Ind. Eng. Chem. Res. 1995,34, 3289-3302
3289
Mathematical Models of Cocurrent Spray Drying Antoine NegizJ Eric S. Lagergren, and Ali Cinar**+ Computing and Applied Mathematics Laboratory, Statistical Engineering Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899
A steady state mathematical model for a cocurrent spray dryer is developed. The model includes the mass, momentum, and energy balances for a single drying droplet as well as the total energy and mass balances of the drying medium. A log normal droplet size distribution is assumed to hold at the exit of the twin-fluid atomizer located at the top of the drying chamber. The discretization of this log normal distribution with a certain number of bins yields a system of nonlinear coupled first-order differential equations as a function of the axial distance of the drying chamber. This system of equations is used to compute the axial changes in droplet diameter, density, velocity, moisture, and temperature for the droplets at each representative bin. Furthermore, the distributions of important process parameters such as droplet moisture content, diameter, density, and temperature are also obtainable along the length of the chamber. On the basis of the developed model, a constrained nonlinear optimization problem is solved, where the exit particle moisture content is minimized with respect to the process inputs subjected to a fixed mean particle diameter at the chamber exit. Response surface studies based on empirical models are also performed to illustrate the effectiveness of these techniques in achieving the optimal solution when a n a priori model is not available. The structure of empirical models obtained from the model is shown to be in agreement with the structure of the empirical models obtained from the experimental studies.
I. Introduction Spray drying transforms a liquid feed (slurry) material t o a dry powder (particles, particle agglomerate) by spraying the feed into a hot drying medium. Spraying is done with an atomizer where the fluid feed is broken into a large number of small droplets. These droplets generally assume a spherical shape due to surface tension effects. This large surface area-to-volume ratio provides the immense driving force for a rapid evaporation of the water from the drying droplets resulting in the formation of particles (powder)which are separated from the hot exhaust air. The spray drylng process has a variety of applications in food, ceramics, chemical, and pharmaceutical industries. It is essential t o have a good understanding of the effects of the process inputs on the final characteristics of the dried product particles for many reasons. Good process knowledge would lead to better productivity, low operational costs, and improved quality of the final product. For example, in the case of ceramic industries, it is important that the powder has a certain mean particle diameter and minimum moisture content in order to ensure a flowable product with uniform bulk density (Lukasiewicz, 1989). It is well-known that these critical product characteristics are functions of slurry properties (such as surface tension, viscosity, density, additives, and solids content), atomization technique, bulk flow and temperature of the drying air (dropletair mixing), and liquid slurry feed rate (Masters, 1985). In addition to the droplets exhibiting evaporation, collision, breakups, and possible agglomeration among themselves, the exchange of heat, momentum, and mass between the droplets and the drying medium complicate the modeling problem. There are mainly two classes of models, the steady state and the dynamic models. The
* Author to whom correspondence should be addressed. t Permanent address: Department of Chemical Engineering,
Illinois Institute of Technology, Chicago, IL 60616. E-mail:
[email protected],
[email protected].
dynamic models attempt to explain the transient behavior of the temperatures, moisture profiles of the drying air,and the droplets in the chamber. Two recent but very similar approaches by Zbicinski et al. (1988) and Clement et al. (1991) treated the drying chamber as a well-mixed continuous stirred tank reactor. In both studies, the resulting system of first-order differential equations with respect to time was solved for the exit temperature, moisture content of the particles, and the drying air. The probabilistic behavior of the droplet size was ignored in both studies in order t o avoid a more intensive computational effort. A more detailed dynamic mathematical model and some comparisons with experimental results were reported by Papadakis and King (1988a,b). In their study, the particle-source-in cell (PSI-CELL) model proposed for gas-droplet flows by Crowe et al. (1977) was adapted to the spray drying system. The model included transient behavior in two spatial dimensions. Although their model accommodated different particle sizes, primary focus was on the temperature and moisture content of the drying air rather than particle size distribution, since only those variables were measurable in their experimental system. OaMey and Bahu (1993) report a three-dimensional simulation using the computational fluid dynamics code FLOW3D enhanced by a discrete droplet and drying model which is an implementation of the PSICELL model. They propose additional research t o be done to verify the performance of their model. They advocate the use of phase Doppler anemometry which is the same method that is used in the experimental studies reported in this manuscript. Dynamic models were considered mainly for process control. However, since a stable operating condition is always desired for the process t o be operated, models regarding the steady state behavior of the process also received attention. An overview of this type of model is given by Crowe (1980). The steady state models were classified as one-dimensional models considering only
0888-5885l95/2634-3289$Q9.0QlQ 0 1995 American Chemical Society
3290 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 the axial direction and two-dimensional models that consider the axial and radial directions of the drying chamber. The present paper considers an axial steady state formulation that is a modified version of the work of Parti and Palhcz (1974). The main modification is the addition of a droplet size distribution from the tip of the atomizer that allows one to predict distributions of droplet diameters, temperatures, moisture content, densities, and velocities along the axial direction. The aim is t o simulate a cylindrical spray dryer that dries ceramic slurries with a twin-fluid atomizer a t the top of the chamber. The model is introduced first. A dual optimization problem is posed based on the derived model where the exit moisture content of the powder is minimized subject to a fEed mean particle diameter at the exit of the drying chamber. This problem is also referred t o as a response surface problem in the area of experimental design (Box et al., 1978)where an apriori detailed model is replaced with an empirical model obtained from statistically designed experiments. The dual response problem is solved with both a nonlinear constrained optimization algorithm and a response surface analysis technique to demonstrate compatibility of the two approaches. The empirical models obtained using the proposed model are also compared to the empirical models obtained from experimental data. 11. Formulation of the Model
The proposed model assumes no distinguished zones inside the chamber. The zones of nozzle and free entrainment are modeled with the same set of momentum balances for a drying droplet. The drying air and the slurry flow are cocurrent. Droplet properties are assumed t o vary only along the axial direction. In the dryer, air temperature and moisture content are constant for a cross-sectional area at a fixed axial location. Models of drying processes that are based on assumptions of constant velocity profiles across axial cross sections are also categorized as flash drying. The moisture and temperature profiles inside a single spherical droplet are neglected. The stochastic description of possible droplet collisions, breakups, and agglomerations is excluded from the model. On the basis of the above assumptions, the mass balance for the moisture content of the droplet W, as kilograms of moisture per kilogram of dry solids can be given as
where h is the axial distance from the tip of the nozzle, n,is the mass of the dry solids in the droplet, d, is the spherical diameter of the droplet, up is the velocity of the droplet with respect to the chamber wall, and mv is the rate of evaporation from the droplet. The rate of evaporation from the droplet is
where B represents the mass transfer coefficient of the water vapor through the air (kg vapor/m2 s) and Xi (kg moisturekg dry air) is the relative humidity of the air with respect to the vapor pressure of water calculated a t the droplet temperature. SimilarlyXb is the relative humidity of the bulk air.
The conservation of energy for the single drying droplet with mass ms becomes u pm
d -[(cp,
+ W,C,JT,
- T ~ ~ ~mi,2a(Tg ) I = - T,)-
[Mref + C,JT, - ~~,~)1nci,2rn, (3) The symbols c,,, c, and c,, represent the heat capacities in kJ/(kg K) of the dry solid, water, and the water vapor, respectively. a is the heat transfer coefficient as kJ/(m2 K SI. The heat and mass transfer coefficients are obtained from the Reynolds and Nusselt numbers as
ad
$=Nu = 2 + 0.6(Re)0.5(Pr)0.33
(4)
Kair
Pr is the Prandtl number
where pair, kair, and c,,, are the viscosity, the thermal conductivity, and the heat capacity of air in their appropriate units. The Reynolds number (Re)is calculated as
Re = QslrdpUp'Pslr
(6)
where esk is the density of the sluny. The mass transfer coefficient is calculated from the heat transfer coefficient according to (7) where D e R is the effective diffisivity of water vapor in the air as m2/s, @air is the density of air at the current conditions obtained using the ideal gas law, Tgis the bulk air temperature, Tmf is an arbitrary reference temperature for the enthalpies to be calculated, and AHrefis the latent heat of vaporization per kilogram of water at the reference temperature. The above equations are written for a certain droplet diameter d,. However, a log normal size distribution on the droplet diameters a t the exit of the twin-fluid atomizer is assumed t o hold with a certain mean and variance. Experimental studies reported by Lefebvre (1991) indicate that the mean of this log normal distribution is related t o the atomization air velocity u as d s , the slurry feed rate F as kg/s, the viscosity of the slurry psi,. as (Pa s), the density of the slurry eslr as kg/m3, the surface tension of the slurry Oslr as kg/s2 and the initial moisture content of the slurry W,,by eq 8.
c=
0.15 0.1 0.15
(F(1+ Wp,,)) Pslroslr 1.05 0.5
(8)
VQslr
denotes the mean droplet diameter at the
where 2, denotes the alOOth percentile point of a standard -(zero mean, unit variance) normal distribution and In d , and qnd, denote the mean and the standard deviation of the logarithm of the diameters, respectively. q, d, is calculated by assuming that 95% of all droplets
have diameters less than 2 c (i.e., dPa=0,9S = 2dp,) and making use of eq 9 with the 95 percentile standard normal variate. The computation requires - a simple quadratic solution for 9 n d p . Having d, and q n d , gives the mean for the logarithm of the diameters as (Dudewicz and Mishra, 1988, p 352)
In d, = ln(d,m)
Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 3291
(13) The droplet diameter at the first stage is calculated using
(10)
(14)
In@,) is normal with mean lnd, and standard deviation a n d p . The histogram of the droplet size distribution is developed by specifying the number of bins. The droplet diameter corresponding to the midrange of each bin is taken to be the representative diameter for that particular bin. The total number of droplets per unit time that enter the chamber can be obtained by averaging over the volumes as
where the d,, designates the initial diameter of the droplet at the nozzle exit. This equation is obtained from the relation between the spherical volumes and the densities. The change in the velocity of the droplet is accounted for by using a momentum balance for the droplet. The net forces acting on the droplet constitute the downward gravity force, the upward friction, and buoyant forces. The resulting net force is balanced with the acceleration of the droplet resulting in
1 2 - pn dp
The distribution for the -
NtQt=
F n 3
es,ro[2(ln(dp))gdp
(11)
d(ln(d,))
The symbol p(ln(d,)) represents the normal point density function with respect t o ln(dp). The term p(ln(d,)) d(ln(d,)) for each bin gives the fraction of the total number of droplets that belong to that specific bin. Furthermore, the frequency distribution for the number of droplets according to their sizes is readily obtained from multiplying those fractions by the total number of droplets entering the chamber (Nt,t). The above integral is approximated by adding the product of the area occupied by each bin under the normal probability curve with its representative volume. With this treatment, the above given mass and energy balance equations are applied for each bin with its representative diameter to assure continuity in the integration process. The shrinkage of the droplet is accounted for by using a shrinking balloon approach. Since the density of each spherical droplet is subjected to change due to different stages in the evaporation process, a mechanism is needed to incorporate those stages in the model (Parti and Palhcz, 1974). The first stage is the shrinking balloon stage, where the rate of evaporation is balanced with the transfer of moisture from the center to the surface of the droplet. During this stage the density of the droplet will increase as the diameter decreases due to the rapid evaporation. The density of the droplet in this stage is given as
dv Qp - epair f @air -- -3 -(uair dhp -g @PUP
(15)
where Vair is the velocity of the drying air, up is the droplet velocity, and f is the friction factor also known as the drag coefficient which is a function of the Reynolds number defined earlier. The above set of equations describes the drying phenomenon as far as a single droplet is concerned. The model is completed by considering the mass and energy balances for the drying medium which in this case is air. The mass balance for the air moisture content is given as
G-=
dh
m&p2 droplets
(16)
Up
where G is the mass flow based on the dry air (kg/s). The summation on the right-hand side indicates the total contribution t o the mass transfer from each individual droplet. Since the droplets are already characterized into groups that are represented with a single droplet diameter, the number of terms in this summation is equal t o the number of bins selected to discretize the continuous log normal distribution. The energy balance for the drying medium with the same reasoning on the summation operation is given as
C where eeoland @,at are the pure densities of solids and water, respectively. After the droplet reaches its critical moisture content W,,, which is an adjustable parameter in the model, the droplet density starts to decrease. W,, is the moisture content when the transport of water from the inner parts of the drying droplet to its outer surface becomes the rate-limiting step. As a result, a solid layer forms at the outermost surface of the droplet. From this point forward, due t o the partial surface solidification, the diameter of the droplet remains essentially constant while its porosity starts to increase, resulting in a decrease in the droplet density. The density of the droplet a t this stage is calculated from
-
dP@PVP
droplets
nd;
-a(Tg
- Tp>(17)
up
The term hair refers to the enthalpy content (kJ/kg) of the moist air at its relative humidity and temperature. This enthalpy can be given as
The resulting equations for the droplets (four equations for each bin) and the two equations for the drying air form a system of coupled first-order differential equations to be solved for the droplet temperature (Tp), moisture (W,), density (ep),and velocity (up) and the air
3292 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 moisture ( x b ) and temperature (T,) along the axial length of the chamber. The droplet diameters are updated explicitly from these variables. The differential equations are solved using Gear's method with backward differentiation which is applicable to both stiff and nonstiff ordinary differential equations. The complete discussion of the numerical algorithm and its implementation as a FORTRAN subroutine (DDRIV) are given in Kahaner et al. (1989). If uniform droplet properties are assumed at the exit of the atomizer, the initial value problem described will require six boundary conditions a t the inlet region of the chamber. The inlet of the drying chamber is considered to be the tip of the twin-fluid atomizer. The boundary conditions required are the moisture content of the slurry feed W,,, the slurry feed temperature Tp,, the moisture content of the drying air fed into the chamber xb,,the temperature of the drying air Tgo,the velocity of the droplets exiting the nozzle upo (up, = Uair 0.04~1,and the density of the droplets exiting the nozzle e,,. The Factors Involved in the Model (Inputs). The drying chamber is cylindrical with both its diameter and length being equal to 0.8 m. For the air: @air = 1.0 kg/ m3, Cpair = 0.24 kcal/(kg K), pdr = 1812 lo-* Pa S, and Kair = 0.717 kcal/(s m K). For the vapor: c,, = 0.46 kcal/(kg K). For the dry powder: c,, = 0.3 kcal/(kg K) and gsol = 2500 kg/m3. For the slurry: the density of the slurry a t the inlet is found from eq 12 as a function of the inlet moisture content (W,,). For a fured sized chamber the following are considered as inputs with their respective ranges to the spray drying model. (1)The moisture content of the slurry feed, W,, (0.25-1.2 kg moisturekg dry solids). (2) The slurry feed temperature, Tpo (20-45 "C). (3) The moisture content of the drying air fed into the chamber, x b , (0.10-0.04 kg moisturekg dry air). (4) The inlet temperature of the drying air, Tgo(200-340 "C). (5) The mass flow rate of the slurry, F (0.010-0.10 kg/s). (6) The mass flow rate of the drying air, G (0.60-1.2 kg/s). (7) The viscosity of the slurry, pslr (0.110-3-l.210-3 Pa 5). (8)The atomization velocity, u (38-100 d s ) . (9) The surface tension of the slurry, oslr (0.01-0.10 kg/s2). Responses (Outputs). From the droplet size, velocity, density, moisture, and temperature distributions as a function of axial distance, the following response variables are derived: (1) Exit mean particle diameter. (2) Exit mean moisture content. (3) Exit standard deviation of the partile diameters. (4) Exhaust air temperature.
+
111. Dual Response Problem In this section the constrained nonlinear optimization problem and its numerical solution is presented. Given the above ranges for the inputs of the process, the aim is to find the optimum operating conditions for producing a product powder with a minimum moisture content subjected t o a required exit mean particle diameter. These types of problems are referred to as dual response problems where one of the primary response variables is minimized while the other is required to be constant. The specification of the product sometimes may require many constraints to be met, in this case the problem becomes more complicated if no physical knowledge is known a priori. If a mathematical model that maps inputs to the response variables is available, then the solution outlined below can be applied to any number and type of constraints to be met without loss of
generality. The area of experimental design and response surface methodologies provide alternatives when such a physical model is not available. In mathematical terms the constrained optimization problem (dual response problem) can be formulated as
-
- 0.00712
(19a)
= 100 pm
(19b)
min [W,,,(X) XE
.@?
subject to
