## One-dimensional transient radiative transfer by lattice Boltzmann method |

Optics Express, Vol. 21, Issue 21, pp. 24532-24549 (2013)

http://dx.doi.org/10.1364/OE.21.024532

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### Abstract

The lattice Boltzmann method (LBM) is extended to solve transient radiative transfer in one-dimensional slab containing scattering media subjected to a collimated short laser irradiation. By using a fully implicit backward differencing scheme to discretize the transient term in the radiative transfer equation, a new type of lattice structure is devised. The accuracy and computational efficiency of this algorithm are examined firstly. Afterwards, effects of the medium properties such as the extinction coefficient, the scattering albedo and the anisotropy factor, and the shapes of laser pulse on time-resolved signals of transmittance and reflectance are investigated. Results of the present method are found to compare very well with the data from the literature. For an oblique incidence, the LBM results in this paper are compared with those by Monte Carlo method generated by ourselves. In addition, transient radiative transfer in a two-Layer inhomogeneous media subjected to a short square pulse irradiation is investigated. At last, the LBM is further extended to study the transient radiative transfer in homogeneous medium with a refractive index discontinuity irradiated by the short pulse laser. Several trends on the time-resolved signals different from those for refractive index of 1 (i.e. refractive-index-matched boundary) are observed and analysed.

© 2013 Optical Society of America

## 1. Introduction

1. A. Majumdar, “Microscale heat conduction in dielectric thin films,” J. Heat Transfer **115**(1), 7–16 (1993). [CrossRef]

2. J. Y. Murthy and S. R. Mathur, “Computation of sub-micron thermal transport using an unstructured finite volume method,” J. Heat Transfer **124**(6), 1176–1181 (2002). [CrossRef]

3. T. Q. Qiu and C. L. Tien, “Short-pulse laser heating on metals,” Int. J. Heat Mass Transfer **35**(3), 719–726 (1992). [CrossRef]

4. F. Liu, K. M. Yoo, and R. R. Alfano, “Ultrafast Laser-Pulse Transmission and Imaging Through Biological Tissues,” Appl. Opt. **32**(4), 554–558 (1993). [CrossRef] [PubMed]

5. M. C. van Gemert and A. J. Welch, “Clinical Use of Laser-Tissue Interactions,” IEEE Eng. Med. Biol. Mag. **8**(4), 10–13 (1989). [CrossRef] [PubMed]

6. K. J. Grant, J. A. Piper, D. J. Ramsay, and K. L. Williams, “Pulsed lasers in particle detection and sizing,” Appl. Opt. **32**(4), 416–417 (1993). [CrossRef] [PubMed]

7. S. Kumar and K. Mitra, “Microscale Aspects of Thermal Radiation and Laser Applications,” Adv. Heat Transfer **33**, 187–294 (1999). [CrossRef]

9. Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. **73**(2-5), 159–168 (2002). [CrossRef]

10. X. D. Lu and P. F. Hsu, “Reverse Monte Carlo method for transient radiative transfer in participating media,” J. Heat Transfer **126**(4), 621–627 (2004). [CrossRef]

11. X. D. Lu and P. F. Hsu, “Reverse Monte Carlo simulations of light pulse propagation in nonhomogeneous media,” J. Quant. Spectrosc. Radiat. Transf. **93**(1-3), 349–367 (2005). [CrossRef]

12. M. Martinelli, A. Gardner, D. Cuccia, C. Hayakawa, J. Spanier, and V. Venugopalan, “Analysis of single Monte Carlo methods for prediction of reflectance from turbid media,” Opt. Express **19**(20), 19627–19642 (2011). [CrossRef] [PubMed]

13. Z. X. Guo and S. Kumar, “Discrete-ordinates solution of short-pulsed laser transport in two-dimensional turbid media,” Appl. Opt. **40**(19), 3156–3163 (2001). [CrossRef] [PubMed]

14. M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf. **73**(2-5), 169–179 (2002). [CrossRef]

16. J. M. Wang and C. Y. Wu, “Transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate,” Int. J. Heat Mass Transfer **53**(19-20), 3799–3806 (2010). [CrossRef]

17. Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” J. Heat Transfer **123**(3), 466–475 (2001). [CrossRef]

18. C. Y. Wu, “Propagation of scattered radiation in a participating planar medium with pulse irradiation,” J. Quant. Spectrosc. Radiat. Transf. **64**(5), 537–548 (2000). [CrossRef]

19. P. F. Hsu, “Effects of multiple scattering and reflective boundary on the transient radiative transfer process,” Int. J. Therm. Sci. **40**(6), 539–549 (2001). [CrossRef]

20. J. C. Chai, “One-dimensional transient radiation heat transfer modeling using a finite-volume method,” Numer. Heat Transf. B **44**(2), 187–208 (2003). [CrossRef]

21. M. Y. Kim, S. Menon, and S. W. Baek, “On the transient radiative transfer in a one-dimensional planar medium subjected to radiative equilibrium,” Int. J. Heat Mass Transfer **53**(25-26), 5682–5691 (2010). [CrossRef]

22. L. M. Ruan, S. G. Wang, H. Qi, and D. L. Wang, “Analysis of the characteristics of time-resolved signals for transient radiative transfer in scattering participating media,” J. Quant. Spectrosc. Radiat. Transf. **111**(16), 2405–2414 (2010). [CrossRef]

22. L. M. Ruan, S. G. Wang, H. Qi, and D. L. Wang, “Analysis of the characteristics of time-resolved signals for transient radiative transfer in scattering participating media,” J. Quant. Spectrosc. Radiat. Transf. **111**(16), 2405–2414 (2010). [CrossRef]

23. S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer **49**(11-12), 1820–1832 (2006). [CrossRef]

24. L. H. Liu and L. J. Liu, “Discontinuous finite element approach for transient radiative transfer equation,” J. Heat Transfer **129**(8), 1069–1074 (2007). [CrossRef]

25. L. H. Liu and P. F. Hsu, “Time shift and superposition method for solving transient radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. **109**(7), 1297–1308 (2008). [CrossRef]

26. X. He, S. Chen, and R. A. Zhang, “Lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh–Taylor instability,” J. Comput. Phys. **152**(2), 642–663 (1999). [CrossRef]

30. S. C. Mishra and H. K. Roy, “Solving transient conduction-radiation problems using the lattice Boltzmann method and the finite volume method,” J. Comput. Phys. **223**(1), 89–107 (2007). [CrossRef]

37. S. C. Mishra and R. R. Vernekar, “Analysis of transport of collimated radiation in a participating media using the lattice Boltzmann method,” J. Quant. Spectrosc. Radiat. Transf. **113**(16), 2088–2099 (2012). [CrossRef]

34. A. F. D. Rienzo, P. Asinari, R. Borchiellini, and S. C. Mishra, “Improved angular discretization and error analysis of the lattice Boltzmann method for solving radiative heat transfer in a participating medium,” Int. J. Numer. Methods Heat Fluid Flow **21**(5), 640–662 (2011). [CrossRef]

35. Y. Ma, S. K. Dong, and H. P. Tan, “Lattice Boltzmann method for one-dimensional radiation transfer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **84**(1), 016704 (2011). [CrossRef] [PubMed]

36. H. Bindra and D. V. Patil, “Radiative or neutron transport modeling using a lattice Boltzmann equation framework,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **86**(1), 016706 (2012). [CrossRef] [PubMed]

37. S. C. Mishra and R. R. Vernekar, “Analysis of transport of collimated radiation in a participating media using the lattice Boltzmann method,” J. Quant. Spectrosc. Radiat. Transf. **113**(16), 2088–2099 (2012). [CrossRef]

## 2. Mathematical formulation

*L*as shown in Fig. 1(a). The medium is assumed to have azimuthal symmetry with constant physical properties. The refractive index of the medium

*n*is homogeneous and could be either equal to or higher than those of the environment (in this paper, the refractive index of the environment is 1). The boundary at

*x*= 0 is exposed to a collimated short-pulse irradiation with an incident angle

*θ*

_{0}and a radiation intensity

*I*

_{0}as illustrated in Fig. 1(a). The wave shape of the short pulse could be either square or Gaussian as shown in Fig. 1(b). Besides, it is assumed that the thermal emission of the medium is negligible as compared with the incident radiation. The propagation of light pulse in the semi-transparent medium is described by the transient radiative transfer equation (TRTE). In the Cartesian coordinate system, the TRTE for one-dimensional problem can be written aswhere

*c*is the propagation speed of light in vacuum,

_{o}*n*is the refractive index of the medium,

*t*is the time,

*μ*is the direction cosine of the polar angle (−1 ≤

*μ*≤ 1),

*I*(

*x*,

*μ*,

*t*) is the radiative intensity, and

*κ*,

_{a}*σ*and

_{s}*β*=

*κ*+ σ

_{a}*are the absorption, scattering and extinction coefficients, respectively. Φ(*

_{s}*μ*,

*μ*) = 1 +

^{′}*aμμ*is the scattering phase function with

^{′}*a*denoting the anisotropically scattering coefficient.

*n*= 1, the collimated radiation penetrates directly into the medium without changing the direction. In this case the intensity

*I*within the medium can be divided into two components, viz., the collimated intensity

*I*and the diffuse intensity

_{c}*I*.

_{d}*I*within the medium is decreased exponentially according to Beer’s law [23

_{c}23. S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer **49**(11-12), 1820–1832 (2006). [CrossRef]

*S*=

_{t}*S*+

_{c}*S*is the total source term.

_{d}*n*= 1), the two boundaries at

*x*= 0 and

*x*=

*L*are considered to be non-reflecting. Thus the radiative boundary condition can be written as

*n*> 1) and the medium surfaces are supposed to be diffusely reflecting and semitransparent, the transient process of radiation transfer is quite different from the case of

*n*= 1. The collimated pulse irradiation on the left boundary is divided into two parts, viz., the diffusely reflected intensity towards the environment and the diffusely transmitted intensity towards the medium inside. Therefore, only the diffuse intensity

*I*

_{d}can be obtained.

*ρ*can be expressed as [38]:

_{O}*n′*=

*n*/1 =

*n*. Considering the effect of total reflection, the internal diffuse reflectivity is [38]

*x*= 0 may then be written aswhere

*t*is the pulse duration,

_{p}*H*(

*t*) is the Heaviside step function,

*δ*is the Dirac delta function, and

*μ*

_{0}is the direction cosine of the angle of incidence.

*t*=

*t*and the half maxima at

_{c}*t*=

*t*±

_{c}*t*/2, where the pulse intensity exceeds one-half of the maximum intensity.

_{p}*n*= 1, the collimated intensity

*I*

_{c}in the medium for the square pulse and the Gaussian pulse are needed and can be derived from Eqs. (3), (9) and (10). The collimated remnant of the square pulse irradiation can be expressed aswhere

*s*=

*x*/

*μ*

_{0}is the geometric distance in the incident direction,

*t**=

*βc*and

_{o}t*t*=

_{p}**βc*is the non-dimensional time and the non-dimensional pulse-width, respectively. The attenuation of the collimated Gaussian irradiation as it travels through the medium is given by

_{o}t_{p}*t**, the RTE given by Eq. (4) is now rewritten as

*Μ*− velocity lattice model in 1D (D1QM) [37

37. S. C. Mishra and R. R. Vernekar, “Analysis of transport of collimated radiation in a participating media using the lattice Boltzmann method,” J. Quant. Spectrosc. Radiat. Transf. **113**(16), 2088–2099 (2012). [CrossRef]

*M*is the total number of discrete directions. It is obviously that the speed of particle propagation along the

*m*th discrete direction is

*m*th discrete direction can be expressed as

**113**(16), 2088–2099 (2012). [CrossRef]

*M*parts, the source term

*S*and

_{c}*S*are computed from the following equations: where

_{d}*m*.

^{′}*q*and transmittance

_{R}*q*, respectively. In the analysis of the transient radiative transfer, the time-resolved reflectance and transmittance provide specific information about the media. Transmittance is defined as the dimensionless net radiative heat flux emerging out of the medium due to transmission, namely the dimensionless net radiative heat flux at the right boundary (

_{T}*x*=

*L*). Reflectance is the dimensionless net radiative heat flux at the boundary which is subjected to the laser irradiation, and in the present case, it is the dimensionless reflected heat flux at the left boundary (

*x*= 0).

*n*= 1, the time-resolved signals can be expressed as where

*n*>1, they can be defined as where

*Step 1*: Set the initial parameters, using appropriate number of lattices to mesh the solution domain.

*Step 2*: Confirm the time step Δ

*t**and total calculation time span.

*Step 3*: Loop at each time step.

- (1) Loop for the global iterations.
- (a) For each discrete direction
*m*, implement the streaming and colliding processes according to Eq. (19), and update the radiative intensity. - (b) Impose boundary conditions on the boundary nodes.
- (c) Terminate the global iteration process if the stop criterion (the maximum relative error of source term
*S*is not bigger than a very small value) is satisfied. Otherwise, go back to step (a)._{t}

- (2) Compute the time-resolved reflectance and transmittance from Eqs. (23a) and (23b) or from Eqs. (24a) and (24b), respectively. If the total non-dimensional time reaches the total calculation time span, terminate the iteration process of the time loop, otherwise, go back to step (1).

## 3. Results and discussion

*L*= 1.0 m. At a given time level, convergence is assumed to have been achieved when the change in source term

*S*value at all points for the two consecutive iterations do not exceed 1 × 10

_{t}^{−7}. The present LBM for transient radiative heat transfer is coded using MATLAB. All runs were taken on Intel(R) Core(TM) i5-2320 processor with 3.00GHz CPU and 6GB RAM.

### 3.1 The correctness and computational efficiency of the LBM

*τ*= 1 and the scattering albedo is

_{L}*ω*= 1. The pulse incident on the left side of the slab is a square pulse with duration

*t*= 1.0. With normal incidence of the collimated radiation (

_{p}**μ*= 0.0), the LBM results for the transient transmittance signal

_{0}*q*are shown in Fig. 2. The non-dimensional computation time span is taken as

_{T}*t**= 10. For grid and ray independent solutions, a maximum of 201 lattices and 12 directions are used. The time interval is chosen as Δ

*t**= 0.05. We can see that the LBM results agree with those obtained by FVM [23

23. S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer **49**(11-12), 1820–1832 (2006). [CrossRef]

### 3.2 Transient radiative transfer in isotropically scattering medium with short square pulse laser irratiation

*t*= 1.0, with normal incidence of the collimated radiation (

_{p}**μ*= 0.0), the curves of

_{0}*q*and

_{T}*q*have been plotted in Figs. 3(a)-3(f) for different scattering albedo and extinction coefficient. The effects of scattering albedo

_{R}*ω*, taken as 0.5 and 0.9, on the results are shown for extinction coefficient

*β*= 1, 5 and 10 m

^{−1}, respectively. It can be seen that the transmittance signals begin to appear just at

*t**=

*τ*and the reflectance signals remain available since the start of the transient process. For a given extinction coefficient

_{L}*β*, with decreases in

*ω*, the peaks of both the signals decrease and they last for a shorter duration. In all these cases, number of 12 rays and the time interval chosen as Δ

*t**= 0.05 are considered. These solutions have been obtained with lattices of 201 for

*β*= 1 m

^{−1}, 501 for

*β*= 5 m

^{−1}and 1001 for

*β*= 10.0 m

^{−1}. It can be seen that for

*β*= 1 m

^{−1}and 5 m

^{−1}with different scattering albedo, results by the LBM agree well with those obtained by the FVM [23

**49**(11-12), 1820–1832 (2006). [CrossRef]

*β*= 10.0 m

^{−1}and

*ω*= 0.9, an obviously deviation from the FVM and DTM solutions is found. While the LBM results agree well with those by Monte Carlo method (MCM) developed by ourselves.

*θ*on

_{0}*q*and

_{T}*q*has been shown in Figs. 4(a)-4(f). For

_{R}*ω*= 1 and

*t*= 1.0, the transmittance and reflectance signals are presented for three values of

_{p}**β*, 1.0, 5.0 and 10.0 m

^{−1}, respectively. For each value of

*β*, results have been illustrated for

*θ*= 0°, 45° and 60°. Mishra et al. have investigated these problems [23

_{0}**49**(11-12), 1820–1832 (2006). [CrossRef]

*ω*= 1,

*β*= 1.0 m

^{−1}and

*θ*= 60° obtained by LBM with those by Monte Carlo method (MCM) developed by ourselves. In Figs. 4(a) and 4(b), it can been seen that the LBM results agree well with those by MCM.

_{0}*t**= 0 to

*t*=

*t*increases faster to a maximum value at

_{p}**t**=

*t*, as a result of which, a higher peak value is obtained. The contribution of the pulse irradiation to the transient process is embodied in the source terms in Eq. (4). It can be concluded from the analytical solutions of the collimated intensity (see Eq. (11)) that as the incident angle increases, the source terms at all the position decrease, and as a result of which, the diffuse intensity

_{p}**I*

_{d}obtained decreases. Thus, the numerator in Eq. (22a) decreases. However, the denominator (

*t*=

*t*the time-resolved reflectance decreases more quickly.

_{p}**q*signals begin to appear at

_{T}*t**=

*τ*, while

_{L}*q*signals remain available from the start of the process. For the cases of oblique incidence, the existence of the

_{R}*q*signals during the time period

_{T}*t**=

*τ*to

_{L}*t*=

_{s}**τ*cos

_{L}/*θ*is owing to the contribution of the diffuse radiation which reaches the right boundary before the collimated radiation. It can be further observed that the

_{0}*q*signals undergo a noticeable change at

_{T}*t*and

_{s}**t**=

*t*+

_{s}**t*. This behavior is owing to the fact that the collimated radiation combined with the diffuse radiation passes through the right boundary. Right after

_{p}**t**=

*t*+

_{s}**t*, only the diffuse radiation is at work.

_{p}*### 3.3 Transient radiative transfer in anisotropically scattering medium with square pulse laser irratiation

*τ*= 10 and the albedo

_{L}*ω*= 0.998. The incident pulse with normal incidence on the left side of the slab is a square signal with a duration of

*t*= 1.0. Three values of the anisotropically scattering coefficient

_{p}**a*are taken for the case. Backward scattering is considered for

*a*= – 0.9, isotropic scattering for

*a*= 0.0 and forward scattering for

*a*= + 0.9. Lu et al. [10

10. X. D. Lu and P. F. Hsu, “Reverse Monte Carlo method for transient radiative transfer in participating media,” J. Heat Transfer **126**(4), 621–627 (2004). [CrossRef]

*t**= 100. With the time interval setting as Δ

*t**= 0.1, it takes about 389.899 CPU seconds for the calculations. It can be seen that in Fig. 5(a) the time-resolved results of transmittance obtained by LBM agree well with the data obtained by the RMCM [10

10. X. D. Lu and P. F. Hsu, “Reverse Monte Carlo method for transient radiative transfer in participating media,” J. Heat Transfer **126**(4), 621–627 (2004). [CrossRef]

### 3.4 Transient radiative transfer in purely scattering medium subjected to Gaussian pulse

*x*= 0) of the slab is exposed normally to a laser Gaussian pulse with an incident radiation:

*τ*= 1.0 and the albedo is

_{L}*ω*= 1.0. The LBM is used to solve the time-resolved reflectance and transmittance for the case of

*t*= 0.4. Lattices of 101 and 12 directions are used in this case. With the time step taken as Δ

_{p}**t**= 0.1, the CPU time cost for the calculation till

*t**= 10 is 17.73661 seconds. The results are shown in Figs. 6(a) and 6(b). The solutions obtained by LBM agree well with the data obtained by the time shift and superposition method in combination with the DFEM [25

25. L. H. Liu and P. F. Hsu, “Time shift and superposition method for solving transient radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. **109**(7), 1297–1308 (2008). [CrossRef]

### 3.5 Transient radiative transfer in two-layer nonhomogeneous media with short square pulse irradiation

*t**= 6. With the time interval setting as Δ

*t**= 0.025, it takes about 72.3591 CPU seconds for the calculations. As shown in Fig. 7(b), our results are in good agreement compared with those in [11

11. X. D. Lu and P. F. Hsu, “Reverse Monte Carlo simulations of light pulse propagation in nonhomogeneous media,” J. Quant. Spectrosc. Radiat. Transf. **93**(1-3), 349–367 (2005). [CrossRef]

*ω*

_{1}= 0.1 and

*ω*

_{2}= 0.9, the special ‘dual peak’ phenomenon occurs in the reflectance signals. Due to the strong scattering of the second layer, the local minimum in reflectance signal can be found at

*t**= 1 moment. Just as is presented by Lu and Hsu [11

11. X. D. Lu and P. F. Hsu, “Reverse Monte Carlo simulations of light pulse propagation in nonhomogeneous media,” J. Quant. Spectrosc. Radiat. Transf. **93**(1-3), 349–367 (2005). [CrossRef]

### 3.6 Transient radiative transfer in isotropically scattering media with refractive index mismatched at the boundary

*n*>1) is investigated. The two surfaces of the slab are diffusely reflecting and semitransparent. The normal incident pulse on the left side of the slab is a square pulse with duration of

*t*= 1.0.

_{p}**n*= 1.5,

*β*= 1.0 m

^{−1}and

*ω*= 1.0. The MCM results developed by ourselves are also presented for the comparison. It can be seen that the LBM results agree well with those obtained by the MCM.

*t**are presented in Figs. 9(a) and 9(b) for

*x*= 0 and

*L*, respectively.

*t**=

*τ*(1.5). It is owing to the fact that, the diffuse radiation produced by the normal irradiation on the semitransparent surface takes

_{L}n*t**=

*τ*at least to reach the right boundary. The

_{L}n*q*increases gradually to the peak value at the time

_{T}*t**=

*τ*(2.5). The noticeable change observed in the case of

_{L}n + t_{p}**n*= 1 doesn’t appear in the case of

*n*= 1.5. This is due to the fact that in the case with the refractive index bigger than one, only diffuse radiation transfers in the medium, and consequently the corresponding transmittance signal changes gradually inside the slab. Referring to Fig. 9(b), we can have an intuitive feeling for the increase of the

*q*. It can be observed that the intensities in the positive directions are great during this period of time

_{T}*t**= 1.5~2.5.

*q*is available from the start of the transient process. During the time span of

_{R}*t**= 1 to

*t*=

*t*it increases gradually, and its values are noticeable higher than those after

_{p}**t*=

*t*. This is because the reflectance signals during this period consist of two parts, viz., the diffuse radiation transmitting through the internal surface and the radiation diffusely reflected by the external surface. Owing to the existence of the pulse irradiation,

_{p}**q*increases during this period of time. Right after the time

_{R}*t*=

*t*, only the diffusely transmitted radiation make contributions to the

_{p}**q*. It can be further observed that,

_{R}*q*increases again during the time interval of

_{R}*t**≈3 to

*t**≈4.5. It is owing to the fact that, the diffuse energy produced by the pulse on the left interface takes time of

*τ*to travel to the right interface, and after being reflected it takes a total of at least time of 2

_{L}n*τ*(

_{L}n*t**≈3) to get back to the left interface . Referring to Fig. 9(a), we can also have an intuitive feeling for the increase of the

*q*in this period. It can be observed that the values of intensities in the negative directions are noticeable during this period.

_{R}*β*= 1.0 m

^{−1}and

*ω*= 1.0, effects of the refractive index

*n*on the time-resolved reflectance and transmittance have been shown in Figs. 10(a) and 10(b). With different

*n*, the diffuse reflectivity on the semitransparent surface and the propagation speed of the light in the medium are different. Consequently, the time-resolved signals for different

*n*are different.

*n*= 1.2, 1.5 and 1.8. It can be seen that, with increases in

*n*, both the internal and external diffuse reflectivity increase. It means that, for a higher

*n*, pulsed energy transmitted through the left internal interface is lower and the energy inside the slab is reflected more by the internal interface.

*t**= 0 to

*t**=

*t*, the

_{p}**q*curve for the case with a higher

_{R}*n*is of higher values but has a lower increasing rate. As it can be seen in Eq. (24a) that the

*ρ*is added to the

_{O}*q*as a constant value during this period. For a higher

_{R}*n*, the value of

*ρ*is higher, which results in the higher value of the

_{O}*q*curve. The rise of the curve is owing to the diffuse radiation inside the slab. For a higher

_{R}*n*, based on the above analysis, the diffuse radiative energy inside the slab is lower, which results in the lower increasing rate of the

*q*curve. (ii) After the time of

_{R}*t*, a second peak value of

_{p}**q*appears for different

_{R}*n*. For a higher

*n*, the peak appears later, which is owing to the slower speed of light. (iii) For the higher

*n*, the diffuse radiative energy inside the slab is lower. However, the curve descends slower than those cases having lower

*n*after the second peak. It is owing to the fact that, the inner diffuse reflectivity is higher for a higher

*n*, which means that the energy could be kept longer inside the slab.

*q*may be observed. (i) As expected, the

_{T}*q*begins to appear just at

_{T}*t**=

*τ*. (ii) With a lower

_{L}n*n*, the peak of the curve is higher. For a lower

*n*, both the

*ρ*and

_{O}*ρ*are lower, and as a result of which, more pulsed energy is transmitted through the two interfaces of the slab. (ii) Similar to the curve of

_{I}*q*, with a higher

_{R}*n*, the

*q*curve decreases slower than those cases having lower

_{T}*n*after the peak.

## 4. Conclusion

*n*= 1) or mismatched (

*n*>1) semitransparent boundary were considered.

## Acknowledgments

## References and links

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7. | S. Kumar and K. Mitra, “Microscale Aspects of Thermal Radiation and Laser Applications,” Adv. Heat Transfer |

8. | H. Schweiger, A. Oliva, M. Costa, and C. D. P. Segarra, “A Monte Carlo method for the simulation of transient radiation heat transfer: application to compound honeycomb transparent insulation,” Numer. Heat Transf. B |

9. | Z. Guo, J. Aber, B. A. Garetz, and S. Kumar, “Monte Carlo simulation and experiments of pulsed radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. |

10. | X. D. Lu and P. F. Hsu, “Reverse Monte Carlo method for transient radiative transfer in participating media,” J. Heat Transfer |

11. | X. D. Lu and P. F. Hsu, “Reverse Monte Carlo simulations of light pulse propagation in nonhomogeneous media,” J. Quant. Spectrosc. Radiat. Transf. |

12. | M. Martinelli, A. Gardner, D. Cuccia, C. Hayakawa, J. Spanier, and V. Venugopalan, “Analysis of single Monte Carlo methods for prediction of reflectance from turbid media,” Opt. Express |

13. | Z. X. Guo and S. Kumar, “Discrete-ordinates solution of short-pulsed laser transport in two-dimensional turbid media,” Appl. Opt. |

14. | M. Sakami, K. Mitra, and P. F. Hsu, “Analysis of light-pulse transport through two-dimensional scattering and absorbing media,” J. Quant. Spectrosc. Radiat. Transf. |

15. | Z. X. Guo and S. Kumar, “Three-dimensional discrete ordinates method in transient radiative transfer,” J. Thermophys, Heat Transfer |

16. | J. M. Wang and C. Y. Wu, “Transient radiative transfer in a scattering slab with variable refractive index and diffuse substrate,” Int. J. Heat Mass Transfer |

17. | Z. M. Tan and P. F. Hsu, “An integral formulation of transient radiative transfer,” J. Heat Transfer |

18. | C. Y. Wu, “Propagation of scattered radiation in a participating planar medium with pulse irradiation,” J. Quant. Spectrosc. Radiat. Transf. |

19. | P. F. Hsu, “Effects of multiple scattering and reflective boundary on the transient radiative transfer process,” Int. J. Therm. Sci. |

20. | J. C. Chai, “One-dimensional transient radiation heat transfer modeling using a finite-volume method,” Numer. Heat Transf. B |

21. | M. Y. Kim, S. Menon, and S. W. Baek, “On the transient radiative transfer in a one-dimensional planar medium subjected to radiative equilibrium,” Int. J. Heat Mass Transfer |

22. | L. M. Ruan, S. G. Wang, H. Qi, and D. L. Wang, “Analysis of the characteristics of time-resolved signals for transient radiative transfer in scattering participating media,” J. Quant. Spectrosc. Radiat. Transf. |

23. | S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer |

24. | L. H. Liu and L. J. Liu, “Discontinuous finite element approach for transient radiative transfer equation,” J. Heat Transfer |

25. | L. H. Liu and P. F. Hsu, “Time shift and superposition method for solving transient radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. |

26. | X. He, S. Chen, and R. A. Zhang, “Lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh–Taylor instability,” J. Comput. Phys. |

27. | S. Succi, |

28. | W. S. Jiaung, J. R. Ho, and C. P. Kuo, “Lattice Boltzmann method for heat conduction problem with phase change,” Numer. Heat Transfer, Part B |

29. | S. C. Mishra and A. Lankadasu, “Analysis of transient conduction and radiation heat transfer using the lattice Boltzmann method and the discrete transfer method,” Numer. Heat Transfer, Part A |

30. | S. C. Mishra and H. K. Roy, “Solving transient conduction-radiation problems using the lattice Boltzmann method and the finite volume method,” J. Comput. Phys. |

31. | S. C. Mishra, T. B. Pavan Kumar, and B. Mondal, “Lattice Boltzmann method applied to the solution of energy equation of a radiation and non-Fourier heat conduction problem,” Numer. Heat Transfer, Part A |

32. | B. Mondal and S. C. Mishra, “Simulation of natural convection in the presence of volumetric radiation using the lattice Boltzmann method,” Numer. Heat Transfer, Part A |

33. | P. Asinari, S. C. Mishra, and R. Borchiellini, “A lattice Boltzmann formulation to the analysis of radiative heat transfer problems in a participating medium,” Numer. Heat Transfer, Part B |

34. | A. F. D. Rienzo, P. Asinari, R. Borchiellini, and S. C. Mishra, “Improved angular discretization and error analysis of the lattice Boltzmann method for solving radiative heat transfer in a participating medium,” Int. J. Numer. Methods Heat Fluid Flow |

35. | Y. Ma, S. K. Dong, and H. P. Tan, “Lattice Boltzmann method for one-dimensional radiation transfer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

36. | H. Bindra and D. V. Patil, “Radiative or neutron transport modeling using a lattice Boltzmann equation framework,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

37. | S. C. Mishra and R. R. Vernekar, “Analysis of transport of collimated radiation in a participating media using the lattice Boltzmann method,” J. Quant. Spectrosc. Radiat. Transf. |

38. | R. Siegel, “Variable Refractive Index Effects on Radiation in Semitransparent Scattering Multilayered Regions,” J. Thermophys. Heat Transfer |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(140.7090) Lasers and laser optics : Ultrafast lasers

(290.7050) Scattering : Turbid media

(010.5620) Atmospheric and oceanic optics : Radiative transfer

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: July 26, 2013

Revised Manuscript: September 15, 2013

Manuscript Accepted: September 30, 2013

Published: October 7, 2013

**Citation**

Yong Zhang, Hongliang Yi, and Heping Tan, "One-dimensional transient radiative transfer by lattice Boltzmann method," Opt. Express **21**, 24532-24549 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-21-24532

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- M. Martinelli, A. Gardner, D. Cuccia, C. Hayakawa, J. Spanier, and V. Venugopalan, “Analysis of single Monte Carlo methods for prediction of reflectance from turbid media,” Opt. Express19(20), 19627–19642 (2011). [CrossRef] [PubMed]
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- P. F. Hsu, “Effects of multiple scattering and reflective boundary on the transient radiative transfer process,” Int. J. Therm. Sci.40(6), 539–549 (2001). [CrossRef]
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- L. M. Ruan, S. G. Wang, H. Qi, and D. L. Wang, “Analysis of the characteristics of time-resolved signals for transient radiative transfer in scattering participating media,” J. Quant. Spectrosc. Radiat. Transf.111(16), 2405–2414 (2010). [CrossRef]
- S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer49(11-12), 1820–1832 (2006). [CrossRef]
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- W. S. Jiaung, J. R. Ho, and C. P. Kuo, “Lattice Boltzmann method for heat conduction problem with phase change,” Numer. Heat Transfer, Part B39, 167–187 (2001).
- S. C. Mishra and A. Lankadasu, “Analysis of transient conduction and radiation heat transfer using the lattice Boltzmann method and the discrete transfer method,” Numer. Heat Transfer, Part A47, 935–954 (2005).
- S. C. Mishra and H. K. Roy, “Solving transient conduction-radiation problems using the lattice Boltzmann method and the finite volume method,” J. Comput. Phys.223(1), 89–107 (2007). [CrossRef]
- S. C. Mishra, T. B. Pavan Kumar, and B. Mondal, “Lattice Boltzmann method applied to the solution of energy equation of a radiation and non-Fourier heat conduction problem,” Numer. Heat Transfer, Part A54, 798–818 (2008).
- B. Mondal and S. C. Mishra, “Simulation of natural convection in the presence of volumetric radiation using the lattice Boltzmann method,” Numer. Heat Transfer, Part A55, 18–41 (2009).
- P. Asinari, S. C. Mishra, and R. Borchiellini, “A lattice Boltzmann formulation to the analysis of radiative heat transfer problems in a participating medium,” Numer. Heat Transfer, Part B57, 126–146 (2010).
- A. F. D. Rienzo, P. Asinari, R. Borchiellini, and S. C. Mishra, “Improved angular discretization and error analysis of the lattice Boltzmann method for solving radiative heat transfer in a participating medium,” Int. J. Numer. Methods Heat Fluid Flow21(5), 640–662 (2011). [CrossRef]
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- S. C. Mishra and R. R. Vernekar, “Analysis of transport of collimated radiation in a participating media using the lattice Boltzmann method,” J. Quant. Spectrosc. Radiat. Transf.113(16), 2088–2099 (2012). [CrossRef]
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