Aug 25, 2013 - 37-41 Mortimer Street, London W1T 3JH, UK ... Department of Chemical Engineering , College of Engineering, Shahid Bahonar University of. Kerman ...... transport in flighted rotary dryers based on physical considerations.
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Mathematical Modeling and Simulation of Drying Using Two Industrial Concurrent and Countercurrent Rotary Dryers for Ammonium Nitrate a
Hamed Abbasfard , Sattar Ghader
a b
a
, Hasan Hashemipour Rafsanjani & Mehdi Ghanbari
c
a
Department of Chemical Engineering , College of Engineering, Shahid Bahonar University of Kerman , Kerman , Iran b
Mineral Industries Research Center, Shahid Bahonar University of Kerman , Kerman , Iran
c
Department of Chemical Engineering , School of Chemical and Petroleum Engineering, Shiraz University , Shiraz , Iran Published online: 25 Aug 2013.
To cite this article: Hamed Abbasfard , Sattar Ghader , Hasan Hashemipour Rafsanjani & Mehdi Ghanbari (2013) Mathematical Modeling and Simulation of Drying Using Two Industrial Concurrent and Countercurrent Rotary Dryers for Ammonium Nitrate, Drying Technology: An International Journal, 31:11, 1297-1306 To link to this article: http://dx.doi.org/10.1080/07373937.2013.791307
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Drying Technology, 31: 1297–1306, 2013 Copyright # 2013 Taylor & Francis Group, LLC ISSN: 0737-3937 print=1532-2300 online DOI: 10.1080/07373937.2013.791307
Mathematical Modeling and Simulation of Drying Using Two Industrial Concurrent and Countercurrent Rotary Dryers for Ammonium Nitrate Hamed Abbasfard,1 Sattar Ghader,1,2 Hasan Hashemipour Rafsanjani,1 and Mehdi Ghanbari3 1
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Department of Chemical Engineering, College of Engineering, Shahid Bahonar University of Kerman, Kerman, Iran 2 Mineral Industries Research Center, Shahid Bahonar University of Kerman, Kerman, Iran 3 Department of Chemical Engineering, School of Chemical and Petroleum Engineering, Shiraz University, Shiraz, Iran
Drying of ammonium nitrate (AN) is accomplished in the Shiraz Petrochemical Complex (SPC) using a concurrent rotary dryer following a countercurrent rotary dryer. A mathematical model for these rotary dryers including heat and mass transfer was developed. The model was checked against industrial-scale data, which showed a good agreement. The average absolute deviation of the simulation results compared to the industrial data for the concurrent dryer was 4.0% for solids moisture, 1.3% for solids temperature, and 1.8% for air temperature and for the countercurrent dryer it was 9.0% for the solids moisture, 2.0% for solids temperature, and 4.6% for air temperature. These simulation results reveal that for outlet solid moisture, inlet AN moisture, and air temperature as well as the outlet temperature of product, the inlet solid and air temperature have major effects for both concurrent and countercurrent flow. Keywords Ammonium nitrate plant; Concurrent and countercurrent; Drying kinetics; Mathematical model; Rotary dryer
INTRODUCTION Ammonium Nitrate Plant The Shiraz Petrochemical Complex (SPC) is a large chemical complex involving manufacture of several products such as ammonia, nitric acid, and ammonium nitrate. The capacity of the ammonium nitrate plant is 750 metric tons per day of AN prills used as a fertilizer or 650 metric tons per day of porous prills of pure AN used to prepare an explosive (ANFO).[1] Many parts of the process are common to both products, as indicated in Fig. 1.
Correspondence: Hamed Abbasfard, Department of Chemical Engineering, College of Engineering, Shahid Bahonar University of Kerman, Kerman 7618891167, Iran; E-mail: h.abbasfard@ eng.uk.ac.ir
Figure 2 shows the process flow diagram of the domestic ANFO plant, which is the theme of present work. Drying is utilized in this grade to dry wet particles and produce the required porosity. The raw materials required for production of AN are nitric acid and ammonia, for which relevant properties at the inlet of the neutralizer are provided in Table 1. These raw materials are introduced to a reaction vessel (neutralizer) in order to produce a 78% solution of AN by means of highly exothermic neutralization reaction: NH3 þ HNO3 ! NH4 NO3 DH ¼ 146 kJ=mol:
ð1Þ
The 78% AN solution is fed to the primary evaporation device, where the AN concentration is increased to 95%. The primary evaporation device consists of four different pieces of equipment through which the liquid runs from the top to the bottom: The The The The
flash tank falling-film-type evaporator primary separator drum constant-level tank
The 95% pure AN solution is sent directly to the homogenizer tank through the constant-level tank by means of a centrifugal pump. Then the homogenizer feeds the prilling headers and showers of prilling tower through which the AN prills are sprayed to the bottom of the tower. The ANFO prills are dried using two rotary dryers in series. The first is a concurrent rotary dryer and the second one is a countercurrent rotary dryer. A belt conveyor discharges the ANFO prills into the fluidized bed cooler using air. Finally, to avoid caking during storage, the prills are coated with an anticaking agent and then sent to storage.
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TABLE 1 Feed properties at the inlet of the neutralizer of the AN plant at the Shiraz Petrochemical Complex[1] Properties
FIG. 1. Process block diagram of the AN plant at the Shiraz Petrochemical Complex (color figure available online).
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The properties of the final product sent to storage are provided in Table 2. Prior Related Works Drying is a process consisting of the removal of water or another solvent by evaporation from a solid, semi-solid, or liquid and is often used as a final production step before selling or packaging the dried product. Despite its importance, in many cases the design and operation of dryers is subject to empiricism, based on prior experience.[2] Rotary drying is one of the many drying methods that are widely employed in various industries. The most
FIG. 2.
Operating temperature Operating pressure Concentration (by weight) Average molecular weight Density Velocity Normal flow Normal flow
Nitric acid
Ammonia
C bar %
55 7 58
10 6 99.9
kg=kg mol kg=m3 m=s kg=h m3=h
63
17
1,300 0.4 56,235.3 27.6
3.85 9 5,749.70 1,493
Unit
commonly used rotary dryers in industry are direct contact type, based on the method of heat transfer between hot air and solid particles.[3] The vast use of rotary dryers is motivated by their ability to handle a wide range of solids such as fertilizers, pharmaceuticals, mineral concentrates, cement, sugar, soybean meal, corn meal, plastics, and many others.[4]
Simplified process flow diagram of the ANFO plant (color figure available online).
DRYING OF AMMONIUM NITRATE
TABLE 2 Properties of the final product from the ANFO plant[1]
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Properties
Unit
Nitrogen content (by weight) Water content (by weight) Size (mainly) Fuel oil absorption (minimum by weight) Apparent density Temperature Type
% % mm %
Normal flow Normal flow
kg=h m3=h
kg=m3 C —
Value 34.5 8 1–3 6 700–850 25 Hard, free-flowing prills 27,085 33.9
Several works are available in the literature on the steady-state modeling of the rotary drying process. Myklestad[5] was the first to obtain an expression to predict product moisture content throughout a rotary dryer. Sharples et al.[6] developed a steady-state model of a concurrent fertilizer rotary dryer including material and energy balances. Thorpe[7] carried out an analysis similar to that of Sharples et al.[6] but used another equation to calculate the residence time. By subdividing the dryer into high enough elemental volumes, he obtained results similar to those of Sharples et al.[6] However, in this case the results were not compared to experimental values. O’Donnell[8] developed a new equation to calculate retention time, which was coupled with heat transfer equations to construct an overall complex and tedious dryer model. A simplified drying model was proposed by Kisakurek,[9] who assumed a constant solids temperature and neglected sensible heat effects. Kamke and Wilson[10] developed a model to predict heat transfer using a retention time equation similar to that of Kelly and O’Donnell[11] and Ranz and Marshall[12] for wood. The model agreed with experimental values and depicted that initial product moisture content and drying air temperature had the greatest effect on outlet product moisture content. In order to improve drying efficiency, another configuration of the rotary dryer known as a roto-aerated dryer was presented by Lisboa[13] and Arruda and coworkers.[14,15] Mathematical modeling of woody biomass drying was developed by Xu and Pang[16] and a new correlation between the theoretical maximum drying rate and the actual constant drying rate for the wood chips was proposed from the drying experiments. Osorno and Hensel[17] performed a test of drying homogeneity of a mixture of white clover and ryegrass dried in a rotary drum. Castan˜o et al.[18] presented a model of a cocurrent dryer, showing the process by which it was
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arrived at, and raised suggestions for some improvements in the determination of the drying rate to obtain a more accurate analytical expression. Generally, the performance of rotary dryers is dictated by three important transport phenomena, namely, solids transportation[19–24] and heat and mass transfer.[14,15,25–29] The ability to estimate each of these transport mechanisms is essential for proper design and operation of rotary dryers.[30] This approach requires constitutive equations for the heat transfer coefficients, drying kinetics, equilibrium moisture content, and fluid dynamic characteristics. Although several models have been proposed, there is no general agreement on the best one to describe the mechanism of rotary drying. It appears that specific models for an equipment and material are more useful than general models. Moreover, there is a major lack of information with respect to model validation with large-scale (industrial-scale) data. The objective of this project is to develop a mathematical model to simulate the two industrial rotary dryers for AN particles and to provide information for optimization as well as the key factors that have the greatest effects on the performance of dryers. In this work, the concurrent and countercurrent rotary dryer will be modeled and the influence of different operating variables on the outlet moisture content and outlet temperature of the solid is studied. Industrial experience suggests that the most significant parameter of AN exiting the rotary dryer that affect the commercial properties of the final product are material moisture content and temperature. These two variables have major effects on caking and the size distribution of particles. Therefore, the effect of various parameters on these variables was analyzed. AN DRYING PROCESS As illustrated in Fig. 3, while molten AN is prilled from the showers, which are used instead of prior nozzles as modification, at the top of the prilling tower, crystallization takes place along the tower height. At the bottom of the prilling tower, the ANFO prill temperature is around 90 C, which is then introduced to the entrance of rotary dryer using belt conveyors. The AN grains have a size ranging 1 to 3.15 mm and moisture content of about 2.5% (db). The rotary dryers used in this plant consist of two 18- and 20-m-long cylinders for concurrent and countercurrent, respectively, rotating upon bearings (3 rpm) and slightly inclined (2.5%) to the horizontal, which is illustrated in Fig. 3a. The feeding of the prills is ensured by a belt conveyor, which extends into the cylinder at the top end. The prills progress through the cylinder by means of rotation, head effect, and slop and are discharged at the other end on the belt conveyor. According to Fig. 3b, the rotary dryer is equipped with flights on the interior for lifting and showering the prills through the hot air stream and
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to predict the changes of water content and temperate of corresponding material along the dryer length. The mathematical model of rotary dryer consists of the following assumptions:
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The product particles are spherical and their dimensions remain unchanged during drying. The operation is assumed to be in steady state. The drying process takes place only in the falling rate period.[16,28] Both the flows of air and solid throughout the dryer follow a plug flow regime. Axial diffusion, dispersion, or back-mixing of the solid is not taken into account. The heat capacities of materials were assumed to be constant and the latent heat changes according to the Clausius-Clapeyron equation throughout the dryer length.
Figure 4 shows a scheme of the infinitesimal volume element of the rotary dryer operating at the concurrent and countercurrent flow. From the conservation of the mass and energy around each element, the following set of differential equations is derived: dX RM ¼ ; dz S
ð2Þ
dY R M ¼ ; dz G
ð3Þ
FIG. 3. (a) Schematic diagram for solid crystallization and drying process and (b) cross-sectional area of rotary dryer including flights and partitions (color figure available online).
partitions to increase the effectiveness of material distribution and reduce dusting during passage through the cylinder. The air is drawn from the atmosphere through a filter by means of a fan and is then heated up by an air heater. Because AN prills are very heat sensitive and must be dried carefully, the hot air and prills flow cocurrently to ensure that the hot air cools rapidly during the initial evaporation of surface moisture. Then the solids enter the second countercurrent rotary dryer for final drying. Dust entrained in the exit air stream is recovered in a washing tower (scrubber) where it is dissolved with water, and the resulting solution is sent back to the neutralizer. MODEL DEVELOPMENT Energy and Mass Balances A one-dimensional mathematical model considering a set of mass and energy balances was developed in order
FIG. 4. Infinitesimal volume element of the rotary dryer: (a) concurrent and (b) countercurrent (color figure available online).
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DRYING OF AMMONIUM NITRATE
dTs Q kRM ¼ ; SðCpd þ XCpw Þ dz
ð4Þ
dTg Q þ Cpv RMðTs Tg Þ Qp ¼ : dz GðCpg þ YCpv Þ
ð5Þ
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Equations (2)–(5) are solved numerically with knowledge of the drying rate curve and the residence time to determine the changes in the air temperature and humidity, AN prills moisture content, and temperature. In the above equations, M is the total load (kg), which is defined as the product of the solid flow rate and average residence time (M ¼ s S). The heat transfer within the control volume and lost through the shell wall is defined by the following equations: Q ¼ Uva V ðTs Tg Þ
ð6Þ
Qp ¼ Up AðTg Tamb Þ:
ð7Þ
The best correlations for the global volumetric heat transfer coefficient (Uva) and heat loss coefficient (Up) determined by Arruda[14] were used; they are Uva ¼ 0:394ðG=AÞ0:289 ðS=AÞ0:541
ð8Þ
UP ¼ 0:022ðG=AÞ0:879
ð9Þ
Equilibrium Moisture and Drying Kinetics A laboratory-scale hot air dryer of the static tray type was used for this study (Fig. 5). The main parts of the dryer
FIG. 6. Drying curves for thin-layer drying of AN at air temperatures of 29, 38, and 44 C. Solid moisture and air absolute humidity profiles along the dryer length (color figure available online).
system consist of an adjustable centrifugal blower, air heating chamber (2.5 kW), drying chamber, system controller, inverter (Testo 625, Testo AG, Lenzkirch, Germany), and tray sample. Experiments were performed at air temperatures of 29, 38, and 44 C. Figures 6 and 7 illustrate experimental data to characterize the solids drying and equilibrium isotherms through the thin-layer drying experiments, respectively. The following correlations show good agreement with experimental data.[31] Table 3 lists the constants of Eq. (10) for relevant temperatures. X ¼ A½B expðCtÞ
FIG. 5. Schematic diagram of laboratory-scale dryer: 1, electromotor; 2, heater compartment; 3, seed tray; 4, control box.
ð10Þ
FIG. 7. Equilibrium isotherms for AN measured by thin-layer drying experiments at air temperature of 29, 38, and 44 C.[31] (color figure available online).
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TABLE 3 Constants of Eq. (10) at temperature of 29, 38, and 44 C Constants Temperature ( C) a
c
R2
0.010495 1.6586 0.026584 0.9989 0.004425 1.6124 0.028389 0.9972 0.004500 1.3777 0.029185 0.9999
29 38 44
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b
Xeq ¼ RHða RH 2 þ b RH þ cÞ
ð11Þ
a ¼ 2:39 106 0:987Tg Tg0:832
ð12Þ
b ¼ 5:76 105 þ 1:306 105 lnðTg Þ
ð13Þ
c ¼ 0:9715 1:024
Tg
Tg2:31 :
dX ¼ RðX Þ ¼ CðX ABÞ: dt
ð15Þ
Regarding Eq. (10), if the drying time tends to infinity in a specified air relative humidity and temperature, the remaining (AB) is equal to equilibrium moisture, then: R ¼ CðX Xeq Þ:
ð16Þ
Drying experiments on an AN sample using air temperatures similar to those used for equilibrium moisture experiments were performed. It was found that the simple exponential equation adequately described the drying kinetic of these particles. In Eq. (16), C is the drying constant and is related to the temperature of the drying air by 7:95 C ¼ 0:0349 exp : Ta
Models Foust et al.[32] Friedman and Marshall[34] Perry and Green[35] Song et al.[22] Thibault et al.[4]
ð17Þ
Residence Time In order to express the residence time as a function of the dryer characteristics, several suggestions are available in the literature. Table 4 lists some of the correlations and compares their deviations to the real residence time measured for the rotary dryer at the AN plant. There is no general model that best describes the residence time of solids in a rotary dryer. However, the relation proposed by Foust et al.[32] provides a better fit to the industrial rotary dryer residence time for concurrent dryer (30 min)
Residence time
Error (%)
31.113 10.581 17.254 20.375 31.971
þ3.76 64.73 42.49 32.086 þ6.56
and is expressed as 13:8L s ¼ 0:9 sn D
ð14Þ
The drying rate expression is derived by differentiating Eq. (10) and substituting t by relation depending X, which is derived from Eq. (10) as follows:
TABLE 4 Comparison of models for residence time based on industrial data for concurrent dryer
! 614:2 LG : D0:5 S p
ð19Þ
The plus or minus sign represents the countercurrent and concurrent flows, respectively.
NUMERICAL SOLUTION The model consists of four ordinary differential equations. The most widely used method of integration for ordinary differential equations is the series of second-, third-, and fourth-order Runge-Kutta methods. We used the fourth-order method to solve the system of equations considering the following boundary conditions for concurrent flow: X ð0Þ ¼ X0 ;
Y ð0Þ ¼ Y0 ;
Ts ð0Þ ¼ Ts0 ;
Tg ð0Þ ¼ Tg0 : ð20Þ
A boundary value problem is solved for the countercurrent flow because the solid and air boundary conditions are at opposite ends of the dryer. The shooting method converts the boundary value problem to an initial value problem. In this method, the unspecified initial conditions of the differential equation system are guessed and the equations are integrated forward as a set of simultaneous initial value differential equations. At the end, the calculated final values are compared with the boundary conditions and the guessed initial conditions are corrected if necessary.[33] This procedure is repeated until the specified terminal values are achieved within a small convergence criterion. Boundary conditions for the countercurrent case are X ð0Þ ¼ X0 ;
Y ðLÞ ¼ Y0 ;
Ts ð0Þ ¼ Ts0 ;
Tg ðLÞ ¼ Tg0 : ð21Þ
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DRYING OF AMMONIUM NITRATE
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MODEL VALIDATION Steady-state model validation was performed between the plant data and the mathematical modeling of concurrent and countercurrent rotary dryers for AN. The model results and the corresponding observed data of the plant are presented in Table 5. The average absolute deviation of the simulation results compared to industrial data for the concurrent dryer was 4.0% for the solids moisture, 1.3% for solids temperature, and 1.8% for air temperature and for the countercurrent dryer it was 9.0% for the solids moisture, 2.0% for solids temperature, and 4.6% for air temperature. Therefore, it can be said that the mathematical model performs well under industrial conditions.
RESULTS AND DISCUSSION Figure 8 shows the profile of material moisture and air absolute humidity and Fig. 9 shows the profile of material and air temperature along the length of the concurrent and countercurrent rotary dryer. The parameters used in the modeling are listed in Table 6 based on the SPC’s AN plant data. Figures 10 and 11 summarize the predicted effects of some selected input variables on outlet product moisture content and temperature, respectively. The reference conditions for all of the comparisons are the same as those shown in Table 6. Simulations were performed for each variation of a reference condition 30%, and all other conditions were held constant.
TABLE 5 Model validity for plant data: (a) concurrent,[31] (b) countercurrent a) Dryer inlet variables
Run no. 1 2 3 4 5 6 7 8
Product moisture (kg water=kg dry)
Dryer outlet variables
Product moisture Product Air temperature Air Product (kg water=kg dry) temperature ( C) ( C) temperature temperature ( C) Measured Predicted Measured Predicted Measured Predicted ( C)
0.0256 0.0243 0.0214 0.0288 0.0225 0.0204 0.0277 0.0204
80 80 81 88 82 90 89 95
73 71 73 75 73 73 75 75
0.0073 0.0068 0.0063 0.0081 0.0064 0.0056 0.0068 0.0055
0.0071 0.0070 0.0062 0.0075 0.0065 0.0059 0.0072 0.0058
73 73 72 77 72 77 77 79
71.91 71.47 71.90 77.11 73.65 79.01 77.91 79.66
69 65 69 68 69 71 72 75
68.55 67.43 68.54 71.87 69.24 71.60 72.20 72.77
b) Dryer inlet variables
Run no. 1 2 3 4 5 6 7 8
Dryer outlet variables
Feed moisture (kg water=kg dry)
Feed temperature ( C)
Air temperature ( C)
0.0073 0.0068 0.0063 0.0081 0.0064 0.0056 0.0068 0.0055
73 73 72 77 72 77 77 79
125 124 126 125 123 125 126 125
Product moisture (kg water=kg dry)
Product temperature ( C)
Air temperature ( C)
Measured Predicted Measured Predicted Measured Predicted 0.0015 0.0014 0.0014 0.0016 0.0013 0.0011 0.0014 0.0010
0.0013 0.0013 0.0012 0.0014 0.0012 0.0011 0.0013 0.0011
85 86 86 83 85 88 86 89
85.37 86.02 88.06 84.95 86.03 91.12 88.75 92.14
83 85 86 83 84 87 86 88
87.79 87.88 89.08 88.45 87.43 91.35 90.46 92.26
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TABLE 6 Industrial rotary dryer characteristics used in the modeling for concurrent and countercurrent flow Value
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Properties
Unit
Concurrent Countercurrent
Figure 10 suggests that within the range of conditions examined, solid moisture and inlet air temperature are the parameters that have the greatest effect on outlet solid moisture for both concurrent and countercurrent dryers. However, it should be noted that a 30% increase in air temperature for countercurrent flow had no effect on outlet solid moisture. Inlet air humidity was found to have no effect on outlet solid moisture content for both dryers. Figure 11 shows that for the outlet temperature of the product, the inlet solid and air temperatures had a major
Inlet air C temperature Inlet solid kg water=kg moisture dry solid Inlet air kg water=kg absolute dry air humidity Solid flow kg=h Air flow kg=h Dryer length m Dryer inside m diameter Dryer slope m=m Dryer RPM rotational speed Average solids mm diameter Solids heat kJ=kg C capacity Air heat kJ=kg C capacity Water heat kJ=kg C capacity Vapor heat kJ=kg C capacity
FIG. 9. Profile of solid temperature and air temperature along the dryer length for concurrent and countercurrent dryers.
effect for both dryers. However, the effect of inlet solid temperature was greater for concurrent flow and the effect of inlet air temperature was greater for countercurrent flow. Considering the relations used for heat transfer and residence time, it was found that the inlet air flow rate had two opposing effects on the outlet product moisture content of the concurrent dryer. An increase in the air flow rate causes an increase in the heat transfer rate (Eq. (8)) between the air and product and the drying rate becomes higher. On the other hand, an increase in the air flow rate results in a decrease in the residence time (Eq. (19)) and therefore the product dries less and the outlet moisture content increases. Figure 10a reveals that for any variation in air flow rate in the existing concurrent flow dryer, the effect of the heat transfer was greater than the effect of residence time on the outlet product moisture content. In other words, for an increase in the inlet air flow the final product moisture decreased accordingly.
FIG. 8. Profile of solid moisture and air absolute humidity along the dryer length for concurrent and countercurrent dryers.
73
123
0.0225
0.0064
0.0223
0.0223
32,251 60,979 18 3.324
32,251 65,640 20 3.324
0.025 3
0.025 3
2
2
1.56
1.56
1.009
1.009
4.18
4.18
1.88
1.88
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DRYING OF AMMONIUM NITRATE
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CONCLUSION A set of differential equations including mass and energy conservation was developed to describe air absolute humidity, solids moisture, and temperature distribution along the length of the rotary dryer used to dry AN prills at a domestic petrochemical complex. The average absolute deviation of the simulation results from the industrial data showed good accuracy for water content and temperatures of the solid and air. Therefore, it can be said that the mathematical model performs well under the industrial conditions and can be used in optimization of this and other similar industrial rotary dryers. Within the range of conditions examined, for outlet solid moisture, inlet AN moisture, and air temperature and for the outlet temperature of the product, the inlet solid and air temperature had a major effect for both concurrent and countercurrent flow.
FIG. 10. Effect of variations in several inlet variable and parameters, 30%, on the predicted outlet moisture content of the product: (a) concurrent and (b) countercurrent.
NOMENCLATURE A Dryer cross-sectional area (m2) Cpd Specific heat of dry solid (kJ kg1 C1) Cpg Specific heat of dry air (kJ kg1 C1) Cpv Specific heat of water vapor (kJ kg1 C1) Cpw Specific heat of pure water (kJ kg1 C1) D Diameter (m) Dp Particle diameter (mm) G Air mass flow rate (kg=s) L Dryer length (m) M Dryer total load (kg) n Dryer rotation speed (rpm) R Drying rate (kg water kg dry solid1 min1) RH Air relative humidity S Dry solid mass flow rate (kg=s) s Dryer slope (m=m) T Temperature ( C) t Time (s) Up Coefficient of heat lost (kWm2 C1) Uva Global volumetric heat transfer coefficient (kWm3 C1) V Dryer volume (m3) X Solid moisture (kg water=kg dry solid) Y Air absolute humidity (kg water=kg dry air) z Dimensionless length (position=L) Greek Letters Latent heat of vaporization (kJ kg1) Average residence time (s) s k
FIG. 11. Effect of variations in several inlet variable and parameters, 30%, on the predicted outlet temperature of the product: (a) concurrent and (b) countercurrent.
Subscripts amb g s
Ambient Air Solid
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