Proceedings of the Asian Conference on Thermal Sciences 2017, 1st ACTS
March 26-30, 2017, Jeju Island, Korea
ACTS-P00050
COUPLING EFFECT OF EVAPORATION AND THERMOCAPILLARY FLOW IN
AN ANNULAR SHALLOW POOL WITH MODERATE PRANDTL NUMBER FLUID
Li Zhang, Chun-Mei Wu, You-Rong Li*
College of Power Engineering, Chongqing University, 174 Shazhengjie, Shapingba, Chongqing 400044, China

Presenting Author: [email protected]
Corresponding Author: [email protected]
*
ABSTRACT
In order to understand the coupling effect of surface evaporation and thermocapillary convection for the fluids with
moderate Prandtl number in an annular shallow pool, this paper presented a series of numerical simulations on
thermocapillary convection of the cold water with the evaporating free surface. The Prandtl number of the cold water
is 12. The radius ratio of the annular pool is 0.5, and the aspect ratio ranges from 0.05 to 0.2. Results show that: (1)
with the increase of evaporation Biot number, the radial temperature gradient on the free surface near the inner wall
decreases obviously, therefore, the thermocapillary force weakens and the flow of the surface fluid slows down. In
this case, the surface fluid from the outer wall begins to return to the outer wall before it reaches the cold wall, which
results in that the thermocapillary flow cell shrinks toward the outer wall and the flow region decreases gradually.
However, the flow strength increases first, and then decreases. (2) The non-dimensional evaporative mass flux
increases with the increase of evaporation Biot number near the outer wall, while it is influenced by the coupling
effect of surface evaporation and thermocapillary convection near the inner wall. (3) The total evaporation mass on
the free surface is enhanced with the increase of evaporation Biot number. Furthermore, with the increase of the
aspect ratio of liquid pool, thermocapillary convection is strengthened, and the dimensionless total evaporation mass
also rises.
KEYWORDS: Thermocapillary convection, Evaporation, Annular shallow pool, Numerical simulation
1. INTRODUCTION
Thermocapillary convection exists widely in engineering technology, such as crystal growth, droplet evaporation,
laser welding, and film spraying and so on. At present, many scholars have extensively researched about
thermocapillary convection instability in the liquid layer without surface evaporation. However, during the practical
engineering process, the interfacial non-equilibrium effect makes the evaporation of the liquid layer inevitable,
which is bound to affect the temperature and velocity distribution on the free surface and in the fluid, and thus
changes the convective flow structure; simultaneously, the heat convection itself will affect the surface local
evaporation rate. Both coupled mutually, and formed a complex convective heat and mass transfer process. Qin et al.
[1,2] simulated numerically thermocapillary-buoyancy convection of volatile fluid in the rectangular pool, and found
that in the atmospheric environment and pure steam environment, the different evaporation rate on the free surface
makes the flow appear different heat transfer characteristics. Yamamura et al. [3] observed thermocapillary
convection in a thin liquid layer subjected to a temperature gradient parallel to the evaporating surface. The results
showed that as the evaporating liquid layer gradually becomes thinner, the flow in the liquid layer will change from
initial thermocapillary convection to the coupled convection of thermocapillary and Marangoni-Bénard, and
eventually evolve into Marangoni-Bénard convection. Li et al. [4] made experimental studies on thermocapillarybuoyancy convection of 0.65cSt silicone oil in the rectangular pool, and the results showed that with the increase of
Marangoni number, the flow converted to the unsteady flow and the critical Marangoni number of flow transition is
closely related to the surface evaporation rate. Liu et al. [5-8] studied on thermocapillary convection of evaporating
thin liquid layer in the rectangular pool. The results show that the interfacial evaporation has a great influence on the
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stability of thermocapillary convection, and the introduction of thermocapillary convection will increase the
interfacial evaporation mass flux. Thermocapillary convection made that the degree of evaporation enhancement is
correlative with the non-equilibrium degree of the evaporation interface, which depends on the interfacial
evaporation Biot number. Sáenz et al. [9] simulated numerically the influence of the evaporating effect on the flow
stability in a rectangular tank heated from the side wall. The results showed that the effect of surface evaporation on
the dynamic instability is twofold: for one thing, the endothermic effect of evaporation has an inhibitory effect on the
stability of thermal convection; for another, with continuing evaporation, the thickness of the liquid layer is thinned,
which increases the deformation of the interface, and can accelerate the flow instability. Li et al. [10-12] carried out
a series of numerical simulations on thermocapillary convection and the thermocapillary-buoyancy convection in an
annular shallow pool, ascertained the critical conditions for the flow instability, and revealed the basic characteristics
of the hydrothermal wave, but they never consider the interfacial evaporating effect. This paper conducted a series of
numerical simulations on thermocapillary convection in an annular shallow pool for moderate Prandtl number fluids
with the surface evaporation, and the main purpose is to explore the coupling mechanism between surface
evaporation and thermocapillary convection.
2. THEORETICAL MODEL AND NUMERICAL METHOD
The annular pool with inner radius ri, outer radius ro and depth d is filled with cold water with Prandtl number of 12,
as shown in Fig. 1. The aspect ratio and the radius ratio of the annular pool are respectively defined as ε=d/(ro−ri)
and =ri/ro. To ensure that the height of the interfacial liquid level is constant during the evaporating process, we
assumed that the bottom of the liquid pool is formed by the insulating porous material, which can allow the liquid to
supply slowly and evenly from the bottom. For simplicity, the following assumptions are introduced: (1) the free
surface is flat, non-deformable; (2) the fluid is an incompressible Newtonian fluid, the velocity is small and the flow
is axisymmetric laminar; (3) all physical property parameters are constant except that the surface tension changes
linearly with temperature; (4) the evaporating heat flux is far greater than that of natural convection on the free
surface.
Fig. 1 physical model
Furthermore, assumed that there is only pure vapor of the liquid above the free surface. At the initial time, the liquid
in the pool, vapor above the free surface and all boundaries of an annular pool are in saturation temperature Ts under
corresponding pressure. Then, the outer wall temperature is suddenly increased to the temperature value Th which is
a little higher than the saturation temperature Ts, and the radial temperature difference T=ThTs will induce
thermocapillary convection from the outer wall to the inner wall along the free surface. Simultaneously, due to the
temperature at the free surface exceeds the saturation temperature under corresponding the vapor pressure, the
thermodynamics non-equilibrium effect will lead to surface evaporation, and the evaporation mass flux j is
determined by the linear approximation Hertz -Knudsen equation [13]:
M
(1)
j   v h fg
(Ti  Ts )
2 RTs3
where,  is accommodation coefficient and determined by the experiment. M is vapor molar mass, R universal gas
constant, hfg latent heat of vaporization, and v vapor density.
Scale quantities for length, velocity, time and pressure are expressed as (rori), (rori), (rori)2and /(rori)2,
respectively. The non-dimensional controlling equations are described as follows:
 V  0
(2)
V
 V  V  P  2V

(3)
2

1 2
 V   


Pr
(4)
Here V is the dimensionless velocity,  the dimensionless time and P the dimensionless pressure, the
dimensionless temperature, =(TTs)/(ThTs), and Prandtl number Pr=/,
The boundary conditions are listed as follows:
At the inner cylinder (R=Ri=ri/(rori)=/(1), 0≤Z≤):
(5)
U  V  0,  0 .
At the outer cylinder (R=Ro=ro/(rori)=1/(1), 0≤Z≤):
U  V  0,  1 .
(6)
At the free surface (Z=, /(1)<R<1/(1)), the radial velocity U depends on force equilibrium effect,
U
Ma 
  ro  ri   v h 2fg
T
Ja
M

,   
,
(7)
Bi 

Bi 
  Biev , V 
Z
Pr R
 h fg ev Z  Pr ev Z 
Z

2 RTs3
where, Ma is Marangoni number, Ma=TT(rori)/(), T surface tension temperature coefficient, and  dynamic
viscosity. Biev is evaporation Biot number and Ja is Jacob number.
At the bottom (Z=0, /(1)<R<1/(1)):

Ro
2 1    Ja
U  0,
0, V  
(8)
Biev   Z  RdR .
R
Z
i
1    Pr
The governing equations and the boundary conditions are discretized by the finite volume method. The diffusion
term is divided into the central difference, and the convection term is the QUICK scheme. The correction for
pressure and velocity term is adopted by the SIMPLE algorithm. In the iterative process, when the maximum
relative error of temperature and velocity is less than 10-5, the solution is convergent. Non-uniform staggered grids of
200R×50Z which are encrypted near the solid wall and the free surface are applied.
3. RESULTS AND ANALYSIS
When evaporation occurs on the free surface, the endothermic effect of evaporation is bound to change the surface
temperature distribution, which can affect the thermal capillary force and change the structure of thermocapillary
convection. When Marangoni number is small, the thermal capillary force on the free surface is small; when the
surface evaporation effect is not considered, counterclockwise thermocapillary flow cells driven by the thermal
capillary force occupies the liquid pool, but the flow intensity is weak. Therefore, the flow field has little influence
on the temperature field, and the isotherms showed almost the conductivity temperature distribution. When
evaporation occurs on the free surface, the surface temperature gradually decreases with the increase of evaporation
Biot number, the temperature gradient near the inner wall is obviously reduced, the thermal capillary force is
weakened, the flow slows down, and the thermal fluid from the outer wall begins to return before it reaches the cold
wall. Therefore, the thermocapillary flow cell shrinks to the outer wall. However, the temperature gradient near the
outer wall increases, the thermal capillary force increases and the flow enhances. At this time, the temperature
gradient in the vicinity of the outer wall continues to increase, but in the narrow flow space, the flow cell intensity is
reduced. It can be known that, when the Marangoni number is smaller, the thermocapillary convection is weak, and
the evaporation plays a leading role.
When Marangoni number is larger, without the evaporation, the thermocapillary convection cells also occupy the
entire pool, and the difference is that the two cells can be formed in the mainstream cell; at this time, the flow is
stronger, therefore, the flow field has a great influence on the temperature field, the isotherm in the vicinity of the
free surface is tilted inward, and the isotherms at the bottom of the liquid pool are inclined to the outside wall. When
evaporation is present on the surface, if the Biot number is small, then thermocapillary convection is still dominant;
for example, when Biev=10, the flow cell is still full of the whole liquid pool, and the flow will be strengthened with
the increase of the temperature gradient near the outer wall. It should be noted that the endothermic effect of
evaporation on the free surface makes the isotherms near the free surface be outward-inclined. When evaporation
Biot number is larger, the evaporation effect will become apparent; for example, when Biev=100, the flow cell will
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shrink to about 1/2 of the width of the liquid pool, and the temperature drop occurs mainly in the vicinity of the outer
wall.
At a fixed aspect ratio, the greater Marangoni number is, the stronger thermocapillary convection from outside to
inside along the free surface is and the larger the inward radial flow speed is, which will take more heat to the free
surface, so the higher the surface temperature is. However, when the evaporation Biot number is large and surface
evaporation is very strong, the evaporation effect is dominant; therefore, the temperature drop and the fluid flow
only occur near the outer wall, while the radial temperature gradient near the inner wall is almost zero, and the flow
is in a static state. When evaporation Biot number is specified, as the aspect ratio increases, the liquid layer thickness
and the flow space increase, and the flow with the same Marangoni number is stronger; therefore, the greater the
inward radial velocity is, the higher the surface temperature is.
Due to the gradual rise of the temperature from the cold wall to the hot wall along the free surface, evaporation is
gradually strengthened, the dimensionless evaporation mass flux increases. Near the outer wall, the dimensionless
evaporation mass flux is mainly affected by the evaporation effect, and therefore, it increases with the increases of
evaporation Biot number; away from the hot wall, the dimensionless mass flux is influenced by the couple of
evaporation and thermocapillary convection, which makes the dimensionless evaporation flux increase with the
increase of evaporation Biot number in small evaporation Biot number; conversely, then decreases. The
dimensionless total evaporation mass rises with the increase of evaporation Biot number, but the growth slows down
gradually. As the aspect ratio increases, thermocapillary convection is strengthened, the dimensionless total
evaporation mass will grow.
4. CONCLUSIONS
Through a series of numerical simulations on thermocapillary convection for the moderate Prandtl number fluids in
an annular shallow pool with the evaporating free surface, the results show that: (1) with the increase of Biot number,
thermocapillary convection cells gradually shrink to the outer wall, the flow area is gradually reduced, but the flow
intensity first increases and then decreases; (2) near the outer wall, the dimensionless mass flux rises with the
increase of evaporation Biot number, and near the inner wall, dimensionless evaporation flux is affected by the
coupling effect of evaporation and thermocapillary convection; (3) with the increase of evaporation Biot number, the
total evaporation mass on the free surface grows; (4) as the aspect ratio of the liquid pool increases, thermocapillary
convection is enhanced, the dimensionless total evaporation mass is amplified.
ACKNOWLEDGMENT
This work is supported by National Natural Science Foundation of China (Grant No. 11532015).
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