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Process synthesis for particle separations using centrifuges
Pergamon
Computerschem.EngngVol.22. No. 3, pp. 351-356. 1998 Copyright© 1998ElsevierScienceLtd.All rightsreserved Printedin GreatBritain PII: S0098-1354(9"/)00249-4 0098-1354/98$19.00+0.00
Process synthesis for particle separations using centrifuges Sabine M. Agena*, I. David L. Bogle and Alan R. H. Cornish Department of Chemical and Biochemical Engineering, University College London, Torrington Place, London WC 1E 7JE, United Kingdom
(Received 1 June 1997; revised 7 July 1997) Abstract Particle separations with centrifuges, i.e. non-sharp separations, are investigated. A methodology is studied which will lead to the creation of a cost optimal particle separation process. The particle separation task examined is specified by fixing the feed flowrate and particle composition of feed and products. According to separation breakpoints for adjacent particle size channels, a superstructure is built which contains all possible processing solutions. The optimal particle separation process is obtained using a non-linear optimisation technique (MINOS, GAMS). This technique is used to minimise an objective function in terms of centrifugation costs, subject to the formulations of the superstucture. It is demonstrated that the methodology of the superstructure can be successfully applied for particle separation processes. Cost optimal operating and design conditions are deduced for three particle separation tasks. © 1998 Elsevier Science Ltd. All rights reserved
Keywords: Sloppy separation; Non-sharp separation; Superstructure; Non-linear programming; Optimisation; GAMS; MINOS
1. Introduction Optimal processes are required for particle separations with centrifuges in order to guarantee competitive products. For a specific separation task different alternatives for the arrangement, design and operating conditions of processing units are possible. A methodology that leads to the optimal synthesis for a particle separation task with centrifuges is studied here and has application in, for example, biochemical processing of intraceUular proteins where centrifugation is an essential operation and a major separation cost factor. Nishida et al., 1981 reviewed early research done in separation process synthesis and showed that work had been directed towards sharp separation with distillation columns. However, for particle separation via eentrifugation, non-sharp separation synthesis needs to be examined. Aggarwai and Floudas, 1990 applied a superstructural approach considering sharp and nonsharp separations with distillation columns. In this work a different kind of separator is approached - - a centrifuge. For a particle separation task the optimal process is assessed via a superstructure approach. As particle separation with centrifuges is * Corresponding author: Sabine Agena, Biophysics ES 76, NASA/MSFC,Huntsville,AL 35812, U.S.A. 351
intended, the particle separation principle is first introduced, followed by the superstructure approach. Four different unit operations, i.e. mixers, splitters, dilution steps and centrifuges are involved in the construction of the superstructure. The integration of these unit operations into the superstructure approach is demonstrated. Following the set up of the superstructure and therefore the introduction of all possible flowsheets for particle separation tasks an optimisation towards minimal process costs is described. For three different particle separation tasks the results for the optimal flowsheet, best design and operating conditions are presented and discussed.
2. Particle separation via centrifugation The forces governing the separation of solid particles in liquids are due to buoyancy and drag. The buoyancy force on a particle is initially greater than the drag force and leads to particle acceleration until the two forces are equal. The particles, which are assumed to be spherical, eventually reach a constant velocity, which is described by:
uo=
D2.Ap.g 18.r/
(1)
352
S. M. AGENAet al.
The Stokes velocity of a particle, u,, is a function of particle diameter, D, the density difference between particle and fluid, Ap, gravitational acceleration, g, and fluid viscosity, r/. The product of the square of the diameter and density difference, D-'.Ap, is referred to as the Stokes parameter, which characterises the driving force of particle separations and is used to define the particle size channels. The Stokes velocity of a particle or particle size channel has to be corrected for the total amount of solid particles present in a system. The hindered settling velocity, vg, is introduced taking account of concentration dependent interactions between particles. The Richardson Zaki correction is used for the description of concentration effects (Middelberg, 1988, Richardson and Zaki, 1954): v~=(l-c,o~) 3.65 "uo
3. Superstructure for particle separation with centrifuges In order to select the optimal process for a specific particle separation task the superstructure approach is applied. A superstructure with all the possible process configurations is created. The number of particle size channels to be dealt with dictates the layout of the superstructure. To determine the task specific superstructure the number of adjacent particle size channels is evaluated and is referred to as separation breakpoints indicating the number of feasible separations. The number of breakpoints indicates the maximum number of separation units possibly needed. For each separation breakpoint one separation unit, i.e. centrifuge, is supplied in the superstructure. A particle separation task with three particle channels would have two adjacent particle channels and therefore a maximum of two separations might occur. Addition of a particle channel would lead to three adjacent channels, which would imply a superstructure with three particle separation units. Figure 1 shows the superstructure for a separation task with three particle channels, A, B and C. Two centrifuges are integrated into the superstructure to deal with the two separation breakpoints, A/B and B/C. In order to guarantee all possible particle separation processes a number of splitters (SP), mixers (M) and streams are introduced. From this structure series and parallel arrangements of centrifuges, as well as single ones with or without a bypass, can be deduced for the optimal particle separation process. Aggarwal and Floudas, 1990 introduced a similar superstructure for separation systems with distillation columns. However, here the separation unit consists of two units, a dilution step (D) and a centrifuge (CENT), and dilution is pursued if the total solid volume fraction, Ctot,is more than 20% for the centrifuge feed. The separation tasks to be dealt with in this work are restricted to systems with one feed stream, two product streams and three particle channels. The superstructure representing these cases is given in Fig. 1.
(2)
The hindered settling velocity of a particle or particle channel is described in terms of the total solid volume fraction, ct,~, which is the sum of all particle channel solid volume fractions, c. Density differences of fluid and particle, and the difference in particle diameters, consequently lead to a difference in particle settling velocities, vs. This difference in particle settling velocities is exploited by centrifuges for solid-solid separation. The settling velocity can be integrated into the centrifuge model (Middelberg, 1988): (3)
So:,t,if, se.v s.p= fp.Q
S is the characteristic settling area of the centrifuge, which is a measure of the centrifugal acceleration acting on the particles. For the various centrifuges the S values represent the design and operating conditions (Belter et al., 1988). The total flowrate at which the centrifuge is fed is represented by Q while the factor fp, for each particle size channel, p, describes the percentage of particles of size p found in the sediment after centrifugation.
I
I°''
?
I
,,
27
<
, ,,
CL_.15
2~_Pmducl2
/
e Row dimc~ons not assigned am tow~udstie right
Fig. 1. Superstructure for separation tasks with three particle channels.
25
Process synthesis for particle separations using centrifuges
4. The unit operations and their integration into the superstructure For any particle separation task the units involved are centrifuges, splitters, mixers and dilution steps. Streams connecting these units form the superstructure. The change of flowrates and solid volume fractions depending on the operating conditions of the various units are modelled. The flow balance of the centrifuge has been developed in terms of partial flowrates of the liquid and the solids. For each particle size channel the volumetric flow, qp, is described by: (4)
qsediment~, = fp'qFeea4,
(4) describes the flow per particle channel leaving the centrifuge with the sediment, qSediment,p, referring to the flow per particle channel to the centrifuge, q r ~ . The remainder of the particles is found in the supernatant. The factor, f, is defined by the centrifuge model ((3)) and indicates the percentage that is found per channel, p, in the sediment. For the liquid flow through the centrifuge 10% is assigned to the sediment while the other 90% is found in the supernatant. The liquid and solid flows add up to overall flowrate balances for the centrifuges. Overall flowrate balances are likewise constructed for the other three units leading to a representation of the superstructure.
5. The objective function and optimi~fion algorithm The objective function for this formulation is represented by centrifuges alone since these are the major cost factors of particulate separation processes. The centrifuge costs, C, are related to the value of S ((3)). For
353
any type of centrifuge the S value is a function of the acceleration and therefore of the operating conditions (operating costs) and of the specific design and size of the centrifuge (investment costs). The following cost function is used: Cc=.mfuge= Clnv,tm~,t(size) + Cope~ng(Operatingconditions) --~Scentrlfug e
(5)
The non-linear constraint optimisation problem to be solved is: 3
Min i=l Z C~
(6)
The costs of the centrifuges applied for a specific particle separation task is minimised subject to the mathematical formulation for the superstucture. The optimisation package used to solve the non-linear optimisation task was the FORTRAN based GAMS/ MINOS software (Brooke et al., 1988).
6. Results and discussion Three tasks are introduced to demonstrate that the superstructure approach can be applied for particle separations. The tasks are specified according to the composition c' of the one feed and two product streams, where c' is the solid volume fraction with respect to qor Also the feed flowrate, Q, has been specified. The specifications for the three tasks are documented in the tables given in Figs 2, 3 and 4 and are in bold typeface. For all tasks the solid compositions of the feed stream are held constant with 30, 50 and 20% for channels A, B and C respectively, while the product stream specifications are different reflecting possible different product requirements. A separation of particle size channel A from particle channel C in product 1 is the overall aim
fOiLI
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0.018 O.064g~o.o151iO.Oe4gio.oosg3.E4~ i
Fig. 2. Optimum synthesis for the first task referring to operating conditions and design.
S.M. AGENA et al.
354
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~
(
..).Produa 1
,>-, ~
- - Optimum Rowdirec~onsnotessigned
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g&18 lO&lg 13&20t 14 t15&23 i 28 27 57 11 42 i 30 I 11 ~ O O 43 55 48 i 50 .i 54 ~ 0 O 0 34 10 i 20 I 30 i O 0 0.00611 0.5727 0.1287 i 0.179"!0.459111 O 0 0.0572 0.0136" 0.0503 i07658~1 i-I)~-0"08"8'"!0.0"045 0.0082 ,
Fig. 3. Optimum synthesis for the second task referring to operating conditions and design. and the degree of separation of A from C is steadily raised over the three separation tasks. For the first task, product 1 should consist of 46% of particles erA and 7% of particles of C. The second task requires an increase in particle size channel A to 50%, and a reduction in particle size channel C to 5%. The last task increases the separation specifications between channel A and C further. It is required that all particles of C are removed from product 1, while channel A should be further increased to 64%. To evaluate the optimal separation process for each
task, some parameters and constraints are introduced. The Stokes parameters are constant for all tasks. They are fixed in order to be able to demonstrate and emphasise effects of separation, and are in accordance with values found in the literature for an inclusion body separation problem (Middeiberg, 1988). For the three particle channels the Stokes parameters are 3.10 -9 , 9.10 -9 and 15.10-9kg/m for channels A, B and C respectively. For the evaluation of the Stokes velocity, pure water viscosity at 25°C is applied (Perry, 1974). For the centrifuges, three inequality constraints are set up. A
IDi. I
~ ~).
m Optimum Rowdirectionsnot ~signad ~ ~rcl= lhe right ic',~ i
=,.). Product2
8 I - 135118 m = 8=-15105111 =
~
i8111mm Or4~T&14 ltg&le i
t--)-Product 1
2
~10 i 04 i 14
ier st12 &1310 &10 15&23 I
43
i 18 1 11 |
i~;~- 7~--r-~--r-~-i---~--!~Fi--~--1 !~,~i~T~b~-b~T .......... ~......... T-~--~--I i~i:i..........~...:..~.T ~ T ~ ; ~ F ~ T ~ ¥ i ~ : ~ I ~'~[~*~iT-~---,-*.~T~,=~T-~~ T o.-Ol-=-t-o.-~-~-1 Fig. 4. Optimum synthesis for the third task referring to operating conditions and design.
Process synthesis for particle separations using centrifuges maximum S value of 100,000m 2 (Perry, 1974), a maximum centrifuge feed flowrate of 0.085 ma/s (Belter et al., 1988) and a maximum total solid volume fraction of the centrifuge feed of 0.2 are introduced.
6.1. First particle separation task The first task requires a major separation of particle channel A and C. Particles of A are mainly to be found in product 1 and particles of C are mainly to be found in product 2. For product 1 almost the same amounts for channels A and B are required, about 46% each, while only 7% of channel C should result. The composition specification for product 2 is of 10, 54, 36% for particle channels A, B and C, respectively. The feed flowrate is specified as 0.1 m3/s (Fig. 2). For the first task, a flowsheet with one centrifuge exploiting the separation breakpoint AIB results and a bypass is created. One centrifuge, one splitter, one mixer and one dilution step are involved in the optimal process. Figure 2 shows the optimal arrangement that results. The whole superstructure is given and the bold typeface streams show the resulting optimal flowsheet. All other streams have flowrates of zero and are not required for the optimal process. Figure 2 shows the optimal flowsheet conditions and hence the optimal design and operating conditions are given. For the optimal process the feed is split into two streams. A stream to centrifuge I (stream 3) and a bypass stream (stream 5) are created from the feed. The stream to the centrifuge is diluted before being fed to the centrifuge. Dilution is performed although the total solid volume fraction of stream 6 is equal to 0.2 and would not violate the constraint for the centrifuge feed if fed directly. After dilution and centrifugation the bottom outlet (stream 10) gives product 2. Product 1 is a result of mixer 3, where the top outlet of centrifuge I is mixed with the bypass (stream 5, i.e. 24). The optimal process resembles a configuration of one centrifuge with a dilution step and a bypass. The separation performed in centrifuge I collects most of the particles from channels B and C. This is indicated by the collection efficiencies, f, which are more than 0.5 for the two channels. For channel A the majority of particles leave through the supernatant while only 19% of A is collected in the bottom, i.e. in the sediment. This behaviour represents a separation breakpoint between channels A and B. The separation is non-sharp as collection efficiencies of the particle channels show only maximum differences of about 40%. For a sharp separation, collection efficiencies of at least 5 and 95% are required for the two adjacent particle channels at the separation breakpoint. For the centrifuge, an S value of about 20,300 m 2 results. This value is equivalent to the value of the objective function and indicates the minimal process costs with which the first particle separation task can be realised.
6.2. Second particle separation task The specifications for the second separation task of product 1 are higher than those used in the first task. The specifications for channel A is higher by 4% while being
355
lower by 2% for both channels B and C. The feed flowrate is 0.16 m3/s and is therefore the highest for all the tasks presented here (Fig. 3). For this task, an optimal flowsheet with two parallel centrifuges, centrifuge feed stream dilution and a bypass, results (Fig. 3). The feed (stream 0) is split into three streams to supply each centrifuge and to create a bypass. The centrifuge streams are both diluted before being fed to the centrifuges. Again, as for the first task, dilution is not necessary but nevertheless performed as it leads to an improvement in the objective function. The top outlets of both centrifuges (stream 9 and 13) are mixed to make product 1. The bottom outlets of the centrifuges join with the bypass, and make product 2. The second task involves the most complex optimal topology of the three tasks in that it has seven units and 14 streams. The two centrifuges each exploit a different separation breakpoint. In centrifuge I, the total collection of particles C is achieved with a collection efficiency of 1 and the majority of B is collected while A is hardly collected. Centrifuge II exploits the other separation breakpoint, B/C, aiming at the separation of particles A and B from C. While more than 70% of particles C are found in the sediment the majority of particles A and B are leaving centrifuge II with the supernatant. Both separations are non-sharp as indicated by the collection efficiencies. The S values of the two centrifuges are about 17,500 and 9,200 m 2 for centrifuges I and II, respectively. The higher S value of centrifuge I indicates that a more difficult separation is performed than in centrifuge II. Overall, a cost characteristic figure with a value of about 26,700 m 2 results for the optimal process. About 65% of the overall cost relates to centrifuge I, whilst 35% relates to centrifuge II. The process costs of the second task are higher than those of the first task. Comparison of the objective function values for the first two tasks demonstrates that the overall costs of the two processes differs by about 30%. The greater degree of separation required between particle channels A and C leads to a 30% cost increase.
6.3. Third particle separation task The specified separation requires the complete removal of particles C from product 1. Compared with the previous tasks the specifications on product 1 for channel A are further increased while those of B and C are decreased. With respect to the first task, A is increased by 8%, and B and C are lowered by 11% and 7%, respectively. Product 2 appears with a composition of 14, 57 and 29% for channels A, B and C. The optimal flowsheet that results is a series configuration of two centrifuges as shown in Fig. 4. No bypass or dilution steps are involved in the optimal process. The total feed (stream 0) is led to centrifuge II and the resulting top product is used as a feed to centrifuge I. The top product resulting after the second centrifugation from centrifuge I has a composition as specified for product 1, and need not be further processed. The two bottom products that result after centrifugation are mixed, and provide the composition required for product
356
S. M. AGEJqAet al.
2. An optimal particle separation process with two centrifuges arranged in a series configuration and one mixer is obtained. For the optimal process, first of all a separation of particle channel C from B and A is performed. About 70% of particles C are separated into the sediment while the remaining 30% leave in the supernatant, which is fed to the second centrifuge, centrifuge I. In the second centrifuge the separation breakpoint AIB is exploited and leads to the total separation of particles C that are not found in the supernatant, which makes product 1. Centrifuge I and centrifuge II have optimal S values of about 15,100 and 13,600 m 2, respectively. In centrifuge II a slightly more expensive separation is performed. Overall a cost related value of about 28,700 m 2 results for this separation task. Comparison with task one indicates that 40% more costs are involved if specification for product 1 are increased to this level, reflecting that this approach may be used to evaluate the cost-profit margin for varying product specifications.
arrangement. For the third task, where total separation of particles C in product I is requited a series configuration of two centrifuges results. Comparing the objective function values of the three optima, it is found that this function increases with greater degree of separation for product 1. The increase of the separation degree leads to rising process costs. This behaviour is expected and confirms the process synthesis methodology and applied optimisation technique. Extension of the separation tasks to larger particle size distribution systems should be possible by adapting the proposed methodology. In order to reduce the mathematical load a heuristic screening of most promising breakpoints could be added. The scheme used is highly flexible and therefore introduction of a different specification procedure or application to other synthesis separation tasks, for example, optimal network of flotation units as required for the mineral industry, or precipitation operations, should be successful. Likewise objective functions that additionally take account of product quality and quantity are also possible.
7. Conclusion It has been demonstrated that the superstructure approach to process synthesis can be applied for particle separation tasks. In this work a separation task is specified by fixing the flowrate of the feed, and the solid volume fractions of feed and products. The superstructure representing all feasible processes is built according to the number of separation breakpoints. These breakpoints can be deduced from adjacent particle channels introduced with the separation task. The optimal process is evaluated by solving a constrained non-linear optimisation problem. The objective function used here is the sum of the S values of all centrifuges since this reflects the major costs arising from centrifugal separations. A minimisation of the S values should secure minimal process costs and hence product costs. For the three tasks, three different topologies were developed. Topologies with single centrifuges, parallel or series arrangements resulted. For the three tasks specifications with increasing requirements for the separation of particles A from particles C in product 1 are chosen. This requirement is augmented with each task and finally no particles C are specified for the last task. For the first task, one centrifuge is set up to fulfil the specifications. For further separation of particles A and C another centrifuge is introduced in a parallel
Acknowledgements Anton P.J. Middelberg is gratefully acknowledged for assistance with the centrifuge model.
References Aggarwal, A. and Floudas, C. A. (1990) Synthesis of general distillation sequences - - nonsharp separations. Comp. Chem. Engng., 631-653. Belter, E A., Cussler, E. L. and Hu, W.-S. Bioseparations, Downstream Processing for Biotechnology. John Wiley and Sons, New York, USA (1988). Brooke, A., Kendrick, D. and Meeraus, A., GAMS: A User's Guide. Boyd and Fraser Publishing Company, Danvers, Mass., USA (1988). Middelberg, A. P. J., Simulation of a Novel Biochemical Process. Honours Thesis, University of Adelaide, Australia. (1988). Perry, J. H., Chemical Engineers Handbook. McGrawHill, New York, USA (1974). Nishida N., Stephanopoulos, G. and Westerberg, A. W., Review of process synthesis. AIChE J. (1981) p. 321-351. Richardson, J.E and Zaki, W. N., Sedimentation and fluidisation: Part I. Trans. lChemE 32, (1954) 35-53.
Computerschem.EngngVol.22. No. 3, pp. 351-356. 1998 Copyright© 1998ElsevierScienceLtd.All rightsreserved Printedin GreatBritain PII: S0098-1354(9"/)00249-4 0098-1354/98$19.00+0.00
Process synthesis for particle separations using centrifuges Sabine M. Agena*, I. David L. Bogle and Alan R. H. Cornish Department of Chemical and Biochemical Engineering, University College London, Torrington Place, London WC 1E 7JE, United Kingdom
(Received 1 June 1997; revised 7 July 1997) Abstract Particle separations with centrifuges, i.e. non-sharp separations, are investigated. A methodology is studied which will lead to the creation of a cost optimal particle separation process. The particle separation task examined is specified by fixing the feed flowrate and particle composition of feed and products. According to separation breakpoints for adjacent particle size channels, a superstructure is built which contains all possible processing solutions. The optimal particle separation process is obtained using a non-linear optimisation technique (MINOS, GAMS). This technique is used to minimise an objective function in terms of centrifugation costs, subject to the formulations of the superstucture. It is demonstrated that the methodology of the superstructure can be successfully applied for particle separation processes. Cost optimal operating and design conditions are deduced for three particle separation tasks. © 1998 Elsevier Science Ltd. All rights reserved
Keywords: Sloppy separation; Non-sharp separation; Superstructure; Non-linear programming; Optimisation; GAMS; MINOS
1. Introduction Optimal processes are required for particle separations with centrifuges in order to guarantee competitive products. For a specific separation task different alternatives for the arrangement, design and operating conditions of processing units are possible. A methodology that leads to the optimal synthesis for a particle separation task with centrifuges is studied here and has application in, for example, biochemical processing of intraceUular proteins where centrifugation is an essential operation and a major separation cost factor. Nishida et al., 1981 reviewed early research done in separation process synthesis and showed that work had been directed towards sharp separation with distillation columns. However, for particle separation via eentrifugation, non-sharp separation synthesis needs to be examined. Aggarwai and Floudas, 1990 applied a superstructural approach considering sharp and nonsharp separations with distillation columns. In this work a different kind of separator is approached - - a centrifuge. For a particle separation task the optimal process is assessed via a superstructure approach. As particle separation with centrifuges is * Corresponding author: Sabine Agena, Biophysics ES 76, NASA/MSFC,Huntsville,AL 35812, U.S.A. 351
intended, the particle separation principle is first introduced, followed by the superstructure approach. Four different unit operations, i.e. mixers, splitters, dilution steps and centrifuges are involved in the construction of the superstructure. The integration of these unit operations into the superstructure approach is demonstrated. Following the set up of the superstructure and therefore the introduction of all possible flowsheets for particle separation tasks an optimisation towards minimal process costs is described. For three different particle separation tasks the results for the optimal flowsheet, best design and operating conditions are presented and discussed.
2. Particle separation via centrifugation The forces governing the separation of solid particles in liquids are due to buoyancy and drag. The buoyancy force on a particle is initially greater than the drag force and leads to particle acceleration until the two forces are equal. The particles, which are assumed to be spherical, eventually reach a constant velocity, which is described by:
uo=
D2.Ap.g 18.r/
(1)
352
S. M. AGENAet al.
The Stokes velocity of a particle, u,, is a function of particle diameter, D, the density difference between particle and fluid, Ap, gravitational acceleration, g, and fluid viscosity, r/. The product of the square of the diameter and density difference, D-'.Ap, is referred to as the Stokes parameter, which characterises the driving force of particle separations and is used to define the particle size channels. The Stokes velocity of a particle or particle size channel has to be corrected for the total amount of solid particles present in a system. The hindered settling velocity, vg, is introduced taking account of concentration dependent interactions between particles. The Richardson Zaki correction is used for the description of concentration effects (Middelberg, 1988, Richardson and Zaki, 1954): v~=(l-c,o~) 3.65 "uo
3. Superstructure for particle separation with centrifuges In order to select the optimal process for a specific particle separation task the superstructure approach is applied. A superstructure with all the possible process configurations is created. The number of particle size channels to be dealt with dictates the layout of the superstructure. To determine the task specific superstructure the number of adjacent particle size channels is evaluated and is referred to as separation breakpoints indicating the number of feasible separations. The number of breakpoints indicates the maximum number of separation units possibly needed. For each separation breakpoint one separation unit, i.e. centrifuge, is supplied in the superstructure. A particle separation task with three particle channels would have two adjacent particle channels and therefore a maximum of two separations might occur. Addition of a particle channel would lead to three adjacent channels, which would imply a superstructure with three particle separation units. Figure 1 shows the superstructure for a separation task with three particle channels, A, B and C. Two centrifuges are integrated into the superstructure to deal with the two separation breakpoints, A/B and B/C. In order to guarantee all possible particle separation processes a number of splitters (SP), mixers (M) and streams are introduced. From this structure series and parallel arrangements of centrifuges, as well as single ones with or without a bypass, can be deduced for the optimal particle separation process. Aggarwal and Floudas, 1990 introduced a similar superstructure for separation systems with distillation columns. However, here the separation unit consists of two units, a dilution step (D) and a centrifuge (CENT), and dilution is pursued if the total solid volume fraction, Ctot,is more than 20% for the centrifuge feed. The separation tasks to be dealt with in this work are restricted to systems with one feed stream, two product streams and three particle channels. The superstructure representing these cases is given in Fig. 1.
(2)
The hindered settling velocity of a particle or particle channel is described in terms of the total solid volume fraction, ct,~, which is the sum of all particle channel solid volume fractions, c. Density differences of fluid and particle, and the difference in particle diameters, consequently lead to a difference in particle settling velocities, vs. This difference in particle settling velocities is exploited by centrifuges for solid-solid separation. The settling velocity can be integrated into the centrifuge model (Middelberg, 1988): (3)
So:,t,if, se.v s.p= fp.Q
S is the characteristic settling area of the centrifuge, which is a measure of the centrifugal acceleration acting on the particles. For the various centrifuges the S values represent the design and operating conditions (Belter et al., 1988). The total flowrate at which the centrifuge is fed is represented by Q while the factor fp, for each particle size channel, p, describes the percentage of particles of size p found in the sediment after centrifugation.
I
I°''
?
I
,,
27
<
, ,,
CL_.15
2~_Pmducl2
/
e Row dimc~ons not assigned am tow~udstie right
Fig. 1. Superstructure for separation tasks with three particle channels.
25
Process synthesis for particle separations using centrifuges
4. The unit operations and their integration into the superstructure For any particle separation task the units involved are centrifuges, splitters, mixers and dilution steps. Streams connecting these units form the superstructure. The change of flowrates and solid volume fractions depending on the operating conditions of the various units are modelled. The flow balance of the centrifuge has been developed in terms of partial flowrates of the liquid and the solids. For each particle size channel the volumetric flow, qp, is described by: (4)
qsediment~, = fp'qFeea4,
(4) describes the flow per particle channel leaving the centrifuge with the sediment, qSediment,p, referring to the flow per particle channel to the centrifuge, q r ~ . The remainder of the particles is found in the supernatant. The factor, f, is defined by the centrifuge model ((3)) and indicates the percentage that is found per channel, p, in the sediment. For the liquid flow through the centrifuge 10% is assigned to the sediment while the other 90% is found in the supernatant. The liquid and solid flows add up to overall flowrate balances for the centrifuges. Overall flowrate balances are likewise constructed for the other three units leading to a representation of the superstructure.
5. The objective function and optimi~fion algorithm The objective function for this formulation is represented by centrifuges alone since these are the major cost factors of particulate separation processes. The centrifuge costs, C, are related to the value of S ((3)). For
353
any type of centrifuge the S value is a function of the acceleration and therefore of the operating conditions (operating costs) and of the specific design and size of the centrifuge (investment costs). The following cost function is used: Cc=.mfuge= Clnv,tm~,t(size) + Cope~ng(Operatingconditions) --~Scentrlfug e
(5)
The non-linear constraint optimisation problem to be solved is: 3
Min i=l Z C~
(6)
The costs of the centrifuges applied for a specific particle separation task is minimised subject to the mathematical formulation for the superstucture. The optimisation package used to solve the non-linear optimisation task was the FORTRAN based GAMS/ MINOS software (Brooke et al., 1988).
6. Results and discussion Three tasks are introduced to demonstrate that the superstructure approach can be applied for particle separations. The tasks are specified according to the composition c' of the one feed and two product streams, where c' is the solid volume fraction with respect to qor Also the feed flowrate, Q, has been specified. The specifications for the three tasks are documented in the tables given in Figs 2, 3 and 4 and are in bold typeface. For all tasks the solid compositions of the feed stream are held constant with 30, 50 and 20% for channels A, B and C respectively, while the product stream specifications are different reflecting possible different product requirements. A separation of particle size channel A from particle channel C in product 1 is the overall aim
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Fig. 3. Optimum synthesis for the second task referring to operating conditions and design. and the degree of separation of A from C is steadily raised over the three separation tasks. For the first task, product 1 should consist of 46% of particles erA and 7% of particles of C. The second task requires an increase in particle size channel A to 50%, and a reduction in particle size channel C to 5%. The last task increases the separation specifications between channel A and C further. It is required that all particles of C are removed from product 1, while channel A should be further increased to 64%. To evaluate the optimal separation process for each
task, some parameters and constraints are introduced. The Stokes parameters are constant for all tasks. They are fixed in order to be able to demonstrate and emphasise effects of separation, and are in accordance with values found in the literature for an inclusion body separation problem (Middeiberg, 1988). For the three particle channels the Stokes parameters are 3.10 -9 , 9.10 -9 and 15.10-9kg/m for channels A, B and C respectively. For the evaluation of the Stokes velocity, pure water viscosity at 25°C is applied (Perry, 1974). For the centrifuges, three inequality constraints are set up. A
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Process synthesis for particle separations using centrifuges maximum S value of 100,000m 2 (Perry, 1974), a maximum centrifuge feed flowrate of 0.085 ma/s (Belter et al., 1988) and a maximum total solid volume fraction of the centrifuge feed of 0.2 are introduced.
6.1. First particle separation task The first task requires a major separation of particle channel A and C. Particles of A are mainly to be found in product 1 and particles of C are mainly to be found in product 2. For product 1 almost the same amounts for channels A and B are required, about 46% each, while only 7% of channel C should result. The composition specification for product 2 is of 10, 54, 36% for particle channels A, B and C, respectively. The feed flowrate is specified as 0.1 m3/s (Fig. 2). For the first task, a flowsheet with one centrifuge exploiting the separation breakpoint AIB results and a bypass is created. One centrifuge, one splitter, one mixer and one dilution step are involved in the optimal process. Figure 2 shows the optimal arrangement that results. The whole superstructure is given and the bold typeface streams show the resulting optimal flowsheet. All other streams have flowrates of zero and are not required for the optimal process. Figure 2 shows the optimal flowsheet conditions and hence the optimal design and operating conditions are given. For the optimal process the feed is split into two streams. A stream to centrifuge I (stream 3) and a bypass stream (stream 5) are created from the feed. The stream to the centrifuge is diluted before being fed to the centrifuge. Dilution is performed although the total solid volume fraction of stream 6 is equal to 0.2 and would not violate the constraint for the centrifuge feed if fed directly. After dilution and centrifugation the bottom outlet (stream 10) gives product 2. Product 1 is a result of mixer 3, where the top outlet of centrifuge I is mixed with the bypass (stream 5, i.e. 24). The optimal process resembles a configuration of one centrifuge with a dilution step and a bypass. The separation performed in centrifuge I collects most of the particles from channels B and C. This is indicated by the collection efficiencies, f, which are more than 0.5 for the two channels. For channel A the majority of particles leave through the supernatant while only 19% of A is collected in the bottom, i.e. in the sediment. This behaviour represents a separation breakpoint between channels A and B. The separation is non-sharp as collection efficiencies of the particle channels show only maximum differences of about 40%. For a sharp separation, collection efficiencies of at least 5 and 95% are required for the two adjacent particle channels at the separation breakpoint. For the centrifuge, an S value of about 20,300 m 2 results. This value is equivalent to the value of the objective function and indicates the minimal process costs with which the first particle separation task can be realised.
6.2. Second particle separation task The specifications for the second separation task of product 1 are higher than those used in the first task. The specifications for channel A is higher by 4% while being
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lower by 2% for both channels B and C. The feed flowrate is 0.16 m3/s and is therefore the highest for all the tasks presented here (Fig. 3). For this task, an optimal flowsheet with two parallel centrifuges, centrifuge feed stream dilution and a bypass, results (Fig. 3). The feed (stream 0) is split into three streams to supply each centrifuge and to create a bypass. The centrifuge streams are both diluted before being fed to the centrifuges. Again, as for the first task, dilution is not necessary but nevertheless performed as it leads to an improvement in the objective function. The top outlets of both centrifuges (stream 9 and 13) are mixed to make product 1. The bottom outlets of the centrifuges join with the bypass, and make product 2. The second task involves the most complex optimal topology of the three tasks in that it has seven units and 14 streams. The two centrifuges each exploit a different separation breakpoint. In centrifuge I, the total collection of particles C is achieved with a collection efficiency of 1 and the majority of B is collected while A is hardly collected. Centrifuge II exploits the other separation breakpoint, B/C, aiming at the separation of particles A and B from C. While more than 70% of particles C are found in the sediment the majority of particles A and B are leaving centrifuge II with the supernatant. Both separations are non-sharp as indicated by the collection efficiencies. The S values of the two centrifuges are about 17,500 and 9,200 m 2 for centrifuges I and II, respectively. The higher S value of centrifuge I indicates that a more difficult separation is performed than in centrifuge II. Overall, a cost characteristic figure with a value of about 26,700 m 2 results for the optimal process. About 65% of the overall cost relates to centrifuge I, whilst 35% relates to centrifuge II. The process costs of the second task are higher than those of the first task. Comparison of the objective function values for the first two tasks demonstrates that the overall costs of the two processes differs by about 30%. The greater degree of separation required between particle channels A and C leads to a 30% cost increase.
6.3. Third particle separation task The specified separation requires the complete removal of particles C from product 1. Compared with the previous tasks the specifications on product 1 for channel A are further increased while those of B and C are decreased. With respect to the first task, A is increased by 8%, and B and C are lowered by 11% and 7%, respectively. Product 2 appears with a composition of 14, 57 and 29% for channels A, B and C. The optimal flowsheet that results is a series configuration of two centrifuges as shown in Fig. 4. No bypass or dilution steps are involved in the optimal process. The total feed (stream 0) is led to centrifuge II and the resulting top product is used as a feed to centrifuge I. The top product resulting after the second centrifugation from centrifuge I has a composition as specified for product 1, and need not be further processed. The two bottom products that result after centrifugation are mixed, and provide the composition required for product
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2. An optimal particle separation process with two centrifuges arranged in a series configuration and one mixer is obtained. For the optimal process, first of all a separation of particle channel C from B and A is performed. About 70% of particles C are separated into the sediment while the remaining 30% leave in the supernatant, which is fed to the second centrifuge, centrifuge I. In the second centrifuge the separation breakpoint AIB is exploited and leads to the total separation of particles C that are not found in the supernatant, which makes product 1. Centrifuge I and centrifuge II have optimal S values of about 15,100 and 13,600 m 2, respectively. In centrifuge II a slightly more expensive separation is performed. Overall a cost related value of about 28,700 m 2 results for this separation task. Comparison with task one indicates that 40% more costs are involved if specification for product 1 are increased to this level, reflecting that this approach may be used to evaluate the cost-profit margin for varying product specifications.
arrangement. For the third task, where total separation of particles C in product I is requited a series configuration of two centrifuges results. Comparing the objective function values of the three optima, it is found that this function increases with greater degree of separation for product 1. The increase of the separation degree leads to rising process costs. This behaviour is expected and confirms the process synthesis methodology and applied optimisation technique. Extension of the separation tasks to larger particle size distribution systems should be possible by adapting the proposed methodology. In order to reduce the mathematical load a heuristic screening of most promising breakpoints could be added. The scheme used is highly flexible and therefore introduction of a different specification procedure or application to other synthesis separation tasks, for example, optimal network of flotation units as required for the mineral industry, or precipitation operations, should be successful. Likewise objective functions that additionally take account of product quality and quantity are also possible.
7. Conclusion It has been demonstrated that the superstructure approach to process synthesis can be applied for particle separation tasks. In this work a separation task is specified by fixing the flowrate of the feed, and the solid volume fractions of feed and products. The superstructure representing all feasible processes is built according to the number of separation breakpoints. These breakpoints can be deduced from adjacent particle channels introduced with the separation task. The optimal process is evaluated by solving a constrained non-linear optimisation problem. The objective function used here is the sum of the S values of all centrifuges since this reflects the major costs arising from centrifugal separations. A minimisation of the S values should secure minimal process costs and hence product costs. For the three tasks, three different topologies were developed. Topologies with single centrifuges, parallel or series arrangements resulted. For the three tasks specifications with increasing requirements for the separation of particles A from particles C in product 1 are chosen. This requirement is augmented with each task and finally no particles C are specified for the last task. For the first task, one centrifuge is set up to fulfil the specifications. For further separation of particles A and C another centrifuge is introduced in a parallel
Acknowledgements Anton P.J. Middelberg is gratefully acknowledged for assistance with the centrifuge model.
References Aggarwal, A. and Floudas, C. A. (1990) Synthesis of general distillation sequences - - nonsharp separations. Comp. Chem. Engng., 631-653. Belter, E A., Cussler, E. L. and Hu, W.-S. Bioseparations, Downstream Processing for Biotechnology. John Wiley and Sons, New York, USA (1988). Brooke, A., Kendrick, D. and Meeraus, A., GAMS: A User's Guide. Boyd and Fraser Publishing Company, Danvers, Mass., USA (1988). Middelberg, A. P. J., Simulation of a Novel Biochemical Process. Honours Thesis, University of Adelaide, Australia. (1988). Perry, J. H., Chemical Engineers Handbook. McGrawHill, New York, USA (1974). Nishida N., Stephanopoulos, G. and Westerberg, A. W., Review of process synthesis. AIChE J. (1981) p. 321-351. Richardson, J.E and Zaki, W. N., Sedimentation and fluidisation: Part I. Trans. lChemE 32, (1954) 35-53.