Ind. Eng. Chem. Fundam. 1981, 20,
102
102-104
L = length of fiber r = aspect ratio = L / d C' = average velocity vX = eddy velocity vo = friction velocity Greek Letters /3 = function of aspect ratio t
= energy dissipation rate
X = eddy scale u = kinematic viscosity u p = eddy viscosity = vX,hp 4 = concentration of fiber in volume fraction
Subscripts
max = maximum p = fiber
Literature Cited 33,L.0 301
-
-
--
09'
~
- 010
@ Plot of In (f/fp) vs. 9, comparison of eq 12 with the results of Kerekes and Douglas (1972). F i g u r e 1.
pension viscosity are not high enough to reverse drag reduction to drag increase. Nomenclature C = constant in Eq 9 DR = fractional drag reduction d = diameter of fiber f = friction factor
Davies, J. T. "Turbulence Phenomena", Academic Press: New York, 1972; p 27. Hoyt, J. W. "Naval Undersea Center Rept. T.P. 299", San Diego, Calif., 1972. Kale, D. D.; Metzner, A. B. AIChE J . 1976, 22, 669. Kerekes, R. J. E.; Douglas, W. J. M. Can. J . Chem. Eng. 1972, 50, 228. Mewis, J.; Metzner, A. B. J. N u M M c h . 1974, 62, 593. Radin, I.; J. L.; Patterson, G. K. AIChE J. 1975, 21, 358. Vaseleski. R. C.:Metzner, A. 6 . AIChE J . 1974, 20, 301.
Department of Chemical Engineering Polytechnic of Wales Pontypridd, Mid- Glamorgan CF37 1DL South- Wales, United Kingdom
M. S. Doulah
Received for review April 15, 1980 Accepted September 26, 1980
Residence-Time Distributions for Systems Having Many Connections with Their Environments The difference reported by Ritchie and Tobgy (1978) between their expression for the residence-time density function for a closed system and that previously established by Buffham and Kropholler (1970) is because Riche and Tobgy's analysis is not consistent with their definitions. The agreement they found between their equation and that of Treleaven and Tobgy (1971) for systems having two inlets and one exit is fortuitous. Functions relating the time spent in the system and the place whence material leaves to the location at which it entered may be defined in different ways. Correct expressions for the residence-timedensity function are given for several definitions and collected into a table for easy comparison.
Introduction In a recent paper, Ritchie and Tobgy (1978) have presented a residence-time analysis for systems having many connections with their environments. They stress that their relation describing the overall residence-time frequency function differs from that previously published by us (Buffham and Kropholler, 1970). Our purpose here is to explain the difference between the two formulations and to indicate the flaws in Ritchie and Tobgy's treatment, lest it be thought that the method is fundamentally unsound. The system under consideration has several connections with its environment through which material passes by bulk flow so that the connections may be classified unambiguously as inlets and outlets. In the steady state, the residence-time distribution is the distribution of ages of material leaving the system or, equivalently, the distribution of life expectancies of material entering the system. Our analysis was in terms of the concentration responses at the outlets to concentration forcing at the inlets. Ritchie 0196-43 13/81/1020-0102$01 .OO/O
and Tobgy's analysis is in terms of frequency functions. The treatment that follows uses frequency functions. We have discussed a somewhat more general case elsewhere (Buffham and Kropholler, 1973). Frequency Analysis Our principal concern is the discrepancy between Ritchie and Tobgy's (1978) eq 17 and eq 9 of our paper (Buffham and Kropholler, 1970). We shall refer to these equations as eq RT17 and eq BK9 and refer similarly to other equations from these papers. Ritchie and Tobgy express eq RT17 in terms of a function fij(t) defined by (definition 0 : f , ( t ) d t = the fraction of the ith inlet stream which leaves via the j t h outlet stream with residence time between t and t + dt. Equations RT17 and BK9 differ, partly because eq RT17 does not follow from definition I and partly because f,,(t) in eq RT17 is not the same as fij(t) in eq BK9. First, we shall develop the correct expression for the residencetime density from definition I; then we shall give frequency-function definitions which lead to residence-time 0 1981 American
Chemical Society
Ind. Eng. Chem. Fundam., Vol. 20,No. 1, 1981
relations of the same form as eq RT17 and BK9. We shall use superscripts to distinguish between several different fij(t) functions which can be used to represent the inletoutlet interactions, and use fi{(t) for fij(t) in definition I. Let Qi. be the flow rate of the ith inlet stream; then Qif$t) dt is the rate of flow of material from the jth outlet which both originated from the ith inlet and also has its residence time in ( t ,t dt). Summing this quantity over all the outlet streams gives the outflow of material with residence time ( t ,t + dt) which entered via the ith inlet. The total flow of material with residence time ( t ,t + dt) is then found by summing over all the inlets and may also be expressed directly in terms of the residence-time density function f(t)
+
CCQJijYt) dt = Qf(t) dt i J
(1)
where Q is the volumetric flow rate through the system
Q = CQi.
(2)
1
Hence f(t) = Q-'CCQi.fi{(t) i
l
(3)
Ritchie and Tobgy claim to have deduced eq RT17 from definition I, but their expression is not equivalent to eq 3. The defect in their development is that eq RT12 does not represent the fact that the fraction of the ith inlet stream with residence time between t and t + dt is the sum over j of the fractions of the ith inlet stream with residence time between t and t + dt. Instead of eq RT12, Ritchie and Tobgy should have fi(t) = Cfij(t) 1
(4)
which when substituted into eq RT14 gives an expression of the same form as eq 3. A different definition of fij(t) is required in order to obtain eq RT17, viz. (definition IT): f@) dt = the fraction of the flow, Qij, from inlet i to outlet j which has residence time in ( t , t + dt). Repeating the steps leading to eq 3 with definition I1 yields f(t) = Q-lCCQijfij"(t) i l
(5)
which is of the same form as eq RT17. Definition I was couched in terms of fractions of the inlet flows; it is equally possible to use a definition in terms of fractions of the outlet flows (definition ZIO: fitl'(t) dt = the fraction of the j t h effluent stream which entered via the ith inlet and has residence time ( t , t + dt). Repeating the analysis again, we find that f(t) = Q-'CCQ.jfij"'(t) i
l
(6)
where is the rate of flow from the j t h outlet. Q.j is Ritchie and Tobgy's (1978) Q,j and the sum of such flows is equal to Q. Equation 6 is of the same form as eq BK9 and definition I11 provides a meaning for our fij(t) in terms of frequencies. Our fij(t) was defined (Buffham and Kropholler, 1970) by (definition I V ) : fi/"(t) = response a t outlet j due to &function forcing at inlet i. By using this definition, it is a simple matter to establish a relation for the residence-time density function for a species, A, which is present in the several feed streams at different concentrations, say Ci. The concentration of the j t h outlet stream is fitV(t)when the concentration of the ith inlet stream is a 6 function. The quantity of substance
103
+
A which both leaves from the j t h outlet in (t,t dt) and which also entered at the ith inlet in (0, d r ) is Cid7.Q,j X fJV(t) dt. Summing over all the outlets and inlets gives &e quantity of A which both enters in (0, dT) and leaves in ( t , t dt), which quantity may also be expressed in terms of the residence-time density function for A, say f(t) again CCCi dT*Q.jfij"(t) dt = (CQi.Ci dT)f(t) dt (7)
+
i
i J
Hence f(t) = (CQi.Ci)-'CC ciQ.jfijIv(t) i
i J
(8)
which corrects a misprint in eq BK10. If all the feed concentrations are the same (or if one chooses to disregard the presence of different species), eq 8 reduces to f(t) = QICCQ.jfijI"(t) i
(9)
l
which is eq BK9 and is equivalent to eq 6 above. Ritchie and Tobgy (1978) show that, for the special case of a system with two inlets and one outlet, eq BK9 leads to a different result from that of Treleaven and Tobgy (1971). These two results are in fact equivalent and the difference is one of notation. Our method (Buffham and Kropholler, 1970) treats the problem in terms of forcing functions. A concentration impulse can be generated a t the two inlets by connecting them to a common source which is itself subject to &function forcing. The outlet concentration in response to this stimulus is fllIv(t) fzlIv(t)where fllN(t) and filIv(t) are the outlet responses to &function forcing of the individual inlets. The residence-time density function is given by the ratio of the tracer leaving the system in (t,t dt) to the total entering.
+
+
f(t) dt = Q[f1lIv(t) + f211V(t)ldt/(Qi. which reduces to
+ Qz.)
(10)
f(t) = fll'V(t) + filI"(t) (11) and is the same as eq RT7. Treleaven and Tobgy's result is in terms of the (conditional) residence-time densities, fl(t) and f2(t),of the material entering a t the two inlets. The probability that material leaving entered a t inlet 1 is Ql,/(Ql. + Q2.) and the probability that it entered at inlet 2 is Q2./(Ql. + Q2.). It follows that the probability that material leaving came from inlet 1 and has residence time ( t , t + dt) is Q1.fl(t) dt/(Ql. + Q2.)and similarly for inlet 2. These categories are mutually exclusive and so
which is eq RT8. The fact that Ritchie and Tobgy obtained eq RT8 from eq RT17 which purports to represent multiple inlet, multiple outlet systems, is merely good luck and is because definitions I and I1 coincide when a system has but one outlet. In the notation of the present paper fi(t), f2(t)are fli"(t) and fzl"(t). Table I summarizes the fij(t) definitions and corresponding f(t) expressions. Discussion A simple example will serve to confiim that the eq RT17 can lead to incorrect results while eq 3 is correct. Our example is a single well-stirred vessel, volume V, provided with two inlets and two outlets. Injection of mass M of tracer into either inlet gives the same outlet concentration from either outlet C.,(t) = C.,(t) =
Me-Qtl" ~
V
(13)
104 Ind. Eng. Chem. Fundam., Vol. 20, No. 1, 1981 Table I. Summary of
fii(t)
Definitions and Corresponding Expressions for the Residence-Time Density Function, f(t)
definition of fii(t) I
11
f$(t) d t = the fraction of the ith inlet stream which leaves via the jth outlet stream with residence time between t and t + dt f$(t) d t = the fraction of the flow, Qij, from inlet i to outletj which has residence time in ( t ,
t I11
IV
remarks
t)
(3)
Ritchie and Tobgy's result, eq RT17, for this case is incorrect because of an error in eq RT12.
Q-'fyQijfijn( t )
(5)
Equation 5 is the same form as Ritchie and Tobgy's eq RT17, but is based on a different definition of fij(t).
Q-'FTQ.jfijm(t)
(6)
Equation 6 is of the same form as eq BK9. Definition I11 provides a frequency interpretation for Buffham and Kropholler's (1970) fij(t).
($. Qi.Ci)-'ccCiQ.jfiiIv(t)
(8)
Equation 8 applies to a component present in the feeds in different concentrations. while eq 9 applies to the flow as a whoie. Equation 9 is equivalent to eq BK9.
Q-'%FQi.fij'( 1 1
+ dt)
f$(t) dt = the fraction of the jth effluent stream which entered via the ith inlet and has residence time ( t , t + d t ) f y ( t ) = response at outletj due to 6 -function forcing at inlet i
11
1
Q-'Z
CQ.jfijIV(t )
It follows from eq RT23 that Ritchie and Tobgy's fij(t) (our fjj'(t)) is
As the vessel contents are well mixed the flow Q, from inlet i to exit j is Q1, = Q,.(Q.,/Q) (15) Substitution of QL,and fJt) from eq 15 and 14 into eq RT17 yields f(t) =
eq no. in text
expression for f(t)
Q.i2Qi. + Q.Z2Qi.
+ Q.1'82.+ Q.22Q2.e-Qt1v
VQ'
(16)
which simplifies to f(t) = (Qv)-1(Q.12 + Q.22)e-Qt/v
(17)
Substitution of f,,(t) from eq 14 into eq 3, however, gives f(t) =
Q,-QtIV
(18)
V which is clearly the correct result: the residence-time distribution of a well-mixed vessel depends only on Q / V and is not changed by splitting the incoming and outgoing flows among several inlets and outlets. Equation 17 is correct only for the trivial cases Q.l = 0 and Q.z = 0: The question arises whether any of the fJt) functions are superior to the others. Our view is that it is preferable to use the definition which is natural in the context. Response experiments involving the rapid injection of quantities of tracer into the inlets are easily described in terms of f,'(t) and eq 3 if the object is to establish the residence-time distribution of the fluid as a whole. On the other hand, if deconvolution of input and output concentration records were to be carried out, the natural result would be f,,Iv(t) functions. Any of the functions may be calculated from mathematical models, but if one likes to think in terms of transfer functions, then the f,Iv(t) are the appropriate time functions. Equation 5 has a pleasing symmetry and the fji"( t ) are
(9)
(conditional) density functions so that their time integrals are unity, but the Qij are less direct experimental variables than Qi. and Q.j, and will seldom be explicit variables in models. Equation 8 has the distinct advantage that it allows for the determination of the residence-time distribution for a species present at different concentrations in the several feeds.
Nomenclature C, = concentration of ith feed stream C, = concentration at jth outlet (used only in example) f(t) = residence time density function fJt) = response function connecting ith feed stream and jth outlet stream f,,I(t): see definition I "(t): see definition I1 b l ( t ) : see definition I11 fil:Iv(t): see definition IV M = mass of tracer injected (used only in example) Q = total throughput flow rate Q,. = flow rate of ith feed stream Q , = flow rate of jth product stream Q , = rate of flow from ith inlet to jth outlet t = residence time V = system volume T = time Literature Cited Buffham, E. A., Kropholler, H. W., Math. Biosci., 8 , 179 (1970). Buffham, E. A., Kropholler, H. W., Chem. Eng. Sci., 28, 1081 (1973). Rltchle, E. W., Tobgy, A. H., Ind. Eng. Chem. Fundam., 17, 287 (1978). Treleaven, C. I?.,Tobgy, A. H., Chem. Eng. Sci., 26, 1259 (1971).
Department of Chemical Engineering University o f Technology Loughborough, England LE11 3TU
B. A. Buffham*
Department of Polymer and Fibre Science University o f Manchester Institute of Science and Technology Manchester, England M60 1QD
H. W.Kropholler
Received for review May 4, 1979 Accepted July 23, 1980
