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Heat and mass transfer in the liquid film on a vertical wall in roll-wave regime
International Journal of Heat and Mass Transfer 55 (2012) 6514–6518
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Heat and mass transfer in the liquid film on a vertical wall in roll-wave regime V.E. Nakoryakov a,c, V.V. Ostapenko b,c, M.V. Bartashevich a,c,⇑ a
Kutateladze Institute of Thermophysics, Siberian Branch, Russian Academy of Sciences, 1, Acad. Lavrentiev Avenue, Novosibirsk 630090, Russia Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, 15, Acad. Lavrentiev Avenue, Novosibirsk 630090, Russia c Novosibirsk State University, 2, Pirogova Str., Novosibirsk 630090, Russia b
a r t i c l e
i n f o
Article history: Received 12 June 2012 Accepted 18 June 2012 Available online 26 July 2012 Keywords: Heat and mass transfer Condensation Self-similar solutions Falling film Roll waves Progressive waves
a b s t r a c t The investigation of thin liquid film flowing down a vertical wall in the roll-wave regime in presence of heat and mass transfer through the free surface is presented. The roll-wave equation taking into account heat and mass transfer through the liquid–vapor interface has been derived. The self-similar solutions of the progressive-wave type for film thickness have been obtained. The families of discontinuous solutions have been constructed, where the progressive waves are conjugated with each other or with ‘‘residual’’ film thickness through the strong discontinuity. As an example, the calculations of the condensate water film flowing down a vertical surface are presented. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction First analytical and experimental investigations of heat transfer at laminar film condensation on a vertical wall were done by Nusselt [1]. However, the Nusselt theory is valid only for low Reynolds numbers. Even for film Reynolds numbers of about several units, the waves appear at the interface and increase the heat transfer coefficient [2]. The pioneering investigations of waves on the surface of a thin layer of viscous liquid films flowing down a wall were made by Kapitsa [3]. The problem of rolling waves in a sheet of fluid flowing in a vertical plane was treated by Pukhnachev [4] on the basis of complete Navier–Stokes equations with conditions on the unknown free boundary. The existence of progressive (capillarygravity) waves in a layer on a vertical wall at indefinitely small Reynolds numbers was proved. The roll waves on the surface of thin liquid films were studied experimentally and theoretically by Nakoryakov et al. [5,6]. The waves obtained in [5] have amplitude greater than the average thickness of the liquid film. The roll wave profiles have the main part and capillary precursor moving ahead with the same speed. This capillary precursor has the form of a damped harmonic wave. The characteristics of two-dimensional steady-state roll waves on a vertical liquid film were investigated. It was shown that the parameters of large-amplitude waves on a thin vertical film are heavily dependent on the viscosity of liquid ⇑ Corresponding author at: Kutateladze Institute of Thermophysics, Siberian Branch, Russian Academy of Sciences, 1, Acad. Lavrentiev Avenue, Novosibirsk 630090, Russia. Tel.: +7 383 330 70 50; fax: +7 383 330 84 80. E-mail address: [email protected] (M.V. Bartashevich). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.06.054
and these waves can have the non-capillary nature. Water, waterglycerol and water-alcohol solutions were used as the working liquids. The viscosity varied from 0.9 106 to 11.2 106 m2/s. The waves were formed at some distance from the distributing slot outlet, their amplitude increased rapidly, and then the steady state regime was achieved. The dependence of the phase velocity of the roll-waves on their amplitude was linear. There was no dependence between phase velocity and the average flow rate. There was also a slight dependence of the phase velocity on the surface tension. The possibility of the existence of progressive waves on the surface of a vertically flowing liquid film with neglected surface tension was shown in [6]. The equation for perturbations of a thin film was obtained. This equation allows the steady periodic discontinuous solutions of the roll-wave type (similar solutions were constructed in [7]). The constructed discontinuous solutions agree qualitatively with the experimental results [6]. The bore theory, which describes the wave breaking, is presented in [8]. The original ‘‘bore model’’ was presented by Gantchev [9]. The model is based on a large volume of experimental data. It describes the ‘‘bulldozer’’ effect, when self-similarity of the profile is saved, but the volume of liquid and the wave height increase, and the thickness of continuous layer decreases. The results of experimental investigations of heat transfer at film condensation of stationary immobile vapor on vertical tubes in a wide range of Reynolds numbers (10 6 Re 6 4300) were presented by Gogonin et al. [10]. The R21-refrigerant was used as the working liquid. The dependence of the ‘‘residual’’ liquid film thickness on the Reynolds number was investigated in [11,12]. It was found that the ‘‘residual’’ thickness of the film remains almost constant over a
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Nomenclature a Cp D g l p r T t u, v V x, y
thermal diffusivity, a = k/qCp, specific heat, shock velocity, gravity, characteristic wavelength, L = 2l, pressure, latent heat, temperature, time, longitudinal and transverse velocity components, velocity of a progressive wave, coordinates.
Ku = r/CpDT Kutateladze number. Greek Symbols pffiffiffiffiffi Ar, a b 1/PrKu, k thermal conductivity, h inclination angle of the wall, q density, m kinematic viscosity, n = x Vt self-similar variable. Indices s w
Dimensionless criteria Ar = gl3/m2 Archimedes number, Pr = m/a Prandtl number,
wide range of Reynolds number, and the large-amplitude waves propagate with some velocity along this ‘‘residual’’ layer. The problem of the flow of a thin layer of viscous fluid over a vertical surface in approximation of shallow water, when surface tension is neglected, was considered by Buchin and Shaposhnikova [13]. The exact solution with periodic system of jumps moving with a constant velocity downward the flow was constructed. In these solutions the distance between the jumps is a free parameter. The modulation equations for periodic roll waves were investigated by Boudlal and Liapidevskii [14]. The self-similar solutions were constructed. It was shown that the parameters of these waves are little sensitive to their shape in the vicinity of the ridge, so the account of capillary effects does not lead to a qualitative change in the rolling wave’s shape. The problems of thin liquid film flowing on a vertical wall in the presence of mass transfer through the free surface were considered in [15,16]. It is necessary to note that the models, used in [13–16], are the special cases of the Shkadov model [17–19]. These days the mathematical models are being developed [2– 19], but the general aspects of the free-surface wave generation in thin liquid films flowing over a vertical or inclined wall in the presence of heat and mass transfer through the liquid–gas interface are still poorly understood. The aim of the present work is investigation of the mathematical model of the thin liquid film flowing over a vertical wall in the regime of roll waves in the presence of heat and mass transfer through the liquid–vapor interface. 2. Equation describing roll waves in the presence of heat and mass transfer through the liquid–vapor interface In long-wave approximation [20], when film thickness h = h(x, t) is much less than the wavelength, the boundary-layer theory [21] is used to describe the slow flow of the thin film of condensate along an inclined wall
@u @u @u @ 2 u 1 @p þu þv ¼ g sin h þ m 2 ; @t @x @y @y q @x
ð1Þ
@u @ v þ ¼ 0; @x @y
ð2Þ
1 @p ¼ g cos h; q @y
ð3Þ
@T @T @T @2T þu þv ¼a 2: @t @x @y @y
ð4Þ
saturation, wall.
The Cartesian coordinate system in Eqs. (1)–(4) is chosen in such a way that x is the direction of the flow, and y is the direction from the solid wall to the free film surface. The point x = 0 is a point of the emergence of condensate, or the initial point of the flow. For the solid surface at y = 0 we have no-slip condition
uð0; xÞ ¼ v ð0; xÞ ¼ 0
ð5Þ
and a constant wall temperature
Tjy¼0 ¼ T w :
ð6Þ
For the interface boundary at y = h(x, t) we have the kinematic condition
v¼
@h @h k þu ðT s T w Þ; @t @x qrh
ð7Þ
the condition of the absence of shear stress
@u ¼0 @y y¼h
ð8Þ
and the saturation temperature
Tjy¼h ¼ T s :
ð9Þ
For the case of condensation of one-component saturated vapor on the isothermal wall, it is performed that k(Ts Tw)/qr = const. After integration of Eq. (2) with respect to h we obtain
v¼u
@h @ @x @x
Z
h
udy:
ð10Þ
0
Then, taking into account (7), we obtain the equation for the condensate film thickness
@h @ þ @t @x
Z
h
udy ¼ 0
k
qrh
ðT s T w Þ:
ð11Þ
The last equation corresponds to the assumption that the heat released at the vapor–liquid interface in a condensation process is consumed only for heating the liquid. The inertia terms can be neglected in Eqs. (1)–(4) for the limiting case of low flow rate. On a vertical wall (h = p/2) this assumption and hydrostatic condition lead to the Nusselt problem [1]
m
@2u þ g ¼ 0; @y2
@2T ¼ 0; @y2 ujy¼0 ¼ uy jy¼h ¼ 0;
ð12Þ ð13Þ
k
Tjy¼0 ¼ T w ;
Tjy¼h ¼ T s ;
ð14Þ
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After integrating Eq. (12) with boundary conditions (14), the parabolic velocity profile can be obtained 2
u¼
gh 2m
2y y2 2 : h h
ð15Þ
Then, the average cross-sectional flow rate can be received by integration over the film thickness
q¼
Z
2
h
udy ¼
0
gh 2m
Z
h
0
3 2y y2 gh 2 dy ¼ : h h 3m
ð16Þ
Substituting flow rate (16) into Eq. (11), we obtain the mass conservation equation for film thickness 3
@h @ gh þ @t @x 3m
! ¼
k Ts Tw : h
ð17Þ
qr
In the absence of source terms, i.e., when we can neglect mass transfer through the free surface, Eq. (17) describes the kinematic waves, which have been investigated in [2]. By using dimensionless variables, we can transform Eq. (17) and obtain 3
@h a @h b þ ¼ ; @t 3 @x ah
ð18Þ
here
pffiffiffiffiffi
a ¼ Ar; b ¼ Ku ¼
r ; C p DT
1 ; PrKu
Ar ¼
gl
3
m2
;
m
Pr ¼ ; a
DT ¼ T s T w ;
ð19Þ
l is a characteristic length of the roll wave. Eq. (18) is the roll-wave equation taking into account heat and mass transfer through the liquid–vapor interface. The exact continuous and discontinuous solutions to Eq. (18) are constructed in Section 3 of the present paper.
We will construct the exact discontinuous solutions using continuous solutions (21) with sign ‘‘+’’. These discontinuous solutions form the system of roll waves. At h > 0 Eq. (18) leads to the Rankine–Hugoniot (RH) condition [22] at the shock x = x(t)
D½h ¼
a 3
3
½h ;
where D = xt is the shock velocity, [f] = f1 f0 is the jump of function f(t, x) at the shock, i.e. f0 = f(t, x(t) + 0), f1 = f(t, x(t) 0). From condition (23) we get the formula for the shock velocity [23]
D ¼ Dðh0 ; h1 Þ ¼
8 ~ > n 6 n1 ðtÞ; > > hðtÞ; > < h ðn; n Þ; n ðtÞ 6 n < n ðtÞ; þ 1 2 hðn; tÞ ¼ ~ > hðtÞ; > n > n2 ðtÞ; > > :
2
ð20Þ
After integrating Eq. (20), two families of solutions can be obtained
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 4bðn n Þ ; h ¼ h ðn; n Þ ¼
a
n P n ;
ð21Þ
here n⁄ is a heuristic parameter. The family of solution, with sign ‘‘+’’ under square root, corresponds to increasing waves, and other family, at V > 0 and with sign ‘‘’’ under square root, corresponds to decreasing waves. Formally the solution of the increasing-wave type h+(n, n⁄) exists for all n P n⁄. The solution of the decreasing wave-type h(n, n⁄) exists for n 6 n 6 ~ n ¼ n þ V 2 =ð4bÞ, where ~ n is defined from eq. hð~ n ; n Þ ¼ 0. The increasing wave at V = 0 changes over to the steady-state solution, given by the Nusselt formula [1]
h ¼ hN ðx; n Þ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 4b ðx n Þ; 2
a
x P n :
ð22Þ
ð25Þ
ð26Þ
where
n2 ðtÞ ¼
dh 2b : ¼ dn a
ð24Þ
where lr = (r/qg)1/2 and lm = (m2/g)1/3 are capillary and viscosity constants. The equation (18) has no solutions h ¼ const, therefore we construct the systems of rolling waves propagating over the increasing ~ ~ coincides with the ‘‘residual’’ depth hðtÞ, where the initial value of hðtÞ ~ It follows from these assumptions that the thickness, i.e. hð0Þ ¼ h. thickness profile of the solitary rolling wave is defined by the formula
The equation (18) has the solutions which are not dependent qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ ¼ 2bt=a þ h 2 . from the space variable and can be written as hðtÞ
2
3
2 2 h1 þ h1 h0 þ h0 ;
3=15 ¼ h ¼ 1:9 lr ¼ hlm ; )h h lm lm l
n1 ¼ n þ
ðah VÞ
a
where h1 > h0. From the experimental results [11] it follows that the rolling waves on the surface of condensate film propagate over the ‘‘resid The characteristic value of this ‘‘residual’’ thickual’’ thickness h. ness is a constant and its dimensionless value is calculated by the empirical formula [12]
3. Exact solutions of progressive-wave type separated by strong discontinuities
This solution is obtained by the integration of the ordinary differential equation ht ¼ b=ah, which follows from the Eq. (18) at hx ¼ 0. The equation (18) also has a solutions, depending on the selfsimilarity variable n = x Vt, where V = const P 0 is a velocity of progressive wave. Eq. (18) can be rewritten as:
ð23Þ
Z
1 2 ~4 a h 2ah~2 V þ V 2 P n : 4b t
~ hþ ðn ðsÞ; n Þ ds þ n n > n : D h; 3 ;
ð27Þ
ð28Þ
0
~ In formula (26) n1(t) is found from equation hþ ðn1 ; n Þ ¼ hðtÞ, and l ¼ n n is an integral wavelength. It follows from Eq. (21) that 2 . In the case solution (26)–(28) can exist if V 6 ah
2 2 () V ¼ a1=3 h V ¼ ah
ð29Þ
we obtain n1 (0) = n⁄. It is follows from formulas (26)–(28) that the averaged thickness of the film increases unlimitedly in time. As a result, this solution can be used for describing the falling condensate film only in the restricted time interval. In Section 4 of the present paper formulas (21), (24) and (26)– (28) have been used for construction of the exact discontinuous solutions, forming the system of roll waves. 4. Simulation of the system of periodic roll waves In this section we present the simulation of the periodic system of rolling waves by solving the Cauchy problem for divergent Eq. (18) with the periodic initial data
hð0; xÞ ¼ HðxÞ;
Hðx þ LÞ HðxÞ;
8 > < hþ ðx; 0Þ; 0 6 x < l; HðxÞ ¼ h; l < x 6 L; > :
ð30Þ
ð31Þ
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where l = L/2 is characteristic wavelength; taking into account (21) and (29) we obtain
hþ ðx; 0Þ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi V þ 4bx
a
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 4bx ¼ h2 þ :
hðt; xÞ ¼ hþ ðx Vt; kLÞ;
a
Figs. 1 and 2 present the solution to this problem (the profiles of falling film thickness) at four consecutive time moments. At the time interval [0, t1], when the shocks propagate over the thickness ~ hðtÞ, solution h(t, x) at the space interval [0, L] has the form (Fig. 1)
8 ~ > hðtÞ; 0 < x 6 g1 ðtÞ > > > < h ðx Vt; 0Þ; g ðtÞ 6 x < g ðtÞ; þ 1 2 hðt; xÞ ¼ ~ > > hðtÞ; g ðtÞ < x 6 L; > 2 > :
ð32Þ
where
g1 ðtÞ ¼
~2 ðtÞ VÞ2 ðah 4b Z
t
0
ð33Þ
When solving Eq. (33), we can find t1 from the equation
Z 0
t1
~ sÞ; hþ ðg ðsÞ; 0Þ ds ¼ l D hð 2
ð34Þ
obtained from Eq. (33) when g2 (t1) = L = 2l. At time t > t1, when the shocks propagate over the surface of the next waves, the solution h(t, x) at the interval (g2(t), g2(t) + L) is continuous and has the form (Fig. 2)
hðt; xÞ ¼ hþ ðx Vt; LÞ
ð35Þ
where g2(t) is defined from integral equation
g2 ðtÞ ¼ L þ
Z
ð38Þ
Pr ¼ 1:76;
Ku ¼ 47:4 ) a ¼ 349:0;
~ sÞ; hþ ðg ðsÞ; 0Þ ds: D hð 2
k2Z
and locations of discontinuity lines are defined by the way of numerical solution to integral Eq. (33) at t 2 [0, t1] and Eq. (36) at t > t1. correspondAs an example, let’s consider the values of a; b; h, ing to the regime of condensate water film flowing down a vertical wall. In this case the ‘‘residual’’ thickness (25) equals h0 = 0.103 mm. The characteristic length of the roll wave corresponds to the long-wave approximation and equals l = 1.03 mm. From ¼ h0 =l ¼ 0:1. The saturation these assumptions it is follows that h temperature is Ts = 100 °C and the constant-wall-temperature is Tw = 90 °C. The parameters a, b and the velocity of progressive wave V have been calculated from formulas (19) and (29). Using these data, we can define
Ar ¼ 1:22 105 ;
and g2(t) is defined from integral equation
g2 ðtÞ ¼ l þ
In the continuity regions the constructed solutions are calculated by exact formulas
b ¼ 0:012;
V ¼ 3:49;
t1 ¼ 27:
5. Conclusion The roll-wave equation taking into account heat and mass transfer through the liquid–vapor interface has been derived. The solutions describing the roll waves on a vertical thin liquid film in the presence of heat and mass transfer through the liquid–vapor interface have been obtained. In these solutions the progressive waves are conjugated with each other or with ‘‘residual’’ film thickness through the strong discontinuity. Complexes Ar, Pr, and Ku are the key criteria characterizing this process. The velocity of progressive wave and the shock velocity have been obtained. As an example, the calculations of the condensed water film flowing down a vertical surface are presented.
t
t1
Dðhþ ðg2 ðsÞ; LÞ; hþ ðg2 ðsÞ; 0ÞÞds:
ð36Þ
At points g2 ðtÞ þ kL; k 2 Z, where Z is a set of integers, the solution has discontinuities with amplitude
DhðtÞ ¼ hþ ðg2 ðtÞ; 0Þ hþ ðg2 ðtÞ; LÞ:
ð37Þ
However, now the existence of the regime (Fig. 2) is not confirmed experimentally.
This work was supported by the Grant of the Government of Russian Federation designed to support scientific research projects implemented under the supervision of leading scientists at Russian institutions of higher education No. 11.G34.31.0046 (Lead Scientist – Kemal Hanjalic, Novosibirsk State University). References
1.04
3
h/h0 1.02
2
1
1
0
2
4
6
8
ξ /l Fig. 1. Profiles of falling film thickness at time moments t = 0 (1 – thin dash line), t = 15 (2 – thick dash line), t = 27 (3-solid line).
2
h/h0
Acknowledgments
1.04
1
1 0
2
4
6
8
ξ /s Fig. 2. Profiles of falling film thickness at time moments t = 27 (1 – dash line), t = 77 (2 – solid line).
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[13] V.A. Buchin, G.A. Shaposhnikova, Flow of shallow water with a periodic system of jumps over a vertical surface, Dokl. Phys. 54 (5) (2009) 248–251. [14] A. Boudlal, V.Yu. Liapidevskii, Modulation equations for wave flows of a fluid film along a vertical wall, Dokl. Phys. 57 (1) (2012) 23–28. [15] E.A. Demekhin, E.N. Kalaidin, A.A. Rastaturin, The effect of wave modes on mass transfer in falling liquid films, Thermophys. Aeromech. 12 (2) (2005) 243–252. [16] V.A. Buchin, G.A. Shaposhnikova, Stability of a thin liquid film flowing on a vertical wall in the presence of mass transfer through the free surface, Dokl. Phys. 50 (10) (2005) 539–543. [17] V. Ya. Shkadov, Wave flow regimes of a thin layer of viscous fluid subject to gravity, Fluid Dynam. 2 (1) (1967) 29–34.
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Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Heat and mass transfer in the liquid film on a vertical wall in roll-wave regime V.E. Nakoryakov a,c, V.V. Ostapenko b,c, M.V. Bartashevich a,c,⇑ a
Kutateladze Institute of Thermophysics, Siberian Branch, Russian Academy of Sciences, 1, Acad. Lavrentiev Avenue, Novosibirsk 630090, Russia Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, 15, Acad. Lavrentiev Avenue, Novosibirsk 630090, Russia c Novosibirsk State University, 2, Pirogova Str., Novosibirsk 630090, Russia b
a r t i c l e
i n f o
Article history: Received 12 June 2012 Accepted 18 June 2012 Available online 26 July 2012 Keywords: Heat and mass transfer Condensation Self-similar solutions Falling film Roll waves Progressive waves
a b s t r a c t The investigation of thin liquid film flowing down a vertical wall in the roll-wave regime in presence of heat and mass transfer through the free surface is presented. The roll-wave equation taking into account heat and mass transfer through the liquid–vapor interface has been derived. The self-similar solutions of the progressive-wave type for film thickness have been obtained. The families of discontinuous solutions have been constructed, where the progressive waves are conjugated with each other or with ‘‘residual’’ film thickness through the strong discontinuity. As an example, the calculations of the condensate water film flowing down a vertical surface are presented. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction First analytical and experimental investigations of heat transfer at laminar film condensation on a vertical wall were done by Nusselt [1]. However, the Nusselt theory is valid only for low Reynolds numbers. Even for film Reynolds numbers of about several units, the waves appear at the interface and increase the heat transfer coefficient [2]. The pioneering investigations of waves on the surface of a thin layer of viscous liquid films flowing down a wall were made by Kapitsa [3]. The problem of rolling waves in a sheet of fluid flowing in a vertical plane was treated by Pukhnachev [4] on the basis of complete Navier–Stokes equations with conditions on the unknown free boundary. The existence of progressive (capillarygravity) waves in a layer on a vertical wall at indefinitely small Reynolds numbers was proved. The roll waves on the surface of thin liquid films were studied experimentally and theoretically by Nakoryakov et al. [5,6]. The waves obtained in [5] have amplitude greater than the average thickness of the liquid film. The roll wave profiles have the main part and capillary precursor moving ahead with the same speed. This capillary precursor has the form of a damped harmonic wave. The characteristics of two-dimensional steady-state roll waves on a vertical liquid film were investigated. It was shown that the parameters of large-amplitude waves on a thin vertical film are heavily dependent on the viscosity of liquid ⇑ Corresponding author at: Kutateladze Institute of Thermophysics, Siberian Branch, Russian Academy of Sciences, 1, Acad. Lavrentiev Avenue, Novosibirsk 630090, Russia. Tel.: +7 383 330 70 50; fax: +7 383 330 84 80. E-mail address: [email protected] (M.V. Bartashevich). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.06.054
and these waves can have the non-capillary nature. Water, waterglycerol and water-alcohol solutions were used as the working liquids. The viscosity varied from 0.9 106 to 11.2 106 m2/s. The waves were formed at some distance from the distributing slot outlet, their amplitude increased rapidly, and then the steady state regime was achieved. The dependence of the phase velocity of the roll-waves on their amplitude was linear. There was no dependence between phase velocity and the average flow rate. There was also a slight dependence of the phase velocity on the surface tension. The possibility of the existence of progressive waves on the surface of a vertically flowing liquid film with neglected surface tension was shown in [6]. The equation for perturbations of a thin film was obtained. This equation allows the steady periodic discontinuous solutions of the roll-wave type (similar solutions were constructed in [7]). The constructed discontinuous solutions agree qualitatively with the experimental results [6]. The bore theory, which describes the wave breaking, is presented in [8]. The original ‘‘bore model’’ was presented by Gantchev [9]. The model is based on a large volume of experimental data. It describes the ‘‘bulldozer’’ effect, when self-similarity of the profile is saved, but the volume of liquid and the wave height increase, and the thickness of continuous layer decreases. The results of experimental investigations of heat transfer at film condensation of stationary immobile vapor on vertical tubes in a wide range of Reynolds numbers (10 6 Re 6 4300) were presented by Gogonin et al. [10]. The R21-refrigerant was used as the working liquid. The dependence of the ‘‘residual’’ liquid film thickness on the Reynolds number was investigated in [11,12]. It was found that the ‘‘residual’’ thickness of the film remains almost constant over a
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Nomenclature a Cp D g l p r T t u, v V x, y
thermal diffusivity, a = k/qCp, specific heat, shock velocity, gravity, characteristic wavelength, L = 2l, pressure, latent heat, temperature, time, longitudinal and transverse velocity components, velocity of a progressive wave, coordinates.
Ku = r/CpDT Kutateladze number. Greek Symbols pffiffiffiffiffi Ar, a b 1/PrKu, k thermal conductivity, h inclination angle of the wall, q density, m kinematic viscosity, n = x Vt self-similar variable. Indices s w
Dimensionless criteria Ar = gl3/m2 Archimedes number, Pr = m/a Prandtl number,
wide range of Reynolds number, and the large-amplitude waves propagate with some velocity along this ‘‘residual’’ layer. The problem of the flow of a thin layer of viscous fluid over a vertical surface in approximation of shallow water, when surface tension is neglected, was considered by Buchin and Shaposhnikova [13]. The exact solution with periodic system of jumps moving with a constant velocity downward the flow was constructed. In these solutions the distance between the jumps is a free parameter. The modulation equations for periodic roll waves were investigated by Boudlal and Liapidevskii [14]. The self-similar solutions were constructed. It was shown that the parameters of these waves are little sensitive to their shape in the vicinity of the ridge, so the account of capillary effects does not lead to a qualitative change in the rolling wave’s shape. The problems of thin liquid film flowing on a vertical wall in the presence of mass transfer through the free surface were considered in [15,16]. It is necessary to note that the models, used in [13–16], are the special cases of the Shkadov model [17–19]. These days the mathematical models are being developed [2– 19], but the general aspects of the free-surface wave generation in thin liquid films flowing over a vertical or inclined wall in the presence of heat and mass transfer through the liquid–gas interface are still poorly understood. The aim of the present work is investigation of the mathematical model of the thin liquid film flowing over a vertical wall in the regime of roll waves in the presence of heat and mass transfer through the liquid–vapor interface. 2. Equation describing roll waves in the presence of heat and mass transfer through the liquid–vapor interface In long-wave approximation [20], when film thickness h = h(x, t) is much less than the wavelength, the boundary-layer theory [21] is used to describe the slow flow of the thin film of condensate along an inclined wall
@u @u @u @ 2 u 1 @p þu þv ¼ g sin h þ m 2 ; @t @x @y @y q @x
ð1Þ
@u @ v þ ¼ 0; @x @y
ð2Þ
1 @p ¼ g cos h; q @y
ð3Þ
@T @T @T @2T þu þv ¼a 2: @t @x @y @y
ð4Þ
saturation, wall.
The Cartesian coordinate system in Eqs. (1)–(4) is chosen in such a way that x is the direction of the flow, and y is the direction from the solid wall to the free film surface. The point x = 0 is a point of the emergence of condensate, or the initial point of the flow. For the solid surface at y = 0 we have no-slip condition
uð0; xÞ ¼ v ð0; xÞ ¼ 0
ð5Þ
and a constant wall temperature
Tjy¼0 ¼ T w :
ð6Þ
For the interface boundary at y = h(x, t) we have the kinematic condition
v¼
@h @h k þu ðT s T w Þ; @t @x qrh
ð7Þ
the condition of the absence of shear stress
@u ¼0 @y y¼h
ð8Þ
and the saturation temperature
Tjy¼h ¼ T s :
ð9Þ
For the case of condensation of one-component saturated vapor on the isothermal wall, it is performed that k(Ts Tw)/qr = const. After integration of Eq. (2) with respect to h we obtain
v¼u
@h @ @x @x
Z
h
udy:
ð10Þ
0
Then, taking into account (7), we obtain the equation for the condensate film thickness
@h @ þ @t @x
Z
h
udy ¼ 0
k
qrh
ðT s T w Þ:
ð11Þ
The last equation corresponds to the assumption that the heat released at the vapor–liquid interface in a condensation process is consumed only for heating the liquid. The inertia terms can be neglected in Eqs. (1)–(4) for the limiting case of low flow rate. On a vertical wall (h = p/2) this assumption and hydrostatic condition lead to the Nusselt problem [1]
m
@2u þ g ¼ 0; @y2
@2T ¼ 0; @y2 ujy¼0 ¼ uy jy¼h ¼ 0;
ð12Þ ð13Þ
k
Tjy¼0 ¼ T w ;
Tjy¼h ¼ T s ;
ð14Þ
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After integrating Eq. (12) with boundary conditions (14), the parabolic velocity profile can be obtained 2
u¼
gh 2m
2y y2 2 : h h
ð15Þ
Then, the average cross-sectional flow rate can be received by integration over the film thickness
q¼
Z
2
h
udy ¼
0
gh 2m
Z
h
0
3 2y y2 gh 2 dy ¼ : h h 3m
ð16Þ
Substituting flow rate (16) into Eq. (11), we obtain the mass conservation equation for film thickness 3
@h @ gh þ @t @x 3m
! ¼
k Ts Tw : h
ð17Þ
qr
In the absence of source terms, i.e., when we can neglect mass transfer through the free surface, Eq. (17) describes the kinematic waves, which have been investigated in [2]. By using dimensionless variables, we can transform Eq. (17) and obtain 3
@h a @h b þ ¼ ; @t 3 @x ah
ð18Þ
here
pffiffiffiffiffi
a ¼ Ar; b ¼ Ku ¼
r ; C p DT
1 ; PrKu
Ar ¼
gl
3
m2
;
m
Pr ¼ ; a
DT ¼ T s T w ;
ð19Þ
l is a characteristic length of the roll wave. Eq. (18) is the roll-wave equation taking into account heat and mass transfer through the liquid–vapor interface. The exact continuous and discontinuous solutions to Eq. (18) are constructed in Section 3 of the present paper.
We will construct the exact discontinuous solutions using continuous solutions (21) with sign ‘‘+’’. These discontinuous solutions form the system of roll waves. At h > 0 Eq. (18) leads to the Rankine–Hugoniot (RH) condition [22] at the shock x = x(t)
D½h ¼
a 3
3
½h ;
where D = xt is the shock velocity, [f] = f1 f0 is the jump of function f(t, x) at the shock, i.e. f0 = f(t, x(t) + 0), f1 = f(t, x(t) 0). From condition (23) we get the formula for the shock velocity [23]
D ¼ Dðh0 ; h1 Þ ¼
8 ~ > n 6 n1 ðtÞ; > > hðtÞ; > < h ðn; n Þ; n ðtÞ 6 n < n ðtÞ; þ 1 2 hðn; tÞ ¼ ~ > hðtÞ; > n > n2 ðtÞ; > > :
2
ð20Þ
After integrating Eq. (20), two families of solutions can be obtained
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 4bðn n Þ ; h ¼ h ðn; n Þ ¼
a
n P n ;
ð21Þ
here n⁄ is a heuristic parameter. The family of solution, with sign ‘‘+’’ under square root, corresponds to increasing waves, and other family, at V > 0 and with sign ‘‘’’ under square root, corresponds to decreasing waves. Formally the solution of the increasing-wave type h+(n, n⁄) exists for all n P n⁄. The solution of the decreasing wave-type h(n, n⁄) exists for n 6 n 6 ~ n ¼ n þ V 2 =ð4bÞ, where ~ n is defined from eq. hð~ n ; n Þ ¼ 0. The increasing wave at V = 0 changes over to the steady-state solution, given by the Nusselt formula [1]
h ¼ hN ðx; n Þ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 4b ðx n Þ; 2
a
x P n :
ð22Þ
ð25Þ
ð26Þ
where
n2 ðtÞ ¼
dh 2b : ¼ dn a
ð24Þ
where lr = (r/qg)1/2 and lm = (m2/g)1/3 are capillary and viscosity constants. The equation (18) has no solutions h ¼ const, therefore we construct the systems of rolling waves propagating over the increasing ~ ~ coincides with the ‘‘residual’’ depth hðtÞ, where the initial value of hðtÞ ~ It follows from these assumptions that the thickness, i.e. hð0Þ ¼ h. thickness profile of the solitary rolling wave is defined by the formula
The equation (18) has the solutions which are not dependent qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ ¼ 2bt=a þ h 2 . from the space variable and can be written as hðtÞ
2
3
2 2 h1 þ h1 h0 þ h0 ;
3=15 ¼ h ¼ 1:9 lr ¼ hlm ; )h h lm lm l
n1 ¼ n þ
ðah VÞ
a
where h1 > h0. From the experimental results [11] it follows that the rolling waves on the surface of condensate film propagate over the ‘‘resid The characteristic value of this ‘‘residual’’ thickual’’ thickness h. ness is a constant and its dimensionless value is calculated by the empirical formula [12]
3. Exact solutions of progressive-wave type separated by strong discontinuities
This solution is obtained by the integration of the ordinary differential equation ht ¼ b=ah, which follows from the Eq. (18) at hx ¼ 0. The equation (18) also has a solutions, depending on the selfsimilarity variable n = x Vt, where V = const P 0 is a velocity of progressive wave. Eq. (18) can be rewritten as:
ð23Þ
Z
1 2 ~4 a h 2ah~2 V þ V 2 P n : 4b t
~ hþ ðn ðsÞ; n Þ ds þ n n > n : D h; 3 ;
ð27Þ
ð28Þ
0
~ In formula (26) n1(t) is found from equation hþ ðn1 ; n Þ ¼ hðtÞ, and l ¼ n n is an integral wavelength. It follows from Eq. (21) that 2 . In the case solution (26)–(28) can exist if V 6 ah
2 2 () V ¼ a1=3 h V ¼ ah
ð29Þ
we obtain n1 (0) = n⁄. It is follows from formulas (26)–(28) that the averaged thickness of the film increases unlimitedly in time. As a result, this solution can be used for describing the falling condensate film only in the restricted time interval. In Section 4 of the present paper formulas (21), (24) and (26)– (28) have been used for construction of the exact discontinuous solutions, forming the system of roll waves. 4. Simulation of the system of periodic roll waves In this section we present the simulation of the periodic system of rolling waves by solving the Cauchy problem for divergent Eq. (18) with the periodic initial data
hð0; xÞ ¼ HðxÞ;
Hðx þ LÞ HðxÞ;
8 > < hþ ðx; 0Þ; 0 6 x < l; HðxÞ ¼ h; l < x 6 L; > :
ð30Þ
ð31Þ
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where l = L/2 is characteristic wavelength; taking into account (21) and (29) we obtain
hþ ðx; 0Þ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi V þ 4bx
a
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 4bx ¼ h2 þ :
hðt; xÞ ¼ hþ ðx Vt; kLÞ;
a
Figs. 1 and 2 present the solution to this problem (the profiles of falling film thickness) at four consecutive time moments. At the time interval [0, t1], when the shocks propagate over the thickness ~ hðtÞ, solution h(t, x) at the space interval [0, L] has the form (Fig. 1)
8 ~ > hðtÞ; 0 < x 6 g1 ðtÞ > > > < h ðx Vt; 0Þ; g ðtÞ 6 x < g ðtÞ; þ 1 2 hðt; xÞ ¼ ~ > > hðtÞ; g ðtÞ < x 6 L; > 2 > :
ð32Þ
where
g1 ðtÞ ¼
~2 ðtÞ VÞ2 ðah 4b Z
t
0
ð33Þ
When solving Eq. (33), we can find t1 from the equation
Z 0
t1
~ sÞ; hþ ðg ðsÞ; 0Þ ds ¼ l D hð 2
ð34Þ
obtained from Eq. (33) when g2 (t1) = L = 2l. At time t > t1, when the shocks propagate over the surface of the next waves, the solution h(t, x) at the interval (g2(t), g2(t) + L) is continuous and has the form (Fig. 2)
hðt; xÞ ¼ hþ ðx Vt; LÞ
ð35Þ
where g2(t) is defined from integral equation
g2 ðtÞ ¼ L þ
Z
ð38Þ
Pr ¼ 1:76;
Ku ¼ 47:4 ) a ¼ 349:0;
~ sÞ; hþ ðg ðsÞ; 0Þ ds: D hð 2
k2Z
and locations of discontinuity lines are defined by the way of numerical solution to integral Eq. (33) at t 2 [0, t1] and Eq. (36) at t > t1. correspondAs an example, let’s consider the values of a; b; h, ing to the regime of condensate water film flowing down a vertical wall. In this case the ‘‘residual’’ thickness (25) equals h0 = 0.103 mm. The characteristic length of the roll wave corresponds to the long-wave approximation and equals l = 1.03 mm. From ¼ h0 =l ¼ 0:1. The saturation these assumptions it is follows that h temperature is Ts = 100 °C and the constant-wall-temperature is Tw = 90 °C. The parameters a, b and the velocity of progressive wave V have been calculated from formulas (19) and (29). Using these data, we can define
Ar ¼ 1:22 105 ;
and g2(t) is defined from integral equation
g2 ðtÞ ¼ l þ
In the continuity regions the constructed solutions are calculated by exact formulas
b ¼ 0:012;
V ¼ 3:49;
t1 ¼ 27:
5. Conclusion The roll-wave equation taking into account heat and mass transfer through the liquid–vapor interface has been derived. The solutions describing the roll waves on a vertical thin liquid film in the presence of heat and mass transfer through the liquid–vapor interface have been obtained. In these solutions the progressive waves are conjugated with each other or with ‘‘residual’’ film thickness through the strong discontinuity. Complexes Ar, Pr, and Ku are the key criteria characterizing this process. The velocity of progressive wave and the shock velocity have been obtained. As an example, the calculations of the condensed water film flowing down a vertical surface are presented.
t
t1
Dðhþ ðg2 ðsÞ; LÞ; hþ ðg2 ðsÞ; 0ÞÞds:
ð36Þ
At points g2 ðtÞ þ kL; k 2 Z, where Z is a set of integers, the solution has discontinuities with amplitude
DhðtÞ ¼ hþ ðg2 ðtÞ; 0Þ hþ ðg2 ðtÞ; LÞ:
ð37Þ
However, now the existence of the regime (Fig. 2) is not confirmed experimentally.
This work was supported by the Grant of the Government of Russian Federation designed to support scientific research projects implemented under the supervision of leading scientists at Russian institutions of higher education No. 11.G34.31.0046 (Lead Scientist – Kemal Hanjalic, Novosibirsk State University). References
1.04
3
h/h0 1.02
2
1
1
0
2
4
6
8
ξ /l Fig. 1. Profiles of falling film thickness at time moments t = 0 (1 – thin dash line), t = 15 (2 – thick dash line), t = 27 (3-solid line).
2
h/h0
Acknowledgments
1.04
1
1 0
2
4
6
8
ξ /s Fig. 2. Profiles of falling film thickness at time moments t = 27 (1 – dash line), t = 77 (2 – solid line).
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