The concern of the present work is the convective drying of empty cocoa shells in an indirect solar dryer. Some drying experiments, using one sample, were carried out. During the experiments, the sample is introduced in the drying chamber. Then at steady time intervals, the sample is withdrawn from the drying chamber, for a rapid weighing. After each weighing, the sample is reintroduced in the dryer. At each time interval, the ambient temperature of the drying chamber and its relative humidity γ are measured by a thermo-hygrometer. From the experimental data, a theoretical determination of the moisture evaporated from the product was performed and a good agreement was found between the theoretical and experimental values, confirmed by the value of the RMSE. Those calculations used the constants in the Nusselt number found in literature. Then those constants were evaluated again, to get new values more suitable with the experimental data. The dimensionless numbers of Nusselt, Grashof and Prandtl were calculated. That allowed the calculation of the average value of the Nusselt number. The average convective heat transfer coefficient was determined.
An old technique used by farmers to preserve their foodstuffs is natural Solar drying. This process of course reduces the moisture content of the foodstuffs. The result is a reduction of the wastes, resulting in a decrease of the microorganism’s activities. Compared with the natural air drying, the indirect solar drying has the advantage that the solar radiation does not directly impinge the drying product. Drying involves heat and mass transfers, which are linked to the specific nature of each product. Regarding cocoa pods, some drying studies have been carried out about the cocoa beans [
Drying is a physical process which involves heat and mass transfers. When heat is transferred to the product surface, water flows from inside the product to its surface. The product surface temperature also increases. As a result, moisture is evaporated from the product surface. The rate of heat used to evaporate water is linked to the heat transfer coefficient which depends on some parameters such as the product temperature, the product surrounding air temperatures and the physical properties of humid air. Those parameters allow the calculation of the rate of heat and the mass of water evaporated from the product. Those calculations are used as basis of the modelling of the drying process. Several modellings are found in literature. Some of them are based on the second law of Fick. Some other modellings don’t use this Fick’s law which allows the calculation of the moisture ratio [
The materials are the electronic compact scaleSF-400C (accuracy of ± 0.01 g) used for weighing small samples of shells for drying experiments, and also materials used for temperature and solar radiations measurements and data logging.
The solar radiation-absorbing surface, i.e. the absorber, of the solar collector is a rectangular steel plate, with dimensions of 0.705 × 1.77 (width × length) painted back. The absorber is covered by a transparent flat glass plate, 4 mm
thick. It has an area of 1.248 m2. There are nine fins soldered along the absorber, 2 m long, which create ten channels. The collector has a rectangular entrance through which the ambient air enters following the ten channels under the absorbing surface and receives thermal energy from the absorber. This thermal energy is transferred in the drying chamber. The drying chamber is made of steel plates. It is insulated with plywood sheets, 10 mm thick. The dimensions of the drying chamber are 0.45 × 0.77 × 0.85 m (width × height × depth). There are six trays inside the drying chamber. The drying chamber is fitted with a chimney 24 cm high. A schematic representation of the solar dryer is given in
For the drying experiments, one sample was used. The sample has a square shape with dimensions 3 cm × 3 cm. Its initial mass was 11.24 g.
During the experiments, the sample is introduced in the drying chamber of the dryer. Then at steady time intervals Δt = 10 min, the sample is withdrawn from the drying chamber, for a rapid weighing. After each weighing, the sample is reintroduced in the dryer. At each time interval Δt, the ambient temperature of the drying chamber and its relative humidity γ are measured by a thermo-hygrometer. The results gotten from the experiment of March 3rd 2022 are given in this paper.
The convective heat transfer coefficient h c is, in general, linked to the physical properties of humid air. Those physical properties are very often gathered to get some dimensionless numbers. Moreover, the theoretical moisture evaporated from a drying product is correlated to the convective heat transfer coefficient, and also to the difference between the partial pressure of water vapor in the air and the partial pressure of water vapor in the saturated boundary layer just near the evaporating surface of the product.
The Nusselt number Nu is one of the dimensionless numbers involved in the evaluation of the convective heat transfer coefficient. It is expressed by the following relation
N u = h c L k f (1)
Then one gets
h c = k f L N u (2)
L is the characteristic dimension which is here L = 0.03 m.
Moreover, the Nusselt number Nu is linked to the Grashof number Gr and the Prandtl number Prby the following relation
N u = C ( G r P r ) n (3)
In the above Equation (3), C and n are experimental constants. Some values of C and n are given in literature. Jannot, Y (2012) [
Finally, the convective heat transfer coefficient h c is expressed as
h c = k f L C ( G r P r ) n (4)
The dimensionless numbers Gr and Pr are expressed as indicated below
G r = g β L 3 ρ f 2 ( T i − T p ) μ f 2 (5)
T p is the temperature of the product surface; T i is defined as: T i = T p + T a 2 .
T a is the temperature of the drying air above the product surface; β is the coefficient of volumetric expansion; ρ f is the density of humid air; μ f is the dynamic viscosity of humid air
P r = μ f C f k f (6)
C f is the specific heat of humid air, k f is the thermal conductivity of humid air.
The rate of heat used to evaporate moisture from the drying product is expressed as [
Q e v = 0.016 h c [ P ( T p ) − γ P ( T i ) ] (7)
P(T) is the partial vapour pressure at temperature T; γ is the relative humidity. From Equations (4) and (7), one gets
Q e v = 0.016 k f L C ( G r P r ) n [ P ( T p ) − γ P ( T i ) ] (8)
The theoretical moisture evaporated from the product, at time t, is expressed as
m t h = Q e v A t λ (9)
A is the area of the product; λ is the latent heat of evaporation of water. From Equations (8) and (9), one gets
m t h = 0.016 k f L λ C ( G r P r ) n [ P ( T p ) − γ P ( T i ) ] A t (10)
In a first step, the values of C and n used were those found in literature: C = 0.59; n = 0.25.
The evaluation of the physical properties of humid air, found in the above equations, was performed, at the temperature T i , by using the following equations [
C f = 999.2 + 0.1434 T i + 1.101 × 10 − 4 T i 2 − 6.7581 × 10 − 8 T i 3 (11)
k f = 0.0244 + 0.7673 × 10 − 4 T i (12)
ρ f = 353.44 T i + 273.15 (13)
μ f = 1.718 × 10 − 5 + 4.620 × 10 − 8 T i (14)
P ( T ) = exp [ 25.317 − 5144 T + 273.15 ] (15)
The latent heat of evaporation is expressed as [
λ = − 2.419 T p + 3164 (16)
In Equation (16), Tp is expressed in K.
The temperature Tp of the product was not measured, but estimated. Some previous works show a link between Tp and Ta, the temperature of the drying air above the product surface [
From those equations, the theoretical moisture evaporated from the product mth was estimated step by step, for each time interval. The calculations were made by choosing, for the constants C and n, C = 0.59; n = 0.25.
In order to evaluate the goodness of fit between the theoretical moistures evaporated mth end the experimental values mex, the Root Mean Square Error (RMSE) is determined according to the following formula
RMSE = [ 1 N ∑ i = 1 N ( m e x − m t h ) 2 ] 1 / 2 (17)
Then, a new calculation of C and n was performed, by using the following procedure [
Z = 0.016 k f L λ [ P ( T p ) − γ P ( T i ) ] A t (18)
Then, from Equation (10), one can write
m t h Z = C ( G r P r ) n (19)
When the logarithm of the two sides of Equation (18) is taken, one gets:
ln ( m t h Z ) = ln C + n ln ( G r P r ) (20)
Let Y, X, m and b be defined as:
Y = ln ( m t h Z ) (21)
X = ln ( G r P r ) (22)
m = n (23)
b = ln ( C ) (24)
Then one gets the following linear equation
Y = m X + b (24)
From the slope m of the regression straight line, one gets the constant n.
Then from the intercept b of the line, one gets C = e b .
From the experimental weighing of the sample at steady time intervals Δt = 10 min, the evolution of the sample mass with time is gotten. It is shown in
Then the calculations of the the theoretical moistures evaporated mth was carried out, using the temperature and relative humidity data. The results are given in
On the whole, there is a good agreement between the experimental and the theoretical moistures evaporated from the product. Significant differences between the experimental and the theoretical moistures evaporated occur for only two points, for which Ta = 52.9˚C and Ta = 51.6˚C. It should be noted that those two points are characterised by a sudden rise of the temperature Ta of the drying air above the product surface, resulting in more moisture evaporated that the model takes into account. The good agreement between the experimental and the theoretical moistures evaporated was confirmed by the value of the Root Mean Square Error. It was found RMSE = 0.0307 g.
The data were also used for the calculation of the constants C and n of the Nusselt number, using the procedure described from Equation (18) to Equation (24).
The equation found is:
Y = 0.2425 − 0.5473 X (25)
From Equation (25), one gets C = 0.5785 and n = 0.2425.
The values of the Nusselt number, the Grashof number, the Prandtlnumber and convective heat transfer coefficient were calculated.
The Grashof number varies from 6.544 × 103 to 8.645 × 103. As for the Prandtl number, it varies from 0.6946 to 0.6974. For the Nusselt number, the following average value found was: Nu = 4.4938. The average value of the convective heat transfer coefficient was: hc = 4.2252 W/m2˚C.
| Ta (˚C) | Tp (˚C) | Ti (˚C) | γ (%) | mth (g) | mex (g) |
|---|---|---|---|---|---|
| 49.6 | 47,0 | 48.3 | 0.25 | 0.1267 | 0.12 |
| 49.8 | 47.2 | 48.5 | 0.32 | 0.115 | 0.14 |
| 49.4 | 46.8 | 48.10 | 0.22 | 0.131 | 0.14 |
| 53.1 | 50.5 | 51.8 | 0.19 | 0.164 | 0.20 |
| 49.1 | 46.5 | 47.8 | 0.21 | 0.131 | 0.14 |
| 52.9 | 50.3 | 51.6 | 0.16 | 0.169 | 0.08 |
| 51.6 | 49.0 | 50.3 | 0.18 | 0.15 | 0.10 |
| 49.8 | 47.2 | 48.5 | 0.19 | 0.139 | 0.14 |
| 49.2 | 46.6 | 47.9 | 0.19 | 0.135 | 0.13 |
| 47.4 | 44.8 | 46.1 | 0.22 | 0.118 | 0.09 |
| 45.9 | 43.3 | 44.6 | 0.23 | 0.107 | 0.08 |
| 43.6 | 41.0 | 42.3 | 0.26 | 0.091 | 0.09 |
| 43.4 | 40.8 | 42.1 | 0.26 | 0.0907 | 0.09 |
| 40.2 | 37.6 | 38.9 | 0.36 | 0.065 | 0.07 |
| 39.2 | 36.6 | 37.9 | 0.32 | 0.066 | 0.06 |
The concern of the present work is the convective drying of empty cocoa shells in an indirect solar dryer. From the experimental data, a theoretical determination of the moisture evaporated from the product was performed. A good agreement was found between the theoretical and experimental values. Then the two constants involved in the Nusselt number were determined. It was found that C = 0.5785 and n = 0.2425. The dimensionless numbers of Nusselt, Grashof and Prandtl were calculated. The average value for the Nusselt number was: Nu = 4.4938. As for the convective heat transfer coefficient, its average value was: hc = 4.2252 W/m2˚C.
The authors declare no conflicts of interest regarding the publication of this paper.
Touré, S., Ogo, A.C. and Sidibé, M. (2024) Experimental Study and Thermal Modelling of Cocoa Shell Convective Drying in an Indirect Solar Dryer. Modeling and Numerical Simulation of Material Science, 14, 69-78. https://doi.org/10.4236/mnsms.2024.142004