CC BY
12
Ruzmetov Raxmatjon Ibodullaevich,
researcher,
Madumarov Ilxonjon Dedaxanovich, docent, candidate of the technical sciences, Mardonov Botir Mardonovich, prof. doctor of sciences, Tuychiev Timur Ortikovich, Ph D., Tashkent Institute of Textile and Light Industry E-mail: [email protected]
MODELING OF HEAT EXCHANGE PROCESSES BETWEEN RAW COTTON AND COOLANT IN A SCREW DRUM
Abstract: a mathematical model is proposed for describing the heat exchange processes between raw cotton and air moving together in sections of the screw, with an open area where the air temperature is constant. To describe the heat exchange processes between cotton and air, as well as between them and the walls of the screw, Newton's law was used.
Keywords: raw cotton, feather, cotton cleaner, model, netlike surface, spiked-drum, pin, blow.
Introduction: Use for small distances (up to 20-30 m) of 2. The lower part of the section is in contact with the mesh
small-sized materials, which include cotton of raw materials, surface, along which, by impact and pulling of the raw with augers and calculations of their productivity are given in [1, the pins, the raw material is purified from weedy impurities; 2, 3]. Consider the process of drying the mass of raw cotton 3. Suction of heat flow, T0 temperature is carried out
through the mesh surface;
4. On the surface of the rotating shaft of the screw, the temperature is zero
5. Each particle of the flow of raw cotton moves along a screw line;
6. In each section, heat exchange takes place between raw cotton and air according to Newton's law [4], as well as contact heat exchange between the raw material and the surfaces of the screw.
T = T
in the sections of the moving screw (Fig. 1). We believe that all sections of the screw are filled with the same amount of cotton raw material. Let's study the drying process in each section of the screw. For this purpose, the process is considered stationary.
Analyses. The initial section takes a section BCD that is filled with moving raw cotton. To simulate the process of drying the raw material, we accept the following assumptions:
1. Each section has an upper open area, where the air temperature is constant and equal Tj ;
Fugire 1. Scheme of the process for drying raw cotton mass in sections of a moving auger
Depending on the screw angle in the above assumptions, each section consists of separate sections, like BCD. If you direct the axis OZ along the axis of the screw, then the coordinate of an arbitrary particle of raw cotton in the first section of the screw is determined by the formulas:
h
x = r cosQ, y = r sin0, z = —9 R0 < r < Rj, 0 <9 < 2n 2n
here r - is the polar radius, h - the pitch of the screw, R0 and Rj the inner and outer radiuses of the screw.
The open part of the screw is determined by the value of the screw angle 6.
0 <6 < 2a (1)
2a - the angle of the screw, determined by the intersection of a cylindrical surface with a plane parallel to the axis of the screw. Let's denote, T1v, T1x - the air and cotton temperatures in the first section of the screw with the open part (R0 < r < R1, 0 <6 < 2a). Through T2v, T2x similar values in the closed part of the section R0 < r < R1, 2a <6 < 2n, where the cotton raw cotton particles contact the mesh surface. We believe that the air temperature Tlv in the section R0 < r < R1, o <9 < 2a postponed and equal T1, the temperature of raw cotton does not depend on the polar coordinate r. The movement of raw cotton particles Oz and air occurs along the axis and their velocities are equal to vx = vv = ha /2n .
According to clause 6 and taking into account the equali-
dTlv dTlv 2n dTlv dTx dTx 2n dTx ties —- = v—- = — and —x = v —x = — v —x,
dt dz h dO dt x dz h x dQ
T1x, T2v and T2x satisfy the stationary heat equation: on areas with a free surface:
k, =
Tlb = T under 0 < 0Z2a
(2)
= a*T -Tlx) under 0 <6Z2a (3)
h dQ
2~ncivi^ = ^(T2X-TJ + h *Sn(T0 -T2Junder 2aZdZ2n(4) h dO
= aT-T^ + PS^T-T2Junder 2aZ9Z2n (5)
h dQ
where, cb, cx - is the specific heat of air and raw cotton, abx -the coefficient of convective heat exchange between air particles and raw cotton, pb and Px - the coefficients of contact heat exchange between the mesh air and raw cotton particles for the air-cotton mixture, respectively, S1 = (1 - m)S, S2 = mS, m surface contacting with the particles of raw cotton S - the total contact area of the screw with the mesh, Equations (3) - (5) are given to the types:
^ + ajlx = aT under 0 < 0Z2a (6)
dO
T
de
dTx
de
+ ClT2x - aiT2b = blT0
^ + c2T2t -a2T2x = b2T0 under 2aZeZ2n (7)
Where c1 = a1 + bv c2 = a2 + b2, a1 = avx / cxa, a2 = avx / cba,
bi =Px/cx®, b2 =Py /cyw
Equations (6) and system (7) are integrated under the following boundary conditions:
Tlx = Tnx under e = 0,, T2X = Tlx (2a), T2v = T, (8) solutions of which have the form Tx = T -(T -Tnx)exp(-afi), Tlv = T under 0<6Z2a (9) T2x = A2exp[k1(e-2a)] + B2exp[(k2(e-2a)] + T0, (10)
T2v = A2Ziexp[ki(0 - 2a)] + B2x2exp[(k2(0 - 2a)] + To under 2aZ0Z2n (11)
Where
cl + c2 c + c)22 -4(clc2 -ala2)
k + cl ; k2 + cl Xl = ; %2 =
A2l: =
B21 =
X2TX(2a0) -Tp]-T + To
X2 — Xl XT(2ap) - Tp]-T + Tp
Xl — X2
Equations in other sections have a similar form (6) and (7), which are integrated under the following conditions
T2x = T2,-ix((2 -+ 2a), T2l+l x = T2l_2x(2in) i = 1....n. The solution of equations (6) and (7) can be determined from the recursive formulas
T3x = T1 -[T1 - T2x(2n)]exp[-a1(0 - 2n)] , T3v = T1 under 2n<Q< 2a + 2n
Tix = Aiexp[k1(6- 2a- 2n)] + B4exp[k2(6- 2a- 2n)],
T4- = A4x, expk(6- 2a-2n)] + X2 exp[k2(6- 2a- 2n)] under 2a + 2n < QZ4n
T5X = Tl-T - Tix(4n)]exp[-al(6 -4n)] , TSx = T under 4n <Q < 2a + 4n
T6x = A6exp[k1(0- 2a- 4n)] + B6exp[k2(6- 2a- 4n)],
T6r = Ax expk(0-2a-4n)] + B6x2exp[k2(0 -2a-4n)] under 2a + 4n < 0Z6n
T2-x = Tl - (T -T2-2x(2i - 2)n]exp[-ai(0 - 2in + 2n)] under 2in - 2n <0 < 2a + 2 in - 2n
T = T
3v Jl
(12) (13)
T2hx = ; exp[kl(0 -2a-2in + 2n)] + +B2i exp[k2(0 -2a-2in + 2n)]
T2i,v = A2iXi exp[kt(0 -2a-2in + 2n)] + +B2iX2 exp[k2(0 -2a -2in + 2n)] under 2a + 2in-2n <0< 2in i = l..n
T2n-lx = T - (T -T2nx(2n - 2)n]exp[-al(0 - 2nn + 2n)], T3v = Tl
T2n,x = A2ni exp[kl(0-2a-2nn + 2n)] + +B2ni exp[k2(0 -2a- 2nn + 2n)]
T2v,v = A2niXi exp[ki(0-2a-2nn + 2n)] + +B2nix2 exp[k2(0- 2a- 2nn + 2n)] under 2a + 2in -2n <0 < 2nn
Figures 2-5 show the curves of the temperature change of raw cotton (Fig. 2, 3) and air (Fig. 4, 5) from the screw angle в for two values of the temperature of the agent Ti (K) on the open surface part and the number of the screw section n. In the calculations it is customary: h = 0.3м, сч = 2.5кдж/кг ■ трад , cv = 1.5кдж/кг ■ град , T0 = 290°K, Txn = 285°K , ю = 25c-,axv = 3-104кдж/м3 • ч■ град ,
вх = 10~5кдж/м1 ■ ч ■ град , а= 30°,
Pv = 1.6 -10-5 кдж/м2 ■ ч ■ град .
Figure 2. Changes in the temperature of raw cotton TxK along the helical screw line Tj = 4100K with the number
of sections n = 6 and n = 12
Figure 3. Changes in the temperature of raw cotton Tx K along the helical screw line T1 = 420 K with the number
of sections n = 6 and n = 12
Figure 4. Changes in the temperature of raw cotton Tx0K along the helical screw line T1 = 4100K with the number
of sections n = 6 and n = 12
Figure 5. Changes in the temperature of raw cotton Tx°K along the helical screw line T1 = 4200K with the number
of sections n = 6 and n = 12
Analysis of the curves of the temperature variation of raw cotton (Fig. 2, 3) shows that with the growth of the screw angle, the temperature of raw cotton first varies according to a law close to linear, and then the intensity of temperature growth decreases. The growth of the temperature of the thermal agent on the free surface of the screw leads to an intensive increase in the temperature of the raw cotton in the initial sections of the screw and then the character of the temperature change from the screw angle remains unchanged.
Conclusions: From the analysis of the curves presented in Figures 4 and 5 it follows that the constancy of the temperature of the air (agent) on the open part of the screw leads to a sudden change in the temperature of the air particles moving along the helical line. With each revolution, the air temperature in the screw sections decreases with the approach to the lower part of the screw and with the increase in the number of sections it approaches the temperature of the agent. Thus, in contrast to the temperature of the raw cotton particle, which continuously increases as it moves along the helix, the air temperature changes periodically with a damped amplitude.
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