Validation study of a turbulence radiation interaction model: Weak, intermediate and strong TRI in jet flames

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Abstract

A TRI methodology developed previously for hydrogen jet flames is applied to carbon monoxide and methane jet flames. The paper demonstrates that the TRI methodology can accurately predict the spectral intensity distribution and heat flux distribution across a range of jet flames. A β probability density function (PDF) together with Reynolds averaging of the instantaneous properties along rays traversing the flames is used to prescribe the instantaneous incident intensity. Unlike most other studies in this area the model is used to predict the heat flux distribution as well as the spectral intensity. The underlying reason why hydrogen jet flames exhibit strong TRI compared to carbon monoxide and methane is also considered and shown to be a complex phenomena deserving of more research. The paper demonstrates that for the data available the heat flux distribution tends to be more sensitive to TRI than the spectral intensity distribution. Another interesting finding is that TRI can reduce as well as enhance the spectral intensity.

Introduction

It is well known that for some turbulent combusting systems the turbulent nature of the flow has the potential to significantly modify the radiation field. As discussed further below this is principally due to the nonlinear dependence of the Plank black body distribution on temperature and the fluctuating nature of the instantaneous temperature field. There has been significant research into TRI in turbulent hydrogen jet flames as TRI has a strong influence on the radiation field [1], [2] and there has been much interest in hydrogen as an alternative green fuel to help combat the green house effect due to the increased CO2 concentration in the atmosphere [3].

In this article a previously developed TRI methodology originally applied to hydrogen jet flames is applied to a number of jet flames with different fuels and source conditions. The fuels considered are hydrogen, methane and carbon monoxide. The performance of the model for different fuels is of interest as it is important that a TRI model can accurately predict the radiation field when TRI has a relatively weak influence (carbon monoxide) or moderate influence (methane) on the radiation field as well as when TRI is more significant (hydrogen) [1].

Over the last 3–4 decades there has been much interest in the computational modelling of turbulent flames [1], [2], [3], [4], [5], [6], [7], [8], [9]. The motivation for this work has been the development and evaluation of sub-models to predict different aspects of this generic flow, such as turbulence modelling [10], turbulence–chemistry interaction [11], [12], depending on the fuel the modelling of soot may be of interest [13], radiation heat transfer [14], [15] and turbulence–radiation interaction [1], [2], [16], [17], [18], [19]. Commercial drivers for these investigations are diverse; reduction of pollution emissions such as NOx prediction, flaring operations and consequence analysis as part of a safety case being some examples. Model sophistication has increased over the years as understanding has grown from experimental investigations, more efficient numerical techniques are developed and relatively cheap computational resources become available.

The first investigations of TRI were completed in the 1970s. Cox [17], based on a grey analysis of a turbulent homogeneous volume showed that the mean emission takes the form,εσT4where ε denotes the emissivity of the gas, σ is the Stefan–Boltzmann constant and T is the temperature. The over bar indicates the term is a Reynolds average. Due to the nonlinearity in temperature, depending on the magnitude of the fluctuations in temperature and participating species, radiation emission is significantly larger than the approximation to the mean emission based on the mean temperature and emissivity evaluated from the mean participating species concentrations.ε¯σTεσT4This is a serious problem for most flame structure models where the mean flow fields are available for calculating the radiation source term in the conservation of energy transport equation or similar and when calculating the heat flux distributions external to the flame.

The first serious attempt to investigate TRI in jet flames, taking account of the spectral nature of flames was Faeth’s group [1], [2], [18], [19]. In [1], [19], the flame structure is calculated using a parabolic Favre averaged flow model closed using a kεg turbulence model. Turbulent combustion is modelled using a laminar flamelet library, combined with a clipped Gaussian PDF. In all of the simulations presented in [1], [19] the primary interest is the external radiation fields rather than the internal structure of the flame, hence the influence of TRI on the flame structure is not considered and the radiation fields are calculated as a post-process to the flame structure simulation. Radiation heat loss is accounted for by adjusting the temperature flamelet to predict the measured fraction of heat radiated for each flame investigated.

Gore et al. [1] use a stochastic methodology to simulate instantaneous realisations of the spectral intensity distribution for pencils of radiation or rays with a horizontal orientation passing through the axis of the jet. The mean spectral intensity is calculated by averaging the instantaneous realisations. In Faeth’s group’s initial stochastic methodology the non-homogeneous path of a ray is separated into a number of segments that are assumed to be well approximated to be homogeneous and each ray segment is taken to be statistically independent. The instantaneous composition and temperature in each homogeneous ray segment is characterised by the instantaneous mixture fraction via a laminar flamelet library. The instantaneous realisation of the temperature and participating species are then input into Grosshandler’s narrow band model, RADCAL [14], [20] to produce the instantaneous spectral intensity distribution for a given ray orientation and receiver location. For each realisation the instantaneous mixture fraction is evaluated from a clipped Gaussian PDF using a pseudo random number generator. Faeth’s group implementation of the clipped Gaussian is such that,z=f̃-finstf2where z satisfies a Gaussian distribution with zero mean and unit standard deviation. Where the above gives physically unrealistic values of the instantaneous mixture fraction a limiter function is imposed to restrict mixture fraction to the unit interval. The clipped Gaussian PDF can reproduce the intermittency at the boundary of jets, Faeth et al. [18] but the main justification for its use is ease of implementation as they found other PDFs difficult to implement. Indeed the advantage of this approach is analytical rational approximations to the Gaussian distribution of the form,finst=Q(z)are readily available [21]. The potential disadvantages of this approach are the cumulative probability density function (CPDF) is based on Favre averaging rather than Reynolds averaging of the instantaneous fields. Faeth et al. [18], when validating their TRI model using the measured spectral intensity distribution state the type of averaging and the shape of the PDF used are unimportant. Previous work by the authors has shown that the predicted heat flux distribution is sensitive to the type of averaging employed, with Reynolds averaging tending to give more accurate results than Favre averaging and the heat flux distribution is sensitive to the shape of the PDF [22].

In spite of these weaknesses in nearly all stochastic simulations of TRI the clipped Gaussian PDF is implemented in this way. This is important as a great deal of research has been conducted into improving the stochastic methodology incorporating spatial [18], [23], [24] and temporal correlations [19], and cross-correlations [25] building on the clipped Gaussian PDF as a foundation to the stochastic methodology, see the papers cited. This should be contrasted with a β PDF being commonly used for modelling the flame structure [6], [7], [13], [26].

Faeth and co-workers [1], [2], [18], [27], [28] investigated TRI in a number of different jet flames with different fuels, finding significant TRI in hydrogen jet flames, (up to 100%) [1], [19] with less significant TRI effects on heat transfer for methane jet flames (20–30%) [2] and carbon monoxide jet flames (10%) [28]. Faeth’s group also studied ethylene jet flames [27] concluding TRI was significant for the continuous radiation part of the spectrum associated with the soot in the flame. It should be noted that this conclusion is difficult to justify as in [27] soot is modelled using a laminar flamelet which many consider to be an oversimplification and further the agreement between the measured and predicted spectral intensity distributions are poor for one of two jet flames considered in [27]. As we shall discuss later [1] proposed that TRI significance is strongly influenced by the shape of the flamelets in the laminar flamelet library. One conclusion of the present study is that although the shape of the flamelet is undoubtedly important the situation is more complicated than this. In all of Faeth’s groups investigations TRI increased the heat transfer from the flame by a factor of 1.1–4.2 compared to a mean analysis where TRI is not included [1]. This conclusion is based on the evaluation of the spectral intensity distribution alone.

Gore et al. [1], measured the spectral intensity distribution and the external radiation heat flux distribution but only modelled TRI when calculating the spectral intensity distribution. Predictions of spectral intensity distribution neglecting TRI were compared with measured data to confirm the importance of TRI when modelling turbulent hydrogen jet flames. Kounalakis et al. [19] extended the stochastic methodology by modelling the temporal correlation,Rt(Δt)=f(t)f(t+Δt)(f)2Including the temporal correlation is required if the temporal properties of the radiation fluctuations are of interest such as in fire detection based on flame flicker, but for evaluating the mean radiation fields is not generally required [19]. Faeth et al. [18] extended the original stochastic methodology to include spatial correlation of the instantaneous mixture fraction field,Rs(ΔS)=f(s)f(s+Δs)(f)2The spatial and temporal correlations are modelled as exponential functions of the local dissipation length scale and a temporal scale estimated using Taylor’s hypothesis [18]. Faeth et al. [18], reported no significant change in the predicted mean spectral intensity distribution with or without including the spatial and temporal correlation, (1), (2). More recently Zheng et al. [29], [30] reported excellent results using a stochastic methodology including temporal and spatial correlation of the instantaneous mixture fraction field using a technique based on tomography to estimate the integral length and timescales. However this approach requires the measured radiation fields to be available so cannot be considered a predictive model.

Chan and Pan [23], investigated the importance of including pre-spatial and temporal correlations in a radiation analysis of a CO–H2 jet flame, using a first order causal stochastic model. They applied the causal model to the spectral intensity at specific wavelengths for horizontal rays passing through the axis of the jet for a number of downstream locations. Instantaneous values of the mixture fraction were evaluated using a clipped Gaussian PDF. Chan and Pan compared their model predictions with Kounalakis et al. [31] original stochastic methodology where the spatial and temporal correlations are assumed to be negligible and found that the average relative difference between model prediction and measured spectral intensity is worse using the more sophisticated causal stochastic model. Chan et al. [25], extended the original causal stochastic model to include a spatial- temporal cross-correlation and applied it to the same jet flames. Comparing the predicted spectral intensity with the measured value, Chan et al. [25] found that including the cross correlation term improved the predictive capability of the causal model but the results are still marginally inferior to Faeth’s group’s original simpler stochastic model. Chan and Pan [24] proposed a final extension to their original approach to produce a general semicausal stochastic model that include both temporal and spatial correlations. The improvement being that the random field includes pre and post spatial correlation. This leads to a linear system to be solved for each realisation. This is a potentially large computational overhead but it can be minimised as the coefficients in the linear system are not time dependent so the inverse matrix or LU decomposition of the matrix for each ray can be calculated once only. This causal model does exhibit better agreement with the measured spectral intensity than Faeth’s group’s original simpler stochastic model for the jet flames considered. This is in contradiction to Faeth et al. [18] as they included both pre and post spatial correlations and found little improvement in accuracy. This is an interesting area of research and looks promising but further validation is required and the requirement for solving a linear system for every realisation makes it unattractive for calculating radiation heat flux distributions or coupled simulations. It is also restricted to a clipped Gaussian PDF which ultimately might not be the most appropriate PDF to use.

A further consideration is the simulation of TRI is a computationally intensive process. It is therefore important to include sufficient sophistication of the model to provide the accuracy required and no more and implement efficient numerical methods to keep computational run-times as short as possible. A further possibility is to exploit parallel computer technology and algorithms [32], although this will not be considered further in this article.

Recently Onokpe and Cumber [22], developed a TRI methodology that could be implemented with any PDF and either Reynolds or Favre averaging. An interesting finding in [22] is that although the spectral intensity distribution is relatively insensitive to the shape of the PDF, the heat flux distribution is sensitive. In [22], an argument is proposed as to why the sensitivity of the predicted heat flux distribution to the PDF occurs. Finally in [33] the TRI methodology presented in [22] incorporated a number of new techniques to improve the efficiency of the methodology. The incident heat flux is evaluated using an adaptive Monte-Carlo algorithm called hot sampling [34], [35] and the use of Sobol sequences to accelerate the evaluation of the mean incident intensity from a smaller sample of the instantaneous incident intensity than is required with the conventional Monte-Carlo method. These changes to the algorithm typically reduced the computational run-time by a factor of 10 [33].

From this short review of the literature it is clear that a significant body of research has been produced relating to the application of stochastic simulation techniques to predicting the external radiation fields surrounding jet flames.

The present paper can be considered a natural extension of [22] where the developed TRI methodology is applied to an extensive range of laboratory scale jet flames with different fuels and thereby different levels of TRI. It is important to demonstrate that such a model is accurate for a wide range of jet flames without the need for recalibration. Having validated the model in this way it is possible to explore the physio-chemical basis for weak through to strong TRI in jet flames.

Section snippets

Mathematical model

The basis of the flow structure model has been presented several times before [13], [26], [36], [37], so only a brief overview will be given here. The main focus of this paper is the TRI methodology so this will be presented in detail.

Experimental jet flames

In previous investigations a number of experimental jet flames have been used to successfully validate the flame structure model described above for hydrogen jet flames [36], [37] and methane jet flames [26], [34]. The validation studies, demonstrate that the flow fields required to predict the radiation fields are accurately reproduced.

Numerical issues

The model described in Section 2 has a number of parameters that must be assigned values such that the numerical or statistical error is sufficiently small, such that it can be neglected. Once this is achieved the predicted spectral intensity distribution and the predicted heat flux distribution can be taken as an indication of the models predictive capability. However this is balanced by the need to produce predictions in a reasonable time and using readily available computational resources.

In

TRI model validation

Having assigned values to all of the numerical parameters such that model predictions are essentially free from numerical or statistical error we can evaluate the models predictive capability and assess the impact of the numerical algorithms implemented to evaluate the spectral intensity distribution and incident heat flux at a receiver.

In all eight jet flames have been used in the validation study, two carbon monoxide jet flames [28], three methane jet flames [2], [46], two hydrogen jet flames

TRI in jet flames for different fuels

As stated above, Faeth’s research group used the predicted spectral intensity distribution with and without TRI to assess the significance of TRI in a number of jet flames for different fuels [1], [2], [28]. From inspection of the predicted spectral intensity distributions in Fig. 4, Fig. 5, Fig. 6 it is clear that the influence of the fuels on TRI is similar to Faeth’s research group’s conclusions in that there is little TRI in CO jet flames with more for the CH4 and H2 jet flames. The

Conclusion

In this paper a radiation model for jet flames including TRI has been presented. The TRI model is based on a stochastic Monte Carlo methodology where the instantaneous realisations of the incident spectral intensity distribution are calculated. The model differs in a number of key aspects to previously published models, a β CPDF for the instantaneous mixture fraction is used as well as Reynolds averaging of the instantaneous temperature and participating chemical species concentrations used as

Conflict of interest

None declared.

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