Fundamental Mathematical Relations of Solar Drying Systems

Abstract

Drying of agricultural products is a widely spread method achieving a physiochemical stabilization of the material by removing part of the moisture content, producing therefore products with new qualitative properties and different nutritional and economical value. Significant amounts of agricultural crops are dried artificially in mechanical drying systems using heated air. Simulation models of the drying process are used either for designing new or improving existing drying systems or for the control of the drying process. All parameters (transfer coefficients, drying constants, etc.) used by the simulation models are directly related to the drying conditions, i.e., temperature and velocity of the drying medium inside the mechanical dryer. As a consequence, the drying conditions, as directly related to the drying time, are affecting the energy demands.

The drying process principles, describing the periods of drying and their modeling (constant and falling-rate periods), are first reported and analyzed, giving the fundamental mathematical relations describing the drying process and the driving forces involved. The concepts of water activity and equilibrium moisture content are therefore introduced, in order to describe the fundamental concept of sorption-desorption isotherms which are the curves that fundamentally determine the way the particular solid can be dehydrated.

In the subsequent chapter, the basic mathematical relations and theories of the drying process involving simultaneous heat and mass transfer models as well as those of the simplified thin-layer and deep-bed models are given.

An overview of solar drying methods (in both thin-layer and deep-bed dryers) along with the principal solar drying systems (direct sun dryers, passive and active dryers) will then be briefly introduced, discussing the respective fields of application and analyzing their advantages and disadvantages. Finally, the basic mathematic equations used for describing and modeling the various physical processes within the most common drying systems and devices are reported. A brief discussion of the recent advances in modeling is finally presented where pertinent.

Nomenclature

aw	Water activity	–
a, b, c, n	Constants	–
A	Area	m2
Ap	Area of a drying product	m2
Bi m	Biot number for mass	–
Bi h	Biot number for heat	–
c p,p	Specific thermal capacity of a product	kJ ⋅ kg−1 ⋅ K−1
c p,a	Specific thermal capacity of air	kJ ⋅ kg−1 ⋅ K−1
C, c o, K, k o	Constants in GAB equation	–
D	Moisture diffusion coefficient	m2 ⋅ s−1
D o	Constant in the Arrhenius equation for diffusion	–
D eff	Effective value of diffusivity	m2⋅ s−1
D eff,ref	Reference value for effective diffusivity	m2⋅ s−1
DR	Drying rate	kg ⋅ kg−1 dry mater ⋅ s−1
Fo	Nondimensional Fourier number	–
h g	Heat transfer coefficient	W ⋅ m−2 ⋅ °C−1
hfg	Latent heat of evaporation of water	kJ ⋅ kg−1
J m	Mass flow	kg⋅ m−2⋅ s−1
j h	Chilton-Colburn coefficient for heat	–
j m	Chilton-Colburn coefficient for mass	–
k g	Mass transfer coefficients	kg ⋅ m−2 ⋅ s−1
k, k o , k 1	Constants	–
Κ v	Vapor diffusion coefficient	m2 ⋅ s−1
K	Drying constant	h−1
L	Characteristic length	m
m	Mass	kg
m p	Mass of product	kg
m d	Mass of dry material	kg
M	Moisture content, wet basis	kg H2O ⋅ kg−1 prod.
Mo	Initial moisture content, wet basis	kg H2O ⋅ kg−1 prod.
M eq	Equilibrium moisture content, wet basis	kg H2O ⋅ kg−1 prod.
MR	Moisture ratio	–
M v	Vapor molecular weight	kg
M g	Drying air molecular weight	kg
Nu	Nusselt number	–
Pr	Prandtl number	–
P v	Vapor pressure	Pa
P v,sat	Saturated vapor pressure	Pa
P t	Total pressure of vapor	Pa
P w	Pressure of vapor over water	Pa
q	Heat flow	W ⋅ m−2
Re	Reynolds number	–
R	Σταθερά αερίων	8.3143 J ⋅ mol−1⋅ K−1
t	Time	h
T, θ	Temperature	oC
T abs	Absolute air temperature	K
T air	Air temperature	oC
T p	Product temperature	oC
T wb	Wet-bulb temperature	oC
Υ	Moisture content of air	kg H2O ⋅ kg−1 dry air
V	Volume	m3
V b	Apparent volume	m3
V ref	Reference volume	m3
vair	Velocity	m ⋅ s−1
W	Weight	kg
W o	Initial weight	kg
W d	Weight of dry matter	kg
Χ	Moisture content, dry basis	kg H2O ⋅ kg−1 dry matter
Χο	Initial moisture content, dry basis	kg H2O ⋅ kg−1 dry matter
Χeq	Equilibrium moisture content, dry basis	kg H2O ⋅ kg−1 dry matter
Χ cr	Moisture content at critical point	kg H2O ⋅ kg−1 dry matter
Χ s	Moisture content, dB, at the surface	kg H2O ⋅ kg−1 dry matter

1.1 Greek Symbols

α	Air or thermal diffusivity of a fluid	m2 ⋅ s−1
ε	Porosity of a solid	–
λ	Thermal conductivity	W ⋅ m −1 ⋅ oC−1
μ	Dynamic viscosity	kg ⋅ m −1 ⋅ s−1
ν	Kinematic viscosity	m2 ⋅ s−1
ρ	Density	kg ⋅ m−3
φ	Relative humidity	%
Φ	Drying parameter	–

1.2 Indices

air	Air
b	Bulk
cr	Critical
d	Dry matter
eff	Effective value
eq	Equilibrium value
h	Heat
in	Initial value
l	Liquid
m	Mass
ο	Initial value
p	Product
ref	Reference value
sat	Saturated value
v ή vap	Vapor
w	Water