Results with 1-4 pulse train - Results and Discussion

5.4 Results and Discussion – Step pulse train

5.4.2 Results with 1-4 pulse train

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

Y ⁼ with its south boundary subjected to either diffuse or collimated pulses. The pulse train is considered consisting of 1-4 pulses.

With the south boundary subjected to a diffuse pulse train, Figs. 5.3a-f show transmittance ^q^t^*

(

^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. With scattering albedoω=1.0, for 1-4 pulses, these results are shown for three values of the extinction coefficientβ. It is seen from Figs. 5.3a-c that the magnitudes of the signal peaks decrease with increasingβ. It is also observed that troughs that are present in 2-4 pulse trains in Fig. 5.3a forβ =1.0, vanish for higher values of β (Figs. 5.3b and 5.3c). Observations of Figs. 5.3a-c show that the peaks of different pulse-trains are more aligned for higher β, and temporal spreads in the signals are more spread for higherβ. A distinct difference in the peak magnitudes of the signals for multiple pulses are observed for β =5.0and 10.0. Further, since radiation takes t^*=βct=βY to reach the opposite (north boundary), withY =1.0, in Figs. 5.3a-c, the transmittance signals start appearing att^*=β.

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Time

Transmittance

0 2 4 6 8 10 12

0 0.05 0.1 0.15 0.2 0.25

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

Time

Reflectance

0 2 4 6 8 10

0 0.05 0.1 0.15 0.2 0.25

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

(a) (b)

Time

Transmittance

0 8 16 24 32 40 48

0 0.005 0.01 0.015 0.02 0.025

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

Time

Reflectance

0 6 12 18 24 30

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

N = 1 N = 2 N = 3 N = 4

β = 5.0 ω = 1.0

Time

Transmittance

0 25 50 75 100

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045

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

Time

Reflectance

0 6 12 18 24 30

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

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

(e) (f)

Figure 5.3: Transmittance ^q^t^*

(

^0.5,1.0,^t^*

)

signals and reflectance ^q^*^r

(

^0.5,0.0,^t^*

)

^signals

for 1-4 pulses The south boundary subjected to diffuse radiation.

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For the corresponding cases, reflectance ^q^*^r

(

^0.5,0.0,^t^*

)

results are shown in Figs.

5.3d-f. Unlike transmittance^q^t^*

(

^0.5,1.0,^t^*

)

^,the peak magnitudes of the reflectance

( )

* 0.5,0.0, *

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

(

^0.5,1.0,^t^*

)

(Fig. 5.3a), the reflectance ^q^*^r

(

^0.5,0.0,^t^*

)

signals (Fig. 5.3d) do not last long forβ =1.0, and for β =5.0and 10.0, they last longer (Figs. 5.3e and 5.3f). However, their temporal spans are shorter than the transmittance^q^*^t

(

^0.5,1.0,^t^*

)

signals (Figs. 5.3b and 5.3c).

Since the south boundary starts getting radiation the time the radiation enters the medium, the reflectance signals are seen to start with t^*=0.0 (Figs. 5.3d-f).

Transmittance^q^t^*

(

^0.5,1.0,^t^*

)

and reflectance ^q^*^r

(

^0.5,0.0,^t^*

)

results with collimated pulses are shown in Figs. 5.4a-f. With scattering albedo ω=1.0, for 1-4 pulses, these results are shown for three values of the extinction coefficient β =1.0,5.0and 10.0.

For β =1.0, trends of the two signals (Figs. 5.4a and 5.4d) are similar to that of diffuse pulse train (Figs. 5.3a and 5.3d). For 1-3 pulses, the transmittance signals with β =5.0have a distinct peak and a sharp decline leading to another well distributed maxima (Fig. 5.4c). 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. Though the troughs in the curves occur at the same time for 1-4 pulses (Fig. 5.4b), a large difference in their magnitude is observed. With β =10.0 (Fig. 5.4c), unlike β =5.0(Fig. 5.4b), distinct crests and troughs at an early stage are not prominent. Like diffuse radiation (Fig. 5.3c), forβ =10.0, the signals are characterized by a single crest for any number of pulses. The reflectance

( )

* 0.5,0.0, *

qr t results with β =1.0,5.0and 10.0 for 1-4 pulses are shown in Figs.5.

4d-f, respectively. Profiles of these signals are the same as that for the diffuse pulse train (Figs. 5.3d-f) except for β =5.0 and 10.0, narrow crests and troughs are visible for a single and a 2-pulse train.

TH-0531_RMUTHUKUMARAN

100

Time

Transmittance

0 2 4 6 8 10 12

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 2 4 6 8 10 12

0 0.025 0.05 0.075 0.1 0.125 0.15

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

(a) (d)

Time

Transmittance

0 10 20 30 40 50

0 0.005 0.01 0.015 0.02

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

Time

Reflectance

0 6 12 18 24 30

0 0.05 0.1 0.15 0.2 0.25 0.3

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 0.0025 0.003

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

Time

Reflectance

0 6 12 18 24 30

0 0.05 0.1 0.15 0.2 0.25 0.3

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

Figure 5.4: 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 subjected to collimated radiation.

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101

1 1 1 1

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

3 3 3 3 3

4 4

6 6

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

2 2 2

3 3 3 3

4 4 4 4

6 6 6 6

8 8 8 8

1 1

2 2 2 2

2 3

3 3

4 4 4 4

6 6 6 6 4

8 8 8 8 6

x/X

Reflectance

0 0.2 0.4 0.6 0.8 1

0 0.05 0.1 0.15 0.2

β = 1.0 ω = 1.0

(a) (d)

1 1 1 1

2 2

2 3

3 3

4 4 4 4

6 6 6 6

8 8 8 8

1 1 1 1 1

2 2

3 3 3 3 3

4 4

6 6 6 6

8 8 8 8 68

x/X

Transmittance

0.2 0.4 0.6 0.8 1

0 0.005 0.01 0.015 0.02

β = 5.0 ω = 1.0

1 1

2 2 2 2

3 3 3 3

4 4 4 4

6 6 6 6

8 8 8 8

1 1

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 0.2 0.4 0.6 0.8 1

0 0.05 0.1 0.15 0.2 0.25 0.3

β = 5.0 ω = 1.0

(b) (e)

1 1 1 1 1

2 2 2

3 2 3

3 3

4 3

4 4 4

6 6 6 6 6

8 8 8 8 8

1 1 1 1

2 2

3 3 3 3

4 4

4 6

6 8

8 8

x/X

Transmittance

0 0.2 0.4 0.6 0.8 1

0 0.001 0.002 0.003 0.004

β = 10.0 ω = 1.0

1 1

2 2 2 2

3 3 3 3

4 4 4 4

6 6 6 6

8 8 8 8

1 1

2 2 2

3 3 3 3 23

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 0.2

β = 10.0 ω = 1.0

Figure 5.5: Time evolution of the distribution of transmittance ^{q x X}^*^t

(

^{/ ,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.

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102

1 1 1 1

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

3 3 3 3 3

4 4 4

6 4

6 6 6

8 6

8 8 8

x/X

Transmittance

0 0.2 0.4 0.6 0.8 1

0.1 0.2 0.3 0.4 0.5

β = 1.0 ω = 1.0

1 1

2 2

3 3 3 3

4 4 4 4

6 6 6 6

8 8 8 8

1 1

2 2 2

2 3

3 3

4 4 4

6 4

6 6 6

8 6

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

2 2

4 4

6 6 6 6

8 8 8 8

2 2

3 3 3 3 3

4 4

6 6 6 6

8 8 8 6

x/X

Transmittance

0 0.2 0.4 0.6 0.8 1

0.002 0.004 0.006 0.008 0.01 0.012

β = 5.0 ω = 1.0

1 1

2 2 2 2

3 3 3 3

4 4 4 4

6 6 6 6

8 8 8 8

1 1

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 0.2 0.4 0.6 0.8 1

0 0.05 0.1 0.15 0.2

0.25 β = 5.0

ω = 1.0

(b) (e)

1 1 1 1

2 2

2 3

3 4

6 6 6 6

8 8 8 8

1 1 1 1 1

2 2

3 3 3 3 3

4 4

6 6

8 8 8

x/X

Transmittance

0 0.2 0.4 0.6 0.8 1

0 0.0005 0.001 0.0015 0.002 0.0025

β = 10.0 ω = 1.0

1 1

2 2 2 2

3 3 3 3

4 4 4 4

6 6 6 6

8 8 8 8

1 1

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 0.2 0.4 0.6 0.8 1

0 0.05 0.1 0.15 0.2

0.25 β = 10.0

ω = 1.0

Figure 5.6: Time evolution of the distribution of transmittance ^{q x X}^*^t

(

^{/ ,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

103

In Figs. 5.5 and 5.6, for a single and a 4-pulse train, distributions of transmittanceq^*_t x ,1.0,t^*

X and reflectance q_r^* x ,0.0,t^*

X results along the boundaries, have been plotted for diffuse and collimated radiations, respectively.

Withω =1.0, these distributions are shown for the extinction coefficient 1.0,5.0

β = and 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. 5.5a and 5.6a that when the medium is less participating

(

^{β =}^1.0

)

, the transmittance _t^* x ,1.0, ^*

q t

X results for a single and a 4-pulse train almost coincide with each other at all times. However, the reflectance

* ,0.0, *

r x

q t

X signals are more distinct at almost all times for a single and a 4-pulse train. For β =5.0and 10.0, the transmittance _t^* x ,1.0, ^*

q t

X signals at all time levels are well separated. However, it is observed that for any value ofβ, at an early stage

* 100

t t =

∆ since the magnitudes of the signals are very small, they are not noticeable.

This can be verified from Figs. 5.3a-c, and 5.4a-c, where the time evolution of the transmittance signals in the middle of the north boundary

(

^0.5,1.0

)

have been plotted.

It is observed from Figs. 5.5c and 5.6c that for β =1.0at t^*_* 100 t =

∆ , for both diffuse and collimated pulses, profiles of the reflectance ^*_r x ,0.0, ^*

q t

X signals coincide for both a single and a 4-pulse train. However, at t^*_* 300

t =

∆ ^{, the} q^*_r x ,0.0,t^*

X ^is

considerably higher for a 4-pulse train. At later times, they are close with each other.

For β =5.0and 10.0, (Figs. 5.5b, 5.5c, 5.6e and 5.6f), profiles of q^*_r x ,0.0,t^*

X ^at

the first two time levels are far too distinct for a 4-pulse train. However, for a single-

TH-0531_RMUTHUKUMARAN

104

pulse, their magnitudes are less. At later time levels, for both a single-pulse and a 4- pulse train, results are close to each other.

Figures 5.7-5.10 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.

(5.7-5.10)a-c, whereas the same for a 4-pulse train are given in Figs. (5.7-5.10)d-f.

For ω=1.0 and β =1.0, heat flux contours are given in Figs. 5.7 and 5.8 for diffuse and collimated pulses, respectively. In Figs. 5.9 and 5.10, heat flux contours are presented for diffuse and collimated pulses, respectively for ω =1.0 and β =10.0.

From Figs. 5.7a and 5.7d it is observed that for a single-pulse, at t^*_* 50 t =

∆ ^{, the}

negative heat flux q^* x y, ,t^*

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

q t

X signals.

For a 4-pulse train, from Fig. 5.7d it is observed that at t^*_* 50 t =

∆ , like Fig. 5.7a, 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. 5.7b and 5.7c, and 5.7e and 5.7f 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. 5.5a and 5.5d.

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105

-0.065 0.022

0.126 0.126 0.000

0.022 0.061

0.126 0.173

-0.065

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.319 0.061

0.089 0.000

0.022 0.061

0.089

0.635

0.089 0.000

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

(a) (d)

-0.008 -0.006

-0.004 -0.002 0.000

0.002 0.003

0.004 0.005 0.006

0.006

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

-0.065 0.011

0.061 0.011

0.022

0.035

0.061 0.074 0.084

-0.030

-0.104

0.022

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

(b) (e)

-0.000039 -0.000030

-0.000018 -0.000012

-0.000007 -0.000002 0.000007

0.000014 0.000020 0.000027 0.000038

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

-0.000295 -0.000250

-0.000192 -0.000138 -0.000102 -0.000059 -0.000021

0.000015 0.000045 0.000076

0.000114 0.000158 0.000210

0.000254 0.000305

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

Figure 5.7: 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 subjected to diffuse radiation. β =1.0,ω=1.0.

TH-0531_RMUTHUKUMARAN

106

0.022038 0.051647

0.087031

0.022038 -0.022983 -0.049595

-0.070001 -0.004492

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.022038

0.051647 0.087031

0.022038 -0.015307

-0.015307

-0.049595

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

(a) (d)

-0.010988 -0.008287

-0.005339 -0.002477 0.000059 0.001986

0.003304 0.005029

0.007217 0.008096

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

-0.049595 -0.020938

-0.005339 0.007217

0.022038 0.051647 0.007217

0.039431 0.022038

0.012280

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

(b) (e)

-0.000048 -0.000035 -0.000021 -0.000005 0.000006 0.000014 0.000027 0.000040 0.000050

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

-0.000214 -0.000165 -0.000120 -0.000017 0.000025 0.000061 0.000105 0.000146

0.000202

-0.000064

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

Figure 5.8: 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 subjected to collimated radiation. β =1.0,ω=1.0.

TH-0531_RMUTHUKUMARAN

107

0.007217 0.012280 0.022038 0.033058

-0.020938 0.037733

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.007217 0.012280

0.022038 0.033058

0.037733

-0.020938 -0.005339

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

(a) (d)

-0.004701 -0.003037 -0.001084 -0.000227 0.000246

0.000516 0.001242 0.001933 0.002356

0.002636

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.012280 0.007217

0.002636 0.000516 -0.004701

-0.020938 -0.012331

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

(b) (e)

-0.000094 -0.000054 -0.000023

-0.000009 0.000010 -0.000001

0.000029 0.000067 0.000096

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.000484 0.000371

0.000243 0.000131 0.000062

-0.000029 -0.000132

-0.000262

-0.000417

-0.000511

0 0.2 0.4 0.6 0.8 1

Figure 5.9: 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 subjected to diffuse radiation. β =10.0,ω=1.0.

TH-0531_RMUTHUKUMARAN

108

0.074569 0.041143 0.026619 0.003554

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

0.074569 0.041143 0.026619 0.003554

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

(a) (d)

-0.004225 -0.001422

-0.000384 0.001396 0.000336

0.001396 0.002213 0.002799

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

-0.010232 -0.002936 0.0010700.002698

0.006016 0.009150 0.010368 0.001070

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

(b) (e)

-0.000125 -0.000088 -0.000036 -0.000014 0.000001

0.000020 0.000053

0.000098 0.000129

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

-0.000125 -0.000088 -0.000036 -0.000014 0.000001

0.000020 0.000053

0.000098 0.000129

x/X

y/Y

0 0.2 0.4 0.6 0.8 1

Figure 5.10: 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 subjected to collimated radiation. β =10.0,ω=1.0.

TH-0531_RMUTHUKUMARAN

109

A comparison of Figs. 5.7b and 5.8b show that the heat flux concentration around the middle of the north boundary is already established at t^*_* 200

t =

∆ in the case of collimated single-pulse (Fig. 5.8b). Though the medium is low absorbing

(

^{β =}^1.0

)

in case of collimated radiation, since radiation is more directional than the diffuse, the heat flux packet is found much earlier at the north boundary (Fig. 5.8b).

Heat flux contours in Figs. 5.9a and 5.9c for diffuse radiation for a single-pulse and a 4-pulse train for β =10.0look similar at t^*_* 50

t =

∆ . A similar trend is also observed in Figs. 5.10a and 5.10c for the collimated radiation.

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.

5.9b, 5.9e, 5.10b and 5.10e. The magnitude of the maximum is much higher in the case of Figs. 5.9e and 5.10e 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. 5.9c, 5.9f, 5.10c and 5.10f, the heat flux concentrations at the boundaries are widely distributed around the middle of the north and the south boundaries. The curves marked 8 in Figs. 5.5e, 5.5f, 5.6e and 5.6f are analogous to this observation.

With extinction coefficientβ =1.0, for 1-4 pulse trains, transmittance ^q^*^t

(

^0.5,1.0,^t^*

)

and reflectance ^q^*^r

(

^0.5,0.0,^t^*

)

signals for three different values of the scattering albedo ωare shown in Figs. 5.11 and 5.12 for diffuse and collimated radiations, respectively. It is seen from these figures (Fig. 5.11a-5.11c and 5.12a-5.12c) 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

TH-0531_RMUTHUKUMARAN

110

Time

Transmittance

0 2 4 6 8 10 12

0 0.05 0.1 0.15

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

Time

Reflectance

0 2 4 6 8 10

0 0.005 0.01 0.015 0.02

N = 1 N = 2 N = 3 N = 4 β = 1.0

ω = 0.1

(a) (d)

Time

Transmittance

0 2 4 6 8 10 12

0 0.05 0.1 0.15

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

Time

Reflectance

0 2 4 6 8 10

0 0.025 0.05 0.075 0.1

N = 1 N = 2 N = 3 N = 4 β = 1.0

ω = 0.5

(b) (e)

Time

Transmittance

0 2 4 6 8 10 12

0 0.05 0.1 0.15 0.2 0.25

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

Time

Reflectance

0 2 4 6 8 10

0 0.05 0.1 0.15 0.2 0.25

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

Figure 5.11: 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 scattering albedoω. The south boundary subjected to diffuse radiation.

TH-0531_RMUTHUKUMARAN

111

Time

Transmittance

2 4 6 8 10 12 14

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 2 4 6 8 10

0 0.005 0.01 0.015

N = 1 N = 2 N = 3 N = 4 β = 1.0

ω = 0.1

(a) (c)

Time

Transmittance

2 4 6 8 10 12 14

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 2 4 6 8 10 12

0 0.01 0.02 0.03 0.04 0.05 0.06

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

(b) (e)

Time

Transmittance

0 2 4 6 8 10 12

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 2 4 6 8 10 12

0 0.025 0.05 0.075 0.1 0.125 0.15

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

Figure 5.12: 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 scattering albedoω. The south boundary subjected to collimated radiation.

TH-0531_RMUTHUKUMARAN

112

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 in ω. A comparison of Figs. 5.11 and 5.12 shows that in case of collimated radiation, the transmittance signals are the replicas of the incident square pulse but with a considerable reduction in magnitude. Since the collimated radiation is more directional, the shape of the transmittance signals (Fig. 5.12a) for a lower value of β and a small value of ω is more or less like that of the incident pulses (Fig. 4.1b).

However, in case of diffuse radiation, the incident energy is equally distributed in all directions, the signal profiles are of diffuse nature.

Dalam dokumen Transient response of a Participating Medium subjected to short-pulse radiation sources (Halaman 129-144)