- Email: [email protected]
Asymptotic solutions for laminar dispersion in circular tubes
Chemical Engineering Science 55 (2000) 849}855
Asymptotic solutions for laminar dispersion in circular tubes J.S. Vrentas*, C.M. Vrentas Department of Chemical Engineering, The Pennsylvania State University, 133 Fenske Laboratory, University Park, PA 16802-4400, USA Received 1 July 1998; accepted 14 March 1999
Abstract A short-time asymptotic solution is developed for the dispersion of a passive solute in the fully developed laminar #ow of a Newtonian #uid through a straight circular tube. The new solution describes the axial dependence of the average concentration both near the leading and trailing edges of the concentration distribution, and it predicts the existence of a pronounced maximum near the trailing edge. ( 1999 Elsevier Science Ltd. All rights reserved. Keywords: Short-time solution; Laminar dispersion
1. Introduction A large number of theoretical investigations have been concerned with the dispersion of a passive solute in the fully-developed laminar #ow of a Newtonian #uid through a straight circular tube. The convective dispersion process, which plays an important role in many transport problems, is described by a linear boundary value problem. Unfortunately, however, no simple, exact, analytical solution has been presented either for the local concentration distribution or for the mean concentration at a particular axial position. As noted previously (Vrentas & Vrentas, 1988), it is convenient to divide the available solutions into three categories: numerical solutions, exact series solutions, and asymptotic solutions. A reasonably detailed discussion of previous theoretical studies has been presented earlier (Vrentas & Vrentas, 1988), but it is useful here to summarize the available solutions in each category and to comment on their general utility. Although numerical solutions can of course be very accurate, they are not particularly convenient since a new solution is required for each di!erent set of conditions. Series solutions achieve the status of exact solutions as the number of terms in the series becomes large. Probably, the most useful exact series solutions are those presented by Yu (1981) and by Shankar and Lenho! (1989). However, the evaluation of the
* Corresponding author. Tel.: 001-814-863-4808; fax: 001-814-8657846.
terms of the series requires a signi"cant numerical e!ort so that such series solutions are also not particularly convenient. These series solutions can be used, however, to assess the range of validity of asymptotic solutions, and the solutions of Yu and of Shankar and Lenho! are very useful in the evaluation of such solutions. Asymptotic solutions to the convection dispersion problem are relatively simple analytical expressions which are valid for short or long times or which give reasonable approximations to the exact solution at low or high values of the Peclet number. The pioneering work of Taylor (1953,1954) and Aris (1956) led to the "rst solution of this type which is valid asymptotically at su$ciently large values of time. Yu (1981) used his series solution to conclude that this long-time solution was valid at values of dimensionless time t'0.7 over the entire range of Peclet numbers. Viable short-time asymptotic solutions have been developed by Lighthill (1966) and by Chatwin (1976). The solution of Lighthill is further restricted to high values of the Peclet number since axial di!usion is neglected. Chatwin derived a short-time solution which is valid for all values of the Peclet number by including the axial di!usion term in the species continuity equation. However, consideration of the average concentration pro"le at a particular instant of time indicates that these solutions describe the dependence of the average concentration on axial position only near the leading edge of the concentration distribution. These solutions thus do not predict the pronounced maximum observed by both Yu (1981) and Shankar and Lenho! (1989) near the trailing edge of the concentration pro"le
0009-2509/00/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 3 5 5 - 3
850
J.S. Vrentas, C.M. Vrentas / Chemical Engineering Science 55 (2000) 849}855
Nomenclature C CH CM C e CM e C f C i C n D ¸ M 0 Pe
dimensionless mass density of solute ("CH!C /C !C ) i f i mass density of solute radial average solute concentration de"ned by Eq. (7) Fourier transform of C Fourier transform of CM ("C #M /pR3) i 0 mass density of solute in mixture before injection nth order approximation for C binary mutual di!usion coe$cient length scale for axial transport at long times mass of injected solute Peclet number ("R; /D) c
at early times. Thus, the solutions of Lighthill and Chatwin are not complete short-time solutions. In a typical laminar dispersion experiment, a pulse of solute is injected into a solvent #owing down a tube. The initial solute pulse is deformed by #ow and di!usion as it is carried down the tube. At early times, the front edge of this pulse can be identi"ed as the leading edge, and the back edge can be called the trailing edge. Vrentas and Vrentas (1988) developed a perturbation series solution for the mean concentration for low values of Pe, the Peclet number. The derived perturbation solution is valid at both short and long times for su$ciently low values of the Peclet number (Pe)15). In addition, the series is valid at any value of the Peclet number for su$ciently small values of time. The time interval for validity of the series decreases as the Peclet number increases. For high values of the Peclet number, the solution can be used only for very short times. The leading terms in this perturbation solution is the solution for pure di!usion in the axial direction. Finally, Taylor (1953) developed a pure convection solution which should of course be valid for large values of Pe where the mass transfer process is dominated by convection e!ects. The objective of this paper is to derive a complete short-time solution for laminar dispersion in a tube which is valid for all values of the Peclet number. This derived solution provides a valid description, for su$ciently short times, of the dependence of the average concentration on axial position both near the leading and trailing edges of the concentration distribution. The laminar dispersion problem is formulated in Section 2, and a new short-time solution is developed in Section 3. The general predictions of the asymptotic short-time solution are presented in Section 4.
q r rH R t tH ; c zH
quantity de"ned by Eq. (21) dimensionless radial distance ("rH/R) radial distance variable radius of tube dimensionless time ("DtH/R2) time velocity at the center of the tube axial distance variable
variable
Greek letters a d e j
Fourier transform variable Dirac delta function small perturbation parameter ("R/¸) dimensionless axial distance variable ("zH!; tH/2/R) c
2. Problem Formulation The equation set used for the dispersion problem considered here is based on the following assumptions: (1) The di!usion process is isothermal. (2) There is laminar, axisymmetric #ow in an in"nitely long circular tube. (3) The #uid is an incompressible, two-component Newtonian #uid. (4) The amount of solute in the system is su$ciently small that all of the physical properties of the system are e!ectively constant. (5) The solute does not undergo any chemical reaction. (6) There is a concentrated initial solute input distributed uniformly over the cross section of the tube at the origin of the stationary coordinate system (zH"0). For these conditions, the convective dispersion process is described by the following form of the species continuity equation for the solute subject to the following boundary conditions:
A
B
A B
LC 1 LC 1 L LC L2C #Pe !r2 " r # , Lt 2 Lj r Lr Lr Lj2
(1)
LC (0, j, t)"0, Lr
(2)
LC (1, j, t)"0, Lr
(3)
C(r, !R, t)"0,
(4)
C(r, #R, t)"0,
(5)
C(r, j, 0)"d(j)
(6)
In these equations, the axial coordinate j is de"ned so that it has an origin moving with the average #ow
J.S. Vrentas, C.M. Vrentas / Chemical Engineering Science 55 (2000) 849}855
velocity. In this study, an equation will be derived for the average solute concentration which is de"ned as follows:
P
1 Cr dr (7) 0 In this paper, the following approach will be used to develop asymptotic solutions to the laminar dispersion problem. First, integration of the above equations produces an exact result for CM in terms of integrals and derivatives of C. Second, short-time and long-time approximations for C are derived using the above set of equations. Finally, the asymptotic approximations for C are substituted into the equation for CM to produce short-time and long-time asymptotic solutions for CM (j, t). The long-time result derived using this approach will of course be the Taylor}Aris solution, and the short-time result is a new, complete short-time asymptotic solution. Multiplication of Eqs. (1) and (4)}(6) by r, integration from r"0 to 1, and utilization of Eq. (2) produce the following equation set for CM : CM "
PA
B
A B
1 1 LCM LC LC L2CM #Pe r !r2 dr" # , Lt 2 Lj Lr Lj2 0 r/1 CM (!R, t)"0,
(9)
d(j) . CM (j, 0)" 2
(11)
These equations can be used to derive asymptotic solutions for CM if short-time and long-time approximations are used for the axial derivative in the integral and for the radial derivative at the wall. In the long-time limit, the solute interacts with the walls, and Eq. (3) must be used as the proper boundary condition at r"1. In this case, the second term on the right-hand side of Eq. (8) must be deleted. At short times, however, it is reasonable to follow Lighthill (1966) and Chatwin (1976) and suppose that the solute interaction with the tube walls is small. In this case, it is not critical to impose the proper boundary condition of zero #ux across the tube wall. The utilization of a short-time asymptotic solution for C which has (LC/Lr) O0 should thus not lead to a signi"cant r/1 amount of error if su$ciently short times are considered. Since the radial derivative is not equal to zero at the tube wall, there of course is a local failure in the approximate solution. However, for the solution that is derived, the mass of solute is conserved in an overall sense so that the principle of mass conservation is not violated. Solution of the equation set for CM is facilitated if an exponential Fourier transform is introduced (Churchill, 1972): =
CM (j, t)e~*aj dj. ~=
PA
B
A B
1 1 dCM LC e#a2CM "!Pe ia r !r2 C dr# e , e e dt 2 Lr 0 r/1 (13) CM (0)"1. (14) 2 e As noted above, substitution of short-time or long-time approximations for C into Eq. (13) leads to short-time or e long-time solutions for CM . Eqs. (13) and (14) are of course e exact results for CM . Since only some of the terms in Eq. e (13) are approximated using short-time or long-time approximations for C , it seems reasonable to expect e that the asymptotic solutions for CM should be more e widely applicable than the asymptotic approximations for C which will be derived under more restrictive e conditions.
3. Problem solution
(10)
P
The transformed equation set for CM takes the following e form:
(8)
CM (#R, t)"0,
CM (a, t)" e
851
(12)
In this section, both short-time and long-time asymptotic solutions are developed. The short-time solution, which is the new result, is derived "rst, and, then, it is shown that the same general approach can be used to derive the well-known Taylor}Aris long-time solution. Chatwin (1976) noted that the interaction between axial di!usion and convection dominates the dispersion process at very short times, but, as time increases, the interaction between radial di!usion and convection is the dominant mechanism in the mass transfer process. Hence, it seems reasonable to derive a short-time asymptotic solution for C by neglecting the radial di!usion term
A
B
LC 1 LC L2C #Pe !r2 " , Lt 2 Lj Lj2
(15)
so that the time dependence of C is described by the interaction of the convective #ow with the axial di!usion. A solution of Eq. (15) subject to Eqs. (4)}(6) can easily be constructed using the exponential Fourier transform. The transformed concentration C can be derived by e solving the equation
A
B
dC 1 e#Pe !r2 iaC "!a2C , e e dt 2
(16)
subject to the initial condition: C (0)"1. e The solution is simply
C G
(17)
A
1 C "exp ! a2#Pe ia !r2 e 2
BH D
t ,
(18)
852
J.S. Vrentas, C.M. Vrentas / Chemical Engineering Science 55 (2000) 849}855
and straightforward inversion yields a short-time approximation for the solute concentration distribution in the tube exp[!Mj!Pe t(1/2!r2)N2/4t] C" . 2p1@2t1@2
(19)
Substitution of the equation for C , Eq. (18), into e Eq. (13) gives the following di!erential equation:
C
D
dCM eqt#e~qt e~qt!eqt e#a2CM "e~a2t # #4qte~a2teqt, e 4t 4qt2 dt (20) Pe ia q" . 2
(21)
Solution of the equation subject to the initial condition, Eq. (14), yields the following expression: e~a2t[eqt!e~qt] CM " e 4qt
CA B D
(22)
This result can be rewritten as follows: 2Pe tCM "2e~a2t e
C
D
sin(Pe at/2) a
#16te~a2t
C
!16te~a2t
sin(Pe at/2) a
D
1!cos(Pe at/2) ia
C D
Pe iat . 2
#8Pe t2e~a2t exp
0.4 t( . (Pe)2@3
(23)
(25)
In addition, a comparison of terms in the equation for C shows that the e!ect of the interaction of the solute with the tube wall should not be signi"cant for times satisfying the following inequality: t(0.001.
1 1 #4e~a t eqt t! # . q q 2
process is dominated by axial di!usion and convection since it is based on a solution for C which neglects radial di!usion. All terms are retained in the exact result for the average concentration, even though some are clearly approximated. Hence, the solution for CM does include an approximate characterization of radial di!usion and can hence be regarded as a theoretical result for which radial di!usion slightly perturbs the basic results of the interaction of convection and axial di!usion. The time interval for which axial di!usion and convection are the dominant mass transfer mechanisms can be estimated using the results of Chatwin (1976):
(26)
This equation is derived by requiring that the radial derivative term at the wall be less than 0.5 per cent of the convective term. Consequently, the new asymptotic short-time solution, Eq. (24), should be valid from t"0 to a time which is the lesser of the times calculated from Eqs. (25) and (26). In the long-time limit, the importance of the individual terms in Eq. (1) can be ascertained by following Taylor (1954) and introducing a length ¸, the distance over which the major part of the change in concentration takes place in the axial direction. If the quantity ¸ is used as the length scale for time and for the axial derivatives, then it follows that the terms in Eq. (1) can be ordered as follows:
A B
Inversion of this expression using the convolution integral for exponential Fourier transforms produces the following short-time asymptotic solution:
1 L LC r "O(1), r Lr Lr
(27)
2Pe tCM
L2C "O(e2), Lj2
(28)
LC "O(e2), Lt
(29)
erf [Pe t1@2/4!j/2t1@2]#erf [Pe t1@2/4#j/2t1@2] " 2 !8terf
C A B C
D
Pe t1@2 j # 4 2t1@2
A
Pe
j #8terf 2t1@2
D
4Pe t3@2 (j#Pe t/2)2 # exp ! . p1@2 4t
(24)
This short-time asymptotic solution for CM should be valid in the initial time period where the dispersion
B
1 LC !r2 "O(e). 2 Lj
(30)
Here, e"R/¸ and e@1 in the long-time limit. Hence, a zero-order approximation C for the concentration is 0 the solution of the equation
A B
L LC r 0 "0, Lr Lr
(31)
J.S. Vrentas, C.M. Vrentas / Chemical Engineering Science 55 (2000) 849}855
subject to Eqs. (2) and (3), and a "rst-order approximation C is the solution of the equation 1 1 L LC 1 LC 0, r 1 "Pe !r2 (32) r Lr 2 Lj Lr
A B A
B
853
The solution of Eqs. (14) and (38) is simply
C A
Pe2 CM "1exp !a2t 1# e 2 192
BD
,
(39)
and inversion gives the following result:
subject again to Eqs. (2) and (3). The solution for C is 0 simply
exp [!j2/4t(1#Pe2/192)] . CM " 4p1@2[t(1#Pe2/192)]1@2
C "f (j, t), (33) 0 0 and the solution for C can be expressed as follows: 1 Lf r2 r4 C "Pe 0 ! #f (j, t). (34) 1 1 16 Lj 8
This is the well-known Taylor}Aris solution. It is shown here how the method proposed for derivation of the short-time asymptotic solution can also be used to recover the proper long-time asymptotic solution.
This expression represents a long-time, "rst-order approximation for C and, hence, for C : e r2 r4 C "Pe ! i a f (a, t)#f (a, t). (35) e 0e 1e 8 16
4. Results and discussion
A
A
B
B
Since the lowest order result for CM is e f CM " 0e, e 2
(36)
the long-time, "rst-order approximation for C can be e expressed as follows:
A
B
r2 r4 C "2Pe ! i a CM #f (a, t). e e 1e 8 16
(37)
The quantity CM must satisfy Eqs. (13) and (14) with the e radial wall derivative set equal to zero in Eq. (13) because of the importance of solute interaction with the wall. Substitution of Eq. (37) into Eq. (13) and integration produce the di!erential equation:
CA
dCM Pe2 e#CM a2 1# e dt 192
BD
"0.
(38)
(40)
The general predictions of the new short-time asymptotic solution are now compared to the predictions of the pure axial di!usion solution, the pure convection solution, and the exact series solutions of Yu (1981) and of Shankar and Lenho! (1989). The pure axial di!usion solution can be expressed as follows: e~j2@4t CM " , 4p1@2t1@2
(41)
and the pure convection solution takes the following form: 2j 2Pe tCM "1, 1' '!1, Pe t 2Pe tCM "0,
K K
2j '1. Pe t
(42) (43)
As noted previously (Vrentas & Vrentas, 1988), axial molecular di!usion predominates over axial convection even for large values of the Peclet number if t is su$ciently small. This is illustrated in Fig. 1 where the axial
Fig. 1. Axial dependence of average concentration for Pe"104 and t"10~8. Curve is the short-time asymptotic solution, and solid points are the predictions of the pure axial di!usion solution [Eq. (41)].
854
J.S. Vrentas, C.M. Vrentas / Chemical Engineering Science 55 (2000) 849}855
Fig. 2. Axial dependence of average concentration for Pe"104 and t"10~5. Solid curve is the pure convection solution [Eqs. (42) and (43)], and dotted curve is the short-time asymptotic solution. Fig. 3. Axial dependence of average concentration for Pe"104 and t"0.001. Complete concentration distribution is shown.
dependence of CM is presented for Pe"104 and t"10~8. For this very low value of time, the short-time asymptotic solution is essentially identical to the pure axial di!usion solution even though the Peclet number is very high. When the time is increased to 10~5 for Pe"104, the average concentration distribution computed from the short-time asymptotic solution is close to the pure convection solution. This is clearly illustrated in Fig. 2 which shows that convection becomes the dominant transport mechanism. As the time is further increased, it is to be expected that the concentration distribution will become asymmetric before approaching the symmetric Taylor}Aris solution at su$ciently long times. For Pe"104, it is evident from Eqs. (25) and (26) that the maximum time for which the asymptotic short-time solution, Eq. (24), is valid is about 0.001. The predicted average concentration distribution for Pe"104 and t"0.001 is presented in Figs. 3 and 4. For this large value of Pe, the concentration distribution is very similar to the pure convection solution with the exception of a pronounced maximum near the trailing edge. The general shape of the average concentration pro"le is similar to the results presented by Yu (1981) for Pe"104 and t"0.02 and by Shankar and Lenho! (1989) for Pe"R and t"0.002. No direct comparison is possible because the lowest time reported for the exact series solutions is t"0.002. However, the height of the maximum reported by Shankar and Lenho! for Pe"R and t"0.002 is comparable to the height of the maximum reported here
Fig. 4. Axial dependence of average concentration for Pe"104 and t"0.001. Concentration distribution near pronounced maximum is shown.
J.S. Vrentas, C.M. Vrentas / Chemical Engineering Science 55 (2000) 849}855
855
Fig. 5. Axial dependence of average concentration for Pe"10 and t"0.001. Curve is the short-time asymptotic solution and solid points are the pure di!usion solution [Eq. (41)].
for Pe"104 and t"0.001 Shankar and Lenho! noted that large series expansions are needed at very short times, and, hence, analytical solutions are more economical. Consequently, the short-time asymptotic solution presented here should be a useful complement to the exact series solutions of Yu and of Shankar and Lenho!. When the Peclet number is low, the transport process is dominated by axial di!usion even for the largest time (t"0.001) for which the asymptotic solution is valid. This is illustrated in Fig. 5 which shows that the shorttime asymptotic solution is virtually identical to the pure axial di!usion solution for Pe"10 even for the maximum allowable time, t"0.001. From the above discussion, it is fair to conclude that the proposed short-time asymptotic solution, Eq. (24), provides a valid description of laminar dispersion in a circular pipe for the time interval given by Eqs. (25) and (26). The proposed solution is a complete solution in that it describes the axial dependence of the average concentration both near the leading and trailing edges of the concentration distribution. Previous short-time solutions (Lighthill, 1966; Chatwin, 1976) provided valid results only near the leading edge of the concentration pro"le.
Acknowledgements This work was supported by funds provided by the Dow Chemical Company. References Aris, R. (1956). On the dispersion of a solute in a #uid #owing through a tube. Proceedings of the Royal Society, A 235, 67}77. Chatwin, P. C. (1976). The initial dispersion of contaminant in Poiseuille #ow and the smoothing of the snout. Journal of Fluid Mechanics, 77, 593}602. Churchill, R. V. (1972). Operational mathematics. New York: McGrawHill. Lighthill, M. J. (1966). Initial development of di!usion in Poiseuille #ow. Journal of the Institution of Maths Applications, 2, 97}108. Shankar, A., & Lenho!, A. M. (1989). Dispersion in laminar #ow in short tubes. A.I.Ch.E. Journal, 35, 2048}2052. Taylor, G. (1953). Dispersion of soluble matter in solvent #owing slowly through a tube. Proceedings of the Royal Society, A 219, 186}203. Taylor, G. (1954). Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular di!usion. Proceedings of the Royal Society, A 225, 473}477. Vrentas, J. S., & Vrentas, C. M. (1988). Dispersion in laminar tube #ow at low Peclet numbers or short times. A.I.Ch.E. Journal, 34, 1423}1430. Yu, J. S. (1981). Dispersion in laminar #ow through tubes by simultaneous di!usion and convection. Journal of Applied Mechanics, 48, 217}223.
Asymptotic solutions for laminar dispersion in circular tubes J.S. Vrentas*, C.M. Vrentas Department of Chemical Engineering, The Pennsylvania State University, 133 Fenske Laboratory, University Park, PA 16802-4400, USA Received 1 July 1998; accepted 14 March 1999
Abstract A short-time asymptotic solution is developed for the dispersion of a passive solute in the fully developed laminar #ow of a Newtonian #uid through a straight circular tube. The new solution describes the axial dependence of the average concentration both near the leading and trailing edges of the concentration distribution, and it predicts the existence of a pronounced maximum near the trailing edge. ( 1999 Elsevier Science Ltd. All rights reserved. Keywords: Short-time solution; Laminar dispersion
1. Introduction A large number of theoretical investigations have been concerned with the dispersion of a passive solute in the fully-developed laminar #ow of a Newtonian #uid through a straight circular tube. The convective dispersion process, which plays an important role in many transport problems, is described by a linear boundary value problem. Unfortunately, however, no simple, exact, analytical solution has been presented either for the local concentration distribution or for the mean concentration at a particular axial position. As noted previously (Vrentas & Vrentas, 1988), it is convenient to divide the available solutions into three categories: numerical solutions, exact series solutions, and asymptotic solutions. A reasonably detailed discussion of previous theoretical studies has been presented earlier (Vrentas & Vrentas, 1988), but it is useful here to summarize the available solutions in each category and to comment on their general utility. Although numerical solutions can of course be very accurate, they are not particularly convenient since a new solution is required for each di!erent set of conditions. Series solutions achieve the status of exact solutions as the number of terms in the series becomes large. Probably, the most useful exact series solutions are those presented by Yu (1981) and by Shankar and Lenho! (1989). However, the evaluation of the
* Corresponding author. Tel.: 001-814-863-4808; fax: 001-814-8657846.
terms of the series requires a signi"cant numerical e!ort so that such series solutions are also not particularly convenient. These series solutions can be used, however, to assess the range of validity of asymptotic solutions, and the solutions of Yu and of Shankar and Lenho! are very useful in the evaluation of such solutions. Asymptotic solutions to the convection dispersion problem are relatively simple analytical expressions which are valid for short or long times or which give reasonable approximations to the exact solution at low or high values of the Peclet number. The pioneering work of Taylor (1953,1954) and Aris (1956) led to the "rst solution of this type which is valid asymptotically at su$ciently large values of time. Yu (1981) used his series solution to conclude that this long-time solution was valid at values of dimensionless time t'0.7 over the entire range of Peclet numbers. Viable short-time asymptotic solutions have been developed by Lighthill (1966) and by Chatwin (1976). The solution of Lighthill is further restricted to high values of the Peclet number since axial di!usion is neglected. Chatwin derived a short-time solution which is valid for all values of the Peclet number by including the axial di!usion term in the species continuity equation. However, consideration of the average concentration pro"le at a particular instant of time indicates that these solutions describe the dependence of the average concentration on axial position only near the leading edge of the concentration distribution. These solutions thus do not predict the pronounced maximum observed by both Yu (1981) and Shankar and Lenho! (1989) near the trailing edge of the concentration pro"le
0009-2509/00/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 3 5 5 - 3
850
J.S. Vrentas, C.M. Vrentas / Chemical Engineering Science 55 (2000) 849}855
Nomenclature C CH CM C e CM e C f C i C n D ¸ M 0 Pe
dimensionless mass density of solute ("CH!C /C !C ) i f i mass density of solute radial average solute concentration de"ned by Eq. (7) Fourier transform of C Fourier transform of CM ("C #M /pR3) i 0 mass density of solute in mixture before injection nth order approximation for C binary mutual di!usion coe$cient length scale for axial transport at long times mass of injected solute Peclet number ("R; /D) c
at early times. Thus, the solutions of Lighthill and Chatwin are not complete short-time solutions. In a typical laminar dispersion experiment, a pulse of solute is injected into a solvent #owing down a tube. The initial solute pulse is deformed by #ow and di!usion as it is carried down the tube. At early times, the front edge of this pulse can be identi"ed as the leading edge, and the back edge can be called the trailing edge. Vrentas and Vrentas (1988) developed a perturbation series solution for the mean concentration for low values of Pe, the Peclet number. The derived perturbation solution is valid at both short and long times for su$ciently low values of the Peclet number (Pe)15). In addition, the series is valid at any value of the Peclet number for su$ciently small values of time. The time interval for validity of the series decreases as the Peclet number increases. For high values of the Peclet number, the solution can be used only for very short times. The leading terms in this perturbation solution is the solution for pure di!usion in the axial direction. Finally, Taylor (1953) developed a pure convection solution which should of course be valid for large values of Pe where the mass transfer process is dominated by convection e!ects. The objective of this paper is to derive a complete short-time solution for laminar dispersion in a tube which is valid for all values of the Peclet number. This derived solution provides a valid description, for su$ciently short times, of the dependence of the average concentration on axial position both near the leading and trailing edges of the concentration distribution. The laminar dispersion problem is formulated in Section 2, and a new short-time solution is developed in Section 3. The general predictions of the asymptotic short-time solution are presented in Section 4.
q r rH R t tH ; c zH
quantity de"ned by Eq. (21) dimensionless radial distance ("rH/R) radial distance variable radius of tube dimensionless time ("DtH/R2) time velocity at the center of the tube axial distance variable
variable
Greek letters a d e j
Fourier transform variable Dirac delta function small perturbation parameter ("R/¸) dimensionless axial distance variable ("zH!; tH/2/R) c
2. Problem Formulation The equation set used for the dispersion problem considered here is based on the following assumptions: (1) The di!usion process is isothermal. (2) There is laminar, axisymmetric #ow in an in"nitely long circular tube. (3) The #uid is an incompressible, two-component Newtonian #uid. (4) The amount of solute in the system is su$ciently small that all of the physical properties of the system are e!ectively constant. (5) The solute does not undergo any chemical reaction. (6) There is a concentrated initial solute input distributed uniformly over the cross section of the tube at the origin of the stationary coordinate system (zH"0). For these conditions, the convective dispersion process is described by the following form of the species continuity equation for the solute subject to the following boundary conditions:
A
B
A B
LC 1 LC 1 L LC L2C #Pe !r2 " r # , Lt 2 Lj r Lr Lr Lj2
(1)
LC (0, j, t)"0, Lr
(2)
LC (1, j, t)"0, Lr
(3)
C(r, !R, t)"0,
(4)
C(r, #R, t)"0,
(5)
C(r, j, 0)"d(j)
(6)
In these equations, the axial coordinate j is de"ned so that it has an origin moving with the average #ow
J.S. Vrentas, C.M. Vrentas / Chemical Engineering Science 55 (2000) 849}855
velocity. In this study, an equation will be derived for the average solute concentration which is de"ned as follows:
P
1 Cr dr (7) 0 In this paper, the following approach will be used to develop asymptotic solutions to the laminar dispersion problem. First, integration of the above equations produces an exact result for CM in terms of integrals and derivatives of C. Second, short-time and long-time approximations for C are derived using the above set of equations. Finally, the asymptotic approximations for C are substituted into the equation for CM to produce short-time and long-time asymptotic solutions for CM (j, t). The long-time result derived using this approach will of course be the Taylor}Aris solution, and the short-time result is a new, complete short-time asymptotic solution. Multiplication of Eqs. (1) and (4)}(6) by r, integration from r"0 to 1, and utilization of Eq. (2) produce the following equation set for CM : CM "
PA
B
A B
1 1 LCM LC LC L2CM #Pe r !r2 dr" # , Lt 2 Lj Lr Lj2 0 r/1 CM (!R, t)"0,
(9)
d(j) . CM (j, 0)" 2
(11)
These equations can be used to derive asymptotic solutions for CM if short-time and long-time approximations are used for the axial derivative in the integral and for the radial derivative at the wall. In the long-time limit, the solute interacts with the walls, and Eq. (3) must be used as the proper boundary condition at r"1. In this case, the second term on the right-hand side of Eq. (8) must be deleted. At short times, however, it is reasonable to follow Lighthill (1966) and Chatwin (1976) and suppose that the solute interaction with the tube walls is small. In this case, it is not critical to impose the proper boundary condition of zero #ux across the tube wall. The utilization of a short-time asymptotic solution for C which has (LC/Lr) O0 should thus not lead to a signi"cant r/1 amount of error if su$ciently short times are considered. Since the radial derivative is not equal to zero at the tube wall, there of course is a local failure in the approximate solution. However, for the solution that is derived, the mass of solute is conserved in an overall sense so that the principle of mass conservation is not violated. Solution of the equation set for CM is facilitated if an exponential Fourier transform is introduced (Churchill, 1972): =
CM (j, t)e~*aj dj. ~=
PA
B
A B
1 1 dCM LC e#a2CM "!Pe ia r !r2 C dr# e , e e dt 2 Lr 0 r/1 (13) CM (0)"1. (14) 2 e As noted above, substitution of short-time or long-time approximations for C into Eq. (13) leads to short-time or e long-time solutions for CM . Eqs. (13) and (14) are of course e exact results for CM . Since only some of the terms in Eq. e (13) are approximated using short-time or long-time approximations for C , it seems reasonable to expect e that the asymptotic solutions for CM should be more e widely applicable than the asymptotic approximations for C which will be derived under more restrictive e conditions.
3. Problem solution
(10)
P
The transformed equation set for CM takes the following e form:
(8)
CM (#R, t)"0,
CM (a, t)" e
851
(12)
In this section, both short-time and long-time asymptotic solutions are developed. The short-time solution, which is the new result, is derived "rst, and, then, it is shown that the same general approach can be used to derive the well-known Taylor}Aris long-time solution. Chatwin (1976) noted that the interaction between axial di!usion and convection dominates the dispersion process at very short times, but, as time increases, the interaction between radial di!usion and convection is the dominant mechanism in the mass transfer process. Hence, it seems reasonable to derive a short-time asymptotic solution for C by neglecting the radial di!usion term
A
B
LC 1 LC L2C #Pe !r2 " , Lt 2 Lj Lj2
(15)
so that the time dependence of C is described by the interaction of the convective #ow with the axial di!usion. A solution of Eq. (15) subject to Eqs. (4)}(6) can easily be constructed using the exponential Fourier transform. The transformed concentration C can be derived by e solving the equation
A
B
dC 1 e#Pe !r2 iaC "!a2C , e e dt 2
(16)
subject to the initial condition: C (0)"1. e The solution is simply
C G
(17)
A
1 C "exp ! a2#Pe ia !r2 e 2
BH D
t ,
(18)
852
J.S. Vrentas, C.M. Vrentas / Chemical Engineering Science 55 (2000) 849}855
and straightforward inversion yields a short-time approximation for the solute concentration distribution in the tube exp[!Mj!Pe t(1/2!r2)N2/4t] C" . 2p1@2t1@2
(19)
Substitution of the equation for C , Eq. (18), into e Eq. (13) gives the following di!erential equation:
C
D
dCM eqt#e~qt e~qt!eqt e#a2CM "e~a2t # #4qte~a2teqt, e 4t 4qt2 dt (20) Pe ia q" . 2
(21)
Solution of the equation subject to the initial condition, Eq. (14), yields the following expression: e~a2t[eqt!e~qt] CM " e 4qt
CA B D
(22)
This result can be rewritten as follows: 2Pe tCM "2e~a2t e
C
D
sin(Pe at/2) a
#16te~a2t
C
!16te~a2t
sin(Pe at/2) a
D
1!cos(Pe at/2) ia
C D
Pe iat . 2
#8Pe t2e~a2t exp
0.4 t( . (Pe)2@3
(23)
(25)
In addition, a comparison of terms in the equation for C shows that the e!ect of the interaction of the solute with the tube wall should not be signi"cant for times satisfying the following inequality: t(0.001.
1 1 #4e~a t eqt t! # . q q 2
process is dominated by axial di!usion and convection since it is based on a solution for C which neglects radial di!usion. All terms are retained in the exact result for the average concentration, even though some are clearly approximated. Hence, the solution for CM does include an approximate characterization of radial di!usion and can hence be regarded as a theoretical result for which radial di!usion slightly perturbs the basic results of the interaction of convection and axial di!usion. The time interval for which axial di!usion and convection are the dominant mass transfer mechanisms can be estimated using the results of Chatwin (1976):
(26)
This equation is derived by requiring that the radial derivative term at the wall be less than 0.5 per cent of the convective term. Consequently, the new asymptotic short-time solution, Eq. (24), should be valid from t"0 to a time which is the lesser of the times calculated from Eqs. (25) and (26). In the long-time limit, the importance of the individual terms in Eq. (1) can be ascertained by following Taylor (1954) and introducing a length ¸, the distance over which the major part of the change in concentration takes place in the axial direction. If the quantity ¸ is used as the length scale for time and for the axial derivatives, then it follows that the terms in Eq. (1) can be ordered as follows:
A B
Inversion of this expression using the convolution integral for exponential Fourier transforms produces the following short-time asymptotic solution:
1 L LC r "O(1), r Lr Lr
(27)
2Pe tCM
L2C "O(e2), Lj2
(28)
LC "O(e2), Lt
(29)
erf [Pe t1@2/4!j/2t1@2]#erf [Pe t1@2/4#j/2t1@2] " 2 !8terf
C A B C
D
Pe t1@2 j # 4 2t1@2
A
Pe
j #8terf 2t1@2
D
4Pe t3@2 (j#Pe t/2)2 # exp ! . p1@2 4t
(24)
This short-time asymptotic solution for CM should be valid in the initial time period where the dispersion
B
1 LC !r2 "O(e). 2 Lj
(30)
Here, e"R/¸ and e@1 in the long-time limit. Hence, a zero-order approximation C for the concentration is 0 the solution of the equation
A B
L LC r 0 "0, Lr Lr
(31)
J.S. Vrentas, C.M. Vrentas / Chemical Engineering Science 55 (2000) 849}855
subject to Eqs. (2) and (3), and a "rst-order approximation C is the solution of the equation 1 1 L LC 1 LC 0, r 1 "Pe !r2 (32) r Lr 2 Lj Lr
A B A
B
853
The solution of Eqs. (14) and (38) is simply
C A
Pe2 CM "1exp !a2t 1# e 2 192
BD
,
(39)
and inversion gives the following result:
subject again to Eqs. (2) and (3). The solution for C is 0 simply
exp [!j2/4t(1#Pe2/192)] . CM " 4p1@2[t(1#Pe2/192)]1@2
C "f (j, t), (33) 0 0 and the solution for C can be expressed as follows: 1 Lf r2 r4 C "Pe 0 ! #f (j, t). (34) 1 1 16 Lj 8
This is the well-known Taylor}Aris solution. It is shown here how the method proposed for derivation of the short-time asymptotic solution can also be used to recover the proper long-time asymptotic solution.
This expression represents a long-time, "rst-order approximation for C and, hence, for C : e r2 r4 C "Pe ! i a f (a, t)#f (a, t). (35) e 0e 1e 8 16
4. Results and discussion
A
A
B
B
Since the lowest order result for CM is e f CM " 0e, e 2
(36)
the long-time, "rst-order approximation for C can be e expressed as follows:
A
B
r2 r4 C "2Pe ! i a CM #f (a, t). e e 1e 8 16
(37)
The quantity CM must satisfy Eqs. (13) and (14) with the e radial wall derivative set equal to zero in Eq. (13) because of the importance of solute interaction with the wall. Substitution of Eq. (37) into Eq. (13) and integration produce the di!erential equation:
CA
dCM Pe2 e#CM a2 1# e dt 192
BD
"0.
(38)
(40)
The general predictions of the new short-time asymptotic solution are now compared to the predictions of the pure axial di!usion solution, the pure convection solution, and the exact series solutions of Yu (1981) and of Shankar and Lenho! (1989). The pure axial di!usion solution can be expressed as follows: e~j2@4t CM " , 4p1@2t1@2
(41)
and the pure convection solution takes the following form: 2j 2Pe tCM "1, 1' '!1, Pe t 2Pe tCM "0,
K K
2j '1. Pe t
(42) (43)
As noted previously (Vrentas & Vrentas, 1988), axial molecular di!usion predominates over axial convection even for large values of the Peclet number if t is su$ciently small. This is illustrated in Fig. 1 where the axial
Fig. 1. Axial dependence of average concentration for Pe"104 and t"10~8. Curve is the short-time asymptotic solution, and solid points are the predictions of the pure axial di!usion solution [Eq. (41)].
854
J.S. Vrentas, C.M. Vrentas / Chemical Engineering Science 55 (2000) 849}855
Fig. 2. Axial dependence of average concentration for Pe"104 and t"10~5. Solid curve is the pure convection solution [Eqs. (42) and (43)], and dotted curve is the short-time asymptotic solution. Fig. 3. Axial dependence of average concentration for Pe"104 and t"0.001. Complete concentration distribution is shown.
dependence of CM is presented for Pe"104 and t"10~8. For this very low value of time, the short-time asymptotic solution is essentially identical to the pure axial di!usion solution even though the Peclet number is very high. When the time is increased to 10~5 for Pe"104, the average concentration distribution computed from the short-time asymptotic solution is close to the pure convection solution. This is clearly illustrated in Fig. 2 which shows that convection becomes the dominant transport mechanism. As the time is further increased, it is to be expected that the concentration distribution will become asymmetric before approaching the symmetric Taylor}Aris solution at su$ciently long times. For Pe"104, it is evident from Eqs. (25) and (26) that the maximum time for which the asymptotic short-time solution, Eq. (24), is valid is about 0.001. The predicted average concentration distribution for Pe"104 and t"0.001 is presented in Figs. 3 and 4. For this large value of Pe, the concentration distribution is very similar to the pure convection solution with the exception of a pronounced maximum near the trailing edge. The general shape of the average concentration pro"le is similar to the results presented by Yu (1981) for Pe"104 and t"0.02 and by Shankar and Lenho! (1989) for Pe"R and t"0.002. No direct comparison is possible because the lowest time reported for the exact series solutions is t"0.002. However, the height of the maximum reported by Shankar and Lenho! for Pe"R and t"0.002 is comparable to the height of the maximum reported here
Fig. 4. Axial dependence of average concentration for Pe"104 and t"0.001. Concentration distribution near pronounced maximum is shown.
J.S. Vrentas, C.M. Vrentas / Chemical Engineering Science 55 (2000) 849}855
855
Fig. 5. Axial dependence of average concentration for Pe"10 and t"0.001. Curve is the short-time asymptotic solution and solid points are the pure di!usion solution [Eq. (41)].
for Pe"104 and t"0.001 Shankar and Lenho! noted that large series expansions are needed at very short times, and, hence, analytical solutions are more economical. Consequently, the short-time asymptotic solution presented here should be a useful complement to the exact series solutions of Yu and of Shankar and Lenho!. When the Peclet number is low, the transport process is dominated by axial di!usion even for the largest time (t"0.001) for which the asymptotic solution is valid. This is illustrated in Fig. 5 which shows that the shorttime asymptotic solution is virtually identical to the pure axial di!usion solution for Pe"10 even for the maximum allowable time, t"0.001. From the above discussion, it is fair to conclude that the proposed short-time asymptotic solution, Eq. (24), provides a valid description of laminar dispersion in a circular pipe for the time interval given by Eqs. (25) and (26). The proposed solution is a complete solution in that it describes the axial dependence of the average concentration both near the leading and trailing edges of the concentration distribution. Previous short-time solutions (Lighthill, 1966; Chatwin, 1976) provided valid results only near the leading edge of the concentration pro"le.
Acknowledgements This work was supported by funds provided by the Dow Chemical Company. References Aris, R. (1956). On the dispersion of a solute in a #uid #owing through a tube. Proceedings of the Royal Society, A 235, 67}77. Chatwin, P. C. (1976). The initial dispersion of contaminant in Poiseuille #ow and the smoothing of the snout. Journal of Fluid Mechanics, 77, 593}602. Churchill, R. V. (1972). Operational mathematics. New York: McGrawHill. Lighthill, M. J. (1966). Initial development of di!usion in Poiseuille #ow. Journal of the Institution of Maths Applications, 2, 97}108. Shankar, A., & Lenho!, A. M. (1989). Dispersion in laminar #ow in short tubes. A.I.Ch.E. Journal, 35, 2048}2052. Taylor, G. (1953). Dispersion of soluble matter in solvent #owing slowly through a tube. Proceedings of the Royal Society, A 219, 186}203. Taylor, G. (1954). Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular di!usion. Proceedings of the Royal Society, A 225, 473}477. Vrentas, J. S., & Vrentas, C. M. (1988). Dispersion in laminar tube #ow at low Peclet numbers or short times. A.I.Ch.E. Journal, 34, 1423}1430. Yu, J. S. (1981). Dispersion in laminar #ow through tubes by simultaneous di!usion and convection. Journal of Applied Mechanics, 48, 217}223.