Chemical Engineering Communications

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MATHEMATICAL MODELLING OF UREA PRILLING PROCESS

A. Alamdari a; A. Jahanmiri a; N. Rahmaniyan a a Department of Chemical Engineering, School of Engineering, Shiraz University, Shiraz, Iran Online Publication Date: 01 March 2000 To cite this Article: Alamdari, A., Jahanmiri, A. and Rahmaniyan, N. (2000) 'MATHEMATICAL MODELLING OF UREA PRILLING PROCESS', Chemical Engineering Communications, 178:1, 185 - 198 To link to this article: DOI: 10.1080/00986440008912182 URL: http://dx.doi.org/10.1080/00986440008912182

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MATHEMATICAL MODELLING OF UREA PRILLING PROCESS A. ALAMDARI*, A . JAHANMIRI and N. RAHMANlYAN Department of Chemical Engineering, School of Engineering. Shiraz Universily, Shiraz, Iran (Received 2 June 1998; Infinal form I0 August 1999) A mathematical model is proposed for the urea prilling process of a commercial plant. In this model the prilling process has been simulated by simultaneous solution of the continuity, hydrodynamic, mass and energy transfer equations. Particle trajectory, temperature and moisture distribution of the particles and of the cooling air along the height of the tower was calculated from the mathematical model. The air temperature profile resulted from the model was compared with the profile from the urea plant. The model predicted data were consistent with the plant data indicating the validity of the model. The result of this work showed that an increase in heat removed from the particles would improve the characteristics of the product urea. Keywords: Prilling; urea; spray crystallization; mathematical modeling; simulation

INTRODUCTION

Prilling is a granulation process in which a melt is sprayed to small size droplets into a cooling medium which is usually air. Under the influence of surface tension the melt drops form a spherical shape. During falling down through the uprising air, droplets will cool until the first outer layer starts to solidify. The solidified layer grows and the boundary of the solid and melt moves towards the center as heat is convected to the air. Fully solidified and dry particles are expected to settle on the floor of the tower. The process is also called spray crystallization [I, 21.

Torresponding author. Fax: 98-71-337491 and 98-71-52725, e-mail: [email protected] a2u.ac.k 185


Schweizer er a/. [3] formulated solidification of a particle using equations of motion and phase change of a droplet. However, moisture transfer and size distribution effects were not considered in their work. Hassanien er al. [4] developed a simple mathematical model in which the moisture content of particles and drying air were considered as variables. However, the temperature variations through the cooling air and along the particle radius and effects of size distribution were not studied. In this work, the proposed model takes into account the variations of moisture and temperature in particles of different sizes and in cooling air and the variations of particle velocity in radial and vertical directions and changes in air density, to simulate a prilling process by considering heat transfer between particles and air in a counter current system.

MODEL DEVELOPMENT In order to analyze the process, equations of continuity, momentum, mass and energy transfer were solved simultaneously. The temperature, moisture and velocity distribution of air and particles a t different heights of the tower was calculated. A feature of the present analysis is that a uniform particle size distribution (PSD) is not assumed but rather particles may have a known size distribution. The PSD produced was determined by measuring the PSD obtained on the tower floor. Steady state conditions were assumed to prill a pure melt of urea feed in which biuret content, a n impurity usually with urea, was less than one percent and the moisture content was in the range of 0.3% to 0.5% by weight. Since the diameter of the tower is usually large, the air velocity profile in the direction of flow was assumed flat [S].The pressure drop of air through the bed (about 20mm H 2 0 ) was low enough to apply a constant presiure condition. It is worth noting that low urea evaporation (0.06%) and low conversion of urea t o ammonia and biuret (0.9% by wt.) were neglected in this study and also due to a large thickness of the tower wall an adiabatic process was considered. Hydrodynamics

Three kinds of forces exert o n the particles during their fall through the tower. Gravity pulls the particles down while drag and buoyancy forces resist against the motion of particles. Since the density of particles is much


MODELLING OF UREA PRlLLlNG PROCESS

187

larger than the density of air the effect of the buoyancy force is negligible. The Newton's law of motion along the height of the tower (z-axis) is m

dv. dl

2=

1 2

+

4

mpg - - CDP,,A,V,,I . v,

where m, is the mass of particle, v, is the vertical velocity of particle, p, is the' air density, A, is the projected area of particle, v,l is the velocity of particle relative to air, and CD is the drag coefficient. A similar equation of motion is written for particles in radial direction. A method of 4th order RungeKutta was used to solve the governing equations numerically. Moisture Analysis of Particles Moisture is continuously removed from the urea particles and is transferred to the air until the equilibrium is attained. In order to find the moisture distribution in particles, a moisture balance on a spherical shell in the particle is written as:

where M is the local moisture content a t radius r, and D is the diffusivity of moisture in the particle. The appropriate initial and boundary conditions are M(r,O) = Miniti,, for initial moisture within the particle; and M(R,, r) = Mequilibrium for the points a t the particle surface; and (dM/dr),=o = 0 for the center point. It is notable that mass transfer resistance in the gas phase was neglected and Mcquilibrium was measured experimentally. Chapman [6] solved Eq. (2) analytically for bodies with specific shapes. Moisture Analysis of Air The water evaporated from the particles equals the change in humidity of the air. A balance for the moisture transfer from particles to air is

where W is the humidity of air, is the average moisture content of each particle, G, is the mass velocity of particles, f(D,) is the weight fraction of particles of average size D, within the size range, and G, is the mass velocity of air.


Thermal Analysis

.

The heat flow from the center of particle towards the surface due to temperature gradient (4,) goes partially for evaporation of moisture diffused to the surface and the remainder is convected to the air stream from the s u r f x e ( q z ) Heat flow for particles can be formulated as:

where /I,-- is the latent heat of vaporization of water a t the temperature of particle surface and R is the radius of the tower. The energy convected from the particles ( C q 2 )to the air is equivalent to the enthalpy change of the air which can be expressed as:

where p,, v,, c, and T , are density, velocity, specific heat and temperature of air respectively, and cp, is the specific heat of vapor. On the other hand, the heat convected to the cooling air is

where h(D,) and a(D,) are the heat transfer coefficient and the area of particles in a unit volume of bed, respectively. T, is the surface temperature of each prill. The heat transfer coefficient, h, for each individual particle can be obtained using the relation [7]

where Nu, Re, and Pr are the dimensionless numbers of Nusselt, Reynolds, and Prandtl, respectively. Three zones of state have been assumed for each particle as it falls from the top to the bottom. In the first zone, droplet loses its sensible heat and cools to a temperature just above the crystallization temperature. In the second zone, the most outer layer of drop starts to crystallize and latent heat of crystallization transfers to the cooling air. This zone for a particle ends as the most inner layer solidifies and the particle totally converts to a solid prill. The prill further cools to a lower temperature as it falls down in the third zone of its path.


MODELLING OF UREA PRILLING PROCESS

ZONE I

189

The conduction mechanism inside the droplet is formulated as

PI:

where pl, cp/,kh TI,vp and r are the density, specific heat, thermal conductivity, temperature of melt, velocity of particle and the radius a t which the energy balance is written, respectively. This equation was solved numerically using the appropriate initial and boundary conditions [8].

Z O N E II At the end of Zone I, crystallization starts a t r = Rp and heat is removed from the center towards the outer surface of the particle by conduction through the melt and solid phases and then to the air by force convection. As the solidified layer grows, the solid-melt interface, S(i), moves towards the center. Equation (8) applies to formulate the energy transfer inside the melt phase of particle in the range 0 < r < S(z). Using the solid properties, this equation may also be used for conduction through the solid phase in the range S(z) < r < Rp. The boundary conditions for the solid-melt interface are [9]

where Tm is the crystallization temperature and X is the latent heat of crystallization of melt. T,, k,, and p, are the temperature, thermal conductivity, and the density of solid urea, respectively. Equation (10) is derived by considering an extremely thin layer in which the phase changes from melt to crystal. The energy conducted from melt to this layer plus the crystallization energy generated by the phase change is conducted from the layer to the outer solid phase. The mathematical formulation of phase change problems involving solidification is represented as Stefan Problem [9, 101.

Z O N E I I I Modeling equations and boundary conditions for a prill in Zone I11 are similar to the equations for that in the Zone I unless the transport properties of the solid urea, p,, k,, and cpsare used. Computational Methods The thermal partial differential equations for Zones I and 111 were solved using the Crank-Nichlson Method [ I I]. Thermal equations in Zone 11 were


solved using the methods specified to moving boundary problems. Crowly [12]; Voller and Cross [13, 141 and Tacke [IS] used the enthalpy method to solve phase change problems for materials in which a distinct solid-liquid interface takes place. In this method, a n enthalpy function, H ( T ) , which is the total heat content of the substance, is used to represent the equation. In order to remove the limitation of stability, an implicit method of solution was used to solve the enthalpy equations. It is worth noting that all the equations used in the model development were solved simultaneously. In order to confirm the model, the results of this model for some limited cases were compared to those produced by the model developed by Schweizer et a/.[3]. The restrictions of uniformity of particle size and no-humidity transfer were made to make the two models comparable. The results of two models were consistent indicating the present model works correctly.

RESULTS AND DISCUSSION The temperature of air along the tower height measured from plant and that predicted by the model were compared as a criterion for validity of the model. First, this comparison was made for the case when the variations of temperature along the particle radius was neglected and whole particle was assumed at a constant temperature (lumped method). Figure I shows that the air temperature profile predicted by the model is not consistent with that

E

40

-Lumped Model 30

FIGURE 1

40 50 60 Temperarure PC)

70

Comparison of data generated by model using a lumped method with plant data.


MODELLING O F UREA PRILLlNG PROCESS

191

of the plant data. The inconsistency proved that the radial temperature variations through the particles could not be neglected. The comparison, on the other hand, showed that the model is very sensitive to the tenns representing the effects of the variations of temperature along the particle radius. Plant data were collected by measuring the temperature of air along the height of the urea prilling tower at a distance of 2.5 m from the wall using a platinum resistance device. The comparison for the case when the variations of temperature along the particle diameter in the model put into effect is shown in Figure 2. The value of parameters used in the model is given in Table I. Product size distribution

_j_

Distributed Mods

30

40

50

60

70

Temperature PC)

FIGURE 2 Comparison of data predicted by model using a distributed method with plant data. TABLE I Operation conditions used in the urea plant Variables

Tower Height (m) Tower Diameter (m) Bucket Angle with Vertical Line (deg) Rotation Speed of Bucket (rpm) Ambient Pressure ( a m ) Volume Flow Rate of Air (m3/hr) Inlet Air Temperature ('C) Relative Humidity of Inlet Air (76) Capacity of Plant (MTPD) Temperature of Urea Feed ("C) Moisture of Urea Feed (% by wt.) Average Prill Diameter ( m m ) Standard Deviation ( m m ) Meter at see level (m.a.s.l.)

Values

50 18

10 275 0.86 540000 29.0 45 1425 139.0 0.5 1.8 0.49 1590


192

A . ALAMDARI

el a/,

was measured using sieve analysis and the function represented the distribution was normal. The physical properties of urea here extracted from the Refs. [I6 and 171 and the program was run with increments along the tower height and particle radius of lOcm and 100p-1, respectively. The temperature variations predicted by the model along the radius of fine and coarse particles at different heights of the tower are shown in Figures 3 and 4, respectively. Figure 3 shows that for fine particles, the variations of temperature inside the particle diminishes after falling a

0.1 0.2 0.3 0.4 Radial Distance (mm)

0.0

0.5

FIGURE 3 Temperature variations along the radius for fine particles of diameter I.Omm.

0.0

0.2

0.4 0.6 0.8 1.0 1.2 Radial Distance (mm)

1.4

FIGURE 4 Tcmperature variations along the radius for large particles of diameter 2.5mm.


MODELLING OF UREA PRILLlNG PROCESS

193

distance of about 25m. This proved that. using a lumped method calculations was valid only for fine particles in the second half of the tower height, where the melt phase had completely changed to solid phase. Coarse particles ( > 1.8 m m ) d o not reach to a thermal equilibrium with the cooling air a t the operating conditions applied to this prilling plant (Fig. 4). The flatness on curves in Figures 3 and 4 a t temperatures around 132.7"C is due to loss of latent heat of crystallization a t the interface of solid and melt layers within the particles. These figures also show that heights of Zones I, 11 and 111 for different sizes of particles are different. Small droplets would crystallize quickly and reach to Zone I11 but the large ones may never approach Zone 111 during their residency in the tower. The average moisture content based on a dry basis for fine and large particles along the height of the tower is shown in Figure 5. It shows that the smaller the particle the more moisture is removed from that particle. This removal mostly occurs a t the top of the tower where particles are still mostly in the melt phase. Air velocity increases along the height of the tower due to a decrease in density of air because of the temperature rise. Change of Reynolds number for particles of different sizes is shown in Figure 6. The trajectory of particles shown in Figure 7 shows that large particles tend to move near the wall of the tower and small particles move near the center. Figure 8 shows changes in heat transfer coefficient for particles of different sizes. For a larger particle, Re increases because of a velocity increase (Fig. 6). Accordingly, Nu increases as Eq. (7) shows, but change of

0.1

0.2 0.3 0.4 0.5 Mohture, % (kg W a r f i g Dry Urea)

FIGURE 5 Moisture distribution in particles of different sizes


A. ALAMDARI et a/.

FIGURE 6 Reynolds number for particles of different sizes.

0

1

2

3

4 5 6 Radius (m)

7

8

9

FIGURE 7 Trajectory of particles of different sizes.

D, in the relation h = (Nu. k/D,) is more than that of Nu. Therefore, the net effect would be a decrease in h value for larger particles. Lower dissipation of heat from larger particles shown in Figure 4 may be due to lower values of heat transfer coefficients. Change in humidity of air along the height of the tower calculated by the model is shown in Figure 9. Humidity of air increases with the height of the tower due to evaporation of moisture from the particles. Rate of change of the humidity at the top of the tower is more than that at the bottom indicating that most of the moisture is removed at the top where particles are still at liquid state. This also may be due to a lower relative humidity of air and therefore a higher potential to absorb moisture at the top (Fig 10).


FIGURE 8 Heat transfer coefficients of particles at different heights

Absolute Humidily (kg Water& Dry Air)

FIGURE 9 Absolute humidity of air at different heights of tower.

5

10

15 20 25 30 Relative Humidily (%)

35

FIGURE 10 Relative humidity of air at dimerent heights of tower.


CONCLUSIONS The prilling process of urea was mechanistically modeled. The significance of the model was considering a temperature distribution along the radius of the urea particles. Humidity and temperature of the cooling air along the tower height calculated by the model and those taken from the urea plant were compared to determine the validity of the model. This comparison showed a good consistency between the model predicted and the plant data. The model could be used to study the effects of process variables such as flow rates, temperature and moisture content of urea melt and inlet air on the product quality. It is also useful for a better control of the currently operating units and in the design of new prilling plants. The results of the study showed that under certain conditions of plant operation it is not possible to remove the necessary amount of heat from the large particles in order to attain thermal equilibrium before particle is reached the bottom of the tower. Based on this finding the problem of cake formation on the scraper of the commercial plant can be easily justified.

SYMBOLS projected area of particle (m2) total surface area of particles in unit volume of tower (m2/m3) specific heat of air at constant pressure (J/kg°C) specific heat of melt urea a t constant pressure (J/kg°C) specific heat of water vapor at constant pressure (J/kg°C) drag coefficient (dimensionless) particle diameter (m) diffusion coefficient of moisture through urea melt or solid within particle (m2/s) weight fraction of particles with average size Dpwithin the size range mass flux of particles (kg/m2s) mass flux of air (kg/m2s) gravitational acceleration (m/s2) convective heat-transfer coefficient between air and particles of diameter Dp ( w / m 2 " c ) latent heat of vaporization of water a t the particle surface temperature (J/kg) thermal conductivity of melt urea (W/m°C) thermal conductivity of solid urea (W/m0C)


MODELLING O F UREA PRlLLlNG PROCESS

197

local moisture content of particle at radius r (kg H20/kg dry urea) average moisture content of particle (kg H20/kg dry urea) mass of a particle (kg) Nusselt number (dimensionless) Prandtl number (dimensionless) rate of heat transfer (W) radius of the tower (m) Reynolds number (dimensionless) particle radius (m) radial distance from particle center (m) temperature in liquid phase ("C) crystallization temperature of urea melt which equals to 132.7"C temperature a t particle surface ("C) temperature in solid phase ("C) temperature of air in the bulk ("C) time it takes a particle falls the distance z from the top (s) velocity of air (m/s) velocity of particle (m/s) vertical velocity of particle (m/s) relative velocity of particle to air (m/s) humidity of air, dry basis (kg H20/kg dry air) distance from top of the tower (m)

Greek Symbols PO p, PS X

density of air (kg/m3) density of particle (kg/m3) density of solid particle (kg/m3) latent heat of crystallization (J/kg)

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Wiliiams, L., Wright, L. F. and Hendricks, R. (1946). US Patent 2402192. Roberts, A. G. and Shah, K. D. (1975). The Chem. Engr., p. 748. Schweizer, P., Covelli, B. and Widmer, F. (1975). Chimia, 29(2), 78 (In German). Hassanien, S. and El safty, M. (1985). The Trans. Egyptian Soc. Chem. Engr., 11(1), 61. Brodkey, R. S. and Hershey, H. C. (1988). Transport Phenomena (McGraw-Hill, Chemical Eng. Series, Singapore). Chapman, A. J. (1984). Hear Trans/er (Macmillan, New York), 4th edn. Ranz, W. E. and Marshall, W. R. (1952). Chen~.Eng. Prog., 48(5), 141. Bird, R. B., Warren, E. S. and Edwin, N. L. (1963). Transport Phenomena (John Wiley & Sons, Singapore). Ozisik, M. N. (1980). Heat Conduction (John Wiley, New York).


[lo] Taran, A. L. and Taran, A. V., Inzh-Fiz.zh., 51, 60 (In Russian). 11 11. Gerald, F. C. and Pattrick, 0.W. (1989). Numerical Analvsis . . . Applied .. . (Addison-Wesley . [I21 1131 i14i [IS] (161 [I71

Publishing Co.), 4th edn. Crowley, A. B. (1978). In!. J. Heat and Mass Transfer, 21, 215. Voller. V. and Cross. M. (1981). Int. J. Heat and Mass Transfir. 24. 545. ~ o l l e r V. ; and cross; M . (1983j. Int. J. Heat and Mass ~ r a n 2 e r ;26(1), 147. Tacke, K. H. (1985). Int. J. Num. Merhods in Eng., 21, 543. Berliner, J. F. T. (1936). Ind. and Eng. Chemistry, p. 517. Gambino, M. and Bros, J. P., Thermo Chim. Acta, 127, 223 (In French).