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Теоретично обгрунтовано процеси радiацiй-но-конвективного тепломасообмту мiж уЫма визначальними об'ектами всередит вiбрацiй-ног сушарки з 1Ч-енерготдведенням. На основi теплового i матерiального баланыв визначено рiвняння, як описують основт динамiчнi характеристики режиму суштня олieвмiсного зернового матерiалу в безперервно дючш Ш-сущарщ. У зв'язку з тим, що точного аналтичного розв'яз-ку представленог математичног моделi у виглядi системи диференщальних рiвнянь у частинних похiдних не ^нуе. Запропоноване наближене ршення дозволяв iдентифiкувати залежностi розподЫу температури i вологовм^ту зернового та олieвмiсного матерiалу за довжиною сушарки для будь-якого моменту часу.
Чисельний розв'язок наведеног математичног моделi можливий лише при наявностi визначе-них взавмопов'язаних ктетичних коефiцieнтiв. Кнетичт коефщенти експериментально прями-ми вимiрами знайти не можливо, тому у стат-тi запропоновано метод подолання цих труд-нощгв. На основi представленого наближеного аналтичногорозв'язку синтезованог математичног моделi, з використанням методу обернених задач визначаються комплекси коефiцieнтiв за результатами експериментальног iдентифiкацiг процесу зневоднення. В подальшому експериментально iдентифiкованi параметричт комплекси моделi можливо використовувати при аналiзi процесу суштня на наближенихрозв'язкахабо для подальшого точного чисельного розв'язку.
На основi експериментальних дослiджень зневоднення зерноматерiалу встановлено, що при збшьшент потужностi 1Ч-джерела вiд 400 до 500 Вт час суштня з початковог вологостi матерiалу 11 % до 8,75 % зменшувться з 9 до 7хвилин. Визначено, що критерш Ребтдера, який характеризув вологотепловi характеристики матерiалу, зменшувться при зменшент його во-логостi з величини 0,04 при 11 % до 0,01 при 9 %. Це в щкавим з практичног точки зору i дав мож-лив^ть використовувати отримат результати i математичну модель при енергетичному удо-сконалент реалiзацiг процеыв термiчного суштня в типових об'вктах тдготовки до переробки олiввмiсних зерноматерiалiв
Ключовi слова: тепломасообмт, тфрачервоне енерготдведення, вiбрацiйна сушарка, олiввмiсне
зерно, параметрична iдентифiкацiя -□ □-
UDC 621.9.048; 664.002.5
[DOI: 10.15587/1729-4061.2018.139314 |
THEORETICAL RATIONALE AND IDENTIFICATION OF HEATAND MASS TRANSFER PROCESSES IN VIBRATION DRYERS WITH IR-ENERGY SUPPLY
V. Bandura
PhD, Professor Department of Processes and equipment processing and food production named after prof. P. S. Bernik* E-mail: bandura_3@ukr.net R. Kalinichenko PhD, Associate Professor Department of electrified technologies in agricultural production Separated Subdivision NULESU «Nizhyn Agrotechnical Institute» Shevchenka str., 10, Nizhyn, Ukraine, 16600 E-mail: rkalinichenko@ukr.net B. Kotov Doctor of Technical Sciences, Professor Department of Energy and Electrotechnical Systems in Agroindustrial Complex State Agrarian and Engineering University of Podilya Shevchenka str., 13, Kamianets-Podilskyi,
Ukraine, 32300 E-mail: kotovboris@ukr.net A. S pi ri n PhD, Associate Professor Department of general technical disciplines and
labor protection* *Vinnytsia National Agrarian University Soniachna str., 3, Vinnytsia, Ukraine, 21008
1. Introduction
About 15 % of the total energy consumption in the agro-industrial complex of developed countries accounts for the processes of drying and heat treatment of agricultural
materials [1]. Potentially, the most energy-efficient dryers are those with infrared energy, because they provide direct impact of the radiant energy on the moisture in the dried material and reduce losses tied to the spent drying agent. However, in Ukraine the majority of available drying systems and
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facilities for highly intensive heat treatment of grain materials with infrared (IR) energy consume an excess specific energy per unit of the obtained product [2, 3].
It is extremely important to introduce innovative solutions aimed at reducing the specific energy consumption of IR equipment for preparation of grain materials for storage or processing, consumption or feeding. The solution is complicated by the lack of unified approaches to the synthesis of the mathematical model of heat and mass transfer processes and the variability of interconnected kinetic coefficients that depend on the physical properties of the interacting substances and energies in analytical models.
Optimal (as to the criteria of energy efficiency and quality) process of material dehydration in dryers with infrared energy supply is impossible without an adequate mathematical model. Therefore, research should be considered relevant if it is aimed at further development and improvement of the mathematical apparatus that characterizes heat and mass transfer processes in IR dryers.
2. Literature review and problem statement
The modes and parameters of equipment for IR heat treatment are substantiated due to the use of mathematical models of the theory of heat and mass transfer, as well as due to physical laws, physical-chemical balance ratios, generalization of experimental data - the theory of similarity, probability, and mathematical statistics [4, 5].
The classical mathematical models of the theory of heat and mass transfer [6] are the most versatile for different types of moisture and heat transfer. However, cumbersome analytical solutions are valid only for constant coefficients, simplified external material form (such as unbounded plate, ball, or cylinder), and uniquely determined boundary conditions. This is possible only with a slow process, or within sufficiently small areas into which the volume of a moist product is divided. The main empirical mathematical models of dehydration of grain material, which are used to describe heat and mass transfer in the grain material, are presented in [7]. But they are considered only for the convective method of heat supply and are applicable only in the range of the investigated parameter values.
Therefore, the engineering calculation of the kinetics of heating and dehydration of moist dispersed materials is based on approximate methods developed with approximate assumptions for specific systems, with taking into account the hydrodynamic state in the apparatus [8, 9]. Thus, studies [10, 11], in particular, use the balance equations for heat and mass transfer to research the kinetics of heating in the drying of grain material under constant or periodic infrared energy supply. Other studies [10, 11], however, simplify analytical calculations when compiling balance ratios and fail to consider all interacting objects engaged in the combined heat exchange.
Studies [12, 13] present the solutions of models shaped as systems of differential equations with partial derivatives, using the grid method or the mathematical computer package. This allows you to determine the length of the process, the moisture and temperature fields of the grain mass and the drying agent in the layer in time for different grain drying conditions. However, it should be noted that numerical simulation always has a partial character (the result is adequate only under certain conditions), which makes it difficult to use the results obtained from the mathematical model.
To determine kinetic coefficients of heat transfer and moisture transfer included in the equations of mathematical models is extremely difficult. This is due to the fact that the coefficients depend on many defining parameters and conditions of the process, and direct measurements can not be obtained.
Scarce references give different values of thermophysical and mass transfer coefficients. Usually, the coefficients are determined approximately with the use of the empirical dependences for the heat and diffusion Nusselt number, which are obtained on the basis of generalizing a large number of experimental studies [14]. Therefore, there are reasons to believe that the lack of certainty of these factors may lead to inaccuracies in simulation.
Another study [15] proves the relevance of the method for solving inverse problems in the theory of heat and mass transfer. The method of inverse problems allows determining the complexes of coefficients included in the solutions of differential equations [16] with the use of experimentally obtained temperatures at several points inside the body or the material layer as well as its average moisture content. Further, the fields of temperature and moisture content of grain materials can be reproduced numerically or analytically. But the use of this method requires at least an approximate solution of a complex mathematical model of processes and indicative values in the complexes of the sought coefficients.
The problem of the theoretical substantiation of the processes of heat and mass transfer in dryers consists in the complex choice between exact but limited in their application empirical models of processes and universal analytical models with insufficiently reliable kinetic coefficients. Therefore, perspective studies must use theoretical and experimental approaches to enhance the versatility of analytical models and the adequacy of empirical dependences. This would allow determining the parameters and operating modes of heat and mass exchangers and synthesizing optimal control systems.
3. The aim and objectives of the study
The aim of this research is to study theoretical and experimental methods of a combined drying process in a vibration dryer with infrared energy supply for its further improvement.
According to the above aim, the following tasks had to be solved:
- to develop an analytical mathematical model of heat and mass transfer processes in a vibration dryer with infrared energy supply, taking into account all interacting surfaces of objects inside the dryer, and obtain its approximate solution;
- to determine experimentally the kinetics of heating and dehydration of the layer of grain material (rapeseed);
- to use the method of inverse problems for identifying the parametric complexes included in the analytical model's solutions, based on the experimental findings on rapeseed drying.
4. Devising a mathematical model of thermal dehydration in the IR energy
Let us consider an experimental plant of our own design. It is a vibrational surface along which there moves grain material from which moisture is removed due to thermal energy transmitted by radiation from infrared (IR) emitters (Fig. 1).
Fig. 1. Experimental infrared dryer: a — appearance; b — calculation scheme; c — technological chart: 1 — infrared heater; 2 — screen; 3 — vibrodiffused layer of grain; 4 — drying agent — air; 5 — side screens; 6 — conveyor;
7 — vibroactivator
Emitters are placed in parallel with the material layer along the entire vibrational transportation surface. A directed flow is ensured due to the screen reflector placed over the emitters and due to the vertical surfaces shielding the side surface of the dryer. The material moves evenly due to the vibrations of the bearing surface of the conveyor belt. In this case, the material perceives the energy of infrared radiation that is partially absorbed and partially reflected, and also by convection transmitted to the air and surrounding surfaces. The thermal energy received by the material is spent on heating the moist material (increasing its enthalpy), evaporation of moisture and is released into the environment: air, side screens, and the conveyor belt. Heat from the material is transmitted both by radiation and convection. The conductive component of the heat transfer to the vibrating surface is incomparably smaller than the convective and radiation components. In the process of drying, the material continuously moves from the place of loading to the place of unloading being heated and dehydrated. The air moves between the belt with the material and the emitters.
In order to facilitate the analysis of this process, we make the following general assumptions in the mathematical description [10, 11]:
- the thermophysical parameters of the material and air do not depend on temperature and in the first approximation are taken as average for the process;
- gradients of temperature and moisture content are viewed regardless of the size of the grain particles (heating and dehydration in individual grains are non-gradient);
- the temperature of the belt, screens and emitters remain the same along the length and in the cross section of the plant (dryer);
- the amount of moisture removed from the material is proportional to the moisture content, and the proportionality factor depends on the temperature linearly.
In accordance with the above description and the assumptions made, we get the following general form of the equation of the heat and material balance for the time dx within the segment dy in the direction of motion of the layer of grain material:
dQo = dQi + dQn + dQ,^ ± dQ14 + dQ15 ; dQn = dQ2 + dQ23 + dQ24 + dQM;
dQi3 + dQi4 = dQ3 + dQ34 + dQ^ + dQw ; dQ35 = dQ5 + dQM; dQ4 = dQi4 + dQ24 + dQ54 ± dQ34 ; dQ36=dQ6+dQ64;
dM4 = -dM3,
(1) (2)
(3)
(4)
(5)
(6) (7)
where dQ0 is the energy from the IR emitter, J; dQ,, dQ2, dQ3, dQ4, dQ5, and dQ6 denote energy consumed to heat the elements of the plant, respectively: emitter (lamp, TEH), reflecting screen, material, air, side screens, and transporting means, J; dQi2, dQi3, dQi5, dQ25, dQ23, and dQ5& stand for energy transmitted by radiation from the surface of one element to another, according to indices, J; dQ14, dQ24, dQ34, dQ36, dQ54, and dQ64 denote energy transmitted by convec-tive (or complex) heat transfer from one element to another, J; dQw is energy absorbed by the material and consumed to evaporate moisture from it, J; M is the mass of moisture, kg.
Having determined the values of individual components in the heat and material balance through the known connections [10, 11], we can develop the following system of differential equations (as a mathematical model):
P = mC dt + C12F12 ( - T4 ) + C13F13 ( - T34 ) +
+C15 F15 (( - T4 W4 F (01 -1 );
C12F12 (( - T24 ) = m2C2 It2 + C25F25 (( - T4 ) +
+C23 F23 (4 - T34 )+a24 F2 (0 2 -1 ) ;
A D13 F« (T4 - T4 )+C23 F23 (t4 - T4 )] =
303 ^ r 303 r / A = m3c3—L + G3c3L-3- + a34F34 (03 -t) + at dy
(8)
(9)
+a6F6 (03 -06 )± C35F35 (T34 - T4 ) + r
dU
dU nT
mo aTGoL ay
;(10)
b
c
dt dt , \ m c — ±G c L— = a;4F (6, -1) +
" p9t " " dy 14 lV 1 '
+«24F2 (62 - t) + «3iF3 (63 - t) + a54F5 (65 - A;
C15F15 ( - T54 ) = m5C5 ^dt5 + a54F54 (65 - t)
96
< F6 (6?-q6 )=m6c6-df+a6 F6 (66-1 ') ;
Rb
aez gzl 9ez
dt m? dy
dU GiL dU dt m70 dy
dU
dU
m--+ GL-
9t
dy
= mfil
dd
dd
— + G,L—
9t
dy
+ GsCsL^r dt d^
(11) (12)
(13)
(14)
(15)
= A
a13F13 (61 -63 ) + a23F23 ( 62 - 63 ) --a34F34 (63 - t)-
-aeF6(6 -68)±a35F35(63-65)
+r
dU ^ dU
m lf+^ ay.
where indices in the symbols of the coefficients indicate the objects of heat and mass transfer in accordance with Fig. 1, b; aj are coefficients of the complex heat exchange; Fij are the surfaces, m2; Ti is temperature of the energy source, K; t is air temperature, K; Q is the temperature of the dried material, K; Cj is a reduced radiation coefficient; P is power of the emitter, W; L is length of the conveyor, m; mi, Gi is the mass of the object and its consumption, kg, kg/s; mi, ci is heat capacity of the objects, J/K; A\ is coefficient of the radiation absorption; r is a reduced heat of vaporization, J/kg; Rb is the Rebinder effect; U is moisture content in the material, kg/kg.
The system can be solved numerically if you set boundary conditions and numerical values of coefficients, which are difficult to calculate. A reduced mathematical description of nonstationary processes allows theoretical analysis of the heat and mass transfer in the existing computer environment and obtaining analytical calculation dependences.
The density of the radiation flux from the radiation surface to the absorbing surface can be represented [10, 11] in the form of the equation similar to the one for convective heat transfer:
^ = v!0-8 (( - T4 )=
= v 10-8 (12 - T2 ) (+T2 )=a j ( - 6 J, (16)
where
a j = V10-8 (( + T2 )( + T2)
is the linear coefficient of heat transfer under radiation.
Linearizing the radiation components in equations (8)-(10), (12), we rewrite the equation in the following form:
96
m1c1 "dt" = P -a12F12 (61 62 )-
-a«F13 (61 - 63 ) - a15^5 (61 - 65 ) - a^ F (61 -t) ; (17)
dt _ rdt , „ \ m c —±G c L— = a14F(61 -1) +
p "dt p p dy 14 11 1 '
+a32F3 (63 - t ) + a54F5 (65 - t) + a24F4 (62 - t) ;
96
m5c5 It5 = a15F15 (61 - 65 ) - a54F54 ( 65 - A
96
m6c6~df = aeF6 (67 - 66 ) - a36F36 ( 66 - t ').
(20) (21) (22)
In equation (19) we replace the last component, which determines the amount of heat to evaporate the moisture, with its thermal analog from equation (14) and obtain the following:
4 (- -% +-4' - Gb =
= A
a13F13 (61 - 63 ) + a23F23 (62 - 63 ) --a34F34 (6? - t) - a'eF6 (6? - 66 ) ± ±a35F35 ( 63 6 5 )
(23)
For the established mode, derivatives of time in equations (17)—(22) will equal to zero, and the system of equations for the stationary mode will look as follows:
P = a12F12 (61 - 62 ) + a13F13 (61 - 63 ) + +a15F15 (61 65 ) + a14F1 (61 - t);
a12F12 (61 62 )=a 25F25 ( 62 - 65 ) + +a23F23 ( 62 - 63 ) + a'24F2 (62 - t)
a15F15 (61 65 ) = a54F54 (65 - A a6F6 (63 -66 ) = a36F36 ( 66 -1,);
cL (g? - g03 = * I ? Rb Jdy
a13F13 (61 - 63 ) + a23F23 ( 62 - 63 ) -= A -a34 F34 (6? - t )-a eF6 (6?-66 )± ±a35F35 ( 63 6 5 )
(24)
(25)
(26) (27)
(28)
96
= a12F12 (61 - 62 ) -
dt
-a25F25 (62 - 65 ) - a23F23 ( 62 - 63 ) - a'liF2 (62 - ^ (18)
G"c"L dy = a'4F1 (61 - t ) + a32F3 (63 - t) +
+a54F5 (65 - t) + a'24F4 (62 - t).
(29)
c
From equations (24)-(27) we determine temperature values for the individual components of the plant (61, 62, 65, 66) by means of the sought values of t and 63 and obtain the following:
e1 = k1e3 + k2t; (30) G d83 (y).
1 dy
82 - k31 83 + k4^ ; (31) ~ dt (y) G 2 =
8 5 = k583 +V; (32) dy
86 — k78 3 + kit , (33) where
Substituting the obtained values (30)-(33) into expressions (28) and (29), after the corresponding transformations, we get the following system of two interrelated equations:
(34)
(35)
where ki-k8 are dimensionless complexes determined from the constructive Fj and the regime azj parameters of the plant, which are taken for calculations as constant values:
k1 = «13F13 («12F12 + «24F2 + «25F25 )(«15F15 + «54F54 )/ K;
«12^2
«15«UF15 (F1 + F2 ) +
«15F15 + «14 (F1 + F2 ) +
+a,
V+«25F25
+F
'54 / /
«54«14F1F54 + + «15F15 ( «14F1 + « 54 F54)
/ K;
+ («.. F2 +«25F25 ) k3 = «13F13 («15«25F\5F25 +«12F12 («15F15 +«54F54 K;
K =
«15«KF15
«14 F1F2 + «13^13^2 + +«12F12
(F1 + F2 ) + «25F1F25
+«54F54
(«K-; F1 + «13F13 + «15 F15 )
x(«k-;F2 +«25F25 ) +
«15F15 + «14 (F1 + F2 ) +
+«12F12
V+«25F25
/ K;
k5 = «13«15F13F15 («12F12 + «; -; F2 + «25F25 ) / K /
(«..-F +«25F25 )
k6 =
«15«14F1F15 + «54F54 X x(«; - F1 +«13F13 +«15F15 )
+«12F12
«15«14F15 (F1 + F2 ) + «54F54 +«14 (F1 + F2 ) + «25F25
«13F13 +
V+«15F15
\ \ +
/ K ;
k7 =
«eF6 ;
« eF6 + « 36 F36
k = «36F36 ;
« eF6 + « 36 F36
where
K =(«K- F2 +«25F25 )
+«12F12
«54F54 («Î4F1 + «13F13 ) + +«15F15 (F1 + «13F13 + «54F54 ) «54F54 («13F13 + «14 (F1 + F2 ) + «25F25 ) + +«15F15 ( «13F13 + «14 (F1 + F2 ) + «54F54 )
G =
A7LG'
A1 («13F13k2 + « 23 F23k4 + «34F34 « 35F35k6 )
G ' = G7
7 £6
«13F13 (k1 1) + «23F23 (k3 1) «34F34 + a = +«35F35 (1 - k5 ) + «eF6 (k7 - 1)_;
«13F13k2 + «23F23k4 + «34F34 - «35F35k6
6 =_«eF6k8t'_;
Al («13F13k2 + «23F23k4 + «34F34 - «35F35k6 )
G2 =
c G L
f f
«14F1k1 + «24F2k3 + «34 F3 + «45F5k5
fl = «14F1 (1 - k2 ) + «24F2 (1 - k4 ) + «54F5 (1 - k6 +) + «34F3 «14F1k1 + «24F2k3 + «34F3 + «45F5k5
Solving equations (34) and (35) with regard to i(^) and 0(y), after transformations we obtain the following:
d ^83 b dq
dy2 di
d 2t „ dt
A^-3 + B—3 + C83 = D1, dy dy
(36)
(37)
a—y+b—+a=d 2,
dy dy
where
A = G1G2 ; B = a2G1 + a1G2 ; C = a1a2 -1; D1 = -61a2, D2 = 61.
General solution of equations, under boundary conditions,
y = 0; 83 = e0 = 81; t = t0 = t1; ddt=dt=0
according to [10] looks as follows: t (y)=y - ^ )+)cL;
83 (y) = c2 ( y - ^ ) + D-;
where
, -B'±VB'2 -4AC
r2 =-
1,2 2A'
(38)
+
c,= t0C - D2 . c, = 603C - D1 1 C (r1 - r2) 2 C(r1 - r2 )
To determine the distribution of moisture in the material during drying along the length of the drying chamber in the direction of motion, we use equation (14) for zero derivatives of time:
c7 G?L 96? G4L dU Rb m? dy m70 dy
(41)
G,
G3 = -—= U3, t t
G
thus, —L = u, and eventually we get the following: m
1 d63 dU
c3 (U = -
Rb(U )
dy dy
To obtain an analytical expression of the change in the grain moisture content over time, it is necessary to have a dependence of the temperature change in the material in some time. Having substituted the expression of the derivative of
grain temperature from the coordinate —-3 from equation
dy
(43) into equation (19), we obtain the following:
The value of the ratio — L characterizes the rate of
m3
movement of the grain material because
(42)
A? \ m
= A
- me ïae? + G - G031 ? Rb dt
+AL\G?IF1 (y ) =
Rb J
a13F13 (61 - 63 ) + a23F23 (62 - 63 ) -
-a34F34 (6? - t) - aeF6 (6? - 66 ) ± c35F35 ( 63 - 65 )
Let us differentiate equation (47): ^¡y = Kt ((e^" -1 er1y) = F2 (y),
.(47)
(48)
where
Having differentiated equation (39), we obtain the following:
d 63 (y)
dy
= KT (y -r2r e y) = F1 (y),
(43)
where
K = 603C - D1 T C(r1 - r2)
Substituting the obtained dependence into equation (41), we get the following:
dU _ c3(U)■ KT
U i) i)
-ARr0JenUdU = KTr1r2 Jer*ydy-Jer ydy
we get the following:
U (y) = ^ln n
enU1 - K
er2y -1 er1y -1
5 C (r1 - r2)
and substitute the resulting dependence into equation (19), and we will have the following:
dt
m"c" aï ± G"c"F2( y)=a" F1 (61-1 )+
+a32F3 (63 - t) + a54F5 (65 - t) + a24F4 (62 - ^
(49)
Equations (47) and (49) as well as (28) and (29) have the same values in the right-hand side, so using substitution (30)-(33) we obtain the equation of the dynamics of change in the parameters 93 (t, y) and t (t, y) in the following form:
(44)
dy Rb(U ) ■ r0 where c3(U ) is heat capacity of the grain material:
c3 (U ) = U ■ c, + (1-U ) ■ c c 3, (45)
where cv is specific heat capacity of water, and cc3 is specific heat capacity of the dry grain material. Provided
RbUU)=A enU
c, (U)= ARe ,
where AR and n are empirical coefficients, having integrated equation (44)
d6
T1~ït=t - fl163+b1 (y);
T2 ¡¡t=63 - a2t+b2 (y);
(50)
(51)
where
t=
T2 = b1 = b =
A (a13F13k2 + a23F23k4 + a34F34 a35F35k6 )
_cm_;
ai4F1k1 + a24F2k3 + a34 F3 + a45F5k5
_b1 - G3c3L ■ F1 (y )_.
A (a13F13k2 + a23F23k4 + a34F34 - a35F35k6 )
_—G7c?L ■ F2 (y)_
A (ul3F13k2 + a23F23k4 + a34F34 - a35F35k6 )
(46)
where
Solution of the system (50), (51) is analogous to the solution of equations (34) and (35) under the following initial conditions:
t = 0; 83 = 80 = 81; t = t0 = t1; dt = fc= 0,
where 60 and t0 are the initial values of temperature of the grain and air, which are equal to their values at the input.
8' (',У)=eCDr(re<•-r;er')+D;(yl; (52) t (t, y ) = "C^Dl) ( - **)♦ (53)
where
<2 ; A1 = ^
1,2 2A 1 2
B' = a1T2 + a2T1; C = (a1a2 -1);
A (y)=b2 (y)- a2b1 (y); d2 (y)=b1 (y)- a1b2 (y).
Equations (50) and (51) describe the change in temperature of the grain and air with time, along the length of the dryer's chamber. To obtain the dependence of the change in moisture content in the material on the time and coordinates, we differentiate equation (52) by time:
^ = Uy)rr( t-e*),
(54)
where
e03c - D1(y)
C (- r'2)
and substitute the obtained value in the equation for determining the Rebinder effect:
c U )fT-R„{U K f,
provided Rb(U) =
c3 (U)= ^ , having integrated equation (52):
-ARr0 J enUdU = Kt (y) r'r2' jer2ydt -Jer'Mx
we get the following:
U(t,y) = -ln enU' -Kc,(y\ n
er2T-1 er,t-1
where
My)=n-^r:r,
AR ' r0
scribed by equations (52) and (55). Dependences (52) and (55) determine the distribution of temperature and moisture content of grain and oil-bearing materials along the length at any moment of time. To determine the parameters of grain and oil-bearing materials at the outlet it is necessary to accept that y=L. Then equations (52) and (55) will determine the change in the final parameters of the material, i. e. moisture content U2 (t) and temperature e2 (t).
Parametric identification of the mathematical model of heat and mass transfer processes in the IR drying of grain materials (with vibrational motion) requires the values of the coefficients substituted in the equation. Theoretically, coefficients can be determined only by iterative calculations by the known reference data.
5. Experimental findings on heat and mass transfer in the IR dryer
The obtained mathematical models can be identified according to experimental data, by comparing the analytical and experimentally obtained dependences [14, 15]. But it is very difficult to determine parameters (such as temperature and moisture content) of the material in the moving layer in the presence of IR radiation and vibration. Therefore, we will determine the final parameters of temperature and moisture content in the grain at the output of the dryer changing the heat treatment exposure. Experimental data have been approximated by polynomials of the second order; the graphic interpretation of the experimental data and calculated curves of the dependence of temperature and moisture content are presented in Fig. 2, 3.
(55)
Thus, the basic dynamic characteristics of the drying conditions of grain and oil-bearing materials in a continuously operating plant with IR energy supply can be de-
Fig. 2. Changes in the dampness of IR-treated rapeseed in different-weight batches (experimental data and curves of polynomial regressions), P=320 W: 1 — 100 g; 2 — 200 g;
3 — 300 g
In order to determine the Rebinder effect, experiments were carried out in a periodic drying mode, and the changes in temperature and mass were recorded in real time. The dependence of the change in the Rebinder effect on the material dampness is shown in Fig. 4.
As you can see from Fig. 4, the value of the Rebinder effect decreases alongside reducing the moisture content in the grain, which indicates that higher moisture content in the grain material involves higher specific heat consumption for its evaporation.
The change in the moisture content of the material with time (see the drying curve) during the dryer operation in a
U,
continuous mode was determined by the method of successive overlay of experiments, i. e. there were recorded changes in the moisture content during the dryer operation within a certain interval of time.
t,°c 80
Wc, % 11
64
48
32
16
1 > n[
fpl
10
, min.
15
20
Fig. 3. Thermograms of IR drying of rapeseed in different-weight batches (experimental data and curves of polynomial regressions), P=320 W: 1 - 100 g; 2 - 200 g; 3 - 300 g
Rb 0.04
0.03
0.02
0.01
2 X / $ /
\ J - 3
( ■ 1 1 * - '
10
Wc,%
11
Fig. 5. Changes in the dampness of grain in the moving layer: 1 - P=500 W, 2 - P=400 W
In this case, the analytical dependence can be used to identify mathematical models by comparison with the experimental data.
10.25
9.5
8.75
V
<Nj
>
8
10
Fig. 4. Dependence of the Rebinder effect on the moisture content in the material, P=320 W: 1 - 100 g; 2 - 200 g; 3 - 300 g
Fig. 5 shows the results of experiments with the graphs of changing moisture content in rapeseed (in a moving layer).
Fig. 6. Experimental data on rapeseed drying and the calculated curves: 1 - P=400 W, 2 - P=500 W
The use of the Leuenberg-Marquard algorithm and the method of parametric identification of heating plants [16] allowed determining the main parametric complexes of models (50) and (53). Thus, in particular, Fig. 6 shows the graphic interpretation of the calculated curves according to the analytical dependence (53) and experimental points.
Fig. 6 shows that the theoretical curves with identified parametric complexes deviate from the experimental points by less than 5 %.
6. Discussion of the results of mathematical modeling and experimental identification of the model
The mathematical model of heat and mass transfer processes is devised by means of analytical modeling based on energy and material balance equations as well as generalizations of experimental data based on the theory of similarity, probability and mathematical statistics. The peculiarity and value of the proposed model consist in the found regularities of heat and material interaction between all objects of heat and mass transfer inside the dryer. The approximate analytical solution presented by this model allows the following:
- to determine the fields of temperature and moisture content of the grain material within time, by the coordinate in the movement direction;
- to determine and clarify the parametric complexes of the model based on the empirical dependences of the kinetics of heating and dehydration of the grain material;
- in the future, to use this model for synthesizing a system of the optimal drying control.
The results of experimental studies are presented in the form of curves obtained by approximation of the experimental data. The found dependences allow summarizing the experimental data in the form of criteria dependences. In particular, Fig. 4 shows the dependence of the Rebinder effect; the use of this dependence allows model (8)-(15) to be solved analytically.
The obtained empirical mathematical models of the kinetics of heating and drying were used for the parametric identification of approximate solutions of the analytical mathematical model. The specified kinetic coefficients of dependences allow further significantly improvement of the accuracy of the numerical solution of the devised analytical model.
The above research findings will be useful to improve plants and technologies of combined IR drying. But their practical use requires rememdering the following: the pre-
sented mathematical models are kinetic, i.e. they do not take into account the distribution of temperature and moisture content inside the seeds, which limits their application when IR energy is highly intensive or the geometric sizes of seeds increase.
In this study, analytical calculations were facilitated due to the linearized coefficients of the radiation heat transfer, regardless the peculiarities of absorption of IR radiation by the material. But the nonlinear or piecewise linear approximation of the coefficients of radiation transfer and taking into account variables of the optical coefficients of the IR-treated material significantly complicate the mathematical calculations and can make obtaining an approximate solution of the system of balance equations impossible.
7. Conclusions
1. The devised mathematical model of heat and mass transfer in the vibration dryer with IR energy supply takes into account the heat transfer by convective and radiation methods between all interacting objects inside the dryer. The proposed approximate solution of the sys-
tem allows predicting the development of temperature and moisture content fields depending on the power of the IR emitter for further calculation of the energy efficiency in the dryer and synthesizing of the system of optimal process control.
2. The results of experimental studies of the kinetics of heating and dehydration of the grain material have laid the basis for determining how parameters of the IR dryer impact the exposure of the drying process. In particular, if the power of the IR emitter increases from 400 to 500 W, the drying time from the initial dampness of the material of 11 % to the moisture content of 8.75 % decreases from 9 to 7 minutes. It is determined that the Rebinder effect, which characterizes the dampness and heat properties of the material, decreases with a decrease in its moisture content from the value of 0.04 at 11 % to 0.01 at 9 %. This proves that higher moisture content of the material involves higher specific heat consumption for the moisture evaporation.
3. The method of inverse problems was used to identify parametric complexes of finite analytical dependences. The study presents the found parametric complexes and shows that experimental data coincide with the theoretical calculations within the range of 95-98 %.
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