93
Table 5.1
Commonly used semi - empirical and empirical thin layer drying Models.
Model Model Equation Ref.
Semi-empirical models
(a) Models derived from Newton’s law of cooling
Lewis (Newton) MRexp(kt) Lewis (1921)
Page MRexp(ktn) Yaldy and Erteky (2001)
Modified Page - 1 MRexp(kt)n Overhults et al. (1973) Modified Page - 2 MRexp ( ) kt n White et al., (1981) Modified Page - 2
exp 2
t n
MR k
l
Diamante and Munro (1993)
(b)Models derived from Fick’s second law of diffusion
Henderson and Pabis (Single term) MRaexp(kt) Henderson (1974)
Logarithmic MRaexp(kt)c Chandra and Singh
(1995)
Midilli-Kucuk MRaexp(ktn)b t* Midilli et al.(2002) Demir et al. MRaexp[(kt)]n b Demir et al. (2007)
Two-term 1
2
exp( ) exp( )
MR a k t
b k t
Henderson and Pabis (1961)
Two-term exponential exp( )
(1 ) exp( ) MR a kt
a kat
Sharaf - Eldeen et al.
(1980) Modified two-term exponential
(Verma et al.)
exp( ) (1 ) exp( o ) MR a kt
a g t
Verma et al. (1985)
Diffusion approach model or approximation of diffusion
exp( ) (1 ) exp( ) MR a kt
a kbt
Akpinar et al. (2003)
Modified Henderson and Pabis model (Three term exponential )
exp( ) exp( )
exp( )
o o
MR a kt b g t c h t
Karathanos (1999)
Empirical models
Thompson taln(MR)b[ln(MR)]2 Thompson et al. (1968) Wang and Singh MR 1 b t* a t* 2 Wang and Singh (1978) Kalemullah MRexp(c T* )b t* (pT n ) Kaleemullah (2002)
95
The thin layer drying equations are easy to use in the modelling of the solar dryers due to less complexity and the requirement of less data, unlike the distributed model. The drying constant of the thin layer drying equation combines all the transport properties and describes the drying phenomenon in a unified way. The detailed review of the thin layer drying equations were undertaken by Erbay and Icier, 2009 and Kucuk et al., 2014. The commonly used thin layers drying equations are listed in Table 5.1.
Development of an appropriate model of a particular product is significant for designing a new drying system or improving an existing dryer and identifying the optimum operating parameters for precise prediction of the simultaneous heat and mass transfer phenomenon in the drying process (Kucuk et al., 2014). Drying systems need to be designed properly to meet the desired operating parameters of a particular product and to get the satisfactory performance in view of the energy requirements and quality of the product (Chavan et al., 2008). The drying air velocity, temperature, relative humidity, and size of the material affect the kinetics of the drying process and model parameters (Rajkumar et al., 2007). Full-scale experimental investigation of the dryer of different configurations under diverse conditions is a costly affair and time-consuming. Therefore, modelling of the drying process of a specific product under the given drying conditions and the simulation model of the dryer are necessary for predicting the performance of dryer.
A thin layer model of the drying process of a particular product can be developed by the experimental method. In the thin layer drying experiment, air at the constant flow rate, temperature, and relative humidity is supplied through a thin layer of food or agricultural product. The mass of the product is measured at a regular interval of time until the equilibrium moisture content is reached. Then the moisture content of the product is calculated at a regular interval of time. The moisture content at any given time on wet basis can be evaluated by applying Eq. (5.1) (Ekechukwu, 1999). Then MR vs time curve is plotted. The MR at constant relative humidity is obtained by Eq. (5.2). However, when the relative humidity of the drying air fluctuates, the MR is obtained by the Eq. (5.3) (El - Sebaii and Shalaby, 2013).
(1 )
1 i ip
t
tp
M M m
m
(5.1)
( )
( )
t e
i e
M M
MR M M
(5.2)
t i
MR M
M (5.3)
where 𝑀𝑡 is the moisture content at a given time (w.b.), 𝑚𝑖𝑝 is the initial mass of the product, 𝑚𝑡𝑝 is the mass of product at a given time, 𝑀𝑖 is the initial moisture content (w.b.), and 𝑀𝑒 is the equilibrium moisture content (w.b.).
The regression analysis is carried out with different thin layer drying models to find out the models constants and the coefficients. The goodness of fit is judged by calculating statistical parameters such as coefficient of determination (R2), reduced chi - square (x2), root mean square error (RMSE), and mean relative deviation modulus. The widely used statistical parameters are 𝑅2,𝑥2 and RMSE. The best model is chosen based on the criteria of the highest value of 𝑅2 and the lowest values of 𝑥2 and RMSE. The values of the 𝑅2, 𝑥2 and RMSE are calculated applying Eqs. (5.4)‒(5.6) (Goyal et al., 2007; Shi et al., 2013; Sharma et al., 2005, Koukouch et al., 2015; Aghbashlo et al., 2009).
2
r, ,
2 1
2 1 ,
( )
1
( )
N
p i ex i
i N
pr ex i
i
MR MR
R
MR MR
(5.4)2
, ,
2 iN1(MRex i MRpr i)
x N n
(5.5)
, ,
21
1 N
ex i pr i
RMSE i MR MR
N
(5.6)Effective moisture diffusivity is an important factor which combines various diffusion process in solids such as the molecular diffusion, capillary flow, Knudsen flow, hydrodynamic flow and surface diffusion. It indicates the flow of moisture in the material. The effective moisture diffusivity can be calculated from Eq. (5.7) by the method of slopes (Vega et al., 2007).
2
2 2
8 exp 4
eff s
D π t
MR π L
(5.7)
Where Deff is the effective moisture diffusivity, Ls is the length of the product thickness and t is the drying time. In the method of slop, a curve is plotted between ln (MR) and t. The slope
97 of the curve gives the rate constant 𝑘 which is equal to
2