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Design and performance of two-stage reverse osmosis plant
119
Design and Performance of Two-Stage Reverse Osmosis Plants* Progetto
e Prestazioni
FRANC0
EVANGELISTA
di un Impianto
di Osmosi Inversa a Due Stadi
Dipartimento di Chimica, Ingggneria Chimica e Materiali, 67100 L’Aquila (Italy) (Received
June 9, 1988; in final form March
Universith degli Studi dell’dquila,
3, 1989)
Abstract Simple explicit equations are derived for the design of permeate-staged and reject-staged reverse osmosis plants. Their accuracy has been tested against iterative numerical procedures. Errors in the membrane area of both stages are below 10% under normal operating conditions. Therefore, the method can be usefully applied for quick initial estimates and for discriminating between competing processes. Two design examples are presented: sea-water desalination and sugar solution concentration. Recoveries of up to 40% can be expected even for highly concentrated sea-water (50 000 ppm). Concentrations of more than 35 wt.% can also be obtained with relatively low energy. In both cases reverse osmosis proves to be competitive with evaporative processes. Sommario Gli impianti di osmosi inversa a due stadi sono usati per ottenere tat rapport0 di separazione non facilmente od economicamente ottenibile in un solo stadio con le membrane attualmente in commercio. Gli impianti possono essere di due tipi, ‘permeate-staged’ (Fig. 2) o ‘reject-staged’ (Fig. 3), in dipendenza de1 prodotto desiderato. Partendo da un modello a due parametri ed usando relazioni ricavate in precedenti lavori [Z, 4J sono state sviluppate delle equazioni (10-12, 13-15) completamente esplicite per il progetto di detti impianti. I risultati di queste equazioni sono stati confrontati con quelli ottenuti con procedimenti numerici iterativi. In condizioni operative piu’ usuali, si sono ottenuti errori minori de1 10% nel calcolo de1 numero di moduli di entrambi gli stadi (Fig. 4 e 7). Per quest0 motivo il presente metodo puo’ risultare utile, sia agli ingegneri processisti nella scelta de1 tipo di process0 e nella stima economica iniziale, sia a quelli addetti alla conduzione degli impianti all’atto di eventuali modifiche di quelli gia’ esistenti. Vengono presentati due esempi applicativi: la dissalazione di un’acqua di mare altamente concentrata (5O.OOOppm) e la concentrazione di sughi zuccherini moderatamente concentrati (14% in peso). Recuperi
*Part of this paper was presented at the 5th YugoslavianAustrian-Italian Chemical Engineering Conference, Portoroz, Yugoslavia, September 1618, 1986.
0255-2701/89/$3.50
fin0 al 40% possono essere ottenuti nel prim0 case (Fig. 5) e concentrazioni oltre il 35% in peso nel second0 case (Fig. 8) con consumi di energia abbastanza content&i (Fig. 6 e 9). In ambedue i casi, I’osmosi inversa risulta piu’ economica di processi evaporativi convenzionali. Introduction The availability of new and better membranes means that membrane processes are being applied increasingly to many kinds of industrial separations, either as an alternative to more expensive processes, or in conjunction with other conventional processes. Reverse osmosis is the most used and reliable of these membrane processes. One of the factors which prevents its further expansion, however, is the osmotic pressures of the solutions facing the membrane whose difference must not exceed the applied pressure or, rather, should be less than 80% of it. This, in turn, means that there is a maximum reject concentration or a minimum permeate concentration obtainable from a plant for given solutions, membranes and operating conditions. Hence, the separation factor of one stage, defined as the ratio between reject and permeate concentration, has an upper limit. Therefore, if we want to extend its range beyond this limit, one more stage must be added. Depending on the purpose of the operation, that is, if we want to decrease the permeate concentration or increase the reject concentration, the plant can be
Chem. Eng. Process., 25 (1989) 119-125
0 Elsevier SequoiajPrinted
in The Netherlands
120 permeate-staged or reject-staged. While the permeate from the first stage must always be repumped and then fed to a second stage, the inter-stage reject pump may be omitted. But, while both stages may be equipped with the same membrane in the first case, in the second case different membranes, with high rejecting characteristics in the first stage and low rejecting characteristics in the second stage, must usually be used. In both cases, if the overall recovery is low, an energy recovery system can be installed in the reject stream. The design of such plants is usually carried out with iterative numerical procedures owing to the presence of recycle streams and of systems of nonlinear equations for the prediction of local membrane performances [ 11. From them, given the flow rates and concentrations of the two streams and the operating conditions of both stages, concentrations and flow rates of the other streams, recovery fractions and membrane area of both stages are obtained. The purpose of the present paper is to develop an explicit procedure using the same input data as are used in the iterative method. Theory In a recent paper [2] Evangelista has shown that, with some simplifying assumptions, the following analytical relationship can be derived between the recovery fraction 4 and the concentration factor f of a single-stage reverse osmosis plant: 1 -C#I =exp(-tfl/2)
(1)
where
0.0 0.0
l/f
1.0
Fig. 1. A plot of eqn. (1) at different rejections.
coordinates (1, 1) and (0,O). If the membrane had been perfectly rejecting, the curve would have been given by the diagonal of the square. As the rejection decreases, its departure from the diagonal increases accordingly. If the rejection had been 0% the curve would have coincided with the straight line l/f = 1. For most of its length, the curve can be considered a straight line whose slope is an averaged value between those calculated at the points ( 1, 1), which represents the feed end of a reverse osmosis plant, and (l/f, 1 - 4). which represents the reject end. This is particularly true if the effective driving force, in terms of pressure difference, is greater than 20 atm. The value of the slope, at any point, is given by the following relationship which is easily worked out from eqn. (34) of ref. 2:
m=f+
v -
2f2(l - 4)
(6)
(v2+ 2uf +f2)"2
From solvent and solute material balances, it has +L2vInf+ulnf+~+(v2+2~f+f2Y2 also been worked out [2] that for single-stage reverse osmosis
1 + Q + (v2 + 2a + 1)“2
- v ln
of/v + v + (v2 + 2af + f2)“2 a/v + v + (v’+
B = Q/AP
20 + 1)1’2 1
plants
the following
-f(l-4)1/$
relationship
holds:
(2)
G=G]l
(3)
Since a straight line is a suitable substitute for the curve over a wide range, the following relationships are derived from eqn. (7):
Here a two-parameter model has been adopted for transport through membranes. It can be considered suitable for rejections not less than 80%. If lower rejecting membranes must be used a threeparameter model should be preferred. In both cases, however, the mathematical derivation which follows is valid since an explicit equation, similar to eqn. (1), has already been developed, starting with a threeparameter model [ 31. Figure 1 shows a plot of eqn. (1) on a ( 1 - c#), l/f reference frame. The curve crosses the points of
(7)
C,=C,(fi-l)/(A-$0
(8)
c, = C,(rn
(9)
- 1)/M
Moreover, from eqn. (9), it can easily be worked out that a constant ratio holds between the permeate and the reject concentrations, independent of the recovery fraction. Permeate-staged
plant
When highly saline waters (more than 45 000 ppm) are fed to reverse osmosis plants to produce drinkable water it is safer to add a second stage so as to get a permeate whose concentration is less than 500 ppm. Such a configuration is shown in Fig. 2.
121 emerge from the calculations. If & is greater than 1, it means that the permeate from the first stage can be wholly ‘recovered’ and, hence, a second stage is not needed for that separation factor C,,/C,,. Vice versa, if r#~~is less than 0, it means that even a second stage is not able to provide the required separation factor. Defining now the overall plant recovery as 4~ = Fig. 2. A permeate-staged reverse (broken lines) energy recovery.
osmosis
plant
with optional
(11)
P2 I&
the recovery fraction
of the first stage is easily found:
1 Four quantities need to be specified in order to define completely a two-stage reverse osmosis plant. Usually the flow rate and concentration of the feed and of the permeate are specified. Then those of the reject stream are calculated from overall material balances. Once the operating conditions of the two stages have been specified and the membrane characteristics are known, the recovery of each stage and, hence, the membrane area can be calculated by iterative procedures [ 11. An alternative explicit procedure can be found in the following way. In a two-stage reverse osmosis plant, the recycle stream is unlikely to change the concentration of the feed significantly, since its concentration is almost equal to that of the feed and, moreover, its flow rate is much less than that of the feed. Hence, for the first stage, we can calculate the slopes of the curve of Fig. 1 corresponding to the inlet, using eqn. (6), outlet, using eqns. (6) and (l), and, afterwards, the arithmetically averaged slope ml. Knowing ti, , from eqn. (9) we can calculate the permeate concentration of the first stage. From this we can calculate the slope at the inlet of the second stage using eqn. (6) and then estimate, through eqn. (9), its reject concentration, since the permeate concentration of the second stage is already known. In this way, the concentration factor of the second stage can be calculated and, hence, the slope at outlet conditions, again using eqns. (6) and (1). An arithmetically averaged slope is, then, easily calculated. From eqn. (1) a first estimate of the recovery fraction of the second stage can also be found; hence, its reject flow rate can be calculated from the permeate flow rate. At this stage, a better estimate of the concentration of the feed of the first stage can be made. If this is considerably different from that of the incoming water, the above procedure can be repeated. Now, as the permeate from the first stage is fed to the second stage, by applying eqn. (8) to the second stage and eqn. (9) to the first stage the following explicit equation is found for the recovery of the second stage:
and
The recovery fraction & is confined between 0 1. However, values outside this range may
41= 1-92++2/+r Once the recovery known, flow rates and can be calculated. The can, then, be calculated ods [2, 41. Reject-staged
(12) fractions of both stages are concentrations of all streams membrane area of each stage by graphical-analytical meth-
plant
A reject-staged plant is when the reject from the first stage, optionally repurnped, is fed to a second stage, as shown in Fig. 3. Usually, membranes with different rejecting characteristics are used to equip each stage: high rejecting membranes for the first and low rejecting membranes for the second stage [5]. This configuration is selected in order to get the highest possible reject concentration and the lowest possible permeate concentration with reasonably low applied pressures. In some cases, the applied pressure may be lower than the osmotic pressure of the reject stream of the second stage. Also in this case we assume that the actual concentration of the feed of the first stage is almost unchanged by the addition of the permeate of the second stage owing to either similar concentration or low flow rate. In this way we can calculate the slope of the curve given by eqn. (1) at the inlet of the first stage and estimate the reject concentration of the first stage from eqn. (9), since the permeate concentration is known. Hence we can calculate the slope at the exit of the first stage using eqns. (6) and ( 1) and, then, an average slope by arithmetically averaging the two. Afterwards we can calculate the slopes at the inlet and outlet of the second stage and its average slope.
1st stage
2nd
h
Stage
I t
Fig. 3. A reject-staged reverse osmosis plant with optional ken lines) inter-stage repumping and energy recovery.
(bro-
122 Once the average slopes of the two stages have been calculated and the separation factor C,,/C,, established, the recovery fraction of the second stage can be got from eqns. (8) and (9) applied to the second and first stage respectively:
~2=m2_+~ RZ
(13) 1
If #2 is less than 0, it means that a second stage is not needed to get the fixed separation factor, whereas, if 42 is greater than 1, even a second stage, with those operating conditions, is not enough to reach the given separation factor. Defining in this case the total recovery fraction as 4~ = P, I&
(14)
the recovery fraction of the first stage is easily calculated by the following relationship: l-42 &
=
(15)
l/&--42
Once the recovery fractions of both stages have been calculated, the concentrations and flow rates of all streams can be evaluated, as well as the membrane area of each stage [2,4].
Results and discussion Two design examples are presented here to assess the reliability of the proposed equations. In the examples chosen a second stage is really needed to get the fixed separation factor for given operating conditions. The first example refers to a permeate-staged plant, that is, when a low solution concentration must be obtained from a highly concentrated solution. The second refers to a reject-staged plant, where a solution with relatively low concentration must be concentrated to get a highly concentrated solution. In Table 1 the membrane characteristics and input data for the calculations are given. For simplicity, the membrane permeability and mass transfer coefficient have been kept constant throughout, both for examples and stages. In fact, industrial plants experience mass transfer coefficients of that order of magnitude or higher. Moreover, permeability will
TABLE 1. Operating conditions of the processes units are given in the Nomenclature)
examined
Quantities
Desalination
Concentration
G
50 X 10” ppm (51) 36.7 (3.72) 30.0 (303.15) 1.5 (0.15)
14 wt.% (148) 18.3 (1.85) 45.0 (318.15) 1.5 (0.15)
1st stage
2nd stage
1st stage
1.0 (0.1) 2.0 (0.2)
3.0 (0.3) 2.0 (0.2)
XI= TF A x 10’
(D,,/KS)x lo5 k, x 10’
0.5 (0.05) 2.0 (0.2)
(the
2nd stage 30.0 (3.0) 2.0 (0.2)
influence only the specific membrane area and not separation if we neglect the secondary effect of concentration polarization. On the other hand, the solute transport parameters have been chosen on the basis of existing commercial membranes suitable for the application at hand. The figures reported refer to high rejecting membranes for both stages in the first case, and to a high rejecting membrane and a low rejecting membrane for the first and second stage, respectively, in the second case. They can be considered representative of tubular, spiral wound and plate and frame modules, though the procedure can also be applied to hollow fiber modules. Actually, higher permeability must be expected for low rejecting membranes, but calculations of the membrane area is outside the scope of this paper. The examples given can be considered representative for the desalination of highly concentrated sea-water of around 50 000 ppm such as that encountered in the Arabian Gulf and for the concentration of sugar solutions of around 14 wt.% produced by the diffusional process of the sugar beet industry. The reported osmotic pressure of the feed also takes into account the presence of dissolved salts. The higher temperature is adopted to lower the viscosity of solutions and, hence, to minimize the pressure drops. Much higher temperatures would be preferable but these must be avoided in order to limit the degradation of currently available commercial membranes. Calculations have been performed by an iterative numerical procedure and by the present simplified method. Different operating pressures and several recovery fractions for both stages have been explored. In the case of desalination, pressures have been chosen so as to be compatible with high pressure membranes for the first stage and low pressure membranes for the second stage. Also, suitable values of the recovery fractions for each stage have been selected. High pressure membranes must be used in the case of reject-staged plants, unless we start from very dilute solutions. While in the first case the recovery of both stages should be kept as high as possible, in the second case the recovery of the second stage should be kept as low as possible and that of the first stage as high as possible. However, this latter is limited more by the leakage of solute through the permeate stream than by the high osmotic pressure of the reject stream. This is particularly true when no inter-stage repumping is used. First the calculations were performed by the iterative method and the flow rate and concentration of all streams were evaluated as well as the membrane area of both stages. Then, the separation factors so obtained were used as input for calculating the recovery fractions by the approximate equations (10) and (12) or ( 13) and ( 15). Knowing the recovery fractions of each stage, the flow rates and concentrations of the recycle and second-stage feed were evaluated first, and then the membrane area of each stage was evaluated by already developed explicit methods [2,41.
123 xl
. . . . . ..st
8-
-1st
%
8
4
_,..”
0.7
-overall
10
6
Stage
’ 0.6
(b)
‘54
stage
---2nd
6-
(a)
plant
_-overall
,o
%
$1 . . . .a3 ..
1
I
0.0
Q2 0.9
plant
“‘“1st
stage
---2nd
stage
i
+, ._.’
,,.... .““&”
2 0 -2 -4 -6 -8 -10 -12 -14 Cc)
0.6
0.7
0.0
q$
-141
0.9
i 0.6
(d)
0.7
0.6
$*
0.9
Fig. 4. Percentage errors in the membrane area fbr the desalination case.
Figure 4 reports the percentage errors in the number of modules of the first stage, of the second stage and of the overall plant in the case of desalination. The errors are defined as
(16) and calculated by comparing the number of modules obtained by the present method with those obtained by the iterative procedure. Errors are below 10% for all the operating conditions presented. These figures are within the range of performance scattering of commercial modules. Errors are reduced further when more usual operating conditions are employed. In most cases the errors in the total number of modules are nearly or just greater than zero, confirming the reliability of the present method. The curves for 4, = 0.5 and P, = 60 atm have not been reported because these operating conditions are not feasible. Figure 5 shows the permeate concentration as a function of the operating conditions. Water with less than 500 ppm of dissolved solid can be produced from sea-water. This will allow the permeate to be processed further and reconditioned in order also to meet WHO standards for composition. In Fig. 6 the specific consumption is reported. Energy required for pretreatments and auxiliaries have not been included; neither has the recoverable energy been subtracted.
Depending on the operating conditions, productivities of 18-25 1m-2 per hour should be expected for the higher pressures and recoveries around 40%. Single-stage-only plants are often employed but with much lower recovery. However, for big installations, a two-stage configuration is preferable, both for economic reasons and for greater reliability.
500 =, PPm 400
300
200
0.6
Fig. 5. Pemeate conditions.
0.7
0.8
+ 2
concentration
0.9
as a function
of the operating
124 20
-
s
-
*PI
+,
*P,-70
-60
_MWS
40
.
-
300
1
_‘.
----mm_______
0.5
0.6
0.7
O.6
Fig. 6. Energy consumption water.
+20.$
0.4
for the production
of 1 m3 of fresh
I
I
I
I
0.1
0.2
0.3
c$, 0.4
0.1 0
Fig. 8. Reject and permeate concentration operating conditions.
5.4
70
-Overell
%
60
‘.“‘lSt
50
‘,
of the
plant stage
---2nd
40
as a function
stage
‘. AP,
30 9 kWh _/
AP2
-55
50
---50
60
*;
30 MWS
.“‘. . . . ..__ __,
-4
-
-6 _,2
_
m3 \
AP, =55
0.a ,
APs=50 I
-16 0.0
I
0.1
I
0.2
0.3
(a)
AP, \
=50atm
,AP2r60atm ‘.
.
-Overall
6
.
plant
9 -‘_ 0.4
-
Fig. 9. Energy consumption trate.
for the production of 1 m3 of concen-
- . . . . .. .*t stage ---2nd
stage
4 2 0 -2 -4
-6 j-6
-/
0.0
I 0.1
I
0.2
I 0.3
0.4
Fig. 7. Percentage errors in the membrane area for the concentration case.
Figure 7 shows the percentage errors in the number of ,modules for the case of concentration with and without repumping. In the latter, 5 atm of pressure drop through modules, valves and fittings are allowed in the first stage. In both cases maximum operating pressures have not been selected because of
the high temperature. Percentage errors for the first stage and the overall plant are within 10% in this case also. Relative errors in the number of modules of the second stage are considerably higher. However, they should not alarm since the second stage is much smaller than the first and errors in the absolute number of modules are very low and always positive. Also, the latter is of the same order of magnitude as the number of modules eventually added for safety. Figure 8 shows the reject concentration and the permeate concentration as a function of the operating conditions of both stages. Concentrations of more than 35 wt.% sugar solution can be obtained with relatively low pressures. Higher concentrations can be obtained if the sugar liquor is first demineralized. On the other hand, the sugar concentration in the permeate stream can always be kept below
125 0.5 wt.%. In this way, the permeate, properly reheated, can be re-used in the extraction process. In Fig. 9 the energy consumed in the high pressure pump for the production of 1 m3 of concentrate is reported. Higher specific consumptions should be expected if more dilute solutions are to be processed. In any case, in this range of concentrations, reverse osmosis is more economical than evaporative processes, as also reported elsewhere [6]. However, multi-effect evaporation must be used to get a 65 wt.% sugar solution and the best operating conditions of reverse osmosis, first and second stage, and evaporation are a question of optimization. Specific productivities of 3&40 1 me2 per hour could be reasonable figures for practical applications.
k, ; P AP R S T
B & V
IL :
mass transfer coefficient, cm s-i (m s-i) slope of operating curve number of modules permeate flow rate, m3 s-’ pressure difference, atm (MPa) reject flow rate, m3 s-’ specific energy consumption, kWh m-’ (MWs m-‘) temperature, “C (K) defined by eqn. (2) percentage error defined by eqn. (4) osmotic pressure, atm (MPa) defined by eqn. (5) = P/F, recovery fraction
Conclusion Subscripts An approximate and completely explicit procedure has been developed for the design of two-stage reverse osmosis plants. Errors in membrane area evaluation of each stage do not exceed the performance scattering of commercial modules under usual operating conditions. The method can be usefully applied for quick initial estimates and economic evaluations of new processes. With some precautions and designing practice it can also be used for practical installation assessments.
Acknowledgement The financial support of the Italian Minister0 della Pubblica Istruzione is gratefully acknowledged.
Nomenclature A B C D,,/IG
membrane permeability, cm s-’ atm-’ (m s-’ MPa-‘) defined by eqn. (3) concentration, ppm, wt. % (kg m-3) transport parameter, cm s-’ solute (m s-i) feed flow rate, m3 s- ’ = C/C,, concentration factor
1 2 F M P R T
first stage second stage feed module permeate reject total
Superscripts a i
averaged quantities approximate iterative
References 1 H. Ohya, S. Kasahara and S. Sourirajan, Desalination, I6 (1975) 375. 2 F. Evangelista, Ind. Eng. Chem., Process Des. Dev., 25( 1986) 366. 3 F. Evangelista, Ind. Eng. Gem., Research, 26 (1987) 1109. 4 F. Evangelista, Ind. Eng. Chem., Process Des. Dev., 24 (1985) 211. 5 H. F. Van Wijk, A. E. Jansen and R. J. M. Creusen, Desalination, 51 (1984) 103. 6 H. Murakami and N. Igarashi, Ind. Eng. Chem., Product Res. Dev., 20 (1981) 501.
Design and Performance of Two-Stage Reverse Osmosis Plants* Progetto
e Prestazioni
FRANC0
EVANGELISTA
di un Impianto
di Osmosi Inversa a Due Stadi
Dipartimento di Chimica, Ingggneria Chimica e Materiali, 67100 L’Aquila (Italy) (Received
June 9, 1988; in final form March
Universith degli Studi dell’dquila,
3, 1989)
Abstract Simple explicit equations are derived for the design of permeate-staged and reject-staged reverse osmosis plants. Their accuracy has been tested against iterative numerical procedures. Errors in the membrane area of both stages are below 10% under normal operating conditions. Therefore, the method can be usefully applied for quick initial estimates and for discriminating between competing processes. Two design examples are presented: sea-water desalination and sugar solution concentration. Recoveries of up to 40% can be expected even for highly concentrated sea-water (50 000 ppm). Concentrations of more than 35 wt.% can also be obtained with relatively low energy. In both cases reverse osmosis proves to be competitive with evaporative processes. Sommario Gli impianti di osmosi inversa a due stadi sono usati per ottenere tat rapport0 di separazione non facilmente od economicamente ottenibile in un solo stadio con le membrane attualmente in commercio. Gli impianti possono essere di due tipi, ‘permeate-staged’ (Fig. 2) o ‘reject-staged’ (Fig. 3), in dipendenza de1 prodotto desiderato. Partendo da un modello a due parametri ed usando relazioni ricavate in precedenti lavori [Z, 4J sono state sviluppate delle equazioni (10-12, 13-15) completamente esplicite per il progetto di detti impianti. I risultati di queste equazioni sono stati confrontati con quelli ottenuti con procedimenti numerici iterativi. In condizioni operative piu’ usuali, si sono ottenuti errori minori de1 10% nel calcolo de1 numero di moduli di entrambi gli stadi (Fig. 4 e 7). Per quest0 motivo il presente metodo puo’ risultare utile, sia agli ingegneri processisti nella scelta de1 tipo di process0 e nella stima economica iniziale, sia a quelli addetti alla conduzione degli impianti all’atto di eventuali modifiche di quelli gia’ esistenti. Vengono presentati due esempi applicativi: la dissalazione di un’acqua di mare altamente concentrata (5O.OOOppm) e la concentrazione di sughi zuccherini moderatamente concentrati (14% in peso). Recuperi
*Part of this paper was presented at the 5th YugoslavianAustrian-Italian Chemical Engineering Conference, Portoroz, Yugoslavia, September 1618, 1986.
0255-2701/89/$3.50
fin0 al 40% possono essere ottenuti nel prim0 case (Fig. 5) e concentrazioni oltre il 35% in peso nel second0 case (Fig. 8) con consumi di energia abbastanza content&i (Fig. 6 e 9). In ambedue i casi, I’osmosi inversa risulta piu’ economica di processi evaporativi convenzionali. Introduction The availability of new and better membranes means that membrane processes are being applied increasingly to many kinds of industrial separations, either as an alternative to more expensive processes, or in conjunction with other conventional processes. Reverse osmosis is the most used and reliable of these membrane processes. One of the factors which prevents its further expansion, however, is the osmotic pressures of the solutions facing the membrane whose difference must not exceed the applied pressure or, rather, should be less than 80% of it. This, in turn, means that there is a maximum reject concentration or a minimum permeate concentration obtainable from a plant for given solutions, membranes and operating conditions. Hence, the separation factor of one stage, defined as the ratio between reject and permeate concentration, has an upper limit. Therefore, if we want to extend its range beyond this limit, one more stage must be added. Depending on the purpose of the operation, that is, if we want to decrease the permeate concentration or increase the reject concentration, the plant can be
Chem. Eng. Process., 25 (1989) 119-125
0 Elsevier SequoiajPrinted
in The Netherlands
120 permeate-staged or reject-staged. While the permeate from the first stage must always be repumped and then fed to a second stage, the inter-stage reject pump may be omitted. But, while both stages may be equipped with the same membrane in the first case, in the second case different membranes, with high rejecting characteristics in the first stage and low rejecting characteristics in the second stage, must usually be used. In both cases, if the overall recovery is low, an energy recovery system can be installed in the reject stream. The design of such plants is usually carried out with iterative numerical procedures owing to the presence of recycle streams and of systems of nonlinear equations for the prediction of local membrane performances [ 11. From them, given the flow rates and concentrations of the two streams and the operating conditions of both stages, concentrations and flow rates of the other streams, recovery fractions and membrane area of both stages are obtained. The purpose of the present paper is to develop an explicit procedure using the same input data as are used in the iterative method. Theory In a recent paper [2] Evangelista has shown that, with some simplifying assumptions, the following analytical relationship can be derived between the recovery fraction 4 and the concentration factor f of a single-stage reverse osmosis plant: 1 -C#I =exp(-tfl/2)
(1)
where
0.0 0.0
l/f
1.0
Fig. 1. A plot of eqn. (1) at different rejections.
coordinates (1, 1) and (0,O). If the membrane had been perfectly rejecting, the curve would have been given by the diagonal of the square. As the rejection decreases, its departure from the diagonal increases accordingly. If the rejection had been 0% the curve would have coincided with the straight line l/f = 1. For most of its length, the curve can be considered a straight line whose slope is an averaged value between those calculated at the points ( 1, 1), which represents the feed end of a reverse osmosis plant, and (l/f, 1 - 4). which represents the reject end. This is particularly true if the effective driving force, in terms of pressure difference, is greater than 20 atm. The value of the slope, at any point, is given by the following relationship which is easily worked out from eqn. (34) of ref. 2:
m=f+
v -
2f2(l - 4)
(6)
(v2+ 2uf +f2)"2
From solvent and solute material balances, it has +L2vInf+ulnf+~+(v2+2~f+f2Y2 also been worked out [2] that for single-stage reverse osmosis
1 + Q + (v2 + 2a + 1)“2
- v ln
of/v + v + (v2 + 2af + f2)“2 a/v + v + (v’+
B = Q/AP
20 + 1)1’2 1
plants
the following
-f(l-4)1/$
relationship
holds:
(2)
G=G]l
(3)
Since a straight line is a suitable substitute for the curve over a wide range, the following relationships are derived from eqn. (7):
Here a two-parameter model has been adopted for transport through membranes. It can be considered suitable for rejections not less than 80%. If lower rejecting membranes must be used a threeparameter model should be preferred. In both cases, however, the mathematical derivation which follows is valid since an explicit equation, similar to eqn. (1), has already been developed, starting with a threeparameter model [ 31. Figure 1 shows a plot of eqn. (1) on a ( 1 - c#), l/f reference frame. The curve crosses the points of
(7)
C,=C,(fi-l)/(A-$0
(8)
c, = C,(rn
(9)
- 1)/M
Moreover, from eqn. (9), it can easily be worked out that a constant ratio holds between the permeate and the reject concentrations, independent of the recovery fraction. Permeate-staged
plant
When highly saline waters (more than 45 000 ppm) are fed to reverse osmosis plants to produce drinkable water it is safer to add a second stage so as to get a permeate whose concentration is less than 500 ppm. Such a configuration is shown in Fig. 2.
121 emerge from the calculations. If & is greater than 1, it means that the permeate from the first stage can be wholly ‘recovered’ and, hence, a second stage is not needed for that separation factor C,,/C,,. Vice versa, if r#~~is less than 0, it means that even a second stage is not able to provide the required separation factor. Defining now the overall plant recovery as 4~ = Fig. 2. A permeate-staged reverse (broken lines) energy recovery.
osmosis
plant
with optional
(11)
P2 I&
the recovery fraction
of the first stage is easily found:
1 Four quantities need to be specified in order to define completely a two-stage reverse osmosis plant. Usually the flow rate and concentration of the feed and of the permeate are specified. Then those of the reject stream are calculated from overall material balances. Once the operating conditions of the two stages have been specified and the membrane characteristics are known, the recovery of each stage and, hence, the membrane area can be calculated by iterative procedures [ 11. An alternative explicit procedure can be found in the following way. In a two-stage reverse osmosis plant, the recycle stream is unlikely to change the concentration of the feed significantly, since its concentration is almost equal to that of the feed and, moreover, its flow rate is much less than that of the feed. Hence, for the first stage, we can calculate the slopes of the curve of Fig. 1 corresponding to the inlet, using eqn. (6), outlet, using eqns. (6) and (l), and, afterwards, the arithmetically averaged slope ml. Knowing ti, , from eqn. (9) we can calculate the permeate concentration of the first stage. From this we can calculate the slope at the inlet of the second stage using eqn. (6) and then estimate, through eqn. (9), its reject concentration, since the permeate concentration of the second stage is already known. In this way, the concentration factor of the second stage can be calculated and, hence, the slope at outlet conditions, again using eqns. (6) and (1). An arithmetically averaged slope is, then, easily calculated. From eqn. (1) a first estimate of the recovery fraction of the second stage can also be found; hence, its reject flow rate can be calculated from the permeate flow rate. At this stage, a better estimate of the concentration of the feed of the first stage can be made. If this is considerably different from that of the incoming water, the above procedure can be repeated. Now, as the permeate from the first stage is fed to the second stage, by applying eqn. (8) to the second stage and eqn. (9) to the first stage the following explicit equation is found for the recovery of the second stage:
and
The recovery fraction & is confined between 0 1. However, values outside this range may
41= 1-92++2/+r Once the recovery known, flow rates and can be calculated. The can, then, be calculated ods [2, 41. Reject-staged
(12) fractions of both stages are concentrations of all streams membrane area of each stage by graphical-analytical meth-
plant
A reject-staged plant is when the reject from the first stage, optionally repurnped, is fed to a second stage, as shown in Fig. 3. Usually, membranes with different rejecting characteristics are used to equip each stage: high rejecting membranes for the first and low rejecting membranes for the second stage [5]. This configuration is selected in order to get the highest possible reject concentration and the lowest possible permeate concentration with reasonably low applied pressures. In some cases, the applied pressure may be lower than the osmotic pressure of the reject stream of the second stage. Also in this case we assume that the actual concentration of the feed of the first stage is almost unchanged by the addition of the permeate of the second stage owing to either similar concentration or low flow rate. In this way we can calculate the slope of the curve given by eqn. (1) at the inlet of the first stage and estimate the reject concentration of the first stage from eqn. (9), since the permeate concentration is known. Hence we can calculate the slope at the exit of the first stage using eqns. (6) and ( 1) and, then, an average slope by arithmetically averaging the two. Afterwards we can calculate the slopes at the inlet and outlet of the second stage and its average slope.
1st stage
2nd
h
Stage
I t
Fig. 3. A reject-staged reverse osmosis plant with optional ken lines) inter-stage repumping and energy recovery.
(bro-
122 Once the average slopes of the two stages have been calculated and the separation factor C,,/C,, established, the recovery fraction of the second stage can be got from eqns. (8) and (9) applied to the second and first stage respectively:
~2=m2_+~ RZ
(13) 1
If #2 is less than 0, it means that a second stage is not needed to get the fixed separation factor, whereas, if 42 is greater than 1, even a second stage, with those operating conditions, is not enough to reach the given separation factor. Defining in this case the total recovery fraction as 4~ = P, I&
(14)
the recovery fraction of the first stage is easily calculated by the following relationship: l-42 &
=
(15)
l/&--42
Once the recovery fractions of both stages have been calculated, the concentrations and flow rates of all streams can be evaluated, as well as the membrane area of each stage [2,4].
Results and discussion Two design examples are presented here to assess the reliability of the proposed equations. In the examples chosen a second stage is really needed to get the fixed separation factor for given operating conditions. The first example refers to a permeate-staged plant, that is, when a low solution concentration must be obtained from a highly concentrated solution. The second refers to a reject-staged plant, where a solution with relatively low concentration must be concentrated to get a highly concentrated solution. In Table 1 the membrane characteristics and input data for the calculations are given. For simplicity, the membrane permeability and mass transfer coefficient have been kept constant throughout, both for examples and stages. In fact, industrial plants experience mass transfer coefficients of that order of magnitude or higher. Moreover, permeability will
TABLE 1. Operating conditions of the processes units are given in the Nomenclature)
examined
Quantities
Desalination
Concentration
G
50 X 10” ppm (51) 36.7 (3.72) 30.0 (303.15) 1.5 (0.15)
14 wt.% (148) 18.3 (1.85) 45.0 (318.15) 1.5 (0.15)
1st stage
2nd stage
1st stage
1.0 (0.1) 2.0 (0.2)
3.0 (0.3) 2.0 (0.2)
XI= TF A x 10’
(D,,/KS)x lo5 k, x 10’
0.5 (0.05) 2.0 (0.2)
(the
2nd stage 30.0 (3.0) 2.0 (0.2)
influence only the specific membrane area and not separation if we neglect the secondary effect of concentration polarization. On the other hand, the solute transport parameters have been chosen on the basis of existing commercial membranes suitable for the application at hand. The figures reported refer to high rejecting membranes for both stages in the first case, and to a high rejecting membrane and a low rejecting membrane for the first and second stage, respectively, in the second case. They can be considered representative of tubular, spiral wound and plate and frame modules, though the procedure can also be applied to hollow fiber modules. Actually, higher permeability must be expected for low rejecting membranes, but calculations of the membrane area is outside the scope of this paper. The examples given can be considered representative for the desalination of highly concentrated sea-water of around 50 000 ppm such as that encountered in the Arabian Gulf and for the concentration of sugar solutions of around 14 wt.% produced by the diffusional process of the sugar beet industry. The reported osmotic pressure of the feed also takes into account the presence of dissolved salts. The higher temperature is adopted to lower the viscosity of solutions and, hence, to minimize the pressure drops. Much higher temperatures would be preferable but these must be avoided in order to limit the degradation of currently available commercial membranes. Calculations have been performed by an iterative numerical procedure and by the present simplified method. Different operating pressures and several recovery fractions for both stages have been explored. In the case of desalination, pressures have been chosen so as to be compatible with high pressure membranes for the first stage and low pressure membranes for the second stage. Also, suitable values of the recovery fractions for each stage have been selected. High pressure membranes must be used in the case of reject-staged plants, unless we start from very dilute solutions. While in the first case the recovery of both stages should be kept as high as possible, in the second case the recovery of the second stage should be kept as low as possible and that of the first stage as high as possible. However, this latter is limited more by the leakage of solute through the permeate stream than by the high osmotic pressure of the reject stream. This is particularly true when no inter-stage repumping is used. First the calculations were performed by the iterative method and the flow rate and concentration of all streams were evaluated as well as the membrane area of both stages. Then, the separation factors so obtained were used as input for calculating the recovery fractions by the approximate equations (10) and (12) or ( 13) and ( 15). Knowing the recovery fractions of each stage, the flow rates and concentrations of the recycle and second-stage feed were evaluated first, and then the membrane area of each stage was evaluated by already developed explicit methods [2,41.
123 xl
. . . . . ..st
8-
-1st
%
8
4
_,..”
0.7
-overall
10
6
Stage
’ 0.6
(b)
‘54
stage
---2nd
6-
(a)
plant
_-overall
,o
%
$1 . . . .a3 ..
1
I
0.0
Q2 0.9
plant
“‘“1st
stage
---2nd
stage
i
+, ._.’
,,.... .““&”
2 0 -2 -4 -6 -8 -10 -12 -14 Cc)
0.6
0.7
0.0
q$
-141
0.9
i 0.6
(d)
0.7
0.6
$*
0.9
Fig. 4. Percentage errors in the membrane area fbr the desalination case.
Figure 4 reports the percentage errors in the number of modules of the first stage, of the second stage and of the overall plant in the case of desalination. The errors are defined as
(16) and calculated by comparing the number of modules obtained by the present method with those obtained by the iterative procedure. Errors are below 10% for all the operating conditions presented. These figures are within the range of performance scattering of commercial modules. Errors are reduced further when more usual operating conditions are employed. In most cases the errors in the total number of modules are nearly or just greater than zero, confirming the reliability of the present method. The curves for 4, = 0.5 and P, = 60 atm have not been reported because these operating conditions are not feasible. Figure 5 shows the permeate concentration as a function of the operating conditions. Water with less than 500 ppm of dissolved solid can be produced from sea-water. This will allow the permeate to be processed further and reconditioned in order also to meet WHO standards for composition. In Fig. 6 the specific consumption is reported. Energy required for pretreatments and auxiliaries have not been included; neither has the recoverable energy been subtracted.
Depending on the operating conditions, productivities of 18-25 1m-2 per hour should be expected for the higher pressures and recoveries around 40%. Single-stage-only plants are often employed but with much lower recovery. However, for big installations, a two-stage configuration is preferable, both for economic reasons and for greater reliability.
500 =, PPm 400
300
200
0.6
Fig. 5. Pemeate conditions.
0.7
0.8
+ 2
concentration
0.9
as a function
of the operating
124 20
-
s
-
*PI
+,
*P,-70
-60
_MWS
40
.
-
300
1
_‘.
----mm_______
0.5
0.6
0.7
O.6
Fig. 6. Energy consumption water.
+20.$
0.4
for the production
of 1 m3 of fresh
I
I
I
I
0.1
0.2
0.3
c$, 0.4
0.1 0
Fig. 8. Reject and permeate concentration operating conditions.
5.4
70
-Overell
%
60
‘.“‘lSt
50
‘,
of the
plant stage
---2nd
40
as a function
stage
‘. AP,
30 9 kWh _/
AP2
-55
50
---50
60
*;
30 MWS
.“‘. . . . ..__ __,
-4
-
-6 _,2
_
m3 \
AP, =55
0.a ,
APs=50 I
-16 0.0
I
0.1
I
0.2
0.3
(a)
AP, \
=50atm
,AP2r60atm ‘.
.
-Overall
6
.
plant
9 -‘_ 0.4
-
Fig. 9. Energy consumption trate.
for the production of 1 m3 of concen-
- . . . . .. .*t stage ---2nd
stage
4 2 0 -2 -4
-6 j-6
-/
0.0
I 0.1
I
0.2
I 0.3
0.4
Fig. 7. Percentage errors in the membrane area for the concentration case.
Figure 7 shows the percentage errors in the number of ,modules for the case of concentration with and without repumping. In the latter, 5 atm of pressure drop through modules, valves and fittings are allowed in the first stage. In both cases maximum operating pressures have not been selected because of
the high temperature. Percentage errors for the first stage and the overall plant are within 10% in this case also. Relative errors in the number of modules of the second stage are considerably higher. However, they should not alarm since the second stage is much smaller than the first and errors in the absolute number of modules are very low and always positive. Also, the latter is of the same order of magnitude as the number of modules eventually added for safety. Figure 8 shows the reject concentration and the permeate concentration as a function of the operating conditions of both stages. Concentrations of more than 35 wt.% sugar solution can be obtained with relatively low pressures. Higher concentrations can be obtained if the sugar liquor is first demineralized. On the other hand, the sugar concentration in the permeate stream can always be kept below
125 0.5 wt.%. In this way, the permeate, properly reheated, can be re-used in the extraction process. In Fig. 9 the energy consumed in the high pressure pump for the production of 1 m3 of concentrate is reported. Higher specific consumptions should be expected if more dilute solutions are to be processed. In any case, in this range of concentrations, reverse osmosis is more economical than evaporative processes, as also reported elsewhere [6]. However, multi-effect evaporation must be used to get a 65 wt.% sugar solution and the best operating conditions of reverse osmosis, first and second stage, and evaporation are a question of optimization. Specific productivities of 3&40 1 me2 per hour could be reasonable figures for practical applications.
k, ; P AP R S T
B & V
IL :
mass transfer coefficient, cm s-i (m s-i) slope of operating curve number of modules permeate flow rate, m3 s-’ pressure difference, atm (MPa) reject flow rate, m3 s-’ specific energy consumption, kWh m-’ (MWs m-‘) temperature, “C (K) defined by eqn. (2) percentage error defined by eqn. (4) osmotic pressure, atm (MPa) defined by eqn. (5) = P/F, recovery fraction
Conclusion Subscripts An approximate and completely explicit procedure has been developed for the design of two-stage reverse osmosis plants. Errors in membrane area evaluation of each stage do not exceed the performance scattering of commercial modules under usual operating conditions. The method can be usefully applied for quick initial estimates and economic evaluations of new processes. With some precautions and designing practice it can also be used for practical installation assessments.
Acknowledgement The financial support of the Italian Minister0 della Pubblica Istruzione is gratefully acknowledged.
Nomenclature A B C D,,/IG
membrane permeability, cm s-’ atm-’ (m s-’ MPa-‘) defined by eqn. (3) concentration, ppm, wt. % (kg m-3) transport parameter, cm s-’ solute (m s-i) feed flow rate, m3 s- ’ = C/C,, concentration factor
1 2 F M P R T
first stage second stage feed module permeate reject total
Superscripts a i
averaged quantities approximate iterative
References 1 H. Ohya, S. Kasahara and S. Sourirajan, Desalination, I6 (1975) 375. 2 F. Evangelista, Ind. Eng. Chem., Process Des. Dev., 25( 1986) 366. 3 F. Evangelista, Ind. Eng. Gem., Research, 26 (1987) 1109. 4 F. Evangelista, Ind. Eng. Chem., Process Des. Dev., 24 (1985) 211. 5 H. F. Van Wijk, A. E. Jansen and R. J. M. Creusen, Desalination, 51 (1984) 103. 6 H. Murakami and N. Igarashi, Ind. Eng. Chem., Product Res. Dev., 20 (1981) 501.