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Mathematical modelling and calculation of membrane separation processes
Journalof Food Engineering 24 (1995) 51 I-528 Copyright Q 1995 Elsevier Science Limited
Printed in Great Britain. All tights reserved 0260-8774/95/$9 SO ELSEVIER
Mathematical
Modelling and Calculation of Membrane Separation Processes
Lev A. Rovinsky” & Victor L. Yarovenco Corresponding (Received 22 December
Institute of Food Industry, Moscow, Russia 1992; revised version received 30 November accepted 10 January 1994)
1993;
ABSTRACT In order to develop a methodology for mathematical modelling of membrane separation processes in continuous systems of the complete mixing and complete displacement type, theoretical and experimental studies were undertaken. The problem of determining a limiting concentration in the processed medium and the conditions for its achievement are described. New correlation results for current concentration and concentrate and permeate flow rates with membrane surjace area are presented. The non-steady (dynamic) states of a membrane separation process, i.e. the start-up period from initial conditions to steady state, and transient responses because of modification of inlet conditions are presented as analytical expressions and formulae. The theoretical results and empirical formulae are tested using experimental data obtained during membrane concentration of biotechnology products.
NOTATION Empirical parameter Auxiliary parameters Coefficients b,,,b,,& Empirical parameter (m’/kg . s) ( Auxiliary parameters Coefficients $32 Membrane surface area (m2) g( x, L );f (x, L ) Empirical functions i,B
*To whom correspondence Moscow, Russia 111395.
should be addressed
511
at: 19-l-284,
Krasny
kazanetz
St,
512
L. A. Rovinsky, I/: L. Yarovenco
Membrane permeability (the same for pure solvent) (m/s) Inlet and outlet flow rate (m”/s) Time (s) Inlet and local product concentration (kg/m3) Local and average permeate concentration (kg/m3) Auxiliary variable Permeate flow rate (m3/s) a
Auxiliary parameter
Subscripts 0 Lim g
Initial values Limiting values Given values
Superscripts * b
Stationary values Boundary values
INTRODUCTION Membrane processes for liquid media separation are calculated on the basis of diffusion or mass transfer phenomena and empirical relationships describe the solute flow and concentration when passing through a semipermeable membrane. The methods of calculation of complete mixing flow apparatus (CMFA) and complete displacement flow apparatus (CDFA) were described in detail in monographs (Hwang & Kammermeyer, 1975; Brock, 1983; Torrey, 1984; Ditnerskiy, 1986). The apparatus type and technology of membrane separation processes were shown in a survey (Niemy et al., 1986) and a handbook (Cheryan, 1986). Even though the above-mentioned methods of calculation are sufficiently complete there are still some aspects unresolved, for instance, the maximum attainable concentration in each process, non-steady state processes (dynamic models) and relationships between the main processing parameters and processing results. A new approach based on mathematical models mentioned above (Hwang & Kammermeyer, 1975; Ditnerskiy, 1986) will be considered below. This paper considers: - mathematical modelling of membrane separation (solution concentration) for non-steady states; - evaluation of limiting (maximum) concentration as a function of process conditions; - correlation between concentration process results and CDFA parameters; - formulae for technological design of membrane separation equipment.
Modelling of membrane separation processes
513
THEORY Methodology
and assumptions
The methodological approach and mathematical conditions and empirical relationships:
models are based on known
( 1) between concentrations of dissolved solutes in the apparatus (x) and in the permeate (x2): x2 = g(x); (2) the membrane permeability G= Go -f( X,L ) where L is the flow rate of concentrate, C$ is the membrane permeability for pure solvent, g and fare empirical functions.
Further consideration involved all the assumptions usually accepted for such processes, namely: temperature and pressure are constant during the process; surface phenomena on the membrane are taken into consideration to the extent that they are reflected in the empirical relationships; the influence of sediment on the filtering properties of the membrane is either absent or included in the empirical performance lumped in with all other effects. Apparatus
and processes
considered
Three types of membrane apparatus will be considered: - CDFA shown in Fig. I( a); - CMFA (single section) shown in Fig. l(b); - CMFA with recirculation and controlled (Fig. l(c)).
removal
of
concentrate
All cases presume an inlet flow rate L, and concentration x0, and an outlet (concentrate) flow rate L and concentration X. For the CDFA system (Fig. l(a)) one must consider local concentration x2 and common (average) permeate concentration X2. Concentration
in CDFA
The basic relationships A piston-form of liquid motion through the apparatus is assumed. According to Ditnerskiy ( 1986) and Hwang and Kammermeyer ( 1975), the equation system describing the separation process can be written as: mass and flow balance equations: Lox,, = wz, + Lx L,,=L+
w
(1) (2)
The differential equations for mass and flow change:
4
Wi,
)
~=g(x,L)
(3)
L. A. Rovinsky, I/ L. Yaroverlco
514
dW z= If the concentration
G= G,,-f(x,L)
x is small, the functions g and fare assumed to be linear:
g(x) = x2
=
f(x) = G= G,, - cx,
ax;
(5)
where parameters a, c, G,, are determined experimentally. There are known relationships (Ditnerskiy, 1986) for an arbitrary section of the apparatus: L= L&/x,,) ‘b”- ‘) w= L,,[ 1 - (x/x,,p-
-5 = 41
(6) “1
1 _
(x/x”)fJ/(l~- ‘1
1
ix,x,,)l;:“-c
_
(64
(7)
The correlation between concentrations X, X1 and flow rates L, W with the current membrane surface area F were presented as differential equations only, without solutions. Suggested approach The relationships (6), (6a) and (7) are not complete and can be developed. Isolating from eqns ( 1) and (2) the term Wi, = Lox,, - x(L, - W) and substituting it into eqn (3) yields d(Wiz) x,,L,, - wiz
=
dW a L,,-
(8)
Integration of eqn (8) with initial conditions W( 0) = 0; X2(O)= x0, assuming that x,, and L are constant during a given process:
Ml, x,, L,, - Wi?
or after transformations
which supplements models below.
( )
=----L,, (’ L-f,- W
(9)
- the function X2(W ):
relationships
(6), (6a) and (7) and is fundamental
for the
The limiting concentration The concentration in the processed liquid moving along the membrane increases from x,) to a certain limiting value (x,,,) at which separation will stop. The conditions of the separation process may be expressed as ax,,S.f,Ix,,Ixlx,,,
(11)
Modeling of membrane separation processes
515
and the value XLimcan be determined from eqn (4) for steady state when dW z=o: The corresponding
(12)
XLim= G,,lc
limiting flow rates are then
and the limiting permeate concentration,
according to eqn (7).
The process separation dynamics
Equation (6a) can be rewritten as
and substitution into eqn (4) gives 0,
i.e. a first-order differential equation for the variable L = L,, - W: dL --cx,,L;,-“L’‘-I dF
+ G,,=()
(14)
which can be solved as a function L( F ). To obtain the correlation x( F ), eqn ( 14) can be transformed to an equation for x. Differentiating eqn (6a): d-=W Lox:i(‘-“’ x(z - Cl!/\ I - 0) d” dF l-a dF and substituting into eqn (4) gives the equation
FF+Ap+i =Bx”
(15)
where
A=cW-4 L,,
I/‘[~- 11.
x0
B_
9
Gil -4 L,,
2-a
I/(rr- II.
x (’
’
*=1-a’
The solution of eqn ( 15) is given in Appendix 1 and is F=L
c
d-‘-x;-
A ,=, xli,,,(a-
i)(xx,,)“-’ ’
i= 1,2 ,...
(16)
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L. A. Rovinsky, V L. Yarovenco
where xLim= G,/c according to (13) and the numerical exactness of F(x) increases with an increase in the number of terms in expression ( 16). The dependence of other variables on F: L( F ), W( F ), X2(F ) (along the apparatus length) is determined from expressions (6), (6a), (7) and ( 10). Complete mixing flow apparatus (CMFA) This ideal type of apparatus is characterized by an identical concentration throughout, due to extensive liquid mixing within the apparatus. Usually such an apparatus models an isolated section of a real apparatus. The concentration process in a CMFA is shown in Fig. l(b). Here the solute concentration at any arbitrary point in the apparatus volume I/ is the same and the permeate concentration x2 = ax is the same all along the membrane surface F. From the initial eqns ( l)-( 4) only expression (2) is retained but eqns ( 1) and (4) become: Lox, = Lx+
wi,
(17)
W=FG=F(G,,-cx)
(18)
and the differential equation for solute quantity in the apparatus can be written instead of eqn (3):
vp=
Lc,x,, - Lx-
wiz .
(19)
The steady state
If the initial process conditions (x0, L,) are invariable then it will become a steady state separation process with stationary concentrations and flow rates. In this case (dx/d t= 0), eqn ( 19) is identical to ( 17) and substitution of eqn ( 18) and x2 = ax into eqn (17) leads to a quadratic equation for x: cF(l-a)x*+x[L,-(l-a)FG,,]-x,L,,=O.
x2
Fig. 1.
(20)
(b)
Membrane apparatus: (a) complete displacing flow apparatus (CDFA); complete mixing flow apparatus (CMFA); (c) CMFA with recirculation.
(b)
Modelling of membrane separation processes
Solution of eqn (20) gives the stationary concentration
517
x*:
+j[L,-(1-a)FG,]‘+IcF(l-a)x,,L,,-L,+(l-a)FG,,
(21)
2cF( 1 - a) and the corresponding
flow rates of concentrate
* L” = L,, (z’)-aTx* ;
(L*) and permeate ( W*) are
* W*=1.,,(li_~~~*=f.iG,,-~~*).
(22)
The limiting concentration from solution (21) that the function x*(L,,) has an inverse dependence. Under constant membrane-medium system properties (a, c, G,) and initial concentration x0, the stationary values x*, L”, W* are dependent on L, only. It can easily be shown from expression (2 1) that if
It can be noted
L,,-0:
x*-xLlm=
G,,/c,
i.e. limiting (maximum) concentration in CMFA under certain conditions. According to (22) the permeate flow rate W* - 0 because membrane permeability G= G,, - cx* -+0. However it is not the only possibility. It can be shown from the first equation of (22) that when x*=x,/a, the concentrate flow rate L”+O. As the membrane-medium system properties (a, c, G,) and inlet flow concentration xg are independent then both variants: xLim, , = ( G,,/c), xLim,Z= (x,/a) are possible. If the concentration x* increases then x* -+xLimbut xLimis the minimum of two values: xLim=min{x,,,,,
= Go/c;
(23’)
XLllJl.2 =x&d
The boundary of these cases is inlet concentration x6 = aG,Jc: if xc,< XI;then xLlm= x(,/a, i.e. does not depend on inlet flow. This is shown schematically in Fig. 2.
XLim
x, %b=aG,/c
Fig. 2.
The limiting concentration
as a function of x,,.
L. A. Rovinsky, V L. Yarovenco
518
The dynamic separation process There are two non-steady states: the process start period (x(t) changes to x*) and the transient response process from one steady state to because of any perturbation in process conditions. The function x(t) non-steady states can be obtained as follows: the dependencies (2) and substituted into eqn ( 19) and transformed into the Riccati equation dx
;=
from x0 another for such (18) are
- b2x2 - b,x+ b,,
(24)
where coefficients: b,=(l-a)cF/K
b,=[L,-(l-a)FG,,]/V;
b, = L,x,,l K
The solution of eqn (24) is given in Appendix 2 and is ( x(t)=
b,
)
[i*+x(0)+~]e2.D;‘-x*+d0!
b,
--
x*+z *
2b2
jx*+x(O)+~]e'.'..+x'-dOi
(25)
where the auxiliary parameter u = x” + (b, /2 b2) and the stationary concentration x* are given by (2 1). If a start-up period of separation is being considered then x(O) = x,, and solution (25) presents the transient x(t) from x0 to x*, i.e. the movement from start-up to operating mode. If, however, there was already a steady state in the CFMA with x*(O) and one or both initial conditions (x0,&) are changed at an arbitrary moment t= 0, then the solution is the transient response and x*(O) changes to x* # x*( 0). In this case the transient function x(t) can be calculated by (25)underx(O)=x*(O). CMFA with recirculation A real multi-section membrane apparatus contains several sections such as the above CMFA with recirculation (Fig. l(c)). In this scheme a circulating pump continually recycles separated (concentrated) material having a flow rate L, to the inlet of a unit. Concentrate with flow rate L can be removed from the unit when the desired concentration xs is reached; simultaneously, new feed is fed to the apparatus. The balance equation in such a case is VS=L’x’-x(L+L,)-
wx*,
where the flow rate (L’) and concentration (x’) through the apparatus are determined by relationships: L’x’ = L,x, + Lx; L’ + L, + L and accordingly condi-
Modelling of membrane separationprocesses
519
tion (2) becomes
dg=W(x,,-x,)-L(x-x,,) Substituting equality (18) and x2 = ax transforms it to the form: V$=ai:Fx’-x(uFG,,+cFx,,+L)+x,,(L+FG,,).
1;26)
Steady state
If the characteristics a, c, G, and initial condition x0 are constant during a real case then the process becomes a steady state one and stationary concentrations and flow rates are dependent on concentrate removal L only. The stationary concentration x* can be obtained from eqn (26) under condition (dx/dt) = 0: acFx’ - x( aFG,, + cFx,, + L ) + x,,(L + FG,,) = 0.
(27)
The solution of the quadratic eqn (27) gives a value x*( L ) analogous to expression (2 1) but usually there is a given value xp which can be obtained under concentrate removal L :
if steady state has been reached in the apparatus. If concentrate removal from the apparatus is absent (L = 0), then the separation process becomes one with a limiting concentration xLlmas derived below. Assuming in eqn (27) L = 0 and dx $=
0
then
F ( G,, - a)( x,)- ax) = 0.
This equality is valid under two cases: G,, - cx= 0 and xg - ax= 0. The case G,, - cx= 0 was described above and corresponds to limiting concentration permeability decreases to 0. The case x0 - ax= 0 XLlrn. I = G,,/c when membrane has a limiting concentration xLim,2 = x0/a and occurs when solute concentration x2-x0, i.e. enriching the medium with solute is completed. Since the general conditions of the separation process are x2 I x0
separation process in such an apparatus
for three cases: ( 1) mode (2) (3)
under start-up (the process moving from initial state (x0) to operating with a given concentration x* = x,); when the concentrate removal rate (L ) is changed; the transient response to a change in initial concentration x,,.
520
L. A. Rovinsky V. L. Yarovenco
The start-up period (from x0 to steady state of operating mode) is determined as above from eqn (26). Its solution (under L = 0) is given in Appendix 3 and has the double form:
1
(1 - a)Dx(O)
-40)I
l(l-a)~~(0)-au]B’-(l-a)f,x(O)~ifgzo
u
x(r)=
1
Dlc-40)- GJ
Lit+ c
f_,[cx(O)G,]+{cD-f,[cx(O)-
Gc,]}e-““if
D-C0
(29)
where D = F( aG,, - cx(,)/ V, f2 = - acF/ V. It can be shown in solution (29) that when t= 0 (initial moment) x I,=,, = x( 0) for both cases but if D20 (i.e. (X,/~)=XLim,2IXLim,] =(G”/c)) then X(t)~XLim,Z=(x,,/a)andx(t)-x,i,,,=(G,/c)whenD
Concentrate
),&il g D
-~)x(O){~D+fz[~xg-x(O)II
[ax,- x(O)][(1 -
u)fzx(O) - CD] ’
(30)
removal
The concentrate is removed from the apparatus with a flow rate L when its concentration reaches the desired value. The calculation of the transient process to steady state for this case can be realized in the same manner as above and its form is the following: x(t)=-
Q-fi+gi %
%h.m+fi - D, h . P.fAO)+h +Dlle”“-[2f2x(0)+fi-DIl
(31)
where coefficients fi = ( aFGo + CFX, + L )/V; f2 - see above, Dl=$7ii.
It is necessary to take the following into account when using expressions (29), (31). (1) If there is a problem in calculation of the start-up period then eqn (29) is used with x( 0) = x0, L = 0. (2) When the separation process is developed and solute concentration in the apparatus x(t) reaches the desired value xg and concentrate removal starts with a flow rate L then eqn (3 1) is used with x( 0) = xg, L# 0 and calculation according to eqn (3 1) gives the transient process from xg to x* where x* is the root of eqn (27). (3) Lastly, consider the steady state of the separation process (with x,(O), L(0) and corresponding value x*(O) according to (27)) subjected to a step
Model@
of membrane separation processes
521
modification to new values x0 # x,(O) or L # L(0). The concentration change x from x*(O) to x* can be calculated by eqn (3 1) where x(0)=x*(o). EXPERIMENTS
AND NUMERICAL
EXAMPLES
The experimental data and examples described below refer to the concentration of micro-organism culture liquid, producing amylolytic enzymes by ultrafiltration in a three-section unit type YP3-3 designed in the All-Russian Institute of Food Biotechnology (Moscow). Materials and methods A culture liquid of yeast-like fungi Asp. awamori-446, End. bisporu and ethers cultivated by the submerged bath method in the biotechnology plant was concentrated. The initial dry substance concentration was 25 to 93 kg/m3 and the outlet concentration of product (concentrate) was 180 to 230 kg/m3. ‘The concentrate was used for further concentration and for ethanol production from starchy raw materials. The concentration values were determined during the process by refractometry and analysis of enzyme activity. Empirical II, c, G,
parameters
were determined (Rovinsky & Yarowenko, 1978) from experimental data (x2 and G) as functions of medium concentration x(t) during the concentration process. Statistical processing of the experimental data gives the parameters: a= 0.27, c= 2.22 X 10-s m”/kg.s, Go = 0.556 X 10e5 m/s. The functions x?(x) and G(x) with these parameters are calculated according to expression (5): x2 =0*27x, kg/m3; G= 0.556 x 10e5 - 2.22 X lo-” x, m/s, and are shown in Fig. 3 with experimental data (points). The difference between calculated values and experimental data is not greater than 10%. This result confirms the adequacy of expression (5) for experimental data and permits its use for calculations in the concentration range x= 26-240 kg/m”.
0.6
20
40
60
80
100 120 140 160 180 200 220 x, kg/m3
Fig. 3.
Permeate concentration
and membrane
permeability.
522
L. A. Rovinsky, V. L. Yarovenco
Experiments and examples Separation
in CDFA
The initial concentration was x0 = 80 kg/m3 and the above parameters a, c, G,, were used. The limiting concentration according to (12) was: xLirn= Go/c= 250 kg/m3 and when substituted into expression (16) gives the formula 2.37 -i
F=1.39x10h
802’37-’
-
5 ’ i=I 250’(2.37-
i)(250x)2’37-’
(32)
’ m2
The result of calculation according to (32) is shown in Fig. 4 with experimental data (dotted). Separation
by CMFA
For the concentration process in a single section of CMFA, the limiting concentration xLim is dependent on the correlation between parameters a, c, G, and to (23) XLim= XLim,I= 250 kg/m3 initial concentration x0. According k /m3 when x,=80 kg/m”, but XLim=XLim.Z=185 < XLim.2 z~Jaz296.3 kg/m3 when x0 = 50 kg/m B. For an initial value x0 = 50 kg/m3, F= 15 m2 and the above parameters a, c, G, give the stationary concentration according to (2 1) as: x*=
&,-6.09x
1O-5)2+4*86~
lo-?,,-&,+6.09~
lo-’
48.62 x 1O-3
, kg/m”
and the flow rates L *, W* according to (22): L$=L,
50 - 0.27x* 0*73x*
’
m’/s;
W*= 15(0*556 X 10e5 - 2.22 X lo-‘x*),
kg/m.
These are shown in Fig. 5 with experimental data (dotted). For the dynamic response of a concentration process in a CMFA, consider two cases: ( 1) start-up period from x0 to x* (operating mode); (2) the transient response from one steady state to another when the flow rate L,, undergoes a step modification. For the first case there is a CMFA with parameters F= 15 m2, V= 0.08 m3 and initial concentration x,, = 80 kg/m3, L, = 5.56 x 10m5 m3/s. From these values, the calculation parameters are: b,/bz = - 21.71; u= 136.14 kg/m3; x* = 147 kg/m3 and expression (25) becomes: 205*3e”‘X3.10-%_
67.o
x(t)= 136.14 205.3J~83.10-3~+67.0 + 10.86, kg/m3
(33)
The process x(t) increases from x0 = 80 kg/m3 to a stationary value x* = 147 kg/m3 calculated according to (33). This is shown in Fig. 6 (curve 1) with experimental data (dotted) and requires a time of approximately t- 1.2 h For the second case there is steady state in the same apparatus with the same x0 = 80 kg/ m3 and the inlet flow rate undergoes a step modification from L,(O)= 5.56 X 10e5 m3/s to L,=22*22 X lo-’ m3/s at time t=O. It evokes the
Modelling of membrane separation processes
Oo
2II
4
6I
8I
10 I
12 I
14 I
16 II
18
523
20 I
F,m*
Concentration process in the CDFA.
Fig. 4.
ran
c
-k ----
0
2
4
-XLim
6
8
10
b,
Fig. 5.
12
14
16
18
20
m3/s(x 105)
Stationary and limiting values for the CMFA.
transient response for concentration from x*(O) = 147 kg/m3 to x* = 92.6 kg/m’ (according to (21)). Substituting these values into eqn (25) gives the calculating formula: 906*5
pri
x 10-2r
+ 50.8 x(t) = 427.8 906.5 e”‘2ax10-2r- 50.8 - 33 1.7, kg/m’
((34)
The transient process according to (34) is shown in Fig. 6 (curve 2) and requires practically one third of an hour from the start of the change.
524
Fig. 6.
L. A. Rovinsky, 1/ L. Yarovenco
Non-steady
t, h states in the CMFA: (1) start-up period from x0 to x*; (2) transient process when L,, undergoes a step modification.
CMFA with recirculation Consider the same apparatus (parameters a, c, G,, F, I/ - see above) with recirculation and controlled removal of concentrate. First, calculate the start-up period of increasing concentration from initial value to operating mode. Assume x,) = 80 kg/m3 and L = 0 (without concentrate removal). Under these conditions D < 0 in eqn (29), limiting concentration XLim= XLim,,= 250 kg/m3 and substitution of numerical values into (29) gives the calculating formula: x(t)=250+
1.95 0.0423 _ 0.0538 e5.153x10-‘r’
The plot of this function is shown in Fig. 7, curve 1. Next, assume that concentrate removal begins at t= 3 h from the process start. It is necessary to prescribe a concentrate with x* = xg= 150 kg/m3. According to (28), determine the value of concentrate removal L(x,)= 1.89 X 10m5 m”/s and substitute all values into (31) to give the calculating formula: 4328 x(t)= 150+ 89.0 1 e5”04x10-Sr+13.79 , kg/m3 (besides here x(O)=x(,=,,=212 kg/m3 - the moment concentrate removal begins). The function x(t) is shown in Fig. 7, curve 2. For comparison calculate the same process under the same conditions if given a stationary concentration x* = 190 kg/m3. Then a similar calculation gives L(x,) = O-525 X 10W5m3/s and x(t)=
190+ 45.82
which is plotted as curve 3 in Fig. 7.
1080 e25-3 x IO-51 +
4.78
2 kg/m3
Modelling of membrane separation processes
525
t-h
Fig. 7. Non-steady states in the CMFA with recirculation: (1) start-up period when L= 0; (2) process with concentrate removal when L= 1.89. lo-” m”/s; (3) process with concentrate removal when L-O-525 X lo-” m-‘/s.
MAIN
The process of liquid medium complete mixing flow apparatus involves the following.
CONCLUSIONS
separation (concentration of solution) in (CMFA) and displacement flow (CDFA)
1 Analytical description of non-steady states and limiting concentration in CMFA and CDFA is possible and the calculation formulae are presented here. 2 New theoretical and practical results are obtained: - evaluation of limiting solute concentration in different types of apparatus; - the existence of two independent values of limiting concentration and boundary conditions for both possible variants; - mathematical models of non-steady states: start-up period (concentration changing from initial to stationary or limiting values); the transient response changes from one steady state to another because of changes in initial conditions (x0, L,, L ). This also describes the process after concentrate removal begins; - correlation between variables of process separation and membrane surface for CDFA. 3 The empirical dependencies x2 = ax and G= Go - cx are evaluated for the degree of concentration and can be used for practical calculations. 4 It can be noted for CDFA that separation rate is changed along the membrane length (as shown in Fig. 4) and a maximum separation rate exists at a certain membrane point. This phenomenon facilitates the
526
L. A. Rovinsky, K L. Yarovenco
arrangement of individual apparatus sections in order to achieve maximum process productivity. 5 The membrane models obtained can be used as a basis for calculation of main apparatus design parameters, choice of membrane properties and initial conditions for a real separation process and the design of the optimization algorithms and control systems.
REFERENCES Brock, T. D. (1983). Membrane Filtration. Publ. of Science Tech., Madison. Cheryan, M. (1986). Ultrafiltration Handbook. Technomic Publ., Lancaster, Base]. Ditnerskiy, U. I. (1986). Baromembrannie processi. Chimiya, Moscow (in Russian). (Baromembrane processes. Chemistry, Moscow). Hwang, S. & Kammermeyer, K. (1975). Membranes in Separation. John Wiley, NJ. Niemi, H., Raimoaho, Y. & Palosaari, S. (1986). Modelling and simulation of ultrafiltration and reverse osmosis processes. Acta Polytechnica Scandinavica, No. 174, Helsinki. Rovinsky, I. A. & Yarovenko, V. L. (1978). Modelirovanie i Optimizaciya Microbiologicheskih processov. Pischevaya Promishlennost, Moscow (in Russian). (Simulation and Optimization of Microbiology Processes. Food Industry, Moscow). Torrey, S. (1984). Membrane and Ultrafiltration Technology. Noyes Data Corp., Park Ridge.
APPENDIX
1: THE SOLUTION
OF EQUATION
( 15)
A consideration of coefficients A and B in eqn (15) shows that B= AG,/ c= AxLim,then eqn ( 15) can be rewritten as
$=Axa(_qi,
- x)
and after separation of the variables: dx
dx
dt ’ Xa( XLi, Integration
of eqn (Al.l)
- X)
=AdF.
(Al.l)
gives
J
dx
x”( XLim -
X)
+ c=AF
(A1.2)
where the integration constant c is determined from the initial conditions. The integral in (A1.2) is tabular and is expressed by a recurrence formula:
I
dx
1
Xa(XLim - X)=X~im((X-l)Xa-1-X~im(a-2)Xa-2-
1 “’
527
Modelling of membrane separation processes
or in the common form dX
=-
c
(A1..3)
1
,= I xllm( a - i)x”-’ ’
I XU(%Il - x)
number of a term in the sum (A1.3). Substitution into (A1.2) where i= 1,2,...gives the constant C:
and its substitution into (A1.2) gives the solution 1
AF= 1
,= I xllm(a - i)xR-’
which, after transformation,
APPENDIX
-1
gives the solution (16).
2: THE SOLUTION
Substitute z = x+ (h,/2b,)
1
I=l xl_,J Q - i)x”-’
OF EQUATION
(24)
and transform eqn (24) to form:
and after substitution u= x* + ( b,/2bz): dz dt=bL(~Z-z’)dt
(A2.1)
where x* is the stationary concentration according to (2 1). After separation the variables eqn (A2.1) is transformed to
dz -= uL-z.!
b2 dt
and integration with initial condition u I,= (,= u( 0) gives the solution Z(t)=
u
[u+z(0)]ezb'"'-
u+ z(0)
[u+z(0)]ezbzU'+
u-z(O)
’
Taking into account the above substitutions leads to the solution (25).
of
L. A. Rovinsky, V. L. Yarovenco
528 APPENDIX 1 The case
3: THE SOLUTION OF EQUATION
(26)
L= 0 (without concentrate removal)
The initial eqn (26) is written in the form dx d,+fix’+/lx+fu=o
(A3.1)
where: J;, = - xoFG,,/ V,f, = F (aG, + c.qJ/ V, f2 = - acF/ V. The substitution of a new variable z= x+ (fi/2fi) transforms eqn (A3.1) to (dz/dr) +f2z - (f,/4fi) +J;, = 0 and substitution of z = ( l/u) + (D/2f,) gives (&A/ dt)=f,kDDuwhereD=F(aG,cxJ/ K After variables separation PCdu
f,kDu
(A3.2)
dt.
The integration of eqn (A3.2) gives:
u(t)=
L jj+
u(O)-; [
1
f7 e - I>1
when D-CO
This means that u(O) equals u(O) = (xo/a) when D2 0 and u( 0) = ( G,,/c) when D < 0. Taking into account the above substitutions leads to solution (29). 2 The case L# 0 (with concentrate
removal)
Here in eqn (A3.1) the coefficients are: A, = - x,(FG,, + L )/ v; f, = (aFG,, + CFX, + L )/v;
fi = -
acF/ V.
Next, eqn (A3.1) is transformed to form (A3.2) but with parameter D, instead of D: D, =,/m. A solution similar to that above under initial condition x I,= ,) = x(0)gives solution (3 1).
Printed in Great Britain. All tights reserved 0260-8774/95/$9 SO ELSEVIER
Mathematical
Modelling and Calculation of Membrane Separation Processes
Lev A. Rovinsky” & Victor L. Yarovenco Corresponding (Received 22 December
Institute of Food Industry, Moscow, Russia 1992; revised version received 30 November accepted 10 January 1994)
1993;
ABSTRACT In order to develop a methodology for mathematical modelling of membrane separation processes in continuous systems of the complete mixing and complete displacement type, theoretical and experimental studies were undertaken. The problem of determining a limiting concentration in the processed medium and the conditions for its achievement are described. New correlation results for current concentration and concentrate and permeate flow rates with membrane surjace area are presented. The non-steady (dynamic) states of a membrane separation process, i.e. the start-up period from initial conditions to steady state, and transient responses because of modification of inlet conditions are presented as analytical expressions and formulae. The theoretical results and empirical formulae are tested using experimental data obtained during membrane concentration of biotechnology products.
NOTATION Empirical parameter Auxiliary parameters Coefficients b,,,b,,& Empirical parameter (m’/kg . s) ( Auxiliary parameters Coefficients $32 Membrane surface area (m2) g( x, L );f (x, L ) Empirical functions i,B
*To whom correspondence Moscow, Russia 111395.
should be addressed
511
at: 19-l-284,
Krasny
kazanetz
St,
512
L. A. Rovinsky, I/: L. Yarovenco
Membrane permeability (the same for pure solvent) (m/s) Inlet and outlet flow rate (m”/s) Time (s) Inlet and local product concentration (kg/m3) Local and average permeate concentration (kg/m3) Auxiliary variable Permeate flow rate (m3/s) a
Auxiliary parameter
Subscripts 0 Lim g
Initial values Limiting values Given values
Superscripts * b
Stationary values Boundary values
INTRODUCTION Membrane processes for liquid media separation are calculated on the basis of diffusion or mass transfer phenomena and empirical relationships describe the solute flow and concentration when passing through a semipermeable membrane. The methods of calculation of complete mixing flow apparatus (CMFA) and complete displacement flow apparatus (CDFA) were described in detail in monographs (Hwang & Kammermeyer, 1975; Brock, 1983; Torrey, 1984; Ditnerskiy, 1986). The apparatus type and technology of membrane separation processes were shown in a survey (Niemy et al., 1986) and a handbook (Cheryan, 1986). Even though the above-mentioned methods of calculation are sufficiently complete there are still some aspects unresolved, for instance, the maximum attainable concentration in each process, non-steady state processes (dynamic models) and relationships between the main processing parameters and processing results. A new approach based on mathematical models mentioned above (Hwang & Kammermeyer, 1975; Ditnerskiy, 1986) will be considered below. This paper considers: - mathematical modelling of membrane separation (solution concentration) for non-steady states; - evaluation of limiting (maximum) concentration as a function of process conditions; - correlation between concentration process results and CDFA parameters; - formulae for technological design of membrane separation equipment.
Modelling of membrane separation processes
513
THEORY Methodology
and assumptions
The methodological approach and mathematical conditions and empirical relationships:
models are based on known
( 1) between concentrations of dissolved solutes in the apparatus (x) and in the permeate (x2): x2 = g(x); (2) the membrane permeability G= Go -f( X,L ) where L is the flow rate of concentrate, C$ is the membrane permeability for pure solvent, g and fare empirical functions.
Further consideration involved all the assumptions usually accepted for such processes, namely: temperature and pressure are constant during the process; surface phenomena on the membrane are taken into consideration to the extent that they are reflected in the empirical relationships; the influence of sediment on the filtering properties of the membrane is either absent or included in the empirical performance lumped in with all other effects. Apparatus
and processes
considered
Three types of membrane apparatus will be considered: - CDFA shown in Fig. I( a); - CMFA (single section) shown in Fig. l(b); - CMFA with recirculation and controlled (Fig. l(c)).
removal
of
concentrate
All cases presume an inlet flow rate L, and concentration x0, and an outlet (concentrate) flow rate L and concentration X. For the CDFA system (Fig. l(a)) one must consider local concentration x2 and common (average) permeate concentration X2. Concentration
in CDFA
The basic relationships A piston-form of liquid motion through the apparatus is assumed. According to Ditnerskiy ( 1986) and Hwang and Kammermeyer ( 1975), the equation system describing the separation process can be written as: mass and flow balance equations: Lox,, = wz, + Lx L,,=L+
w
(1) (2)
The differential equations for mass and flow change:
4
Wi,
)
~=g(x,L)
(3)
L. A. Rovinsky, I/ L. Yaroverlco
514
dW z= If the concentration
G= G,,-f(x,L)
x is small, the functions g and fare assumed to be linear:
g(x) = x2
=
f(x) = G= G,, - cx,
ax;
(5)
where parameters a, c, G,, are determined experimentally. There are known relationships (Ditnerskiy, 1986) for an arbitrary section of the apparatus: L= L&/x,,) ‘b”- ‘) w= L,,[ 1 - (x/x,,p-
-5 = 41
(6) “1
1 _
(x/x”)fJ/(l~- ‘1
1
ix,x,,)l;:“-c
_
(64
(7)
The correlation between concentrations X, X1 and flow rates L, W with the current membrane surface area F were presented as differential equations only, without solutions. Suggested approach The relationships (6), (6a) and (7) are not complete and can be developed. Isolating from eqns ( 1) and (2) the term Wi, = Lox,, - x(L, - W) and substituting it into eqn (3) yields d(Wiz) x,,L,, - wiz
=
dW a L,,-
(8)
Integration of eqn (8) with initial conditions W( 0) = 0; X2(O)= x0, assuming that x,, and L are constant during a given process:
Ml, x,, L,, - Wi?
or after transformations
which supplements models below.
( )
=----L,, (’ L-f,- W
(9)
- the function X2(W ):
relationships
(6), (6a) and (7) and is fundamental
for the
The limiting concentration The concentration in the processed liquid moving along the membrane increases from x,) to a certain limiting value (x,,,) at which separation will stop. The conditions of the separation process may be expressed as ax,,S.f,Ix,,Ixlx,,,
(11)
Modeling of membrane separation processes
515
and the value XLimcan be determined from eqn (4) for steady state when dW z=o: The corresponding
(12)
XLim= G,,lc
limiting flow rates are then
and the limiting permeate concentration,
according to eqn (7).
The process separation dynamics
Equation (6a) can be rewritten as
and substitution into eqn (4) gives 0,
i.e. a first-order differential equation for the variable L = L,, - W: dL --cx,,L;,-“L’‘-I dF
+ G,,=()
(14)
which can be solved as a function L( F ). To obtain the correlation x( F ), eqn ( 14) can be transformed to an equation for x. Differentiating eqn (6a): d-=W Lox:i(‘-“’ x(z - Cl!/\ I - 0) d” dF l-a dF and substituting into eqn (4) gives the equation
FF+Ap+i =Bx”
(15)
where
A=cW-4 L,,
I/‘[~- 11.
x0
B_
9
Gil -4 L,,
2-a
I/(rr- II.
x (’
’
*=1-a’
The solution of eqn ( 15) is given in Appendix 1 and is F=L
c
d-‘-x;-
A ,=, xli,,,(a-
i)(xx,,)“-’ ’
i= 1,2 ,...
(16)
516
L. A. Rovinsky, V L. Yarovenco
where xLim= G,/c according to (13) and the numerical exactness of F(x) increases with an increase in the number of terms in expression ( 16). The dependence of other variables on F: L( F ), W( F ), X2(F ) (along the apparatus length) is determined from expressions (6), (6a), (7) and ( 10). Complete mixing flow apparatus (CMFA) This ideal type of apparatus is characterized by an identical concentration throughout, due to extensive liquid mixing within the apparatus. Usually such an apparatus models an isolated section of a real apparatus. The concentration process in a CMFA is shown in Fig. l(b). Here the solute concentration at any arbitrary point in the apparatus volume I/ is the same and the permeate concentration x2 = ax is the same all along the membrane surface F. From the initial eqns ( l)-( 4) only expression (2) is retained but eqns ( 1) and (4) become: Lox, = Lx+
wi,
(17)
W=FG=F(G,,-cx)
(18)
and the differential equation for solute quantity in the apparatus can be written instead of eqn (3):
vp=
Lc,x,, - Lx-
wiz .
(19)
The steady state
If the initial process conditions (x0, L,) are invariable then it will become a steady state separation process with stationary concentrations and flow rates. In this case (dx/d t= 0), eqn ( 19) is identical to ( 17) and substitution of eqn ( 18) and x2 = ax into eqn (17) leads to a quadratic equation for x: cF(l-a)x*+x[L,-(l-a)FG,,]-x,L,,=O.
x2
Fig. 1.
(20)
(b)
Membrane apparatus: (a) complete displacing flow apparatus (CDFA); complete mixing flow apparatus (CMFA); (c) CMFA with recirculation.
(b)
Modelling of membrane separation processes
Solution of eqn (20) gives the stationary concentration
517
x*:
+j[L,-(1-a)FG,]‘+IcF(l-a)x,,L,,-L,+(l-a)FG,,
(21)
2cF( 1 - a) and the corresponding
flow rates of concentrate
* L” = L,, (z’)-aTx* ;
(L*) and permeate ( W*) are
* W*=1.,,(li_~~~*=f.iG,,-~~*).
(22)
The limiting concentration from solution (21) that the function x*(L,,) has an inverse dependence. Under constant membrane-medium system properties (a, c, G,) and initial concentration x0, the stationary values x*, L”, W* are dependent on L, only. It can easily be shown from expression (2 1) that if
It can be noted
L,,-0:
x*-xLlm=
G,,/c,
i.e. limiting (maximum) concentration in CMFA under certain conditions. According to (22) the permeate flow rate W* - 0 because membrane permeability G= G,, - cx* -+0. However it is not the only possibility. It can be shown from the first equation of (22) that when x*=x,/a, the concentrate flow rate L”+O. As the membrane-medium system properties (a, c, G,) and inlet flow concentration xg are independent then both variants: xLim, , = ( G,,/c), xLim,Z= (x,/a) are possible. If the concentration x* increases then x* -+xLimbut xLimis the minimum of two values: xLim=min{x,,,,,
= Go/c;
(23’)
XLllJl.2 =x&d
The boundary of these cases is inlet concentration x6 = aG,Jc: if xc,< XI;then xLlm= x(,/a, i.e. does not depend on inlet flow. This is shown schematically in Fig. 2.
XLim
x, %b=aG,/c
Fig. 2.
The limiting concentration
as a function of x,,.
L. A. Rovinsky, V L. Yarovenco
518
The dynamic separation process There are two non-steady states: the process start period (x(t) changes to x*) and the transient response process from one steady state to because of any perturbation in process conditions. The function x(t) non-steady states can be obtained as follows: the dependencies (2) and substituted into eqn ( 19) and transformed into the Riccati equation dx
;=
from x0 another for such (18) are
- b2x2 - b,x+ b,,
(24)
where coefficients: b,=(l-a)cF/K
b,=[L,-(l-a)FG,,]/V;
b, = L,x,,l K
The solution of eqn (24) is given in Appendix 2 and is ( x(t)=
b,
)
[i*+x(0)+~]e2.D;‘-x*+d0!
b,
--
x*+z *
2b2
jx*+x(O)+~]e'.'..+x'-dOi
(25)
where the auxiliary parameter u = x” + (b, /2 b2) and the stationary concentration x* are given by (2 1). If a start-up period of separation is being considered then x(O) = x,, and solution (25) presents the transient x(t) from x0 to x*, i.e. the movement from start-up to operating mode. If, however, there was already a steady state in the CFMA with x*(O) and one or both initial conditions (x0,&) are changed at an arbitrary moment t= 0, then the solution is the transient response and x*(O) changes to x* # x*( 0). In this case the transient function x(t) can be calculated by (25)underx(O)=x*(O). CMFA with recirculation A real multi-section membrane apparatus contains several sections such as the above CMFA with recirculation (Fig. l(c)). In this scheme a circulating pump continually recycles separated (concentrated) material having a flow rate L, to the inlet of a unit. Concentrate with flow rate L can be removed from the unit when the desired concentration xs is reached; simultaneously, new feed is fed to the apparatus. The balance equation in such a case is VS=L’x’-x(L+L,)-
wx*,
where the flow rate (L’) and concentration (x’) through the apparatus are determined by relationships: L’x’ = L,x, + Lx; L’ + L, + L and accordingly condi-
Modelling of membrane separationprocesses
519
tion (2) becomes
dg=W(x,,-x,)-L(x-x,,) Substituting equality (18) and x2 = ax transforms it to the form: V$=ai:Fx’-x(uFG,,+cFx,,+L)+x,,(L+FG,,).
1;26)
Steady state
If the characteristics a, c, G, and initial condition x0 are constant during a real case then the process becomes a steady state one and stationary concentrations and flow rates are dependent on concentrate removal L only. The stationary concentration x* can be obtained from eqn (26) under condition (dx/dt) = 0: acFx’ - x( aFG,, + cFx,, + L ) + x,,(L + FG,,) = 0.
(27)
The solution of the quadratic eqn (27) gives a value x*( L ) analogous to expression (2 1) but usually there is a given value xp which can be obtained under concentrate removal L :
if steady state has been reached in the apparatus. If concentrate removal from the apparatus is absent (L = 0), then the separation process becomes one with a limiting concentration xLlmas derived below. Assuming in eqn (27) L = 0 and dx $=
0
then
F ( G,, - a)( x,)- ax) = 0.
This equality is valid under two cases: G,, - cx= 0 and xg - ax= 0. The case G,, - cx= 0 was described above and corresponds to limiting concentration permeability decreases to 0. The case x0 - ax= 0 XLlrn. I = G,,/c when membrane has a limiting concentration xLim,2 = x0/a and occurs when solute concentration x2-x0, i.e. enriching the medium with solute is completed. Since the general conditions of the separation process are x2 I x0
separation process in such an apparatus
for three cases: ( 1) mode (2) (3)
under start-up (the process moving from initial state (x0) to operating with a given concentration x* = x,); when the concentrate removal rate (L ) is changed; the transient response to a change in initial concentration x,,.
520
L. A. Rovinsky V. L. Yarovenco
The start-up period (from x0 to steady state of operating mode) is determined as above from eqn (26). Its solution (under L = 0) is given in Appendix 3 and has the double form:
1
(1 - a)Dx(O)
-40)I
l(l-a)~~(0)-au]B’-(l-a)f,x(O)~ifgzo
u
x(r)=
1
Dlc-40)- GJ
Lit+ c
f_,[cx(O)G,]+{cD-f,[cx(O)-
Gc,]}e-““if
D-C0
(29)
where D = F( aG,, - cx(,)/ V, f2 = - acF/ V. It can be shown in solution (29) that when t= 0 (initial moment) x I,=,, = x( 0) for both cases but if D20 (i.e. (X,/~)=XLim,2IXLim,] =(G”/c)) then X(t)~XLim,Z=(x,,/a)andx(t)-x,i,,,=(G,/c)whenD
Concentrate
),&il g D
-~)x(O){~D+fz[~xg-x(O)II
[ax,- x(O)][(1 -
u)fzx(O) - CD] ’
(30)
removal
The concentrate is removed from the apparatus with a flow rate L when its concentration reaches the desired value. The calculation of the transient process to steady state for this case can be realized in the same manner as above and its form is the following: x(t)=-
Q-fi+gi %
%h.m+fi - D, h . P.fAO)+h +Dlle”“-[2f2x(0)+fi-DIl
(31)
where coefficients fi = ( aFGo + CFX, + L )/V; f2 - see above, Dl=$7ii.
It is necessary to take the following into account when using expressions (29), (31). (1) If there is a problem in calculation of the start-up period then eqn (29) is used with x( 0) = x0, L = 0. (2) When the separation process is developed and solute concentration in the apparatus x(t) reaches the desired value xg and concentrate removal starts with a flow rate L then eqn (3 1) is used with x( 0) = xg, L# 0 and calculation according to eqn (3 1) gives the transient process from xg to x* where x* is the root of eqn (27). (3) Lastly, consider the steady state of the separation process (with x,(O), L(0) and corresponding value x*(O) according to (27)) subjected to a step
Model@
of membrane separation processes
521
modification to new values x0 # x,(O) or L # L(0). The concentration change x from x*(O) to x* can be calculated by eqn (3 1) where x(0)=x*(o). EXPERIMENTS
AND NUMERICAL
EXAMPLES
The experimental data and examples described below refer to the concentration of micro-organism culture liquid, producing amylolytic enzymes by ultrafiltration in a three-section unit type YP3-3 designed in the All-Russian Institute of Food Biotechnology (Moscow). Materials and methods A culture liquid of yeast-like fungi Asp. awamori-446, End. bisporu and ethers cultivated by the submerged bath method in the biotechnology plant was concentrated. The initial dry substance concentration was 25 to 93 kg/m3 and the outlet concentration of product (concentrate) was 180 to 230 kg/m3. ‘The concentrate was used for further concentration and for ethanol production from starchy raw materials. The concentration values were determined during the process by refractometry and analysis of enzyme activity. Empirical II, c, G,
parameters
were determined (Rovinsky & Yarowenko, 1978) from experimental data (x2 and G) as functions of medium concentration x(t) during the concentration process. Statistical processing of the experimental data gives the parameters: a= 0.27, c= 2.22 X 10-s m”/kg.s, Go = 0.556 X 10e5 m/s. The functions x?(x) and G(x) with these parameters are calculated according to expression (5): x2 =0*27x, kg/m3; G= 0.556 x 10e5 - 2.22 X lo-” x, m/s, and are shown in Fig. 3 with experimental data (points). The difference between calculated values and experimental data is not greater than 10%. This result confirms the adequacy of expression (5) for experimental data and permits its use for calculations in the concentration range x= 26-240 kg/m”.
0.6
20
40
60
80
100 120 140 160 180 200 220 x, kg/m3
Fig. 3.
Permeate concentration
and membrane
permeability.
522
L. A. Rovinsky, V. L. Yarovenco
Experiments and examples Separation
in CDFA
The initial concentration was x0 = 80 kg/m3 and the above parameters a, c, G,, were used. The limiting concentration according to (12) was: xLirn= Go/c= 250 kg/m3 and when substituted into expression (16) gives the formula 2.37 -i
F=1.39x10h
802’37-’
-
5 ’ i=I 250’(2.37-
i)(250x)2’37-’
(32)
’ m2
The result of calculation according to (32) is shown in Fig. 4 with experimental data (dotted). Separation
by CMFA
For the concentration process in a single section of CMFA, the limiting concentration xLim is dependent on the correlation between parameters a, c, G, and to (23) XLim= XLim,I= 250 kg/m3 initial concentration x0. According k /m3 when x,=80 kg/m”, but XLim=XLim.Z=185 < XLim.2 z~Jaz296.3 kg/m3 when x0 = 50 kg/m B. For an initial value x0 = 50 kg/m3, F= 15 m2 and the above parameters a, c, G, give the stationary concentration according to (2 1) as: x*=
&,-6.09x
1O-5)2+4*86~
lo-?,,-&,+6.09~
lo-’
48.62 x 1O-3
, kg/m”
and the flow rates L *, W* according to (22): L$=L,
50 - 0.27x* 0*73x*
’
m’/s;
W*= 15(0*556 X 10e5 - 2.22 X lo-‘x*),
kg/m.
These are shown in Fig. 5 with experimental data (dotted). For the dynamic response of a concentration process in a CMFA, consider two cases: ( 1) start-up period from x0 to x* (operating mode); (2) the transient response from one steady state to another when the flow rate L,, undergoes a step modification. For the first case there is a CMFA with parameters F= 15 m2, V= 0.08 m3 and initial concentration x,, = 80 kg/m3, L, = 5.56 x 10m5 m3/s. From these values, the calculation parameters are: b,/bz = - 21.71; u= 136.14 kg/m3; x* = 147 kg/m3 and expression (25) becomes: 205*3e”‘X3.10-%_
67.o
x(t)= 136.14 205.3J~83.10-3~+67.0 + 10.86, kg/m3
(33)
The process x(t) increases from x0 = 80 kg/m3 to a stationary value x* = 147 kg/m3 calculated according to (33). This is shown in Fig. 6 (curve 1) with experimental data (dotted) and requires a time of approximately t- 1.2 h For the second case there is steady state in the same apparatus with the same x0 = 80 kg/ m3 and the inlet flow rate undergoes a step modification from L,(O)= 5.56 X 10e5 m3/s to L,=22*22 X lo-’ m3/s at time t=O. It evokes the
Modelling of membrane separation processes
Oo
2II
4
6I
8I
10 I
12 I
14 I
16 II
18
523
20 I
F,m*
Concentration process in the CDFA.
Fig. 4.
ran
c
-k ----
0
2
4
-XLim
6
8
10
b,
Fig. 5.
12
14
16
18
20
m3/s(x 105)
Stationary and limiting values for the CMFA.
transient response for concentration from x*(O) = 147 kg/m3 to x* = 92.6 kg/m’ (according to (21)). Substituting these values into eqn (25) gives the calculating formula: 906*5
pri
x 10-2r
+ 50.8 x(t) = 427.8 906.5 e”‘2ax10-2r- 50.8 - 33 1.7, kg/m’
((34)
The transient process according to (34) is shown in Fig. 6 (curve 2) and requires practically one third of an hour from the start of the change.
524
Fig. 6.
L. A. Rovinsky, 1/ L. Yarovenco
Non-steady
t, h states in the CMFA: (1) start-up period from x0 to x*; (2) transient process when L,, undergoes a step modification.
CMFA with recirculation Consider the same apparatus (parameters a, c, G,, F, I/ - see above) with recirculation and controlled removal of concentrate. First, calculate the start-up period of increasing concentration from initial value to operating mode. Assume x,) = 80 kg/m3 and L = 0 (without concentrate removal). Under these conditions D < 0 in eqn (29), limiting concentration XLim= XLim,,= 250 kg/m3 and substitution of numerical values into (29) gives the calculating formula: x(t)=250+
1.95 0.0423 _ 0.0538 e5.153x10-‘r’
The plot of this function is shown in Fig. 7, curve 1. Next, assume that concentrate removal begins at t= 3 h from the process start. It is necessary to prescribe a concentrate with x* = xg= 150 kg/m3. According to (28), determine the value of concentrate removal L(x,)= 1.89 X 10m5 m”/s and substitute all values into (31) to give the calculating formula: 4328 x(t)= 150+ 89.0 1 e5”04x10-Sr+13.79 , kg/m3 (besides here x(O)=x(,=,,=212 kg/m3 - the moment concentrate removal begins). The function x(t) is shown in Fig. 7, curve 2. For comparison calculate the same process under the same conditions if given a stationary concentration x* = 190 kg/m3. Then a similar calculation gives L(x,) = O-525 X 10W5m3/s and x(t)=
190+ 45.82
which is plotted as curve 3 in Fig. 7.
1080 e25-3 x IO-51 +
4.78
2 kg/m3
Modelling of membrane separation processes
525
t-h
Fig. 7. Non-steady states in the CMFA with recirculation: (1) start-up period when L= 0; (2) process with concentrate removal when L= 1.89. lo-” m”/s; (3) process with concentrate removal when L-O-525 X lo-” m-‘/s.
MAIN
The process of liquid medium complete mixing flow apparatus involves the following.
CONCLUSIONS
separation (concentration of solution) in (CMFA) and displacement flow (CDFA)
1 Analytical description of non-steady states and limiting concentration in CMFA and CDFA is possible and the calculation formulae are presented here. 2 New theoretical and practical results are obtained: - evaluation of limiting solute concentration in different types of apparatus; - the existence of two independent values of limiting concentration and boundary conditions for both possible variants; - mathematical models of non-steady states: start-up period (concentration changing from initial to stationary or limiting values); the transient response changes from one steady state to another because of changes in initial conditions (x0, L,, L ). This also describes the process after concentrate removal begins; - correlation between variables of process separation and membrane surface for CDFA. 3 The empirical dependencies x2 = ax and G= Go - cx are evaluated for the degree of concentration and can be used for practical calculations. 4 It can be noted for CDFA that separation rate is changed along the membrane length (as shown in Fig. 4) and a maximum separation rate exists at a certain membrane point. This phenomenon facilitates the
526
L. A. Rovinsky, K L. Yarovenco
arrangement of individual apparatus sections in order to achieve maximum process productivity. 5 The membrane models obtained can be used as a basis for calculation of main apparatus design parameters, choice of membrane properties and initial conditions for a real separation process and the design of the optimization algorithms and control systems.
REFERENCES Brock, T. D. (1983). Membrane Filtration. Publ. of Science Tech., Madison. Cheryan, M. (1986). Ultrafiltration Handbook. Technomic Publ., Lancaster, Base]. Ditnerskiy, U. I. (1986). Baromembrannie processi. Chimiya, Moscow (in Russian). (Baromembrane processes. Chemistry, Moscow). Hwang, S. & Kammermeyer, K. (1975). Membranes in Separation. John Wiley, NJ. Niemi, H., Raimoaho, Y. & Palosaari, S. (1986). Modelling and simulation of ultrafiltration and reverse osmosis processes. Acta Polytechnica Scandinavica, No. 174, Helsinki. Rovinsky, I. A. & Yarovenko, V. L. (1978). Modelirovanie i Optimizaciya Microbiologicheskih processov. Pischevaya Promishlennost, Moscow (in Russian). (Simulation and Optimization of Microbiology Processes. Food Industry, Moscow). Torrey, S. (1984). Membrane and Ultrafiltration Technology. Noyes Data Corp., Park Ridge.
APPENDIX
1: THE SOLUTION
OF EQUATION
( 15)
A consideration of coefficients A and B in eqn (15) shows that B= AG,/ c= AxLim,then eqn ( 15) can be rewritten as
$=Axa(_qi,
- x)
and after separation of the variables: dx
dx
dt ’ Xa( XLi, Integration
of eqn (Al.l)
- X)
=AdF.
(Al.l)
gives
J
dx
x”( XLim -
X)
+ c=AF
(A1.2)
where the integration constant c is determined from the initial conditions. The integral in (A1.2) is tabular and is expressed by a recurrence formula:
I
dx
1
Xa(XLim - X)=X~im((X-l)Xa-1-X~im(a-2)Xa-2-
1 “’
527
Modelling of membrane separation processes
or in the common form dX
=-
c
(A1..3)
1
,= I xllm( a - i)x”-’ ’
I XU(%Il - x)
number of a term in the sum (A1.3). Substitution into (A1.2) where i= 1,2,...gives the constant C:
and its substitution into (A1.2) gives the solution 1
AF= 1
,= I xllm(a - i)xR-’
which, after transformation,
APPENDIX
-1
gives the solution (16).
2: THE SOLUTION
Substitute z = x+ (h,/2b,)
1
I=l xl_,J Q - i)x”-’
OF EQUATION
(24)
and transform eqn (24) to form:
and after substitution u= x* + ( b,/2bz): dz dt=bL(~Z-z’)dt
(A2.1)
where x* is the stationary concentration according to (2 1). After separation the variables eqn (A2.1) is transformed to
dz -= uL-z.!
b2 dt
and integration with initial condition u I,= (,= u( 0) gives the solution Z(t)=
u
[u+z(0)]ezb'"'-
u+ z(0)
[u+z(0)]ezbzU'+
u-z(O)
’
Taking into account the above substitutions leads to the solution (25).
of
L. A. Rovinsky, V. L. Yarovenco
528 APPENDIX 1 The case
3: THE SOLUTION OF EQUATION
(26)
L= 0 (without concentrate removal)
The initial eqn (26) is written in the form dx d,+fix’+/lx+fu=o
(A3.1)
where: J;, = - xoFG,,/ V,f, = F (aG, + c.qJ/ V, f2 = - acF/ V. The substitution of a new variable z= x+ (fi/2fi) transforms eqn (A3.1) to (dz/dr) +f2z - (f,/4fi) +J;, = 0 and substitution of z = ( l/u) + (D/2f,) gives (&A/ dt)=f,kDDuwhereD=F(aG,cxJ/ K After variables separation PCdu
f,kDu
(A3.2)
dt.
The integration of eqn (A3.2) gives:
u(t)=
L jj+
u(O)-; [
1
f7 e - I>1
when D-CO
This means that u(O) equals u(O) = (xo/a) when D2 0 and u( 0) = ( G,,/c) when D < 0. Taking into account the above substitutions leads to solution (29). 2 The case L# 0 (with concentrate
removal)
Here in eqn (A3.1) the coefficients are: A, = - x,(FG,, + L )/ v; f, = (aFG,, + CFX, + L )/v;
fi = -
acF/ V.
Next, eqn (A3.1) is transformed to form (A3.2) but with parameter D, instead of D: D, =,/m. A solution similar to that above under initial condition x I,= ,) = x(0)gives solution (3 1).