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Spray Drying: Influence of Developing Drop Morphology on Drying Rates and Retention of Volatile Substances. 2. Modeling John P. Hecht† and C. Judson King* Department of Chemical Engineering, University of California, Berkeley, California 94720
Two numerical models simulating the drying of a drop containing dissolved solids are presented. The first is a simple model to predict the drying rate and temperature history of a drying drop during the period of morphological development. The model assumes that the drying rate is controlled by the rate of heat transfer from the drying gas to the drop. The second model utilizes the convective diffusion equations for a drop containing SF6 (a model volatile component), water, and sucrose. Concentration profiles in the liquid are treated as radially symmetric. A single, centrally located bubble containing water vapor, SF6, and an inert gas is included to simulate the effects of morphological development. This bubble never ruptures. The model predicts a net inward flux of SF6 from a sufficiently negative value of the cross-diffusion coefficient after the drop surface is sufficiently dry. The model predictions are in good agreement with experimental data. Introduction The goals in modeling the drop-drying process are to predict how drying conditions will affect the time of drying and/or the quality of the dried product and to construct a tool useful in spray dryer design. The quality of a dried drop of food or beverage material is affected by the loss of volatile flavor and aroma components, the extent to which heat-degradation reactions occur, and the particle size, among other factors. Such a model can eventually be used in conjunction with models of the flow and mixing patterns in a spray dryer to rationalize the relationship between dryer operation and product attributes. Many researchers have undertaken modeling of the drop-drying process. Invariably, drops have been modeled as stagnant spheres, meaning that convective activity from atomization and internal circulation caused by the drag of the drying gas are neglected. Morphological development is usually neglected and is treated in a very simplistic fashion if it is considered. Van der Lijn et al.1 treated the drop-drying problem numerically for a stagnant sphere. They performed a coordinate transformation in which they altered the independent “distance” variable so that it was stated in terms of the mass of the solute (e.g. sucrose or coffee) rather than actual distance. This work was extended by Kerkhof and Schoeber2 to include the transport of a volatile aroma component. The same solute-fixed coordinate system was employed. They used a StefanMaxwell treatment of the ternary-diffusion problem with estimated values for the diffusivities. In addition, reaction kinetics for a thermal degradation reaction was included. Wijlhuizen et al.3 used the same coordinate transformation to simulate the drying of a drop of skim milk. The reaction kinetics for the thermal degradation of phosphatase was included in the model as well. Their * To whom correspondence should be addressed. † Present address: The Procter & Gamble Company, Corporate Engineering Technologies, 8256 Union Centre Blvd., West Chester, OH 45069.
model drop also contained a centrally located air bubble, which was permitted to expand and contract due to changes in the drop temperature. Sano and Keey4 modeled the drying of skim milk droplets without the use of solute-fixed coordinates. Like Wijlhuizen et al.3 they also included a centrally located air bubble, but they allowed it to rupture once during the simulation. Furuta et al.5 modeled the drying of a drop containing dissolved ammonium chloride by solving the convective-diffusion equation in spherical coordinates to predict the time of formation of crystals on the drop surface. Schoeber6 developed a short-cut method for the calculation of drying rates for systems with a strongly concentration-dependent diffusion coefficient. This method has the advantage that no solution to the diffusion equations is necessary. However, the treatment is empirical in nature and may not give accurate predictions over a wide range of drying conditions. Two models are presented in this work. The first treats the drying rate during morphological development in a simplistic fashion. The second is a more complicated numerical model that solves the complete convective diffusion equations for a drying drop. Simple Model for Drying during Morphological Development It has been shown in Part 1 and by other experiments7 that morphological development occurs at the boiling temperature of the drop. The drying rate of a drop during this period is governed by gas-phase heat transfer with a known ∆T driving force (Tdrying gas Tdrop). This allows for the development of a simple model using an energy balance about the drop:
hA(Tgas - Tdrop) + ∆Hvap
dmdrop dTdrop ) mdropCp,drop dt dt (1)
Heat transfer by radiation and the heat of mixing are neglected. The rate of change of the drop mass can be related to the average mass fraction of water in the drop:
10.1021/ie990464+ CCC: $19.00 © 2000 American Chemical Society Published on Web 05/12/2000
Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000 1767
dωavg dt avg)
ms dm ) dt (1 - ω
(2)
2
Equation 2 can be derived by using the Quotient Rule to evaluate dωavg/dt, where ωavg ) mw/(mw + ms), recognizing that ms is a constant.7 The boiling temperature of the drop is estimated from the average water content of the drop, where the drop temperature is a function of the average water composition.
Tdrop ) f(ωavg)
(3)
This corresponds to an assumption that the liquid is well mixed and saturated. The equilibrium pressure of water over a solution is typically lowered, and hence the equilibrium temperature is raised, by the presence of solutes. Next, the energy accumulation term, the term on the right side of eq 1, can be neglected, yielding eq 4, the final equation of the model.
Tgas - Tdrop(ωavg) )
ms -∆Hvap hA (1 - ω
dωavg (4) dt avg) 2
This equation can be integrated numerically to predict the drying rate and temperature of a drop as a function of time during morphological development. This model may also be used for drops following a trajectory through drying gas of varying temperature and velocity by using time-dependent functions for Tgas and h. Figure 1 compares the prediction of the model to data obtained experimentally for a 30 wt % sucrose drop. An initial value of the average weight fraction of water was needed, and the experimental value at the onset of morphological development was used. A fitting parameter was used to match the experimental data to the model prediction. This was necessary to account for the heat added by conduction to the drop from the solid support, the increased surface area, and the increased value of the gas-phase heat-transfer coefficient due to droplet inflation. The best-fit parameter, R, as shown in eq 5, had a lower value (680 °C/W) than the calculated product of 1/hA (1180 °C/W). The heat-transfer coefficient was calculated using the RanzMarshall correlation,8 and the area used for the 1180 °C/W estimate was the initial interfacial area of the drop, despite the fact that this area was changing during morphological development.
∆Hvapms dωavg Tgas - Tdrop ) -R (1 - ωavg)2 dt
(5)
The value to this model is its simplicity. No information is needed regarding the specific types of bubbling and bursting occurring in order to predict the overall drying rate and drop temperature history. In other words, the overall drying rate is, to a first approximation, independent of the details of the morphological development. These details affect the overall drying rate only insofar as they affect the product hA. Numerical Drop-Drying Model A program was written in FORTRAN and run on a 300 MHz Pentium II computer (donated by Intel Cor-
Figure 1. Comparison of experimental data with the drying model prediction for the temperature history and moisture content for a 30 wt % sucrose drop during morphological development.
poration) to simulate the drying of a single drop of aqueous sucrose solution. A run typically lasted 4-20 h, depending on input conditions. The long run time stemmed from the need to include many distance increments and short time increments for conditions near the boiling temperature. Sucrose was chosen as a solute, since many of the relevant physical properties are known. The program was originally donated to our group by Keey,4 but it has now been drastically modified and bears little resemblance to the original. The model drop contained three components: SF6 as a highly volatile component (1), water (2), and sucrose (3). The drop also contained a single, centrally located bubble that was allowed to change size, depending on the drop temperature and water activity of the inside drop surface. The rate of growth or shrinkage of the internal bubble was limited by viscous resistance to flow. The bubble was not allowed to burst; instead, the bubble inflation was always taken to be ruptureless (see Part 1). The drying process was assumed to be one-dimensional, with concentration gradients occurring in only the radial direction. Likewise, single average values of the gas-phase mass- and heat-transfer coefficients were used for the entire drop surface. The small size of the drops (R ∼ 1 mm) and the relatively high thermal diffusivity allowed for the assumption of a uniform drop temperature.9 Calculation of the Drop Temperature. The drop temperature is calculated from an energy balance about the drop.
Tt+K ) Tt + 4πKR2(h•(Tgas - Tdrop) - F∆Hvap)/ h p,drop) - (∆nw,bubble∆Hvap)/(mdropC h p,drop) (6) (mdropC
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Equation 6 contains two latent heat terms. One is for evaporation of water from the outside surface of the drop, and the other is for evaporation of water from the inside surface of the drop into the bubble. The variable ∆nw,bubble represents the change in the number of moles of water in the bubble. Heat- and mass-transfer coefficients were calculated using the Ranz-Marshall correlations:8
Nu ) 2.0 + 0.6Re0.5Pr0.33 )
h2R kcond-gas
Sh ) 2.0 + 0.6Re0.5Sc0.33 )
kc2R Dgas
(7)
The radial concentration profiles of water and SF6 were calculated by solving the appropriate differential equations. During each time step, the water profile was calculated first, followed by the SF6 profile. The current water profile was necessary to calculate the waterconcentration-dependent diffusion coefficients for SF6, while the concentration of SF6 was low enough (∼1 ppm) so that its profile would not affect the water-profile calculation. Both profiles were calculated using a Crank-Nicholson finite-difference scheme10 with an uneven distance grid of 125 increments, each of which was made to contain an equal mass of sucrose. Details of the numerical methods are given elsewhere.7 The convective diffusion equation7 was solved for the calculation of the water profile in solute-fixed coordinates. Since SF6 was taken to be a trace component, it was assumed not to affect the transport of water.
[
]
∂u ∂ ∂u ) D F 2r4 ∂t ∂z 22 3 ∂z
(8)
The following initial and boundary conditions were used:
jv1 ) -D11∇F1 - D12∇F2 jv2 ) -D21∇F1 - D22∇F2
Three of the four diffusion coefficients above are independent. The fourth one may be calculated from the other three by application of the Onsager Reciprocal Relations, using eq 10, which was derived for a ternary system with a trace concentration of component 1:12
(
[
For z ) 0; t > 0 ∂u )0 ∂z For z ) Z; t > 0
[(
jr1 ) -D11∇F1 - D12∇F2 - F1
)
) ]
F1 1 1 + D F∇ω2 F°2 F°3 F3 22 (11)
Equation 11 gives the impact of the water gradient on transport of the volatile component. Inserting eq 11 into the convective diffusion equation results in the differential equation integrated by the program.
[ [
∂F1 ∂F2 ∂u1 ∂ F3r4 -D11 )- D12 ∂t ∂z ∂z ∂z
[(
F1
2 2
The boundary condition at z ) 0, the inside surface of the hollow sphere, reflects the assumption that the water diffusing into the bubble is negligible when compared to the amount of water in the drop. The superscripted dot on the mass-transfer coefficient indicates that a correction has been made for the finite mass-transfer rate using film theory.11 Calculation of the SF6 Concentration Profile. Diffusion and convection of SF6 through the drop are complicated, since all three components must be considered. Multicomponent diffusion equations for the treatment of the ternary system were written in the form of Fick’s Law. Two independent fluxes are given below.
(
The concentration and temperature dependences of D22, the mutual sucrose/water diffusivity, are given by Etzel,13 who fit the data of English and Dole14 and Henrion.15 D11 and D21 for the sucrose/water/trace volatile system are given by Chandrasekaran and King.12 Their data for the ethyl acetate system were taken to apply to SF6 as the trace volatile component. This seemingly gross assumption is mitigated by the fact that D11 values for different volatile components measured by various authors do not differ greatly, and there are no other multicomponent diffusion data available for systems of solute/sucrose/water. D12 was calculated from D11, D22, D21, and thermodynamic data, using eq 10. The dependence of D12 on the weight fraction of sucrose at 298 K is shown in Figure 2. The value of D12/C1 dips below zero at about 45% sucrose. Equations 9 represent fluxes relative to the volumeaverage velocity. The flux of SF6 relative to sucrose is given below:7
•
∂u kp (Pwater,0 - Pwater,gas) -F3 r D22 ) ∂z P - Pwater,0
)]
V h1 ∂ ln γ2 D12 1 ∂ ln γ1 ) + 1+ (D11 C1 C2 ∂ ln C2 1 - C2V h2 ∂ ln C2 ∂ ln γ2 1 1+ D (10) D22) + ∂ ln C2 21 C2(1 - C2V h 2)
For 0 < z < Z; t ) 0 u ) u0
(9)
) ]
]]
∂ω2 1 1 1 + D22F F°2 F°3 F3 ∂z
(12)
The following initial and boundary conditions were used:
For 0 < z < Z; t ) 0 u1 ) u1,0 For z ) 0; t > 0 u1 ) 0 For z ) Z; t > 0 u1 ) 0 A similar result, but using Stefan-Maxwell diffusivities, is given by Schoeber.16
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The second method allows the bubble to change pressure at constant volume. The pressure difference between the gas in the bubble and the drying gas is then converted into an expansion or contraction velocity, using fluid mechanics. This velocity is multiplied by the magnitude of the time increment to determine how much the bubble grows or shrinks. The pressure drop across the drop liquid, which drives the growth or shrinkage of the bubble, is written as
∆P ) P -
3(nw + ninert + nSF6)Rgas constT 3 4πPbubble
(16)
The radial component of the equation of motion11
∂vr τθθ + τϑϑ ∂p 1 ∂ ) - 2 (r2τrr) + - Fvr ∂r ∂r r ∂r r
(17)
where τrr ) -2µ(∂vr/∂r), τθθ ) τϑϑ ) -2µ(vr/r), and, from continuity, Figure 2. D12/C1 as a function of sucrose content for the SF6 (1), water (2), sucrose (3) system at 298 K.
The boundary conditions at both the inner and outer surfaces are that the concentration of SF6 there is zero, due to the high volatility of SF6. The internal bubble is assumed to contain no SF6 initially; this is a realistic assumption in practice, since bubbles present in drops drying in a spray dryer originate during the atomization process and contain the drying gas.17 In the experiments described in Part 1, the initial bubbles were air bubbles that originated from the syringe. Calculation of New Bubble and Drop Size. The internal bubble initially contains an inert gas (the drying gas) and water vapor. The initial bubble volume and temperature are given as input parameters. The pressure within the bubble is given by
Pbubble ) P +
2σ R
(13)
The ideal-gas law is used to determine the number of moles of gas in the bubble. The partial pressure of water vapor is calculated by assuming thermodynamic equilibrium of the bubble contents with the inside surface of the drop:
Pw ) γwxwPw sat
(14)
This internal bubble changes size during drying because of changes in drop temperature and the activity of water on the inside surface and hence the partial pressure of water. The new bubble size is calculated using two methods. Either calculation method requires the number of moles of gas in the bubble to be recalculated, since water vapor may be evaporating or condensing, and SF6 is diffusing into the bubble. The first method allows the bubble to change volume at constant pressure, giving the bubble volume (and the radius) at thermodynamic equilibrium. Equation 15, below, is used.
Rbubble,new )
(
)
3(nw + ninert + nSF6)Rgas constT 4πPbubble
1/3
(15)
2 vr,bubbleµ ∆P ) 4Rbubble
[
R 1 ∫R,bubble r2
]
∂(µ/r) µ + 4 + ∂r r R F 2 4 2vr,bubbleRbubble R,bubble 5 dr (18) r
∫
was integrated to determine the relationship between this pressure difference and the velocity of the bubble surface. The inertial resistance, the second term on the right, was neglected, since it was small. Equation 18 was then solved for vr,bubble, and the new bubble radius was calculated:
Rbubble,new ) Rbubble,old + vr,bubbleK
(19)
The program selects the new value of the bubble radius by choosing whichever of the two methods predicts the smaller change in bubble size. Therefore, the change in bubble size predicted by thermodynamic equilibrium cannot be exceeded, and at the same time, the equilibrium size cannot be reached if the fluidmechanical resistance is too great. More details on the model, including derivations of the flux and differential equations, a more complete description of the physical property correlations used, and a complete listing of the program, are given elsewhere.7 Numerical Modeling Results Comparison of Model Results with Experimental Data. The model was run several times under the same conditions as those for some drying experiments. A comparison of the predicted (left) and measured drying rates (right) is shown in Figure 3. The apparatus and procedures used to obtain experimental data are described in Part 1. The conditions under which the model was operated are given in Table 1. The general shapes of the water-loss profiles and their changes with initial sucrose concentration have been captured effectively by the model. As the initial concentrations of the drops become higher, the falling-rate period of drying is entered earlier. Also, droplet inflation produces a similar increase in the magnitude of the drying rate between the model prediction and the experiment.
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Table 1. Conditions for the Model Runs Shown in Figures 3 and 4 initial % sucrose initial drop radius, m initial bubble radius, m Tgas, K Tdrop (initial), K Re initial g SF6/g sucrose (from solubility)
15 8.95 × 10-4 5.0 × 10-5 430 298 14 1.92 × 10-5
Figure 3. Comparison of model prediction (left) of the drying rate of 3 µL sucrose drops drying at 315 °F with experimental data (right).
The early drying-rate profile has a more rounded appearance in the experiment than in the model. This difference is more profound for the 15% and 30% sucrose drops. This is probably due to the internal circulation of the experimental drops, which enhances the mass transfer of water and shortens the drying time. Heat conduction via the syringe in the experiments may have affected the drying rate as well. A comparison between the model predictions and the experimental data for SF6 loss is shown in Figure 4. Two profiles are shown for the calculated loss rate of SF6 for each drop: the loss rate from the outside surface of the drop (prebubbling loss) and the loss rate into the bubble (loss due to morphological development). The magnitude of the prebubbling loss decreases as the initial concentration of sucrose increases, in accordance with the experimental data and selective diffusion theory.18 A difference between the model prediction and the experimental data is the initial loss of SF6 at t ) 0. The calculated peaks are higher for the model drop, since the experimental drop is noninstantaneously formed on the tip of a syringe and gas-phase axial dispersion in the experimental system acts to dull sharp peaks. The calculated rate of loss of SF6 into a bubble is not as easily compared with the data. The model predicts
30 8.95 × 10-4 5.0 × 10-5 430 298 14 6.48 × 10-6
45 8.95 × 10-4 5.0 × 10-5 430 298 14 2.67 × 10-6
60 8.95 × 10-4 5.0 × 10-5 430 298 14 1.39 × 10-6
Figure 4. Comparison of model prediction (left) of SF6-loss rates of the drying of sucrose drops at 315 °F with experimental data (right). Two plots are shown for each model run: the loss from the outside of the drop and the loss into the bubble.
that the loss into the bubble occurs during droplet inflation, but SF6 losses during morphological development appear in the experimental data only when bubbles rupture. The 45% drop never burst. Therefore, no experimental data are available on the quantity of SF6 inside that bubble. Calculated Concentration Profiles of SF6 and Water. Figure 5 shows calculated SF6 and water concentration profiles in a 15% sucrose drop. For both the SF6 and water profiles, the radial position of the leftmost point of each curve represents the location of the bubble interface and the rightmost point represents the location of the outer drop surface. The movements of these points show the changes in the bubble and drop sizes with time. The SF6 concentration profile begins to sharpen at the outer surface after 70 s, and the concentration reaches a maximum value close to the outer surface. On the inner side of this maximum, the contribution to mass transfer from the SF6 diffusion coefficient, D11, causes a flux of SF6 inward. SF6 concentrations later exceed the initial concentration, largely due to the removal of water. To factor out this effect, local ratios of the mass of SF6 to the mass of sucrose are plotted in the top portion of Figure 6. The SF6 concentration profiles are
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Figure 5. Concentration profiles of SF6 and water for a 15% sucrose drop. The positions of the bubble and the outer drop surfaces are shown at the bottom of the top plot as dots. Run conditions are given in Table 1.
Figure 7. Variation of the parameter W with sucrose content at 298 K given in the range of available diffusion-coefficient data.
Figure 6. Ratio of concentrations of SF6 to that of water and values of the parameter W for the same run as that in Figure 5.
not affected by the bubble until the bubble begins to grow. The quantity of SF6 in the bubble becomes noticeable after about 2 min of drying, as seen in the top left-hand plot of Figure 4, while Figure 5 shows that the drop is inflated after 130 s. After 70 s, the SF6 profiles begin to assume a new shape. The maximum moves inward and increases. The region near the surface becomes depleted of SF6, and the loss from the outer surface is negligible (see Figure 4, top left plot). The development of these profiles shows that there is a considerable flux of SF6 inward. Equation 12 gives the flux of SF6 relative to the flux of sucrose. The second and third terms show how a water gradient can affect transport of SF6. The second term may act either to retain SF6 by causing an inward flux from a negative value of D12 or to cause an outward flux if D12 is positive. The third term gives the effect of the convective flow of water on SF6 transport. Since water flows continually outward, it always contributes to SF6 loss. Therefore, a water gradient may have the net effect of either accelerating loss of SF6 or aiding in its retention. Taking the ratio of the second and third terms in eq 12 and simplifying gives the dimensionless parameter W, below. A negative sign has been added for a clearer interpretation.
W)-
F°2D12 u1F°3D22
(20)
A value of W > 1 means that the value of D12 is sufficiently negative so that the net effect of the water
Figure 8. Concentration profiles of SF6 and water for a 60% sucrose drop. The positions of the bubble and outer drop surfaces are shown at the bottom of the top plot as dots. Run conditions are given in Table 1.
gradient is to cause an inward flux of SF6. A negative value of W means that the cross-diffusion effect is causing an outward flux of SF6 and therefore accelerating its loss. The bottom plot in Figure 6 shows W plotted against time and position. Some values of W are plotted against the sucrose content in Figure 7. The value of W is negative at low sucrose contents but later rises above 1 as the sucrose concentration increases. These are in the range of available diffusion-coefficient data given by Chandrasakaran and King,12 again with the assumption that the diffusivities for the SF6 system are the same as those for the ethyl acetate system. Figure 8 shows SF6 and water concentration profiles in a 60% sucrose drop. The general characteristics of the profiles are the same as those for the 15% drop. However, sharper profiles develop more quickly in this case. Likewise, the retention of SF6 by the water gradient occurs sooner in the 60% drop, as can be seen by the depletion of SF6 near the outer surface. Changes in Drop and Bubble Size and Temperature History. Figure 9 shows the changes in drop and
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Figure 9. Changes in drop and bubble sizes and temperature history for a 45% drop drying at 460 K.
bubble size and temperature history predicted by the model for a 45% drop drying at 460 K. The other run parameters are listed in Table 1. After a steady decrease in size from the loss of water, the drop inflates rapidly and then deflates, as the drop becomes very dry. Meanwhile, the temperature rises to the boiling temperature and reaches a plateau. It then rises again as the boiling temperature increases and then eventually reaches the temperature of the drying gas. The drop and bubble sizes and temperature dip slightly at about 55 s, resulting from the enhanced water transport rate due to stretching, as discussed in Part 1. Larger oscillations are observed to occur in experimental drops, as seen in Figure 8 of Part 1. There are several reasons for this difference. The model neglects heat- and mass-transfer limitations in the bubble and conduction of heat through the liquid, which will cause a time delay between a change in the conditions of the drop liquid and the size of the bubble. These delays may encourage oscillation by failing to reduce the bubble size immediately after the drop surface has cooled from a surface-stretch-enhanced water flux. The bubble may continue to grow briefly, even as the surface of the drop is cooling. Likewise, drops that are heating may heat for a longer time, as bubble growth will be slowed by these delays. Effects of Drying Gas Temperature and Reynolds Number. Figure 10 shows the effect of increasing the temperature and the Reynolds number of the drying gas on the predictions of the SF6-loss rate and the drying rate. Increasing the drying-gas temperature from 430 to 460 K causes the drop to dry faster and gives a slight reduction in the prebubbling SF6 loss. Increasing the Reynolds number from 14 to 140 also decreases the prebubbling SF6 loss and increases the drying rate. Both the higher gas temperature and the higher Reynolds number increase the drying rate and decrease the
Figure 10. Model predictions of the effects of changing the drying gas temperature and the Reynolds number on mass-transfer rates from 45% sucrose drops. See Table 1 for other input parameters.
Figure 11. Model predictions of the effects of changing the initial drop and bubble sizes on mass-transfer rates from 45% sucrose drops. The time axis is scaled with the square of the difference of the drop and bubble radii. See Table 1 for other input parameters.
prebubbling SF6-loss rate by shortening the time before the onset of the falling-rate period. Effects of Drop Size and Bubble Size. Figure 11 shows the SF6 and water loss profiles as affected by changes in the initial drop and bubble sizes. The profiles are plotted against a reduced time, t/(Rdrop - Rbubble)2.
Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000 1773 Table 2. Losses of SF6 from the outside Surface and into the Central Bubble As a Function of Initial Drop and Bubble Sizea
loss from outer surface loss into bubble final retention a
R ) 8.45 × 10-4 m Rbubble ) 5.0 × 10-5 m
R ) 5.0 × 10-4 m Rbubble ) 5.0 × 10-5 m
R ) 8.45 × 10-4 m Rbubble ) 1.0 × 10-4 m
10.8% 56% 33.2%
10.9% 55.7% 33.4%
10.9% 56.1% 33.0%
The external conditions, Tgas and Re, were held constant.
Scaling the time axis by the square of the drop radius has been proposed by Kerkhof and Schoeber2 and Zakarian and King.19 From this similarity analysis, the fractional loss of water and volatile components is a unique function of this scaled time as long as the external conditions are held constant. The scaling factor used in this work includes the radius of the bubble. This was done by analogy with the characteristic length used by Barrer20 in the Fourier number used to describe diffusion in a hollow sphere. For the results shown in Figure 11, the Reynolds number was held constant, and the loss profiles for different initial drop and bubble sizes have a nearly identical appearance. In addition, the fractional losses of SF6 from the drop surface and into the bubble are essentially the same. These results are summarized in Table 2. Conclusions The first model presented, the simple model for drying during morphological development, has potential use in the design of spray dryers. With proper adjustment of the fitted constant, R, it enables a rapid calculation of the time needed to complete drying of a drop once droplet inflation occurs. The second, numerical, model may be used in conjunction with a fluid-dynamical model of a spray dryer to examine the effects of operating conditions on the characteristics of the product powder, such as retention of volatile components. Although the drops modeled in this study were simulated to dry in an isothermal gas stream of a constant velocity and humidity, the program may easily be modified to simulate any trajectory through a field of changing conditions. “Selective diffusion” describes the retention of highly volatile components on the basis that the diffusion coefficient D11 decreases rapidly with increasing dryness relative to D22. Selective diffusion is actually a multicomponent phenomenon. The inward flux of SF6 from cross-diffusion was calculated to be up to several times the outward flux by the convection of water. This result explains the experimental observations (Part 1) that the SF6 was not lost from the mixing of the drop liquid due to bubbling, while the drying rate was enhanced. Even as fresh SF6-bearing liquid was brought to the surface, the loss of SF6 was very small, owing to the rapid development of a water gradient, causing an outward water flux and an inward SF6 flux. The same reasoning may be used to rationalize why the prebubbling losses of SF6 from drops with and without the addition of CMC were not significantly different from one another. After the drop had dried partially, the mean value of the parameter W rose above 1, causing retention of SF6 despite internal circulation. This finding suggests that if the liquid inside drops in a spray dryer could be mixed without the bursting of bubbles, then the drying rate could be enhanced without additional losses of volatile components.
Acknowledgment This research was supported by the National Science Foundation. Nomenclature A ) surface area of drop Ci ) molar concentration of component i Cpi ) specific heat of pure i Dgas ) diffusivity of water vapor through nitrogen Dij ) diffusivity of component i due to a gradient in component j F ) mass flux of water from outside surface of drop h ) heat-transfer coefficient j ) mass flux K ) magnitude of time increment kcond-gas ) thermal conductivity of drying gas kc ) gas-phase mass-transfer coefficient based on a concentration difference kp ) gas-phase mass-transfer coefficient based on a partial pressure difference mi ) total mass of component i in drop ni ) number of moles of component i Nu ) Nusselt number P ) pressure of drying gas Pi ) partial pressure of component i Psat ) saturation vapor pressure Pr ) Prandtl number r ) radial coordinate R ) drop radius Re ) Reynolds number Rgas const ) gas constant Sc ) Schmidt number Sh ) Sherwood number t ) time T ) temperature u ) mass of water/mass of nonvolatile solute u1 ) mass of SF6/mass of nonvolatile solute v ) velocity Vi ) pure component molar volume of i W ) dimensionless parameter; ratio of second and third terms in eq 12 xi ) mole fraction of i in liquid phase z ) solute-fixed mass coordinate Greek Letters ∆Hvap ) latent heat of vaporization of pure water ∆nw,bubble ) change in the number of moles of water vapor in the bubble R ) fitting parameter F ) mass density Fi ) mass concentration of component i ωi ) mass fraction of component i σ ) surface tension γ ) activity coefficient µ ) viscosity τ ) normal stress Subscripts 1 ) component 1: SF6 2 ) component 2: water
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3 ) component 3: nonvolatile solute avg ) average bubble ) of the bubble drop ) of the entire drop i ) component i gas ) of the drying gas far from the drop 0 ) at the drop surface r ) in the r direction R)r ) outside surface of drop s ) sucrose SF6 ) SF6 t ) last time increment t+K ) current time increment w ) water Superscripts • ) corrected for high mass-transfer rates r ) with respect to the reference component velocity v ) with respect to the volume-average velocity ° ) pure component
Literature Cited (1) Van der Lijn, J.; Kerkhof, P. J. A. M.; Rulkens, W. H. Droplet Heat and Mass Transfer Under Spray-Drying Conditions. In International Symposium on Heat and Mass Transfer Problems in Food Engineering; University of Wageningen: Wageningen, The Netherlands, 1972. (2) Kerkhof, P. J. A. M.; Schoeber, W. J. A. H. Theoretical Modelling of the Drying Behavior of Droplets in Spray Drying. In Advances in Preconcentration and Dehydration of Foods, Spicer, A., Ed.; Applied Science Publ.: London, 1974. (3) Wijlhuizen, A. E.; Kerkhof, P. J. A. M.; Bruin, S. Theoretical Study of the Inactivation of Phosphatase During Spray Drying of Skim Milk. Chem. Eng. Sci. 1979, 34, 651. (4) Sano, Y.; Keey, R. B. The Drying of a Spherical Particle Containing Colloidal Material Into a Hollow Sphere. Chem. Eng. Sci. 1982, 37, 881. (5) Furuta, T.; Okazaki, M.; Toei, R.; Crosby, E. J. Formation of Crystals on the Surface of Non-Supported Droplet in Drying. In Drying 82; Mujumdar, A. S., Ed.; Hemisphere Publ. Co.: New York, 1982. (6) Schoeber, W. J. A. H. A Short-Cut Method for the Calculation of Drying Rates in Case of a Concentration Dependent
Diffusion Coefficient. In First International Symposium on Drying; Mujumdar, A. S., Ed.; McGill University: Montreal, Canada, 1978. (7) Hecht, J. P. Influence of the Development of Drop Morphology on Drying Rates and Loss Rates of Volatile Components during Drying, Ph.D. Dissertation, University of California, Berkeley, CA, 1999. (8) Ranz, W. E.; Marshall, W. R. Evaporation from Drops. Part I. Chem. Eng. Prog. 1952, 48 (3), 141. (9) Van der Lijn, J. Simulation of Heat and Mass Transfer in Spray Drying. Ph.D. Dissertation, University of Wageningen, Wageningen, Netherlands, 1976. (10) Lapidus, L. Digital Computation for Chemical Engineers; McGraw-Hill: New York, 1962. (11) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960. (12) Chandrasekaran, S. K.; King, C. J. Multicomponent Diffusion and Vapor-Liquid Equilibria of Dilute Organic Components in Aqueous Sugar Solutions. AIChE J. 1972, 18, 513. (13) Etzel, M. R. Loss of Volatile Trace Organics during Spray Drying. Ph.D. Dissertation, University of California, Berkeley, CA, 1982. (14) English, A. C.; Dole, M. J. Diffusion of Sucrose in Supersaturated Solutions. J. Am. Chem. Soc. 1950, 72, 3261. (15) Henrion, P. N. Diffusion in the Sucrose + Water System. Trans. Faraday Soc. 1964, 60, 72. (16) Schoeber, W. J. A. H. M. Sc. Thesis (in Dutch), Eindhoven University of Technology, Eindhoven, Netherlands, 1973. (17) Verhey, J. G. P. Vacuole Formation in Spray Power Particles 1. Air Incorporation and Bubble Expansion. Neth. Milk Diary J. 1972, 26, 186. (18) Thijssen, H. A. C.; Rulkens, W. H. Retention of Aromas in Drying Food Liquids. De Ingenieur, JRG 1968, 80, Nr. 47. (19) Zakarian, J. A.; King, C. J. Volatiles Loss in the Nozzle Zone during Spray Drying of Emulsions. Ind. Eng. Chem. Process Des. Dev. 1982, 21, 107. (20) Barrer, R. M. Diffusion in Spherical Shells, and a New Method of Measuring the Thermal Diffusivity Constant. Philos. Mag. 1944, 35, 251, 802.
Received for review June 28, 1999 Revised manuscript received October 12, 1999 Accepted October 15, 1999 IE990464+
