of water pollution in Trans Boundary Ecosystems, and environmental risk assessment and emission control.[7] Apart from air and water quality studies, the models must be also capable of being used in other environmental applications as well, for example, in porous media or, in general, in the subsurface.[8] Most of these problems pertain to the management of water resources. They include traditional topics of the science of hydrogeology, such as modeling of aquifers to estimate their capacity to determine the origin and impact of chemical pollution or to predict the intrusion of salty water.[5] They also include more recent topics such as modeling of underground waste disposal sites and, in particular, nuclear waste deposit sites or modeling for the underground sequestration of CO2. The subsurface is a porous medium through which air and water flow close to the surface (in the unsaturated zone), whereas further down, only water flows
Asia-Pacific Journal of Chemical Engineering
(in the saturated zone). Transport is modeled with partial differential equations, which can be linear (in the case of one-phase flow or a saturated medium) or nonlinear (in the case of two-phase flow, or an unsaturated medium). In more complex situations, more than two phases can be present. Solutes are dissolved in water and are transported by it. They may interact chemically among themselves or with the rock. One specific issue of flow in the subsurface is that the latter is a porous medium with strong heterogeneities at several scales. Indeed, the scales of interest vary from pore scale (which is millimeter) to the geologic scale (which is multi-kilometer and is the one in which we are more interested). At the large scale, the porous medium is homogenized and looked at as a continuous medium. It can still be heterogeneous, with physical properties such as permeability varying over several orders of magnitude. For the simulation of CO2 sequestration, the situation is more complex. There are three phases, exchange of chemical species between the phases in addition to chemical reactions, and geo-mechanics may play an important role. In this situation, the physical complexity of the problem is about the same as for petroleum reservoir simulation, where similar physical processes take place. Similarly, modeling what happens in and around a nuclear repository is a very complex task because of the high heterogeneities, the coupling between several complex physical phenomena, and the occurrence of multiphase flow (air and water). Chemistry must be taken into account to assess properly the transport of solutes. Phase exchange occurs when gasses produced from corrosion are taken into account. Also, for some simulations, the geo-mechanics of the rock may play an important role. In most situations dealing with subsurface models, one has to take into account that the measurements are difficult to obtain, so that very little actual data is available. This field is thus a rich source for inverse problems, and this is one of the areas where optimization is needed. The previous examples illustrate the diversity and complexity of environmental problems, which an ideal general mathematical model must address. In summary, the mathematical model we present can simulate environmental problems ranging from simple pollutant-diffusion assessments to complex multidimensional, multiphase applications, for the purposes of: • Prediction/evaluation of environmental dispersions/ spills • Evaluation of environmental changes • Evaluation of thermal discharges • Temperature modeling • Evaluation of atmospheric pollutants transport • Evaluation of water and soil contaminant transport • Determination of maximum ground-level concentrations • Computation of effective stack height © 2012 Curtin University of Technology and John Wiley & Sons, Ltd.
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• Prediction of particulate distributions outdoors and indoors • Prediction of deposition of particulate matter from stacks, and so on • Estimation of industrial emissions impact • Modeling of emissions from landfills • Modeling of atmospheric impacts from mobile sources. • Computation of the release of volatile organic compounds from petroleum storage tanks and from wastewater treatment plants • Modeling contaminants fate in aquatic ecosystems • Modeling CO2 sequestration in geological formation • Performing risk assessments in cases, for example, of fires in buildings/forests/public transport, and so on, pool fires or/and explosions in industrial storage tanks, and so on. It is obvious from the previous arguments that the process of modeling as described previously can (and must) serve also a consensus-building function. It can help build mutual understanding, solicit and test input from a broad range of stakeholder groups, and sustain dialog between members of these groups by presenting objective scientific findings rather than individual beliefs that may be biased and cause disagreements. Dynamic models and decision making Most decision problems are no longer well-structured problems that are easily solved by intuition or experience with the support of relatively simple calculations. Even the same kind of problem that was once easy to define and solve has now become much more complex, because of globalization and of a much greater awareness of its linkages with various environmental, social, and political issues. Modern decision makers typically want to integrate knowledge quickly and reliably from various areas of science and practice, and need tools to assist them to answer the most important question, which a conscious being can ask, namely ‘What will happen if. . ..?’. An important class of such tools comprises models that, at least, must smooth out the great heterogeneity of knowledge representations in various scientific and technical disciplines. A critical element of model-based decision support is a mathematical model, which represents data and relations that are too complex to be adequately analyzed based solely on experience and/or intuition of a decision maker.[9] Properly analyzed models will help the decision maker extend his/her knowledge and intuition. However, models can also mislead users by providing wrong or inadequate information. Such misinformation results not only from flaws or mistakes in model specification and/or implementation, the data used, or unreliable elements of software, but also by misunderstandings between model users and developers about underlying assumptions, limitations of applied methods Asia-Pac. J. Chem. Eng. 2012; 7: 182–205 DOI: 10.1002/apj
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of model analysis, and differences in interpretation of results, and so on. Therefore, the quality of the entire modeling cycle determines to a large extent the quality of the decision-making process for any complex decision problem. Decision making concerning environmental problems is particularly challenging. The environment can be likened to a giant chemical reactor. Gasses and particles are emitted into the atmosphere by industrial and other man-made processes, as well as by a variety of natural processes such as photosynthesis, volcanism, wildfires, and decay processes. These gasses and particles can undergo chemical reactions, and they, or their reaction products, can be transported by the wind, mixed by atmospheric turbulence, and absorbed by water droplets. Ultimately, they either remain in the atmosphere indefinitely or reach the earth’s surface. For example, the hazes of polluted atmosphere consist of submicron aerosols of inorganic and organic compounds, which are formed by chemical reaction, homogeneous nucleation, or condensation of gasses. Models of atmospheric phenomena are similar to those of combustion, and involve the coupling of exceedingly complex chemistry and physics with three-dimensional hydrodynamics. The distribution and transport of chemicals introduced into groundwater also involve a coupling of chemical reactions and transports through porous solid media. The development of groundwater models is critical to understanding the effects of land disposal of toxic waste. Combustion is one of the oldest and most basic chemical processes. Its accurate mathematical modeling can help avoid explosions and catastrophic fires, promote more efficient fuel use, minimize pollutant formation, and design systems for the incineration of toxic materials. For example, modeling the initiation and propagation of fires, explosions, and detonations requires the ability to model combustion phenomena. Models of the internal combustion engine can shed light on the influence of combustion chamber shape or valve and spark plug placement on pollutant generation. Mathematical models of combustion must incorporate intricate fluid mechanics coupled with the kinetics of many chemical reactions among a multitude of compounds and free radicals. They must also consider that those reactions are taking place in turbulent flows. The previous arguments demonstrate the need for dynamic models. To model and simulate major environmental problems, scientists need more than the most powerful computers. They also need software algorithms that express the fundamental physics involved in ways appropriate to modern processing, such as parallel processing. Software of this type exists.[10–14] It should be emphasized that dynamic modeling is distinct from statistical modeling by building into the representation of a phenomenon those aspects of a system that we know actually exist—such as the physical laws of material, momentum, and energy conservation.[14] © 2012 Curtin University of Technology and John Wiley & Sons, Ltd.
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Dynamic modeling, therefore, starts with a considerable advantage over the purely statistical or empirical modeling schemes. It does not rely on historic or crosssectional data to reveal those relationships. This advantage also allows dynamic models to be used in more related applications than empirical models—dynamic models are more transferable to new applications because the fundamental concepts on which they are built are present in many systems. Computers have come to play a large role in developing dynamic models for decision-making support in complex systems. Their ability to solve numerically for complex nonlinear relationships among system components and to deal with time and space lags and nonequilibrium conditions make obsolete the use of linear functional relationships or the restriction of the analysis to equilibrium points.[15] The analysis that follows emphasizes the treatment of multiphase phenomena, as the single-phase ones are just a special case of the former, reached when all of the phases but one disappear, irrespective of which one remains. In general, there are two distinct approaches to the numerical simulation of multiphase fluid dynamics. One is based on the Lagrangian treatment, where the more dense of the phases is followed throughout the flow field on Lagrangian trajectories.[16,17] Although it has the advantage that dense-phase boundaries are not diffused and it is valuable for flows in which the less dense phase is in large excess, it cannot be recommended for the majority of industrial two-phase flows. The second is based on the Eulerian approach, predicting dispersed two-phase flows, on the assumption that there are two space-sharing interspersed continua present. Definition of a volume concentration gives the fraction of a control volume, which is occupied by the phase.[18] The transport of heat, mass, and momentum is weighted by this volume fraction. Although numerical dissipation of phase discontinuities may arise in strongly nonhomogeneous cases, this approach appears more generally and practically useful, particularly when some mechanisms to prevent the above-mentioned smearing of discontinuities are devised.[19] It is therefore the approach followed in this work. The finite-domain techniques are among the most successful, and their recommended practices are summarized in this work. More details may be found in the references cited.
THE MATHEMATICAL MODELING The differential equations A convenient assumption, on the basis of the concepts of time and space-averaging, is that more than one phase can exist at the same location at the same time. Then, any small volume of the domain of interest can be imagined as containing, at any particular time, a Asia-Pac. J. Chem. Eng. 2012; 7: 182–205 DOI: 10.1002/apj
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volume fraction ri of the ith phase. As a consequence, if there are n phases in total, n X
ri ¼ 1
(1)
i¼1
When flow properties are to be computed over finite time intervals, a suitable averaging over space and time must be carried out. Following the previous notion that treats each phase as a continuum in the domain of interest, we can derive the following balance equations: • Conservation of phase mass @ V i ¼ m_ i ðri ri Þ þ div ri ri ! (2) @t ! where ri is the density, V i is the velocity vector, ri is the volume fraction of phase i, m_ i is the mass per unit volume entering the phase from all sources per unit time, and div is the divergence operator (i.e., the limit of the outflow divided by the volume as the volume tends to zero). Summation of (2) over all phases leads to the ‘over-all’ mass-conservation equation: n X i¼1
@ ! ðr ri Þ þ div ri ri V i ¼ 0 @t i
(3)
which has of course a zero on the right-hand side. • Conservation of phase momentum @ ! ðri ri uik Þ þ div ri ri V i uik @t ! ¼ ri kgrad p þ Bik þ Fik þ lik
(4)
where uik is the velocity component in the direction k of phase i, p is pressure assumed to be shared between the ! phases, k is a unit vector in the k-direction, Bik is the k-direction body force per unit volume of phase I, Fik is the friction force exerted on phase i by viscous action within that phase; and lik is the momentum transfer to phase i from interactions with other phases occupying the same space. • Conservation of phase energy
@ ! ðri ðri hi pÞÞ þ div ri ri V i hi @t ¼ ri Qi þ Hi þ Ji
(5)
where hi is stagnation enthalpy of phase i per unit mass (i.e., the thermodynamic enthalpy plus the kinetic energy of the phase plus any potential energy), Qi is © 2012 Curtin University of Technology and John Wiley & Sons, Ltd.
the heat transfer to phase i per unit volume, Hi is heat transfer within the same phase, for example, by thermal conduction and viscous action, and Ji is the effect of interactions with other phases. • Conservation of species-in-phase mass @ V i mil ðri ri mil Þ þ div ri ri ! @t ¼ divðri Γil grad mil Þ þ ri Ril þ m_ i Mil
(6)
where mil is the mass fraction of chemical species l present in phase i, Ril is the rate of production of species l, by chemical reaction, per unit volume of phase i present, Γil is the exchange coefficient of species l (diffusion), and Mil is the l-fraction of the mass crossing the phase boundary, that is, it represents the effect of interactions with other phases. All of the previous equations can be expressed in a single form as follows: ! @ ðri ri ’i Þ þ div ri ri V i ’i @t ¼ div ri Γ’i grad ’i þ m_ i Фi þ ri s’i total source of ’i
(7)
where ’i is any extensive fluid property; the first term on the right-hand side expresses the whole of that part of the source term, which can be so expressed, with Γ’i being the exchange coefficient for ’i, s’i is the source/sink term for ’i, per unit phase volume; and m_ i Фi represents the contribution to the total source of any interactions between the phases, such as any phase change (with Фi being the value of ’i in the material crossing the phase boundary, during phase change). Distribution of effects between m_ i Фi and s’i is sometimes arbitrary, reflecting modeling convenience. For single-phase situations, the previous equations are valid by setting the r’s to unity. For turbulent flow, averaging over times which are large compared with the fluctuation time leads to similar equations for time-averaged values of ’i with fluctuating-velocity effects usually represented by enlargement of Γ’i. More details on the previous concepts and equations may be found in literature.[1–3,10,11,14,20,21] There is much experience to prove that the mathematical model, as formulated previously, does have unique solutions to which the solutions of the relevant finite-domain equations do converge as the time step and grid spacing become small. The present paper provides further confirmation of this aspect of the model. Auxiliary relations and boundary conditions The previous set of differential equations is solved normally together with auxiliary relations that express physical laws of various kinds. For example, equations of state for ri, temperature-enthalpy relations, Asia-Pac. J. Chem. Eng. 2012; 7: 182–205 DOI: 10.1002/apj
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THE NUMERICAL MODEL
• +(sum for all six faces) of [(face area) (area-andtime average for the faceof the outward-normal ! component of ri ri V i ’i ] • =(sum for all six faces) of [(face area) (area-andtime average for the face of the • outward-normal component of riΓ’igrad ’i)] • +(the volume average of m_ i Фi þ ri s’i )
‘Finite-domain’ equations are formulated by integrating the partial differential equations over control volumes of
The quantities appearing previously are expressed by way of interpolation rules,[10] resulting in an equation for ’iP, which can be written in the form
heat-transfer relations, inter-phase friction correlations, and so on. The partial differential equations and the auxiliary relations are also supplemented by appropriate initial and boundary conditions that express the particular situation to be studied.
’iP ¼
aN ’i;N þ aS ’i;S þ aE ’i;E þ aW ’i;W þ aH ’i;H þ aL ’i;L þ aP ’i;P þ Si;P bN þ bS þ bE þ bW þ bH þ bL þ bP þ S′i ;P
finite size that taken together fully cover the entire domain of interest (Fig. 1). These control volumes (or ‘cells’) are topologically Cartesian, they can be either strictly Cartesian or polar cylindrical, or generally curvilinear (orthogonal or nonorthogonal), but will always have six sides and eight corners in the three-dimensional case. For the variables ri, hi, mil, and p, these control volumes are the ones centered on grid nodes such as P (in Fig. 1) and having faces that bisect the lines joining P to N, S, E, W, and so on. For the velocity components ui, vi, and wi, the control volumes are displaced in the directions x, y, or z, respectively, by distances that place the relevant velocity locations in their centers. This practice is conventional.[1,10,22] The integration of Eqn (7) over a control volume and over a finite time step gives the following: (Cell volume divided by time step) • [(volume average of ri ri’i for end of time step) • –(volume average of ri ri’i for start of time step)]
(8)
where P denotes conditions at point P at the earlier time. The coefficients a consist of an inflow-convection contribution (according to term ririVi’i) and a diffusion contribution (according to term riΓ’igrad’i). For example, Γi;n ri;n aN ¼ 〚 ri;n ri;N vi;n 〛 þ (9) An yN yP where 〚A〛 equals A when the latter is positive and zero otherwise, An is the cell north area, (yN yP) is the internodal distance, and Γi, nri, n is the harmonic mean or Γi, Nri, N and Γi, Pri, P. The b’s consist of an outflow contribution and a diffusion contribution, thus, Γi;n ri;n An (10) bN ¼ 〚ri;p ri;p vi;n 〛 þ yN yP The case for aP and bP is different in detail because the ‘velocity through time’ is always positive and there is no ‘diffusion in time’. The formulas are as follows: ri;P ri;P Vp dt ri;P ri;P Vp bP ¼ dt
aP ¼
Figure 1. The control volume (topologically Cartesian). © 2012 Curtin University of Technology and John Wiley & Sons, Ltd.
(11) (12)
where Vp is the volume of the control cell and dt is the time interval. Finally, Si, P and S′i, P are two parts of the linearized source term. There are innumerable details concerning for example the finite-domain equations for ri, the Asia-Pac. J. Chem. Eng. 2012; 7: 182–205 DOI: 10.1002/apj
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corrections required for cases of layered flows (rather than the previous dispersed flows), and so on. They cannot be presented here, and the interested reader is referred to the original work by Pantankar and Spalding.[1] However, the ‘pressure-correction’ procedure will be described briefly as it is a crucial part of the solution procedure. The pressure is not known a priori, it must be computed by a guess-and-correct procedure. Let the continuity equations for a two-phase case be added after division by weighting factors w and W as follows: " # X X (13) q ðrn qÞ m_ =w rP o
"
þ RP
i
X o
Q
X
# _ =W ¼ 0 ðRn QÞ M
i
where subscripts P, n, o, and i denote, respectively, the ‘in-cell’ value, the ‘neighbor-cell’ value, outflow, and inflow; r and R are the volume fractions of phases 1 and 2; q and Q are the fluxes of phases 1 and 2 (e.g., products of density, normal velocity, and cell-face area or products of density and cell volume divided by time _ represent the mass-transfer rates into step); and m_ and M their respective phases from the other, for a given cell _ ¼ 0 . From Eqn (13), the pressure can be comm_ þ M puted by calculating the changes of all terms, except the volume fractions caused by pressure changes. Thus, a pressure-correction equation can be derived from: " # X X ′ ′ ′ rP q rn q m_ =w o i " # X X ′ ′ ′ _ =W þ RP Q Rn Q M o i " # (14) X X q ðrn q Þ m_ =w ¼ rP o i " # X X _ =W þ RP Q ðRn Q Þ M o
i ′
_ can all be expressed linearly in where q′, Q′, m_ ′, and M terms of the pressure corrections, P′ p and P′ n.[20] The result is a system of simultaneous equations for the P′s. The weighting factors w and W are taken to be the corresponding densities, implying that it is the ‘volumetric joint continuity equation’, which is corrected at each state of pressure-field adjustment. Then, the individual ‘phasemass continuity equations’ are satisfied by the next stage of adjustment of r’s and R’s.
SOLUTION PROCEDURE The equations are both nonlinear and strongly coupled, necessitating an iterative solution procedure. There are © 2012 Curtin University of Technology and John Wiley & Sons, Ltd.
DYNAMIC COMPUTER MODELLING OF ENVIRONMENTAL SYSTEMS
several such procedures for single-phase flows. An early publication described the Marker-And-Cell (MAC) procedure,[23] which has subsequently been refined.[24] The ‘SIMPLEu.c’ procedure of [1] combined some of the features of MAC with new ones and was also subsequently refined into ‘SIMPLER’[25] and ‘SIMPLEST’.[11,26] An extension of ‘SIMPLE’ (and ‘SIMPLEST’) to multiphase flows is IPSA (InterPhase Slip Algorithm),[26] a version of which is used for the present work. An outline of the algorithm, which has features in common with a procedure emanating from Los Alamos,[27] is explained below: • Solve the finite-domain equations for R at all grid points by way of the Jacobi point-by-point method. • Obtain the r values from Eqn (1) and ensure that neither exceeds the range from zero to unity (when the solution is far from convergence). • Guess the pressure distribution. • Using this pressure distribution, solve for the velocities of the two phases individually, by application of the tridiagonal matrix algorithm (TDMA) or the Stones algorithm.[28] • Insert the r, R and velocities into the joint continuity equation and compute the consequent errors. Thus, generate the ‘source term’ in the pressure-correction equation. • Solve for the pressure corrections by TDMA or Stones algorithm and apply corrections to pressures and to velocities (proportional to pressure-correction differences). • Return to step (a) and repeat the computational cycle until the continuity errors computed at stage (e) are ‘sufficiently’ small. • Proceed to the next time interval, and start the cycle of operations again from step (a). The previous procedure was found to converge satisfactorily in most circumstances, and it has been embodied in a computer code, PHOENICS,[10,11] which was used for the results presented in the following case studies. Parallel computing is currently the only way to obtain the performance level required for simulating most of realistic problems. As an example, a (still simplified) model for CO2 dissolution, including only the minimal number of chemical species can have 12 million grid cells and 12 chemical species. Many applications, as for example, porous media simulations, require unstructured grids to be able to represent correctly the complex subsurface geometries, and they require implicit methods to solve efficiently the diffusive parts of the models. Parallelizing implicit methods, especially on unstructured meshes, is still considered a difficult problem.[4,13] Indeed, most of the success stories in large-scale parallel computing are related to explicit methods or to implicit methods on structured grids. There have been several quite successful attempts at parallelizing implicit, unstructured grid models. However, building a parallel platform for solving a range of Asia-Pac. J. Chem. Eng. 2012; 7: 182–205 DOI: 10.1002/apj
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different models (in our context, this could range from simple one-phase flow to multiphase, multi-component reactive transport in a geometrically complex medium) on today’s heterogeneous platforms is still a challenge. All the same, many useful environmental applications benefit today from parallel computing.
node
zone boundary
zone
In+ 1 In
L
MODEL OF TURBULENCE AND RADIATION For the problems under consideration, turbulence and radiation are important and must be modeled appropriately. Models that are currently considered as adequate are presented below. The interested reader is referred to the appropriate literature for further details and different models. The radiation model The basis of all methods for the solution of radiation problems is the radiative transfer equation: srI r; s ¼ k r I r; s þ Q r; s
(15)
which describes the radiative intensity field, I, within the domain, as a function of location vector (r) and direction vector (s), Q represents the total attenuation of the radiative intensity due to the gas emission and to the inscattered energy from other directions to the direction of propagation, and k is the total extinction coefficient. The discrete transfer model The discrete transfer model discretizes the radiative transfer equation along rays. The path along a ray is discretized by using the sections formed from breaking the path at zone boundaries. Assuming that the physical properties remain constant inside a zone, Eqn (15) can be integrated from zone entry to zone exit (Fig. 2) to yield: Inþ1 ¼ In e
tn
1 etn þ Ln Qn ; tn ¼ kLn tn
(16)
where Ln is the path length in the nth zone, In and In+1 are the intensities at zone entry and zone exit, respectively, and k a Qn r; s ¼ sTn4 þ ks Jn r p
(17)
coeffiwhere ka, ks, are the absorption and scattering R 1 In ðr; sÞdΩ is cients for a gray medium, Jn ðrÞ ¼ 4p s is the mean intensity of the in-scattered radiation, the Stefan-Boltzmann constant, and dΩ is the element of solid angle containing s. The rays are chosen by fixing nodes to all the physical surfaces, dividing up the interior hemisphere into © 2012 Curtin University of Technology and John Wiley & Sons, Ltd.
fire p h y s ic a l s u r fa c e
Figure 2. Subdivision of the solution domain
into zones.
elements of equal solid angle and projecting one ray into each solid angle. For gray surfaces, integration of Eqn (15) yields the required boundary conditions: e 1e W W sΤ4 r þ R r T r; s ¼ p p R where eW is the wall emissivity, Rðr Þ ¼
(18)
ðsnÞI ðr; sÞdΩ is the radiation flux on a surface, and n is the pointing unit vector normal to the surface atr [29,30] inward . s n
