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PHYSICO-MATHEMATICAL SCIENCES
EFFECT OF VARIABILITY OF THERMOPHYSICAL COEFFICIENTS FOR A CYLINDRICAL CALORIMETER IN THE CONDITIONS OF THE REGULAR HEAT MODE OF THE FIRST KIND Naziyev J.Ya. (Republic of Azerbaijan) Email: [email protected]
Naziyev Jeyhun Yashar - Doctor of technical sciences, Professor, DEPARTMENT OF PHYSICS, AZERBAIJAN STATE UNIVERSITY OF OIL AND INDUSTRY, BAKU, REPUBLIC OF AZERBAIJAN
Abstract: the method of regular thermal regime of the first kind for determining the thermophysical properties of substances has several advantages. However, in the calculation equations it is necessary to take into account the inconsistencies in the heating rate and the coefficients of thermal conductivity and thermal diffusivity. In the present work, the temperature field of the studied annular layer of a bicolorimeter is determined for the variable thermophysical properties of liquids and gases and the rate of cooling (heating) by solving the nonlinear differential heat equation.
Keywords: thermophysical properties, calorimeter, heating rate, temperature field, method of successive approximations.
ВЛИЯНИЕ ПЕРЕМЕННОСТИ ТЕПЛОФИЗИЧЕСКИХ КОЭФФИЦИЕНТОВ ДЛЯ ЦИЛИНДРИЧЕСКОГО КАЛОРИМЕТРА В УСЛОВИЯХ РЕГУЛЯРНОГО ТЕПЛОВОГО РЕЖИМА ПЕРВОГО РОДА Назиев Дж.Я. (Азербайджанская Республика)
Назиев Джейхун Яшар - доктор технических наук, профессор, кафедра физики,
Азербайджанский государственный университет нефти и промышленности, г. Баку, Азербайджанская Республика
Аннотация: среди прочих методов по определению теплофизических свойств веществ метод регулярного теплового режима первого рода отличается рядом преимуществ. Однако в расчетных уравнениях необходимо учитывать непостоянства темпа нагрева и коэффициентов теплопроводности и температуропроводности. В настоящей работе определено температурное поле исследуемого кольцевого слоя бикалориметра при переменных теплофизических свойствах жидкостей и газов и темпе охлаждения (нагревания). С учетом переменности теплофизических коэффициентов необходимо решать нелинейное дифференциальное уравнение теплопроводности.
Ключевые слова: теплофизические свойства, калориметр, скорость нагрева, температурное поле, метод последовательных приближений.
UDC 621.1.016.7
An analysis of the experimental data available in the literature obtained by various methods — the method of regular cooling and classical methods — shows that there is often a discrepancy between them. One of the reasons may be the neglect the variability
of the heating rate and thermal conductivity and thermal diffusivity coefficients in the calculation equations.
In this paper, we consider corrections to the temperature field of the annular cylindrical layer, which determine the disturbing effect of the parameters kx, Ka, Km of relative thermal conductivity, thermal diffusivity, and cooling rate, respectively. It is known solutions to such a problem for the methods of monotonous heating [1-3]. But there are no such solutions for the case of regular cooling (heating) . The solution of temperature distribution in a hollow cylinder with constant thermophysical parameters and a regular thermal condition is given, and this problem is solved only for a solid cylinder with variables m, a, X.
Thus, to estimate the accuracy of measuring X of liquids and gases by the method of a bicalorimeter and, if necessary, to make corrections to the calculation equation, it is important to obtain the temperature field equation for a cylindrical annular layer taking into account the constancy m, a, X. Since the permissible error of calorimeters in measuring X of liquids and gases is currently 1-1.5%, the amendments to the basic equation, even of the order of tenths of a percent, are essential. Calculations show that the magnitude of the correction for the variability of thermal constants can be in the range of 0.5-1% if the temperature difference in the layer is equal to 20C. With large drops, the error increases.
The available calculation equations for the X-calorimeter and the c-calorimeter of the regular thermal condition of the first kind are based on the regularities of the linear theory of thermal conductivity. Taking into account the variability of thermophysical coefficients, it is necessary to solve the nonlinear differential heat conduction equation. Using the method of successive approximations method, we will determine the temperature field of the ring layer of the bicalorimeter at varying heating rates and thermophysical properties.
The temperature field is found from a differential equation
V20 + v2o0 = v2o(ka -kmr)v0-k^d0j , (1)
under boundary conditions
rO = g(r) ; (0)Si = 2 , (2)
where 0 = t — c is the temperature difference between the annular layer and the autoclave; ta = const for calorimeter; = b(r,r) = —m0(r,r) ; b - the rate of
2 m /
temperature change; v0 = y . Here it is assumed that the parameters m, a, X are
/ a2
represented as power functions within the temperature difference V. In equation (1), we restrict ourselves to second order smallness, and for solving, we use
1 j
kmr = km,r + ka ; kbr = krn,r ; v = — g2r +0 i 0=0h exp(-mr) ,
12 x2
dr
where x0 = v0 R is the first root of the equation J0 (x ) = 2.
The exact solution of the Bessel equation (1) in the presence of a free term is associated with great difficulties associated with actions with Bessel functions of the first and second kind and zero and first order.
Solving equation (1) step by step
0(r, z) = 00 (r, z) + 0 (r,z) + 0 (r,z) =
= gR ln x - 1v20 gR [r2(ln x - ln s -1) - Rf(21n s + l)ln x - Rf(ln s +1)]-
1 1 r2 1 r 2 ( - - kAg 2 R12 ln2 x + + — kv2 g 2 Ri2 — (ln x -1) + - kv2 g 2 Ri2 — I ln2 x - 21n x + -1 + 2 x0 2 2 2 V 2 у
1r
2
+ - kv2 g 2 Ri — — -1 ln x ++т к g 2 Ri4-
1r
V xo 2,
4 1 2
V 0 y V 0 y
(3)
where (r, r) is the main function; 0X (r, r), (r, r) - additional functions defining the correction terms of the first and second approximations, respectively.
Thus, taking into account all approximations, we obtain the general solution
(4)
0(r,r) = gR lnX - 1vlgR [r 2(lnX - ln5 -1) - R^ (2ln5 + l)lnX - R2 (ln5 +i)]-
1 1 r2 1 r2 ( 3^
- - kAg 2 Ri2 ln2 X + — kv02 g 2 Ri2 — (ln X -1) + - kv2 g 2 Ri2 — (ln2 X - 2ln x + - 1 +
2 X0 2 2 2 V 2 y
+4g •R t (t')h X+-* g0 R14 t V f -3
where X = ^^ ; 5 = R2 /R is the ratio of the radii; k = kmr - ka .
From the general equation (4) it is possible to determine the total temperature difference in the annular cylindrical layer with r = R2
0(R2,t) = g0Ri ln 5 - -f02g0Ri [(r2 - Ri2) - 2R-2 ln 5]- -f02g02R-2 ln2 5 - -kAg02R- ln2 5 +
. (5)
+-L kv2 g 2 R12 (ln s -1)+1 kv2 g 2r2 r2 [ ln2 s - 2ln s+- J+1 kv2 g 2r1 [Д- -11 ln s
+4 g2 R' (i- 2
Then corrections for variable thermophysical characteristics under the conditions of a bicalorimeter
A0 = 0-0o =-1 k^g 2 R12 ln2 s + -1 kv2 g 2 R12 R22(ln s -1) +
2 x0 , (6)
1 ( 3^1 1 3
+1 kv2g2R12R221 ln2 s-2lns + - l+ — kv2g2R14(lns +1) + -kv2g2R14(lns + -) 4 V 2 J 2x 0 4 2
where 0O = 00 + A0O; A0O - сorrection value that takes into account the heat capacity
of the investigated annular layer. When A0 it is possible to evaluate the influence of the variability of thermal parameters on the result of measuring the coefficient of thermal conductivity of liquids and gases by the method of regular cooling (heating).
References / Список литературы
1. Платунов Е.С. Теплофизические измерения в монотонном режиме. Энергия, 1973. 144 с.
2. Назиев Д.Я. Теплопроводность многокомпонентных смесей углеводородов при высоких параметрах состояния. Дисс. док. тех. наук. Баку, 1997. 454 с.
3. Экспресс методы определения теплофизических свойств различных типов материалов в температурном интервале от -1500C до +18000C. Труды 31-ой Межд. конференции по теплопроводности и 19-го симпозиума по тепловому расширению. Канада. 2011, 325 с.
References in English / Список литературы на английском языке
1. Platunov E.S. Thermophysical measurements in monotonous mode. Energy, 1973. 144 p.
2. Naziyev J. Ya. Thermal conductivity of multicomponent mixtures of hydrocarbons at high parameters of state. Diss. doc those. sciences. Baku, 1997. 454 p.
3. Express methods for determination of thermophysical properties of different types of materials within a temperature range of -1500C to +18000C. - Proceedings of 31-th Int. Thermal conductivity conference and 19-th Thermal Expansion Symposium. Canada. 2011, 325 p.
