DEVELOPMENT OF THE PHYSICAL AND MATHEMATICAL MODEL OF THE BAKING PROCESS OF THE DOUGH PIECES IN BAKERY OVENS

Об’єктом дослідження є фізична та математична моделі, призначені для опису тепломасоперенесення всередині пористого матеріалу під час випікання. З метою підвищення якості та одночасного зниження енергоспоживання у виробництві, а також покращання техніко-економічних показників роботи печей, тривалості і безпеки їхньої експлуатації ведеться удосконалення конструкцій пічних агрегатів, розробка нових і оптимізація теплових режимів їх роботи. Однією з найбільших проблем є задача по заміні застарілих конструкцій печей новими, з автоматичним регулюванням теплового режиму випікання, що забезпечить високу якість хліба при зниженні витрат палива, пари, електроенергії та людських ресурсів. Оскільки якість виробленої продукції, зокрема смак, аромат, пористість, глянець, зовнішній вигляд та інші показники хлібобулочних виробів в значній мірі залежать від конструкції пічного агрегату, теплового та гігротермічного режимів робочої камери, а також правильної його експлуатації. Ці фактори впливають на втрати при випіканні, які можуть змінюватись від 6 до 12 %, що впливає на вихід хліба. У даній роботі наведено фізичну і математичну модель процесу випікання тістових заготовок в хлібопекарських печах на прикладі розробленої автором промислової печі К-ПХМ-25 (Україна). Приведена математична модель процесу випікання хліба в газових каналах пекарної камери з врахуванням радіаційно-конвективного теплообміну, масообміну з врахуванням введення водяної пари для зволожування тістових заготовок та турбулентності багатофазного потоку. Залежність турбулентності багатофазного потоку сформульована на основі осереднених за Рейнольдсом системи рівнянь Ейлера. Дана модель дозволяє з достатньою точністю і детальністю враховувати технологічні режими та конструктивні особливості сучасних конвеєрних хлібопекарських печей. А також дозволяє проводити широкі параметричні дослідження сполученого теплообміну в них з виходом на кінцевий показник – якість готових виробів. Ключові слова: промислова піч К-ПХМ-25, радіаційно-конвективний тепломасообмін, математична модель процесу випікання хліба. Kovalyov A.


Introduction
Baking bread is the most important process in the production of bakery products. Taste, aroma, porosity, gloss and other quality indicators of finished products is the result of a number of physicochemical changes inside the product during baking, which depend primarily on the thermal regime of baking and steam humidification [1][2][3]. The oven is the main equipment of the bakery, it determines the type and capacity of the enterprise, the range and quality of products. The oven is not only a thermal, but primarily a technological unit, the main purpose of which is producing high-quality products while ensuring high technical and economic indicators -product output with minimal energy costs. To ensure high-quality performance of the furnace, it is necessary to use models of a real system and conduct experiments based on a mathematical model for analysis, design or redesign, control and forecasting of a specific real process. This is relevant, so many researchers have analyzed the baking process in order to obtain an accurate mathematical model for modeling bread baking. So, the authors of [4]

Methods of research
The design of the К-BOM-25 baking oven developed by the author (Fig. 1) includes the following components: -an all-metal construction assembled from separate modules and insulated from the outside with mineral wool; -gas channels in which bread is baked; -a movable bucket conveyor for moving dough pieces (DP) in gas channels; -input and output nodes of the furnace, including the gas duct for the supply of green gas; -steam humidification system; -loading and unloading device. That is, in the physical model of the baking oven under consideration, the furnace-igniter unit is not considered, and in accordance with this, the combustion processes of the fuel also.
The environment of the gas channels of the baking oven is considered to be two-phase, selective emitting and absorbing, and is surrounded by diffuse boundaries and consists of green gases and water vapor, which is used to moisten the dough pieces.
Dough pieces and bread are considered wet capillaryporous media with effective physical properties.
Dough pieces move with the bucket conveyor.

Research results and discussion
In the work, a mathematical model of the process of baking bread in the gas channels of the oven's baking chamber, taking into account radiation-convective heat transfer, mass transfer by introducing water vapor (with water droplets) to moisten the dough pieces, is baked, the multiphase flow turbulences can be formulated using the system of Euler equations averaged over Reynolds [9,10]. The equations written for each of the phases: continuity, conservation of momentum, kinetic turbulent energy and its dissipation and energy conservation: where ρ ri -the average density of the i-th phase, kg/m 3 ; α i -the volume fraction of the i-th phase in the flow ; ρ i -the density of the i-th phase, kg/m 3 ; tthe time, s; V i -the vector of the speed of the i-th phase averaged beyond Reynolds or Favre, m/s; ∇ -Hamilton operator, m -1 ;  m ji and  m ij -mass transfer rate from phase j to phase i and vice versa, respectively (moreover), kg/(s⋅m 3 ); n -the number of phases in the stream.
where p i -the partial pressure of the i-th phase, Pa; μ i and l i -the shear and bulk viscosity of phase i, respectively, Pa⋅s; -stress tensor of the 2nd rank (or the physical equation of state of the medium that relates stress to strain rate), Pa; I -the unit tensor of the 2nd rank; g -the acceleration vector associated with gravity, m/s 2 ; K ji = K ij -coefficient of exchange of momentum between phases, depends on friction, pressure and other factors, kg/(m 3 ⋅s); -interphase surface velocity, m/s; F ithe external mass force related to the volume, N/m 3 ; -lifting volumetric force, N/m 3 ; g -index of heating gases; p -water vapor index with the inclusion of water droplets; where k i -the mass turbulent kinetic energy of the i-th phase, where s 0 corresponds to the boundary of the medium. For solid structural elements of the furnace and dough pieces move together with a bucket conveyor, the system of equations (1)-(5) is simplified to the heat equation of the form: where q n -the volumetric heat source associated with loading the DP and unloading the baked bread (BB) from the oven, W/m 3 , the specific power of which is determined from the system of equations: where   m m DP BB , -the mass flow rate of the DP/BB in the loading and unloading unit, respectively, kg/s; c p(DP) , c p(BB) -mass heat capacity of the DP/BB, respectively, J/(kg•K); V DP , V ВB -volume of DP/BB, respectively, m 3 ; X (x, y, z) -the Cartesian coordinate system of the solid elements of the furnace, including the fixed -X im mov (x, y, z) and the moving part, which refers to the bucket conveyor and dough pieces -X im mov (x, y, z, t); V conv -the conveyor speed vector along with the bread that is baked, m/s; W -the calculated area of the baking oven.
The coefficient of exchange of momentum between the phases in equations (2)-(5) depends on the selected model of hydraulic resistance, Reynolds number, viscosity, etc. and is determined depending on the physical state of the phases interacting: liquid-liquid or liquid-gas, or gas-gas. So, for example, the coefficient of exchange of momentum between water droplets p and gas g can be determined as [4]: where t ρ μ -the relaxation time of water droplets, s; d p -diameter of water droplets, m; μ eff = μ g /(1 -α p ) 2.5 -effective dynamic viscosity of a gas-liquid droplets, Pa⋅s; μ eff = μ g /(1 -α p ) 2.5hydraulic resistance function; C D -the coefficient of hydraulic resistance.

ISSN 2226-3780
The Reynolds number of a two-phase medium is determined by the ratio: The coefficient of hydraulic resistance is determined depending on the Reynolds number: ( ) where f * = (1-α p ) 3 ; l RT = (σ/(gDρ pg )) 0.5 -wavelength of the Rayleigh-Taylor instability, m; σ -the coefficient of surface tension, N/m; g -the acceleration of gravity, m/s 2 ; Dρ pg -the density difference between the phases p and g, kg/m 3 .
The initial conditions for the system of equations (1)-(6) at t = 0: where i n = 1, -the phase index in the stream; T i 0 , k i 0 , ε i 0initial temperature (K), mass turbulent kinetic energy (J/kg) and its dissipation rate (J/(kg s)); L 0 -the initial length of the zone of water vapor in the gas channels of the baking oven, m; α 0 -the volume fraction of water vapor together with droplets of length L 0 .
The boundary conditions for the system of equations (1)-(8) with: -in the inlet sections of the furnace, parameters are set for green gas (i = g) and water vapor (i = p), mass flow rate and DP temperature: where n -the vector of the external normal to the surface of the input furnace sections; V i in -normal speed, m/s; T i in -temperature of gas phases, K;  m DP in -DP mass flow rate, kg/s; T DP in -DP temperature, K; k i in -mass turbulent kinetic energy, J/kg; ε i in -rate of its dissipation, J/(kg⋅s); α i in -volumetric part of the i-th phase at the entrance to the furnace; -in the outgoing sections of the oven, parameters are set for the mixture (14), gas phases and mass flow rate, as well as the temperature of the baked bread (15): ; where p mix -the overpressure of the gas mixture, Pa; T i out , α i out ,  m BB out , T BB out , k i out , ε i out -temperature (K), volume fraction of the i-th phase, BB mass flow rate (kg/s), BB temperature (K), mass turbulent kinetic energy (J/kg) and its dissipation rate (J/(kg s)); -at the contact boundary between solids, the conditions for absolute contact are set: where n -the number of phases in the stream; where α -the heat transfer coefficient; T p -the ambient temperature.

Conclusions
A mathematical model of the process of baking bread in the gas channels of the baking chamber is developed TECHNOLOGY AUDIT AND PRODUCTION RESERVES -№ 3/3(47), 2019 ISSN 2226-3780 taking into account radiation-convective heat transfer, mass transfer taking into account the introduction of water vapor to moisten the dough pieces and turbulence of the multiphase flow. It is theoretically grounded that this model will allow with sufficient accuracy and detail to take into account all the operational and design features of modern conveyor baking ovens. And it will also allow for extensive parametric studies of conjugate heat transfer in them with access to the final indicator -the quality of finished products. But this theoretical justification requires empirical evidence.