Results with 1-4 pulse train

5.5 Results and Discussion – Gaussian pulse train

5.5.2 Results with 1-4 pulse train

In the following pages, results for a square medium X 1

Y = with its south boundary subjected to either diffuse or collimated pulses are provided. The pulse train consists of 1-4 pulses.

With the south boundary subjected to a diffuse pulse train, Figs. 5.14a-f show transmittance ^qt*

(

^0.5,1.0,t*

)

and reflectance ^q*r

(

^0.5,1.0,t*

)

at the middle of the south and the north boundaries, respectively. Results are shown for 1-4 pulses. With scattering albedoω=1.0, these results are presented for the extinction coefficientβ =1.0,5.0and 10.0.

It is seen from Figs. 5.14a-c that the magnitudes of the transmittance

( )

* 0.5,1.0, *

qt t signal peaks decrease by an order of magnitude with increasingβ. Observations of Figs. 5.14a-c show that the peaks of different pulse-trains are time- wise more aligned and further signals last longer for higherβ. Peaks of increasing magnitudes are observed for multiple pulses for β =5.0and 10.0. Relative difference in magnitudes of the successive peaks increases with increase inβ. Further, since radiation takes t^*=βct=βY to reach the opposite (north boundary), withY =1.0, in Figs. 3a-c, the transmittance signals ^qt*

(

^0.5,1.0,t*

)

start appearing aftert^*=β. Since the magnitude of the incident pulse is Gaussian function, a gradual increase in the signal is noticed.

Reflectance ^q*r

(

^0.5,0.0,t*

)

results for the corresponding cases of Figs. 5.14a-c are shown in Figs. 5.14d-f. Unlike transmittance^q*t

(

^0.5,1.0,t*

)

, the peak magnitudes of the reflectance ^q*r

(

^0.5,0.0,t*

)

signals increase with increase inβ, and also different crests and troughs are distinct for all values ofβ. Like transmittance^qt*

(

^0.5,1.0,t*

)

TH-0531_RMUTHUKUMARAN

115

Time

Transmittance

5 10 15 20 25 30

0 0.05 0.1 0.15

0.2 N = 1

N = 2 N = 3 N = 4 β = 1.0 ω = 1.0

Time

Reflectance

0 5 10 15 20 25 30

0 0.05 0.1 0.15 0.2

N = 1 N = 2 N = 3 N = 4 β = 1.0 ω = 1.0

(a) (d)

Time

Transmittance

0 5 10 15 20 25 30 35 40 45 50 55 60 0

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

N = 1 N = 2 N = 3 N = 4 β = 5.0 ω = 1.0

Time

Reflectance

0 5 10 15 20 25 30 35 40

0 0.05 0.1 0.15 0.2

N = 1 N = 2 N = 3 N = 4 β = 5.0 ω = 1.0

(b) (e)

Time

Transmittance

10 20 30 40 50 60 70 80 90 100 0

0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005

N = 1 N = 2 N = 3 N = 4 β = 10.0 ω = 1.0

Time

Reflectance

0 10 20 30 40 50 60

0 0.05 0.1 0.15 0.2 0.25

N = 1 N = 2 N = 3 N = 4 β = 10.0 ω = 1.0

(c) (f)

Figure 5.14: Transmittance ^qt*

(

^0.5,1.0,t*

)

signals and reflectance ^qr*

(

^0.5,0.0,t*

)

signals for 1-4 pulses for three different values of extinction coefficientβ. The south boundary is subjected to diffuse radiation.

TH-0531_RMUTHUKUMARAN

116

(Fig. 5.14a), the reflectance ^q*r

(

^0.5,0.0,t*

)

signals (Fig. 5.14d) do not last long forβ =1.0. Both of them exist for approximately the same time, t^*=25.0. With β =5.0and 10.0, they last longer (Figs. 5.14e and 5.14f). However, their temporal spans are shorter than the transmittance^qt*

(

^0.5,1.0,t*

)

signals (Figs. 5.14b and 5.14c).

Since the south boundary starts getting radiation much earlier than the north boundary, the reflectance ^q*r

(

^0.5,0.0,t*

)

signals are seen to start earlier than the transmittance^q*t

(

^0.5,1.0,t*

)

signals (Figs. 5.14d-f).

Figs. 5.15a-f show transmittance^q*t

(

^0.5,1.0,t*

)

and reflectance ^q*r

(

^0.5,0.0,t*

)

^results

with collimated pulses. With scattering albedoω=1.0, these results are shown for the extinction coefficient β =1.0,5.0and 10.0. For β =1.0, trends of the two signals (Figs. 5.15a and 5.15d) are similar to that of diffuse pulse train (Figs. 5.14a and 5.14d). However, magnitudes of ^q*t

(

^0.5,1.0,t*

)

are higher and that of

( )

* 0.5,0.0, *

qr t lower than the diffuse pulse-train. Since the direction of incidence is normal, a higher magnitude of the ^qt*

(

^0.5,1.0,t*

)

signals in case of collimated radiation is attributed to increased concentration of energy with less attenuation that reaches the north boundary.

The transmittance signals with β =5.0 for a single pulse is observed to have a distinct peak and a sharp decline leading to another well distributed maximum (Fig.

5.15b). The sharp peak is attributed to the arrival of the collimated component of the incident radiation much earlier than the diffuse radiations from the medium. For 2-4 pulses, a trend similar to diffuse radiation (Fig. 5.14b) is noticed. Though the troughs in the curves occur at the same time for 1-4 pulses (Fig. 5.15b), a considerable difference in their magnitude is observed.

With β =10.0 (Fig. 5.15c), unlike β =5.0(Fig. 5.15b), distinct crests and troughs corresponding to 1-4 pulses are not observed. Like diffuse pulse (Fig. 5.14c), forβ =10.0, the signals are characterized by a single crest for any number of pulses.

TH-0531_RMUTHUKUMARAN

117

Time

Transmittance

0 5 10 15 20 25 30

0.05 0.15 0.25 0.35 0.45 0.55

N = 1 N = 2 N = 3 N = 4 β = 1.0 ω = 1.0

Time

Reflectance

0 5 10 15 20 25 30

0.01 0.03 0.05 0.07 0.09 0.11 0.13

N = 1 N = 2 N = 3 N = 4 β = 1.0 ω = 1.0

(a) (d)

Time

Transmittance

0 10 20 30 40 50 60

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

N = 1 N = 2 N = 3 N = 4 β = 5.0 ω = 1.0

Time

Reflectance

0 5 10 15 20 25 30 35 40

0.01 0.03 0.05 0.07 0.09 0.11 0.13

N = 1 N = 2 N = 3 N = 4 β = 5.0 ω = 1.0

(b) (e)

Time

Transmittance

0 20 40 60 80 100

0 0.0005 0.001 0.0015 0.002

N = 1 N = 2 N = 3 N = 4 β = 10.0 ω = 1.0

Time

Reflectance

0 5 10 15 20 25 30 35 40 45 50 0

0.025 0.05 0.075 0.1 0.125 0.15

N = 1 N = 2 N = 3 N = 4 β = 10.0 ω = 1.0

(c) (f)

Figure 5.15: Transmittance ^q*t

(

^0.5,1.0,t*

)

signals and reflectance ^q*r

(

^0.5,0.0,t*

)

signals for 1-4 pulses for three different values of extinction coefficientβ. The south boundary is subjected to collimated radiation.

TH-0531_RMUTHUKUMARAN

118

1 1 1 1

2 2 2 2

3 3 3 3

4 4

4 4

4 4

6 6 6 6

8 8 8 8

1 1

1 1 1

1 1

2 2 2 2 2

3 3 3 3 3

4 4

4 4 4

4 4

6 6 6 6 6

8 8 8 8 8

x/X

Transmittance

0.2 0.4 0.6 0.8

0 0.05 0.1 0.15 0.2

β = 1.0 ω = 1.0

1 1 1 1

2 2 2 2

3 3

3 3

3

4 4

4

4

4

6 6 6 6

8 8 8 8

1 1

1 1

1

1

2 2 2 2 2

3 3 3 3 3

4 4

4 4

4

4

6 6 6 6 6

8 8 8 8 8

x/X

Reflectance

0 0.2 0.4 0.6 0.8 1

0 0.05 0.1 0.15

β = 1.0 ω = 1.0

(a) (d)

1

1

1

1

1 2

2 2

2

3 3 3 3

4 4 4 4

6

6

6

6

8 6

8

8

8

8 1

1 1

1 1

1 1

1 2

2 2

2 2

2

3 3

3 3

3

4

4 4 4

4

6 6 6 6 6

8 8 8 8 8

x/X

Transmittance

0 0.2 0.4 0.6 0.8

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

β = 5.0 ω = 1.0

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

6 6 6 6

8 8 8 8

1 1

1 1 1

1

1

2 2

2 2

2

3 3 3 3

4 4 4 4 34

6 6 6 6 6

8 8 8 8 8

x/X

Reflectance

0.2 0.4 0.6 0.8

0 0.05 0.1 0.15 0.2

β = 5.0 ω = 1.0

(b) (e)

1 1 1 1

2

2 2

3 2

3

3

3 4

4 4

4

6 6 6 6

8 8 8 8

1 1 1 1 1

2 2

2 2 2

2

3 3 3 3 3

4 4

4 4

4

4

4

4

6 6

6 6

6

6

6 8

8

8 8

8

x/X

Transmittance

0.2 0.4 0.6 0.8

0 0.001 0.002 0.003 0.004

β = 10.0 ω = 1.0

1

1 1

1

2 2 2 2

3 3 3 3

4 4 4 4

6 6 6 6

8 8 8 8

1 1

1 1 1

1

1

2 2

2 2

2

3

3 3 3

3

4 4 4 4 4

6 6 6 6 6

8 8 8 8 8

x/X

Reflectance

0.2 0.4 0.6 0.8

0 0.05 0.1 0.15 0.2

β = 10.0 ω = 1.0

(c) (f)

Figure 5.16: Time evolution of the distribution of transmittance ^{q x Xt*}

(

^{/ ,1.0,t*}

)

^and

reflectance ^{q x X*r}

(

^{/ ,0.0,t*}

)

signals along the boundaries. The digit n on any curve indicates the distribution of the signal at n×100^thtime step. Solid and dash lines are results for 1- and 4-pulse diffuse radiation.

TH-0531_RMUTHUKUMARAN

119

Observation of Figs. 5.15b and 5.15c shows that the magnitudes of the peak

( )

* 0.5,1.0, *

qt t signals are less than that for the ^q*t

(

^0.5,1.0,t*

)

signals resulting from diffuse pulse cases given in Figs. 5.14b and 5.14c. Although in this case, the collimated energy is more concentrated, unlike β =1.0 in Fig. 5.15a, for β =5.0 and 10.0 in Figs. 5.15b and 5.15c, because of very high attenuation, the magnitudes of

( )

* 0.5,1.0, *

qt t signals are lower than those given in Figs. 5.14b and 5.14c. In Figs.

5.14b and 5.15b, and also Figs. 5.14c and 5.15c, although the extinction coefficient βvalues are the same, overall attenuation is less in case of Figs. 5.14b and 5.14c because of diffuse nature of the pulse.

The reflectance ^q*r

(

^0.5,0.0,t*

)

results with β =1.0,5.0and 10.0 for 1-4 pulses are shown in Figs. 5.15d-f, respectively. Profiles of these signals are the same as that for the diffuse pulse train (Figs. 5.14d-f).In Figs. 5.15 and 5.17, for a single and a 4-pulse train, distributions of transmittanceq^*_t x ,1.0,t^*

X ^and reflectance

* ,0.0, *

r x

q t

X results along the boundaries, have been plotted for diffuse and collimated pulses, respectively. Withω=1.0, these distributions are shown for the extinction coefficient β =1.0,5.0and 10.0. For a givenβ, these distributions are plotted at 6 time levels, viz. t^*_* 100, 200,300, 400,600

t =

∆ and 800.

It is seen from Figs. 5.16a and 5.17a that when the medium is less participating

(

^{β =1.0}

)

, the transmittanceq_t^* x ,1.0,t^*

X results for a single and a 4-pulse train almost coincide with each other at all times. The same is the case with the reflectance

* ,0.0, *

r x

q t

X signals. For β =5.0and 10.0, the transmittanceq^*_t x ,1.0,t^*

X signals

at all time levels are well separated. This can be verified from Figs. 5.14a-c, and 5.15a-c, where the time evolution of the transmittance signals in the middle of the north boundary

(

^0.5,1.0

)

have been plotted.

TH-0531_RMUTHUKUMARAN

120

1 1 1

2 2 2

3 3 3

4 4 4

6 6 6

8 8 8

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

6 6 6 6

8 8 8 8

x/X

Transmittance

0.2 0.4 0.6 0.8

0 0.1 0.2 0.3 0.4

β = 1.0 ω = 1.0

1 1 1 1

2 2 2 2

3 3

3 3

3

4 3 4

4 4

4

6 6 6 6

8 8 8 8

1 1

1 1 1

1

2 2 2 2 2

3 3

3 3 3

3

4 4 4 4 4

6 6 6 6 6

8 8 8 8 8

x/X

Reflectance

0 0.2 0.4 0.6 0.8 1

0 0.02 0.04 0.06 0.08

β = 1.0 ω = 1.0

(a) (d)

1 1 1 1

2 2

2 2

2

3 3

3

3

3 4

4 4

4

6 6 6 6

8 8 8 8

1 1

1

1 1

1

1

2

2 2 2

2

3 3 3 3 3

4 4 4 4 4

6 6 6 6 6

8 8 8 8 8

x/X

Transmittance

0.2 0.4 0.6 0.8

0 0.001 0.002 0.003 0.004 0.005

β = 5.0 ω = 1.0

1 1

1 1

1

1

2

2 2

2

3 3 3 3

4 4 4 4

6 6 6 6

8 8 8 8

1 1

1 1

1

2 2 2 2 2

3 3 3 3 3

4 4 4 4 4

6 6 6 6 6

8 8 8 8 8

x/X

Reflectance

0.2 0.4 0.6 0.8

0 0.01 0.02 0.03 0.04 0.05 0.06

β = 5.0 ω = 1.0

(b) (e)

1 1 1 1

2

2 2

3 2

3 3

3 4

4 4

4

6 6 6 6

8 8 8 8

1 1 1 1 1

2

2 2 2

3 3 3 3 23

4 4

4

4 4

4

4

4 6

6

6 6

6

6

8 8

8 8

8

x/X

Transmittance

0.2 0.4 0.6 0.8

0 0.0005 0.001 0.0015 0.002

β = 10.0 ω = 1.0

1

1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

6 6 6 6

8 8 8 8

1 1

1 1 1

1

1

2

2 2 2

2 3

3 3 3

4 4 4 4 34

6 6 6 6 6

8 8 8 8 8

x/X

Reflectance

0.2 0.4 0.6 0.8

0 0.02 0.04 0.06 0.08 0.1 0.12

β = 10.0 ω = 1.0

(c) (f)

Figure 5.17: Time evolution of the distribution of transmittance ^{q x Xt*}

(

^{/ ,1.0,t*}

)

^and

reflectance ^{q x X*r}

(

^{/ ,0.0,t*}

)

signals along the boundaries. The digit n on any curve indicates the distribution of the signal at n×100^thtime step. Solid and dash lines are results for 1- and 4-pulse collimated radiation.

TH-0531_RMUTHUKUMARAN

121

Figures 5.18-5.21 provide heat flux contours in the medium for a single and a 4-pulse train. In each of these figures, the contours are plotted at three time levels, viz

*

* 50, 200 t

t =

∆ and 800. In these figures, results for a single-pulse are given in Figs.

(5.18-5.21)a-c, whereas the same for a 4-pulse train are given in Figs. (5.18-5.21)d-f.

For ω=1.0 andβ =5.0, heat flux contours are given in Figs. 5.18 and 5.19 for diffuse and collimated pulses, respectively. In Figs. 5.20 and 5.21, heat flux contours are presented for diffuse and collimated pulses, respectively for ω=1.0 andβ =10.0.

From Figs. 5.18a and 5.18d it is observed that for a single-pulse, at t^*_* 50 t =

∆ ^{, the}

negative heat flux ^* x y, , ^*

q t

X Y appears near the south boundary. It is to be noted that the negative heat flux contributes towards the reflectance q^*_r x ,0.0,t^*

X signals.

For a 4-pulse train, from Fig. 5.18d it is observed that at t^*_* 50 t =

∆ , like Fig. 5.18a, for a single-pulse, the radiation has not reached the north boundary, but the positive heat flux towards the north boundary is more prominent.

At t^*_* 200 t =

∆ , it is observed from Figs. 5.18b and 5.18c, and 5.18e and 5.18f that both negative and positive heat fluxes remain in the medium for a single and a 4-pulse train. The region of negative heat flux is observed to be more for a single-pulse.

Further it is observed that the peak magnitudes of the heat flux are still found inside the medium in the neighborhood of the north boundary. But as time progresses t^*_* 800

t =

∆ , the maximum magnitude of the heat flux is found to be concentrated around the middle of the south and the north boundaries. This observation is substantiated with Figs. 5.16a and 5.16d.

A comparison of Figs. 5.18b and 5.19b show that the heat flux concentration around the middle of the north boundary is already established at t^*_* 200

t =

∆ in the case of

TH-0531_RMUTHUKUMARAN

122

0.022

0.061 0.126

0.000 -0.065

0.160 0.098

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

0.02160 0.06088 0.12602

0.02160

-0.06527 -0.00220

0.15954

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

(a) (d)

0.00000

-0.00181 -0.00089

-0.00031 0.00028 0.00077

0.00147 0.00202

-0.00220 0.00112

-0.00128

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

-0.00368

-0.03235 -0.01881 0.00702

0.01268 0.02160 0.02689 0.02999

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

(b) (e)

-3.99999998E-08 -1.99999999E-08

-9.99999994E-09 9.99999994E-09 9.99999994E-09 3.99999998E-08 5.00000006E-08

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

-9.11110339E-07 -1.40349325E-06

-1.89537359E-06

-2.37442725E-06 -1.38774481E-07 2.25666652E-07 8.21179563E-07 1.50429591E-06 2.15830903E-06

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

(c) (f)

Figure 5.18: Heat flux contours in the medium at time t^*_*

∆t (a) = 50, (b) = 200, (c) = 800 for a single-pulse, (d) = 50, (e) = 200, (f) = 800 for a 4-pulse train. The south boundary is subjected to diffuse radiation. β =5.0,ω=1.0.

TH-0531_RMUTHUKUMARAN

123

0.000 0.000

0.005 0.001

x/X

y/Y

0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

0.126020 0.000000

0.021599

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

(a) (d)

0.004 0.005

0.002 0.001

-0.001 -0.003

-0.008

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

0.000661 0.001035

0.000367

0.000050 -0.000311

-0.000674

-0.001094

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

(b) (e)

-0.000026 -0.000017

-0.000006 0.000013 0.000024 0.000029

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

8.50810156E-07

4.09116586E-07 1.90684172E-07 0.00000000E+00

-1.44411230E-07 -3.84130306E-07

-7.52331851E-07

-1.11202401E-06 1.14013758E-06

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

(c) (f)

Figure 5.19: Heat flux contours in the medium at time t^*_*

∆t (a) = 50, (b) = 200, (c) = 800 for a single-pulse, (d) = 50, (e) = 200, (f) = 800 for a 4-pulse train. The south boundary is subjected to collimated radiation. β =5.0,ω=1.0.

TH-0531_RMUTHUKUMARAN

124

0.043537 0.126020

0.057376 -0.065270

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

0.006395 0.043537

0.126020

0.006395

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

(a) (d)

-0.005626 -0.001858 -0.000311

0.000573 0.001219

0.002436 0.003497

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

0.006395

-0.045652 0.035892

-0.029001 0.022367

0.011322 0.006395 0.002285 0.000604

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

(b) (e)

-0.000126 -0.000092 -0.000038

-0.000007 0.000028 0.000071 0.000118

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

-0.000858 -0.000669

-0.000377 -0.000082

0.000008 0.000140

0.000391 0.000604 0.000858

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

(c) (f)

Figure 5.20: Heat flux contours in the medium at time t^*_*

∆t (a) = 50, (b) = 200, (c) = 800 for a single-pulse, (d) = 50, (e) = 200, (f) = 800 for a 4-pulse train. The south boundary is subjected to diffuse radiation. β =10.0,ω=1.0.

TH-0531_RMUTHUKUMARAN

125

0.000521 0.010842 0.043537

-0.000377

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

0.010842 0.010842

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

(a) (d)

0.001579 0.001355

0.000743 0.000365 0.000149

-0.001969 -0.000377

-0.001128

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

-0.014753 0.010842

0.015089 0.010842 0.006547 0.003153

0.001330 0.000475 0.000181

-0.008767

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

(b) (e)

-0.000051 -0.000042 -0.000030 -0.000015 0.000000 0.000012 0.000023

0.000037 0.000052

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8

1 _0.0

00181

-0.000381 -0.000332 -0.000254 -0.000112 -0.000041 0.000012 0.000074 0.000181 0.000288 0.000357

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

(c) (f)

Figure 5.21: Heat flux contours in the medium at time t^*_*

∆t (a) = 50, (b) = 200, (c) =800 for a single-pulse, (d) = 50, (e) = 200, (f) = 800 for a 4-pulse train. . The south boundary is subjected to collimated radiation. β =10.0,ω=1.0.

TH-0531_RMUTHUKUMARAN

126

diffuse single-pulse (Fig. 5.18b). Though the medium is highly absorbing

(

^{β =5.0}

)

^,

in case of diffuse radiation, the heat flux packet is found much earlier at the north boundary (Fig. 5.18b).

It is observed from Figs. 5.20 and 5.21 that heat flux contours for both diffuse and collimated radiation for a single-pulse and a 4-pulse train for β =10.0look similar at various time levels.

At t^*_* 200 t =

∆ , a smooth distribution of heat flux varying from a minimum near the boundaries to a maximum near the geometric centre of the medium is seen from Figs.

5.20b, 5.20e, 5.21b and 5.21e. The magnitude of the maximum is higher in the case of Figs. 5.20e and 5.21e due to more energy contained with a 4-pulse train. The low gradient of heat flux inside the medium accounts for a longer life of both the signals withβ =10.0.

At t^*_* 800 t =

∆ , in Figs. 5.20c, 5.20f, 5.21c and 5.21f, the heat flux concentrations at the boundaries are widely distributed around the middle of the north and the south boundaries.

Figures 5.22 and 5.23 show transmittance ^q*t

(

^0.5,1.0,t*

)

and reflectance

( )

* 0.5,0.0, *

qr t signals for the scattering albedo ω=0.1,0.5and 1.0. With extinction coefficientβ =1.0, these results are given for 1-4 pulse trains for diffuse and collimated radiations, in Figs. 5.22 and 5.23, respectively. It is seen from these figures (Fig. 5.22 a-c and 5.23 a-c) that when the scattering is less

(

^ω=0.1

)

^,

transmittance ^q*t

(

^0.5,1.0,t*

)

signals for a N-pulse train lasts for a shorter duration, their peaks are flattened and magnitudes of the peaks are less. A comparison of reflectance ^q*r

(

^0.5,0.0,t*

)

signals show, no visible effect of ωon the shape of the peaks and temporal spans for any pulse train. Like transmittance, the magnitudes of the reflectance signals too are found to increase with increase inω. A comparison of

TH-0531_RMUTHUKUMARAN

127

Time

Transmittance

0 6 12 18 24 30

0 0.05 0.1 0.15

N = 1 N = 2 N = 3 N = 4 β = 1.0 ω = 0.1

Time

Reflectance

0 5 10 15 20 25 30 35 40

0 0.005 0.01 0.015

N = 1 N = 2 N = 3 N = 4 β = 1.0 ω = 0.1

(a) (d)

Time

Transmittance

0 6 12 18 24 30 36

0 0.05 0.1 0.15

N = 1 N = 2 N = 3 N = 4 β = 1.0 ω = 0.5

Time

Reflectance

0 5 10 15 20 25 30 35 40

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

N = 1 N = 2 N = 3 N = 4 β = 1.0 ω = 1.0

(b) (e)

Time

Transmittance

5 10 15 20 25 30

0 0.05 0.1 0.15

0.2 N = 1

N = 2 N = 3 N = 4 β = 1.0 ω = 1.0

(c) (f)

Figure 5.22: Transmittance ^q*t

(

^0.5,1.0,t*

)

signals and reflectance ^qr*

(

^0.5,0.0,t*

)

signals for 1-4 pulses for three different values of scattering albedoω. The south boundary is subjected to diffuse radiation.

TH-0531_RMUTHUKUMARAN

128

Time

Transmittance

0 3 6 9 12 15 18 21 24 27 30

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0.45 N = 1

N = 2 N = 3 N = 4 β = 1.0 ω = 0.1

Time

Reflectance

0 5 10 15 20 25 30 35

0 0.002 0.004 0.006 0.008

N = 1 N = 2 N = 3 N = 4 β = 1.0 ω = 0.1

(a) (d)

Time

Transmittance

0 4 8 12 16 20 24 28 32 36 40 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0.45 N = 1

N = 2 N = 3 N = 4 β = 1.0 ω = 0.5

Time

Reflectance

0 5 10 15 20 25 30

0 0.01 0.02 0.03 0.04

0.05 N = 1

N = 2 N = 3 N = 4 β = 1.0 ω = 0.5

(b) (e)

Time

Transmittance

0 5 10 15 20 25 30

0.05 0.15 0.25 0.35 0.45 0.55

N = 1 N = 2 N = 3 N = 4 β = 1.0 ω = 1.0

Time

Reflectance

0 5 10 15 20 25 30

0.01 0.03 0.05 0.07 0.09 0.11 0.13

N = 1 N = 2 N = 3 N = 4 β = 1.0 ω = 1.0

(c) (f)

Figure 5.23: Transmittance ^q*t

(

^0.5,1.0,t*

)

^{signals and} reflectance

( )

* 0.5,0.0, *

qr t signals for 1-4 pulses for three different values of scattering albedoω^. The south boundary is subjected to collimated radiation.

TH-0531_RMUTHUKUMARAN

129

Figs. 5.22 and 5.23 shows that in case of both diffuse and collimated pulses, the transmittance signals are the replicas of the incident pulses having Gaussian temporal profile but with a considerable reduction in magnitude. Each segment of the signal is seen to last for t^*=6t^*_p.