As the simplest model, a first-order differential equation is used, reflecting the heat balance as a whole for the caryopsis, in the form as follows:

(2) CMd ( ΔT ) /dt   =   P E   +   P k   -   P d ΔT   =   T   -   T 0

Where:

Radiation power absorbed by the product:

(3) P E (t) = K s S 1 AE(t)

Where:

Convective heat transfer power with the environment:

(4) Pk   =   K α S 1 [ Δ T c ( t )   -   K g Δ T] Δ Tc ( t )   =   T c ( t )   -   T0

Where:

Power spent on desorption of moisture (evaporation):

(5) P d   =   K d ( λ   +   ε/μ ) dm B /dt

Where:

In the general case, during the heating process, neither the mass of caryopsis M (as a result of moisture evaporation), nor the heat capacity C, nor the heat transfer coefficient K_α, nor the coefficient Kg remains constant due to their dependence on the current temperature and moisture content.

The moisture content of the product undergoing treatment also affects the absorption factor А, with the adsorption potential ε.

The processes of heat and moisture exchange are interrelated and, in the general case, are described by a system of corresponding non-linear differential equations. Essentially, in the case of taking into account heat losses due to evaporation, and in the general case, other exothermic reactions as well, occurring when the product is heated, to determine the temperature in a function of time, it is necessary to solve the system of non-linear differential equations:

(6) CMd(ΔT)/dt   =   S 1 {K s AE(t)   +   K α [ΔT c   -   K g ΔT]}-             Kd(λ+ε/μ)dm B /dt},             dm B /dt=-f W [T(t),m B ]

(7) C(T)[M+m B (t)]d(ΔT)/dt   =   S 1 {K s AE(t)   +   K α [ΔT c   -             K g ΔT]   -   Kd(λ+ε/μ)dm B /dt},             dm B /dt = -W[T(t),m B ]

(8) d(ΔT)/dt   +   S 1   K α KgΔT/C(T)[M+m B (t)]   =   {S 1 [K s AE(t)   +     K α ΔT c T   -   Kd(λ+ε/μ)dm B /dt]}/C(T)[M+m B (t)]

The function f_W[T(t),m_в] is non-linear and reflects the process of moisture evaporation. Usually, it is not possible to solve the system of equations (6) in an analytical form. The use of numerical methods is also practically impossible since the number of coefficients included in the equations is not determined. Therefore, we have to resort to various kinds of assumptions and simplifications, often rather rough ones.

Suppose that:

(9) S 1 K α Kg/C(T)[M+m B (t)] = Kt = const,   {S 1 [K s AE(t) +  K α ΔT c T - Kd(λ+ε/μ)         dm B /dt]}/C(T)[M+m B (t)] =                a 1 =const

Where:

ΔТ = Т- Т₀, Т – temperature, °С; Т₀ – initial temperature, °С; Т_∞ – temperature, °С when t → ∞; t – time, s;

ΔT ∞   =   T ∞   − T 0

In the case of the constancy of these quantities, we have for the initial conditions ΔТ(0) = 0, the model of heating can be represented as:

(10) ΔT(t) = K 0 [1-exp(-K t t)]

Or

(11) t(ΔT) = -ln[1 - ΔT/K 0 ]/K t

Where:

The coefficients are taken as constant (independent of time), but dependent on the heating conditions (initial moisture content, irradiance, and the ambient temperature in the treatment zone). The condition of constancy (weak change) of the coefficients is not satisfied in the full temperature range. It is known from experiments that close to a product temperature of 100 ºС, intensive dehydration begins, and the rate of moisture loss sharply increases. Therefore, the proposed dependence can be considered to be an initial, rather rough approximation.

A similar solution can be obtained using the solution obtained by A.V. Lykov in the form of the exponential series, taking the first two terms (Lykov and Mikhailov, 1972)

(12) ΔT/(T ∞ -T 0 )    = 1 + a 1   exp(-k 1 t)   +   ai   exp(-k i t)

Where:

a and k are coefficients.

Note that when identifying the parameters of models (10) for the temperature increment and (11) for time, according to the same experimental data, the numerical values of K₀ and K_t in the expressions will differ.