Journal of Engineering Physics and Thermophysics, Vol. 86, No. 3, May, 2013
HEAT TRANSFER KINETICS AND EXPERIMENTAL METHODS FOR CALCULATING THE MATERIAL TEMPERATURE IN THE DRYING PROCESS A. I. Ol′shanskii
UDC 66.047.37
This paper presents methods for calculating the integral-average temperature of the material for the period of decreasing drying rate on the basis of generalized variables of the process. Keywords: integral-average temperature, temperature coefficient of drying, Rehbinder number, generalized drying time, relative temperature, relative moisture content. Introduction. The temperature of a material is one of the main parameters determining its technological properties, the rate and duration of its drying, the expenditure of energy, and the quality of finished products. Establishing the dependence of the temperature of the moist material on its moisture content in drying is of great practical importance since it determines the choice of optimal thermal conditions of drying and the type of drier. Drying of a moist material is a complex thermophysical process proceeding under the action of various factors complicating the heat and moisture transfer in it. Therefore, the possibility of using analytical heat and mass transfer equations for real drying conditions is limited. In this connection, of substantial interest are methods of calculating the drying kinetics based on the use of the minimal number of constants determined from experience. Formulation of the Problem. The heat transfer between the moist material and the environment and the laws of change in its temperature in the period of decreasing drying rate are investigated on the basis of the A. V. Luikov drying kinetics equation, G. M. Kondrat′ev′s regular regime theory, and the results of solving the A. V. Luikov heat transfer equation [1–3]. The basic drying kinetics equation has the form q* = N * (1 + Rb) .
(1)
The relative drying rate N* is determined by the following dependence established in drying various materials by different techniques [4, 5]: N * = exp (− a0 N τ) ,
(2)
where Nτ is the generalized drying time. Integrating Eq. (2), one obtains the drying time of the moist material in the period of decreasing drying rate [4, 5]: τ II = −
1 ln (1 − a0 (Wcr − W )) . a0
(3)
The coefficient a0 is determined from the relation common for a number of moist materials: a0 =
0.8 , Wcr
(4)
which was obtained as a result of processing data on the drying under different conditions of more than 20 moist materials: asbestos sheets, boards, porous ceramics, clay, felt, peat plates (under convective heat input); yeast, bread, macaroni, vegetables, and other materials (drying in a fluidized bed with radiation heat input) [4, 5]. The central characteristic of the drying kinetics is the criterion Rb = cb/r, where b = dt du is the temperature coefficient of drying. The relative temperature coefficient of drying B is determined as Vitebsk State Technological University, 72 Moskovskii Ave., Vitebsk, 210035, Belarus; email:
[email protected]. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 86, No. 3, pp. 584–594, May–June, 2013. Original article submitted November 27, 2012. 622
0062-0125/13/8603-0622 ©2013 Springer Science+Business Media New York
B = b
ucr . Tmed
(5)
The dependences Rb = f( u ), B = f( u ), and b = f( u ) are found experimentally and are described by the empirical relations [1, 2] Rb = A exp − n0 (u − ueq ) ,
(
)
(6)
B = a exp (− m(u − ueq )) ,
(7)
b = b0 exp (− m0 (u − ueq )) .
(8)
The values of the constants A, n0, m, b0, and m0 in the formulas are obtained by processing the experimental data by the least square technique. The product of two generalized constants in Eq. (1) N*Rb can also be given in the form of the exponential dependence analogous to (6)–(8) [5]: N * Rb = D exp (− aN τ) .
(9)
In [6, 7], we investigated the regular regime of drying moist materials, in which the heating of the body is only determined by the conditions of external heat exchange, as well as by its sizes, shape, and thermal properties. According to A. V. Luikov [3], heating of a moist material in drying in the period of decreasing drying rate is realized in the regular regime, since in this case the temperature and the moisture content of the material behave exponentially. The regular regime of drying is determined by the condition
−
dt = mt ( tmed − t ) . dτ
(10)
The heating rate of a moist body mt is found from the experimental relation tan γ =
ln (t med − t0 ) − ln (t med − t ) = mt = const . τ − τ0
(11)
The parameter mt determines the rate of change with time in the logarithm of excess temperature which is equal for all points of the body at the integral-average temperature. The average temperature of the material t0 in the first period of drying is determined by the relation [6] common for various materials: t0 = 10 + 0.29t med .
(12)
A.V. Luikov [3] obtained from the solution of the heat transfer equation for the period of constant drying rate at tmed = const equations for determining the relative temperature of the material: θ=
θ=
t med − t w , t med − tenv rj0
α cr (t med − tenv )
(13)
.
(14)
In the first period of drying, the temperature of the wet-bulb thermometer tw is assumed to be equal to the temperature of the material surface (tw = ts). Equations (13) and (14) can be used to calculate the period of decreasing drying rate. The evaporation rate of moisture j is determined by the weighting method with the use of the drying curve described by the relation
j =
G0 du , F dτ
(15)
where du d τ is the drying rate in the second period determined with the use of Eq. (2) by the known drying rate in the first period:
du = N exp( − a0 N τ II ) . dτ
(16) 623
For the period of decreasing drying rate, on the basis of numerous experimental data on the drying of various materials the following equation has been obtained [8]:
T* = N*
α cr (1 + Rb) . α
(17)
The relative temperature T* is determined from the relation T* =
t med − ts . t med − t0
(18)
The values of the criterion Rb ≈ 0.1–0.3 in drying point to the necessity to terminate the process in order to avoid overdrying of the material, disturbance of its technological properties, and overexpenditure of energy. The above maximum values of this criterion are observed at the end of drying at a moisture content u = 0.01–0.04 close to the hygroscopic content. At small values of Rb the relation T* = N*
α cr α
(19)
holds. To plot the drying temperature curve of the investigated materials, we used the relations [8] n
⎛ u ⎞ α cr α = N *0.57 and = ⎜ ⎟ . α α cr ⎝ ucr ⎠
(20)
Analysis of the experimental data on the drying of porous ceramics, clay, asbestos sheets, and sole leather points to the possibility of using for the calculation of the average temperature of a moist material the relations T* =
θs =
⎛ u ⎞ t med − t = ⎜ ⎟ t med − t0 ⎝ ucr ⎠
m1
⎛ u ⎞ t med − ts = ⎜ ⎟ t med − tenv ⎝ ucr ⎠
and T * = f ( τ II ) ,
(21)
and θs = f ( τ II ) .
(22)
m2
The material temperature T* is counted off from its temperature in the first drying period t0 , and the temperature θs is counted off from the environment temperature tenv. Therefore, the dimensionless temperatures determined by these equations vary as follows: T* varies from 0 to 1, and θs varies from 0 to θs < 1. Derivation of Equations for Calculating the Temperature Curve. Let us write Eq. (9) in the following form:
N * Rb =
1 du c dt = D exp ( − a0 N τ) , N d τ r du
dt r = N D exp ( − a0 N τ) . du c
(23)
(24)
Integrating relation (24) yields the equation of the temperature curve [5] t = t med −
Dr exp ( − a0 N τ II ) . ca0
(25)
The constant a0 is determined by relation (4). The coefficient D characterizing the properties of a particular material is independent of the regime parameters of drying and is calculated by the formula [5] D = C exp ( − d (ucr − u )) . The coefficients C and d depending on the kind of the moist body are given in Table 1. In view of Eqs. (2) and (25), we write Eq. (1) in the following form [5]:
624
(26)
TABLE 1. Values of Constants in the Calculation Equations for Certain Materials Material and drying regime
A (6)
n0 (6)
m (7)
m0 (8)
mt, min–1 (10)
n (20)
m1 (21)
m2 (21)
C·103 (26)
d (26)
Asbestos sheets tmed = 90–150oC, ϑ = 3–10 m/s
0.5
15
20
14
0.06
0.5
0.4
0.8
4.3
4
Sole leather tmed = 40–60oC, ϑ = 3–5 m/s, φ = 15 %
0.5
8.5
7.5
4
0.04
0.25
0.9
1.9
1.3
4.5
Porous ceramics tmed = 90–150oC, ϑ = 3–10 m/s
0.5
20
45
16
0.1
0.5
0.7
1.2
6
14
Wool felt tmed = 90–150oC, ϑ = 3–10 m/s
0.1
6
5
2.5
0.02
0.5
0.15
0.9
4
3
q* = (1 + D )
(tmed
− t ) ca0 . Dr
(27)
The value of the complex (1 + D) at all possible values of the moisture content is equal to unity, which permits simplifying Eq. (27). The drying temperature coefficients B = f( u ) and b = f( u ) introduced by A. V. Luikov into calculations of the heat transfer kinetics make it possible to calculate the average bulk temperature of the material in the second period. Write expressions (5) in the form t = Tmed
t =
∫
B (u ) du + C1 , ucr
∫ b(u )du + C2 ,
(28)
(29)
where the constants C1 and C2 are given by the material temperature in the first period of drying t0. Substituting relations (7) and (8) into (28) and (29) and integrating the obtained expression between the given limits, we obtain the equations for the temperature curves: t = t med −
aTmed (1 − exp ( − m(u − ueq ))) , mucr
t = t0 +
b0 exp ( − m0 (u − ueq )) . m0
(30)
(31)
The constants a, b0, m, and m0 are found by experimental data processing by the least square technique. The solutions of Eqs. (10) and (11) give relations for calculating the average temperature of the moist material and the drying time in the second period: t med − t0 , exp (mt τ II )
(32)
1 t − t0 ln med . mt t med − t
(33)
t = t med − τ II =
625
Fig. 1. Dependences of the relative drying rates N* and N*Rb on the generalized drying time Nτ in different regimes: 1) tmed = 90oC; ϑ = 3 m/s; 2) 90, 5; 3) 120, 3; 4) 120, 5; 5) 120, 10; 6) 150, 3; 7) 150, 5. It is convenient to count off the drying time in the second period for all regimes from the time τ = 0 and the moisture content ucr , and determine the average temperature t0 by the average temperature of the material in the first period. The heating rate of the moist body mt does not depend on the drying conditions and is a constant mt = const [6, 7]. Consider the method for calculating the temperature of the moist material with the use of Eqs. (13) and (14) obtained by A. V. Luikov [3]. From the solution of these equations, the temperature on the material surface is determined:
ts = t med −
rj . α
(34)
The heat transfer kinetics in the drying process can be determined on the basis of the moisture exchange data. We write the balance equation of heat input to the surface of the moist material in the drying process:
q = r
G0 du G dt G du + cm 0 = 0 ( r + ( c0 + c m u ) b ) , F dτ F du F dτ
(35)
G0 du = j is the moisture evaporation rate in the second period, kg/(m2·oC). Equation (34) on the basis of Eq. (35) F dτ can be written in the form rj ts = t med − . (36) q (t med − ts ) where
The following two methods for calculating the average temperature of the moist material are based on numerous experiments on drying various capillary-porous moist materials by different techniques [8]. In view of Eqs. (19) and (20) we can write the relations n
T* =
⎛ u ⎞ α = T * ⎜ ⎟ = N *, α cr ⎝ ucr ⎠
(37)
(u ucr ) n /0.57 , (u ucr ) n
(38)
T* =
from which we obtain the equation of the temperature curve
626
T* =
⎛ u ⎞ t med − ts = ⎜ ⎟ t med − t0 ⎝ ucr ⎠
0.755n
.
(39)
The exponent n takes into account the decrease in the heat transfer coefficient α in the second period of drying. From the solution of Eqs. (21), (22), and (39) we obtain the equations of the temperature curves for the second drying period: m1 ⎛ u ⎞ (40) t = t med − (t med − t0 ) ⎜ ⎟ , ⎝ ucr ⎠ ⎛ u ⎞ ts = (t med − tenv ) ⎜ ⎟ ⎝ ucr ⎠
⎛ t = t med − ⎜ (t med − t0 ) ⎜ ⎝
⎛ u ⎞ ⎜ ⎟ ⎝ ucr ⎠
m2
,
0.755n ⎞
⎟ (1 + Rb) . ⎟ ⎠
(41)
(42)
Experimental Data Processing. Analysis of a large amount of experimental data on the drying of various materials by different techniques has shown that it is expedient to process results of experimental studies with the use of relations (2) and (9) [5]. Figure 1 shows the dependences N* = f(Nτ) and N*Rb = f(Nτ) for the process of convective drying of porous ceramics in a wide range of regime parameters. It is seen that the experimental points fall well on parallel lines. Analysis of the values of the average temperature calculated by formula (25) has shown that at all possible values of ucr and the generalized time Nτ for the investigated materials given in Table 1, exp (–a0Nτ) ≈ 1. Then Eq. (25) is simplified and takes on the form Dr (43) t = t med − . a0 c Figures 2 and 3 presents the results of processing the experimental data on the drying of an asbestos plate and porous ceramics in a wide range of parameters with the use of Eqs. (7) and (8). Analysis of the experimental data on the drying of a number of materials has made it possible to establish the laws of change in the coefficients a and b0 in Eqs. (7) and (8). Figure 4 shows the dependences b0 = f( ucr ) and a = f(Tmed) plotted with the use of the expressions
a = 6.62 ⋅ 10 −3 Tmed − 1.86 ,
(44)
b0 = 103 (1.11 − 1.45ucr ) .
(45)
Figure 5 shows the dependences ln (tmed – t ) = f(τII) for drying sole leather and porous ceramics under different conditions, by which the heating rate of a moist body mt can be determined. The values of all constants in the calculation equations are presented in Table 1. Calculation of the average temperature of a moist body t by the mt value requires knowledge of the drying time in the second period τII, which can be calculated by the known drying rate in the first period N. Figures 6 and 7 present the results of processing the experimental data on the drying of porous ceramics and asbestos ⎛ u ⎞ sheets with the use of relations (21), (22), and q* = f ⎜ ⎟ . As a result of the experimental data processing, we obtained the ⎝ ucr ⎠ calculation equations for porius ceramics: ⎛ u ⎞ T* = ⎜ ⎟ ⎝ ucr ⎠ 1.2
⎛ u ⎞ θs = ⎜ ⎟ ⎝ ucr ⎠
0.7
, T * = exp (−β t τ II ) = exp (−0.1τ II ) ,
(46)
, θs = 0.9 exp (−1.8β t τ II ) = 0.9 exp (−1.8 ⋅ 0.1τ II ) ,
(47)
and for asbestos sheets: ⎛ u ⎞ T* = ⎜ ⎟ ⎝ ucr ⎠
0.4
, T * = exp (−β t τ II ) = exp (−0.06τ II ) ,
(48)
627
Fig. 2. Dependence of the relative temperature coefficient of drying B on the moisture content u in the process of drying wool felt (a) and porous ceramics (b) at a temperature tmed = 90 (1), 120 (2), and 150oC (3) and an air velocity ϑ = 3–10 m/s. Fig. 3. Dependence of the temperature coefficient b on the moisture content ( u – ueq) for the process of drying porous ceramics (a) and asbestos sheets (b) at tmed = 90 (1), 120 (2), and 150oC (3) and an air velocity ϑ = 3–10 m/s.
Fig. 4. Dependences b0 = f( ucr ) and a = f(Tmed) for drying porous ceramics (1), asbestos sheets (2), sole leather (3), and wool felt (4) in different regimes: 1, 2) tmed = 90–150oC, ϑ = 3–10 m/s; 3) 40–60, 3–5; 4) 9–120, 3–10.
⎛ u ⎞ θs = ⎜ ⎟ ⎝ ucr ⎠
0.8
, θs = 0.9 exp (−2.3β t τ II )
= 0.9 exp (−2.3 ⋅ 0.06τ II ) .
(49)
⎛ u ⎞ The relative heat flow density q* = f ⎜ ⎟ in the process of drying porous ceramics, asbestos sheets, sole leather, ⎝ ucr ⎠ and wool felt in a wide range of regime parameters is determined by the relation ⎛ u ⎞ q* = 0.12 exp ⎜ 2.5 ⎟. ucr ⎠ ⎝
628
(50)
Fig. 5. Dependences ln (tmed – t ) = f(τ) for drying sole leather (a) and porous ceramics (b) in different regimes: a) tmed = 40 (1), 50 (2), and 60oC (3) and ϑ = 3–5 m/s; b) tmed = 490 (4), 120 (5), and 150oC (6) and ϑ = 3–10 m/s. Fig. 6. Dependences T* = f(τII) and T* = ( u ucr ) for drying porous ceramics (a) and asbestos sheets (b) and dependence q* = f( u ucr ) for drying porous ceramics, asbestos sheets, and wool felt (c) at tmed = 90 (1), 120 (2), and 150oC (3) and an air velocity ϑ = 3–10 m/s.
Fig. 7. Dependences θs = f(τII) and θs = ( u ucr ) for drying porous ceramics (a) and asbestos sheets (b) at tmed = 90 (1), 120 (2), and 150oC (3) and an air velocity ϑ = 3–10 m/s. Fig. 8. Dependence of the relative drying time (τI/τII) on the relative moisture content ( u ucr ) for drying porous ceramics and asbestos sheets at tmed = 90 (1), 120 (2), and 150oC (3) and an air velocity ϑ = 3–10 m/s. It should be noted that the coefficients mt and βt have the same physical meaning and are practically equal for different methods of experimental data processing with the use of relations (10), (11), and (46)–(49). The differences between them are within the accuracy of experiments, which confirms the reliability of the obtained results. In investigating the drying kinetics of moist materials, it is necessary to know the drying rate in the first period N. The value of N depends on the regime parameters of drying, the method of heat input, the moisture content of the material, and other factors influencing the drying process. Consequently, the drying rate in the first period can be considered to be the generalized variable taking into account the influence of all factors affecting the drying process. The drying rate in the first du period can be determined from Eq. (2) if N* = 1 and = N = const: dτ 629
TABLE 2. Comparison of the Values of Average Integral Temperatures Calculated by Formulas (30), (31), (32), and (36) with the Experimental Values Porous ceramics δ = 5 mm (30)
Asbestos sheets δ = 6–8 mm (30) Drying regime
tmed = 90oC, ϑ = 5 m/s u
tcalc , oC
0.08 0.06 0.04 0.02
52 56 61 69
texp , oC
tmed = 120oC, ϑ = 5 m/s tcalc , oC
51 63.5 59 67 64 78 71 89 Sole leather δ = 4 mm (31)
tmed = 90oC, ϑ = 5 m/s
texp , oC
u
64 69 86 92
0.16 0.1 0.08 0.04
tcalc , oC
texp , oC
tmed = 120oC, ϑ = 3 m/s tcalc , oC
43.5 41.5 52 47 46 59 51 48.5 63.5 63 61.5 81 Asbestos sheets δ = 6–8 mm (31)
texp , oC
49 59 65 79
Drying regime tmed = 40oC, ϑ = 3 m/s, φ = 15% tcalc , oC
u 0.6 0.42 0.3 0.27 0.24
texp , oC
tmed = 50oC, ϑ = 3 m/s, φ = 15% tcalc , oC
25.6 24.5 27 29 28 30.5 33 33 34.5 35 35 36 36.5 37 38.5 Porous ceramics δ = 5 mm (32)
tmed = 90oC, ϑ = 5 m/s
texp , oC
u
tcalc , oC
26.5 30 35 36.5 40
0.2 0.16 0.12 0.08 0.04
38.3 40.8 42.7 52.8 64.4
texp , oC
tmed = 120oC, ϑ = 5 m/s tcalc , oC
37.5 46.4 38.5 48.8 41.5 50.7 50.5 60.9 64.7 76 Sole leather δ = 4 mm (32)
texp , oC
44.8 47.6 50.1 60 76
Drying regime tmed = 120oC, ϑ = 3 m/s
tmed = 90oC, ϑ = 10 m/s
tmed = 50oC, ϑ = 3 m/s, φ = 15%
tmed = 60oC, ϑ = 3 m/s, φ = 15%
u
tcalc , oC
texp , oC
tcalc , oC
texp , oC
u
tcalc , oC
texp , oC
tcalc , oC
texp , oC
0.07 0.05 0.03 0.02
60 74.5 91.5 101.5
61.5 75.5 91 100
50.4 58 69.3 73
51 60 71 74.5
0.51 0.42 0.3 0.24
27 31.5 38 41
26.5 30 37.5 40.5
32 34 42 48
30.5 33 40.5 46.5
Porous ceramics δ = 5 mm (36)
Asbestos sheets δ = 6–8 mm (36) Drying regime
tmed = 120oC, ϑ = 3 m/s
tmed = 120oC, ϑ = 3 m/s
u
j·103, kg/(m2·oC)
q/Δt, kg/(m2·oC)
ts,calc, oC
ts,exp, oC
u
j·103, kg/ (m2·oC)
q/Δt, kg/(m2·oC)
ts,calc, oC
ts,exp, oC
0.1 0.08 0.06 0.04
1 0.78 0.57 0.34
42.4 39.6 35.9 30.1
56.6 72 83 93
55 70 82 92
0.16 0.12 0.08 0.04
0.71 0.49 0.38 0.216
26 19 16 14
52 59 68 86
51 60 70 88
630
TABLE 3. Comparison of the Values of Average Integral Temperatures Calculated by Formulas (42), (40), and (43) with Those Obtained Experimentally Porous ceramics (42) Drying regime tmed = 120oC, ϑ = 5 m/s, φ = 5%
tmed = 150oC, ϑ = 5 m/s, φ = 5%
u
Rb
tcalc , oC
texp , oC
u
Rb
tcalc , oC
texp , oC
0.08 0.06 0.04 0.02
0.11 0.16 0.23 0.32
58.5 68.5 84.8 105
60 70 84 103.5
0.08 0.06 0.04 0.02
0.13 0.19 0.26 0.35
77.5 89 106.5 127
81 91 105 124
Porous ceramics δ = 5 mm (40) Drying regime tmed = 120oC, ϑ = 5 m/s, φ = 5%
tmed = 90oC, ϑ = 5 m/s, φ = 5%
tmed = 150oC, ϑ = 5 m/s, φ = 5%
u
tcalc , oC
texp , oC
tcalc , oC
texp , oC
tcalc , oC
texp , oC
0.08 0.06 0.04 0.02
60 67.5 80.5 95.5
62 68 81.5 95
48.5 52.5 63 74
50 54 65 75
77 84 100.5 119
80 86 99 120
Sole leather (43)
Asbestos sheets (43) Drying regime
tmed = 60oC, ϑ = 5 m/s, φ = 15%
tmed = 120oC, ϑ = 5 m/s, φ = 5%
u
τ, min
tcalc , oC
texp , oC
u
τ, min
tcalc , oC
texp , oC
0.1 0.08 0.06 0.04
11 17 22 28
32 39 40.5 44
31.5 37 41 45
0.16 0.12 0.1 0.08
14 13 15 17
45 50 54 61.5
44.5 49 55 63.5
N =
1 . a τI
(51)
⎛ u ⎞ τI = f ⎜ ⎟ for the process of drying porous ceramics and asbestos sheets in the τ II ⎝ ucr ⎠ range of change in the regime parameters tmed = 90–150oC, ϑ = 3–10 m/s, which is expressed by the equation
Figure 8 shows the dependence
⎛ τI u ⎞ = 0.32 exp ⎜ 2.5 ⎟, τ II ucr ⎠ ⎝
(52)
where the drying time in the second period τII is found from Eq. (33). Analysis of the Equations of Temperature Curves. The calculated and experimental values of the integral-average temperature t are given in Tables 2 and 3. Comparison of these data has shown a fairly good agreement of the calculated values of the above temperature with its measured values. The discrepancy between these values is within the accuracy of measurements.
631
It may be noted that formula (42) gives a higher spread of temperature values as compared to the other formulas. This is explained by the fact that in deriving the above equation the value of the coefficient 0.755n was determined with two errors instead of one. The advantage of Eq. (36) obtained by A. V. Luikov [3] theoretically is the fact that it brings the drying theory and practice closer together and permits using results of analytical solutions for calculating the heat transfer kinetics by the moisture exchange data without recourse to the calculation of the heat transfer coefficients α by criteria equations. The disadvantage of the calculation by Eq. (36) is the difficulty of experimental data processing in determining the moisture evaporation rate j and the heat flow density q. As an example, consider the calculation of the N value in the period of constant drying rate for porous ceramics at tmed = 90oC, ϑ = 10 m/s, and for asbestos sheets at tmed = 120oC and ϑ = 5 m/s. The drying time in the second period at fixed values of the moisture content (0.07 for the ceramics and 0.12 for the asbestos) is determined by Eq. (33). The temperature of the ceramics and the asbestos in the second period in drying to fixed moisture contents is calculated by relation (31). Then the ratio τI/τII is determined by Eq. (52) and the drying time in the first period N is determined by relation (51). Calculations by the above method gave the following results for the first drying period: Nexp = 1.65% /min and Ncalc = 1.49%/min for the porous ceramics, Nexp = 4.48% /min and Ncalc = 4.9%/min for the asbestos sheets. Conclusions. Investigations of the drying process have been carried out and the results on the drying of various capillary-porous moist materials are presented. Equations for determining the integral-average temperature of the material in the second period have been obtained. An approximate method for calculating the drying rate in the first period is proposed. A comparison between the calculated and the experimental values of the average integral temperatures has been made. The heat exchange for the second period has been investigated. The character of the change in the heat flow density has been determined. The results of the investigations made permit using the approximate experimental equations containing the minimum number of experimentally determined constants for calculating the kinetics of the drying process.
NOTATION a, thermal diffusivity, m2/s; Bi, Biot criterion; c0 and cm, heat capacities of the dry and moist material, kJ/(kg·oC); F, surface area of the body, m2; G0, mass of the dry body, kg; j, moisture evaporation rate in the second period, kg/(m2·s); Kn, Kondrat′ev criterion; N, drying rate in the first period, s–1; N*, relative drying rate; Nu, Nusselt criterion; qI and qII, heat flow densities in the first and second drying periods, W/m2; q*, relative heat flow, %; Rb, Rehbinder criterion; r, heat of vaporization, kJ/kg; Re, Reynolds criterion; Tmed and Tw, absolute temperatures of the medium and the wet-bulb thermometer, K; T*, relative drying temperature; t0 , tmed, t , and ts, material temperature in the first drying period, medium temperature, current temperature of the material, and surface temperature of the material, oC; u0 , ucr , ueq , and u, initial, critical, equilibrium, and current moisture content of the material; W, moisture content of the material, %; α cr and α , heat transfer coefficients in the first and second periods, W/(m2·oC); δ, thickness of the material, m; ϑ , thermophore velocity, m/s; λ, heat conductivity coefficient of the moist body, W/(m2·oC); μ, roots of the characteristic equation; ρ0 and ρm, density of the dry and moist material, kg/m3; τI, τII, and τ, drying time in the first and second periods and current drying time, min; φ, relative humidity of air, %; χ, relative drying coefficient, 1/%; ψ, nonuniformity coefficient of the temperature distribution in the body. Subscripts: cr, critical; w, wet; s, surface; m, moist; med, medium; 0, initial value of a parameter; calc, calculation; exp, experimental; eq, equilibrium; env, environment.
REFERENCES 1. 2. 3. 4. 5.
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A. V. Luikov, The Theory of Drying [in Russian], Énergiya, Moscow (1973). V. V. Krasnikov, Conductive Drying [in Russian], Énergiya, Moscow (1973). A. V. Luikov, Heat-Conduction Theory [in Russian], Vysshaya Shkola, Moscow (1967). A. I. Ol’shanskii and P. S. Kuts, Some laws of food drying kinetics, Izv. Vyssh. Uchebn. Zaved., Pishch. Tekhnol., No. 5, 97–101 (1977). A. I. Ol’shanskii, E. F. Makarenko, and V. I. Ol’shanskii, Some laws of the kinetics of moisture and heat exchange in drying moist materials, Inzh.-Fiz. Zh., 81, No. 6, 1102–1110 (2008).
6. 7. 8.
A. I. Ol’shanskii and V. I. Ol’shanskii, Investigation of the heat transfer in the process of drying by the regular regime method, Inzh.-Fiz. Zh., 85, No. 2, 385–391 (2012). A. I. Ol’shanskii and V. I. Ol’shanskii, Investigation of the process of drying thin materials by the regular regime method, Vestsi NAN Belarusi, Ser. Fiz.-Tékh. Navuk, No. 2, 75–81 (2011). A. V. Luikov, P. S. Kuts, and A. I. Ol’shanskii, Heat transfer kinetics in the process of drying moist materials, Inzh.-Fiz. Zh., 23, No. 3, 401–406 (1972).
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