The general theory of heat and mass transfer in capil- lary porous and other dispersed media belonged to Prof. Lykov. It was based on the thermodynamics of irrever- sible processes and the theory of generalization of vari- ables. According to Prof. Lykov, sets of equations are to be solved as a single complex process [1].The theory of heat and mass transfer is based on solving sets of linear equations with boundary condi- tions, which corresponded to constant and variable po- tentials in a medium that varied according to established laws. Thus, it was intended for stationary material and medium [2].To intensify and improve technological processes, one needs reliable, physically valid modelling metho- ds [3]. This is especially important for energy-inten- sive drying processes of wet solids in a fluidized bed that is moving along a cylindrical surface, e.g. drum dryers [4]. In this paper, modelling means a physical  ana- lysis of heat and mass transfer, as well as the hydrody- namics of the processes that occur a rotary drum dryer, their mathematical description,  and  possible  solutions by analytical or numerical methods involving various software [5]. The analytical and numerical methods were based on preliminary data on the kinetics of drying and heating of individual particles. Such information was obtained either from the available model representations or from experimental data. In most cases, preference was given to direct experimental data, which took into account possible effects of anisotropy of heat and mass transfer properties and irregular geometric shape of par- ticles of real solids.An adequate description of the continuous drying technology of wet solids in a fluidized bed requires an integrated approach to the problem. Such an approach requires a system analysis of hydrodynamic, diffu- sion, and thermal processes complicated by an overlap  Copyright © 2019, Antipov et al. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), allowing third parties to copy and redistribute the material in any medium or format and to remix, transform, and build upon the material for any purpose, even commercially, provided the original work is properly cited and states its license.   of various phenomena. A complete theoretical picture of a continuous drying process should be based on a mathematical model that would link a set of typical structures, or idealized models, each of which reflects a particular type of transfer or transformation. The opti- mal way to develop a new drying technique is to com- bine the multimodel system of drying processes with experimental studies on the kinetics of moisture remo- val in a fluidized bed [6].The research objective was to develop a multimo- del system for the continuous process of drying and heat transfer in a fluidized bed of wet solids. STUDY OBJECTS AND METHODSThe study was based on a complex of general and specific scientific methods. The general scientific me- thods involved analysis and synthesis, testing a theory with practice, interpretation of the results obtained, etc. The specific scientific methods included the abstract logical method, the method of modelling, the empiri- cal method, the method of statistical  probability,  etc. The theoretical and methodological foundation of the research included studies conducted by Russian and fo- authentic trial equipment were used to conduct the ex- periments and test the physical and mathematical models of the drying and steam treatment processes. S  F T FS STThe systematic approach in modelling the convec- tive drying processes. The process of continuous dry- ing is an original research subject that can be marked as a certain system S, while the model of the drying pro- cess M represents a different system. According to the systematic approach principle, the drying process inte- racts with the external environment E. Depending on the research objective, the study may feature different ratios between the subject and the external environment. These relations represent a model of relations of the external environment with the subject. Since our ideas of the su- bject and the external environment are also models, the following models can be proposed: M is the model of the subject S, M is the model of external environment at the inlet, M is the model of external environment at the output, M is the model of the connections between the external environment and the subject at the inlet, and M is the model of the connections between the external environment and the subject at the output.The combination of all these models reign experts in the field of drying, such as Ginzburg, M   = (M , M , M , M , M ),                    (1) Frolov, and Lykov. RS               F         FS          S ST         T The research featured sunflower oil seeds.The thermophysical characteristics of the vege- table raw material were determined according to the non-stationary thermal mode method of two tempera- ture-time points developed by Volkenstein. The method of differential thermal analysis was used to identify the intervals of temperature zones of moisture evaporation with different forms and energy of moisture  binding with the material. It was accompanied by the method of differential scanning calorimetry, which was used for quantitative measurement of heat flows that occurred when the trial sample and the control sample were un- dergoing a simultaneous programmed heating. The methods of high-performance gas chromatography, atomic absorption spectroscopy, infrared spectroscopy, capillary electrophoresis, and acid hydrolysis were used to determine the content of vitamins, amino acids, and other quality indicators of the wet solids. The measure- ment errors did not exceed the values established in the current standards for quantitative analysis of the quality indicators of the wet solids. The main part of the theo- retical and experimental research was carried out on the premises of the Voronezh State University of Engine- ering Technologies (Voronezh, Russia) and the Bo- brovsky Vegetable Oil Plant (Bobrov, Russia).The research objective was achieved by the synthesis and analysis of classical and novel analytical and empi- rical methods in the sphere of heat and mass transfer and food dehydration studies. The obtained relations, the ap- proximating equations, and the simulation results corre- sponded with the experimental data. The measurement results underwent statistical processing. The procedures and design solutions did not contradict the established methods  of  rational  design  and  engineering.  Moderncomputer mathematical programmes, instruments, and forms  a  multimodel  system  of  transfer  phenomena(Fig. 1).A drum dryer as a subject of system modelling of non-stationary processes. Fig. 2 shows the experimen- tal drum dryer used in the study of the continuous dry- ing process.The pressure type fan (2) was fixed on the angular steel frame (1). The fan was designed to supply atmo- spheric air through the duct (4) into the heater (6). The heater consisted of several sections. The flow rate of the air supplied to the heater was regulated by means of the gate valve (5). The temperature and the relative humidi- ty of the air entering the drying chamber were measured with the dry (7) and the wet (8) thermometers.At the core of the whole design there was a steel drum (11) with a diameter of 0.3 m and a length of1.3 m. On the inner surface of the drum there was the channel nozzle (12). The nozzle contained longitudinal slots to feed the drying agent to the bed of wet granu- lar material. The drying drum (11) was supported by two  Fig. 1. Multimodel system of transfer phenomena.   view B   Fig. 2. Experimental drum dryer with a channel nozzle.  pairs of rollers (13). It was driven by the electric motor(26) and the gearbox (24), which were kinematically in- terconnected by the chain transmission (25).During the drying process, the drying agent was sup- plied directly to the zone of the channel nozzles under the dryable material.The pipe (9) was assembled on the body of the loa- ding chamber (10). The pipe fed the wet bulk material to the drying drum. The spiral rotation of the drum moved the material to the discharge chamber (14).The retaining ring (15) was assembled on the end surface of the drum. The ring slid in the groove of the fixed flange (16), through which the bulk material was continuously unloaded. During the drying process, the interior of the drum (11) was under a slight negative pressure due to the fact that the performance of the fan (21), which took away the waste drying agent, was seve- ral times higher than that of the delivery fan (2).The temperature control of the drying agent was car- ried out directly in the bed of the wet solids. The ther- mocouples (17, 18, and 19) were installed on the bracket(20) (Fig. 2, view B).The temperature was recorded with the electronic au- tomatic self-recording potentiometer (23).The relative humidity of the waste drying agent was measured with the hygrometer (22), a sorption-frequency single-channel digital device. The semiconductor ther- moanemometer (27) measured the speed of the drying agent at the inlet and the outlet of the drum, as well as above and below the surface of the bed.The angle of the drum, the frequency of its rotation, and the speed and temperature of the drying agent were set at the initial stage of the experiments. After that, there began a continuous supply of proportioned  wet bulk material. Over the next 60–70 min, the material was sampled to measure its thermophysical characteris- tics, while the process of wet material feeding continued in the same mode.95 After the sampling, the feeding of the wet bulk ma- terial to the drying drum and the movement of the drum stopped simultaneously. The speed and the temperature of the drying agent remained constant, according to the experimental conditions. After that, all the material was discharged from the drum into the receiving container, and its volume and weight were measured.Taking into consideration the design feature of rota- ry drum dryers and the methods of continuous drying of wet solids in a fluidized bed, the model of continuous drying process can be represented as a multimodel sys- tem of transfer phenomena (Fig. 3).The continuous drying process as a system of transfer phenomena. The model of the continuous dryi- ng process of wet solids in a fluidized bed displays the properties of the material and the coolant and the nature of non-steady processes that occur in the fluidized bed. It also makes it possible to improve the drying at various initial parameters of the material and the coolant.The model of the product subjected to drying and the coolant connects the model of the system with the exter- nal environment and control actions. It also calculates the properties of the material and the coolant.The following models of transfer phenomena cor- respond with the continuous process in a rotary drum dryer (Fig. 3):the model of the product subjected to drying;the coolant model;the model of the movement of the wet solids on the cy- lindrical surface;the model of the coolant fed to the fluidized bed;the hydrodynamic model of the fluidized bed;– the model of the complex heat and mass transfer;– the model of the drying process in the fluidized bed;and– the model of the technical and economic characteristics.The model of the product subjected to drying. If we consider the wet product as a subject of drying with   Fig. 3. Multimodel system of the continuous drying process of wet solids in a fluidized bed.  its moisture content, temperature, density of individual particles, and average linear dimensions, we can con- struct a mathematical model of the technological prope- rties of the bulk material [8]. The model can describe the physical properties necessary for hydrodynamic and heat transfer processes. In this case, the linear dimen- sions of the bulk material determine its volume, surface area, equivalent diameter, sphericity (non-sphericity) co- efficient, and the specific surface area. The equilibrium moisture, angle of repose, and the bulk density of the wet solids are determined depending on their humidity and the moisture of the drying agent. The specific heat is de- termined depending on the temperature of the product.The model of the product subjected to drying. Let us consider the wet solids as a subject of drying, chara- cterized by moisture content, temperature, density of individual particles of the product, and average linear dimensions. By doing so, we can construct a  mathe- matical model of the technological properties of the wet solids [8]. The model describes  the  physical  proper- ties necessary for conducting hydrodynamic and heat transfer processes. In this case, the linear dimensions of the bulk material are determined by its volume, sur- face area, equivalent diameter, sphericity coefficient (non-sphericity), and the value of specific surface. Equi- librium moisture, angle of repose, and the bulk density of the wet solids are determined depending on their hu- midity and the moisture of the drying agent. The specific heat depends on the temperature of the product.The coolant model. The wet and heated atmospheric air is considered as a binary mixture of the components of dry air and steam [9]. The parameters of the drying are  calculated  by  the  method  of  superposition  of  thecomponent  parameters.  Tables  and  various  functional and empirical relations are used to calculate the parame- ters of the components.The model of the movement of wet solids on the cylindrical surface. In drum dryers, the material is dried in a fluidized bed while it is purged with a drying agent [5]. Fig.4 shows the flow chart of the bulk material in the transverse direction.The angle of the bed in the cross section is determined by the angle of repose of the bulk material. It is assumed that the trajectory of the particles passes along a radial arc, whose radius equals the distance from the axis of the drum. When the particle gets to the end of the arc and sur- faces, it falls down along the chordal surface of the flow.The particle moves in the axial (translational) direc-tion only if the flow surface has a certain axial angle, i.e.   Direction of rotation               Fig. 4. Scheme of the movement of the wet solids in the rota- ting drum (cross section).   when there is a difference in the levels of the bed at the inlet and the outlet holes of the drum.During translational motion, the trajectory of the particle follows the chordal plane. In this case, the tra- jectory is not perpendicular to the axis of the drum: it forms a < 90° angle with it and is perpendicular to the plane passing through the middle of the chords.The model of the coolant fed to the fluidized bed. The drying agent enters the drying chamber of  the drum dryer through the channels formed by the channel nozzles [6] and the outer cylinder of the drum through the side cavity of the drum (Fig. 5).The side cavity of the drum was divided by a par- tition in such a way that the drying agent entered the channels under the nozzles that are situated under the material while the drum is rotating. The channel nozzle has a gap of constant width. The drying agent enters the drying chamber through this gap and contacts the layer of dispersed material.Thus, the task is reduced to the calculation of gas distribution with an outflow through the lateral perme- able surface, i.e. the layer of the dispersed material. It is possible to make an assumption that the parameters of area of the product or the specific surface area of the bed and the speed with which the drying agent flows around the particles.When solving the problem of the hydrodynamics of a bed, one has to determine the hydraulic resistance of the bed of the dispersed material. It is necessary to know the characteristics of the pore channel of the bed, its coeffi- cient of the hydraulic resistance, and the flow rate of the drying agent in the channel.For generality, it is assumed that the flow velocity around the particle and the flow speed in the channel are equal and related to the flow velocity in the direction of filtration in the whole liquidized bed by the ratio: w  0= ξw.                                        (2) The tortuosity coefficient ξ is calculated as follows: ξ = 1 + (π/2 – 1)(1 – ε)2/3.                       (3) The bed voidage ε is the ratio of voids between the particles in the layer and the volume of the bed. It can be expressed as the following ratio: the drying agent and the height of the bed are constant.Thus, one can calculate the hydrodynamics of the flow ε = 1 – ρ / ρ .                                (4)  p  dof the drying agent. In this case, the bed height may be its height in the middle section of the material flow. The parameters of the drying agent, such as density and ki- nematic viscosity, are assumed to be constant.Two models were considered when choosing the mathematical model for the coolant supply from the channel nozzle slit through the bed of dispersed material. The first was the model for calculating the distribution of velocity and pressures along the z-shaped collectors; theThe specific surface of the material particles in thebed can be calculated through the specific surface of the sparticles S and layer porosity ε:  sg            sS   = S (1 – ε).                                 (5) The loss of pressure during the movement of the dry- ing agent through the granular layer can be calculated similarly to the pressure loss in pipelines: second was the model of the constant-section air distribu- tor with a longitudinal slit of constant width. Δ = λ S hρw2 / (2ε2).                       (6)  g  sgThe hydrodynamic model of the fluidized bed. Ac- cording to the hydrodynamics of the fluidized bed of the wet solids in rotating drum dryer, the drying agent flows around the particles in the fluidized bed of the granular product and between the particles of the material in the channels [6].In this connection, we can point out external (flow-a- round), internal (filtration), and mixed hydrodynamics problem.When solving the problem of heat and mass trans- fer,  it  is  necessary  to  determine  the  active  surface   Fig. 5. Scheme of the coolant flow in the rotating drum dryer.The model of the complex heat and mass trans- fer. In the general case,  when  organizing  continu- ous drying of wet solids in a fluidized bed, the coolant is supplied to the drying chamber at the expense of an external heat source (mechanical energy). As  a  re- sult, there is a continuous forced flow of the heat transfer agent particles around the product.  The  pro- cess of continuous evaporation of moisture  from  the free surface of the moving material occurs,  in  this case, in the  boundary layer and  is caused  by veloci- ty and temperature gradients. As a  result,  there  oc- curs a continuous diffusion flow of the medium. This continuous flow is constantly directed from  the  sur- face of the product into the depth of the coolant flow. The difference in the concentration of the vapor-gas mixture near the evaporation surface and in the main heat-wave flow of the coolant leads to a density diffe- rence in the vapor-gas mixture. It results in a continuous free natural heat and mass transfer. Fig. 6 demonstrates the external heat and mass trans- fer under conditions of continuous dehydration as a com- bination of simple (marginal) modes of motion: forced, free, and diffusive.To calculate the simultaneous continuous heat and mass transfer processes, we used the method based on the superposition of the absolute values of Nusselt num-                                                 er Fig. 6. Motion modes during continuous drying: Re – forced; Ar – free natural; PW – diffusive.Fig. 7. The motion of the medium near the heat exchange surface in mixed convection.   ber [9]. In a complex continuous process, the absolute The effective heat transfer coefficient α is deter- value of the transfer rate is determined according to the projection values of the transfer rate of simple processes onto two mutually perpendicular planes.The velocity attitude of the forced motion of the me- mined by a nonlinear differential equation, presented in dimensionless form:dy/dx = A(y/x) + Rb,                          (13)  Re dium Umakes an arbitrary angle φ with the planes, in where y = t – T ; x = u – u ; A = αS / c K; Rb = r(u – u  /  Ar which the velocity vectors of the free motion Ulie. i                            e                       s        m                                 i         e Fig. 7 shows that the calculated dependence of the forced and the free motion with an arbitrary mutual ori- entation is [c (t – T)] is Rehbinder number. m The solution for the equation (13) is determined by the following power function: Nu = [(Nu4   ORe cos4φ ± Nu4 )0,5) + Nu2   ORe sin2φ]0.5.  (7) Rb = c[(t – T ) / (T – T )]m (αS / c K)n.         (14)  OAr  i             f           i t sWhen φ = 0, the forced and the free motions happenin the same plane:The effective heat transfer coefficient for the flu- idized bed of wet solids under continuous dehydration conditions is determined by the following relation: Nu = (Nu4  ORe ± Nu4 )0.25.                     (8) α  = (c K / S )[lnRb – lnc + mln(t – T ) / (T – T )] / n. (15)  OAr When φ = π/2, the forced and the free motions hap-er             m              s i             f           i pen in mutually perpendicular planes: The model of the drying process in the fluidizedbed. Let us consider the general case of an approximate Nu = (Nu2  ORe ± Nu2 )0.5.                       (9) mathematical  description  of  a  continuous  process  of  OAr If the critical geometrical dimensions of the forcedmoisture transfer: and the free motions are the same (L = L = L  ), formu- dx /dτ = –Kx (τ), Re            Ar                                                                                                    1                          1la (8) coincides with the following relation: dx /[G / U(τ)]Kx (τ),                         (16)4             4                      4                                                                                                                           2                                   1 Nu = NuORe  + NuOAr .                       (10) where, in terms of optimal control methods [8], the pa- In accordance with the nature of the flow of the me-dium in the liquidized bed, the calculated relation of the rameters x 1 control actand x 2  .are phase variables, and U(τ) is the intensity of the continuous transfer of the complex pro- cess meets conditions (8) and (10).The absolute value of Nusselt number in forced mo- ion opt The system of equations (16) is a combination of the kinetics of the continuous drying and the balance rela- tion of moisture in the material and coolant. tion conditions is calculated by the formula for the ball: Let us define the optimal control U (τ) which trans-  Nu OAr= 2 + 0.56(Ar × Pr)0.25[Pr / (0.846 + Pr)]0.25,        (11) fers the continuous process from 1                 1  2                  2– the given initial state: where 1 < Ar Pr < 105. In conditions of free motion it is calculated by the formula for the sphere: x (0) = x  Nu ORe= 2 + 0.03(Re0.54Pr0.33  + 0,35Re0.58Pr0.36,       (12) 0 and x (0) = x 0                              (17) – and in the specified final state:  1    f             1  f and x (τ ) = x f                              (18) where 0,6 < Pr < 8×103 and Re < 3×105.In formulae (11) and (12), the critical value is the x (τ ) = x 2    f             2 equivalent particle diameter of the wet solids. so that the functional assumes the minimum value:t к  er The effective heat transfer coefficient αis used as a I = òU (t )dt .                            (19) characteristic of heat transfer intensity. It takes into ac-count the total amount of heat spent on continuous dry- 0Let us establish a function with auxiliary variables ing in a rotating drum dryer. λ , λ and λ : 0      1                 2   H = λ U – λ Kx  + λ (GK/U )x (20) In rotary drum dryers, the warm-up periods are short, 0               1        1          2                  t     1 and the drying rate is constant. Hence, the drying rate  1 and write the system of equations for the function λ :dl1 / dt = -¶H / ¶x1 = l1K - l2 (GK / UТ );dl2  / dt = ¶H / ¶x2  = 0.    (21) can be represented as a function of moisture content [10]: p  du/dτ = – K(u – u ).                           (27) For rotary drum dryers, u = dz/dt. In this case, if we The optimal control of the continuous process results from the condition that the function H has the maximum value:2 take into account Eq. (25) and Eq. (26), the drying rate can be represented as follows: p du/dz = – (KSρ/q)(u – u ).                    (28)   Thus, ¶H / ¶UТ = l0 - l2 (GK/ UТ ) x1 = 0, (22)   The  heat  is  continuously  supplied  to  the  dryable product. It heats the material and evaporates the mois- Uopt (t ) = (l2  / l0 )GKx1 . (23) ture: The following function solves the equations of the system (7), which describe the kinetics of continuous drying with the boundary condition (8): dT/dz = [(αS (t – T) / v) + r(du/dz)] / (c  + c u),  (29)  s                                                                    m           w S = S (1 – ε) / (V ρ ).                         (30) 0    - Kt s           p                            p   m x1 (t ) = x1 e     . (24) In Eq. (29), we point out, first, the heat supplied to In the rotating drum dryer, the dryable product is, as a rule, continuously loaded into the drying chamber on one side and unloaded on the other. The height of the material,    q  s 1= αS (t – T),                                (31) the bed at the input exceeds the height of the bed at the output, due to the rotation frequency and the horizontal angle of the drum. In general, the coolant is fed to the drying chamber and passes through the fluidized bed of the bulk material [5].To study the dynamics of the continuous dehydra- tion mechanism [9], the original scheme of interaction of the coolant flow and the fluidized bed of the wet solids moving along the cylindrical surface can be presented as (Fig. 8):For practical calculations, the surface of the fluidized (sliding) bed is represented as a plane, and the cross sec- tion of the material flow is represented by a segment of a second, the heat spent on heating the mass of absolutely dry material and moisture contained in it, 2               m           w q  = v(c  + c u)(dT/dz),                       (32) third, the heat spent on the evaporation of moisture from the material, 3  q  = vr(du/dz).                             (33) Using the continuity equation (25), we express the equation for the temperature of the material through its moisture content: circle. To describe the continuous drying mode, the con-tinuity equation for the flow of the dryable material can dT/du = – [αS (t – T)] / c K(u – u )] + r / c ,    (34)  s  e am ambe represented as where c   = c+ c u.  w  am mq = ρvS,                                      (25) Note that the density of the dryable material is the function of its moisture content u:ρ(u) = a – bu.                                (26)   Fig. 8. Diagramme of interaction of the coolant flow and the fluidised bed of the dryable product moving along the cylindri- cal surface.To formulate the balance of the coolant, we divide the fluidized bed of the material of length l into a bed of infinitely small sections of length Δz. Let us assume that, within each of such moving sections, the material is perfectly mixed and has a constant temperature and moisture content. When moving from section to section, the temperature and the moisture contents of the mate- rial change to infinitely small values.Let us single out one of these sections and consider the material flow in the moving element of the wet solids (Fig. 9).The flow G(1 + u) of the wet solids enters the volume of the moving element, and the flow G(1 + u + du) comes out with its moisture content being u + du. The coolant supplied to the fluidized bed is determined by the expen- diture function L(z). When they enter the fluidized bed, the moisture content and the temperature of the coolant are constant throughout the whole layer. They are func- tions of the z coordinate at the output. The amount of moisture evaporated from the elementary volume of thematerial equals the amount of moisture absorbed by the    s  amdT/du = – [αS (t – T) / c where dT/du = ν(dT/dτ).K(u – u )] + r / c,   (43)  e  amBy  integrating  Eq.  (43)  with  the  initial  condition iT(0) = T , we get: T(u) = T φ(u) + {t + [rK(u – u ) / αS ]}[1 – φ(u)],  (44)i                                                       e               s where φ(u) = (u – u ) / (u – u ).e             i          e        Fig. 9. Material flows in the moving element of the wet solids. coolant that enters the elementary volume:  s dL(z) / V = – Gdu.                                (35) The heat balance equation is similar to the moisture balance equation (26) for the elementary volume:– d[L(I – I )] / V = G(di + rdu),                  (36) Formula (44) gives us values that are close to thosewe obtain after integrating Eq. (37), while the ratio error does not exceed 2.0%. This allows us to declare Eq. (43) and its analytical solution (44) applicable for analytical studies of continuous dehydration in a fluidized bed of bulk material.Eqs. (37), (38), (40), and (41) make it possible to cal- culate the moisture content and the temperature of the dryable product in the fluidized bed along the length of the drying drum. We know the values of moisture con- tent and coolant temperature only at the surface of the material. To calculate the parameters of the coolant at the outlet of the drying drum, we use the formulae for coolant mixing [11].The equations for the moisture content and the tem-perature of the coolant at the outlet from the elementary i             svolume of the fluidized bed (Fig. 9) can be formulated  w  , c m iswhere i = (c + c u) is the enthalpy of the material;on the basis of  Eqs. (31) and (32), if we assume that the I = c t + (r + c t)x is the enthalpy of the coolant; c  hc      s temperature tj  and the moisture content xj  of the coolant  s  hcspecific heat capacity of the coolant and the steam.Thus, the continuous drying process on a cylindricalat the outlet from the elemental volume are constant, and the coolant rate through this elementary volume is the surface can be approximately described by a system of difference in rates dL = L – L , where L is the coolant ordinary differential equations: jrate when it is suppli j + 1           j                          jto         flu dized be  in the section   p du/dz = – (KSρ / q) (u – u ),                (37)  j  [0, z ]:ed      the        i               d   s  mdT / dz = [(αS (t – T) / ν) + r(du/dz)] / (c with initial conditionsu(0) = u , T(0) = T  w+ c u), (38)  (39)x = x + (GV / dL )(u       – u ),                  (45)  j          i                  s            j        j + 1          j i                 s             j        j         i                 j + 1           jIj = I – (GV / dL )[(i – i ) + r(u        – u )],      (46)  j            j             j             hc           s   jt = (I – rx ) / (c   + c x ),                        (47) i                         i and  a  system  of  balance   relations   obtained   after Eq. (35) and Eq. (36) were integrated with initial condi- tions x(0) = x and t(0) = t :i                                i x = x + (GV / L)(u – u),                      (40)i                 s                 i where i = c T ; j = c T ; I = c  t + (r + c t )x . i          m   i                m   j     s           hc  i                      s  j     j The values of moisture content and  the  tempera- ture of the coolant at the outlet of the drying chamber are calculated from the parameter values of the coolant flows that are coming out of all the elementary volumes of the fluidized bed, according to using the mix formu- t = {I – rx – (GV / L) [(i – i ) + r(u – u )]} / (c + c x). (41) i                              s                           i i                  hc          s lae of the flows:n Upon integrating Eq. (37), we receive an analyticalexpression for the moisture content of the material in the coordinates of the length of the rotating drum: xê  = (1 / L) åj=1 n xjdLj; (48)   p u(z) = (u + me–a) / (1 + ue–a),                   (42) Iк  = (1/ L)å I jdLj;j=1   (48)  where   m   =   a(u  –   u )/ρ(u );   n   =   b(u  –   u )/ρ(u ); t = (I – rx ) / (c   + c x ).                          (49)  f            f             f             hc           s   f i                 e             ii                 e             i α  =  Kρ(u )V(z)/q ;  q =  G(1  +  u );  ρ(u )  =  a  –  bu ; e                  i          i e                 e                                    p If we use Eq. (42) and take into account that the ki- ρ(u ) = a – bu ; V(z) is the volume of the fluidized bede                              i                                                                                                                              netics of continuous drying is described by Eq. (27), from the upload point to the coordinate z.Unlike Eq. (37). Eq. (38) makes it possible to obtain an analytical solution only with the assumption that the heat capacity of the product remains average. If we agree while the flow of the bulk material moving on the cylin- drical surface is presented as an ideal extrusion model (25), we can determine the equation of the effective con- tinuous drying coefficient:  m that c = c+ c u , where u is the average moisture  w  am amcontent of the material, we obtain the following equation: K  = {q / [V ρ(u )]}ln[ρ(u )(u – u ) / ρ(u ) (u – u )]. (50) e                         c         e f        i          e i          f           e   Relation (50) differs from others that determine thecoefficient of drying. It is based on the analysis of the For a convective drying process, the energy efficien- cy can be expressed by the following relation: mechanics of a fluidized bed of wet solids on a cylindri- η = (t – t ) / (t – t ).                          (52) cal surface with regard to the continuous dehydration.The model of the technical and economic charac- teristics. To assess the energy performance of a drying unit with the convective method of heat supply, one can assess the use of the drying agent. The energy losses are determined by the difference between the amount of the supplied and the usable energy [11, 12–15].Various coefficients of efficiency are used as energycriteria. In the general case, they are defined as the ratio t           1         2           1         0 The thermoeconomic analysis combines exergy ana- lysis and economic optimization. The criterion for the thermoelectric optimization is a composition of additive functions. These functions should quantify the exergy, the equipment costs, etc.The most general formula for the so-called thermo- electric criterion is ì      éå cei ei + å kn ùü of the usable energy E to the expended energy E : {min C}= ïmin ê j               n úï.               (53) 1  1  / E 2.η = E 2 (51)í      ê îï      êë å enkkúý þúûï  Table 1. Baseline input  Input parameter                                                                                                   Variants of the computational experiment 1                 2                  3                4                 5                 6 1.1. Rotary drum dryerDryer drum length, m1.21.21.21.21.21.2Outer radius of the drum, m0.150.150.150.150.150.15Channel nozzle radius, m0.1150.1150.1150.1150.1150.115Width of the slits of the channel nozzle, m0.020.020.020.020.020.02Number of slits channel nozzles, pcs.121212121212Number of slits in the channel nozzle through which the coolant is supplied, pcs.444444Coefficient of local resistance0.160.160.160.160.160.16 1.2. Product     Density of the material particles, kg/m3770770770770770770Geometrical dimensions of the material particles, m:      length10.710.710.710.710.710.7width5.05.05.05.05.05.0thickness3.33.33.33.33.33.3Initial moisture content, kg/kg0.1050.1050.1050.1050.1050.105Initial temperature, °C192014171314Specified final moisture content, kg/kg0.05470.07050.05460.08280.05740.0387Set final temperature, °C565559425566 1.3. Coolant     Barometric pressure, kPa100.5100.5100.4100.3100.3100.3Outside temperature, °C161514161416Outside air humidity, %73.073.081.882.581.972.0Temperature of the drying agent at the inlet of the drying chamber, °C180210240180210240Moisture content of the drying agent at the inlet of the drying chamber, kg/kg0.0084130.0078860.0082950.0095500.0083140.008314Temperature of the drying agent at the outlet of the drying chamber, °C149167187134165185Specified moisture content of the drying agent at the outlet ofthe drying chamber, %0.02430.02750.03320.03050.02940.03421.4. Process parametersDum angle, rad 0.034900.034900.052360.052360.017450.01745Drum rotation frequency, 1/sec 0.02500.05830.02500.05830.04170.0417Product feed rate, kg/sec 0.01060.01740.01670.03480.01590.0121Drying agent consumption, m3/sec 0.040100.038680.045590.043960.045290.04199Radius of the circle touching the bed at the input point, m 0.030.010.020.030.020.01Consumption coefficient of the drying agent 0.60.60.60.60.60.6Absolute roughness of the air duct wall, m 0.1×10–30.1×10–30.1×10–30.1×10–30.1×10–30.1×10–3Filling rate of the drum, % 253530253035  Table 2. The results of the computational experiment (output)  Input parameter                                                                                                                                                                   Variants of the computational experiment 1                           2                           3                           4                           5                             6 2.1. Calculation results for the product model subjected to drying Volume of the particle, m3 Surface area of the particle, m2 Sphericity coefficient0.7506×10–70.1608×10–30.535070.7506×10–70.1608×10–30.535070.7506×10–70.1608×10–30.535070.7506×10–70.1608×10–30.535070.7506×10–70.1608×10–30.535070.7506×10–70.1608×10–30.53507Aspheric coefficient1.871.871.871.871.871.87Equivalent particle diameter, m0.5234×10–20.5234×10–20.5234×10–20.5234×10–20.5234×10–20.5234×10–2Equilibrium moisture content in the material, kg/kg0.021830.021800.021830.021900.021930.02193Angle of friction, rad0.73450.73450.73450.73450.73450.7345Loose weight density, kg/m3398.5398.5398.5398.5398.5398.5Specific heat capacity of the material, kJ/(kg×K)1.51601.51641.51401.51521.51361.5140Specific surface of the particles, m2/m32142.622142.622142.622142.622142.622142.622.2. Calculation results for the coolant modelMoisture, %1.334481.252071.315971.511161.318891.31894Wet thermometer temperature, °C45.210947.819250.288545.497947.915450.2922Specific heat of dry air, kJ/(kg×K)1.0221.0281.0351.0221.0281.035Specific heat capacity, kJ/(kg×K)2.7103.2003.8802.7103.2003.880Specific evaporation heat, kJ/K2015.201900.501766.002015.201900.501766.00Coefficient of kinematic viscosity of the drying agent, m2/sec0.3029×10–40.3136×10–40.3032×10–40.3000×10–40.3111×10–40.3029×10–4Drying agent density, kg/m30.843730.847220.904960.853000.854590.90661Specific volume of wet air, m3/(kg·sec)1.315261.401131.490531.320221.404881.49211Heat conductivity coefficient of the drying agent, W/(m·K)0.3773×10–10.3998×10–10.4207×10–10.377×10–10.3998×10–10.4207×10–1Prandtl number0.70170.69460.69060.70410.69560.6907Schmidt number0.56380.52020.45120.55840.51600.45072.3. Calculation results for the model of the movement of wet solids along the cylindrical surfaceSpecific consumption of the drying agent, kg/kg62.943550.985640.151847.718247.422938.6292Minimum design airflow per drying, kg/sec3.0371×10–22.7698×10–23.0584×10–23.3362×10–23.2481×10–22.8045×10–2Minimum estimated volume flow rate of the drying agent m3/sec0.039950.038810.045590.044050.045630.04185Fictitious speed of the drying agent through the material bed, m/sec0.22050.19410.23850.24170.23690.2108Speed of the drying agent, reduced to the full cross section of the bed, m/sec0.45700.40240.49430.50100.49100.4368Porosity of the bed0.48250.48250.48250.48250.48250.4825Specific surface of the material in the bed, m2/kg1.59131.59131.59131.59131.59131.5913Equivalent pore channel diameter, m0.1740×10–20.1740×10–20.1740×10–20.1740×10–20.1740×10–20.1740×10–2Tortuosity coefficient of the channels1.36791.36791.36791.36791.36791.3679Length of the pore channels, m0.093750.11930.10700.093750.10700.1198Equivalent Reynolds number78.971467.139585.308787.408182.601275.4855Hydraulic resistance coefficient of the bed1.00801.14180.95160.93470.97461.0431Bed resistance, Pa507.731447.649601.354572.002573.984515.488Heat transfer coefficient, kW/(m2 × K)0.2064190.1863650.2242800.2234570.2201310.201593Residence time in the drying chamber, min7.8996.7106.0162.3896.3159.723Drying coefficient, 1/sec0.1959×10–20.1330×10–20.2580×10–20.2168×10–20.2242×10–20.2734×10–22.4. Results of the calculation for the model of the coolant supplied to the fluidized bedCross-sectional area of the air distributor, m20.9712×10–20.9712×10–20.9712×10–20.9712×10–20.9712×10–20.9712×10–2Perimeter of the air distributor, m0.83500.83500.83500.83500.83500.8350Equivalent diameter of the channel nozzle, m0.046530.046530.046530.046530.046530.04653Speed of the drying agent at the beginning of the drum, m/sec4.12873.98524.69404.52614.66314.3233Reynolds number6342.555907.687202.197020.146973.786641.71Average flow rate of the drying agent from the channel nozzle, m/sec0.41770.40290.47490.45790.47180.4374Coefficient of friction of the air nozzle0.037050.037610.036090.036280.036330.03670Coefficient of friction of the air nozzle friction1.141021.141661.139921.140141.140191.14061Total resistance of the air nozzle, Pa8.27611.49.610.69.7Slit parameter5.9315.9315.9315.9315.9315.931Air duct parameter5.0825.0975.0585.0625.0645.0732.5. Results of the calculation for the hydrodynamics model of the fluidized bedVolumetric capacity of the dryer for wet material, m3/sec0.266×10–40.4366×10–40.4191×10–40.8733×10–40.3990×10–40.3036×10–4Dryer productivity according to absolutely dry material, kg/sec0.9593×10–20.1575×10–10.1511×10–10.3149×10–10.1439×10–10.1095×10–1Dryer productivity according to the evaporated moisture, kg/sec0.483×10–30.543×10–30.762×10–30.6991×10–30.6849×10–30.7260×10–3Angle between the surface and the axis of the drum, rad0.027750.029110.028190.027750.028180.02913Radius of the circle touching the bed at the output of the product, m0.063310.044940.053830.063310.053820.04497Volume of the drum occupied by the bed, m30.012460.017450.014960.012460.014960.01745Area of the middle section of the bed, m20.010390.014540.012460.010390.012460.01454Radius of the circle touching the bed in the middle section, m0.046460.027420.036770.046460.036760.02744Thickness of the bed in the middle section, m0.068550.087590.078230.068530.078240.08756Width of the bed in the middle section, m0.15160.16600.15930.15160.15930.1660  The rest Table 2 Effective area of the bed, m20.18190.19320.19120.18190.19120.1992Distance between the beginning of the drum and the middle section of the bed, m0.59310.59820.59450.59300.59450.5982Influence coefficient of the flow rate of the drying agent in the dense blownbed on the performance of the dryer2.242001.410402.520002.400902.747701.983002.6. Results of the calculation for the model of complex heat and mass transferEquivalent Reynolds number71.179367.230581.962379.890579.362975.5839Archimedes number455.991479.825565.384463.839487.412455.778Limit value of Nusselt number for natural convection conditions2.41072.43552.53382.42312.44592.5354Nusselt number for simultaneous processes5.56785.40715.89275.81495.78965.7035Heat transfer coefficient, kW/(m2×K)0.040140.041310.047370.041910.044230.04585Specific heat flow, kW/m25.41076.69988.98665.63747.16918.6980Specific mass flow, kg/(m2×sec)0.2685×10–20.3525×10–20.5089×10–20.2797×10–20.3772×10–20.4925×10–22.7. Results of the calculation for the model of drying in a fluidized bedThe final design value of the moisture content in the material, kg/kg0.05460.07040.05450.08280.05730.03856The final design value of the material temperature, °C565559425566The final calculated value of the moisture content in the drying agent,kg/kg0.02430.02760.03320.03060.02960.0342The final design value of the temperature in the drying agent, °C156.664171.018191.130138.289171.987193.168Effective heat transfer coefficient, kJ/(m2×K)1.4338×10–31.0687×10–31.3248×10–32.4044×10–31.4310×10–31.0144×10–32.8. Results of the calculation for the model of technical and economic characteristicsDryer productivity for moisture removal, kg/sec0.4834×10–30.5441×10–30.7631×10–30.7001×10–30.6862×10–30.7276×10–3Exergy of the drying agent at the inlet to the drying chamber, kJ35.3248.1562.7835.3948.7461.56Specific exergy, kJ/kg2227.662443.202515.931683.072289.722380.89Energy efficiency7.26.05.010.56.45.3Capacity for evaporation according to evaporated moisture, kg/m30.9696×10–21.0913×10–21.5307×10–21.4041×10–21.3763×10–21.4593×10–2  RESULTS AND DISCUSSIONThe research  was  based  on  the  informational and structural scheme of a convective drying unit model. For all its components, we developed the mathematical mod- els in accordance with the analytical multimodel system for the continuous drying process of wet solids in a flu- idized layer. As a result, we constructed an automated calculation system for the continuous process of convec- tive drying, which can be applied to a rotary drum dryer (Tables 1 and 2). The practical result of the study consisted in assess- ing and comparing the quality indicators of sunflower seeds of natural moisture. First, one sample of sunflower seeds was dried on an experimental drum dryer with a channel nozzle. Second, another sample was dried on an industrial drum dryer with a lifting vane system at the vegetable oil plant ZAO ZRM Bobrovsky. Finally, the experimentally obtained results were compared with the The effect of the drying mode on the change in the quality of oil in the sunflower seeds (Table 4) was mea- sured by changing the acid, peroxide, and iodine num- bers at a different initial seed moisture. The heating temperature did not exceed the maximum permissible temperature for the particular humidity. It ensured the inactivation of enzymes, i.e. lipase and lipoxygenase.Table 4 shows that the acid values of the oils in the studied modes were somewhat reduced. This can be ex- plained by the fact that low molecular organic acids were distilled together with the water steam during the dry- ing process. The peroxide numbers somewhat increased with increasing temperature, which can be explained by the catalytic effect of temperature on fat oxidation due Table 4. Effect of drying process of sunflower seeds in a drumdryer with a channel nozzle on the quality of vegetable oil computed results obtained from the mathematical model. Drying agent Acid number, Peroxide Iodine The experimental data show (Table 3) that after thesunflower seeds were dried in a drum dryer with a chan-  temperature, °C     mg KOH                number, % I     number, g I       2                                 210.56% moisture  nel nozzle, the difference between the maximum and01.800.016151.6minimum  humidity  of  individual  seeds  decreased  by1301.710.021148.12.34 times. This can be explained by the same residence1501.680.024145.6 time in the drying zone and the uniform distribution ofthe coolant flow in the fluidized bed.  170                        1.65                   0.035                142.2            14.45% moisture   01.820.015149.81301.710.017149.21501.650.019147.91701.630.030144.5  Table 3. Change in humidity of single sunflower seeds duringthe drying process in the drum dryer with a channel nozzle, %    Before drying                               After drying 18.42% moisture minmaxminmax 01.850.014146.314.0214.695.195.51 1301.790.017143.812.2812.684.194.34 1501.760.018141.210.0411.083.654.10 1701.720.025140.5   Table 5. Physical and chemical indicators of sunflower seedsand oil rial in a rotary drum dryer, most researches determinethe average values of heat transfer coefficients. The pro-                                                                                                                 posed approach for calculating the effective heat transfer  Parameters                                            Drying method coefficient in a fluidized bed provides the required repro- In a drum dryer with In an industri- ducibility and differs from the experimental data by no                                          a channel nozzle          al drum dryer     more than 2.0% (Table 2). Drying agent tempera-ture, °CSeed moisture, % 150                              280 The energy performance of rotary drum dryers with a convective method of heat supply can be assessed ac- initialfinal 14.357.15 14.357.20 cording to the degree of the coolant use. The energylosses  are  determined  by  the  difference  between  the Oil content on abso- lutely dry matter, % 55.82                           56.36 amount of supplied and usable energy. It is more diffi-cult to determine the optimal variant if it is necessary Damaged seeds, %          4.15                             6.15 to satisfy several efficiency conditions. In this case, one Phosphatides in the oil, % Peroxide number, % I20.0190.025 Iodine number, g I2147.8160.6Nomenclature  Acid number, mg KOH0.050                           0.0491.65                             1.87 should use compromise criteria, e.g. capital and energy costs, capacity, quality of the finished product, reliabi- lity of the management system, level of environmental safety, etc. to the presence of oxygen in the air. The iodine numbers decreased with increasing temperature. This resulted from the chemical reactions of breaking double bonds in    τ  –  residence  time  of  the  bulk  material  in  the  dryerdrum, sec;G – dryer capacity, kg/sec; 3 ρdb – dry bulk density, kg/m ; the carbon chain of the fats and the addition of organic compounds and radicals that were present in the air.Table 5 features some results of the comparative pro- duction tests. They confirm the fact that the temperature of the drying agent destroys protein structure. The num- V – dryer volume, m3; d  1G  – amount of material in the drum, kg; 2G  – the amount of the material leaving the drum per unit of time, kg/sec; mV  – volume of material unloaded from the drum, m3;x (τ), x (τ), – the moisture content of the product and the ber of damaged sunflower seeds when dried in a dense             1             2 ventilated  bed  of  moving  seeds  is  significantly  lower than in the fluidized bed.The analysis of the physicochemical parameters of the oil suggests that the structure of the drying agent largely determined the quality of the dried sunflower seeds: acid, peroxide, and iodine numbers decreased by 12, 24, and 9% respectively.If we compare the data obtained from the practical tests and from the model (Table 1), we can conclude that the results are reproducible. The following optimal va- lues were also obtained while solving the problem of convective drying optimization: the initial moisture con- tent of sunflower seeds was 16–17%; the temperature ofthe drying agent in the bed was 66–67%; the consump- coolant, respectively, kg/kg;K – drying ratio, 1/sec;G – consumption of the dryable product, kg/sec;(τ), L(z) – coolant flow rate, kg/sec;q – moving mass flow, kg/sec;ρ – density of the dryable product, kg/m3;u – material speed, m/sec;S – section area of the fluidized bed, m2;a, b – constants determined experimentally; u  e– equilibrium moisture content of the material, kg/kg;T – product temperature, °C;α – heat transfer coefficient, kW/(m2×K);t – coolant temperature, °C; S s– specific surface area of the material, m2/kg;r   specific heat of vaporization, J/kg; tion of drying agent was (3.2–3.4)×10–2  m3/sec; the angle of the drum was 0.61–0.70 rad; the drum rotation fre- – m  wc , c  – specific heat capacity of dry material and water, kJ/(kg×K); quency was 3.6–4.2 min. These results agreed with thedata of the model presented in Table 2. CONCLUSIONThe proposed multimodel system of non-stationary drying processes for bulk materials has a number of ad- vantages. First, it leads to a block-modular construction and expedient aggregation of rotary drum dryers. Seco- – surface of the particle, m2; V  – volume of the particle, m3;  ε – porosity of the fluidized bed; p  s S p– specific volume of the coolant, m3/kg;x(z) – moisture content of the coolant, kg/kg; fV – the volume of the fluidized bed of the bulk materialin the rotating drum, m3;ρ(u ), ρ(u ), ρ(u ) – product density corresponding to the e               i               f nd, it optimizes the allowances on the inputs and out-puts of technological operations and links them toge- equilibrium, initial, and final  moisture content of thematerial, kg/m3; ther. Third, it develops requirements for the quality of   Nu ORe,  Nu  OAr –  Nusselt  numbers  for  forced  and  free raw materials and environmental conditions, in terms ofthe high efficiency of the organization of its processing. Thus, when studying the specifics of heat transfer be-tween the coolant and the solid particles of bulk mate- movement forms, respectively;c, m, n – constants of the equation; n – drum speed per minute, min-1; φ – drum angle, rad;   ψ – the angle between the surface of the bed and the axisof the drum, rad;Θ – friction angle of the material, rad;R – the radius of the channel nozzle of the drum, m; the outlet from the drying chamber and the temperature of the outside air, °C.C – unit exergy value; i e – energy of the consumed raw materials and exergy;  r  0– the radius of the circle touching the bed at the input, m;A – coefficient that takes into account the effect of the average flow rate in the bed of the wet solids on the throughput of the dryer, A = f(Re) is determined exper- imentally;ξ – tortuosity coefficient of the channel;w – flow rate in the direction of filtration, m/sec;c  – unit cost of exergy of the raw materials and energy; ei nke   – exergy of the products; nk   – capital and other associated expenditures for then-subsystem. CONFLICT OF INTERESTThe authors declare that there are no conflicts of in- terest related to this article.  d  pρ , ρ – bulk density and particle density, kg/m3;  λ  g– the resistance coefficient of the granular bed; w w– flow rate in the direction of filtration, m/sec;h – thickness of the bed, m;ρ – density of the drying agent, kg/m3;FUNDINGThe research was conducted by the authors as a part of their work at the Voronezh State University of Engi- neering  Technologies  and  ZRM  Bobrovsky  vegetable t , t , t – temperature of the drying agent at the inlet and oil plant.