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9
ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ
ON CONVECTION STABILITY OF HYDROCARBON-OXYGEN GAS MIXTURE AND EXPLOSION
SAFETY
Palymskiy Igor B.
Dr. Sc., Professor of Physics Department, Novosibirsk city, Russia Fomin Pavel A.
Dr. PhD, Senior Researcher of Lavrentyev Institute of Hydrodynamics, SB RAS, Novosibirsk city, Russia Latifi Chahandachti Anouchah Professor of Physics Department, Qom University of Technology, Qom city, Iran
ABSTRACT
A new physical and mathematical model of Rayleigh-Benard convection in a gas-vapor mixture of oxygen and cyclohexane is proposed, where the dependence of density on temperature has its maximum. Linear analysis of stability is performed, two threshold cases are analyzed when the density inversion parameter is large or small enough. Formulae for the growth increment and attenuation, the neutral curve and the boundary of the instability domain on the wave plane have been obtained. The convection of gas-vapor mixture is considered in connection with explosion safety of technology processes.
АННОТАЦИЯ
Предложена новая физико-математическая модель конвекции Рэлея-Бенара в газо-паровой смеси кислорода и циклогексана, где зависимость плотности от температуры имеет максимум. Выполнен линейный анализ устойчивости, аналитически исследованы два предельных случая, когда параметр инверсии стремится к своим предельным значениям. Получены формулы для инкремента нарастания и затухания, нейтральной кривой и границы области неустойчивости на волновой плоскости. Конвекция газо-паровой смеси рассматривается применительно к проблеме взрывобезопасности технологических процессов.
Keywords: Rayleigh-Benard convection, Rayleigh number, Prandtl number, linear theory, increment.
Introduction
Convection in the flows with monotonous dependence of density on temperature has been analyzed and described in details in [1,2]. However, in natural conditions and numerous important applications, such as transportation of hydrocarbon through pipelines and its store, convection of cold water in the process of glacial melting [3-7], convection of inert and reactive gas-vapor mixture in chemical reactors and processing systems etc., the density of the convective flows is characterized by a non-monotonous, nonlinear function of the temperature with a clear maximum at its critical point. These applications explain the necessity of studying convection in such flows. The most complicated thing in terms of theory and practical application is to study a whole range of convection regimes in the vicinity of the maximum density point, where the buoyancy force changes its sign [6,7].
We would like to emphasize that a physically natural hypothesis about the cyclohexane condensation on the walls and, hence, formation of a more viscous liquid film on them allows us to consider the boundaries impermeable and free from viscous shear [5,9]. Our analysis of the dependence of the heat expansion coefficient on temperature shows that it is reasonable to consider the coefficient to be a piecewise constant temperature function as in [8].
We see that the problem of Rayleigh-Benard convection in such a gas-vapor mixture asymptotically
turns into the classical Rayleigh problem of convective instability in incompressible fluids with impermeable horizontal boundaries in the absence of viscous shear [1], if the density maximum is reached at one of the horizontal boundaries.
We also see an evident analogy with the penetrating convection of cold water close to the point of maximum density, where the heat expansion coefficient goes through zero but is a linear temperature function [3,4], which allows us to compare and verify the model proposed [5].
In particular, the analogy with the penetrating convection of cold water suggests that if the critical temperature is higher than that of the cold boundary and lower than that of the heated one, the layer is divided into two sublayers, and the instability may develop only in the lower sublayer, whereas the upper one is always stable. In upper sublayer the internal waves is observed. The sublayer boundary is trigger generating the motions there. The situation does not change depending on whether we heat the lower or the upper layer [4].
Similar to the convection of cold water, a decrease in the relative thickness of the lower, unstable layer is likely to cause flow stabilization [3,4].
In this article, we aim at describing a new physical and mathematical model of Rayleigh-Benard convection in a gas-vapor mixture of oxygen and cyclohexane taking into account the evaporation and condensation cyclohexane processes, and provide the results of the linear analysis of stability.
Physical properties of a gas-vapor mixture
Convection of a gas-vapor mixture with the oxygen O2 and liquid-vapor cyclohexane CH12 is considered in view of a possible condensation of the latter on the boundaries. To be specific, the total cyclohexane mass fraction, both in condensed and vapor forms, is set as ft0 = 0.524.
At the critical temperature Tr, all the cyclohexane added to the system evaporates. Thus, when T < Tcr, the cyclohexane is present as a liquid vapor condensed on the boundaries and as a saturated vapor, with the rise of temperature leading to a rapid growth of the saturated cyclohexane pressure according to (1) and a corresponding growth of the density of the gas-vapor
P (T) = 9.87 • 102
mixture. When T > Tr, the cyclohexane is present only as an unsaturated vapor with a relatively small change in its partial pressure, and a further increase in temperature leads to a decrease in the density, according to the ideal gas-state equation.
Due to a small temperature change in the Boussinesq approximation used, the partial pressure of the oxygen is considered fixed and equal to 1 atm. The total pressure P includes the pressure of the oxygen and the vapor: P = 1 + Pv.
When T < Tcr, the pressure of the saturated cyclohexane (atm) can be calculated according to the known absolute temperature:
2.9764-
1206.5
T -273.15+223.14
(1)
When T > Tcr, the cyclohexane is unsaturated and has a relatively week dependence on temperature, which can be neglected.
p The molar weight of the gas mixture p (kg/kmol) is determined by
ju(T) = (1 • 32 + Psv(T) • 84) / (1 + Psv(T).
Here, 32kg/kmol and 84kg/kmol are molar weights of the oxygen Q2 and cyclohexane CHn, respectively. When T > Tcr, the molar weight of the gas-vapor mixture in the approximation described is constant. The ideal gas density can be determined according to the equation of state for the ideal gas (Fig. 1): Here R is the is the universal gas constant. The temperatures of the cold and heated boundaries in Fig. 1 are arbitrary, shown just to explain the formulation of the problem, and do not correspond to the real values. The thermal expansion coefficient ft can be calculated from
p = (1 + Psv )» (T )/ RT.
2.5
P,kg/m3
2.0
1.5
ß-1 dl¥-i ^ • B = (1 + P- )(8 + 21 • p. )/T.
p dT B dT
/
^ /
Pcr — 2.47 kg/m-*
/Tcr = 54 °C
- / h
20
40
60
80
T,°C
Fig. 1. Density of the gas-vapor mixture.
In our case, the density maximum is achieved inside the layer. The analysis of the thermal expansion coefficient dependence on the temperature proves that the piecewise-constant approximation with a discontinuity at the critical temperature is correct, with the coefficient ranging in the interval between 0.003054 in the lower unstable layer and -0.01583 in the upper stable one. We should notice that at the point close to the critical temperature the kinematic viscosity v and the thermal diffusivity x are continuous functions
and in terms of the Boussinesq approximation can be considered constant.
The critical (maximum) value of the saturated cyclohexane pressure is determined by its total mass fraction, both in condensed and vapor forms, ft0 and the corresponding critical temperature is found from reversed Eq. (1). Thus, the given ft0 = 0.524 corresponds to the critical values of the saturated gas pressure Psv = 0.4191atm and temperature Tcr = 54.32°C (327.5K).
As it was stated above, the temperature of the upper cold horizontal boundary Tc is lower than the critical temperature Tcr, and the temperature of the lower heated horizontal boundary Th is greater than that, hence Tc < Tcr < Th. Similar to the case with cold water convection, the maximum density line where T = Tcr divides the layer into two sublayers, the instability can develop only in the lower layer with the upper one being always stable [4]. In upper stable sublayer the internal waves is observed. The sublayer boundary is trigger generating the motions there. Relative Linear analysis
By analogy with [1], in order to investigate convective stability in linear approximation, we obtain a set of equations:
thicknesses of the sublayers are characterized by the inversion parameter t = d/H = (Th - Tcr)/(Th - Tc), 0 < t < 1, which shows the ratio of the height of the unstable sublayer d to the thickness of the whole layer H. In the ultimate cases when t = 1 and 0, the set of the problem turns to the classical Rayleigh problem with ft = 0.003054 and -0.01583 respectively [1]. Similar to the case with cold water convection, the analysis of the density profile form (Fig.1) shows that the situation does not change at heating from above [4].
щ = Аю + С(т,y) • Ra • Q, A^ = —œ, Qt = (AQ - )/Pr.
(2)
Here, factor C(z,y) is determined as:
and
С(т,y) = 1,0 —y — т —5.183, т < у — 1.
The stream function y and vorticity m were derived from u = yy, v = - y and m = vx - uy, and Q is the deviation of temperature from the linear profile. Characteristic values are chosen as the layer thickness H for the length, x/H for velocity, H2/v for time, povx/H2 for pressure, 1/T„ (in the unstable sublayer) for the coefficient of thermal expansion and ST = Th - Tc for temperature, where the density inversion parameter t = d/H = (Th - T„)/(Th - Tc), 0 <
t < 1 is the ratio of the height of the unstable sublayer d to the height of the whole layer H. Here Ra = gH3ST/(xvTcr) and Pr = v/x are the Rayleigh and Prandtl numbers.
In order to solve system (2), we use the Galerkin method provided that t is not too small [1,10] and the orthogonalization method if t is small.
According to the Galerkin method, an approximated solution of (2) is considered as:
œ(t,x,у) = exp(—It)sin(ax) ^ rnmsin(m^y), y/(t,x,у) = exp(—It)sin(ax) ^ — sin(m;ry), ( .
m = 1 m = 1 Sm ( )
N
Q(t, x, y) = exp(—Àt)cos(ax) ^ Q sin(m^y), Sm = a2 + m2n2.
m = 1
Here X is the eigenvalue (increment), and a is a By substituting (3) into (2) and following the
wave number. Negative 1 values correspond to standard Galerkin method [1,10], we obtain the instability, while positive ones to the stable flow. condition of non-trivial solution existence in the form
of determinant:
det( A — D) = 0,
(4)
where A is a square matrix with the components Akm
Amk = Akm = J С(т, У) sin(m^y) sm(kлy)dy, Amk = —(a + 1)(
A = ■
Amk
0
a +1
sin(2m^T) 2mn
— т)/2 — a/2, m = k,
(m cos(mn т) sin(k^ т) — к cos(hn т) sin(m^ т)), m Ф к.
n(k2 - m2)
and D is a diagonal matrix with the components dmm.
d Sm (Sa -WSa -APT)
mm 1n 2 '
2Ra •a
1 < a < N.
Consider now asymptotic ratios with t close to 1, asymptotically turns into the classical Rayleigh
when the density maximum is achieved at the upper problem [1].
horizontal boundary and the problem considered From (5), one can find Amk:
11 1
Amk = ---m2n2(1 +a)(1-T)3, m = k; Amk = -mkn2(1 + a)(1 - x)3(-1)m+k+1, m^k.
The equations show that matrix A at t close to 1 Test calculations demonstrate that at t close to 1
has a diagonal dominance, and the equations for the neutral curve and perturbation growth increments
dominant harmonic components can be considered separately. However, it is due to these harmonic components that the flow stability and other characteristics are determined [1,2,10].
When N = 1, the neutral curve has the form:
can be calculated accurately taking into account only one harmonic component (N = 1).
Ra =
2A11a2
, Ra=^(1+-n2(1 +a)(1- t)3) = 1 + 40.68 ■ (1 - t)3), S = a2 + n
Hence, one can obtain the critical value of the Rayleigh number (minimum of Ra versus a):
(5)
Racr = 6.75 ■ — = 65±t±= 657.511 ■(1+~n2(1 + a)(1-T)3),
Cr 2A11 2A11 v 3 v j j
Racr = 657.511 ■ (1 + 40.68(1 - t)3), acr = 2.221 . For the growth increment A, we find that
(6)
A = — ( 2
S1 + Pr. IS2 ,1 - —
Pr
Pr
2 Raa2 A.. )2 + 1,1
S Pr
„ S ,1 + PrN /— 7i (a +1)Raa
A = — (-) -V D +—^- ,—
2K Pr 3Pr^VD
(1 -r)3, D = — (
S2 1 - Pr
4 Pr
)2 +
Raa S Pr
(7)
According to (7), one can see that the increment values obtained differ from those for the classical Rayleigh problem by about O(I-t)3 [1], and decrease in t leads to a more stable flow.
Fig.2 shows the function of the wave number a as a growth increment A, with numbers on the curves showing corresponding t values when Ra = I0 Racr, Racr = 657.511 and Pr = 0.71. Fig.3 shows the
instability domain on the wave plane, with solid curves showing the boundary of the instability domain for Rayleigh problem (t = 1), and dashed curves showing boundaries at t = 0.8 and 0.85. Figures 2 and 3 show flow stabilization as t decreases.
Consider now asymptotic ratios with t close to 0, when the density maximum is achieved at the lower horizontal boundary.
Fig. 2. Growth increment. Fig. 3. Instability domain.
One can obtain for Amk
1 1 1
Amk = - — a+—m2n2(1 + a)T3, m = k; Amk = —mkn2(1 + a)T3, m±k.
These equations show that matrix A at t close to 0 above for the neutral curve (7) that given t close to 0 all has a diagonal dominance, and the equations for the disturbances attenuate. dominant harmonic components can be considered separately. It follows from the first of the equations
3
3
3
s
We can show that attenuating disturbances at t Let us write down the equation for increment k at
close to 0 are typically oscillatory, which appears if t close to 0 with Pr = 1 for simplicity: Rayleigh number exceeds the threshold value 3.757 [1].
A = 5-^7^.(1-^+^3), A = 5 — 2.277ia^RaJS -(1 - 3.925x3).
The equation shows that the attenuation rate, which is determined by the real part of the increment k, does not depend on inversion t, and its growth (or attenuation) leads to a small decrease (increase) in the damped oscillation frequency.
Let us now compare critical Rayleigh numbers and corresponding wave numbers obtained in the present paper for the convection in cold water [3 -5] and for the classical Rayleigh problem [1]. Figures 4 and 5 show the scale of length as the height of the unstable
sublayer, and our data were recalculated as RacrT and acrT, respectively.
The solid curves in Fig. 4 and 5 show our results (line 1) and the data for convection in cold water ([4] -line 2 and [3] - dots), whereas the dotted line corresponds to the classical Rayleigh problem [1]. The dot-and-dash line in Figure 4 shows the result of the experiment with the convection of cold water poured on ice [5].
a,.r-1.8 -
1.4 -
o 0.2 0.4 0.6 0.8 x
Fig. 5. The critical value of the wave number.
According to the data in Figures 4 and 5, we can see that at 0.8 < t < 1 the results of our work are close to those of convection in cold water, and at t = 1 they coincide with the solution of the classical Rayleigh problem, while in the domain of infinitesimal t they asymptotically tend to the constants corresponding to the asymptotes of Racr = 526.4/t4 and a.cr = 2. 122/t.
We should notice that according to the critical Rayleigh number value, the results of experimental studying for convection in cold water [5] are essentially close to our results (with a 9.6% deviation), especially if we compare the former with the theoretical results [3,4], which give a deviation of about 43%. A possible reason for this might be the fact that in both experiment [5] and our work the heat expansion coefficient ft was considered constant and equal to its average value on horizontal boundaries. Due to this, our results are also much closer to the results of solving the classical Rayleigh problem with constant ft [1] than the theoretical results for convection in cold water [3,4].
About explosion safety problem
An explosion safety problem arises when a pipeline or tank using for transportation and storage of hydrocarbons is damaged. The flow in the pipeline can be considered in the coordinate system moving with an average flow velocity, and in this approximation, the flow in the pipeline and in a stationary tank can be considered in the framework of one approach.
In fact, the formation of an explosive mixture requires the mixing of hydrocarbon and air. If the wall of the pipeline or tank is damaged, the evaporation of hydrocarbon leads to its cooling. Moreover, practically any temperature difference leads to the development of convective mixing [1].
The following dependence of the critical temperature Tcr on the mass concentration of cyclohexane ft0 can be obtained (Fig. 6). Moreover, ft0 monotonically changes from its maximum value 1 to the minimum 0. In Fig. 6 temperature values of cold and hot walls are shown conditionally.
300 '■crfC 200
Th
ч
о
-100
У
—~—
f Domain 1 ....................................i............. Domain 2 j. i Domain 3 i
0 0,2 0.4 0,6 0.8 jio
Fig. 6. The critical temperature versus mass concentration.
A few words about a possible scenario for the development of an emergency.
Immediately after damage of the wall, the mass concentration of hydrocarbon is about во = 1 and convection is possible if the pipe is damaged from above (Domain 3).
With a further decrease of the mass concentration of hydrocarbon во, the critical temperature is into the temperature range of the cold and hot walls Tc <Tcr <Th and convection develops both when the pipe is damaged above and below (Domain 2).
With a further decrease in во (at Tcr <Tc), convection develops when the pipe is damaged from below (Domain 1).
Note that during the transportation and storage of cyclohexane, the formation of explosive areas is observed when cyclohexane mass concentration falls in the range with approximate boundaries from 0.0334 to 0.186, which corresponds to a critical temperature range from -0.16.9°C to 16.7°C.
Conclusion
In present work we have described a new physical and mathematical model of Rayleigh-Benard convection in a gas-vapor mixture of oxygen and cyclohexane taking into account cyclohexane evaporation and condensation on the boundaries of the domain. It is shown that at a certain (critical) temperature, when all the added liquid cyclohexane has evaporated, the density of the gas-vapor mixture reaches a local maximum, where the heat expansion coefficient and the buoyancy force in the equation of motion change their sign going through zero.
We conducted linear analysis of stability and analyzed data obtained through calculating nonlinear steady conditions of Rayleigh-Benard convection.
A physically natural supposition about cyclohexane condensation on solid walls and, hence, a more viscous liquid film on the walls allows us to consider the boundaries of the domain horizontal and free from shear strength, which simplified the task.
We showed that the problem of Rayleigh-Benard convection in the gas-vapor mixture considered turns
asymptotically into the classical Rayleigh problem if the density maximum is achieved on one of the horizontal boundaries. In both threshold cases we obtained analytical asymptotic formulae for stability characteristics. In terms of one mode approximation, expressions for the growth increment (or attenuation) of the first major mode were obtained. The boundary of the instability domain was also studied on the wave plane.
At t close to 1, a decrease in the inversion parameter t leads to a growth in the flow stability, and at t close to 0 it leads to a growth in the damped oscillation frequency, but the rate of decay is constant.
Having solved the linear stability problem, we see that a decrease in the relative thickness of the lower, unstable sublayer or a corresponding growth in the relative thickness of the upper stable sublayer cause stabilization of the flow.
We draw a quality analogy between the convection in the gas-vapor mixture considered and the penetrating convection of cold water at close to maximum density point, where the heat expansion coefficient goes through zero but is a linear function of the temperature.
It is shown that the considered problem of convection of a gas-vapor carbon-air mixture is directly related to the problem of explosion safety during the transportation and storage of hydrocarbons.
References
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2. Palymskiy, I.B. Turbulentnaya konvektziya Rehleya-Benara. Chislennyi metod i resultaty raschetov [Turbulent Rayleigh-Benard Convection. Numerical Method and Results of Simulation]. Germany: LAP, 2011. -232p. (in Russian).
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[On Penetrative Convection in the Isothermally Incompressible Fluid Approximation] // Izv. RAN, ser.: MZhG [RAS Bulletin, Series Mechanics of Liquid and Gas].-1996. - № 2. - P.40-52 (in Russian).
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APPLICATION OF THE THEORY OF OPTIMAL SET PARTITIONING BEFORE BUILDING MULTIPLICATIVELY WEIGHTED VORONOI DIAGRAM WITH FUZZY PARAMETERS
DOI: 10.31618/ESU.2413-9335.2020.6.71.615
Kiseleva E.M., Prytomanova O.M., Padalko V.H.
Oles Honchar Dnipro National University
ABSTARCT
An algorithm for constructing a multiplicatively weighted Voronoi diagram involving fuzzy parameters with the optimal location of a finite number of generator points in a limited set of «-dimensional Euclidean space £n has been suggested in the paper. The algorithm has been developed based on the synthesis of methods of solving the problems of optimal set partitioning theory involving neurofuzzy technologies modifications of N.Z. Shor r -algorithm for solving nonsmooth optimization problems.
Keywords: multiplicatively weighted Voronoi diagram, optimal set partitioning problem, optimal location of generator points, neurofuzzy technologies, N.Z. Shor r-algorithm, nonsmooth optimization problems.
Introduction. Currently, hundreds of literary sources on Voronoi diagrams and their application in various fields [1]. Voronoi diagrams for two- and three-dimensional spaces are used in many different areas of applied sciences. Quite apart from the fact that a large amount of known algorithms for constructing Voronoi diagrams of given finite Msets of a plane (space) points, called generator points, have 0(|M| Zog( |M|)) complexity, all these algorithms are quite complicated. In addition, in practical problems the parameters of the Voronoi diagrams often may be fuzzy.
In the papers [2, 3], the algorithms for constructing a standard (classical) Voronoi diagram with deterministic parameters and various generalizations are suggested, based on the application of methods of optimal set partition (OSP) theory and having several advantages over the known ones, which are described in the scientific literature [1, 4, 5]. Particularly, they do not depend on the £„ space dimension containing a limited set to be partitioned, which is independent of the geometry of the sets to be partitioned; the complexity of algorithms for plotting Voronoi diagrams based on the approach described above does not increase with the growing number of generator points; they can be used to construct not only Voronoi diagrams of a given number of generator points of fixed locations, but also with the optimal location of these
points in a limited amount of £„ space, and other advantages. The result of such a versatile approach is the ability to easily construct not only already known Voronoi diagrams but also the new ones.
The versatility of the approach proposed in studies [2, 3] in the construction of Voronoi diagrams is further supported by the fact that models and methods for solving continuous problems of optimal set partitioning can be generalized in the case of fuzzy initial parameters of the problem or a requirement of a fuzzy set partitioning, resulting in a fuzzy nature of Voronoi diagrams.
In this paper, there has been developed the algorithm of constructing a multiplicatively weighted Voronoi diagram involving fuzzy parameters with optimal positioning of N finite number of generator points in n limited set in n-dimensional £n(n > 2) Euclidean space. The algorithm has been developed based on the synthesis of methods of solving problems of the theory of OSP [7]) with neuro-fuzzy technologies [8]) and modifications of N.Z. Shor r -algorithm for solving non-smooth optimization problems [9, 10].
Problem setting. A classical Voronoi diagram of a finite set M = {t1,t2,...,tw} c £n of generator points Tj = (T(1),Tl(2),...,rl(n)), ¿ = 1,2,...,W in n-dimensional Euclidean space £„ (n > 2) is Voronoi polyhedra set
