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The optimum plate-spacing for the best performance in flat-plate thermal diffusion columns
The optimum plate-spacing for the best performance in flat-plate thermal diffusion columns (Firstreceived16 September 1993; accepted in revised Jiwm 23 December 1993)
INTRODUCTION A more detailed study of the mechanism of separation in
thermal diffusion column indicates that the convective currents, in addition to the desirable cascading effect, also produce an undesirable remixing effect (Clusius and Dickel, 1938). Therefore, proper control of the convective strength might effectively suppress this undesirable remixing effect while still preserving the desirable cascading effect, and thereby lead to improved separation. Based on this concept, some improved columns have been developed in the literature, (Sullivan et al., 1955; Ramser, 1957; Powers and Wilke, 1957; Lorenz and Emery, 1959, WashaIl and Molpolder, 1962; Chueh and Yeh, 1967). In developing these improved columns, a number of studies of the operating variables in the thermal diffusion column have been made. There is still an important term, plate spacing, which affects separation ef&iency and which has hardly ever been discussed. COLUMN THEORY Furry et al. (1939) obtained the equation ofseparation for a binary mixture in a contjnuous-flow thecmal diffusion column, applicable only to equifraction solution (0.3 < c f 0.7):
Maximum separation The optimum plate spacing (2c0)* for a maximum separation A,,, is obtained by partially differentiating eq. (4) with respect to (20) and setting aAja(2w) = 0. After diffecentiation and simplification we obtain 8A0/8(2w) = 0 and thus, from eq. (1) 8=1+3x
Solving
for (2w)*, one obtains (2w)* =
pfr$g*NAT/24* = 9!&.2
Ko f(Z&.
(3)
The equation of separation applicable to whole range of’ concentration (0 6 C G 1) was iater derived by Yeh and Yeh (1982) as A=
+ 12C,(l-
C,)
IL? 1
i 2.16b
119
1
a
(7)
where
where
=
UL
Consequently, the maximum separation may be obtained from eqs (1) and (4) by the substitution of eq. (7). The result is
A,.,.*
b
(3
where
(4)
The plate spacing (ZUJ) in a thermal diffusion column is generally so small that changing (20) will not cause any additional fixed charge. The expenditure of making a separation by thermal diffusion essentially includes two parts: a fixed charge and an operating expense. The fixed change is roughly proportional to the equipment cost, while the operating expense is chiefly heat. The transfer rate is obtainable from the expression, kBL (A i”,Gwj. Based on these terms, we shall take account of the influence of plate-spacing change on the degree of separation, the output and the column length with the consideration of fixed operating expense (i.e. AT/20 as well as o and h ace constants).
= 0.215
(o/a)(aLlh)J’~ .
(9)
It should be noted from eqs (7)-(9) that whereas AO,_ as well as A,,, depends on the thermal diffusion constant a, (2(o)* is independent of a. The problem of finding the maximum separation A,., and the best plate spacing (2c0)+ for a specified flow rate (r can readily be solved by using eqs (7H9) since a, b, and L are known constants for a given column and system.
Maximum output The plate spacing
for maximum separation is also the plate spacing required to obtain the maximum pr~uetjon rate c,., for giLen column which is to give a specified degree of separation A. Although eq. (4) cannot be put into a form explicit in o, it is nevertheless possible to maximize o with respect to (2~) at constant A, & and I.. The maximization yields an expression which is identical with eq. (5) and consequently the solution for the optimum plate spacing (Zw)* for maximum output is identical to that given by eqs 17H9) when TVis represented by o,,, and when Am_ and Ao, meXare replaced by A and A,, respectively. The msutts are (2u)* = (~,.L/2.16b)“~ CT mps = 0.0315(u/Ao)“‘“(L~b)‘~~.
(10)
(11)
Minimum column length To find the minimum column length L,, required to accomplish a specified degree of separation A and produc-
2027
2028
Shorter
Table 1. Comparison
of separations,
A,,,
Communications
and A obtainable at the best plate spacing respectively C, = 0.1 or 0.9
B x 10s (kg s-i)
(2)
0.817 1.634 3.268
8.90 8.40 7.60
(20)*x10” (m)
AT (K)
0.697 0.753 0.886
Table 2. Comparison
of production
(2w)* and at (2~) = 9 x 10m4 m,
CF = 0.3 or 0.7
CF = 0.5
A
A (0;;
VSX (h)
29.7 32,l 37.7
15.83 11.64 8.55
3.19 3.04 2.74
5.68 4.14 3.06
7.46 7.05 6.38
13.20 9.74 7.16
rates, (T,,, and 6, obtainable at the best plate spacing(2o)* respectively
8.90 8.40 7.60
15.83 11.64 8.55
and at (20) = 9 x 10e4 m,
A (%) Cr = 0.1 or 0.9
Cr = 0.3 or 0.7
c, = 0.5
(2,
bX 10s (kg s-‘)
(2w)* X 103 (m)
?IG
UnrS,X 10s (kg s-‘)
3.19 3.04 2.74
7.46 7.05 6.38
8.90 8.40 7.60
8.90 8.40 7.60
0.817 1.634 3.268
1.096 1.080 1.053
46.1 46.0 44.9
2.985 3.402 4.269
Table
3. Minimum
column
length
Lmi,
obtainable at (2w)=9x10m4m
the
best
plate
spacing
(2w)*.
L = 1.85m
at
A (%) CF = 0.1 or 0.9
CF = 0.3 or 0.7
CF = 0.5
(2)
OX105 (kg s- ‘)
(2w)’ x 103 (m)
3.19 3.04 2.74
7.46 7.05 6.38
8.90 8.40 7.60
8.90 8.40 7.60
0.817 1.634 3.268
0.619 0.705 0.794
tion rate Q, we rearrange column length L=
eq. (1) into a form explicit
in the
- 26(2w)9 0
ln[l--$$I.
(12)
Minimization of L with respect to (20) at constant A, A0 u yields an expression identical with eq. (5). Therefore, solution for this optimal condition is identical with (7H9) when L is replaced by &. and when A,.,.. A,,,_ are replaced by A,, and A, respectively. The results
and the eqs and are
(2w)* = (~L,;./2.16b)“~
(13)
L,,,, = 15.91 (~A,/L@‘~ (b/o). DlSCUSSlON
(14)
AND CONCLUSlONS
The improvement in performance resulting from operating at the best plate spacing with fixed operating expense, may be illustrated numerically by using the experimental data of Chueh and Yeh’s work (1967). The conditions are: benzene and n-heptane system; AT = 69°F = 38.3 K; (2w)=9~10-~m; L= 1.85m; Ho= 1.4~10~skgs-‘; K0 = 6.98 x low5 kg m s- ‘. If the operating expense is kept AT/(2c0) = 38.3/(9x 10-4) = 4.26 x IO’ i.e., unchanged Km-‘, then o = H,/(~uJ)~
= 2.39 x 10’ kgm-’
b = I&/(~w)~
= 1.8 x 10z3 kgm8s-‘.
g 26.4 30.0 33.8
L (r;;i” 0.632 1.029 1.496
feed concentrations, is shown in Table 1. It is seen from Table 1 that the best plate spacing for maximum separation increases as the flow rate increases. The improvement in separation is really obtained, especially for lower flow-rate operation. The comparison of production rates, o,,,.. and tr, obtainable at the best corresponding plate spacing (20)* and at (2~) = 9 x IO-‘m, respectively, under various feed concentration and degree of separation is presented in Table 2. It is shown in Table 2 that the best plate spacing for maximum production rate increases slightly when the specitied degree of separation increases. The improvement in production rate is really obtained, especially for higher degree of separation. The improvement in production rate under higher specified degree of separation will lead to increasing the required (optimum) plate spacing. Although the plate spacing in a thermal diffusion column is generally so small that changing (2~) will not cause any additional fixed charge. However, increasing (2~) will also lead to increasing AT in order to maintain the AT/2a constant and, therefore, some additional cost may be needed to maintain the higher AT. Table 3 shows the minimum column length &, and the optimum corresponding plate spacing (2w)* under various flow rates, feed concentrations and specified degree of separation. The differences of plate temperatures AT needed to maintain the constant operating expense are also presented.
s-’
The comparison of separation, Amal and A, obtainable at the best corresponding plate spacing (2w)* and at (2~) = 9 x lo-* m, respectively, under various Row rates and
Department ofChemical Engineering Tamkang Uniuersity Tamsui Taiwan, ROC
HO-MING
YEH
Shorter
2029
Communications
maximum value of b, kg s- ’ one half of the plate spacing of the column, m optimum plate spacing for the best performance, m
NOTATION (1
b B c C& CT C, D 9 HO k & L L,i” T
AT
defined by eq. (2), kg II- ’ s-l system constant defined by eq. (3), kg m-as_1 column width, m fractional mass concentration of component 1 C in the product stream existing from the stripping, enriching section C in the feed stream ordinary diffusion coefficient, m2 s- ’ gravitational acceleration, m se2 system constant defined by eq. (2), kg s- ’ thermal conductivity, W m _ I K- ’ system constant defined by eq. (3), kg m s-l column length, m minimum value of L, m reference temperature, K difference in temperature of hot and cold surfaces, K
system
constant
thermal diffusion constant -(l/p)(ap/ar),,kgm-“K~’ CT--c, A obtained when 0.3 i C i 0.7 maximum value of A, A0 viscosity, CP mass density, kg mm3 mass flow rate, kg s- ’
REFERENCES
Chueh, P. L. and Yeh, H. M., 1967, Thermal diffusion in a flat-plate column inclined for improved performance. A.1.Ch.E. J. 13, 37.
Clusius, K. and Dickel, G., 1938, New process for separation of gas mixtures and isotopes. Noturwiss. 26, 546(L). Furry, W. H., Jones, R. H. and Onsager, L., 1939, On the theory of isotope separation by thermal diffusion. Phys. Rev. 55, 1083. Lorenz, M. and Emery, A. M., Jr., 1959, The packed thermal diffusion column. Chem. Engng Sci. 11, 16. Powers, J. E. and Wilke, C. R., 1957, Separation of liquid by thermal diffusion. A.J.Ch.E. J. 3, 213. Ramser, J. H., 1957, Theory of thermal diffusion column under Linear fluid shear. Ind. Engng Chem. 49(l), 155. Sullivan, L. J., Ruppel, T. C. and Willingham, C. B., 1955, Rotary and packed thermal diffusion fractionating columns for liquids. Ind. Engng Chem. 47, 208. Washall, T. A. and Molpolder, F. W., 1962, Improving the separation efficiency of liquid thermal diffusion columns, Ind. Engng Chem. Pmt. Des. Dev. 1, 26. Yeh, H. M. and Yeh, Y. T., 1982, Separation theory in improved thermal diffusion columns. The Chem. Engng J. 25, 55.
Pergamon
Chemuol Enginwring Science,Vol. 49, No 12, pp. 2029-2031.1394 Copyri~t 0 1994 Elswia Sciena Ltd Printed in Orcal Bribin. AU rights rcaervcd @xl!-2509/94 $7.00 + 0.00
0009-2509(94)EOO43-C
Continuous
flow stirred tank reactor with two inflows of reactants: a versatile tool for study of bifurcation in chemical systems (Received
25 November
1993; accepted for publication
The use of the continuous-flow stirred tank reactor (CSTR) has had a revolutionary impact on the study of various bifurcation phenomena occurring in chemical reatiing systems. Many exotic dynamic phenomena, such as the multistability of stationary states and isothermal chemical oscillations, were discovered only after the CSTR mode was employed in experimental study and theoretical modelling. So far, the most extensively used CSTRs in the study of nonequilibrium chemistry are those with single inflow of reactants, and very rich bifurcation phenomena have been shown to occur under both isothermal and nonisothermal conditions (Uppal et al., 1974; Balakotaiah and Luss, 1983; Gray and Scott, 1983). In this paper, we will show by analyzing a single isothermal autocatalytic reaction that additional bifurcations may occur in CSTR if an additional inflow of reactants is introduced. Consider the reversible trimolecular autocatalytic reaction 2X+Ys3X taking
place in a CSTR
(1)
kI
with two inflows
of reactants.
The
governing
mass-balance
dx/dt dyldr
17 December
= k,x’y
1993)
equations
- k2x3 + j,x,
= - klx”y
+ U’
for the concentrations + jlxl
+.ilyt
- ( jl + jz)x
+jzyz
- (j, +_i&
are (2) (3)
where x and y are the concentrations of X and Y in the reactor, xi, x2 and yi, yz are their concentrations in the two inflows, respectively, andj, and jl are the inverse of the mean residence time of two inflows. To analyze the behavior of eqs (2) and (3). we notice first that d(x + y)/dr = (jl + j,)Cc - (x + Y)I
(4)
where c = Cjl(xl
+ yl) +j2(x2
+ y2)l/(j1
+j2).
(5)
By solving eq. (4), one may find that (x + y) will decay exponentially to c. We thus assume that x + Y = Cjl(xl is satisfied
+ ~~1 +h(xl
at all time if it is satisfied
f y2)l/(j1
+j,)
at the initial time.
(6)
INTRODUCTION A more detailed study of the mechanism of separation in
thermal diffusion column indicates that the convective currents, in addition to the desirable cascading effect, also produce an undesirable remixing effect (Clusius and Dickel, 1938). Therefore, proper control of the convective strength might effectively suppress this undesirable remixing effect while still preserving the desirable cascading effect, and thereby lead to improved separation. Based on this concept, some improved columns have been developed in the literature, (Sullivan et al., 1955; Ramser, 1957; Powers and Wilke, 1957; Lorenz and Emery, 1959, WashaIl and Molpolder, 1962; Chueh and Yeh, 1967). In developing these improved columns, a number of studies of the operating variables in the thermal diffusion column have been made. There is still an important term, plate spacing, which affects separation ef&iency and which has hardly ever been discussed. COLUMN THEORY Furry et al. (1939) obtained the equation ofseparation for a binary mixture in a contjnuous-flow thecmal diffusion column, applicable only to equifraction solution (0.3 < c f 0.7):
Maximum separation The optimum plate spacing (2c0)* for a maximum separation A,,, is obtained by partially differentiating eq. (4) with respect to (20) and setting aAja(2w) = 0. After diffecentiation and simplification we obtain 8A0/8(2w) = 0 and thus, from eq. (1) 8=1+3x
Solving
for (2w)*, one obtains (2w)* =
pfr$g*NAT/24* = 9!&.2
Ko f(Z&.
(3)
The equation of separation applicable to whole range of’ concentration (0 6 C G 1) was iater derived by Yeh and Yeh (1982) as A=
+ 12C,(l-
C,)
IL? 1
i 2.16b
119
1
a
(7)
where
where
=
UL
Consequently, the maximum separation may be obtained from eqs (1) and (4) by the substitution of eq. (7). The result is
A,.,.*
b
(3
where
(4)
The plate spacing (ZUJ) in a thermal diffusion column is generally so small that changing (20) will not cause any additional fixed charge. The expenditure of making a separation by thermal diffusion essentially includes two parts: a fixed charge and an operating expense. The fixed change is roughly proportional to the equipment cost, while the operating expense is chiefly heat. The transfer rate is obtainable from the expression, kBL (A i”,Gwj. Based on these terms, we shall take account of the influence of plate-spacing change on the degree of separation, the output and the column length with the consideration of fixed operating expense (i.e. AT/20 as well as o and h ace constants).
= 0.215
(o/a)(aLlh)J’~ .
(9)
It should be noted from eqs (7)-(9) that whereas AO,_ as well as A,,, depends on the thermal diffusion constant a, (2(o)* is independent of a. The problem of finding the maximum separation A,., and the best plate spacing (2c0)+ for a specified flow rate (r can readily be solved by using eqs (7H9) since a, b, and L are known constants for a given column and system.
Maximum output The plate spacing
for maximum separation is also the plate spacing required to obtain the maximum pr~uetjon rate c,., for giLen column which is to give a specified degree of separation A. Although eq. (4) cannot be put into a form explicit in o, it is nevertheless possible to maximize o with respect to (2~) at constant A, & and I.. The maximization yields an expression which is identical with eq. (5) and consequently the solution for the optimum plate spacing (Zw)* for maximum output is identical to that given by eqs 17H9) when TVis represented by o,,, and when Am_ and Ao, meXare replaced by A and A,, respectively. The msutts are (2u)* = (~,.L/2.16b)“~ CT mps = 0.0315(u/Ao)“‘“(L~b)‘~~.
(10)
(11)
Minimum column length To find the minimum column length L,, required to accomplish a specified degree of separation A and produc-
2027
2028
Shorter
Table 1. Comparison
of separations,
A,,,
Communications
and A obtainable at the best plate spacing respectively C, = 0.1 or 0.9
B x 10s (kg s-i)
(2)
0.817 1.634 3.268
8.90 8.40 7.60
(20)*x10” (m)
AT (K)
0.697 0.753 0.886
Table 2. Comparison
of production
(2w)* and at (2~) = 9 x 10m4 m,
CF = 0.3 or 0.7
CF = 0.5
A
A (0;;
VSX (h)
29.7 32,l 37.7
15.83 11.64 8.55
3.19 3.04 2.74
5.68 4.14 3.06
7.46 7.05 6.38
13.20 9.74 7.16
rates, (T,,, and 6, obtainable at the best plate spacing(2o)* respectively
8.90 8.40 7.60
15.83 11.64 8.55
and at (20) = 9 x 10e4 m,
A (%) Cr = 0.1 or 0.9
Cr = 0.3 or 0.7
c, = 0.5
(2,
bX 10s (kg s-‘)
(2w)* X 103 (m)
?IG
UnrS,X 10s (kg s-‘)
3.19 3.04 2.74
7.46 7.05 6.38
8.90 8.40 7.60
8.90 8.40 7.60
0.817 1.634 3.268
1.096 1.080 1.053
46.1 46.0 44.9
2.985 3.402 4.269
Table
3. Minimum
column
length
Lmi,
obtainable at (2w)=9x10m4m
the
best
plate
spacing
(2w)*.
L = 1.85m
at
A (%) CF = 0.1 or 0.9
CF = 0.3 or 0.7
CF = 0.5
(2)
OX105 (kg s- ‘)
(2w)’ x 103 (m)
3.19 3.04 2.74
7.46 7.05 6.38
8.90 8.40 7.60
8.90 8.40 7.60
0.817 1.634 3.268
0.619 0.705 0.794
tion rate Q, we rearrange column length L=
eq. (1) into a form explicit
in the
- 26(2w)9 0
ln[l--$$I.
(12)
Minimization of L with respect to (20) at constant A, A0 u yields an expression identical with eq. (5). Therefore, solution for this optimal condition is identical with (7H9) when L is replaced by &. and when A,.,.. A,,,_ are replaced by A,, and A, respectively. The results
and the eqs and are
(2w)* = (~L,;./2.16b)“~
(13)
L,,,, = 15.91 (~A,/L@‘~ (b/o). DlSCUSSlON
(14)
AND CONCLUSlONS
The improvement in performance resulting from operating at the best plate spacing with fixed operating expense, may be illustrated numerically by using the experimental data of Chueh and Yeh’s work (1967). The conditions are: benzene and n-heptane system; AT = 69°F = 38.3 K; (2w)=9~10-~m; L= 1.85m; Ho= 1.4~10~skgs-‘; K0 = 6.98 x low5 kg m s- ‘. If the operating expense is kept AT/(2c0) = 38.3/(9x 10-4) = 4.26 x IO’ i.e., unchanged Km-‘, then o = H,/(~uJ)~
= 2.39 x 10’ kgm-’
b = I&/(~w)~
= 1.8 x 10z3 kgm8s-‘.
g 26.4 30.0 33.8
L (r;;i” 0.632 1.029 1.496
feed concentrations, is shown in Table 1. It is seen from Table 1 that the best plate spacing for maximum separation increases as the flow rate increases. The improvement in separation is really obtained, especially for lower flow-rate operation. The comparison of production rates, o,,,.. and tr, obtainable at the best corresponding plate spacing (20)* and at (2~) = 9 x IO-‘m, respectively, under various feed concentration and degree of separation is presented in Table 2. It is shown in Table 2 that the best plate spacing for maximum production rate increases slightly when the specitied degree of separation increases. The improvement in production rate is really obtained, especially for higher degree of separation. The improvement in production rate under higher specified degree of separation will lead to increasing the required (optimum) plate spacing. Although the plate spacing in a thermal diffusion column is generally so small that changing (2~) will not cause any additional fixed charge. However, increasing (2~) will also lead to increasing AT in order to maintain the AT/2a constant and, therefore, some additional cost may be needed to maintain the higher AT. Table 3 shows the minimum column length &, and the optimum corresponding plate spacing (2w)* under various flow rates, feed concentrations and specified degree of separation. The differences of plate temperatures AT needed to maintain the constant operating expense are also presented.
s-’
The comparison of separation, Amal and A, obtainable at the best corresponding plate spacing (2w)* and at (2~) = 9 x lo-* m, respectively, under various Row rates and
Department ofChemical Engineering Tamkang Uniuersity Tamsui Taiwan, ROC
HO-MING
YEH
Shorter
2029
Communications
maximum value of b, kg s- ’ one half of the plate spacing of the column, m optimum plate spacing for the best performance, m
NOTATION (1
b B c C& CT C, D 9 HO k & L L,i” T
AT
defined by eq. (2), kg II- ’ s-l system constant defined by eq. (3), kg m-as_1 column width, m fractional mass concentration of component 1 C in the product stream existing from the stripping, enriching section C in the feed stream ordinary diffusion coefficient, m2 s- ’ gravitational acceleration, m se2 system constant defined by eq. (2), kg s- ’ thermal conductivity, W m _ I K- ’ system constant defined by eq. (3), kg m s-l column length, m minimum value of L, m reference temperature, K difference in temperature of hot and cold surfaces, K
system
constant
thermal diffusion constant -(l/p)(ap/ar),,kgm-“K~’ CT--c, A obtained when 0.3 i C i 0.7 maximum value of A, A0 viscosity, CP mass density, kg mm3 mass flow rate, kg s- ’
REFERENCES
Chueh, P. L. and Yeh, H. M., 1967, Thermal diffusion in a flat-plate column inclined for improved performance. A.1.Ch.E. J. 13, 37.
Clusius, K. and Dickel, G., 1938, New process for separation of gas mixtures and isotopes. Noturwiss. 26, 546(L). Furry, W. H., Jones, R. H. and Onsager, L., 1939, On the theory of isotope separation by thermal diffusion. Phys. Rev. 55, 1083. Lorenz, M. and Emery, A. M., Jr., 1959, The packed thermal diffusion column. Chem. Engng Sci. 11, 16. Powers, J. E. and Wilke, C. R., 1957, Separation of liquid by thermal diffusion. A.J.Ch.E. J. 3, 213. Ramser, J. H., 1957, Theory of thermal diffusion column under Linear fluid shear. Ind. Engng Chem. 49(l), 155. Sullivan, L. J., Ruppel, T. C. and Willingham, C. B., 1955, Rotary and packed thermal diffusion fractionating columns for liquids. Ind. Engng Chem. 47, 208. Washall, T. A. and Molpolder, F. W., 1962, Improving the separation efficiency of liquid thermal diffusion columns, Ind. Engng Chem. Pmt. Des. Dev. 1, 26. Yeh, H. M. and Yeh, Y. T., 1982, Separation theory in improved thermal diffusion columns. The Chem. Engng J. 25, 55.
Pergamon
Chemuol Enginwring Science,Vol. 49, No 12, pp. 2029-2031.1394 Copyri~t 0 1994 Elswia Sciena Ltd Printed in Orcal Bribin. AU rights rcaervcd @xl!-2509/94 $7.00 + 0.00
0009-2509(94)EOO43-C
Continuous
flow stirred tank reactor with two inflows of reactants: a versatile tool for study of bifurcation in chemical systems (Received
25 November
1993; accepted for publication
The use of the continuous-flow stirred tank reactor (CSTR) has had a revolutionary impact on the study of various bifurcation phenomena occurring in chemical reatiing systems. Many exotic dynamic phenomena, such as the multistability of stationary states and isothermal chemical oscillations, were discovered only after the CSTR mode was employed in experimental study and theoretical modelling. So far, the most extensively used CSTRs in the study of nonequilibrium chemistry are those with single inflow of reactants, and very rich bifurcation phenomena have been shown to occur under both isothermal and nonisothermal conditions (Uppal et al., 1974; Balakotaiah and Luss, 1983; Gray and Scott, 1983). In this paper, we will show by analyzing a single isothermal autocatalytic reaction that additional bifurcations may occur in CSTR if an additional inflow of reactants is introduced. Consider the reversible trimolecular autocatalytic reaction 2X+Ys3X taking
place in a CSTR
(1)
kI
with two inflows
of reactants.
The
governing
mass-balance
dx/dt dyldr
17 December
= k,x’y
1993)
equations
- k2x3 + j,x,
= - klx”y
+ U’
for the concentrations + jlxl
+.ilyt
- ( jl + jz)x
+jzyz
- (j, +_i&
are (2) (3)
where x and y are the concentrations of X and Y in the reactor, xi, x2 and yi, yz are their concentrations in the two inflows, respectively, andj, and jl are the inverse of the mean residence time of two inflows. To analyze the behavior of eqs (2) and (3). we notice first that d(x + y)/dr = (jl + j,)Cc - (x + Y)I
(4)
where c = Cjl(xl
+ yl) +j2(x2
+ y2)l/(j1
+j2).
(5)
By solving eq. (4), one may find that (x + y) will decay exponentially to c. We thus assume that x + Y = Cjl(xl is satisfied
+ ~~1 +h(xl
at all time if it is satisfied
f y2)l/(j1
+j,)
at the initial time.
(6)