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Proceedings of the Asian Conference on Thermal Sciences 2017, 1st ACTS
March 26-30, 2017, Jeju Island, Korea
ACTS-P00162
A MODIFIED MODEL FOR THE PREDICTION OF BUBBLE
DEPARTURE DIAMETER AND BUBBLE LIFT-OFF DIAMETER IN
FLOW BOILING
Xiao Renjie1, Yan Xiao1, Zan Yuanfeng1
1
CNNC Key Laboratory on Nuclear Reactor Thermal Hydraulics Technology, Nuclear Power Institute of
China, Chengdu, 610041, PR China

Xiao Renjie: [email protected]
*
Yan Xiao: [email protected]
ABSTRACT
Force analysis on a bubble at heated wall is performed in this paper, which shows that growth force is the
key factor to influence bubble detachment. A modified model is developed for the prediction of bubble
departure diameter and bubble lift-off diameter by changing growth force model. Predictions are
validated against the databases of Tomio Okawa and Sugrue, and satisfactory accuracy is obtained for
bubble detachment size at heated wall in flow boiling. Furthermore, with the increase of wall superheat,
growth force increases to restrain bubble lift-off, which increases the bubble growth time, and bubble
grows faster, causing bubble lift-off diameter increases with increasing wall superheat. Also, the quasisteady drag force increases with increasing mass flux to push bubble departure, which decreases the
bubble residence time, and bubble grows slower at the same time, resulting in decreasing bubble
departure diameter along with the increase of mass flux.
KEYWORDS: Flow boiling; Growth force; Bubble departure diameter; Bubble lift-off diameter;
Theoretical analysis
1. INTRODUCTION
Nucleate boiling and two-phase flow has achieved comprehensive applications in nuclear energy,
aerospace, oil and chemical industry, and the micro process of boiling gets more and more attention from
science and industry fields. The process of boiling involves bubble nucleation, bubble growth, bubble
departure, bubble sliding and lift-off, which disturb the flow boundary layer and thermal boundary layer
and influence the heat transfer. In this paper, departure means the movement of bubble departs from
nucleation site, and lift-off represents bubble leaving the heated surface. Bubble departure diameter (Dd)
and bubble lift-off diameter (Dlo) correspond the size of bubble when departure and lift-off happens,
respectively, which have been the most important sub-models in theory analysis and computational fluid
dynamics recently. Fritz[1] first established bubble detachment diameter model by solving the equation of
forces acting on a bubble. Klausner et al[2], Zeng et al[3], Al-Hayes et al[4] and JianJun Xu et al[5] has
studied the departure diameter and lift-off diameter of a bubble growing at heated wall. In this paper,
attention is focused on the force analysis of a bubble growing at heated wall, and a mechanistic model of
bubble detachment diameter is developed by evaluating the magnitude of different forces and modifying
the bubble growth force model. Comparisons between predictions and experiments are carried out to
verify the validity of mechanistic model.
2. THEORETICAL ANALYSIS OF BUBBLE DETACHMENT DIAMETER
Consider bubble growth at a heated surface and the schematic diagram of the forces acting on the bubble
is shown in Fig.1. The net forces in x-and y-are given as following:
(1)
 Fx  Fqs  Fsx  Fb sin   Fdu sini
F
y
 FsL  Fh  Fcp  Fsy  Fb cos   Fdu cosi
1
(2)
Fig. 1 Schematic diagram of the forces acting on a bubble in flow boiling
where Fqs, Fs, Fb, Fdu, FsL, Fh and Fcp are the quasi-steady drag force, the surface tension force, the
buoyancy, the bubble growth force, the shear lift force, the hydrodynamic pressure force and the contact
pressure force, respectively. The surface inclination angle is , and bubble inclination angle is θi.
At the moment of departure, the bubble will move at the direction of x+, which indicates that the equation,
 Fx  0 , is just achieved. As a result, we make the assumption that bubble departure happens when
F
 F becoming zero is the condition of bubble lift-off. Dd and Dlo can
be obtained in solving the equations,  Fx  0 and  F  0 , respectively.
x
becomes zero. Similarly,
y
y
2.1 EVALUATION AND PREDIGESTION OF THE FORCES
According to Cole’s research, the bubble growth force and the buoyancy are the most significant forces
when bubble is leaving the nucleation site[6]. And that bubble growth force is the only model relating to
heat flux and wall superheat, which indicates the influence of heated wall on bubble behavior directly.
Hence, the validity of bubble growth force model always decides the prediction accuracy of bubble
departure diameter and bubble lift-off diameter. In order to confirm the significance of bubble growth
force, simplify the forces model acting on bubble and elide the unimportant forces, comparisons between
the forces acting on bubble have been made using the departure condition of Sugrue’s experiment[7] and
lift-off condition of Tomio Okawa’s work[8]. The bubble upstream contact angle and downstream
contact angle take the value of JianJun Xu[5], and the bubble contact diameter is given[9] by dw=Rb/30.
In consideration of bubble departure and lift-off, bubble radius is written as Rb=Dd/2 or Rb=Dlo/2.
When bubble lift-off happens, bubble is leaving the heated wall, which leads to dw~0, so Fsy, Fh and Fcp
are elided. The values of remaining forces are calculated in Table 1.
Table 1 Forces acting on a bubble for bubble detachment.
Bubble condition
Fqs/10-6N
Fsx/10-6N
Fb/10-6N
Fdu/10-6N
FsL/10-6N
Bubble departure
0.4591
-0.0899
1.1603
3.4434
0.1286
Bubble lift-off
1.5721
-0.1105
2.2694
119.8490
1.0837
According to Table 1, the bubble growth force is larger than any other force obviously, which means
bubble growth force is very important for bubble departure and lift-off. Comparing the ratio of each force
at bubble lift-off to the force at bubble departure, the ratio of bubble growth force is about 35, which is so
great that Zeng’s bubble growth force model may be inaccurate and need to be improved.
2.2 THE IMPROVEMENT OF BUBBLE GROWTH FORCE
n2
The acceleration of bubble moving at heated wall is given by a  nB d (2n  1) . The constant n is the
exponent of time. Based on [10], it has been fixed to 0.5, which means the total bubble acceleration is
zero and growth force is relation to the liquid around bubble. According to the virtual mass force theory
of Chen[11], the bubble growth force is the drag force of the virtual mass around bubble, and Chen gives
3
the virtual volume of a spherical bubble as: Vl  11 Rb /12 .
2
According to the theorem of momentum in the direction of bubble growing,  i , bubble growth force is
Fdu  d ( lVlU i ) / dt .where Ui is the velocity in bubble growing direction, Ui=2dRb/dt, arrange to get
bubble growth force as:
11
 11

Fdu  l Rb2  Rb2  Rb Rb 
2
6


(3)
As to the bubble growth rate model, Rb(t)=f(t), what is commonly used in bubble force analysis is
Rb (t )  AJa t , or Rb (t )  AJa Bt , which is derived from the growth rhythm of the bubble in
superheated liquid. The original form of Jacob number is:
Ja  l c pl (Tl  Tsat ) /  g h fg
(4)
With regard to the bubble growing at heated wall, suppose Tl≈Tw, which takes consider wall temperature
into bubble growth rate model. In flow boiling, the above theory just refers to the thermal effect which
heated wall acts on growing bubble, but do not involve the dynamic effect. The process of bubble growth
is the result of wall flow boundary layer and thermal boundary layer combined action to, so it is not
accurate to make the assumption, Tl≈Tw, to represent the effect that heated wall acts on bubble. Prandtl
number (Pr) is the dimensionless number which reflects the relative value between flow boundary layer
and thermal boundary layer directly, which also reflects the comparison between momentum diffusion
and thermal diffusion of the liquid layer bubble grows in.
Marco Colombo[12] thinks about the evaporation of micro fluid layer at bubble bottom, analyzes on the
basis of Zuber’s model, and takes into account Pr number to express the influence of flow boundary layer
and thermal boundary layer. The expression is:
2
Rb (t )  Pr 0.5 Ja  t
(5)
c2
Here, the constant c2 is between 0.8~1.78. Using the bubble growth force model and bubble growth rate
model established in this section, we can calculate the bubble growth force in Table 1, which are
1.6837×10-6N 和 1.2253×10-5N respectively. They are still the maximal forces, and the ratio is about 7.3,
which is more reasonable than Zeng’s model.
3. MODEL VALIDATION
According to the analysis of previous sections, we make the assumption that the bubble contact diameter
is zero at lift-off, so Fsy, Fh and Fcp are omitted at lift-off direction. At the same time, the bubble
inclination angle is zero at lift-off, 10 degree at departure. This section takes the experimental data of
Tomio Okawa and Sugrue o validate the modified model.
Fig.2 compares the modified model predictions, Zeng’s model predictions against the experimental data.
Fig. 2 Dlo diameter predictions of the modified model (a) and Zeng’s model (b) compared against
experimental data of Tomio Okawa.
According to Fig. 2, the error of modified model predictions is almost within 30% , while Zeng’s
model is bigget relatively.
In Fig. 3, bubble lift-off diameter predictions are shown for an increasing value of wall superheat. The
modified model predictions conform to experimental data quite well, while Zeng’s model overestimates
bubble lift-off diameter obviously. According to the trends of predictions, bubble lift-off diameter
3
increases with increasing wall superheat. Fig. 1 shows that bubble growth force restrains bubble departure
and lift-off. Bubble growth force increases with increasing wall superheat, which intensifies the inhibiting
effect, so it will take longer time for bubble to lift-off. At the same time, the increase in Jacob number
with increasing wall superheat is evident, which causes bubble growth rate increase and bubble grows
faster. Finally, bubble lift-off diameter increases.
As to some error of modified model predictions are beyond 30% , the reason maybe the omission of Fsy,
Fh and Fcp. When bubble lift-off, taking the dw=Rb/30, the value of Fsy is -5.07893×10-6N and 6.24138×10-6N in Table 1, and the order of Fh, Fcp is 10-7N, which indicates these forces can’t be omitted
simply when bubble lift-off.
Fig. 3 Predicted Dlo diameter of modified model
and Zeng’s model variation with wall superheat.
Fig. 4 Comparison between modified model
predictions and experimental data of Sugrue.
Fig.4 compares the modified model predictions against the experimental data of Sugrue. The error of
modified model predictions is within 30% completely, which shows the predictions of bubble
departure diameter are very well. We can conclude the reason why the predictions of bubble departure
diameter are better than that of bubble lift-off diameter is that the predicted value of departure diameter is
derived from thinking over all forces at departure diction, while some forces are omitted at lift-off
direction.
In Fig. 5, bubble departure diameter predictions of the modified model are shown for an increasing value
of mass flux. The decrease in bubble departure diameter with increasing mass flux is evident, and the
decreasing trend is slowing down. Liquid velocity increases with the increase of mass flux, which leads to
the rise of speed difference between bubble centroid and liquid. As a result, the drag force increases and
accelerates the bubble departure, which reduces the bubble residence time at nucleation site. At the same
time, the temperature of superheated liquid layer where bubble grows depresses with increasing mass flux,
which reduces the bubble growth speed. Hence, bubble departure diameter decreases.
Fig. 5 Predicted bubble departure diameter of modified model variation with mass flux.
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4. CONCLUSIONS
Establishing theoretical model of bubble departure diameter and lift-off diameter is one of the core tasks
to research bubble behavior. In this paper, force analysis of a single bubble at heated wall is carried on.
Comparisons between different forces are made, which indicates that bubble growth force is one of the
most important factors resulting in bubble departure and lift-off. A modified model for the prediction of
bubble departure and lift-off diameter is established by applying the modified bubble growth force model
and bubble growth rate model. Quantitative confirmation between predictions of bubble detachment size
and two experiment data is carried out, and satisfactory accuracy is reached that the average relative error
is within ±30%.
Furthermore, with the increase of wall superheat, growth force increases to restrain bubble lift-off, which
increases the bubble growth time. At the same time, the increase in Jacob number with increasing wall
superheat is evident, which causes bubble growth rate increase and bubble grows faster. Finally, bubble
lift-off diameter increases with increasing wall superheat. Also, the quasi-steady drag force increases with
increasing mass flux to push bubble departure, accelerating the bubble departure, which reduces the
bubble residence time at nucleation site. At the same time, the temperature of superheated liquid layer
where bubble grows depresses with increasing mass flux, which reduces the bubble growth speed. Hence,
bubble departure diameter decreases along with the increase of mass flux.
NOMENCLATURE
Dd
F
θi
Rb
cp
hfg
Ja
bubble departure diameter
force
bubble inclination angle
bubble radius
specific heat
latent heat
Jacob number
(m)
(F)
(°)
(m)
(J/kgK)
(J/kg)
(-)
Dlo


dw
Tw
η
Pr
bubble lift-off diameter
(m)
surface inclination angle
(°)
density
(kg/m3)
bubble contact diameter
(m)
wall superheat
(K)
thermal diffusion coefficient (m2/s)
Prandtl number
(-)
REFERENCE
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