Modelling (2).docx

D. Determination of the Drying Models that Best Describe the Drying of Sand in this Experiment Thin-layer equations are often used for a description of the drying kinetics for various types of porous materials which fall mainly into three categories, namely, theoretical, semitheoretical and empirical. Henderson and Pabis model and Lewis model are both semi-theoretical models which are used in this experiment to compare the curve on the drying characteristic of the sand. The Henderson and Pabis model is the first term of a general series solution of Fick’s second law [CITATION Hen69 \l 1033 ].

MR=

M−M e M 0 −M e

=a exp (-kt ) eq (1)

This model was used successfully for modelling a drying corn [CITATION Hen69 \l 1033 ], wheat [CITATION Wat \l 1033 ] and peanuts [ CITATION Mos89 \l 1033 ]. The slope of this model, coefficient k, is related to effective diffusivity when the drying process takes place only in the falling rate period and liquid diffusion controls the process [ CITATION Mad96 \l 1033 ]. The Lewis (Newton) model ( [ CITATION Lew21 \l 1033 ] is a special case of the Henderson and Pabis model where intercept is unity. Lewis described the moisture transfer from agricultural materials as analogous to the flow of heat from a body immersed in cold fluid. Comparing this phenomenon with Newton’s law of cooling, the drying rate is found to be proportional to the difference in moisture content between the material being dried and the equilibrium moisture content in the drying air condition. This can be depicted as,

dM =−k ( M−M e ) dt

eq (2)

Or after integrating yields

MR=

M −M e M 0 −M e

=exp (-kt ) eq (3)

a. Henderson and Pabis Drying Curve Model Using the values of moisture content of sand obtained from the data gathered in the experiment, the equation of Henderson and Pabis model (Eq 1) is used to determine the drying characteristic of the sand in oven at 80⁰C and 120⁰C. The resulting data is graphed and is shown in Figure 1.4 a and b. Henderson and Pabis Model at 80⁰C 0

0

ln(MR)

-0.2

f(x)20= - 0.01x40+ 0.06 60 R² = 0.98

80

100

120

140

160

-0.4 -0.6 -0.8 -1 -1.2

Time,t(min) SERIES 1 Li near (SERIES 1)

Li near (SERIES 1)

Figure 1.4.1 Curve of Henderson and Pabis Model at 80⁰C Handerson Pabis model at 120°C 0

ln(MR)

-0.5

0

f(x) =10- 0.05x + 0.220 R² = 0.96

30

40

50

-1 -1.5 -2 -2.5

Time,t(min) SERIES 1

Li near (SERIES 1)

Figure 1.4.2 Curve of Henderson and Pabis Model at 120⁰C

60

Both the graph of 80⁰C and 120⁰C are described by plotting the natural logarithmic of moisture content versus time. From the figure, it is clear that moisture ratio decreased considerably with increasing drying time. This is due to the escape of the moisture content of the sand as it evaporates. The time required to reduce the moisture ratio to any given level was dependent on the drying temperature, increasing at 80°C and decreasing at 120°C. It is because the main factor influencing drying kinetics was the drying temperature, as noted in other studies [ CITATION Bel00 \l 1033 ]. Thus, a higher drying temperature produced a higher drying rate and consequently the moisture content decreased faster. The coefficient of determination, R2, of 80°C is 0.9808, which is relatively higher to the R2 of 120°C, which is 0.958. This means that at 80°C, the Henderson and Pabis model has a better fit compared to that of 120°C. b. Newton Curve Model

Newton model at 80⁰C 0 -0.2

0 f(x) = - 0.01x 20 R² = 0.99

40

60

80

100

ln(MR)

-0.4 -0.6 -0.8 -1 -1.2

Time,t(min) Li near ()

Figure 1.4.3. Newton Model at 80°C

120

140

160

Newton Model at 120⁰C 0

ln(MR)

-0.5

0 f(x) = - 0.04x 10 R² = 0.97

20

30

40

50

60

-1 -1.5 -2 -2.5

Time,t(min) Li near ()

Figure 1.4.4 Newton Model at 120 °C The figures above show the graph of Newton model at 80

° C and 120◦C. The graph is

described by plotting the natural logarithmic of moisture ratio versus time. The time for the drying process to be carried out at 80 at 120

° C needed longer time compared to the newton model

° C due to the fact that the rate of evaporation is actually driven by relative humidity

in relation to temperature. As the temperature increases, the relative humidity decreases therefore it can absorb more fluid. In the figure, it clearly stated that moisture ratio decreases with increasing drying time with 120°C as lower compared to 80°C. With temperature as one factor in determining the drying kinetics, therefore at higher temperature, the drying rate will increase while moisture ratio decreases. The coefficient of determination, R 2 and k for 80°C are 0.9678 and 0.0064 while at 120°C, 0.9383 and 0.0404 respectively. The mathematical model derived for 80°C is MR = EXP(-0.0064t) and MR = EXP(0.0404t) for 120°C. c. Tabulation of Handerson and Pabis Model and Newton Model

Table 1. R^2, k, a and model equation of Handerson and Pabis Model and Newton Model temp 80

120

model Newton Handerson and Pabis Newton Handerson and Pabis

R^2 0.9678

k

a 0.0064

0.9808 0.9383

0.0071 0.0404

0.958

0.0458

1.86824 6 1.21993 8

mathematical model EXP(-0.0064t) 1.868246EXP(0.0071t) EXP(0.0404t) 1.219938EXP(0.0458t)

Conclusion The sand drying experiment was carried out determine the drying kinetics and determine the suitable drying model based on data gathered. In modeling, Handerson and Pabis Model and Newton model were used and based on the data, Handerson and Pabis model has a highest coefficient of determination with R2 = 0.9808 and 0.958 at 80°C and 120°C. The mathematical model developed for Handerson and Pabis model differ at different temperature with moisture ration, MR = 1.868246EXP(-0.0071t) for 80°C and MR = 1.219938EXP(-0.0458t) for 120°C. This model developed can be used to predict variations of moisture ratio at specific time of the same material at the same temperature with reasonable accuracy using the said model.

References Belghit, A., & Kouila, M. a. (2000). Experimental Study of Drying Kinetics by Forced Convection of Aromatic Plants. Energy COnservation and Management, 1303-1321. Henderson, S. M., & Pabis, S. (1969). Temperature Effect on Drying Coefficient. Journal of Agriculture Engineering Research, 169-174. Lewis, W. K. (1921). The Rate of Drying of Solid Materials. Industrial Engineering Chemistry, 13,427. Madamba, P. S., Driscoll, R. H., & Buckle, K. A. (1996). Thin Layer Drying Characteristics of Garlic Slice. Journal of Food Engineering, 29,75-97.

Moss, J. R., & Otten, L. (1989). A relationship Between Color Development and Moisture Content During Roasting of Peanut. Canadian Institute of Food Science and Technology Journal, 22,34-39. Watson, E. L., & Bhargava, V. K. (1974). Thin Layer Studies on Wheat. Canadian Agricultural Engineering, 16,18-22.