Simple Method to Calculate the Parameters of Sphere Drying Model

Abstract

In the exponential model proposed by Henderson (eq. (1)), the parameter M_e is simply obtained from eq. (5). There existed no such method for Sphere Drying Model (eq. (2)). The author proposed a simple method to calculate the parameters M_e and K of the sphere drying model, defining a function φ_1/n(X) expressed in eq. (6).
The calculation procedure is as follows:
(1) Choose the time t₁ and t₂, where t₂=nt₁, and measure the moisture content M₁ and M₂ at time t₁ and t₂ respectively.
(2) Calculate the value φ*=M₀-M₂/M₀-M₁
(3) Find the value X which satisfies φ_1/n(X)=φ* from the φ_1/n(X) and Φ(X) table, a part of which is shown in Tab. 1.
(4) Find the value of Φ(X) for the X determined above.
(5) The value of parameter K is given by K=X/t₂
(6) M_e is given by M_e=M₂-M₀Φ(X)/1-Φ(X).
This method was applied to the experimental data of grain drying and showed a remarkable stability with calculated parameters, especially for drying data of beans.
The author also computed ξ and μ the relative and normalized deviations of calculated values of K and M_e from “true” values, if the measured moisture contents M₁ and M₂ deviated from “true” values described with ε₁ and ε₂ in eq. (14), where “true” means “of the model with the most suitable parameters for the experimental data determined by e. g. least square method”. As an example, ξ and μ values with ε₁=±0.01, ε₂=-ε₁, are shown in Fig. 3 and Fig. 4.
The partial derivatives of ξ and μ with respect to ε₁ and ε₂ at ε₁=ε₂=0, which represent the contribution of ε₁ and ε₂ to ξ and μ, are shown in Fig. 5 and Fig. 6. Considering the results of Fig. 3-Fig. 6, the following points must be noticed if the deviations ξ and μ are to be small.
i) Both ξ and μ become smaller with greater n.
ii) ξ has a minimum within the non-dimensional time X=3-6, while μ is expected to decrease as X increases.