Ind. Eng. Chem. Fundam. 1980, 79, 117-121
of one of the characteristic loci. The technique used here to suppress interation between the two controlled variables is perhaps not the final solution of this aspect of the design procedure. A minimum in the interaction was imposed at a single (high) frequency. More appropriately, one may want to specify a frequency dependent degree of suppression to accommodate any physical insights that one has about the feasibility of noninteraction. Here, for example, such a gradation in suppression of controlled temperature interaction might have proved beneficial, particularly since there is a fairly clear frequency separation in the control authority of Q and TQ.Such a modification would likely result in reduced control effort, a desirable outcome. The methods used to shape the characteristic loci by selection of dynamic compensators for the commutative controllers have in this study relied heavily on the designer's appreciation of the relationship between the shape of the characteristic loci and the transient response of the closed loop. There is an art in this aspect of the design even for single-loop systems, and without a keen ability at it, one can spend a considerable effort in trial and error. What seems to be needed are easily understood quantitative relationships between the desired closed-loop transient and frequency-dependent eigenproperties of the system matrix that can be used as guides in selecting the compensators. Such relationships will differ, of course, for cases of input tracking and disturbance rejection.
117
One of the major objectives of this investigation was to determine the feasibility of simultaneous control of product concentration and temperature. It was found that there is indeed a chance of accomplishing this but at a cost of control efforts that are about eight times those needed for concentration control alone. These control efforts probably can be reduced by a relaxing of the degree of noninteraction, but most of the additional control effort may be attributed to a near singular system matrix. Such circumstances prompt one to search for new sets of control input variables.
Literature Cited Edmunds, J. M., CUED/F-CAMS/TR 170, Engineering Department, University of Cambridge, 1978. Edmunds, J. M. Kouvarltakis, B., Int. J. Control, to be published, 1979. Kouvarltakis, B., Edmunds, J. M., National Engineering Consortium Symposium on Multivariable System Design, Chicago, Ill., 1977. MacFarlane, A. G. J., Kouvaritakis, B., Int. J. Control, 25(E), E37 (1977). MacFarlane, A. G. J., Kouvaritakis, B., Edmunds, J. M., National Englneerlng Corsortium Symposium on Multivariable System Design, Chicago, Ill., 1977. MacFarlane, A. G. J., Postlethwaite, I., Int. J. Control, 25(1), 81 (1977). Michelsen, M. L., Vakil, H. B., Foss, A. S., Ind. Eng. Chem. Fundam., 12, 323 (1973). Silva, J. M., Ph.D. Thesis, University of California, Berkeley, 1978. Silva, J. M., Wallman, P. H., Foss, A. S., I n d . Eng. Chem. Fundam., 18, 383 (1979). Wallman, P. H., Ph.D. Thesis, University of California, Berkeley, 1977. Wallman. P. H., Silva, J. M., Foss. A. S., Ind. Eng. Chem. Fundam., 18, 392 (1979).
Received for review April 16, 1979 Accepted October 11, 1979
Succinic Acid Crystal Growth Rates in Aqueous Solution J. W. Mullin' and M. J. L. Whiting Department of Chemical and Biochemical Engineering, University College London, London, WC 1E 7J€, England
The effects of supersaturation, temperature, solution velocity, and crystal orientation_on the individual face growth rates of succinic acid crystals in aqueous solution have been measured. For the (111) and (010) faces, the growth process is diffusion controlled and approximately first order with respect to supersaturation. The growth order of the (001) face is about 1.5. Overall growth and dissolution rates of single crystals, located within a well defined hydrodynamic environment, have been measured by a semicontinuous weighing technique.
Introduction The 0 form of succinic acid [(CH2COOH)2,mol wt = 118.09, crystal density 1572 kg/m3] crystallizes from aqueous solution in the monoclinic prismatic class. The unit cell parameters and general crystallography are well established (Broadley et al., 1959). A typical symmetrically formed crystal of succinic acid grown from aqueous solution a t room temperature and low supersaturation, illustrated in Figure 1,resembles a six-sided prism with a large predominant basal plane (001)bounded by smaller faces of the type (OlO], Illi), and (110). The solubility of succinic acid in water over the temperature range 20 < 8 < 40 "C may be represented by the equation c* = 5.502 X
-
1.157 X 10-38+ 9.4307
X
(1)
with a standard deviation of 5.7 x kg of succinic acid/kg of water. This second-orderpolynomial represents the best least-squares fit of 18 data points and is in good 0019-7874/80/1019-0117$01.00/0
agreement with other recorded measurements (ICT, 1926; Seidell, 1941). The densities of saturated aqueous solutions of succinic acid over the temperature range 20-40 "C, measured by the pycnometric technique suggested by Findlay (1955), were correlated by the equation = 1.01780 x 103 - 2.51262 x 10-18 + 1.59298 x 10-202 (2) with a standard deviation of 0.171 kg/m3. The viscosities of saturated aqueous solutions of succinic acid at 18, 25, and 27.3 OC are 1.17, 1.03, and 1.00 Pa s, respectively. Face Growth Rate Measurements Individual face growth rates of single crystals of succinic acid were measured by a technique previously described (Mullin and Amatavivadhana, 1967; Mullin and Garside, 1967). Briefly, a crystal was mounted on a wire in a glass cell through which a solution flowed under carefully controlled conditions. Any chosen face or edge could be ob0 1980 American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980
0
View along D
Plan view
Figure 1. Diagrammatic plan and elevation of a typical crystal of succinic acid grown from aqueous solution.
-e >-
I , , OO
served with a travelling microscope, reading to f 0.01 mm, and its rate of advance measured. Solutions were prepared from recrystallized pure-grade succinic acid and deionized water and filtered through a no. 4 (30-pm) sintered glass filter. Seed crystals were carefully selected from batches grown by slow cooling in unstirred solutions. Crystals with well-defined faces, free from inclusions and out-growths, were chosen. The mounted seed crystal was dipped into deionized water to remove any surface impurities and introduced into the cell when the circulating solution temperature had become constant. About 15 min was allowed before making measurements to enable the washed seed crystal to develop sharp edges and corners. The duration of a run varied from about 1 to 4 h depending on the solution supersaturation and velocity. Measurements were made a t suitable regular intervals. The initial and final solution concentrations were determined by titration against standard NaOH solution using phenolphthalein as indicator. The arithmetic mean concentration was taken as being representative of conditions during the growth process. Crystals of succinic acid grown from aqueous solution develop four principal crystallographic faces of the type (OOl},(OlO},(ill}, and (110) (see Figure 1). A large proportion of the crystal surface is taken up by the (001)face whereas the (010) face tended to be rather small and in many cases did not appear at all. Effect of Solution Velocity. The growth rate of the (111) face was measured at 27.3 (f0.05) OC in aqueous solution at two different supersaturations (Ac = 0.0015 and 0.0033kg/kg water). The empty-cell solution velocity was varied from 0.02 to 0.27 m/s and crystals were mounted with the (111)face perpendicular to the direction of flow. Growth runs were usually continued for sufficient time to ensure that the growth rate was independent of time and that steady-state conditions had been reached. Occasionally (about 1 in 10 crystals) the (111)face did not develop evenly and it was not possible to measure its growth rate. When this occurred, the crystal was removed from the cell, dissolved slightly in water, and replaced. If irregular growth still persisted, the crystal was discarded. This type of erratic behavior was undoubtedly caused by the presence of substantial internal defects in the crystal, which would interfere with the normal pattern of growth. Several growth runs were repeated to see if the seed crystal itself had any influence on the reproducibility of the growth rate. No effect was found when the measured face was well formed and growth took place under steady-state conditions. Crystals from previous runs could be reused provided that they were not larger than about 7 mm in length and the (111) face was still visible. It was possible, within certain limits, to reduce the overall size of the seed crystal by dissolution before commencing a run. The form of the (111)growth rate-solution velocity relationship is shown in Figure 2 for two levels of supersa-
,
,
0.08
,
Solution velocity
,
j
0.24
046
(mlrl
Figure 2. Effect of solution velocity and supersaturation on the (111)face growth rate of succinic acid at 27.3 O C : X, Ac = 0.0015, u = 0.016; 0, Ac = 0.0033, u = 0.035.
>-
Ac, Supersaturation ( g acid / kg water)
Figure 3. Effect of temperature on the (lli)face growth rate of succinic acid at a solution velocity of 0.2 m/s: 0,35.0 O C ; V,27.3 O C ; 0,21.0 “C. Table I temp, e,”C
slope, g
intercept, K ( m / s ) x 10‘’
35.0 27.3 21.0
1.08 f 0.03 1 . 1 5 r 0.03 1.14 i 0.07
1.2 f 0.9 1.4 f 1 . 2 1 . 2 k 0.8
turation, expressed both as a concentration driving force, Ac, and as a dimensionlessratio, r~ = Ac/c*. For all levels of supersaturation studied, the crystal growth rate is clearly solution-velocity dependent. No obvious “plateau” is reached, even at the high velocity of 0.28 m/s. The form of the curves suggests that the growth is primarily influenced by bulk diffusion. Effects of Temperature and Supersaturation. In view of the significant dependence of growth rate on solution velocity (Figure 2), the effects of temperature and supersaturation on the (111)face growth rate were measured at one velocity only (0.2m/s). The (111)face was mounted perpendicular to the solution flow. The data were fitted to a power law equation of the type u = KG (C - c*)B (3) where g is the so-called “order” of the growth process and KG is the overall growth rate coefficient. A logarithmic plot (Figure 3) gave the slopes and intercepts listed in Table I at Ac = 1 kg/kg. Although the standard deviation of g is relatively small (3-6%), the small deviations in slope introduce considerable errors in the estimation of KG (65-85%). The activation energy, E, for growth is usually estimated from the slope of an Arrhenius plot (log KG vs. T1), but the data in Table I are nonlinear, which is not surprising in view of the potential error involved in evaluating KG. However, by assuming a value of g = 1 for all three tem-
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980
2
0.8 I
4
b
810
Ac, Supersaturation ( g acid / kg water)
1
0
,
I n--A---r-_---- ________-
I-
,
(020)
(0 01
1 02 * 0 1 L
erpe"d
1 03
0 03
ilril1r.l
6 0L 11 Perpc-d c u ar (r r;-Tr7m3T+-l .___ ____ 1
1 I
-
Figure 4. Effect of crystal orientation relative to the solution flow (0.2 m/s) for the (010) and 001) faces of succinic acid at 27.3 "C.
peratures, and using a minimization subroutine program to fit the data to the function u = KGAc,values of KG = 7.42 X and 5.28 X 10" m/s were obtained 6.28 X for 35.0,27.3, and 21.0 "C, respectively. From these values an activation energy for the (111)face growth of 18.3 f 1.0 kJ/mol was calculated. This value, although not very reliable in view of the simplifying assumption made above, does tend to confirm that the growth of the (111)face of succinic acid is diffusion controlled. Effect of Crystal Orientation. The boundary-layer characteristics around a solid object are greatly influenced by its shape and orientation with respect to the flowing liquid (Schlichting, 1960). Diffusion-controlled crystal growth rates should therefore be orientation dependent. Such an effect has been reported (Clontz et al., 1972) for the (110) face of MgS04-7H20growing in aqueous solution: the mass transfer coefficient was increased by about 30-40% when the crystal face was changed from a parallel to a perpendicular orientation with respect to the solution flow. Growth rates of the (010) and (001) faces of succinic acid were measured over a range of supersaturation and at two different orientations in a constant solution velocity of 0.2 m/s at 27.3 "C (Figure 4). The growth orders, g, of the (010) and (001) faces are virtually independent of crystal orientation (1.02, 1.03, and 1.46, 1.48, respectively). However, when these faces are aligned perpendicular rather than parallel to the direction of solution flow, their overall growth rate coefficients, KG, are increased by about 20 and 10070, respectively. This is further evidence to support the view that the growth process is diffusion (mass transfer) controlled.
Overall Growth Rate Measurements Overall growth rates of large populations of crystals have been measured in fluidized beds and agitated vessels of many different types (Mullin, 1972),but it is generally very difficult to quantify the hydrodynamic conditions, and the mean total surface area of the suspended crystals cannot be estimated accurately. On the other hand, the overall growth rate of a single crystal can be measured quite sim-
119
ply in a flow cell, under reasonably well-defined hydrodynamic conditions, and the crystal is not subjected to any collision effects. A simple method of measuring the overall growth rate of a suspended single crystal is to remove it periodically from the solution in which it is growing, dry and weigh it, record its dimensions, and then return it to the solution (Mullin, Osman, 1973). A mean overall mass deposition rate can thus be estimated from the mean crystal surface area, but the method is unsatisfactory because growth is discontinuous and the crystal surface may be subjected to contamination and mechanical and thermal shock. Alternatively, the crystal may be suspended from a balance arm and its weight monitored over a period of time (Smythe, 1967; Bennema, 1967). This method presents a number of experimental difficulties, and correction must be made for the changing upthrust. In the present work, a semicontinuous weighing technique was developed for a crystal suspended in a flow cell similar to that used for the single face growth rates. Full experimental details are given elsewhere (Whiting, 1976). A crystal of succinic acid was mounted on the tip of a 1 mm diameter tungsten rod with the a axis (see Figure 1) of the crystal aligned parallel to the axis of the rod. The rod was suspended from one arm of an automatic balance (Stanton, Ltd.) fitted with a chart recorder. To reduce the effect of surface tension at the point where the rod emerged from the solution, this particular section was replaced with a short length of a thinner (0.23 mm) stainless steel wire. In normal operation, the rod was clamped against a soft rubber pad by a solenoid-operated piston located above the growth cell. In this way the crystal was held centrally within the growth cell. To record the weight of the crystal, the flow of solution was stopped and the solenoid activated to allow the rod and crystal to hang freely from the balance arm. The balance was then actioned for about 2 min and the weight was recorded on a chart. The rod was then reclamped and the solution flow restarted. The weighing operations were carried out automatically at 10-min intervals. The sensitivity of the balance was f0.1 mg. Growth Rate Calculations. The overall mass deposition rate, R G , is given by 1dM R =-.(4) Adt where A is the average crystal surface area over the measurement period. The apparent mass increase of an immersed crystal, Am, in time At is smaller than the actual increase, Ah4, by an amount equal to the liquid upthrust Am = A M P , - P s ) / P c (5) where pc and ps are the crystal and solution densities, respectively. Allowance can be made for the increase in crystal surface area over the growth period from
A = fWf3 (6) where f is a crystal shape factor. Over a small interval of time, eq 4-6 may be combined to give
RG = [ P , ~ / ~ / ~-W ~ , /) '~ / ~(I P ( d ,m / d t ) (7) Integrating eq 7 over a time interval tl to t2,during which the apparent weight of the crystal increases from ml to m2, results in m-21i3- milf3= R ~ f ( t-2 t l ) ( p c - ps)1/3/3p,"3 (8) Under steady-state conditions, a plot of m1I3vs. t should therefore yield a straight line and hence a value of the growth rate, R G .
120
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980
Table 11. A Comparison of Face and Overall Growth Rates of Succinic Acid Crystals in Aqueous Solution overall growth rate contribution (kg/m2 s) crystal face ( h k l )
face growth correlation, u (mls)
1.4,
I
I
1
I
re1 surf area of crystal faces 0.694 0.150 0.101 0.055 R , estimated: RG measured: I
ConcenIration diiierence (9 acid I kg woIer
I
Figure 5. Overall growth and dissolution rates of succinic acid in aqueous solution at 27.3 OC. Solution velocity = 0.2 m/s. Growth: RG = 0.20 f O . ~ ~ A C (kg/m* ~ . ~ * s)~ (kg/kg)”=; .~ dissolution: RD = (3.28 f 0.08) X 10-2Ac (kg/m2 s)/(kg/kg).
Effect of Supersaturation. Overall growth rate measurements were made at 27.3 “Cand a relative crystal-solution velocity of 0.2 m/s. The effect of supersaturation on growth rate is shown in Figure 5. When plotted on logarithmic coordinates (not given here) the data are correlated by RG = (0.20 f 0 . 1 5 ) A ~ ’ . ~ ~ * ~ . ~ (9)
Comparison of Face and Overall Growth Rates It was not possible to make a detailed comparison between the measured face and overall growth rates since this would have required data from many more individual crystal faces at a large number of different crystal orientations. However, the following attempt indicates how such a comparison could be made and at the same time highlights the difficulty of making such comparisons. For a crystal composed of faces (hkl},the mass flux is given by =
C[A{hkl} Rlhklll
(10)
CIAlhkPIRlhklJl/CA[hkl}
AC
By a least-squares fit, kd was estimated to be (3.28 f 0.08) X loT2(kg/m2s) (kg/kg). The close similarity between the growth and dissolution rates further illustrates the importance of diffusion effects upon the growth of succinic acid crystals. Surface Integration Kinetics If the crystal growth process may be assumed to be represented by a simple two-step procedure, the growth rate may be expressed by R G = kd’(C - Ci) (13) for diffusion of solute from the bulk solution to the crystal-solution interface and RG
(11)
The relative surface areas of the four predominant faces of succinic acid were estimated from measurements on ten of the crystals used in the growth experiments. It was assumed that the (111)face was perpendicular to the flow of solution and the (001) and (010) faces parallel. It was further assumed, merely to facilitate calculation, that the unmeasured (110) face growth rate was the same as that of the (11T). The overall growth rate comparisons at three different supersaturations (Table 11) are inconclusive since the estimated rates are more than twice the measured, but it should be noted that no allowance has been made for (a)
0.005 kglkg 1.16 x 10-4 7.72 x 10-5 5.49 x 10-5 2.83 x 10-5 2.76 X l o T 4 1.34 X AC =
the interruption in solution flow during the weighing operations (an overall reduction in the growth rate would be expected), (b) the different hydrodynamic conditions at the leading and trailing faces of the crystal (growth at a leading face would be expected to be higher than at a trailing face, and the present analysis assumes that all (110) and (111)faces are leading faces), and (c) the possibility that the (110) face growth rate could have been lower than that used in the calculation. However, the calculations in Table I1 are instructive because they clearly demonstrate the difficulties of estimating overall crystal growth rates arising from the fact that the individual faces of a crystal have different, often very different, growth kinetics and controlling mechanisms. Overall Dissolution Rates The dissolution rate of succinic acid was measured at 27.3 “Cusing the semicontinuous weighing technique described above. Short stubby crystals were selected for dissolution runs because they presented a large surface area to the flow of solution. To allow comparison with the growth experiments, each crystal was fixed to the rod in the same orientation as that adopted for growth. To prevent individual crystals from becoming too misshapen, the duration of a dissolution run was restricted to about 2 h. The data from six runs (Figure 5) were correlated in the form RD = kdhc (12)
and the mean overall growth rate by RG =
= 0.003 kglkg 5.44 x 10-5 4.29 x 10-5 3.24 x 10-5 1.57 x 10-5 1.45 X 6.59 x
0.001 kgkg 1.07 x 10-5 1 . 2 1 x 10-5 1.04 x 10-5 4.45 x 1 0 - 6 3.77 X 1.45 x AC =
= k,(ci - c*)‘
(14)
for the integration of solute molecules into the crystal lattice. If it may be assumed that the mass transfer coefficient for dissolution, k d , is applicable to the diffusion step for growth, i.e., kd = kd’, eq 13 becomes ci = c - RG/kd (15)
so that Rc = k,(Ac - RG/kd)r
(16)
where Ac = c - c*. Therefore, utilizing growth and dissolution data obtained under comparable conditions, eq 16 should enable the
Ind. Eng. Chem. Fundam., Vol. 19,
-
. n E
II
Y 0,
0
a
0.2
0.5
I
2
(Ac -ao/k, 1 (kg / kg 1 XIO3
Figure 6. Surface integration rates f o r succinic acid in aqueous solution a t 27.3 " C a n d a solution velocity of 0.2 m/s. T h e d a t a represented by black symbols are n o t included in t h e correlation.
surface integration exponent, r, and the rate coefficient to be evaluated. Figure 6 is a plot of log R G vs. log (Ac R ~ / k d in ) accordance with eq 16. There is an obvious scatter in the data for growth rates higher than about 6 X kg/m2 s, i.e. for supersaturations Ac > 3g/kg), but correlation of the remaining seven experimental points gives r = 1.8 f 0.2. This value is consistent with that expected (r = 2) from the BCF surface diffusion model of crystal growth (Burtonet al., 1951). The scatter of the data at the higher supersaturations most probably indicates a change in the mode of growth, e.g., from a surface diffusion (BCF) to a surface nucleation mechanism. Nomenclature A = crystal area, m2 c = solution concentration, kg of succinic acid/kg of HzO c* = equilibrium saturation concentration, kg of succinic acid/kg of H20 Ac = supersaturation = (c - c*), kg of succinic acid/kg of HzO
No. 1, 1980 121
ci = crystal-solution interfacial concentration, kg of succinic acid/kg of H20 f = crystal shape factor (m3/kg)2/3 g = "order" of the overall crystal growth process (eq 3) k i , kd =-mass transfer coefficient for diffusion/dissolution, kg/mz s (Ac) KG = overall growth rate coefficient, kg/m2 s ( A C ) ~ k , = surface integration rate coefficient, kg/m2 s ( A c ) ~ m = apparent mass of suspended crystal, kg &f = actual mass of crystal, kg M = mass deposition rate, kg/s r = "order" of the surface integration process (eq 13) RD = overall crystal dissolution rate, k /m2 s RG = overall crystal growth rate, kg/mQ s t = time, s u = linear crystal growth rate, m/s ( h k l ) = crystal face or lattice plane (hkl)= crystal form, comprising all faces equivalent by symmetry to (hkl) qs = solution viscosity, Pa s 0 = temperature, "C pc = crystal density, kg/m3 ps = solution density, kg/m3 u = supersaturation (= Ac/c*)
Literature Cited Bennema, P., J. Cryst. Growth, 1. 278, 287 (1967). Broadley, J. S . , Cruickshank, D. W. J., Morrison, J. D., Robertson, J. M., Shearer, H. M. M., Proc. R . SOC. London, Ser. A , 251, 441 (1959). Burton, W. C., Cabrera, N., Frank, F. C., Phil. Trans., A243, 299 (1951). Clontz, N. A,, McCabe, W. L.,Rousseau, R. W., Ind. Eng. Chem. Fundam., 11, 368 (1972). Findlay, A., "Practical Physical Chemlstry", 8th ed, Longmans, London, 1955. "International Critical Tables", Vol. 4, pp 251, 253, McGraw-Hill, New York, N.Y., 1926. Mullin, J. W., "Crystallization", 2nd ed, Butterworths, London, 1972. Mullin, J. W., Osman, M. M., J. Chem. Eng. Data, 18, 353 (1973). Mullln, J. W., Amatavivadhana, A., J. Appl. Chem., 17, 151 (1967). Mullln, J. W., Garside, J., Trans. Inst. Chem. Eng., 45, 285 (1967). Schllchtlng, H., "Boundary Layer Theory", 4th ed,McGrawHIII, New York, N.Y., 1960. Seldell, A., "SolubllRles of Organic Compounds", 3rd ed, Van Nostrand, New York, N.Y., 1941. Smythe, B. M., Aust. J. Chem., 20, 1087, 1097, 1115 (1967). Whiting, M. J. L., Ph.D. Thesis, University of London, 1976. R e c e i v e d for r e v i e w
Accepted
December 11, 1978 September 28, 1979
