Optimal Control of Batch Raw Sugar Crystallization John A. Frew C.S.I.R.O., Division of Chemical Engineering, Clayton, Victoria,dustralia
Batch raw sugar crystallization i s studied using optimal control theory. Control policies are obtained that enable crystals of specified size to be produced in minimum time without forming "false grain" b y nucleation. Feeding with both syrup and molasses i s considered. The optimal control policies for feed and evaporation rates can b e rationalized into operating principles that form the basis for a readily implemented algorithm. One of these principles, that of withholding feed less pure than the pan contents, represents an innovation in the field.
T h e final step in the manufacture of raw sugar is the transformation of the dissolved sucrose into crystalline form ready for transportation to the refinery. The crystallization is performed stagewise in a series of vacuurn evaporatioii pans, and since this is frequently the limit,ing step in the sugar production there is considerable incentire for improving the pans' performance. This paper describes the application of optimal control theory to the determination of feeding and evaporation policies for the final batch. The studies are based on a mathematical niodel of sugar c tallization developed to represent practice in the Queenslarid sugar industry. The subject is introduced by a brief description of the crystallization process with special attention focussed on the final batch. The model used to describe tlie crystallization is outlined leading to a formulation of the problem and tlie enunciation of the necessary conditions for a n optimum using Pont'ryagin's maximum priiiciple. The requirement of producing crystals of specified size in minimum time is equivalent to maximizing the growth rate m-hile still honoring the constraints of the system. Results are presented covering a range of feed purities and pan evaporative capacities representative of industrial conditions. The form of the results gives rise to a set of operating principles which specify the optimal control in terms of the current state of the pan. These principles are compared with current practice. The emphasis in t,his study is on finding t'he best controls for a given pan rather than on designing the best' pail for t'he given conditions. Consequently it is assumed that the pan size, the maximum feed rate, and the maximum evaporation rate are specified. Different maximum evaporation rates are considered only as far as t,liey elucidate different operating conditions. Crystallization Stage
Sucrose enters the crystallization stage along with a small quantity of impurities dissolved in water. Typically t'his syrup is 607, sucrose: iY0 impurities, and 337, tvater. The objective is to maximize the conversion of the dissolved sucrose into regularly shaped crystals of specified size with 110 conglomeration or false grain. An upper limit' on the crystal content within a single pan forces the conversion to be spread over several batches. The unused sucrose and attendant impurities are recycled as a more concentrated, lower purity feedstock called molasses. The molasses coming from the final batch is typically 527, sucrose, 287, impurities, and 20'7, water. The final batch grows the crystals from an iiiterniediate size up to the specified size. I t is carried out in a vacuum pan 460
Ind. Eng. Chern. Process Des. Develop., Vol. 12, No. 4, 1973
that' is a vert'ical single-st'age evaporator fed by either fresh syrup or molasses and heated by steam condensing inside a calandria. The vapor from tlie pan is condensed by direct contact n-ith cooling water in a system that also provides the vacuum control. The batch begins n-heii t,he footing, which is a mixture of sucrose, impurities, water, and crystals to be grown, is sucked into the pan. The footing covers the calandria by a depth of about' 6 ill. Steam is admitted to the calandria and water is allowed to boil off uiitil the contents reach the desired level of supersaturation. This value is below that a t which false grain appears through secondary nucleation yet high enough to proride the driving force for growth. Feed, first, later molasses, is added and the supersaturation and crystal content are niaiiitaiiied below their limits. During feeding the level in the pan rises and when the pan becomes full the feeding is stopped. Growth continues and the contents of tlie pan "heavy up" as the crystal content increases. I t the end of the batch the pan contents are dropped into a holding vessel and pass into centrifugal separators n-here the reniainiiig liquor is spun off and the crystals are washed. I t h x i l d be noted that t'he footing conditions coupled with the product size specification largely determine the final crystal content' in the pan. I t is assumed that the choice of initial conditiolis precludes a conflict with the maximum available crystal content. Model of the System
The contents of the pan can be described by the water, impurities, dissolved sucrose, and crystals present and the temperature. Operating under constant aacuum, the temperature rises slowly through the batch. However, isot'hernial operation is assumed as the rise is too small to affect the results. The dissolved inipurit,ies are lumped together and, as for x t t e r and sucrose, are characterized by their mass. The cryst,als, because t'liey exist in a range of sizes, must be thought of in terms of their size distribut,ion which can be erpressed in terms of moments having physical significance. The zeroth moment is the number of crystals while the first three moments are related to the length, the surface area, and tlie mass, respectively. Without nucleation, agglomeration, or breakage the number of crystals remains constant throughout the batch. I t is assumed that although the size distribution may spread, is does not skew so that the third central moment remains constant. Experimental evidence (Kright and Khite, 1969) supports t h k assumption and thus the third nionient call be expressed in terms of lower moments.
This means that the dynamic behavior of the size distribution caii be characterized by two quantit'ies: the mass of crystals, C, and their radius, T , are chosen. Thus including the masses of water, impurities, and sucrose the state vector has five components. The quantities that c m be manipulated are the feed flow rate, the flow rate of steam, and the rate of addition of "movement" water to t,lie pan. The latter two caii be combined into the net evaporation of water from the pan and so there are two controls. State Equations. T h e equations for water, impurit,ies, and dissolved sucrose are derived from mass balances (IVriglit and TI-hite, 1988; Evans, et al., 1970). dTQ/dt
dZ/tlt dS/dt
==
(1
=
- B)F - E
(1)
(1 - P)BF
(2)
P*B*F - (dC/dt)
(3)
=
The crystal dynamics are derived from consideration of the crystal population balance (Hulburt and Katz, 1964; Ciolan, 1966) and empirical relationships for growth rate and size dispersion fitted t o industrial practice (Wright and White, 1968, 1969; Kright, 197 1). The details are given in Appendis
I. dCjdf
=
[(C/T)
+ 2 K . V ~ ~ ] ( d ~ / d+t ) 0.OOi5KSr exp ( -09 2 Z j TT')
dr/dt
=
2.75(A
(4)
- 1.02) esp[(-1.577Z/W) - Q ] ( 5 )
A is the supersat~urationwithin the pan which is defined as A
=
( s / V ) / ( S / W )a t sat'uration
(6)
For pure sugar tlie ratio of sucrose to water at sat'uration is a functioii of temperat,ure (Charles, 1960) ; while for impure solutions this is modified by the introduction of a solubility coefficient (Meade, 1963) which can be represented by (Kright, 1971)
D
=
1
- 0.088(Z/VT7)
(7)
Thus
(S/T.I') a t saturation
=
- Z)
(8)
+ 9.035 X 10-T3
(9)
DZ/(100
where
Z
=
64.407
+ 0.072511' + 0.002057T2
Q in eq 5 is a temperature-dependent berm which becomes a constant for isothermal operation. From eq 5 it is evident that growth rate is promoted by higher supersaturation and higher purity. State Constraints. T h e nucleation boundary has been determined esperimeii-tally (Penklis and K r i g h t , 1963; 13roadfoot aiid JT-right, 1972). A
=
1.129 - 0.284[1/(1 (2.333
-.
+ S)]+ 0.0709(T - 6 0 ) [ Z / ( I
+ S)l2
(10)
For the mass in the pail and the crystal content
+ I + s + c I xu G = C / ( V + Z + S + C) 5 G,(t) JlL
I TI-
(11)
(12)
where the time dependence of G, indicates a higher limit during heavying up. Control Constraints. T h e feed rate is limited by either t,hc pipe yize or the sup:)l?-. T h e maximum net, evaporation
is achieved by having the movement water off and it depends on the rate of heat transfer through t h e calaiidria, which has been correlated with the maqs 111 the pan and the crystal content.
E,
= dl{l
- &(TI7
+ Z + S + C) [dsC/ ( T I -
+ I + S)]
(13)
The lower limit is given by maximum movement natcr flow rate less the minimum evaporation t o maintain circulation. Formulation and Solution of the Optimal Control Problem
The primary objective is to minimize the time taken to grow the crystals from their footing size until a given percentage are above a specified size. Allthoughthe spread of the crystal size distribution does depend upon the coiitluct of the batch it' is found that this effect is small and so the batch can be considered finished when t'he mean radius reaches a desired value, ~ d This , represents a final constraint on oiie of the states which for coniputatioiial reasons is coii~eniently handled by Kelly's penalty function approac-li (Kelly, 1962). The objective function becomes @(Z,tf) =
tf
+
16(T(tf)
-
(14)
Td
The same approach is used t o eiiforce the st'ate constrailits so that,, for esample, when the niass in the pan exceeds Mus n penalty
LAM"
=
/1(W -k
z + s + c - -11,,)?
(15)
is imposed. K i t h all these penalt'y functioiis the full objective function becomes (16)
Jo
This approach a l l o w the constraint3 to be violated but adds a penalty if t'hey are. The solution will approach that of tlii. actual problem lvheri the weighting functioni, I , through I s , become very large; although too rapid a n increase will hinder convergence. The optimization problem is to choose, from the allonable set of controls, values for the entire batch that miiiimize the objective function, J above, for the model of tlie c r y tallizer. The necessary conditions for the optimal solutioii to this problem are provided by Pontryagiii's niasimuni princilile (Pontryagin, et al., 1962). The general complesity of the problem precludes an analytical solutioii aiid so iterative procedures must be reqorted to. The gradient procedure is used as it has the advantage of approaching the optimum through a set of allowable profiles, so, if convergeiice difficulties prevent the attainment of an optimum. the pre*ent solution is the best available. Further, maiiiteriaiice of 11liysically meaningful profiles has increased insight iiito the problem. The basic gradient procedure is well kiion-11 (Kopp and McGill, 1964) and some acceleration in convergence was gained by use of a conjugate gradient procedure (Pagurek aiid Koodside, 1968). =It tlie beginning of the iterative procedure fluctuations in t f caused difficulties in improving the control profiles:.This v a s overcome by changing tlie optimizatioii to that of maxiniizing T over a fised interval chosen to be larger than the optimal time. =\t tlie end of the procedure, f f is given by the time a t which r reaches Td. Beiiig a state-constrained optimization, convergence was sloiv, however, some relief was gained by iiit,eracting with the automatic procedure and anticipating logical profile changes. In the q - r u p fed batches it was beneInd. Eng. Chem. Process Des. Develop., Vol. 1 2 , No.
4, 1973
461
Table 1. Numerical Data Initial Conditions (common to all cases) W/X" 0.04818 r 0.25 mm I/lWu 0.02422 m2 0,00025 mm2 SIMU 0.13540 T 60' C CIM" 0.1239C Q 0 Feed Properties Case 1
Case 2
Case 3
B P
0.68 0.895
0.68 0.895
0.8 0.65
Fu/Mu MI./Mu d3
1.25 0.3 0.3 0.3 0.3
rd
0.4
di/Jf" dz
1 0 '
'
'
'
'
'
'
'
'
012
013
n4
oi
d6
0,
ds
0 52
"It1
oll
'
'
'
'
os
la
i,
112
r----
0 58
TIME (Hrs'
Figure 1. Feed, net evaporation, supersaturation, crystal content, and pan purity profiles for case 1 (note: broken lines for F,, E, A, and G, indicate maximum allowable Iimits)
ficial to use a control variable transformation, replacing the net evaporation by
E,
=
(1
- B)F - E
(17)
Further improvement in rate of convergence was gained by scaling; the masses were divided by Ill, and the controls by F,. Results of the Optimization
The optimal control profiles and the behavior of supersaturation, crystal content, and pan purity, for three sets of operating conditions given in Table I, are shown in Figures 1-3, where minor oscillations have been removed in the in462 Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. A, 1 9 7 3
Limits 1.25 0.3 0.25 0.3 0.3
Stopping Criterion 0.4
1.25 0.3 0.3 0.3 0.3 0.33
terest of clarity. The solutions traverse the conditions existing in industrial practice. Feeding with syrup and the lower purity molasses are covered separately, and since syrup has a high water content, two levels of evaporative capacity are investigated for this feedstock. It should be noted that the batch times obtained are heavily dependent on the initial conditions and pan characteristics and should not be taken out of context. Case 1. T h e syrup fed batch having adequate evaporative capacity exhibits three distinct regions. Initially with the feed shut off the evaporation is held a t its maximum value. Because of discretization effects it was necessary to rework the initial part of the batch with a time step one-tenth the original to establish this. Figure 1 is a composite of these runs. Under this control policy the supersaturation rises rapidly from its initial value up to the nucleation boundary. The short duration of the period precludes any large change in either crystal content or purity while the level drops slightly. I n the second region feed is introduced a t the maximum allowable rate while the evaporation drops sharply from the maximum. As the batch progresses the evaporation requirement increases while the capacity drops slightly so that once again the maximum is reached. At this point the feed rate falls from its maximum. Throughout this region the supersaturation remains pressed against the nucleation boundary. The introduction of feed overshadows the water lost by evaporation and so the mass in the pan rises throughout the region until the filling of the pan terminates it. The crystal content drops quickly a t the start but then levels off and is rising at the end of the period. The purity after an initial rise falls away. Once the pan is full both the feed and evaporation rates drop to the same low level. This keeps the pan full and the supersaturation on the nucleation boundary. During this final region the crystal content rises rapidly while the purity declines. The end comes when the crystals reach the final specification. Case 2. I n case 1 the crystal content is always loT7er than the allowable limit due to the ability to feed the pan a t a high rate while maintaining the supersaturation on the nucleation boundary. This is possible only when there is ample evaporative capaclty avallable. However, the amount of evaporation is limited by the design of the pan and so it is
1 36L
" 3
2
L
i
4 c5
oa
TIME
c.
C8
c9
13
1
12
Hrs,
1
0 : 8 P i 1 01
'
02
03
' 04
'
05
'
06
'
07
' 08
'
09
"
10
'
11
'
13
TIME (Hrs)
Figure 2. Feed, net evaporation, supersaturation, crystal content, and pan purity profiles for case 2
Figure 3. Feed, net evaporation, supersaturation, crystal content, and pan purity profiles for case 3
of interest to irivestigati? the case of a much lower capacity. The results are presented. in Figure 2 . The beginning of the batch is similar to the first case although the loner evaporation rate requires a more rapid decrease in the feeding rate. This is reflected in the crystal content aiid before the pan becomes full it reaches the masimum alio~vablevalue at 0.94 hr. This signals the beginning of a new region. The evaporation remains on the maximum value but the feed rate jumps sharply to maintain the crystal content on its limit. This higher feed rate causes the supersaturation to fall away from the iiucleation boundary. The other effects are an acceleration in the rise of pan mass and a tempering of the purit,y 'droli. This new region ends when the pan becomes full.
At this point the feed is shut off while the evaporation rate remains at' its upper limit. As a result the supersaturation and crystal content rises sharply, the purity recommences its decline, and the total mass falls slightly. The batch ended in this region. Case 3. T h e third batch, fed entirely with recycled molasses, is shown in Figure 3 and consists of five regions. The first is identical ivith those of the previous batches. However, once the supersaturation reaches the nucleation boundary the evaporation drops to a low value arid in contrast to cases 1 and 2 the feed remains off, since the pan purity is higher than that of the feed. The level of evaporation maintains the supersaturation on the nucleation boundary and the slow drop in the mass in the pan continues. The purity falls Ind. Eng. Chern. Process Des. Develop., Vol. 12, No. 4, 1973
463
Table It. Operating Principles
A. Major Principles 1. With no constraints active the supersaturation should be raised using maximum evaporation with the feed shut off. 2 . If the nucleation boundary alone is active, the controls should keep the supersaturation on the boundary. When the feed is purer than the contents of the pan it should be admitted a t the maximum rat'e consistent with maintaining the supersaturation on the nucleation boundary. But if it is less pure i t should be withheld and the evaporation used t o control the supersaturation. 3. When crystal zontent is the sole constraint, t h e controls should keep it on the upper limit while the evaporation rate is made as high as possible in order to raise the supersaturation. 4. If both the nucleation boundary and crystal content constraints are active the supersaturation should be kept as high as possible without exceeding the crystal content limit. The controls are given by the second principle provided they would maintain or reduce the crystal content. If they would raise it, the feed and evaporation are determined so that both constraints are maintained simultaneously. Should these controls be infeasible the supersaturation cannot be kept on the nucleation boundary and the controls are given by the third principle. d. Should the pan full constraint alone be active the controls are given by the first principle. 6. With the pan full and the supersaturation on the nuleation boundary, the feed, and evaporation rates are set a t equal values sufficient to maintain the supersaturation on the boundary. B. Minor Principles 7 . If both pan full and crystal content constraints are active feed and evaporation rates should be set to zero. The same controls apply if the nucleation boundary is also active. Under these conditions the crystal content cannot be controlled. This situation can lead to the premature termination of a batch. 8. If the pan mass is on its lower limit the feed and evaporation rates are set equal to the masimum evaporation so that the supersaturatioii is raised without exposing the calandria. 9. With both the nucleation boundary and the lower pan mass constraints active the supersaturation should be kept as high as possible without exposing the calandria. The controls are given by the second principle provided that the feed rate exceeds the evaporation rate. If this is not so the controls are determined by the sixth principle. 10. If both crystal content and lower pan mass are limiting the coiitrols are given by the third principle. 11. When the nucleation boundary, crystal content, and lower pan mass constraints are all active the controls are determined by the fourth principle.
while the crystal content' rises and, after 0.12 hr the crystal content limit is reached and the third region begins with feed on and a rise in evaporation. The two controls, feed and evaporation, are such that the supersaturation remains on the nucleation boundary and the crystal content on its limit. The introduction of the feed accentuat'es the purity drop so that the pan purity reaches that of the molasses feed. This marks the start of the fourth region which is identical with the second region of the first batch while the fifth and final region, when the pan becomes full, follows the same pattern as the third region of the first batch. In this case the evaporation required is much lower and the batch time longer. The low evaporation requirements suggest that this type of feed would not normally be subject to any evaporation limitations. 464 Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. A, 1973
Table 111. Occurrence of Operating Principles Case
Batch description
I
II
Region 111 IV
1 2
Syrup Syrup with limited evaporation Molasses
1
2
6
1 1
2 2
4
3
4
3 2
V
VI
5 6
1
Operating Principles
The results presented in the previous section are not, in a form useful for a practical control algorithm as they specify the controls only for the particular conditions chosen. What is required is a set of rules that specify the two controls in terms of the conditions esisting in the pan a t the time control is to be applied. It is possible to derive this set of rules, or operating principles, which are classified in terms of the state constraints active and divided into major and minor groups on the basis of the likelihood of their occurrence. When no state constraints are active it is necessary to specify both controls. However, one constraint, being active and remaining active implies a relationship between the controls and the states of the pan so that only one control and this relationship need be specified. These relationships are presented in Appendix 11.When two constraints remain active both controls are specified. For these reasons the operating principles are expressed in terms of the state constraints active. The principles are presented in Table 11, their occurrence in the three batches discussed is shown in Table 111, and the mathematical ramifications are summarized in flow diagram form in a\ppendix 11. With no constraints active it is seen that the evaporation is held a t a maximum while the feed is shut off. This can be understood by noting that the growth rate expression (eq 5 ) is maximized by being on the nucleation boundary for pan purities greater than 0.5. For higher grade batches this purity restriction is always met. The rate is also enhanced by raising the purity but this effect is secondary to that of supersaturation. Hence feed with it's undersaturating effect should be restrained and the supersaturation raised by applying maximum available evaporation. When the nucleation boundary is the only constraint active the supersaturation remains on the boundary and thus eq -4.14 must apply. However, there is a marked difference between the second region of the molasses feed batch and the other regions in that the feed remains off. I t is seen that for this case the purity within the pan is greater than that of the feed. Consequently, the remaining control specification is dependent on the relationship between the pan and feed purities. Once on the nucleation boundary growth is accelerated by increasing the purity of the contents of the pan and thus purer feed should be admitted while less pure material should be withheld. The precise form of the control laws depend on the maximum values of the controls as well as the crystallization rate and for very high growth there may be insufficient evaporation capacity to remain on the boundary. When crystal content is the lone limitation it is evident that the supersaturation must be raised toward the nucleation boundary to enhance the growth rate. However, restraining the feed and evaporating the excess water raises the crystal content and so the compromise is to have the highest evaporation possible while maintaining t'he crystal content on its limit. The latter implies that eq A.20 must hold. This policy
achieves the highest supersaturation possible without exceeding the crystal content h i i t . Three possibilities a.re observed when both the nucleation boundary and the crystal content constraint,s become act'ive. The most desirable is when the controls to maintain the supersaturation on its boundary, as computed by the second principle, do not raise the crystal content. I n this case the pari comes off the crystal content limit i in case 3 when the pari purity dropped below the feed . If this is not possible t,he next alteriiat'ive is t o try and niaintairi both constraints in which case the controls are uniquely specified by eq AI.14and -1.20. The least desirable alternative is when the controls so calculated esceed their limits. It is no longer possible to maintain the supersaturation on the nucleation boundary aiid so the crystal content is maintained on its liiiiit by controls deterinined according t'o the third principle. Thus the operating principle is concerned first wit'h whether the constraints should remain active aiid second whether it is possible to achieve this with t,he limit,ed controls available. It is seen t,liat n-hile it is always preferable t'o keep the supersaturction 011 its limit it is riot always deFirable to remain on the crystal cont,ent limit. ;\lso if it is impossible to remain on both limits theii the su1)ersaturatioiimust decrease. *.Ifter 0.95 hr of the iiecoiid batch the pan becomes full with no other constraint active due to the step up in allowable tal coiiteiit. The region is momentary as the groivth rat'e is improved by shutting off the feed aiid masimizing the evaporation, a policy that immediately relieves t,he constraint moving the pan iiito ui unconstrained condition where the same controls apply. Kheii the pan becomes full while the supersaturation is on its limit' both coiist,raints are niaintaiiied aiid the controls are given by eq A.14 and -1.21. The feed so calculated will always be feasible but should the evaporatioii required esceed the masiinuni available t,li.en both feed and evaporation are set to E,,. The pail should be kept full because the purity will be less than that of the feed and, by the second principle, feed should be admitted a t a greater rate than the evaporatioii removes water from the pan. Consequently maximizing the growth creates pressure to overfill t'he pan. If the crystal content reaches its limit with the pan full the batch should be a t an end, because it is no longer possible to control the crystal content. If the crystals are not up to size then the footing conditions were poorly chosen and the best that can be done is set both cont,rols to zero. This policy applies irrespective of d i e t h e r the supersaturation limit' was active or riot. There are 110 examples of t,he lower limit on the pan mass being active although the const,raint could be important a t the begiiiniiig of a batch if insufficient footing was charged. This is especially true of the molasses fed batch. The prim ciples are estrapolated. from the ideas expounded above aiid presented iii Table 11.
Withholding feed until the pan contents are less pure appears to be a n innovation. Since the crystal content rises under this policy its implementation is feasible only when the crystal content is lower than the allowable value and hence it is not surprising that it has not been practiced to date. I n higher grade batches the amount' of molasses fed is not great so t,his refinement is more likely to be important, in lower grade batches. For case 3 the improvement gained by withholding feed was 5y0.However, this rises when running at' lower supersaturations. Feeding t'o match the evaporation once the pan is full is not done industrially. .Isthe pan full limit is vague a little more is added in the feeding period aiid with the feed off tlie level drops back during heavyirig up. Whether masimum growth rate and hence feeding is really the objective tit this stage of the batch is a question best deferred to coilsideration of the paii stage as a whole. ,1chiering faster batch time:: through use of the operating principles requires the determination of scpersaturatioii, cryatal content, total mass in the p a n , a i d the purities of both the pan arid the feed. In addition it is necessary to know the state and control coiistraiiits, of which the most important is the nucleation boundary. Conclusions
Optimal control theory has been applied to the operation of batch crystallization of raw sugar in vacuum pans fed by both syrup and molasses with the object of minimizing batch time. The optimal solutions obtained, covering a range of conditions typical of the high-purity elid of industrial operation, may be rat,ionalized into a set, of 11 operating principles. These iirincililes are classified in trrnis of the state coiistraints that are active and can form the basis for a readily inipleIneiited control algorithm. One of the principle- the optimal policy where tlie crystal content is allon-ed to drop well belo!?its initial value.
The total mass of crystals is given by
C ( f )=
4/37rpAI-p3(t) =
K-Ypdf)
where iY is the total number of crystal. and
dC, dt
=
p
(A.3)
the density. So
K.V(dp~3,'dt)
(AL4)
for a nonikea ed diqtribution w z 3 is zero and hence p3 = 31n2pi
+
pi3
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 4, 1973
(A.5) 465
atives with respect t o time must also hold which upon substitution yield t h e following relationships between the controls. Nucleation Boundary
r --
hiF
- hsE
=
hs(dC/dt)
where
-2
((100
2 0,284)
y 8 ~
A, I + S
- l.O88A,} - PB(
(loo
2 ('>
-
A, I + X
2
hz
h3 =
(loo 2- 2'( 1
=
(100
)'(- -
2YI
I+S
- 2)
- 0.088A.
+
(z - 0.284)) I+S
(A.15)
A,
(916)
+ -----(1 IS A, (I
(A.14)
I
2YI
+ 8)' I + s
- 0.284)) (A.17)
with
A,
= 1.129
- [0.2841/(1
+ S)]+ Y [ I / ( I+ S)lZ (A.18)
Y = 2.333 - O.O709(T - 60) Figure 4. Flow sheet of mathematical ramifications of operating principles
Crystal Content
Differentiating (A.5) and eliminating mz
P a n Full and P a n Empty
F - E = (l/G,)(dC/dt)
F-E=O For a well-mixed batch crystallizer in the absence of nucleation, agglomeration, breakage, and size dispersion, the crystal population balance gives
(A. 7 )
For size-independent growth af(v)/rt
=
(A.8)
(dr/dt) [ a f ( u ) / a r ~
which, when combined with ( A I ) becomes
dp,/dt = &-l(dr/dt)
(A.9)
dkl/dt = dr/dt
(X.10)
dmz/dt = 0
(All)
and in particular but also
which is at variance with observed behavior for raw sugar and so ( A l l ) is replaced by a n experimental correlation (Wright, 1971)
dmn/df
=
0.0025 exp(-O.92I/W)
(A. 12)
Combining (A.4), (A.6), (A.10), and (A12) gives dC/dt
=
[(C/r)
+ 2KiVr2](dr/dt) + 0.0075KNr exp(-0.92I/FV7)
(A.13)
Appendix II. Mathematical Ramifications of the Operating Principles
A. Control Relationships when t h e State Constraints a r e Active. K h e n t h e state constraints are active, their deriv466
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 4, 1973
(A.19)
(X.20)
(A.21)
B. Flow Diagram of Operating Principle Ramifications. The flow diagram is shown in Figure 4. Where there are no lines leaving a box, a return is assumed. Nomenclature
A = supersaturation in pan (eq 6) A , = nucleation boundary (eq A.18) B = fraction of solids in feed material (dry solids) C = mass of crystal in pan (Jf) D = solubility coefficient (eq 7 ) dl-d3 = coefficients in eypression for E, (eq 13) E = net evaporation rate E N = transformed evaporation rate (J1T-I) (eq 19) E L = minimum allowable net evaporation ( M T - l ) E , = maximum allowable net evaporation (JfT-l) F = feed rate of syrup or molasses to pan (JITdl) F , = maximum allowable feed rate (-1fT-I) j' = crystal number size distribution G = crystal content (eq 12) G, = maximum allonable crystal content hrh3 = coefficients in eq -1.14 I = mass of dissolved impurities in pan (-11) J = objective function (eq 18) K = constant related to sugar density (eq A.3) L = component of objective function (eq 18) 11-15 = weighting coefficients in objective function J1 = total mass in the pan (21) J I L = minimum allowable mass in the pan (;If) Mu = maximum allowable mass in the pan (-TI) m, = moments about mean of crystal number size distribution (eq A.2) S = numbers of crystals P = ratio of sucrose to impurities and sucrose in feed (purity) Q = temperature-dependent tern1 in crystal growth rate r = volume equivalent crystal radius ( L ) (hppendis I) rd = specified final crystal radius ( L ) S = mass of dissolved sucrose in pan (M)
T
pan temperature (“C) batch time ( T ) tf final time ( T ) W = mass of water in pan (-If) z = state vector with components ( W , I , S , C , r ) Y = temperature dependent term in A , (eq X.19) 2 = per cent sugar at saturation in a pure syrup (eq 9) p = density of sugar (JlL-3) ,,tl = moments of rrystal number size distribution (eq -4.1) = final time component of objective function (eq 16) t
=
= =
literature Cited
Broadfoot, R., Wright, P. G., Proc. Queensl. SOC.Sugar Cane T~chnol.. - .. . .~ , 39. 353 - - - 11975!’1. - - 7
* - - - - I
Charles, D. F., Znt. Sugar J., 62, 126 (1960). Ciolan, I., Genie Chim., 95, 1381 (1966). Evans, L. B., Trearchis, G. P., Jones C., Sugar Azucar, Oct 19, 37, Dec 19 (1970). Hulburt, H. RI., Katz, S., Chem. Eng. Sei., 19, 555 (1964).
Kelly, H. J., in “Optimization Techniques,” G. Leitman, Ed., Academic Press, New York, N.Y., 1962. Kopp, R. E., McGill, R., in “Computing Methods in Optimization Problems,” A. V. Balakrishnan and L. W.Neustadt, Ed., Academic Press, New York, N.Y., 1964. Meade, G. P., “Cane Sugar Handbook,” 9th ed, Wiley, New York, N.Y., 1963, p 209. Pagurek, B., Woodside, C. AI., Automatzca, 4, 337 (1968). Penklis, P., Wright, P. G., Proc. Queensl. SOC.Sugar Cane Technol., 30, 177 (1963). Pontryagin, L. S., Boltyanskii, T.’. G., Gamkielidze, 11. V., Jlishchenko, E. F., “The LIathematical Theory of Optimal Control Processes,” Interscience, Sew York, K.Y., 1962. White, E. T., Wright, P. G., Can. J. Chem. Eng., submitted for publication. Wright, P. G., White, E. T., Proc. Znt. Soc. Sugar Cane Technol., 13,1697 (1968). Wright, P. G., White, E. T., PTOC.Queensl. Soc. Sugar Cane Technol.. 36. 299 11969). Wright, P.’G.,’private communication, 1971. RECEIVED for reviex December 4, 1972 ACCEPTED April 9, 1973
Vapor-Liquid Equilibria for Five-Carbon Atom M uIticomponent Hy drocarbon Systems Charles C. Peiffer, James D. Schlegel, and Robert H. McCormick* L)epartment of Chemical Engineering, The Pennsylvania State Cniversity, University Park, Pennsylvania 16802
The effects of hydrocarbon type bonding and concentrations on the relative volatility between the components in mixtures were studied b y determining vapor-liquid equilibrium data in a 20-stage unit on binary, ternary, and quaternary systems which included 2-methylbutanef n-pentane, 2-methyl-2-butenet and 2rnethyl-lf3-butadiene. Although only relatively minor effects were ascertained for the change in relative volatility of 2!-methylbutane-n-pentane due to the presence of olefinic and/or diolefinic hydrocarbons, more discernible variations were evident for 2-methylbutane-2. methyl-2-butene and 2-methylbutane-2-methyllf3-butadiene in the presence of the diolefin, particularly in concentrations above 50%.
T h e present investigation is part of a continuing program to supply the needs of irdustry with accurate fundamental design data for physical separational processes. I n this particular study the effects of hydrocarbon structural types, as well as composition, on the relative volatility between the components in multicomponent mixtures were investigated for fivecarbon atom hydrocarbon systems. Ailthough the literature is inundated with vapor-liquid equilibria for binary hydrocarbon systems, very little data can be found for close boiling systems such as isopentane (2methylbutane)-n-pentane. Even less data have been published for multicomponent systems, especially those systems composed of various structural hydrocarbon types, e.g., normal paraffins, branched paraffins, olefins, and diolefins. Lack oi equilibrium data involving these types of systems has been largely a result of the difficulties encountered in determining accurately the small changes in concentration between vapor and liquid samples obtained in the conventional one-stage equilibrium devices, due to proximity of boiling points, and the problems encountered with analyzing hydrocarbon systems containing more than t n o components.
The latter of these problems has been overcome with the advent of gas chromatographic techniques- for analyzing multicomponent liquid and/or gaseous hydrocarbon mixtures. Similarly, che development of multistage equilibrium units has reduced the effect of inaccuracies of analytical procedures resulting from small concentration changes in the equilibrium samples. Two examples of such units are a six-stage equilibrium unit operable a t pressures ranging from atmospheric to 400 psia (McCormick, et al., 1962) and a 20-stage equilibrium unit operable a t pressures ranging from atmospheric to 45 psia (Peiffer, et al., 19i2). This reduction in analytical error is due to the capability of calculating the relative volatility from the vapor and liquid compositions resulting from a separation over n stages ( n = 2 or more), as illustrated in the equations (Fenske, 1932)
The relative volatility, a, between two components can be Ind. Eng. Chem. Process Des. Develop., Vol. 12, No.
4, 1973 467
