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On first order reactants in homogeneous turbulence—numerical results
ON FIRSTORDER REACTANTS IN HOMOGENEOUS TURBULENCE-NUMERICAL RESULT% S. R. PATEL Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi-l 10029,India Abstract-In the previous paper multi-point, multi-time correlations of concentration of a dilute contaminant undergoing a first order chemical reactant is studied by Kumar and Patel. Here the numerical work of that paper is carried out. INTRODUCTION
INTHEprevious paper Kumar and Pate1 [l] have studied the two-point, two-time correlations by considering the equations for the concentration of a dilute contaminant undergoing a first order chemical reaction. The general simplifying restrictions, under which the discussion has been carried out in [l] are as follows: (i) the turbulence and the concentration fields are homogeneous, (ii) the chemical reaction and the local mass transfer have no effect on the velocity field and (iii) the reaction rate and the diffusivity are constant. The correlation equations are then converted into spectral form by taking their Fourier transforms. This operation also converts the partial differential equations into ordinary differential equations. In order to solve the equation for the final period the triple order terms are neglected in comparison to the second order ones. The decay law for the final period is proportional to T-“* with correction factor of the type exp (-2CT) over the pure mixing iesults. A correlation coefficient is also obtained and it is seen that the destruction of the contaminant is more rapid than that in the case of pure mixing. The analysis is extended to three-point, and three-time correlation equations. The quadruple correlation terms have been neglected in comparison to the third order correlations and the correlation equations have been solved on the lines of Deissler [2]. These third order correlations have been used to solve the two-point, two-time correlation equations to give the result valid for times before the final period. The decay law comes out to be 2’ = (ATi3’* + BTi5) exp (-2CT,,,) where ,ij’ denotes the concentration fluctuation energy. With different technique Kumar and Pate1 [3] have studied the same problem for the case of multi-point and single time consideration. There the results were obtained much before the final period of decay. This analysis is still extended for the case of multi-point and multi-time concentration correlations by Pate1 [4]. Here in this note the numerical work of Ref. (1) is carried out. RESULTS AND DISCUSSIONS
In Ref. (1) the concentration fluctuation energy for the case of final period of decay is obtained as
1-t
jxx
NoD-3’2
=
exp{-2C(t-t,-y)]
sd+,+;)
where x and x’ are the concentration fluctuations at point P and P’ at different times t and t’ respectively, D is the diffusivity to is the initial time, At = t’ - t, C is the reaction rate constant, NOis the constant depending on the initial conditions. The space time correlation for the case of homogeneous turbulence is obtained as xxi (r, L) =
NOD-“’ 46
(t, - toy* exp
2C(L - to) + r2 8D(L - to)
where r is the distance between the points P’ and P and t,,, = t + At/2. YThiswork is
supportedby Council of Scientific and Industrial Research (India). 75
(2)
76
S. R. PATEL
The space time correlation coefficient
for the final period becomes
In Fig. 1 the plot of R vs At/(t,,, - t,) is shown. The solid curves corresponds to the case of pure mixing i.e. when C = 0 and the dashed curves corresponds to the case when the reactant is present. The dashed curves are drawn for a particular values of C, say O-5. The result (4) alongwith the figure shows that the effect of the reactant is to accelerate the decay of turbulence, i.e. the decay is faster than in the case of pure mixing. In Fig. 2, the time correlation is plotted when r = 0 and C = 0. The values of time correlation coefficient decrease with time. For the case of three-point, three-time consideration the concentration fluctuation energy is found to be [see Ref. (1). eqn (42)].
62(At,tm)=
NO (
2
gD”‘d%
),,.ex,
T +t
[-2C(T+T)]
2
+ D6(1 t N&
t 2Ns)“’ exp
9
9
X
512 t
16T5’* T++&$AT s ( [ t
s/2
16(T+AT)“’
)
5Ns(7Ns - 6)
16(1+2Ns)T3”
(
16(1t2Ns)(TtAT13” 35Ns(3N:8(1 t 2Ns)T”’
(
T
(
(
Tt&AT)
s
7/z
T+$AT)
s
5Ns(7Ns - 6)
+
t
{-2C(T+y)}$
7/z
T+&AT)
2Ns + 3)
s 35Ns(3N:-2Ns
+ 3)
++$ s AT) 9’z+g(lt2N~)(T+AT)“‘(T+$&AT)9’*
+8Ns(3N’,-2N, +3)(1+2N~l~‘~ = 1.3.5...(2n+9) c .+n!(2n + 1)2’“(1+ Ns)” 3 .2=‘*(1t Ns)“” (51
where Ns = v/D, is Schmidt number and T = t - to. In order to make the computations convenient the following dimensionless numbers are defined [51 N = P
N, = c (u 1
(iiyh
-,
D
a Peclet number
, a Damkohler number of first kind
where (ri’)“* is the root mean square turbulent velocity and A is the length scale typical of that
On lirst order reactants in homogeneous turbulence-numerical results
n
mixing
reactant
0’
I
I
0.5
1.0
.
1.5
20
At it,,.-to
Fig. 1. Variation of space-time correlation coefficient.
Fig. 2. Variation of the time correlation coefficient.
energy containing components of both the turbulent and spectral spectra. Substituting these in eqns (12) and (13) and in the corresponding three-point, three-time equations of Ref. (l), eqn (42) of Ref. (1) can be written in slightly different form
If in eqn (6) we put (following f2, 61)
where a0 is constant depending upon the initial conditions, then we get the concentration energy for the case of pure mixing as
(7)
S. R. PATEL
a
0
12
19
20
AT
Fig. 3. Variation of time correlation coefficient before final period, Ns =
1.
,*77rlo-5 2
3
Fig. 4(a). Variation of double-concentration
4
AT
correlation, N, = 1; T,,,= 2; ND = 0.04.
a.wxd_ 2.8
5.0
0.0
6.0
AT
Fig. 4(b). Variation of double-concentration
correlation, Ns = I ; T,,, = 3; ND = 0.04.
Onikstorderreactantsinhomogeneous ~b~en~nurne~c~
results
79
The time correlation coefficient is defined as folIows R’=
_
xx)W,Kn)
[,yx(Tm+;A.)&-,
-;AT)]“2’
(8)
A plot of the time correlation coefkient R ’ is plotted in Fig. 3 for Schmidt No. 1. The values of R’ decreases with the dimensionIess time separation, which type of property is already seen in the case of final period. It is of interest that the value of xx’ do not exhibit the behavior as that of R’, in fact, they increase rather than decrease with time separation as shown in Figs. 4(a)-(c). This unusual variation is apparently due to the nonlinear decay of the turbulence with time and
Fig. 4(c). Variation of doubie~oncen~~ion correlation, Ns = 1;‘I’,,= 4; ND = OG.
Fig. S(a). Variation of dooble~n~en~ation correlation, N, = 1; ND = 0.0.
80
S. R. PATEL
Fig. j(b). Variationof double-concentrationcorrelation, N, = 10. ~=[(“~~:‘)“‘(“?““)]~_=i
B=[
A= 4; C=[ ITS=6.
would not be observed for stationary turbulence. Equation (7) is plotted in Fig. 5(a)-(b) which shows that values of xx’ increase with time seperation. Comparison of the Figs. 2 and 3 indicates that the general effect of the higher order inertia terms in the correlation equations is to decrease the correlation coefficient at a given value of time separation. This is opposite to the corresponding effect for the space correlation. It is possible that the reduction of the correlation coefficient by inertia terms is caused by the nonhomogeneity of the turbulence with time. CONCLUSION
AND APPLICATION
The effect of the higher order inertia terms in the correlation equations is to reduce the value of the correlation coefficient at a given time interval below that for the final period of decay. In the case of the pure mixing, the concentration fluctuation decays with time in a natural manner. This study shows that if the concentration selected is the chemical reactant of first order, then the effect is that the decomposition of the concentration fluctuation is much more rapid and the faster rate of decomposition is governed by exp (-2CT,,,). In a natural manner, it takes a lot of time to get rid of a pollutant in the air. If it is required to get rid of a pollutant as quickly as possible then the pollutant will have to be mixed with some type of chemical reactant as stated in this study. This will help to eliminate the pollutant much faster. REFERENCES [l] [2] [3] [4] [5] [6]
PREM KUMAR and S. R. PATEL, Int. J. Engng Sci. 13, 305 (1975). R. G. DEISSLER, NASA Tech. Rep. R-96 (1961). PREM KUMAR and S. R. PATEL, Phys. Fluids 13, 1362 (1974). S. R. PATEL, Int. J. Engng Sci. 12, 159 (1974). E. E. O’BRIEN and G. C. FRANCIS, J. Fluid Mech. 13, 369 (1%2). A. L. LOEFFLER Jr. and R. G. DEISSLER, Int. J. Heat Mass Transfer 1, 312 (l%l). (Received 8 Nooember 1974)
INTHEprevious paper Kumar and Pate1 [l] have studied the two-point, two-time correlations by considering the equations for the concentration of a dilute contaminant undergoing a first order chemical reaction. The general simplifying restrictions, under which the discussion has been carried out in [l] are as follows: (i) the turbulence and the concentration fields are homogeneous, (ii) the chemical reaction and the local mass transfer have no effect on the velocity field and (iii) the reaction rate and the diffusivity are constant. The correlation equations are then converted into spectral form by taking their Fourier transforms. This operation also converts the partial differential equations into ordinary differential equations. In order to solve the equation for the final period the triple order terms are neglected in comparison to the second order ones. The decay law for the final period is proportional to T-“* with correction factor of the type exp (-2CT) over the pure mixing iesults. A correlation coefficient is also obtained and it is seen that the destruction of the contaminant is more rapid than that in the case of pure mixing. The analysis is extended to three-point, and three-time correlation equations. The quadruple correlation terms have been neglected in comparison to the third order correlations and the correlation equations have been solved on the lines of Deissler [2]. These third order correlations have been used to solve the two-point, two-time correlation equations to give the result valid for times before the final period. The decay law comes out to be 2’ = (ATi3’* + BTi5) exp (-2CT,,,) where ,ij’ denotes the concentration fluctuation energy. With different technique Kumar and Pate1 [3] have studied the same problem for the case of multi-point and single time consideration. There the results were obtained much before the final period of decay. This analysis is still extended for the case of multi-point and multi-time concentration correlations by Pate1 [4]. Here in this note the numerical work of Ref. (1) is carried out. RESULTS AND DISCUSSIONS
In Ref. (1) the concentration fluctuation energy for the case of final period of decay is obtained as
1-t
jxx
NoD-3’2
=
exp{-2C(t-t,-y)]
sd+,+;)
where x and x’ are the concentration fluctuations at point P and P’ at different times t and t’ respectively, D is the diffusivity to is the initial time, At = t’ - t, C is the reaction rate constant, NOis the constant depending on the initial conditions. The space time correlation for the case of homogeneous turbulence is obtained as xxi (r, L) =
NOD-“’ 46
(t, - toy* exp
2C(L - to) + r2 8D(L - to)
where r is the distance between the points P’ and P and t,,, = t + At/2. YThiswork is
supportedby Council of Scientific and Industrial Research (India). 75
(2)
76
S. R. PATEL
The space time correlation coefficient
for the final period becomes
In Fig. 1 the plot of R vs At/(t,,, - t,) is shown. The solid curves corresponds to the case of pure mixing i.e. when C = 0 and the dashed curves corresponds to the case when the reactant is present. The dashed curves are drawn for a particular values of C, say O-5. The result (4) alongwith the figure shows that the effect of the reactant is to accelerate the decay of turbulence, i.e. the decay is faster than in the case of pure mixing. In Fig. 2, the time correlation is plotted when r = 0 and C = 0. The values of time correlation coefficient decrease with time. For the case of three-point, three-time consideration the concentration fluctuation energy is found to be [see Ref. (1). eqn (42)].
62(At,tm)=
NO (
2
gD”‘d%
),,.ex,
T +t
[-2C(T+T)]
2
+ D6(1 t N&
t 2Ns)“’ exp
9
9
X
512 t
16T5’* T++&$AT s ( [ t
s/2
16(T+AT)“’
)
5Ns(7Ns - 6)
16(1+2Ns)T3”
(
16(1t2Ns)(TtAT13” 35Ns(3N:8(1 t 2Ns)T”’
(
T
(
(
Tt&AT)
s
7/z
T+$AT)
s
5Ns(7Ns - 6)
+
t
{-2C(T+y)}$
7/z
T+&AT)
2Ns + 3)
s 35Ns(3N:-2Ns
+ 3)
++$ s AT) 9’z+g(lt2N~)(T+AT)“‘(T+$&AT)9’*
+8Ns(3N’,-2N, +3)(1+2N~l~‘~ = 1.3.5...(2n+9) c .+n!(2n + 1)2’“(1+ Ns)” 3 .2=‘*(1t Ns)“” (51
where Ns = v/D, is Schmidt number and T = t - to. In order to make the computations convenient the following dimensionless numbers are defined [51 N = P
N, = c (u 1
(iiyh
-,
D
a Peclet number
, a Damkohler number of first kind
where (ri’)“* is the root mean square turbulent velocity and A is the length scale typical of that
On lirst order reactants in homogeneous turbulence-numerical results
n
mixing
reactant
0’
I
I
0.5
1.0
.
1.5
20
At it,,.-to
Fig. 1. Variation of space-time correlation coefficient.
Fig. 2. Variation of the time correlation coefficient.
energy containing components of both the turbulent and spectral spectra. Substituting these in eqns (12) and (13) and in the corresponding three-point, three-time equations of Ref. (l), eqn (42) of Ref. (1) can be written in slightly different form
If in eqn (6) we put (following f2, 61)
where a0 is constant depending upon the initial conditions, then we get the concentration energy for the case of pure mixing as
(7)
S. R. PATEL
a
0
12
19
20
AT
Fig. 3. Variation of time correlation coefficient before final period, Ns =
1.
,*77rlo-5 2
3
Fig. 4(a). Variation of double-concentration
4
AT
correlation, N, = 1; T,,,= 2; ND = 0.04.
a.wxd_ 2.8
5.0
0.0
6.0
AT
Fig. 4(b). Variation of double-concentration
correlation, Ns = I ; T,,, = 3; ND = 0.04.
Onikstorderreactantsinhomogeneous ~b~en~nurne~c~
results
79
The time correlation coefficient is defined as folIows R’=
_
xx)W,Kn)
[,yx(Tm+;A.)&-,
-;AT)]“2’
(8)
A plot of the time correlation coefkient R ’ is plotted in Fig. 3 for Schmidt No. 1. The values of R’ decreases with the dimensionIess time separation, which type of property is already seen in the case of final period. It is of interest that the value of xx’ do not exhibit the behavior as that of R’, in fact, they increase rather than decrease with time separation as shown in Figs. 4(a)-(c). This unusual variation is apparently due to the nonlinear decay of the turbulence with time and
Fig. 4(c). Variation of doubie~oncen~~ion correlation, Ns = 1;‘I’,,= 4; ND = OG.
Fig. S(a). Variation of dooble~n~en~ation correlation, N, = 1; ND = 0.0.
80
S. R. PATEL
Fig. j(b). Variationof double-concentrationcorrelation, N, = 10. ~=[(“~~:‘)“‘(“?““)]~_=i
B=[
A= 4; C=[ ITS=6.
would not be observed for stationary turbulence. Equation (7) is plotted in Fig. 5(a)-(b) which shows that values of xx’ increase with time seperation. Comparison of the Figs. 2 and 3 indicates that the general effect of the higher order inertia terms in the correlation equations is to decrease the correlation coefficient at a given value of time separation. This is opposite to the corresponding effect for the space correlation. It is possible that the reduction of the correlation coefficient by inertia terms is caused by the nonhomogeneity of the turbulence with time. CONCLUSION
AND APPLICATION
The effect of the higher order inertia terms in the correlation equations is to reduce the value of the correlation coefficient at a given time interval below that for the final period of decay. In the case of the pure mixing, the concentration fluctuation decays with time in a natural manner. This study shows that if the concentration selected is the chemical reactant of first order, then the effect is that the decomposition of the concentration fluctuation is much more rapid and the faster rate of decomposition is governed by exp (-2CT,,,). In a natural manner, it takes a lot of time to get rid of a pollutant in the air. If it is required to get rid of a pollutant as quickly as possible then the pollutant will have to be mixed with some type of chemical reactant as stated in this study. This will help to eliminate the pollutant much faster. REFERENCES [l] [2] [3] [4] [5] [6]
PREM KUMAR and S. R. PATEL, Int. J. Engng Sci. 13, 305 (1975). R. G. DEISSLER, NASA Tech. Rep. R-96 (1961). PREM KUMAR and S. R. PATEL, Phys. Fluids 13, 1362 (1974). S. R. PATEL, Int. J. Engng Sci. 12, 159 (1974). E. E. O’BRIEN and G. C. FRANCIS, J. Fluid Mech. 13, 369 (1%2). A. L. LOEFFLER Jr. and R. G. DEISSLER, Int. J. Heat Mass Transfer 1, 312 (l%l). (Received 8 Nooember 1974)