MODELLING OF STATIONARY FLOWS OF A LIQUID-GAS SYSTEM IN AN INCLINED CHANNEL SUBJECT TO EVAPORATION Текст научной статьи по специальности «Физика»

EDN: RPCLDC УДК 536.25

Modelling of Stationary Flows of a Liquid-Gas System in an Inclined Channel Subject to Evaporation

Evgeniy E. Makarov*

Altai State University, Barnaul, Russian Federation

Received 28.06.2022, received in revised form 06.08.2022, accepted 20.10.2022 Abstract. Two-layer flow of liquid and gas-vapor mixture in an inclined channel is considered. The flow is described by the system of the Oberbeck-Boussinesq convection equations with the effects of evaporation and thermodiffusion. A new exact solution of the problem of evaporative convection is constructed under conditions of non-deformable interface and zero vapour flux on the upper channel wall. The analytical form of the solution is presented for the case when the channel boundaries are heated linearly with respect to the longitudinal coordinate. Calculation of the integration constants are described in detail. Examples of flows are provided for the ethanol-nitrogen fluid system.

Keywords: exact solution, two-layer flow, convection, evaporation, interface, inclined channel.

Citation: E.E. Makarov, Modeling of Stationary Flows of a Liquid-Gas System in an Inclined Channel Subject to Evaporation, J. Sib. Fed. Univ. Math. Phys., 2023, 16(1), 110-120. EDN: RPCLDC

Introduction

Mathematical modelling of two-layer systems with interfaces is motivated by intensive development of knowledge-intensive technologies and experimental approach to study the features of joint convective flows of liquids and gases [1]. Most of such convective processes are quite difficult to study due to existence of a great many factors affecting the flow nature.

The problems with evaporation or condensation are of particular interest. The Ostroumov-Birikh type solutions (see [1-3]) are the best known exact solutions of evaporative convection problems since they are realized in reality. One of the feature of these solutions is that they allow one to test various types of boundary conditions for vapour concentration and temperature functions.

Historically, the problem of unidirectional two-layer flow induced by gravitational and Marangoni forces was first considered in [4]. The results of the study of flows with evaporation in a two-layer system based on an analogue of the Ostroumov-Birikh solution were presented in [5].

The impact of the reciprocal thermodiffusion effects on the parameters of convective regimes in the two-layer system was studied on the basis of the Ostroumov-Birikh type solution [6-8]. The Soret effect (or thermodiffusion effect) is related to the molecular transport of matter in the presence of a temperature gradient. The Dufour effect (or diffusive thermal conductivity effect)

*evgeniimakarov1995@gmail.com © Siberian Federal University. All rights reserved

determines the occurrence of temperature differences due to differences in the concentrations of components (see [9,10]).

In the present work a two-layer flow with evaporation in an inclined channel is considered. Various factors that affect the flow structure, temperature and vapour concentration distribution are taken into account.

1. Governing equations and the form of exact solution

The joint flow of viscous incompressible fluid and gas-vapour mixture in an infinite channel is considered (see Fig. 1). The fluid and gas-vapour layers have constant thicknesses l and h. The upper and lower walls of the channel are solid impenetrable boundaries. Vapour is a passive admixture in the upper layer containing gas. The Cartesian coordinate system is oriented so that the non-deformable interface is given by the equation y = 0 and the gravity force vector g is directed at an angle y to the substrate (g = (g cos y, —g sin y)). The system of the Navier-

Fig. 1. The scheme of flow

Stokes equations in the Oberbeck-Boussinesq approximation is used as a mathematical model to describe flow in the bilayer system. The Soret and the Dufour effects are taken into account in the gas phase. The system of equations that determines velocity, temperature, pressure and vapour concentration is written in the following form:

du du

1 dp

( d2u d2i

1t + ^ = —-£: + "(in* + - gcos +yc )>

dx

dy

dv dv + vtt dx dy

p dx 1 dp

\dx2 'd2v

dy2

1 dp' fd2v d2v\ , m

- Pdy + <dx2 + ay2) + g sin + 7e

d 2v

dx2 dy2 du dv o dx dy '

dT dT

+ v

dx dy

( dT

X\dx2

d2T x( + dh2 + s\

dC d C

+

dx2 dy

(1)

(2)

(3)

(4)

The vapour transfer in the gas phase is governed by the convective diffusion equation, which is the result of the Fick's law [11]:

'd2C d2C (d2T d2T \ V dx2 dy2 J

dC dC

dC

+ = ^ + tt^ + a dx dy V dx2

dy2

(5)

The following notations are used in equations (1)-(5): u, v are projections of the velocity vector on the axes of the Cartesian coordinate system Ox and Oy, respectively; p defines the deviation

from the hydrostatic pressure (p' = p — pg • x, x = (x,y), p is the pressure), T is the temperature, C is the vapour concentration, p is the density (relative density value), v is the coefficient of kinematic viscosity, 3 is the coefficient of thermal expansion, 7 is the concentration coefficient of density, x is the coefficient of heat diffusivity, D is the coefficient of vapour diffusion in the gas, parameters a and S are the Soret and Dufour coefficients, respectively. The underlined terms in equations (1), (2) and (4) as well as in equation (5) are used only in the modelling of heat and mass transfer.

System (1)-(5) admits an exact solution of the special form [2,3]:

= Ui(y), Vi =0, Ti = Ax + êi(y), C = -Bx + y), pi = pi(x,y).

(6)

Here, A and B are constant longitudinal temperature and vapour concentration gradients; êi and ^ are functions that depend on the variable y. Index i = 1 corresponds to the lower layer and i = 2 corresponds to the gas-vapour mixture in the upper layer.

2. Conditions on the solid walls and at the interface

The no-slip conditions on the rigid walls are valid for the velocity functions:

ui\y=-i =0, U2 \y=k =0. (7)

The temperature distribution is linear with respect to the longitudinal coordinate:

Ti\y=-i = Ax + ti-, T2\y=h = Ax + ti+. (8)

Here, ti- and ti+ are given constant values.

The vapour concentration satisfies the condition of zero vapour flux at the upper boundary

y =h: (— ad—L) =0 (9)

V dy dy J y=h '

The conditions of continuity of longitudinal velocities and temperature should be fulfilled at the thermocapillary interface y = 0:

ui\y=0 = U2 |y=0, Ti|y=0 = T2|y=0.

(10)

Let us note that equality of the longitudinal temperature gradients and condition êi(0) = = ê2(0) provide the temperature continuity condition.

Kinematic and dynamic conditions have also to be set at the interface. Kinematic condition (v1 = 0 and v2 = 0) is fulfilled automatically due to the type of exact solution (6). Dynamic condition is written as follows

PlVlUiy = p2V2U2y + &T-

dT1

dx

y=0

(11)

Dynamic condition expresses the tangential stress balance at the interface. The constant aT is the temperature coefficient of the surface tension a, aT < 0. The linear dependence of surface tension on temperature is assumed: a = a0 + aT(T - To), a0 is the surface tension at some initial temperature T0.

The heat transfer condition at the interface, including the terms corresponding to the diffusive mass flux M and the Dufour effect, is set in the form:

dT d T dC /

Kidl - K2dy - dy \y=0 = -LM, M = -Dp2\

m

dy

dC

1,2 dy ly

dC dy

dT2

y=0 dy

)■ (12)

y=0

U

Here, L is the latent heat of evaporation, M is the mass flux of liquid evaporating from the unit surface area per unit of time (M = const), k1 and k2 are thermal conductivity coefficients.

The condition for C on the phase boundary defines the saturated vapour concentration. It is the linearised form of the equation that follows from the Clapeyron-Clausius and Mendeleev -Clapeyron equations [6]:

C|tf=0 = C,[1+ e(T2\y=0 — To)], (13)

where e = L^0/(RT2), ¡i0 is the molar mass of evaporating liquid, R is the universal gas constant, C* is the saturated vapour concentration at T2 = T0.

Let us note that equations of fluid motion (1), (2), heat transfer (4) and vapour transfer (5) as well as temperature continuity condition (see the second relation in (10)), dynamic condition (11) and heat transfer condition (12) admit substitution of the temperature function in the form T = Ti-T0. It corresponds to introduction of modified pressure p (p = p — pftT0g■ x). Condition (13) can be written in the form

C|y=0 = C*(1+ eT2 |y=0). (14)

The corresponding substitution in conditions (8) leads to relations rTi = Ax + â-, T2 = = Ax + â+, where â- = â- — T0, â+ = â+ — T0. For convenience, the symbol "tilde" over Ti, pi, â-,â+ is omitted in the next sections.

The problem is solved at given gas flow rate Q defined by relation

Q = / P2u2(y)dy, (15)

0

and under assumption of the closed flow condition in the liquid layer:

J ui(y)dy = 0. (16)

3. Exact solutions under condition of zero vapour flux on the upper solid wall of the channel

Solution of equations (1)-(5) in form (6) can be derived in explicit form for longitudinal velocity ui(y), temperature Ti(x,y) and vapour concentration in the gas phase C(x, y). In the case A > 0 the following relations were obtained

ui (y) = Ci sin(kiy) + C2 cos(kiy) + C3 sh(kiy) + C4 ch(kiy),

U2 (y) = C1 sin(miy) sh(miy) + C2 cos(miy) sh(miy) + C3 sin(miy) ch(miy)+ + C4 cos(miy) ch(miy), Fi

Ti (x,y)= Ax + ( — Ci sin(kiy) — C2 cos(kiy) + C3 sh(kiy) + C4 ch(kiy)) + C5y + Cfj,

F2 ( ^ , , , , , . 7= . , , w , (17)

F2 ( — —

T2 (x, y) = Ax + -—2 ( - C1 cos(miy) ch(miy) + C2 sin(miy) ch(miy)-1

- C3 cos(miy) sh(miy) + C4 sin(miy) sh(miy^ + C5y + Ce

G ( ___

C (x, y) = -Bx + 2m2\ - Ci cos(miy) ch(miy) + C2 sin(miy) ch(miy) - C3 cos(miy) sh(miy) + C4 sin(miy) sh(miy^ + C7y + Cs .

Functions ui and u2 are solutions of the corresponding equations u^ + Aiui = 0 and uiy1 + A2u2 = 0 [12]. They follow from equations (1), (2) as a result of sequence of consecutive actions: substitution of relations (6), cross differentiation (in order to exclude pressure function), differentiation with respect to y. If A > 0 then the inequality Ai < 0 is valid; inequality A2 > 0 is fulfilled for liquid - gas system like "ethanol - nitrogen" because E < 0.

When condition A < 0 is satisfied then the required functions (6) take the form

ui (y) = Ci sin(&2y) sh(fc2y) + ^2 cos(&2y) sh(&2y) + C3 sin(&2y) ch(k2y) + C4 cos(&2y) ch(k2y),

U2 (y) = C1 sin(TO2y) + C2 cos(m2y) + C3 sh(m2y) + C4 ch(m2y), F1 (

Ti (x, y) = Ax - Ci cos(k2y) ch(k2y) + C2 sin(k2y) ch(k2y)-

2k2 v (18)

- C3 cos(k2y) sh(k2y) + C4 sin(k2y) sh(k2y)) + C5y + Ce ,

F2 ( — — — —

T2 (x,y) = Ax+--2 -Ci sin(m2y)-C2 cos(m2y) + C3 sh(m2y) + C4 ch(m2y))+C5y+Ce,

m22

G , ________\ _ _

C (x, y) = -Bx+--2 (-Ci sin(m2y) -C2 cos(m2y)+C3 sh(m2y)+C4 ch(m2y )) + C7y+C8.

m2

Here, ui and u2 also satisfy the equations u^ + Aiui = 0 and u^ + A2u2 = 0. The inequalities Ai > 0 and A2 < 0 hold if A < 0.

Coefficients Ai,A2,ks,ms,Fi,F2,G,E are calculated via geometric, physical and chemical parameters of the problem. Here, index s denotes the solution for positive (s = 1) or negative (s = 2) values of the longitudinal temperature gradient A. Coefficients C and Ci (i = 1,..., 8) are the integration constants. They are different for each solution. Exact representations of the listed parameters {Ai, ks,ms, Fi, G, E} (i =1,2; s = 1, 2) are given in Appendix 1.

One should note that relation for the saturated vapour concentration at the interface (14) dictates the compatibility condition for A and B: B = -C^sA. The pressure functions pi are found on the basis of their partial derivatives from (1), (2).

4. Determination of integration constants

Let us assume that gas flow rate Q (see (15)) and parameters A, are given. From

boundary conditions (7)-(12) and relationships (15), (16) we obtain the system of linear algebraic equations for integration constants. After calculating the constants one can determine the velocity profiles, temperature and distribution of vapour concentration in relation to the sign of parameter A (see (17) or (18)).

An algorithm for finding all the unknown integration constants is given below.

(i) Conditions on the solid walls and the interface (7)-(12), conditions (16) and (15) result in the system of linear algebraic equations for constants Ci,C2,C3,C4,C5,Ce and Ci, C2, C3, C4, C5, Ce, C7. The resulting systems contain 11 equations for 13 unknowns.

(ii) The integration constants Ce, Ce are set equal to zero in order to close the system of linear algebraic equations.

(iii) The system of equations is solved by the Gauss elimination method using program code which also provides a data set for velocity profiles, temperature and distribution of vapour concentration .

(iv) Coefficient C8 is expressed in terms of coefficient C1 (when A > 0) or in terms of coefficients C2 and C4 (when A < 0) using condition (14) to define the vapour concentration at the interface.

The systems of equations for the unknown parameters of integration are presented in Appendix 2 (see (19) and (21)).

5. Examples of two-layer flows

The effect of changing channel inclination angle, the interface temperature gradient and transversal temperature drop on the structure of the flow, temperature and distribution of vapor concentration was studied. The ethanol and nitrogen were chosen as working liquid and gas, respectively. The ethanol evaporates from the lower layer so a mixture of nitrogen and ethanol vapour is formed in the upper layer. The physical parameters of the media are given below [13] for {ethanol, nitrogen} or for ethanol only: p = {7.89 • 102, 1.2} kg/m3, v = {0.15 • 10"5, 0.15 • 10"4} m2/s, / = {1.079 • 10"3, 3.67 • 10"3} K"1, x = {8.9 • 10"8, 0.3 • 10"4} m2/s, k = {0.1672, 0.02717} W/(m-K), aT = -0.8 • 10"4 N/(m-K), D = 0.135 • 10"4 m2/s, L = 8.55 • 105 W^s/kg, C* = 0.1 (corresponds to equilibrium temperature To = 20°C), y = -0.62, e = 0.059 K"1.

The value of the gas flow rate Q is assumed to be equal to 3.6 • 10"5 kg/(m-s). The following parameters are fixed: thickness of the liquid layer l = 5 mm and thickness of the gas layer h = 5 mm, gravity acceleration g = 9.81 m/s2. The values of the Soret and Dufour coefficients are assumed to be equal to a = 10"4 K"1 and S = 10"4 K, respectively. Parameters of external thermal loads applied to the channel walls are the longitudinal temperature gradient A and , (see condition (8)). Let us note that functions T and ft", in Figs. 2-4 refer to the deviation from the reference temperature T0 due to substitution made in Section 2 (here TO = 20 °C). Figure captions contain values of these deviations.

Fig. 2. Velocity profiles in the ethanol - nitrogen system: Q = 3.6 • 10 5 kg/(m-s), A =10 K/m, tf- = -4°C, = 3°C; y = 20° (blue line), 45° (red line), 60° (green line)

Velocity profiles for inclination angle y = 20°, 45°, 60° are presented in Fig. 2. The reverse flow near the interface intensifies at larger values of the angle y.

The effect of intensification of the thermal regime created on the channel walls on the character of the flow was studied. The change of coefficients tf-, (see Fig. 3) results in the formation of

(d) (e) (f)

Fig. 3. Velocity profiles (a, d), thermal field (b, e), distribution of vapour concentration (c, f) in the ethanol - nitrogen system: Q = 3.6 • 10-5 kg/(m-s), A = 5 K/m, p = 80°; (a, b, c) - = 4C, = -3C, M = 1.026 • 10-e kg/(m2^s), (d, e, f) - = -4C, = 3C, M = 1.062 • 10-e kg/(m2^s)

transverse temperature drop and, therefore, results in the change of flow characteristics. In the first case = 4C, = -3C (Fig. 3 (a, b, c)) and in the second case = -4C, = 3C (Fig. 3 (d, e, f)). The maximum values of velocity are observed in the fluid layer in the second case (Fig. 3 (d)). The distributions of vapour concentration are presented in Fig. 3 (c, f). Values of the evaporation mass flow rate M are equal for both cases.

The relationship between velocity field, temperature, distribution of vapour concentration and longitudinal temperature gradient A is shown in Fig. 4. Parameter A takes the following values {5, 10, 15} K/m. The velocity field is restructured due to the change in the intensity of the thermocapillary effect. At the same time more complicated temperature distributions and the growth of the evaporation mass flow rate are observed with increase in A (see Fig. 4 (b, e, h)).

Conclusions

New exact solution of a special type of convection equations was constructed. It describes a two-layer flow in an inclined channel. Evaporation at a non-deformable thermocapillary interface, the Soret and Dufour effects in the gas-vapour layer are taken into account. The condition of zero vapour flux on the upper solid wall of the channel is set. Exact solution was obtained for the cases of positive and negative longitudinal temperature gradients at the channel boundaries. Algorithm

(g) (h) (i)

Fig. 4. Velocity profiles (a, d, g), thermal field (b, e, h), distribution of vapour concentration (c, f, i) in the ethanol- nitrogen system: Q = 3.6 • 10-5, kg/(m-s), p = 45°, = 0 °C, = —7C; (a,b,c) - A = 5 K/m, M = 1.062 • 10-6 kg/(m2-s), (d,e,f) - A =10 K/m, M = 2.124 • 10-6 kg/(m2-s), (g,h,i) - A =15 K/m, M = 3.186 • 10-6 kg/(m2^s)

for determining parameters of the problem and unknown constants is presented. Examples of characteristics for bilayer flows in the ethanol-nitrogen system are presented. The influence of the channel inclination angle and the thermal load at the boundaries on the flow pattern was studied.

The work was carried out in accordance with the State Assignment of the Russian Ministry of Science and Higher Education entitled 'Modern methods of hydrodynamics for environmental management, industrial systems and polar mechanics' (Govt. contract code: FZMW-2020-0008).

Appendix 1. Formulas for calculating parameters in relations (17) and (18)

Coefficients ks ,ms, Fi, G:

J Ag cos J E Ag cos y^i

ki = \ -, mi = y -Ag cos y—, Ai =--,

V Xi vi V 4 XiVi

, J Ag cos yfii ./—.-— ^ . ^

k2 = V--, m2 = v Ag cos yE, A2 = -Ag cos yE,

V 4xivi

F= A F= A(D - ÔX2C* e) G = A(aD - X2Ce) E = D(fo - an ) - X2C*e(Sfa - y) Xi' X2D(1 - aS) X2D(aS - 1) ' X2V2D(1 - aS)

Appendix 2. Systems of linear algebraic equations for integration coefficients in (17) and (18)

System in the case of negative value of parameter A:

sin(k2l)sh(k2l)Ci - cos (k2l) sh(k2l)C2 - sin(k2l) ch(k2l)C3 + cos(k2l) ch(k2l)C4 = 0 ,

sin(m2h)Ci + cos(m2h)C2 + sh(m2h)C3 + ch(m2h)C4 = 0 , Fi

—2 ( - cos(k2l) ch(k2l)Ci - sin(k2l) ch(k2l)C2 + cos(k2l) sh(k2l)C3 + 2k2

+ sin(k2l)sh(k2l)C4) - lC5 = tf",

F2 ( _______) _

—2 ( - sin(m2h)Ci - cos(m2h)C2 + sh(m2h)C3 + ch(m2h)C4 ) + hC5 = , m2

G+ aF2 ( — — —

- - cos(m2h)Ci + sin(m2h)C2 + ch(m2h)C3 + sh(m2h)C4 + aC5 + C7 = 0 ,

m2

C4 - C2 - C4 = 0 , (10)

Fi F2 _ N (10)

- Fi Ci + -2 C 2 - C4 =0,

2k2 m2 V /

PiVik2C2 + pivkC3 - p2V2'm,2Ci - p2V2'm,2C3 = aTA,

—

«i2| (C2 - C3) + «iC5 + *+ ^ + Ci - C3) .

-(K2 + LDp2a)C5 - (SK2 + LDp2)C7 = 0,

(1 - cos(m2h))Ci + sin(m2h)C2 + (ch(m2h) - 1)C3 + ch(m2h)C4 = —— ,

p2

(sin(k21) ch(k2l) - cos(k2l) sh(k2l))Ci + (1 - sin(k2l) sh(k2l)-- cos(k2l) ch(k2l))C2 + (-1 - sin(k2l) sh(k2l) + cos(k2l) ch(k2l))C3+ + (cos(k2l) sh(k2l) + sin(k21) ch(k2l))C4 = 0 .

Coefficient C8 is expressed in terms of coefficients C2 and C4 using condition (14):

C 8 = C* +--2-2 C 2--2—2 C 4. (20)

22

2

System in the case of positive value of parameter A:

- sin(k1l)C1 + cos(k1l)C2 - sh(k1l)C3 + ch(k1h)C4 = 0,

sin(mih) sh(m1h)C 1 + cos(m1h) sh(m1 h)C2 + sin(m1h) ch(m1h)C3+ + cos(m1h) ch(m1h)C4 = 0 ,

sinkl)C1 - cos(k1l)C2 - sh(k1l)C3 + ch(k1l)C^j - lC5 = ê- ,

F2 ( — — —

-2 - cos(m1h) ch(m1h)C 1 + sin(m1h) ch(m1h)C2 - cos(m1 h) sh(m1h)C3+

2m1 V

+ sin(m1h) sh(m1h)C4 j + hC5 = ê+ , G + aF2 ( —

—-- (- cos(m1 h) sh(m1h) + sin(m1h) ch(m1h))C 1 + (sin(m1h) sh(m1h)+

2m1 V

+ cos(m1h) ch(m1h))C2 + (- cos(m1h) ch(m1h) + sin(m1h) sh(m1h))C3+ + (sin(m1h) ch(m1h) + cos(m1h) sh(m1h))C^ + aC5 + C7 = 0 ,

C2 + C4 — C 4 = 0 , Fi / „ . „ \ F2 —

(21)

k2(- C2+ mC1 = °: _

P1V1k1C1 + p1vkC3 - p2V2m1C2 - P2V2m1C3 = A,

K1 £ ( - C1 + C^ + K1C5 - K2+ aF2> (C2 - C3) -

-(K2 + aLDp2)C5 - (<K2 + LDp2)C7 = 0,

(sin(m1 h) ch(m1h) - cos(m1h) sh(m1h))C 1 + (-1 + sin(m1h) sh(m1h)+ + cos(m1h) ch(m1h))C2 + (1 + sin(m1h) sh(m1h) - cos(m2h) ch(m1 h))C3+

_ 2m\Q

+ (cos(m1h) sh(m1h) + sin(m1h) ch(m1h))C4 = -,

P2

(-1 + cos(k1l))C1 + sin(k1l)C2 + (1 - ch(k1l))C3 + sh(k1l)C4 = 0 . Coefficient C8 is expressed in this case in terms of coefficient C1 using condition (14):

c 8=c* +—m—2 c 1. (22)

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Моделирование стационарных течений системы жидкость-газ в наклонном канале с учетом испарения

Евгений Е. Макаров

Алтайский государственный университет Барнаул, Российская Федерация

Аннотация. Двухслойные течения жидкости и газопаровой смеси в наклонном канале моделируются на основе системы уравнений конвекции Обербека-Буссинеска с учетом эффектов испарения и термодиффузии. Построено новое точное решение задачи испарительной конвекции в постановке с недеформируемой границей раздела и при условии отсутствия потока пара на стенке канала. Представлен аналитический вид искомых функций в случае линейного по продольной координате нагрева границ канала. Подробно описаны алгоритмы расчета констант интегрирования. Для системы жидкостей этанол-азот приведены примеры типов течений.

Ключевые слова: точное решение, двухслойное течение, конвекция, испарение, граница раздела, наклонный канал.