Ind. Eng. Chem. Process Des. Dev. 1983, 22, 665-672 Ludtner, R. C.; McConchle. 0. E.; Wills, 0. B. J . Catel. 1973. 28, 63.
Parekh, B. S.; WeHer, S. W. J. Catel. 1977, 47, 100. Stork, W. H. J.; Poit, G. T. R e / . Trav. Chlm. 1977, 96, M105. Thomas, R.; MlttelmelJer-Hazeieger,M. C.; Kerkhoff, F. P. J. M.; Moulijn, J. A.; Medema, J.; de Beer, V. H. J. 3rd Intemetionel Conference on Chembtry and Uses of Molybdenum, 1979; Clhnax Molybdenum Co.: Ann Arbor, p 05.
885
Van Roosmalen. A. J.; Koster, D.; Mol, J. C. J. phvs. Chem. 1980. 84, 2090. Westhoff, R.; MoullJn, J. A. J . Catel. 1977, 46. 414.
Received for review January I, 1983 Accepted March 28, 1983
Limiting Analytical Relationships for Prediction of Spray Dryer Performance Rodger A. Ooffredlt and E. Johansen Crosby" Department of Chemical Engineering, The University of Wisconsin; Madison, Wisconsin 53 706
Tractable analytical relatlonshlps that allow prediction of performance trends for existing spray dryers with well-insulated chambers in which sprays are monodispersed and fine have been developed for certain limiting conditions. Drying mechanisms based on the evaporation of water drops and the diffusion of water in solid particles in combination with cocurrent and completely mixed gas flow were considered. The sensitivity of the drying process to changes In operating variables for directly fired dryers with outlet temperature control is illustrated.
Introduction A major obstacle in the development of predictive procedures for use in the design, scale-up, and performance description of spray dryers is the complexity of the flow patterns and interactions between the drying medium and droplets. As a rault, most of the more detailed procedures require lengthy numerical computations. The work of Edeling (1950)is illustrative of first attempts to consider such flow patterns and interactions. He observed experimentally that high initial drop velocities caused substantial penetration into the dryer before sufficient deceleration occurred to allow the swirling motion of the gas to manifest itself. Relationships were developed to describe the spiral motion of drying drops. However, drop motion and drying were not coupled. Many of the more recently proposed procedures to describe the simultaneous exchange of momentum, mass, and thermal energy between droplets and the drying medium have been reviewed by Crowe (1980).A number of these procedures take the zone of droplet penetration into consideration and allow for coupling of droplet drying and motion. Such techniques yield exact results within the limitations of the models as illustrated by the computations of Keey and Pham (1976).However, lengthy numerical analyses do not give rapid insight relative to expected dryer performance under variable operating conditions. Consequently, it would be advantageous to have a simple system of equations which gives a clear understanding of the drying principles involved, is relatively easy to use, and is sophisticated enough to predict reasonably accurately the interrelationships among the most important design and operating parameters. Gluckert (1962)proposed such relationships for the design of spray dryers equipped with a single atomizer of either rotary, pneumatic, or pressure nozzle variety. These models were based on particle trajectory from the point of drop formation with drying described by likening the largest drop in the spray to a water drop of constant diameter evaporating under stationary 'Power Division, Stone & Webster Engineering Corp., Denver,
co.
0196-4305/83/1122-0665$01.50/0
conditions when exposed to the outlet drying gas conditions. The results were formulated as expected overall heat-transfer rates, and reasonable agreement was reported between prediction and experimental results for small laboratory dryers. The object of this study was to develop a set of limiting but relatively simple and tractable analytical models which would bracket the performance of large, existing spray dryers equipped with single or multiple atomizers which produce sprays consisting of droplet diameters less than ca. 100 pm. Analysis Regardless of the prevailing conditions within a spray dryer, the total transfer of moisture between the spray and the drying medium and of heat between the drying medium, the spray and the surroundings can be determined from the overall mass and energy balances, respectively. However, the drying chamber geometry and size in addition to the operating conditions necessary to meet any given set of product specifications determine the rates at which transfer occurs. The appropriate rate expressions can be combined with the differential forms of the mass and energy balances which upon integration yield a design or performance equation that relates the most important operating parameters to the dryer size. A complete analytical model is, then, a set of three equations composed of a mass balance, an energy balance, and a design or performance relationship. In the present formulation of the performance relationship, the complex mixing effects in the vicinity of the atomizer are neglected and the spray is assumed to be dispersed ideally and instantaneously into the drying medium because of the smallness of the drops. Further, the drops are assumed to be uniform in size and to have negligible velocity relative to the drying medium. Two limiting mechanisms of drying are considered, viz., the evaporation of pure water drops by adiabatic humidification and diffusion-controlleddrying of solid particles. The manner in which each of these mechanisms influences the performance of a well-insulated spray dryer is examined for both cocurrent and mixed-flow operation. 0 1983 American Chemical Society
666
Id.Eng. Chem. Process Des. Dev., Vol. 22, No. 4, 1983
CDRYING GASES IN FEED "
Since the drops are assumed to have no velocity velative to the drying gas, the heat-transfer coefficient is expressed
REFERENCE PLANE "I"
by (Ranz and Marshall, 1952) h = 2kf/D
(9)
at any position in the tower. The surface area available for heat transfer per unit volume of drying gas, a,, is predicted from the number of drops per unit volume of gas and the surface area per drop . N(rD2) a, = (10) ~
Q,
With the volumetric flow rate of the gas, Q,, given by Q
DRYING GASES OUT DRY PRODUCT OUT
Figure 1. Idealized cocurrent spray dryer.
Cocurrent Flow For the idealized cocurrent spray dryer, plug-flow is assumed to prevail with the drying gas experiencing minimal axial mixing and the drops being carried along with the gas. Temperatures, moisture contents, and velocities do not vary with radial position. Such an idealized cocurrent unit is shown in Figure 1. Adiabatic Humidification. In this operation, the drops enter at the top of the dryer with an initial flow rate wL1, an initial diameter D1 and at the adiabatic-saturation temperature of the inlet drying gas, T,. The drying gas enters cocurrently with the drops at a moisture-free rate wG, an inlet temperature T,, and an inlet humidity of Y1. The steady-state overall water balance is w G ( y 2 - y1) = W L l - wL2
At the low pressures ordinarily encountered in spray
wrRT[i~ wJIT
=
P
-
+
-
L :]
= - PMg
(11)
and the number of drops per unit time, N, passing through a control surface at any location in the tower being constant and defined by N = WLl/VlP (12) with V1= ?rD13/6,eq 8 can then be written as
The droplet diameter, D, is related to the gas temperature, T , by application of the overall energy balance, in the form of eq 6, relative to the bottom of the tower M v a p , T , ( w L - w1.2) = WGC,(T - 7'2) (14) in combination with an extension of eq 1 2 wL = wLl(D/DJ3
(15)
M v a p , ~ , [ ~ ~ 1 ( D / -DW1 L) 3~ ] = WGC~(T - 7'2)
(16)
(1)
When the reference temperature, on which the enthalpies are based, is assigned to be the adiabatic-saturation temperature, the steady-state overall energy balance for an adiabatic dryer is wflG2 = wflGl (2)
g
in the form Next it is assumed that the outlet liquid rate, wL2, is zero, i.e., the drops disappear just as they reach the bottom of the drying chamber, and eq 16, with the aid of eq 6, can be simplified to
drying, the gas-phase enthalpy can be estimated reliably by the relationship (3) HG * Cs(T- Ts) + YMvap,T. and eq 2 can be written as C,1(T1 - TS) - Cs2(T2- T J (4) M v a p , T , ( y 2 - Y1) The use of some average value for the humid heat allows eq 4 to be simplified to AHVap,T,(Y2 - Yi) Ca(Ti- T,) (5) Combination of eq 1 and 5 then gives M v a p , T , ( W L l - W L ~ ) WGC,(T, -Tz)
Substitution of eq 17 into eq 13 then gives PM~S ( T , - T2)lI3TdT - ~ ~ ( w L ~ / w G )dz~ ~(18) (T - T2)'I3(T,- T j wGCaRpD12 Equation 18 can now be integrated from T = T I at z = 0 to T = T2at z = 2 to give the performance equation for complete drying when some average value for the molecular weight of the drying medium is assumed.
(6)
To obtain an expression which describes the rate of spray evaporation, an energy balance is applied only to the gas phase; viz. W&G2
- w&Gl
=
W G ( Y Z - Yl)Mvap,T,- JZ(hav)(T- T J S dz (7)
Application of eq 3 as before, use of an average humid heat, and differentiation with respect to the tower height yield
where 13 is a dimensionless temperature defined by
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 4, 1983 667
and the corresponding outlet gas humidity, Y i , results similarly from a mass balance. WL1
ACTUAL
ACTUAL
DRYER
DRYER
VOLUME
VOLUME
Y i = Y1+ WG
i
FICTITIOUS
I I
COMPLETE DRYING
I
I
DRYER
I
EXTENSION
Diffusional Drying. Here the spray entering the top of the dryer is assumed to be composed of spherical particles whose drying rate is governed by the solid phase only. If the droplet diameter, D1, and the time-average, volume-average diffusivity, Deff,are constant, the volumeaverage moisture content of a drop at any time t is (Crank, 1975)
I
FICTITIOUS DRYER EXTENSION SCHEME FOR INCOMPLETE DRYING
Figure 2. Adiabatic humidifier simulation of cocunent spray dryer performance.
Examination of eq 19 reveals that such major industrially controllable operating parameters as mean inlet drop size, inlet and outlet gas temperatures, ratio of feed to gas flow rates, and tower volume are contained in this result. The inlet gas humidity, which is generally not controllable, is implicit in the adiabatic-saturation temperature, T,. Although eq 19 is valid only for complete drying, it may be applied for incomplete drying when a fictitious extension is added to the dryer as shown in Figure 2. The drops would disappear completely at the bottom of this extension while they would have a finite diameter at the bottom of the dryer proper. Hence, incomplete droplet evaporation in the dryer can be described by the difference between application of eq 18 over the dryer plus the fictitious extension and that over the fictitious extension alone. In both applications, the outlet gas temperature is the outlet temperature from the fictitious dryer extension, Ti, and, therefore, the final design equation for the idealized cocurrent adiabatic-humidification is 12(W~i/W~)kfpM,oMg(Sa
-
WGCsRPDl
For drying times required to give an unaccomplished moisture level of ca. 0.7 or less, only the first term of the series in eq 23 is significant and the drying rate is given by
The effect of the moisture content of the drying medium in which the particles are suspended manifests itself through the equilibrium-moisture content, Re. Unlike the adiabatic-humidification process, the temperature of the drops in diffusional drying is not constant. In order to obtain a tractable solution, the droplet temperature is assumed to equal that of the drying medium surrounding the drop. The overall moisture balance for the dryer is given by
- yl) = wS(Q1 - Q2)
wG(y2
(25)
and when heats of solution, adsorption, etc. are neglected, the overall energy balance for the dryer can be written as
If the solid and liquid heat capacities are assumed to remain constant, and the reference temperature is set equal to the inlet gas and drop temperature, eq 26 reduces to c
and combination of eq 25 and 27 gives c
where
e,,) and 6 ( 2 ) are defined by (m) (m) T1 T i
=
T2
=
T2'
To follow the drying progress, an expression relating the operating temperature to axial location in the tower results from
'1'
The outlet gas temperature from the fictitious dryer extension follows from an energy balance written over the dryer plus fictitious extension
The solids moisture content can be obtained by replacing T2and Q2 with T and Q in eq 28 which, upon differentiation, yields d T - m v a p , T 1 + CpL(T1- T ) _ dQ - ;cs + c,s + QC,L
(
(30)
668
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 4, 1983
d e D IN
-B(T1 - 2'2)' -E(T1 - TJ F
2B
+F
1-
1
+ E - G1/2](E+ [2B(T1 - T2) + E + G'I2](E - G1/2) [2B(T1 - T2)
-
b
D
R
-REFERENCE PLANE "2"
(36)
where B, E, F, and G are constants defined by B = CpL[
DRYING GASES OUT Y PRODUCT OUT
( 2+
m) x
Figure 3. Idealized mixed-flow spray dryer.
From the local volumetric flow rate and the tower crosssectional area it results that
( t : )+ (
mCpL 2Y1+
C, + -C,S
If the equilibrium-moisture curve is assumed to be linear, the equilibrium-moisture content of the particle can be defined by 52, = mY
(32)
where m is somewhat temperature dependent. Equation 32, when combined with eq 25, becomes R,=m
( 1: ) 1: Y1+-Ql
-m-Q
(33)
and subsequent substitution of this result into eq 24 gives
(34) The mositure content, 52, is eliminated from eq 30 and 34 by application of the energy balance, eq 28, written in the form used to obtain eq 30. Substitution of eq 30,31, and 34 as modified into eq 29 then gives
-dT- - - 4 ~ ~ 0 , f f P MX~ S dz
w$TD12
Hvap,T1
+ CpL(T1 - T )
and G = 4BF
+ E2
(36d)
All the important operating parameters are contained in the design equation or can be calculated from the mass and energy balances.
Mixed Flow For an idealized mixed-flow spray dryer, perfect mixing is assumed; i.e., the outlet gas-phase conditions are identical with those inside the dryer where no temperature or humidity variations exist. This idealized dryer is known in Figure 3. Adiabatic Humidification. In this case, the evaporative load and energy requirements are the same as for the cocurrent case with the overall mass and energy balances being identical. Therefore eq l through 6 are again applicable. Even though the drops are initially uniform in size, their diameters will vary throughout the dryer as some drops will stay in the dryer a short period of time while others will remain until they have evaporated completely because of the mixing process. For a drop having an initial diameter D1and a temperature constant at T,, experiencing zero velocity relative to the gas and being exposed to an environment of constant temperature T2and humidity Y2, the evaporation history varies according to the relation (Marshall, 1954)
+ CpS + nlcpL (37)
The mean residence time, 7,of any element of gas in a well-mixed dryer operating at steady state is defined by (35) Integration of eq 35 over the dryer yields
j-=-
sz =- PSZ Qg
w&Tz
1
(38)
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 4, 1983
and the residencetime distribution for the gas in the dryer is given by (Levenspiel, 1972) E(t) = 1 (39) 7
The mean emerging drop diameter, &, is the average of all the drop diameters found in the effluent gas of which each element has a unique residence time. The mass of liquid effluent is related to the average of the drop volumes bY
669
The average moisture content of all the particles leaving the dryer, can be obtained by integrating the moisture content of any given particle over the entire residence-time distribution of the drying gas, viz.
G,
n2 =
(47) Evaluation of eq 47 yields
Summation of the drop volumes over the gas residencetime distribution then gives
After a drop has been in the dryer for a certain time, t,, it will disappear and will no longer contribute any mass to the outlet liquid flow rate, wL2. It then follows that eq 41 can be written as
(42) with the right-had integral equa to zero. Elimination of the droplet diameter between eq 37 and 42 gives
which, upon integration, yields
where A is defined by
The average inlet drop size, dryer operating temperature, extent of drying, gas flow rate, and dryer volume are contained explicitly in this performance equation. Again, the inlet temperature and humidity of the drying gas are implicit in the adiabatic-saturation temperature. Diffusional Drying. As the evaporative load and energy requirements are the same as for cocurrent drying, the overall mass and energy balances are given by eq 25 through 28. The evaporation rate, however, is different because of the differences in the equilibrium-moisture levels during the courses of drying. As the residence times of the particles within the dryer vary, the outlet moisture contents of the individual particles also vary. Integration of eq 24 gives a moderately conservative estimate of droplet moisture content as a function of time.
Since the drops are dispersed in a drying gas of constant temperature T2and humidity Y2,the equilibrium-moisture content is also constant and according to eq 32 Re = my2 (46)
Substitution of the expression for the average gas-phase residence time in terms of eq 38 gives the performance equation in the form 4r2D&'Mgz(Sz) WJID12T2
521
-
02
-ne
=-
(49)
The average outlet and equilibrium-moisture content of the solids may be eliminated with the aid of eq 25,26, and 46 with the reference temperature set equal to the outlet gas and drop temperature, and eq 49 becomes 4r2DeffPMg2 (Sz) -
B(T1- 7'2) mY1)AHvap,T2 - BE(T1 - T2) W&Dl2T2 (50) The constants B and E are the same as defined in eq 36a and 36b. Performance Predictions The predictive nature of the previously developed relationships is illustrated by examination of the performance to be expected for direct fired dryers operating under somewhat extreme summer and winter conditions. Summer conditions are taken as 38 "C (100 OF) and 0.03 kg of water/kg of dry air corresponding to ca. 70% relative humidity, and winter conditions are taken as -23 "C (-9 OF) and 0.003 kg of water/kg of dry air. The fuel is considered to be pure methane which, for complete combustion, raises the humidity of the resulting drying gases about 0.0048 kg of water/kg of dry gas for every increase of 100 O C in gas temperature (0.0027 lb, of water/lb, of dry gas for every increas of 100 OF). For purposes of comparison, a dryer volume of 28.3 m3 (1000 ft3) is considered. For convenience, the dry gas rate is specified, rather than the inlet or outlet total gas rate, at 9.1 kg/s (20 lb,/s) and the outlet temperature is maintained at 79 O C (175 OF). The physical properties of the drying gas are assumed to be those of an air-diluted combustion gas. Adiabatic Humidification The spray drying of moderately dilute slurries and non-film-forming solutions closely approximates the adiabatic humidification process. For these feedstocks, the drop temperature will attain the adiabatic saturation temperature of the dry gas for aqueous-aerial systems after only very minor drying has occurred. The mass and energy balances which must be satisfied for any dryer configuration in conjunction with the appropriate performance equation are given by eq 1 and 5, respectively. For complete evaporation in an adiabatic humidifier with a cocurrent flow pattern, the design equation is given by eq 19. The inlet drying gas conditions determine the adiabatic-saturation temperature. Knowledge of the outlet
-
(QI -
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 4, 1983
670
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INLET GAS TEMPERATURE, T I ,
2
01 02 03 04 05 06 07 RESIDUAL MOISTURE LEVEL,wu/wLI, DiMENSiONLESS
Figure 6. Predicted conditions for droplet evaporation in mixedflow spray dryer.
OC(OFI
Figure 4. Predicted conditions for complete droplet evaporation in cocurrent spray dryer.
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,
,
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Oi 02 03 04 05 06 07 RESIDUAL MOISTURE LEVEL,w&wLl, DIMENSIONLESS
Figure 5. Predicted conditions for incomplete droplet evaporation in cocurrent spray dryer.
gas temperature, dry-gas flow rate, and tower volume then allows calculation of the outlet humidity, feed rate, and inlet drop diameter with the energy balance, mass balance, and design equation, respectively. Illustrative performance predictions are presented in Figure 4. Because the humidity is lower in the winter than in the summer, larger drops can be evaporated at either equal inlet gas temperatures or equal evaporative leads. For a constant evaporative load, lower inlet gas temperature is possible in the summer if the drop size is reduced somewhat. This is a consequence of a higher humid heat resulting from a higher humidity and a lower heat of vaporization owing to a higher adiabatic saturation temperature. For incomplete evaporation of water sprays in a cocurrent unit, the design equation is given by eq 20. The outlet gas humidity and initial drop size are calculated as in the case of complete evaporation. In this case, however, the feed rate is specified at the beginning of the calculations and the mass balance is used to determine the amount of residual moisture leaving the dryer. The performance predictions shown in Figure 5 suggest again that larger drops can be evaporated while higher inlet gas temperatures are required during the winter as compared to summer for any specified evaporative load. As for complete evaporation, an increase in evaporative load requires an increase in inlet air temperature and simultaneously allows an increase in drop size because the evaporation rate increases faster than the available evaporation time decreases. Complete evaporation in an adiabatic humidifier with well-mixed flow is not to be expected because of the negligible residence time experienced by some of the drops. For this flow pattern, the design equation is given by eq 44. The computational strategy is the same as that for
' 0
0.1 01 0.2 0 2 0.3 0 3 0.4 0 4 0 .5 0.6 0 6 0.7 0 7 RESIDUAL 01,DIMENSIONLESS RESIDUAL MOISTURE MOISTURE L LE EV VE EL L ,, 02/ 02/ 01, DIMENSIONLESS
0.0 0 0
Figure 7. Predicted conditionsfor particle drying in cocurrent spray dryer.
incomplete evaporation with cocurrent flow and some results of these computations are shown in Figure 6. Comparison of Figures 5 and 6 indicates that smaller drops are required for mixed flow than for cocurrent flow and this requirement increases markedly as the specified residual moisture level decreases. An increase in evaporative load requires a decrease in drop size because the evaporation rate decreases relative to the available evaporation time. Diffusional Drying When concentrated non-film-forming solutions and solution/slurry feedstocks are spray dried, the drying mechanism can approach diffusional drying. As the drying rates are rather low in comparison to the heat-transfer rates, the temperature of the drying particles will rapidly approach and remain near that of the drying gas within the dryer. It seems reasonable, therefore, to consider that the drops and the drying gases quickly and nearly attain thermal equilibrium before any major portion of the drying is accomplished. The resulting temperature then will be that of both the inlet drying gas and atomized feedstock. For this drying mechanism, the forms of the mass and energy balances used to predict the changes in humidity and temperature of the drying gas are given by eq 25 and 28, respectively. A feed having a temperature of 49 OC (120 O F ) and containing 70% solids is considered for illustration. The effect of the humidity of the drying gas manifests itself through the equilibrium-moisture content as indicated by eq 32, and a value of m for a weakly hygroscopic material of 0.39 kg of dry gas/kg of dry solid is considered. The heat capacity of the solid is taken as 0.4 kcal/kg "C and the heat capacity of the liquid is taken as that of water. A volume-averagediffusivity of 9.3 x m2/s (1.0 x ft2/s) is assumed as being reasonable (Crosby and Weyl, 1977). The rate equation for diffusional drying as given by eq 23 prescribes that either an infinite time or a zero inlet particle diameter is required to reduce the volumeaverage moisture content of the solids to its equilibrium value. Complete drying is, therefore, not to be expected.
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 4, 1983 871
0 i I
g
5100, O l
00 01 0.2 03 04 RESIDUAL MOISTURE LEVEL,
0.5
06
07
n2 nl, DIMENSIONLESS
Figure 8. Predicted conditions for particle drying in mixed-flow spray dryer.
Results for cocurrent drying, for which the performance equation is given by eq 36, are shown in Figure 7. The computational strategy was the same as that used for cocurrent adiabatic humidification. These results reflect the seasonal variations in evaporative load predicted by the energy balance and show that somewhat higher inlet gas temperatures must be used in the winter to dry the solids to the same degree as in the summer. Somewhat larger drops can be dried in the winter to any specific residual moisture level because the humidity of the drying gas is lower. The drops must be very small if the moisture content of the final product is to approach the equilibrium value. At such moisture levels, any net transfer of moisture from the solids to the gas is extremely small. Any increase in feed rate, i.e., increase in evaporative load, requires a reduction in drop size and restricts the minimum moisture level of the product to a higher value. The performance equation for mixed flow operation is given by eq 50 and results for conditions identical with those for cocurrent drying are shown in Figure 8. Performance results similar to those given in Figure 7 are indicated. However, considerably smaller drops are required for mixed flow operations than for cocurrent flow. Summary Tractable design equations based on limiting drying mechanisms and idealized flow patterns have been proposed for use in conjunction with the mass and energy balances to describe the operational characteristics of existing spray dryers. The manner in which these relationships are used to evaluate and compare dryer performance under different controllable and noncontrollableconditions was illustrated. Inspection of the derived relationships indicates the manner in which the operating variables are interrelated and the general inflexibility encountered in spray dryer operation. Dryer operation is quite sensitive to the atomization process and a close interdependence exists between drop size and all other operating variables. Rather modest changes in the drop size can have a significant effect on the final moisture content of the product. The uncontrolled increase in humidity during summer operation requires some reduction in drop size if the product moisture content is to remain unchanged. In this case, a simple increase in the inlet temperature of the drying gas is not adequate as the mean residence time of the drops and particles in the drying chamber is reduced more than the drying rate is increased. The relationships which have been developed are concerned only with the moisture removal capability of a dryer. All product properties other than moisture content are subject to the conditions required for adequate drying. Consequently, variation in particle size and density, bulk density, particle texture, certain physicochemical properties and/or degree of thermal degradation would be
expected to vary as the necessary drying conditions are varied. In addition to prediction of the performance trends of existing spray dryers, the developed relationships may be useful in the scale-up and initial deisgn of new units. Laboratory and many pilot-plant dryers require a finer degree of atomization and operate under quite different drying conditions than desired for a commercial unit. Even though the flow patterns may differ considerably between small pilot dryers and large commercial dryers, with said flow patterns in neither case being ideally cocurrent or mixed flow, judicious application of the two limiting flow patterns might allow a reasonable estimation of the dryer volume and required operating conditions. Acknowledgment The authors wish to express their appreciation to the Shell Development Company, the Walter B. Schulte Trust, and the University of Wisconsin-Madison Engineering Experiment Station for financial assistance during the course of this study, Professor R. C. Reid for his meticulous review, and G. Renno for checking the analyses. Nomenclature Dimensions are given in terms of mass (M), length ( L ) ,time ( t ) ,and temperature (T). A = constant defined by eq 44a, dimensionless a, = surface area available for heat transfer per unit volume of dryer, 1 / L B = constant defined by eq 36a, L 4 / t 4 P C = heat capacity at constant pressure, per unit mass, L 2 / t 2 T = gas-phase humid heat, L 2 / t 2 T D = water droplet or solid particle diameter, L Defi= effective volume-averaged diffusivity of moisture in solid particles, L 2 / t E = constant defined by eq 36b, L 4 / t 4 T E ( t ) = residence-timedistribution function defined by eq 39, dimensionless F = constant defined by eq 36c, L4 t4 G = constant defined by eq 36d, L / t g P H = enthalpy per unit mass, L21t2 AH,,,, = heat of vaporization, L / t 2 h = heat-transfer coefficient, M / t 3 T Izf = thermal conductivity of gas evaluated at average “film” temperature, M L / t 3 T M = molecular weight, M/mol m = slope of equilibrium-moisture curve, eq 32, M / M N = number of drops per time crossing a control surface, l / t n = index, dimensionless P = total pressure, M / L t 2 Q = volumetric flow rate, L 3 / t R = gas constant, M L 2 / t 2 Tmol S = spray-dryer cross-sectional area, L2 T = temperature, T t = time, t t , = critical residence time, t V = droplet volume, L3 w = mass flow rate, M / t Y = gas-phase absolute humidity, M / M 2 = spray-dryer tower height between reference planes, L z = height, L Greek Letters 8 = dimensionless temperature defined in eq 19a, 20a, and 20b, dimensionless p = liquid density, M / L 3 T = mean residence time, eq 38, t 0 = volume-average moisture content, dimensionless 0, = equilibrium-moisture content, dimensionless Overbars _ -- time-averaged value
4
Q
672
Ind. Eng. Chem. Process Des. Dev. 1983, 22, 672-676
Superscripts
Crosby, E. J.; Weyl, R. W. AICMSymp. Ser. 1977, 7 3 , 82-94. Crowe, C. T. "Advances In Drying"; Mujumdar, A. S., Hemisphere Publishing Corp.: New York, 1980; Vol. 1, pp 63-99. Edellng. C. Angew. Chem. 1850, Beiheffe, N r . 57, 24-51. Gluckert, F. A. A I C M J. 1862, 8 , 480-466. Keev. R. B.: Pham. 9. T. Cbem. €no. (London) 1976. No. 311. 516-521. Levenspiel, 0. "Chemical Reaction Eigineerlng"; 2nd ed.; Wiley: New York, 1972; pp 255-260. Marshall, W. R., Jr. Chem. Eng. Prog. Mormgr. Ser. 1954, 50, 89. Ranz, W. E.; Marshall, W. R., Jr. Chem. Eng. h o g . 1852, 4 8 , 141-146, 173-180.
' = outlet condition from fictitious extension to dryer volume in a cocurrent adiabatic humidification
Subscripts G = pertaining to bone-dry drying medium g = pertaining to gas-vapor drying medium
L = pertaining to pure liquid phase S = pertaining to bone-dry solid phase s = quantity evaluated at saturation 1 = quantity evaluated at inlet conditions 2 = quantity evaluated at outlet conditions Literature Cited
Receiued for review January 13, 1981 Reuised manuscript received January 10,1983 Accepted February 15, 1983
Crank, J. "The Mathematics of Diffusion", 2nd ed.;Clarendon Press: Oxford, 1975: p 96.
A Method for the Calculation of Gas-Liquid Critical Temperatures and Pressures of Multicomponent Mixtures Amyn S. Tela,' Kul 6. Garg, and Richard L. Smlth School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332
The critical properties of 75 binary systems have been correlated by use of a modified two-parameter Wilson equation. The great advantage of using the Wilson equation is that Its generalization to multicomponent systems is straightforward and requires binary parameters only. Extensive tables of binary parameters are given in this work. Comparisons between experimental and predicted critical properties for 6 1 multicomponent systems are shown. The new method is accurate, it is simple to use, and it requires considerably less computational effort than rigorous methods which solve the equations for the Gibbs criteria at the critical point.
The method reported in this paper makes use of the concept of an excess critical property introduced by Etter and Kay in 1961. Etter and Kay demonstrated that this concept is useful in studying the effect of shape, size, and the chemical nature of the components on the critical properties of mixtures. They also obtained correlations for the excess critical properties for various classes of mixtures. We have chosen to fit a modified Wilson (1964) equation to sets of binary critical data obtained from the literature. The great attraction of the method is that it can very easily be extended to multicomponent mixtures using parameters obtained from the binary pairs only. No ternary or higher mixture data are required in the calculations. Moreover, once the binary parameters have been evaluated, the prediction of multicomponent critical properties requires no iterations. There is therefore a considerable saving in computation time. A possible further advantage is that the method may prove amenable to a group contribution approach (analogous to the use of ASOG or UNIFAC to obtain excess Gibbs energies). This would result in a powerful technique for the prediction of critical states from data on a limited number of binary mixtures.
A knowledge of the gas-liquid critical states of mixtures is of great practical importance, especially in hydrocarbon processing applications. Many hydrocarbon processing operations take place at high pressures and involve retrograde phenomena which are characteristic of the behavior of mixtures in the critical region. In addition, phase equilibrium calculations and fluid property predictions are difficult to make in the critical region. It is therefore often necessary to locate the critical point of a multicomponent system prior to carrying out other calculations in this region. Attempts to predict critical properties have relied primarily on two approaches: (i) empirical methods involving the use of excess properties (Etter and Kay, 1961) or fitted correction factors (Chueh and Prausnitz, 1967);(ii) rigorous methods involving the solution of Gibbs criteria for the critical point in a mixture. These latter methods have used either equations of state (Peng and Robinson, 1977; Heidemann and Khalil, 1980; Michelsen, 1980) or the Corresponding States Principle (Teja and Rowlinson, 1973) to obtain the Gibbs energy and its derivatives required in the calculations. Although the second approach is preferable to the first because of its basis in thermodynamics, the rigorous methods do not always lead to a solution and they may sometimes require a large amount of computing time. This is likely to prove disadvantageous in some simulation calculations. Moreover, methods which use simple equations of state do not predict all critical properties accurately, with the predicted critical volume showing the greatest deviations. 0196-4305/83/ 1122-0672$01.50/0
The Modified Wilson Equation for Excess Critical Properties Etter and Kay (1961) defined an excess critical property 4: as 4: = 4 c - 4cIDEAL where 4 is T, P, or V , & is the critical property of the 0
1983 American Chemical Society
