In the Laboratory
W
A Laboratory Experiment for Measuring Solid–Liquid Mass Transfer Parameters
Sonia Dapía, Carlos Vila, Herminia Domínguez, and Juan Carlos Parajó* Department of Chemical Engineering, University of Vigo (Campus Ourense), Polytechnical Building, As Lagoas, 32004 Ourense, Spain; *
[email protected] Many real-life and biochemical processes take place in heterogeneous media, but little attention has been traditionally paid to them in graduate courses (1), probably as a result of the complexity of the involved phenomena. One of the simplest mass transfer experiments is the dissolution of a solid in an agitated liquid. Miller (2) studied the mechanisms involved in mass transfer from benzoic acid to solutions, whereas Boon Long et al. (3) considered the dissolution of the same solute in water. Sensel and Myers (4) studied the dissolution of a candy (composed of sugar, citric acid, and color additives) in water. These authors measured the diameters of candy pieces as experimental variables, and developed a simplified model for data interpretation. The laboratory experiment described in this article starts from the same principles developed by Sensel and Myers, but the experimental procedure was modified to provide a more reliable experimental assessment. The experimental data are analyzed according to nonsimplified theoretical models. The course of candy dissolution is followed by the dinitrosalycilic acid (DNS) spectrophotometric method (5), which measures the reducing power of the liquid phase. The absorbance readings are used to calculate the fraction of undissolved candy by means of material balances. The mass transfer equation is solved and all the involved parameters are calculated by a simple, numerical method.
where Wsat and W are the solid to solvent mass ratios corresponding to the saturation and to the considered situation, respectively, and kW is the mass transfer coefficient in terms of the mass ratio (versus the concentration in eq 1). International units are used exclusively in this experiment: M in kg; t in s; A in m2; kW in (kg solvent)兾(s m2); Wsat and W in (kg solid)兾(kg solvent). When n solid particles with initial mass, M0, are solubilized in given solvent mass, MD, the conservation law takes the form:
Mass Transfer Kinetics
then:
The instantaneous rate of mass transfer from a solid to a liquid phase can be calculated as the product of the mass transfer coefficient, kLS, the interfacial area, A, and the driving force (defined as the difference between the saturation concentration Csat and the actual concentration C ). In the transient dissolution of n solid pieces, a material balance to the solid phase takes the form, −d(nM ) = kLS A ( Csat − C ) dt
(1)
where M is the undissolved mass of each piece of solid at the considered dissolution time. For material balance purposes, the driving force can be expressed in terms of the mass ratio W, defined as (mass of dissolved solid)兾(mass of solvent). It should be noted that the mass of solvent, and not the mass of solution, is constant through the whole dissolution process and that the utilization of the liquid phase density is avoided when W is used to measure the concentration of the dissolved solid. The material balance to the solid phase can be rewritten as, −d(nM ) = kW A (Wsat − W ) dt 1502
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nM0 = nM + WMD
(3)
Combining eqs 2 and 3,
−d ( nM ) n ( M0 − M ) = kW A W sat − dt MD
(4)
The time-dependent mass transfer area corresponding to n spherical particles of solid with radius R is:
A = n4πR 2
(5)
If the density of solid particles is ρS, M =
ρS 4 πR 3 3
R =
3M ρS 4 π
1
(6)
3
(7)
Substituting R in eq 5: 3M A = n4π ρS 4 π
2
3
(8)
Substituting A in eq 4 and simplifying: 1
dM 36 π M 2 = −k W dt ρS2
3
Wsat −
n ( M0 − M ) MD
(9)
This expression can be adapted for an approximate numerical solution as follows: M (t + ∆t ) − M ( t ) ∆tt 2
≈ −k W
36π M ( t ) ρS2
(2)
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1
(10) 3
Wsat − n
M0 − M (t ) MD
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In the Laboratory
Experimental The candies used in this experiment (Chupa Chups, Spain) are spherically shaped and composed of sugars, glucose syrup, food acid (citric), colorant E-162, and flavoring. The candies are sold attached to a stick, a particular feature facilitating their implementation in this experiment. The initial mass and diameters of the candies were measured carefully for each experiment. Sticks (free from candy) were weighed at the end of experiments, and the corresponding masses were deducted from the initial candy weights. Candies (1, 2, 3, or 4 pieces, measured by variable n) were placed in a flask containing 1 kg of water thermostated at 30.0 ⬚C with magnetic stirring (Figure 1). At selected contact times, samples were withdrawn from the liquid phase, diluted, and analyzed by the DNS method (5). In each experiment, the last sample was collected when the candies were totally dissolved (M = 0), and the corresponding concentrations of solids were calculated by a material balance. Once the Beer’s coefficient giving the interrelationship between absorbance and concentration was known, all the absorbances were directly converted into concentrations. A comparison between the Beer’s coefficients obtained in the various runs provided a first indication on the precision of the experimental work. Hazards
PID/TC control 30.0 °C
Figure 1. Experimental setup.
The DNS reagent employed in the spectrophotometric determinations contains potassium sodium tartrate, which is not hazardous, sodium hydroxide, and 3,5-dinitrosalicylic acid. Sodium hydroxide is caustic and causes burns; in case of contact with eyes, rinse immediately with plenty of water and seek medical advice. 3,5-Dinitrosalicylic acid may cause skin irritation or respiratory and digestive tract irritation and is harmful if swallowed; in case of contact with eyes, rinse immediately with plenty of water and seek medical advice. Students are instructed to follow general safety laboratory rules. Results and Discussion
Improvements in the Experiment The dissolution of spherical candy is a well established laboratory experiment in which mass transfer is followed by particle diameter measurements. With this operational procedure, the experimental error increases significantly when the particles become small, limiting the suitable operational zone. For example, Sensel and Myers (4) covered 55% of mass dissolution with four experimental data. For theoretical interpretation, these authors assumed that the concentration of dissolved solids, C, was low and consequently eq 1 can be simplified assuming that (Csat – C ) ≈ Csat. With this approach, the resulting equation can be integrated, but just the product CsatkLS can be calculated: no separate values can be obtained for Csat and kLS. This article deals with a more sophisticated approach to the same problem. The dissolution of solids is measured spectrophotometrically. The suitability of this method for assessing the whole process of candy dissolution (measured by the constant of proportionality between dissolved solid and absorbance readings) was experimentally confirmed. The specwww.JCE.DivCHED.org
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trophotometric DNS method shows favorable features, including ease of handling, stability of chromogenic reagents, and lack of harmful chemicals. No calibration is needed, because the interrelationship between concentration and absorbance was directly established from the results obtained after total solid dissolution. The main advantages of the conventional spherical candy dissolution experiment (particularly low cost and utilization of standard laboratory equipment) are retained in our approach. The experimental procedure proposed in this article allows substantial improvements in both experimental work and interpretation of results: (i) the experimental error does not increase here with the course of solid dissolution, (ii) the process can be followed until the solid is completely dissolved, (iii) repetitive and reliable data are obtained under all the operational conditions, (iv) wider concentration ranges can be explored, (v) nonsimplified models can be used for data interpretation, and (vi) all the mass transfer parameters involved (Csat and kLS) can be separately calculated in each experiment.
Experimental Development Experiments were carried out with 1, 2, 3, or 4 candy pieces in a thermostated, nonbaffled flask (Figure 1) with magnetic agitation at a fixed speed. The agitation speed and the magnetic bar were the same in all the experiments. The situation for a single candy is qualitatively different from the situation with multiple candies: when two or more candies are employed, the circulation of the fluid around a given sphere is affected by the presence of the rest, which act as baffles. This difference can be avoided either by using baffled flasks or by placing a nonsoluble solid sphere in the liquid
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In the Laboratory
Solving the Mathematical Model Equation 9 can be integrated by variable substitution and polynomial decomposition, but the resulting equation is very complicated and shows a complex dependence on both kW and Wsat, which cannot be calculated by linear regression. Because of this, numerical solving is considered a better approach for teaching purposes. The Runge–Kutta method can be easily employed for solving eq 9 for trial values of kW and Wsat. If the students know this method or are able to understand its development (for example, following the useful indications provided in ref 6 ), the differential equation can be solved directly. Alternatively, as the slopes of the curves show smooth variations, eq 10 provides a valid and easy approach for solving the problem. In our case, the equation was solved with a commercial spreadsheet for trial values of kW and Wsat using ∆t = 1 s. With this operational procedure, the average variation in the parameters´ values versus the results achieved with the Runge–Kutta method was about 5%. When the dissolution profiles have been calculated by by any of the methods cited above, the sum of deviation squares can be calculated, and both experimental and calculated results can be presented in a single plot. Optimal values can be then found for kW and Wsat by just modifying by hand the initial values (for example, by the sectioning method) and looking for a minimum sum of deviation squares. Additionally, if the spreadsheet includes a built-in optimization routine, the sum of deviation squares can be minimized with respect to both kW and Wsat to calculate the corresponding best-fit values. The calculated time courses of solid dissolution shown in Figure 2 have been obtained in this way. The goodness of fit can be then checked by calculating the corresponding regression or significance statistical parameters, and the effective driving forces can be calculated for each experiment. Even if the parameters kW and Wsat are expected to be independent from the number of candy pieces, the above procedure leads to different individual results for both parameters owing to the incidence of the experimental error. The deviations with respect to the mean will measure the reliability of the experimental work. If (as expected) small differences are found for each variable, the possibility of using a single
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n=4 0.05
Mass of n Candies / kg
when a single candy is employed, to cause the expected hydrodynamic effect. The time course of candy dissolution obtained for the various situations is shown in Figure 2. Using this information with the experimental values of ρS (1751 kg m᎑3) and the diameter of every piece of candy (in the range 0.025– 0.028 m), the mass transfer parameters can be calculated.
0.04
n=3
0.03
n=2 0.02
n=1 0.01
0.00 500
0
1000
1500
2000
Time / s Figure 2. Time dependence of the undissolved mass of n pieces of candy in experiments carried out with 1, 2, 3, or 4 candies.
value for each parameter can be easily assessed by optimization of the overall sum of deviation squares corresponding to the various experiments. In our case, satisfactory fitting was obtained for Wsat = 0.259 and kW = 0.026 International units (see previous discussion). Even if these values depend on a variety of operational parameters (including the type of candy, the shape and dimensions of the agitation bar, and the agitation speed), these results can be employed as trial values for the corresponding variables in the calculation procedure described above. WSupplemental
Material
Instructions for the students and notes for the instructor are available in this issue of JCE Online. Literature cited 1. Rodríguez, M. F.; Ríos, M. C.; Mosquera, M.; Ríos, A. M.; Mejuto, J. C. J. Chem. Educ. 1995, 72, 662–663. 2. Miller, D. N. Ind. Eng. Chem. Proc. Des. Dev. 1971, 10, 365– 375. 3. Boon Long, S.; Laguerie, C.; Couderc, J. P. Chem. Eng. Sci. 1978, 33, 813–819. 4. Sensel, M. E.; Myers, K. Chem. Eng. Educ. 1992, 26, 156– 159. 5. Miller, G. L. Anal. Chem. 1959, 31, 426–428. 6. Perry, R. H.; Green, D. W.; Maloney, J. O. Perry’s Chemical Engineers´ Handbook; McGraw Hill: New York, 1997; Vol. 1, Chapter 5, pp 53–97.
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