Theoretical Model for Predicting Moisture Ratio during Drying of Spherical Particles in a Rotary Dryer

Abstract

A mathematical model was developed for predicting the drying kinetics of spherical particles in a rotary dryer. Drying experiments were carried out by drying fermented ground cassava particles in a bench scale rotary dryer at inlet air temperatures of 115–230°C, air velocities of 0.83 m/s–1.55 m/s, feed mass of 50–500 g, drum drive speed of 8 rpm, and feed drive speed of 100 rpm to validate the model. The data obtained from the experiments were used to calculate the experimental moisture ratio which compared well with the theoretical moisture ratio calculated from the newly developed Abowei-Ademiluyi model. The comparisons and correlations of the results indicate that validation and performance of the established model are rather reasonable.

1. Introduction

Rotary drying is a very complicated process that can be applied not only to thermal drying but also movement of particles within the dryer. Several authors have carried out investigations on the steady state modeling of the rotary drying process. Static models are in general differential equations, and they are suitable for investigation of static distributions. Myklestad [1] was the first to obtain an expression to predict product moisture content throughout a rotary dryer based on drying air temperature, initial moisture content, and product feed rate. Thin layer drying equations contribute to the understanding of the heat and mass transfer phenomena in agricultural products and computer simulations for designing new and improving existing commercial drying processes [2]. They are used to estimate drying times of several products and also to generalize drying curves. In thin layer drying model, the rate of change in material moisture content in the falling rate drying period is proportional to the instantaneous difference between material moisture content and the expected material moisture content when it comes into equilibrium with the drying air [3].

Many authors have developed semiempirical models based on the diffusion theory to predict the drying kinetics of moist substances in thin layer as shown in Table 1 (where MR is the moisture ratio). The constants 𝑎, 𝑏, 𝑐, 𝑘, 𝑘_𝑜, 𝐾₁, and 𝑛 in eight models by most authors have been found to be functions of inlet air temperature, inlet air velocity, humidity, and so forth, the mass of feed was not accounted for by all the authors and in the drying of substances with high moisture content like fermented ground cassava, dairy products, and some pharmaceutical product in rotary dryer, and the mass of feed should be accounted for in the thin layer drying equation. It was observed that although several models have been proposed, there is not a general theory to describe the mechanism of rotary drying, and it seems that specific models for an equipment and material are more useful than general models [4].

Table 1 

Mathematical models given by various authors for the drying curves.

Therefore the objective of this study is to develop a theoretical model for predicting the drying kinetics of spherical particles in a rotary dryer accounting for the quantity of materials to be dried in the model.

2. Materials and Method

2.1. Theoretical Development of Thin Layer Drying Equation

Fick’s diffusion equation (1) has been accepted for describing the drying characteristics of biological and chemical products in the falling rate period [16] as follows: 𝜕𝑀=𝜕𝜕𝑡𝐷𝜕𝑥𝜕𝑀+𝜕𝜕𝑥𝐷𝜕𝑦𝜕𝑀+𝜕𝜕𝑦𝐷𝜕𝑧𝜕𝑀,𝜕𝑧 (1) where 𝐷 is the diffusion coefficient, 𝑀 is moisture content (dry basis) at any time 𝑡, and 𝑡 is drying time. The equation of diffusion for a spherical particle at constant diffusivity and radial (as shown in Figure 1) flux takes the following form: 𝜕𝑀𝜕𝜕𝑡=𝐷²𝑀𝜕𝑟²+2𝑟𝜕𝑀.𝜕𝑟 (2)

Figure 1 

Showing hypothetical profile of moisture diffusion from a spherical particle in a rotary dryer.

In order to solve (2) the following assumptions were adopted:(1)moisture movement is only diffusion and unidirectional;(2)diffusion coefficient 𝐷 is independent of moisture concentration;(3)drying process is isothermal, that is, adiabatic dryer;(4)material to be dried is spherical in shape;(5)shrinkage is neglected.Using −𝜆² as a separation constant we obtain from (2) 1𝐷𝑑𝑀𝑑𝑡=−𝜆²,𝜕²𝑀𝜕𝑟²+2𝑟𝜕𝑀𝜕𝑟=−𝜆². (3)(4) Integrating (3) using separation of variables gives 𝑇=𝐶₁𝑒^{−𝜆2𝐷𝑡}. (5) Equation (4) is of the form 𝑟²𝑅^′′+2𝑟𝑅^′+𝜆²𝑟²𝑅=0. (6) Equation (6) is a Bessel equation of order zero, the solution of which is [17] 𝑅(𝑟)=𝐶₂(𝜆𝑟)^1/2𝐽_1/2(𝜆𝑟)+𝐶₃(𝜆𝑟)^1/2𝐽_−1/2(𝜆𝑟). (7) But 𝐽_1/2(𝜆𝑟)=2𝜋()(),𝐽𝜆𝑟sin𝜆𝑟_−1/2(𝜆𝑟)=2𝜋()(),𝑅(𝑟)=𝜆𝑟cos𝜆𝑟2𝐶𝜋𝜆𝑟²()+𝐶sin𝜆𝑟³𝑟cos(𝜆𝑟). (8)(9) Combining (5) and (9) gives 𝑀(𝑟⋅𝑡)=𝑅(𝑟)𝑇(𝑡), so that 𝑀(𝑟,𝑡)=𝐶₁𝑒^{−𝜆2𝐷𝑡}2𝐶𝜋𝜆𝑟²()+𝐶sin𝜆𝑟³𝑟cos(𝜆𝑟) (10) and applying the boundary conditions in (11). The solution to (10) in the case of a sphere is expressed as 𝜕𝑀𝜕𝑟=0,𝑟=0,𝑟≥0,𝑀=𝑀_𝑒,𝑟=𝑎,𝑡≥0,𝑀=𝑀_𝑜𝑀,0≤𝑟≤𝑎,𝑡=0,MR=^𝑖−𝑀_𝑒𝑀_𝑜−𝑀_𝑒=6𝜋^2∞_𝑛=11𝑛²exp−𝑛²𝜋²𝐷𝑟²𝑡, (11)(12) where 𝑟 is the radius of sphere, MR is moisture ratio, 𝑀_𝑜 is initial moisture content (% db), 𝑀_𝑒 is equilibrium moisture content (% db), 𝑀_𝑖 is moisture content at time 𝑡 (% db), and 𝑡 is drying time (hr).

From the work of Abowei [18], the mass of hydrocarbon 𝑀 was accounted for when modeling one-dimensional diffusion of oil spill in water and obtain a general solution (see (13)) to predict the diffusion of known quantity of crude oil in water. This equation is analogous to the diffusion equation (12) describing diffusion of moisture in porous spherical particles as follows: 𝐶_𝑝=𝑀_𝑝𝐴4𝜋𝐷_𝑚𝑡^1/2𝑒^{−𝑥2/4𝐷𝑚𝑡}g/cm³, (13) where 𝑀_𝑝 is the quantity of oil spilled and 𝐶_𝑝 is the concentration of oil spilled at any time. 𝐴 is the area where oil is spill; 𝐷_𝑚 is the diffusion coefficient, and 𝑡 is the time.

Comparing (12) with (13), the term 𝑀_𝑝/𝐴[4𝜋𝐷_𝑚𝑡]^1/2 in (12) is analogous to the term (6/𝜋²)∑^∞_𝑛=1(1/𝑛²)in (13), and hence (12) can be rewritten as 𝑀MR=^𝑝𝜌𝐴4𝜋𝐷_𝑚𝑡^1/2exp−𝑛²𝜋²𝐷_𝑚𝑟²𝑡, (14) where MR is the moisture ratio. 𝐴 is surface area available for moisture transfer which for the rotary dryer is 𝜋(𝑅²+𝑅𝐿), 𝑀_𝑝 in equation (14) is now the mass of fermented ground cassava which is analogous to the 𝑀_𝑝 in equation (13), 𝜌 is the average density of sample to be dried, and the density is added 𝑡_𝑝 equation (14) to make the equation dimensionless, since moisture ratio MR is dimensionless, so that (14) becomes 𝑀MR=^𝑝𝜌𝜋𝑅²+𝑅𝐿4𝜋𝐷_𝑚𝑡^1/2exp−𝑛²𝜋²𝐷_𝑚𝑟²𝑡, (15) where 𝑟 is the average radius of particle to be dried 𝑅 is the radius of rotary dryer drum, 𝐿 is the length of the rotary dryer, and 𝐷_𝑚=𝐷 is diffusion coefficient = 𝐷_𝑜exp(𝐸_𝑎/𝑅𝑇), where 𝐸_𝑎 is the activation energy. Equation (15) is the new theoretical Abowei-Ademiluyi model for predicting the drying of any spherical particles in a rotary dryer. Equation (15) was simulated to obtain the theoretically determined moisture ratio.

2.1.1. Dimensional Analysis Approach

In order to remove moisture from a moist material in a rotary dryer, the moisture ratio (MR) can be taken to be a function of the change in temperature Δ𝑇, the quantity of fermented ground cassava to be dried 𝑀_𝑝, the latent heat 𝜆, diameter of particle 𝐷 to be dried, inlet air velocity 𝑣, and drum speed 𝑁, so that mathematically the moisture ratio MR is dimensionless as MR=𝜙Δ𝑇,𝑀_𝑝,𝜆,𝐷,𝑉,𝑁, (16) where 𝜙 is a correction factor.

Applying dimensional analysis we have 𝑀𝑀=𝜙𝑇^𝑎,𝑀^𝑏,𝐹𝐿𝑀^𝑐,𝐿^𝑑,𝐿𝜃^𝑒,1𝜃^𝑓. (17) Applying the Buckingham 𝜋 method gives 𝑀∶0=𝑏−𝑐,𝑇∶0=𝑎,𝐿∶0=𝑑+𝑒+𝑐,𝜃∶0=−𝑒−𝑓,𝐹∶0=𝑐. (18)(19)(20)(21)(22) Solving (18) to (22) gives 𝑎=0, 𝑏=0, 𝑐=0, 𝑑=−1, 𝑒=1, and 𝑓=−1 which MR=𝜙𝑉𝑁𝐷. (23) So that()𝐷𝜙=MR.𝑉𝑁 (24) The dimensionless constant 𝜙 can be evaluated theoretically and experimentally by substituting for MR in (14) into (24) to give the correction factor as 𝑀𝜙=^𝑝/𝜌𝐴4𝜋𝐷_𝑚𝑡^1/2exp−𝑛²𝜋²𝐷_𝑚/𝑟²𝑡𝐷.𝑉𝑁 (25)

2.2. Experimental Work

In order to validate the model, fermented ground cassava particles was dried in a bench scale rotary dryer (Figure 2). The developed theoretical model (15) was simulated with Microsoft Excel 2007 using the following data: (i)average density product (kg/m³) = 400, [19],(ii)drying time (120–1200 secs)-step 60, (iii)𝑟=0.0175 m, 𝑅=0.0508 m, 𝐿=0.46 m. The diffusion coefficients (𝐷_𝑇, 𝐷_𝑀, and 𝐷_𝑉 in m²/s) in (26)–(28) were obtained experimentally at different inlet air temperature (𝑇 in °C), inlet air velocity (𝑉 in m/s), and mass of feed (𝑀 in kg) from previous work [15] on fermented ground cassava as follows: 𝐷_𝑇=9.747×10⁻⁸exp−13892𝑟8.314𝑇²𝐷=0.994,_𝑀=8.938×10⁻¹⁰+5.937×10⁻¹¹∗log(𝑀)𝑀𝑟²𝐷=0.986,_𝑉=4.702×10⁻⁹+−8.49×10⁻⁹exp(−𝑉)𝑟²=0.990. (26)(27)(28)

Figure 2 

Schematic diagram of bench scale rotary dryer: (1) cyclone, (2  and  4) probe connections, (3) rotary drum, (5) feed hopper, (6) feed drive, (7) electric heater arrangement, (8  and  15) sight glasses, (9) air blower and orifice plate control, (10) support, (11) control box, (12) chain drive, (13  and  14) dried product receivers, and (16) steel table.

2.2.1. Sample Preparation

The cassava cultivar used in this study is TMS 30572 obtained from Rivers State Agricultural Development Project farm (ADP) at Rumuokoro, Port Harcourt. The choice of this cassava cultivar TMS 30572 was based on its preference by farmers, because of its high yield and suitability for gari processing [20]. The cassava cultivar was peeled, washed, grated, and packed in sack for pressing. The dewatered mash was allowed to ferment naturally for 72 hrs; sieved with a mesh of 3.5 mm, and then dried in a bench rotary dryer (Figure 2).

2.2.2. Experimental Procedure

At the beginning of each experiment, the dryer was allowed to reach steady state at the desired airflow rate, inlet air temperature, feed drive speed, and drum drive speed. When steady state condition had been attained, the fermented ground cassava mash of known moisture content was introduced into the dryer feed hopper. The drying conditions used in the experiments are inlet air temperatures of 115°C, 140°C, 190°C, and 230°C, air velocities of 0.83, 1.02, 1.397, and 1.55 m/s, mass of feed of 50 g, 100 g, 200 g, and 500 g, feed drive speeds 100 rpm, and drum drive speeds of 8 rpm. The decrease in mass of fermented mash was monitored with time per pass. The initial moisture content of samples was determined separately before start of experiment. The weight loss during drying was used to calculate the moisture content. The drying data obtained were used to calculate the experimental moisture ratio (MR) to predict the kinetics of drying fermented ground cassava.

3. Results and Discussion

3.1. Simulated Theoretical Results

The theoretical moisture ratio results are presented in Figures 3, 5, and 7. The theoretical moisture ratio decreases with drying time as inlet air temperature, inlet air velocity, and mass of feed increases. A similar profile is also exhibited in Figures 4, 6, and 8 for the experimental moisture ratio. The theoretical moisture ratio plots show a typical drying curve generally obtained during drying of moist materials [3, 21].

Figure 3 

Variation of moisture ratio with time at different air inlet temperatures using the theoretical model.

Figure 4 

Experimental moisture ratio at different inlet temperature.

Figure 5 

Variation of moisture ratio with time at different inlet air velocity using the theoretical model.

Figure 6 

Experimental moisture ratio at different inlet air velocity.

Figure 7 

Variation of moisture ratio with time at different mass of feed using the theoretical model.

Figure 8 

Experimental moisture ratio at different mass of feed.

It can be observed from the theoretical moisture ratio plots that the Abowei-Ademiluyi model does not give values for moisture ratio at 𝑡=0, and this will not be a problem, since the initial moisture content from which the moisture ratio at 𝑡=0 (i.e., 𝑀_𝑜) was calculated is always known at start of drying. Hence the Abowei-Ademiluyi model can be used to predict the drying kinetics of spherical particles at any known particle diameter, rotary drum diameter, and dryer length once the diffusion coefficient is known.

3.2. Comparison of Theoretical and Experimented Results

The simulated theoretical result compared favorably with those of the experimental results. The similarity is shown from the high value (𝑟 close to 1) obtained for the coefficient of multiple determinations 𝑅² at different inlet air temperature and inlet air velocity as shown in Figures 9 and 10. However, better fit could be obtained if the average density particle is correctly chosen. The theoretical (Abowei-Ademiluyi model) moisture ratio also compared well with experimentally moisture ratio at different mass of feed as shown in Figure 11.

Figure 9 

Theoretical (Abowei-Ademiluyi model) and experimental moisture ratios at different inlet air temperature.

Figure 10 

Theoretical (Abowei-Ademiluyi model) and experimental moisture ratios at different inlet air velocity.

Figure 11 

Theoretical (Abowei-Ademiluyi model) and experimental moisture ratios at different mass of feed.

4. Conclusion

The new theoretical, Abowei-Ademiluyi model has been developed for predicting drying kinetics of spherical particles at any known particle diameter, rotary drum diameter, and dryer length. The new model also account for the mass of feed. Model validation was carried out by drying fermented ground cassava particles in a bench scale rotary dryer at inlet air temperatures of 115–230°C, air velocities of 0.83 m/s–1.55 m/s, feed mass of 50–500 g, drum drive speed of 8 rpm, and feed drive speed of 100 rpm. The theoretical moisture ratio calculated from the model compared favorably with experimental moisture ratio.