# Mathematical Model and Numerical Method of Calculating the Dynamics of High-Temperature Drying of Milled Peat for the Production of Fuel Briquettes

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*Computation*—Computational Heat and Mass Transfer (ICCHMT 2023))

## Abstract

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## 1. Introduction

_{d.a.}= 60–100 °C towards the perforated partition, the space below which is connected to the air suction fan. The power of the fan limits the thickness of the layer of wet material and, accordingly, the productivity of the dryer.

_{d.a.}= 120–170 °C), and a highly intensive course (T

_{d.a.}= 300–500 °C) can be ensured. In the latter case, flue gases are involved as a drying agent. The high-temperature drying of milled peat particles can be accompanied by the thermal decomposition of the solid phase. The temperature interval at the beginning of the thermal destruction of peat is 160–210 °C [3]. At the same time, the decomposition of hemicellulose begins with the release of oxygen-containing gases and pyrogenetic moisture, which helps to reduce the weight and increase the caloric content of the dry residue [4,5,6,7]. The next two stages are characterized by the decomposition of cellulose and lignin and begin [3] in the temperature ranges of 260–340 °C and 315–400 °C, respectively. In the presence of air, the processes of the second and third stages of decomposition are exothermic, accompanied by a rapid increase in temperature and a significant loss of the combustible component of peat; these are undesirable in obtaining quality peat briquettes.

_{V}intensity function of phase transitions between the liquid and vapor phases in the internal points of the porous body. In general, the number of transfer equations in a mathematical model is determined by the number of system state parameters to be determined, as well as interphase interaction parameters, in particular the I

_{V}.

_{V}is determined by differentiating the sorption equation. With this approach, the I

_{V}for the case of stationary heat and mass transfer is zero, while during stationary processes, the value of I

_{V}can be of the same order as for non-stationary processes.

_{V}function is determined through the phase transition criterion. It represents the ratio of the change in moisture content of the material due to evaporation to the total change in moisture content due to mass transfer and phase transitions. However, these processes can take place independently of each other. That is why such a technique is represented by the introduction of an additional unknown function, the estimation of which requires conducting physical experiments. It should be noted that the model of A.V. Luikov is quite widely used for the theoretical description of the drying process [17,18,19,20,21,22]. Therefore, the phase transition criterion is defined for a sufficiently wide range of materials. In [17,18,19,20,21], the solution of the differential equations system of heat and mass transfer is carried out by an analytical method under several assumptions. In [22,23], the expediency of using numerical methods for the simultaneous solution of nonlinear differential transfer equations included in the mathematical model of the drying dynamics of porous materials is demonstrated.

- Darcy’s law for calculating the filtration velocities of the phases;
- the equation of state for calculating the pressures of the gas phase components;
- the expression for the capillary pressure of the liquid;
- the formula for the contact surface area of the liquid and gas phases in the pores of the body;
- the thermal-concentration deformation equation;
- the formulas for the intensity of the phase transitions on the outer and inner surfaces of peat particles and the diffusion coefficients of the liquid and gas phases.

_{ef}, of the bound substance particles. The A

_{ef}value [6] for crushed milled peat was included in the mathematical model. A numerical method for calculating a mathematical model has been developed. It allows for the determination of the dynamics of changes in temperature and volume concentrations of the bound substance components in a porous particle, depending on its thermophysical, structural, and geometric characteristics, as well as the parameters of the drying agent. The development of drying modes of porous materials is based on specific calculations. They ensure a reduction in the time of the process and, accordingly, energy resources for its implementation while maintaining the high quality of the final product.

## 2. Materials and Methods

#### 2.1. Mathematical Model

_{fl}, U

_{v}and U

_{ai}—volume concentrations of liquid, steam, and air; t—time; c

_{ef}—effective specific volumetric heat capacity of a peat particle, ${c}_{\mathrm{ef}}={c}_{b}{\rho}_{b}{\Psi}_{b}+{c}_{\mathrm{fl}}{U}_{\mathrm{fl}}+{c}_{v}{U}_{v}+{c}_{\mathrm{ai}}{U}_{\mathrm{ai}}$, c

_{b}, c

_{fl}, c

_{v}and c

_{ai}—specific mass heat capacities of the solid phase, water, steam and air, ρ

_{b}and Ψ

_{b}—density and volume fraction of the solid phase; λ

_{ef}—effective thermal conductivity: ${\lambda}_{\mathrm{ef}}={\lambda}_{b}{\Psi}_{b}+{\lambda}_{\mathrm{fl}}{U}_{\mathrm{fl}}/{\rho}_{\mathrm{fl}}+{\lambda}_{v}{U}_{v}/{\rho}_{v}+{\lambda}_{\mathrm{ai}}{U}_{\mathrm{ai}}/{\rho}_{\mathrm{ai}}$; D

_{fl}, D

_{v}and D

_{ai}—diffusion coefficients of liquid, vapor and air phases; I

_{V}—the intensity of phase transitions in the body pores; L—latent heat of phase transitions; ${\mathsf{\epsilon}}_{V}$—relative volumetric strain; ${w}_{\mathrm{ef}r}$—effective rate of filtration of the bound substance along the coordinate r, ${w}_{\mathrm{ef}r}=\left[{w}_{\mathrm{fl}r}{c}_{\mathrm{fl}}{U}_{\mathrm{fl}}+{w}_{gr}\left({c}_{v}{U}_{v}+{c}_{\mathrm{ai}}{U}_{\mathrm{ai}}\right)\right]/{c}_{\mathrm{ef}}$, where w

_{flr}, w

_{gr i}—filtration rates of liquid and gas phases.

_{u}—universal gas constant, A

_{D}—activation energy of diffusion transfer, γ

_{D}

_{fl}—diffusion coefficient. This formula is present in boundary cases, when ${A}_{D}/\left({R}_{u}T\right)>>1$ it becomes the empirical Arrhenius formula for solid bodies, and, when ${A}_{D}/\left({R}_{u}T\right)<<1$ it becomes the Einstein formula for liquid medium. Diffusion coefficients for gas phase components are found according to the well-known formula [32]: ${D}_{v}={D}_{\mathrm{ai}}={\mathsf{\gamma}}_{Dv}{T}^{3/2}/{P}_{\mathrm{g}}$, where P

_{g}—gas phase pressure, γ

_{D}

_{fl}—diffusion coefficient.

_{fl}and gas w

_{g}phases are proportional to the pressure gradients of the corresponding phase and can be calculated with Darcy’s Equation (5):

_{0}—total permeability of the medium; K

_{ψ}—relative permeability of the phase ψ; η

_{ψ}—dynamic viscosity coefficient of the phase ψ.

_{fl}and gas P

_{g}phases are calculated through functions U

_{fl}, U

_{v}, U

_{ai,}and T. For this, the volume fractions of the body Ψ

_{b}, liquid Ψ

_{fl}, and gas Ψ

_{g}in the material are determined according to the following relations: Ψ

_{b}= 1 − Π, Ψ

_{fl}= U

_{fl}/ρ

_{fl}and Ψ

_{g}= 1 − Ψ

_{b}− Ψ

_{fl}, where Π—peat porosity and ρ

_{fl}—liquid phase density. The partial densities of steam and air are equal, ρ

_{v}= U

_{v}/Ψ

_{g}and ρ

_{ai}= U

_{ai}/Ψ

_{g}, respectively. Partial pressures are found using the equation of state for diluted gases P

_{v}= ρ

_{v}R

_{u}T/μ

_{v}and P

_{ai}= ρ

_{ai}R

_{u}T/μ

_{ai}. The pressure of the gas mixture will be P

_{g}= P

_{v}+ P

_{ai}. Liquid phase pressure will be P

_{fl}= P

_{g}+ P

_{cap}, where the capillary pressure P

_{cap}is calculated [8,23] as the average capillary pressure of the liquid. The volume of liquid dV(r), contained in capillaries with a radius from r to r + dr in a unit volume of the body is proportional to the differential function f(r) of the capillary size distribution and the volume fraction θ(r) of the capillary, which is occupied by liquid: dV(r) = θ(r)f(r)dr. Then, the average value of the capillary pressure at a given point of the body is represented by an expression similar to the Laplace Equation (6):

_{min}, r

_{max}—minimum and maximum pores radius and r*—characteristic parameter of pore size dispersion.

_{Π}/P

_{H}(T); δ*—the average displacement length of an activated particle in the liquid layer, ${\mathsf{\delta}}^{*}=A/\left(\mathsf{\xi}n\right)$. Where A is activation energy at which a liquid particle can move, $\xi =$ const—coefficient of resistance to the particle movement to the free surface of the body, n—density of evaporating molecules, $n={U}_{\mathrm{fl}}{N}_{A}/\mathsf{\mu}$, μ—molar mass, N

_{A}—the Avogadro number; $\overline{\mathsf{\delta}}=\mathsf{\delta}/\mathsf{\delta}*$ when $0<\mathsf{\delta}<\mathsf{\delta}*$ and $\overline{\mathsf{\delta}}=1$ when $\mathsf{\delta}>\mathsf{\delta}*$. The value δ* can be considered as the thickness of the boundary layer. It is adjacent to the free surface of a rather massive, condensed body in which the evaporation process takes place.

_{P}can be found as a result of the solution of the system of two equations obtained by writing Equation (8) for two points on the saturation line. They correspond to the values of T

_{1}, P

_{s}

_{1}and T

_{2}, P

_{s}

_{2}in the table of saturated steam and water, respectively. Formula (8) is valid for liquids of various natures and is quite accurate. This is the case when dividing the temperature interval of the existence of water into two areas 0 < T < 100 °C and 100 < T < 374 °C. The maximum errors of calculated and tabular data were: Π

_{max}= 3.4% when A’ = 0.4206 × 10

^{8}J/mol and N = 0.4361 × 10

^{10}kg/(m·s

^{2}·K

^{0.5}) for the first interval, and Π

_{max}= 2.6% when A’ = 0.3689 × 10

^{8}and N = 0.8514 × 10

^{9}for the second interval. Here is A’ = AN

_{A}.

_{c}—coefficient of evaporation from the surface, ${\mathsf{\gamma}}_{\mathrm{c}}=\mathsf{\epsilon}{\mathsf{\rho}}_{\mathrm{fl}}\mathsf{\delta}*/4$; $\mathsf{\epsilon}$—coefficient of evaporation; φ

_{b}—moisture content of the body, which can be considered as the relative moisture content of the steam and gas mixture, which, according to the sorption isotherm, corresponds to the volume concentration of the liquid (7): ${\mathsf{\phi}}_{\mathrm{b}}=\overline{\mathsf{\delta}}(2-\overline{\mathsf{\delta}})$.

_{max}[33] freed from the liquid. This function can be determined based on the equation of the sorption isotherm [23]. According to the equation φ = f

_{φ}(U

_{fl}) the relative moisture content of the air φ is found, which corresponds to the volume concentration of the liquid U

_{fl}at the considered point of the porous body. Formula (7) determines the average thickness δ of the condensate layer on the surfaces of partially-filled capillaries. During the time interval dt, as a result of the processes of evaporation and heat and mass transfer, the thickness δ will change by the value dδ. This value is found when differentiating (7): $d\mathsf{\delta}={\mathsf{\delta}}^{*}d\mathsf{\phi}/(2\sqrt{1-\mathsf{\phi}})$. The volume concentration of the liquid will change in the same period of time to $d{U}_{\mathrm{fl}}={\rho}_{\mathrm{fl}}Sd\mathsf{\delta}$. Here, the expression for the contact area of the phases S is represented by the Equation (11):

_{fl}/∂φ

_{b}is determined by differentiating the desorption isotherm equation. For wood peat, the results on the equilibrium moisture content [10] are rather well-approximated by the equation ${U}_{\mathrm{fl}}=0.3{U}_{\mathrm{max}}{\left({\mathsf{\phi}}_{\mathrm{b}}/(1-{\mathsf{\phi}}_{\mathrm{b}})\right)}^{1/3}$.

_{V}is based on the differential equation of the thermal-concentration deformation [23]. In spherical coordinates, for the case of uniform blowing of the body, when there is axial symmetry of the deformation and the radial displacement u

_{r}depends only on the radius r, and there are no displacements u

_{φ}, u

_{θ}in the directions of φ and θ, the thermal-concentration deformation equation is represented in the following form (12):

_{1}—Lamé coefficients, $G={E}_{y}/[2(1+{\mathsf{\nu}}_{\Pi})]$, ${G}_{1}={E}_{y}{\mathsf{\nu}}_{\Pi}/[(1-2{\mathsf{\nu}}_{\Pi})(1+{\mathsf{\nu}}_{\Pi})]$; ${\mathsf{\nu}}_{\Pi}$—Poisson’s ratio; ${E}_{y}$—modulus of elasticity; N—thermal-concentration function [34,35], which determines the change in the specific volume of a body during its free expansion caused by the processes of thermal conductivity, diffusion, filtration, phase, and chemical transition, $N={\mathsf{\beta}}_{T}(T-{T}_{0})+{\displaystyle \sum _{\mathsf{\psi}}{\mathsf{\beta}}_{\mathsf{\psi}}({\mathsf{\omega}}_{\mathsf{\psi}}-{\mathsf{\omega}}_{\mathsf{\psi}0})}$. There ${\mathsf{\beta}}_{T}=\left(\partial x/\partial T\right)/x$ and ${\mathsf{\beta}}_{\mathsf{\psi}}=\left(\partial x/\partial {\mathsf{\omega}}_{\mathsf{\psi}}\right)/x$ are average coefficients of thermal and concentration expansion in the intervals of temperature [T, T

_{0}] and mass concentration of the component ψ [ω

_{ψ}, ω

_{ψ0}]. The function ε

_{V}is related to the normal components ε

_{rr}, ε

_{φφ}, ε

_{θθ}of the deformation tensor ε

_{ij}(i$,j$ = 1, 2, 3) by the relation ${\mathsf{\epsilon}}_{V}(t)=[1+{\mathsf{\epsilon}}_{rr}(t)][1+{\mathsf{\epsilon}}_{\mathsf{\varphi}\mathsf{\varphi}}(t)][1+{\mathsf{\epsilon}}_{\mathsf{\theta}\mathsf{\theta}}(t)]-1]$, where ε

_{rr}= du

_{r}/dr, ε

_{φφ}= ε

_{θθ}= u

_{r}/r. If the body is capillary-porous, its shrinkage during drying can be neglected, and ε

_{V}= 0.

_{t}of the beginning of the thermal decomposition of peat and the values of the effective energy of activation A

_{ef}are obtained. Thus, in the process of calculating the dynamics of high-temperature drying based on the mathematical model (1)–(4), as the temperature T

_{t}is reached in individual points of the peat particles, the values of A

_{ef}should be included in the formulas for D

_{fl}, P

_{s}, I

_{v}and I.

#### 2.2. Single-Valued Condition

_{0}and pressure P

_{g}of the steam and gas mixture in the internal points of the body are equal to the environmental temperature T

_{e.m.}and pressure P

_{e.m.}, respectively.

_{v}= ρ

_{v,e.m.}, ${T|}_{r=R}={T}_{\mathrm{e}.\mathrm{m}.}$.

^{0.33}Re. The similarity criteria Nu and Re include a diameter of the peat particle that depends on time due to shrinkage.

#### 2.3. Numerical Method of Solution

_{i}= ih, (i = 0, 1, …, IK; h ≠ const), t

_{n}= nl (n = 0, 1, …, l > 0). There, h is the spatial coordinate step, and l is the time step. At the initial time, the following applies: h = d/(2∙IK), where d is the peat particle diameter (Figure 1).

_{ψ}(ψ = T, fl, v, ai) allows to increase in the time step, Ω

_{ψ}≥ 0. The necessary conditions for the stability of the difference equations, obtained based on the method of a conditional set of some required functions of the system [22], have the form of inequality (24):

_{r}= w

_{efr}, ν

_{ψ}= λ

_{ef}/c

_{ef}, Ω

_{ψ}= Ω

_{T}when ψ = T; w

_{r}= w

_{flr}, ν

_{ψ}= D

_{fl}, Ω

_{ψ}= Ω

_{fl}when ψ = fl; w

_{r}= w

_{gr}, ν

_{ψ}= D

_{v}, Ω

_{ψ}= Ω

_{v}when ψ = v; w

_{r}= w

_{gr}, ν

_{ψ}= D

_{ai}, Ω

_{ψ}= Ω

_{ai}when ψ = ai. The calculated time step for the solution of the system (1)–(4) is selected from the condition $l\le min\left({l}_{T};{l}_{\mathrm{fl}};{l}_{v};{l}_{\mathrm{ai}}\right)$. In accordance with (24), Ω

_{T}, Ω

_{fl}, Ω

_{v}and Ω

_{ai}are determined after the optional sampling of the difference grid h steps.

#### 2.4. Approbation

_{e.m.}= 120 °C, at velocity of w

_{e.m.}= 1 m/s and moisture content of d

_{e.m.}= 10 g/kg of dry air. The shape of the particles was close to spherical with a diameter of 7 mm, the initial temperature was T

_{0}= 291 K, the initial moisture content—W

_{0}= 0.97 kg/kg, the porosity—Π = 0.6, the specific surface area S

_{max}= 8 × 10

^{5}m

^{2}/kg. For the calculation, thermophysical parameters of peat were taken [37]: λ

_{b}= 0.08 W/(m·K); c

_{b}= 1970 J/(kg·K); ρ

_{b}= 700 kg/m

^{3}. The activation energy was A = A

_{D}= 0.4350 × 10

^{8}J/kmol [5]. The constants used in the calculation are presented in Table 1. The constants for the coefficients of diffusion and evaporation from surface are determined by the experimental verification.

_{fl}, take place at the outer boundary. Due to shrinkage, the surface area of the external forces tends to decrease. The inner layers of the particle, in which the volume concentration changes more slowly under the action of compressive forces, somewhat reduce their area. At the same time, they increase in their thickness in such a way that their volume is practically unchanged. As the equilibrium moisture content is approached, a more uniform displacement of the coordinates of the nodal points along the radius of the particle is observed. The shrinkage of the particle was 28%, which corresponds to the data given in [10].

_{ef}= 0.370 × 10

^{8}J/kmol. Accordingly, the intensity of reaching the final peat moisture content with the simultaneous thermal decomposition process will increase. In order to account for the influence of thermal destruction on the dynamics of peat particle dehydration, the calculation program assumed the condition of a local change in the activation energy. The program itself was developed on the basis of the mathematical model (1)–(4) and the numerical method (18)–(22). The abovementioned means that when a temperature of 175 °C is reached at this nodal point of the particle at the given moment of time, in expressions (8)–(10), and for D

_{fl}, the value of the activation energies A and A

_{D}changed to the effective A

_{ef}. The results of calculating the change over time in the average values of temperature and moisture content of the particle are presented in Figure 4, Figure 5 and Figure 6. Figure 4, Figure 5 and Figure 6 also show the results of calculating the temperature change of the surface in contact with the high-temperature flow without taking into account thermal decomposition and taking into account its effect on the duration of dehydration. To exclude the self-ignition of particles, there was a flow of flue gases. When conducting numerical experiments, the temperature of the flow T

_{e.m.}and particle size were varied. The flue gases flow velocity was w

_{e.m.}= 4 m/s.

## 3. Discussion

_{fl}of the liquid phase, the intensity of evaporation on the outer and inner surfaces of peat particles, and for the saturation pressure. The formulas take into account the dependence of these parameters on temperature and activation energy at each spatial and temporal calculation step. The first stage of thermal destruction is the decomposition of hemicellulose. It is accompanied by the removal of oxygen-containing gases and pyrogenetic moisture and is characterized by a sharp change in the effective activation energy. When peat particles are in contact with a high-temperature flow, the temperature of their outer surface can reach the temperature of thermal decomposition. However, the internal pores will still contain a liquid phase. In this case, the removal of bound water will occur together with pyrogenetic water. This allows us to consider that the proposed approach, when the mathematical model takes into account the simultaneous processes of drying and thermal destruction of peat by means of a local change in the activation energy at the nodal points of the particles, is physically justified.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Stolbikova, G.E.; Kuporova, A.V. Milled peat drying in case of different size of feed and conditions. Gorn. Inf. Anal. Byulleten
**2018**, 5, 65–73. Available online: https://cyberleninka.ru/article/n/osobennosti-sushki-frezernogo-torfa-razlichnoy-udelnoy-zagruzki-i-rezhimov-sushki/viewer (accessed on 12 November 2022). (In Russian). [CrossRef] - Kindzera, D.P.; Khanyk, Y.a.M.; Atamanyuk, V.M.; Duleba, V.P. Kinetic of filtration drying of peat. NU Lviv. Polytech. Chem. Technol. Subst. Appl.
**2002**, 447, 179–182. Available online: https://ena.lpnu.ua:8443/server/api/core/bitstreams/10659275-4d3c-4960-baf9-d8ae59a11013/content (accessed on 13 November 2022). - Smolyaninov, S.I.; Lozbin, V.I.; Ikryn, V.M.; Bblichmaer, Y.A. Features of thermal decomposition of peat according to derivatographic data. Bull. Tomsk. Polytech. Univ. Geo Assets Eng.
**1976**, 274, 56–60. Available online: https://cyberleninka.ru/article/n/osobennosti-termicheskogo-razlozheniya-torfa-po-derivatograficheskim-dannym (accessed on 1 December 2022). - Mykhailyk, V.A.; Snezhkin, Y.u.F.; Oranska, O.I.; Korinchevska, T.V.; Korinchuk, D.M. Study of the thermal properties of solid residues of milled peat after the humus substances extraction. Ind. Heat Eng.
**2015**, 37, 54–64. (In Ukrainian) [Google Scholar] [CrossRef][Green Version] - Korinchuk, D. Non-isothermal analysis of components of composite fuels based on peat and biomass. Energy Autom.
**2018**, 1, 56–71. Available online: http://nbuv.gov.ua/UJRN/eia_2018_1_8 (accessed on 1 December 2022). (In Russian). [CrossRef] - Leroy-Cancellieri, V.; Cancellieri, D.; Leoni, E.; Filkov, A.I.; Simeoni, A. A global mechanism for the thermal degradation of peat. In Proceedings of the 4th Fire Behavior and Fuels Conference, St. Petersburg, Russia, 1–4 July 2013; International Association of Wildland Fire: Missoula, MT, USA, 2013. [Google Scholar]
- Chen, W.-H.; Kuo, P.-C. Isothermal torrefaction kinetics of hemicellulose, cellulose, lignin and xylan using thermogravimetric analysis. Energy
**2011**, 36, 6451–6460. [Google Scholar] [CrossRef] - Nikitenko, N.I.; Snezhkin, Y.F.; Sorokovaya, N.N. Mathematical simulation of heat and mass transfer, phase conversions, and shrinkage for optimization of the process of drying of thermolabile materials. J. Eng. Phys. Thermophys.
**2005**, 78, 75–89. [Google Scholar] [CrossRef] - Fil’kov, A.I.; Gladkii, D.A. Mathematical modeling of low-temperature drying of a peat layer. Tomsk. State Univ. J.
**2012**, 3, 93–106. (In Russian) [Google Scholar] - Luikov, A.V. Drying Theory; Energy: Moscow, Russia, 1968; p. 472. Available online: https://www.twirpx.com/file/1241798/ (accessed on 10 December 2022). (In Russian)
- Wang, Z.; Wang, Q.; Lai, J.; Liu, D.; Hu, A.; Xu, L.; Chen, Y. Numerical Simulation of Heat and Mass Transfer in Sludge Low-Temperature Drying Process. Entropy
**2022**, 24, 1682. [Google Scholar] [CrossRef] - Nikitenko, N.I. Problems of the radiation theory of heat and mass transfer in solid and liquid media. J. Eng. Phys. Thermophys.
**2000**, 73, 840–848. [Google Scholar] [CrossRef] - Aleksanyan IYu Titova, L.M.; Nugmanov, A.K. Simulation of the process of drying a dispersed material in a fluidized bed. Tech. Technol. Food Prod.
**2014**, 3, 96–102. Available online: https://cyberleninka.ru/article/n/modelirovanie-protsessa-sushki-dispersnogo-materiala-v-kipyaschem-sloe/viewer (accessed on 15 November 2022). (In Russian). - Strumillo, C.; Grinchik, N.N.; Kuts, P.S.; Akulich, P.V.; Zbicinski, I. Numerical modeling of nonisothermal moisture transfer in biological colloidal porous materials. J. Eng. Phys. Thermophys.
**1994**, 66, 181–190. [Google Scholar] [CrossRef] - Huzova, I.O.; Atamanyuk, V.M. Dynamics of drying processes of plant raw material in the period of decreasing speed. J. Chem. Technol.
**2022**, 30, 419–430. Available online: http://chemistry.dnu.dp.ua/article/view/259694 (accessed on 9 December 2022). - Luikov, A.V. Heat and Mass Transfer in Capillary Porous Bodies; Pergamon Press: Oxford, UK, 1966; pp. 233–303. [Google Scholar] [CrossRef]
- Akulich, P.V. Thermohydrodynamic Processes in Drying Technique; ITMO: Minsk, Belarus, 2002; p. 268. [Google Scholar]
- Kotov, B.I.; Bandura, V.N.; Kalinichenko, R.A. Mathematical modeling and identification of heat and mass transfer in plant dispersed material during drying and heating by an ultra-high frequency electromagnetic field. Energy Autom.
**2018**, 6, 35–50. Available online: http://nbuv.gov.ua/UJRN/eia_2018_6_6 (accessed on 3 December 2022). - Akulich, P.V.; Slizhuk, D.S. Heat and Mass Transfer in a Dense Layer during Dehydration of Colloidal and Sorption Capillary-Porous Materials under Conditions of Unsteady Radiation-Convective Energy Supply. Theor. Found. Chem. Eng.
**2022**, 56, 152–161. [Google Scholar] [CrossRef] - Rudobashta, S.P.; Kartashov, E.M.; Zueva, G.A. Mathematical modeling of convective drying of materials with taking into account their shrinking. Inzh. Fiz. Zh.
**2020**, 93, 1394–1401. [Google Scholar] [CrossRef] - Rudobashta, S.P.; Dmitriev, V.M. Investigation of the Diffusion Properties of Plant Capillary-Porous Colloidal Materials with Regard to Their Shrinkage. J. Eng. Phys. Thermophys.
**2022**, 95, 1357–1365. [Google Scholar] [CrossRef] - Narang, H.; Wu, F.; Mohammed, A.R. An Efficient Acceleration of Solving Heat and Mass Transfer Equations with the First Kind Boundary Conditions in Capillary Porous Radially Composite Cylinder Using Programmable Graphics Hardware. J. Comput. Commun.
**2019**, 7, 267–281. [Google Scholar] [CrossRef][Green Version] - Nikitenko, N.I.; Snezhkin, Y.u.F.; Sorokovaya, N.N.; Kolchik Yu, N. Molecular Radiation Theory and Methods for Calculating Heat and Mass Transfer; Naukova Dumka: Kyiv, Ukraine, 2014; p. 744. Available online: http://ittf.kiev.ua/wp-content/uploads/2016/12/nikitenko.pdf (accessed on 3 December 2022). (In Russian)
- Sneszkin, Y.F.; Korinchuk, D.N. Modeling of high-temperature drying of peat and biomass in biofuel production technologies. Sci. Work.
**2017**, 81, 125–130. Available online: https://sciworks.ontu.edu.ua/en/site/archives/81-1?page=2 (accessed on 15 January 2023). (In Russian). - Leibenzon, L.S. Variational Methods for Solving Problems in the Theory of Elasticity; Gostekhizdat: Moscow, Russia, 1943; p. 286. (In Russian) [Google Scholar]
- Nikitenko, N.I. Investigation of dynamics of evaporation of condensed bodies on the basis of the law of spectral-radiation intensity of particles. Inzh. Fiz. Zh.
**2002**, 75, 128–134. [Google Scholar] [CrossRef] - Sorokovaya, N.N.; Snezhkin, Y.F.; Shapar, R.A.; Sorokovoi, R.Y. Mathematical Simulation and Optimization of the Continuous Drying of Thermolabile Materials. J. Eng. Phys. Thermophys.
**2019**, 92, 1180–1190. [Google Scholar] [CrossRef] - Sorokova, N.; Didur, V.; Variny, M. Mathematical Modeling of Heat and Mass Transfer during Moisture–Heat Treatment of Castor Beans to Improve the Quality of Vegetable Oil. Agriculture
**2022**, 12, 1356. [Google Scholar] [CrossRef] - Broido, A. A Simple, sensitive graphical method of treating thermogravimetric analysis data. J. Polym. Sci. Part B Polym. Phys.
**1969**, 7, 1761–1773. [Google Scholar] [CrossRef] - Oostindie, K. A Simulation Model for the Calculation of Water Balance, Cracking and Surface Subsidence of Clay Soils; Rep. 47; Winand Staring Centre for Integrated Land, Soil and Water Research: Wageningen, The Netherlands, 1992; p. 65. Available online: https://books.google.com.ua/books/about/FLOCR.html?id=D39jHAAACAAJ&redir_esc=y (accessed on 12 January 2023).
- Garnier, P.; Perrier, E.; Angulo, A.J.; Baveye, P. Numerical model of 3-dimensional anisotropic deformation and water flow in welling soil. Soil Sci.
**1997**, 162, 410–420. [Google Scholar] [CrossRef][Green Version] - Rudobashta, S.P. Mass Transfer in Systems with a Solid Phase; Chemistry: Moscow, Russia, 1980; p. 248. Available online: https://www.libex.ru/detail/book796756.html (accessed on 11 January 2023). (In Russian)
- Keltsev, N.V. Fundamentals of Sorption Technology, 2nd ed.; Chemistry: Moscow, Russia, 1984; p. 590. (In Russian) [Google Scholar]
- Nikitenko, N.I. A method for calculating the temperature field by the data of measurement of the deformation of a body. Inzh. Fiz. Zh.
**1980**, 39, 281–285. [Google Scholar] - Nikitenko, N.I. Radiation heat conduction mikromechanism. In Proceedings of the First International Conference on Transport Phenomena in Processing, Lancaster, PA, USA, 22–26 March 1992; pp. 1580–1588. [Google Scholar]
- Kutateladze, S.S. Fundamentals of the Theory of Heat Transfer; Atomizdat: Moscow, Russia, 1979; p. 416. (In Russian) [Google Scholar]
- Voznyuk, S.T.; Moshinsky, V.S.; Klymenko, M.O.; Lyko, D.V.; Gneushev, V.O.; Lagodnyuk, O.A.; Voznyuk, N.M.; Kucherova, A.V. Peat Land Resource of the North-Western Region of Ukraine; NUWEE: Rivne, Ukraine, 2017; p. 117. Available online: http://ep3.nuwm.edu.ua/id/eprint/7506 (accessed on 11 October 2022). (In Ukrainian)

**Figure 2.**Graphs of changes in the average values of moisture content W and temperature T, as well as the temperature on the surface T

_{IK}of a spherical particle of peat with a diameter of d = 7 mm during its drying with the flow uniform washing with the following parameters: T

_{e.m.}= 120 °C, w

_{e.m.}= 1 m/s, d

_{e.m.}= 10 g/kg of dry air.

**Figure 3.**Curves of changes in time of nodal points coordinates ${r}_{i}=f({r}_{i}^{0}{,}_{}t)$ along the radius of the particle, which at the initial moment of time had the value ${r}_{i}^{0}=i{h}^{0}$, (i = 0, 1, …, IK; IK = 5).

**Figure 4.**Changes over time in the average moisture content W and temperature T, temperature T

_{IK}on the surface of a spherical particle of peat with a diameter of d = 10 mm during drying with and without taking into account thermal destruction (W1, T1, T

_{IK}1) in the flow of flue gases with the parameters: T

_{e.m.}= 300 °C, w

_{e.m.}= 4 m/s, d

_{e.m.}= 12 g/kg of dry gas.

**Figure 5.**Changes over time in the average values of moisture content W and temperature T, temperature T

_{IK}on the surface of a spherical particle of peat with a diameter of d = 10 mm (

**a**) and a diameter of d = 13 mm (

**b**) during drying with and without taking into account thermal destruction (W1, T1, T

_{IK}1) in the flow of flue gases with parameters: T

_{e.m.}= 400 °C, w

_{e.m.}= 4 m/s, d

_{e.m.}= 12 g/kg of dry gas.

**Figure 6.**Changes over time in the average values of moisture content W and temperature T, temperature T

_{IK}on the surface of a spherical particle of peat with a diameter of d = 10 mm (

**a**) and a diameter of d = 13 mm (

**b**) during drying with and without taking into account thermal destruction (W1, T1, T

_{IK}1) in the flow of flue gases with parameters: T

_{e.m.}= 500 °C, w

_{e.m.}= 4 m/s, d

_{e.m.}= 12 g/kg of dry gas.

Name | Meaning |
---|---|

Constants for coefficients of diffusion, | γ_{D}_{fl} = 0.9 × 10^{−8} m^{2}/s; γ_{D}_{v} = 0.134 × 10^{−4} m^{2}/s; |

Coefficient of evaporation from surface, | γ_{c} = 0.2578 × 10^{−4} kg/(m^{2}∙s); |

Total permeability of the medium, | K_{0} = 1 × 10^{−5}; |

Relative permeability of the fluid, | K_{fl} = 0.2 × 10^{−14}; |

Relative permeability of the gas, | K_{g} = 1.1 × 10^{−8}; |

Characteristic parameter of pore size dispersion | r* = 1 × 10^{−8} m. |

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## Share and Cite

**MDPI and ACS Style**

Sorokova, N.; Variny, M.; Pysmennyy, Y.; Kol’chik, Y.
Mathematical Model and Numerical Method of Calculating the Dynamics of High-Temperature Drying of Milled Peat for the Production of Fuel Briquettes. *Computation* **2023**, *11*, 53.
https://doi.org/10.3390/computation11030053

**AMA Style**

Sorokova N, Variny M, Pysmennyy Y, Kol’chik Y.
Mathematical Model and Numerical Method of Calculating the Dynamics of High-Temperature Drying of Milled Peat for the Production of Fuel Briquettes. *Computation*. 2023; 11(3):53.
https://doi.org/10.3390/computation11030053

**Chicago/Turabian Style**

Sorokova, Natalia, Miroslav Variny, Yevhen Pysmennyy, and Yuliia Kol’chik.
2023. "Mathematical Model and Numerical Method of Calculating the Dynamics of High-Temperature Drying of Milled Peat for the Production of Fuel Briquettes" *Computation* 11, no. 3: 53.
https://doi.org/10.3390/computation11030053