Formulation

The pulse width t_pof the radiation is of the order of 10⁻⁹s. The time interval between two consecutive pulses in the case of a pulse train is a multiple of t_p. To analyze the transmittance and the reflectance signals caused only by the short-pulse radiation, the homogeneous medium and its diffuse-gray boundaries are considered cold. The incident pulse travels with the speed of light^c

(

^{= ×3 10 m/s8}

)

. In case of the step-pulse

TH-0531_RMUTHUKUMARAN

32

(Fig. 3.1c), at any location in the medium, the part of the radiation source remains available only for the duration of the pulse-width t_p, whereas the Gaussian-pulse (Fig. 3.1d) remains available for the duration of 6t_p. When the short-pulse radiation propagates through participating medium, the time-dependent diffuse radiation manifests and its life span in the medium is of the order of the pulse-width of the radiation source.

In the present case in which radiation transport is a time-dependent phenomenon, the radiative transfer equation (RTE) in any direction ˆs is given by [1, 80, 103]

( )

4

1 ,

4

a b s

I I I I I p d

c t s _π

β κ σ

π

∂ + ∂ = − + + Ω Ω′ Ω′

∂ ∂ (3.1)

where s is the geometric distance in the direction ˆs, k_a is the absorption coefficient, β is the extinction coefficient, σ_s is the scattering coefficient and p is the scattering phase function.

Since in the present work, we are dealing with both diffuse as well as collimated radiation, in the following pages we provide a general formulation for the latter case.

The changes in the formulation for the diffuse radiation will be highlighted wherever its need is felt.

When the collimated radiation encounters the medium and passes through it, its decay gives rise to the diffuse radiation. Thus within the medium, the intensity I is composed of two components, viz., the collimated intensity I_c and the diffuse intensityI_d.

c d

I = I +I (3.2)

The variation of the collimated intensity I_cwithin the medium is given by

1 _{c c}

c

I I I

c ^∂t + ^∂s = −β

∂ ∂ ^(3.3)

Substituting Eq. (3.2) in Eq. (3.1), we get

TH-0531_RMUTHUKUMARAN

33

( )

( )

4

4

1 1

4 ,

4 ,

c c d d

c d a b s d

s c

I I I I

c t s c t s

I I I I p d

I p d

π

π

β β κ σ

π σ

π

∂ + ∂ + ∂ + ∂

∂ ∂ ∂ ∂

′ ′

= − − + + Ω Ω Ω

′ ′

+ Ω Ω Ω

(3.4)

From Eqs. (3.3) and (3.4), we get

1 _{d d}

d c d d t

I I I S S I S

c ^∂ t + ^∂s = −β + + = −β +

∂ ∂ (3.5)

where S_c and S_d are the source terms resulting from the collimated and the diffuse components of radiation, respectively. In Eq. (3.5), S_t =S_c+S_d is the total source term. The source term S_c resulting from the collimated radiation I_c is given by

( )

⁴

( ) ( )

0

, ,

4

c s c

S t I t p d

σ π

π _{Ω =′} ^{′ ′}

= Ω Ω Ω Ω (3.6)

For a linear anisotropic phase function^p

(

^{Ω Ω = +, ′}

)

^{1 acos cosθ θ′}, the source term S_c in terms of the incident radiation G_c and heat flux q_c is written as

( )

₄ ^s

( )

^{c o s}

( )

c c c

S t σ G t a θq t

= π + (3.7)

Since the collimated intensity Ic

( )

θ^,t for a step (Fig. 3.1c) and a Gaussian (Fig. 3.1d) pulse are defined by Eqs. (3.9) and (3.10), respectively, G_c and q_c in Eq. (3.7) are given by

( ) (

^,

)

c c

G t = I θ t (3.8a)

( ) (

^,

)

^{c o s}

c c

q t = I θ t θ (3.8b)

( ) ( ) ( ) { ^{( )} }

( )

{ }

( )

, ,max , exp

( 1)

c

c c

c p p

c c

H ct s

I t I t s

H ct s ct N cT

θ θ β β

β β β

δ θ θ

= − × −

− − − − −

× −

(3.9)

( ) ( ) ( )

( )

2

,max

( 1)

, , exp exp 4 ln 2

, 0 2

c c p

c c

p

c c

c

t s t N T

I t I t s c

t t t

θ θ β

δ θ θ

− − − −

= − × −

× − < <

(3.10)

TH-0531_RMUTHUKUMARAN

34

where in Eqs. (3.9) and (3.10), I_c,maxis the collimated intensity at the north boundary, cos

c c

s ⁼ z θ is the geometric distance in the direction θ_cof the collimated radiation, δ is the Dirac-delta function and H is the Heaviside function defined as

( )

^1,_0, ^y ₀⁰

H y y

= >

< (3.11)

In Eqs. (3.9) and (3.10), the Dirac-delta function δ takes care of existence of collimated radiation in θ_cdirection while the Heaviside function Hguarantees that the short-pulse radiation is available at any location in the medium only for the time duration t_pfor a step-function. Since the Gaussian-pulse is a continuous one, it is always available for the time span of 0≤ ≤t 6t_p and the cut-off period t_c =3t_p

In case of a diffuse radiation (Fig. 3.1a), whose temporal variations could either be a step (Fig. 3.1c) or a Gaussian function (Fig. 3.1d), the diffuse intensity at the boundary is given by

( )

^{, ,max}

( )

^,

( ) (

^{( 1)}

)

d d p p

I θ t =I θ t × H βct −H βct−βct −βc N− T (3.12)

( ) ( )

2 ,max

( 1)

, , exp 4 ^{c p} ln 2 , 0 2

d d c

p

t t N T

I t I t t t

θ = θ × − ^{− −}t ⁻ < < (3.13)

It is to be noted that Eqs. (3.9) and (3.10) are valid for all locations 0.0≤ ≤z Zwhereas Eq. (3.12) and (3.13) are applicable to the boundary of incidence, in the present case z=0.0, the north boundary. Equations governing the variation of the diffuse intensity Id

( )

θ^,t are given afterwards.

If t^*=βct, t^*p =βctp and t^*c =βctcare the dimensionless times, Eqs. (3.9) - (3.13) can be written as

( ) ( ) ^{( )} ( )

( ) ^{( )}

*

* *

,max * * *

, , exp

( 1)

c c c

p p

c c

c

H t s

I t I t s

H t s t N T

θ θ β β δ θ θ

β

= − × − × −

− − − − − (3.14)

TH-0531_RMUTHUKUMARAN

35

( ) ( ) ^{( )}

( )

* * * 2

* *

,max *

* *

( 1)

, , exp exp 4 ln 2

, 0 2

c c p

c c

p

c c

c

t s t N T

I t I t s

t

t t

θ θ β β

δ θ θ

− − − −

= − × −

× − < <

(3.15)

( )

^{, * ,max}

( )

^{, *}

( ) (

^{* * * ( 1) *}

)

d d p p

I θ t =I θ t × H t −H t − −t N− T (3.16)

( )

^{* ,max}

( )

^{* * *} * ^{* 2 * *}

( 1)

, , exp 4 ^{c p} ln 2 , 0 2

d d c

p

t t N T

I t I t t t

θ = θ × − ^{− −}t ⁻ < < (3.17)

In Eq. (5), for linear anisotropic phase function^p

(

^{Ω Ω = +, ′}

)

^{1 acos cosθ θ′}, the source term S_d in terms of incident radiation G_d and heat flux q_d resulting from the diffuse radiation I_d is given by

( )

^*

( )

^* ₄ ^s

( )

^{* c o s}

( )

^*

d a b d d

S t κ I t σ G t a θ q t

= + π + (3.18)

In case of a planar medium in which radiation is azimuthally symmetric, G_d and q_d are given by and numerically computed from [103]

( )

^{* *}

0

* 1

2 ( , ) s in

4 ( , ) s in s in

2

d d

M

d k k k

k

G t I t d

I t

θ

π θ

π θ θ θ

π θ θ θ

=

=

=

≈ ∆

(3.19)

( )

^{* *}

0

* 1

2 ( , ) c o s s in

2 ( , ) s in c o s s in

d d

M

d k k k k

k

q t I t d

I t

θ

π θ

π θ θ θ θ

π θ θ θ θ

=

=

=

≈ ∆

(3.20)

where M_θ is the number of discrete points considered over the complete span of the polar angle θ

(

^{0≤ ≤θ π}

)

^.

For a boundary having temperature T_w and emissivityε_w, the boundary intensity

( )

^{, *}

d w

I r t is given by and computed from

( )

^{* 4 2 ,}

( ) ( )

^{* , *}

1

, 1 2 , , sin cos sin

M

w w w

d w d w k c w k k k k

k

I r t T I t I t

ε σ ε π θ θ θ θ θ θ

π π ₌

≈ + − + ∆ (3.21)

TH-0531_RMUTHUKUMARAN

36

where in Eq. (3.21), the first and the second terms represent emitted and reflected components of the boundary intensity, respectively.

It is to be noted that if the boundary is subjected to only the diffuse radiation, in Eq.

(3.21)I_{c w,} =0.0.

In terms of non-dimensional time t^*, the RTE given in Eq. (3.5) is now written as

*

d d

d t

I I

I S

t s

β ^∂ +^∂ +β =

∂ ∂ (3.22)

Using backward differencing scheme in time, Eq. (22) becomes

( ) (

^* * ^{* *}

) ( )

^*

( ) ( )

^{* *}

d d d

d t

I t I t t I t

I t S t

t s

β − − ∆ ∂ β

+ + =

∆ ∂ ^(3.23)

Equation (23) can be written in simplified form as

( )

^*

( )

^*

( )

^*

(

^{* *}

)

d

d t d

B I t I t BS t CI t t

s β

∂ + = + − ∆

∂ (3.24)

where ^*_*

(1 )

B t

t

= ∆

+ ∆ and _*

C 1

t

= β

+ ∆ .

With expressions for the source terms, incident radiation and heat flux given above, Eq. (3.24) is the resulting radiative transfer equation to be used in the present analysis.

Below we briefly present formulation and methodology to solve Eq. (3.24) using the FVM. Details of the FVM formulation for the steady-state radiative transfer in general can be found in Mishra and Roy [103] and for the transient radiation study, the same can be found in Mishra et al. [97].

Writing Eq. (3.24) for a discrete direction Ω^m and integrating it over the elemental solid angle∆Ω^m, we get

( )

^*

( )

^*

( )

^*

(

^{* *}

)

m m m

dm m m m

d t d

I t

B d I t d BS t CI t t d

s β

∆Ω ∆Ω ∆Ω

∂ Ω + Ω = + − ∆ Ω

∂ (3.25)

TH-0531_RMUTHUKUMARAN

37

In case of a 1-D planar medium, Eq. (3.25) can be written as

( )

^*

( )

^*

( )

^*

(

^{* *}

)

dm m m m m m

z d t d

I t

B D I t BS t CI t t

z β

∂ + = + − ∆ ∆Ω

∂ (3.26)

where D_z^m and ∆Ω^m are given by

( )

cos 2 sin cos sin

m

m m m m

Dz θd π θ θ θ

∆Ω

= Ω = ∆ (3.27)

4 sin sin

m 2

m d π θm θm

∆Ω

∆Ω = Ω = ∆ (3.28)

Integrating Eq. (3.26) over the 1-D control volume

(

^{dV = × ×1 1 dz}

)

^{we get}

( )

^*

( )

^*

( )

^*

(

^{* *}

)

, , , , ,

m m m m m m m

d N d S z d P t P C d P

I t I t D I t S I t t dz

B B

− = −β + + − ∆ ∆Ω (3.29)

where I_{d N}^m_, and I_{d S}^m_, are north and south control surface average intensities, respectively and I_{d P}^m_, and S_{t P}^m_, are the intensities and source terms at the cell centre P, respectively. In any discrete direction Ω^m, the cell-surface intensities are related to the cell-centre intensity I_{d P}^m_, as

, ,

, 2

m m

d N d S d Pm

I I

I +

= (3.30)

From Eq. (3.29) and (3.30), while marching from the north boundary for which 2

θ<π , we get

( )

^*

( )

^*

(

^{* *}

)

, , ,

,

2

2

m m m m m m

z d N t P d P

d Pm

m m

z

D I t S t dz Cdz I t t

I B

D dz

B β

+ ∆Ω + ∆Ω −∆

=

+ ∆Ω

(3.31a)

and while marching from the south boundary for which 2

θ>π , we get

TH-0531_RMUTHUKUMARAN

38

( )

^*

( )

^*

(

^{* *}

)

, , ,

,

2

2

m m m m m m

z d S t P d P

d Pm

m m

z

D I t S t dz Cdz I t t

I B

D dz

B β

+ ∆Ω + ∆Ω − ∆

=

+ ∆Ω

(3.31b)