A few weeks back I posted a blog post regarding S2S-radiation (surface to surface) [S2S radiation in Simcenter STAR-CCM+ – VOLUPE Software]. Some tips where given, especially considering patches for view factor calculations. This week we will discuss radiation when the volume takes part, meaning that your fluid has radiative properties that needs to be considered. There are of course many ways to this. One theory that is commonly used, not only in CFD, but in general is the assumption that your gases consist of “Gray” gases. This allows for several simplifications and assumptions. What can then be employed in Simcenter STAR-CCM+ is WSGG (Weighted Sum of Gray Gases). Information about the user input OPL (Optimal Path Length) and what that is needs, I feel, require further explanation. The information provided on these topics in the Simcenter STAR-CCM+ documentation could be considered ambiguous, and the articles on Support center are scattered with somewhat conflicting recommendations. Note that this model (WSGG) is a sub model of the DOM radiation model (Discrete Ordinates Method).

# Gray gases and Kirchhoff’s law

What we need to understand first is what a Gray gas is. In Gray gas theory radiation properties (Absorbtivity, emissivity, reflectivity) of a gas are wavelength independent. In the same way the radiation properties of surrounding surface are also wavelength independent. In Simcenter STAR-CCM+, Kirchhoff’s law of thermal radiation is active by default. The law states that emissivity and absorptivity is equal. This law is generally valid when a system is in thermal equilibrium, or at least close to thermal equilibrium. Meaning that the surfaces in a system are all the same, or near the same temperature. This can be deactivated in Simcenter STAR-CCM+, but that applicability will not be further discussed in this article.

It is important not to confuse thermal equilibrium with steady state conditions. Those are not necessarily the same. Take the earth and the sun as an example, the steady state conditions (more or less) are experienced at all times, while at the same time the system is not in thermodynamical equilibrium. The average temperature of earth is around 14°C, while the sun surface is slightly higher at above 5500 °C. The steady state conditions are when time derivatives are zero, while equilibrium refers to equality of temperatures.

The assumption of Gray bodies, Gray surface and Gray gases, where absorptivity is equal to emissivity and radiative properties are wavelength independent, allows for simplifications that are useful from an engineering point of view. Note though, that properties are temperature dependant, as will be evident from the section on Hottel charts.

# WSGG

Now that we understand the definition of grey gases, we can go on to describe the theory of Weighted Sum of Grey Gases Model. The WSGG model is an accurate technique for modelling the radiative behaviour of combustion gases. The theory of WSGG is generally considered an accurate way of modelling a media containing H2O and CO2, and sometimes the addition of soot. In Simcenter STAR-CCM+ the WSGG model is limited to the inclusion of specifically H2O and CO2. In Simcenter STAR-CCM+ the input you are expected to provide when selecting the WSGG-model is called OPL (Optical Path Length).

OPL is the term which is used as the only input when deciding the total absorptivity of your system from a user point of view. But what is it and what does it describe? It is referred to in the documentation and there are articles on the knowledge base that is touching up on the subject. However, after reading these articles at least I felt that a summary is necessary. Both on its definition and the use of different geometrical relationships that are presented for the term.

# OPL

First know that OPL in Simcenter STAR-CCM+ is strictly user input and is different for different systems. I hope this discussion regarding its definition and formulas to express it, can help you in determining what value to use for your specific system.

The OPL is an extremely important input because it is used several times in your CFD simulation. It is used first as an input to interpolate the total emissivity for your mixture of H2O and CO2 in a typical combustion case. The interpolation can be described in a Hottel chart, but more on that in the section “Hottel emissivity charts and correction”. It is then also used to evaluate the absorption coefficient used in the RTE (Radiative Transfer Equation).

### Definition

Optical path length (OPL) is the same thing as Mean Beam Length (MBL). Hottel introduced the MBL/OPL as; *“a length scale to account for the effect of geometry in the evaluation of radiative heat transfer between an isothermal gas volume and its boundary”*. In that sense it is strictly a geometrical property if, and only if, the gas within that geometry is isothermal. Because if you have temperature variations your radiative properties will vary locally and the strict interpretation of OPL/MBL as a geometric property will be incorrect. Another way of describing the OPL/MBL is saying that is the distance over which the radiative properties vary negligibly. This could in theory also mean that if you have widely varying concentrations of CO2 and H2O, even if the system is isothermal, the radiative properties calculated from OPL is incorrect, because the OPL is interpreted incorrectly.

Physically the OPL/MBL can be described as; “the required radius of an equivalent hemisphere of a medium such that the flux received by the center of its base is equal to the average flux radiated to the area of interest by the actual volume of the medium.” This again implies homogenous properties of the gas.

### Estimated values

If you turn to the documentation of Simcenter STAR-CCM+ it is stated that for an arbitrary geometry the value of OPL can be approximated as;

Where the volume is the total volume of the gas, and the surface area is the total surface area of the gas. This is considered on many occasions to be a “good enough” estimate of the OPL, and hopefully it would lead you to a good solution in your simulation. We could leave it at that and simply say use that expression. But what I want to do here is discuss other formulations of OPL.

As stated earlier, OPL is a user input, and my hope is that more information regarding this topic of OPL will make it easier to define a more accurate value in your simulations. You are generally referred to literature, empirical data, and quality guesswork when it comes to deciding your OPL. There are some articles on the knowledge base that discuss OPL, and I will summarize them and adding some more notes from literature.

One article on Support center that is called “When using the WSGG model, how do I choose the Optical Path Length (OPL)?”, states that the above expression is not a good method for combustion problems in general. It instead proposes a different equation:

The volume here is the volume of your combustion zone, the volume in which your temperature is higher than a threshold value, signifying that combustion is taking place, a suggested value is 1000K. Using this definition for OPL will require an iterative procedure, as the volume of your combustion zone will depend on your input of OPL. If indeed your entire geometry is falling under the definition of combustion zone, your OPL has strict geometric coupling.

Below is a table of further definitions from literature, the references trace back to Hottel himself, and is mostly dealing with geometrical relationships between OPL and relatively basic shapes.

Shape | OPL |

Sphere | 2/3 x diameter |

Infinite cylinder | 1 x diameter |

Space between infinite parallel planes | 1.8 x diameter |

Cube | 2/3 x side |

The difference in formulations comes from looking at your medium as optically thick or optically thin. Let us try to understand what that means also.

### Optically thick or optically thin medium?

An optically thick medium is one for which the mean free path of a photon is low. The photon will not be able to travel far before interacting with the medium. On the contrary, an optically thin medium will allow for longer travel of the photon before interacting with the media. Optical depth or optical thickness is not a distance per se but is in fact monotonically dependant of OPL. Consequently, your notion to optical depth will impact your selection of OPL.

In terms of visibility, a medium is optically thick if the average photon cannot pass through the medium without being absorbed. Using water as an example, you can see the bottom of a pool at 2m depth, but you cannot see the bottom of the ocean at 20m depth. The depth here decides if the medium is optically thick or thin.

Considering, for instance, a case where OPL could not be considered do be a definition of strict geometrical properties. Let us use a pool fire as an example. Where the combustion uses only about 1/10^{th} of the total volume. You will in fact only have a combustion zone in the actual fire. You should consider the fire as the volume of interest to decide the OPL for the emissivity that comes from the Hottel charts.

Note that the recommendations regarding optical thickness could be considered separate from those of geometrical properties. In the pool fire example one part of the domain, the fire, could be considered optically thick, at least thicker than the surrounding (9/10^{th} of the total domain). It is again up to the user to decide what is optically thick or thin. There is also a recommendation regarding if your medium is optically thick, then you could use the cell dimensions within your simulation to define OPL. That would indicate only short distances over where the absorption properties are homogenous. While an optically thin medium again is referred to the 3.6 * (Domain Volume/Domain Area) equation.

I hope this is not confusing you as a reader. My desire here is to highlight that there are several factors to consider when calculating OPL. Both geometrical and factors based on your media. You are encouraged to form your own idea of the correct description of OPL of your system.

### Back to OPL

There are suggestions in literature that OPL can be approximated “exactly” for any medium, with arbitrary optical thickness using the formulation:

Where V is the medium Volume and A is the Boundary surface of the medium. The C here is estimated to be in the range of 0.8-1 and is case specific.

As you can see from all the formulations available to calculate OPL, there is not necessarily one that can describe any given system completely. But hopefully, this summary can give you a point in the direction when deciding which value to use for this parameter.

So far, a lot of effort has been spent on selecting an OPL, which incidentally is the only user input for the absorption coefficient using the WSGG model. Let us see what happens once the OPL has been selected. How does the OPL relate to actual radiative properties used in the simulation?

# Hottel emissivity charts and correction

The Hottel charts are diagrams that Hoyt Clarke Hottel (1903-1998) worked up to describe the emissivity of H2O and CO2. The charts can be seen below. Input to the Hottel chat is temperature on the x-axis. And what line you select in the actual chart is decided by what is called the “pressure path length”, in mathematical formulations, simply the product of OPL and partial pressure of the gas. What you read from the y-axis then is the emissivity of the component.

I WSGG model you add together the contributions from your different gases, H2O and CO2. The components emissivity is then a function of partial pressure and OPL leading the total emissivity to:

Where the correction effects are evaluated, from separate graphs, at:

As that formulation account for overlap in wavelength. The C_H2O and C_CO2 are correction factors read for each component read from a graph like the one below (this graph is for CO2). The correction factor is applied when the system total pressure varies from atmospheric pressure. To clarify, there is a constant multiplied with each component’s contribution, to account for pressure differing from atmospheric pressure, and there is a correction term subtracted from the total summation of emissivity that accounts for spectral overlap.

In Simcenter STAR-CCM+ all this happens in the background using interpolation. In your simulation you have the field functions of Temperature and Pressure together with the user input of OPL that calculates the total emissivity and, since Kirchhoff’s law is applied, the total absorptivity.

# Absorption coefficient

Previous description of how the total absorptivity is calculated leaves the question about the use of absorption coefficients? In the documentation of Simcenter STAR-CCM+ it is stated that the relationship between the total gray gas absorption coefficient, k, and total absorptivity, a, is the Bouguer-Lambert law:

And the total absorptivity is approximated with a weighted sum of several gray gases as:

Where N is the total number of gases. After computing total absorptivity, a, from the later of the two equations, the summation! The Bouguer-Lambert law is used to calculate the absorption coefficient, k. This value is then used solving the RTE to calculate heat transfer.

# Summary

This text is supposed to help you consider the different definitions and formulations for mainly OPL, when using the WSGG model in Simcenter STAR-CCM+. Since OPL is user input, this text hopefully helps you in deciding you specific OPL-value. There is a lot of information on OPL to find, some is somewhat conflicting. You are encouraged to do a study of your own and to form your own idea.

I hope this text has been useful and that it can help you in your use of Simcenter STAR-CCM+ or in this case to understand the more general theory of WSGG modelling. As usual, do not hesitate to reach out to support@volupe.com if you have any questions.

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