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Theoretical approach for a pressure drop in two-phase particle-laden flows

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Theoretical approach for a pressure drop in two-phase particle-laden flows
☆
Seyun Kim
a
, Kye Bock Lee
b,
⁎
, Chung Gu Lee
b
, Seong-O Kim
a
a
Fluid Engineering Division, Korea Atomic Energy Research Institute, 150 Dukjin-dong, Yuseong-gu, Daejeon, 305-353, Korea
b
School of Mechanical Engineering, Chungbuk National University, 12 Gaeshin-dong, Heungduk-gu, Cheongju, Chungbuk, 361-763, Korea
Available online 15 December 2006
Abstract
The purpose of this research is to develop an analytical model for a pressure drop per unit pipe length due to the turbulencemodulations of a carrier phase which results from the presence of a dispersed phase in various types of diluted two-phase flows.The wake behind a particle, a particle size, the loading ratio and the density difference between two phases of a particle-laden flowwere considered as significant parameters, which have an influence on the turbulence of a particle-laden flow, and the relativevelocity of the laden particles was calculated by using a terminal velocity. The frictional pressure drop was formulated by using theforce balance in the control volume by considering the shear stresses due to the presence of particles and an analogy of the shear stresses in the solid surfaces. The numerical results show a good agreement with the available experimental data and the modelsuccessfully predicted the mechanism of the pressure drop in the particle-laden flows.© 2006 Elsevier Ltd. All rights reserved.
Keywords:
Particle-laden; Gas-particle; Pressure drop; Turbulence modulation
1. Introduction
A discrete flow means the state at which the second phase forms a flow in a continuous first phase flow. Continuousliquid and discrete gas form a bubbly flow, continuous gas and discrete liquid create a droplet flow, and a solid particleflow in a gas composes a particulate flow. In a liquid metal reactor the reaction products of supercritical carbon dioxideand sodium form gas-particle flows. Since a particle-laden shear flow has occurred frequently in industrial processes,many researchers have paid considerable attention to this phenomenon and lots of experimental researches andnumerical analyses have been performed. In spite of the many researches, a model that can simulate precisely the particle-laden flow characteristics has not been developed as yet due to the complexity of the physical phenomena.Thus, there are certain peculiarities and inconsistencies in numerical researches about the turbulence aspects of two- phase flows.The early experimental studies were mostly focused on a measurement of the pressure drop and the velocitydistribution to develop the relevant correlations. Recently, by using up-to-date technologies, i.e. LDVand PIV, studiesto analyze the fluid
–
particle interactions and correlations have been performed by turbulence measurements.
International Communications in Heat and Mass Transfer 34 (2007) 153
–
161www.elsevier.com/locate/ichmt
☆
Communicated by Dr. W.J. Minkowycz
⁎
Corresponding author.
E-mail address:
kblee@chungbuk.ac.kr (K.B. Lee).0735-1933/$ - see front matter © 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.icheatmasstransfer.2006.11.002
These theoretical studies have been conducted to develop a turbulence model to explain the particle-laden effect on the turbulence flow by using experimental results. Several of the analysis models proposed by a few researchersfor a particle-laden flow, based on correlations of the experimental data, show limitations in their applications because of an insufficient comprehension of the physical phenomena of the particle-laden flows. To examine theeffect of a laden particle size, Gore and Crowe [1] adopted a length scale ratio between the particle diameter,
d
p
andturbulent length scale,
l
, which can be considered as the eddy size of an energy transportation. They showed arelationship between the ratio and the turbulence by using previous experimental data. When the ratio,
d
p
/
l
is smaller than 0.1, the turbulence is attenuated. And when the ratio is larger than 0.1, the turbulence is augmented. Hetsroni [2] proposed that a turbulence is increased in the condition where the particle Reynolds number,
Re
p
is about O (0.1),and a turbulence is suppressed when the particle Reynolds number is about O (1000), and the intermediate effect occurs in
Re
p
∼
O (100), by utilizing the experimental data of Tsuji et al. [3]. The particle Reynolds number isdefined by the particle diameter and the relative velocity between the phases. Kenning and Crowe [4] suggested asimple model for a generation and an extinction of a turbulence kinetic energy by considering a turbulence lengthscale for the space between particles. However, these models need a lot of experimental data. Kim et al. [5] derivedsource terms of a turbulence kinetic energy based on the physical phenomena and developed, not a correlation of theexperimental data but a physical model for turbulence intensity by using the mixing length model and the equilibriumflow conditions.For industrial applications, a pressure loss (or drop) is a significant design parameter. To calculate a pump capacityand to design a pipe system, a precise analysis of a system pressure distribution is essential. A pressure gradient isexpressed as a summation of the frictional, gravitational and acceleration components in a single phase flow [6]. Thesecomponents are attributed to different physical effects respectively. For a two-phase flow, the gravitational andacceleration pressure gradient terms contain a void fraction á through an analysis of the experimental data. For theinternal two-phase flow, the frictional pressure loss term is usually dominant when compared to the other terms. Toobtain a pressure loss, a simple modification of the correlation coefficient and a regression of the experimental datahave been carried out such as in the study of Beattie and Whalley [7]. For this reason, because of the experimentalconditions, the reliability of the correlation is decreased. Therefore, instead of merely a curve fitting of the limitedexperimental data of a particle-laden flow, a theoretical model of a pressure drop based on an investigation of the physical mechanisms of a turbulence variation due to laden particles is required.In the present research, to obtain the pressure drop per unit length in a particle-laden flow, a theoretical basis for the prediction of a pressure drop in two-phase flows has been proposed and shown to be useful in providing the design datafor gas-particle flow systems.
2. Theoretical analysis
To obtain a pressure drop due to laden particles, the force balance of each phase in a unit volume of a particle-ladenflow is arranged as follows. The conceptual drawing is shown in Fig. 1.
s
e
i
þ
a
D
p
D
l
−
aq
p
g
¼
0
ð
1
Þ
−
s
e
i
þð
1
−
a
Þ
D
p
D
l
−
ð
1
−
a
Þ
q
f
g
−
s
e
w
¼
0
ð
2
Þ
Where,
α
is the volume fraction of a particle, and
τ
˜
i
is the shear stress between the phases in a unit volume, definedas follows.
s
e
i
¼
2
C
d
aq
f
d
p
j
U
r
j
U
r
ð
3
Þ
In Eqs. (1) and (2), the shear stresses are eliminated as follows.
D
p
D
l
−
aq
p
g
−
ð
1
−
a
Þ
q
f
g
þ
s
e
w
¼
0
ð
4
Þ
154
S. Kim et al. / International Communications in Heat and Mass Transfer 34 (2007) 153
–
161
The relationship between the wall shear stress per unit area,
τ
w
and the wall shear velocity,
u
w
⁎
can be expressed asEq. (5) by a definition.
u
*
w
¼
ﬃﬃﬃﬃﬃ
s
w
q
f
r
ð
5
Þ
Total shear stress is a summation of the wall shear stress and the shear stress between the phases. However, when theladen particles are diluted, the total shear stress can be approximated by the wall shear stress and the wall shear stresscan be expressed by using the turbulent fluctuation velocity in turbulent flows [8].
u
*
w
c
u
V
ð
6
Þ
Therefore,
s
w
¼
q
f
u
V
2
ð
7
Þ
This result is substituted into the equation for the force balance per unit pipe length, Eq. (4) as follows.
D
p
D
l
¼
aq
p
g
þ ð
1
−
a
Þ
q
f
g
þ
C
s
4
D
q
f
u
V
2
ð
8
Þ
Awall shear stress is generated not only in a pipe wall but also on a particle surface where a continuous flow passesover it. Accordingly, by considering the total wall surface area and the particle surface area in a unit volume, theequation can be rearranged.
D
p
D
l
¼
aq
p
g
þ ð
1
−
a
Þ
q
f
g
þ
C
s
q
f
u
V
2
1
þ
32
a
Dd
p
−
1
4
D
ð
9
Þ
where,
C
τ
is an empirical constant. From Eq. (9), the relationship between the turbulent fluctuation velocity and the pressure drop can be derived.To develop an analytical model for an estimation of the turbulent fluctuation velocity in a particle-laden flow,an approximation of the equilibrium flow is adopted. In a fully developed particle-laden flow in a straight pipe,the lift force of a particle due to a velocity gradient is relatively small in the volume-averaged turbulence kineticenergy equation, and the effect is negligible. The approximation of an equilibrium flow is valid in a core region of a fully developed pipe flow. In this case, the convection term and the diffusion term can be ignored, and the
Fig. 1. Particle-laden flow in a pipe.155
S. Kim et al. / International Communications in Heat and Mass Transfer 34 (2007) 153
–
161
source term and the sink term should be balanced. Consequently, an equation of the turbulence kinetic energy can be written as
aq
p
f
s
r
j
U
i
−
V
i
j
2
−
ð
1
−
a
Þ
q
f
u
i
u
j
P
A
U
i
A
x
j
þ
aq
p
f
s
r
ð
m
i
m
i
P
−
u
i
m
i
P
Þ
−
ð
1
−
a
Þ
q
f
e
¼
0
ð
10
Þ
The first term is a work for the carrier fluid by the drag caused by an acceleration and a turbulence kinetic energysource due to velocity defects in the wakes behind the particles, and this term is expressed as a velocity difference between the particles and the carrier flow [4]. To model the relative velocity between the phases, to calculate therelative velocity, the terminal velocity, which is derived from the kinetic equation of a free-falling particle, is supposedto be an approximated relative velocity between the phases [5].
U
r
¼
V
T
¼
g
s
r
f
¼
q
p
d
2
p
g
18
l
d
f
ð
11
Þ
f
is the drag factor. When a drag is not a Stokes drag, this factor supplements the differences between a Stokes drag anda particle drag. In the present study, a general use of the Schiller model for the drag factor was adopted [8].
f
¼
1
:
0
þ
0
:
15
Re
0
:
687
p
ð
12
Þ
Accordingly, the first term of Eq. (10) can be modeled as follow.
aq
p
f
s
r
j
U
i
−
V
i
j
2
i
aq
p
f
s
r
V
2
T
ð
13
Þ
The second term of the left hand side of Eq. (10), which is the source term of the velocity gradient, must contain theeffect of the fluctuations of a flow caused by the wake behind the laden particles. The schematic of the wake flow and anotation are depicted in Fig. 2.Velocity profiles in the wake made behind a particle in a uniform flow have a similarity function in the downstreamof a particle [9]. The source term due to the velocity gradient can be expressed as Eq. (14) with the mixing lengthmodel.
−
u
i
u
j
P
A
U
i
A
x
j
¼
l
2
j
A
U
i
A
x
j
j
A
U
i
A
x
j
2
ð
14
Þ
Fig. 2. Schematic of a wake flow.156
S. Kim et al. / International Communications in Heat and Mass Transfer 34 (2007) 153
–
161
Where, the velocity gradient is described with a similarity function,
U
d
−
U U
d
−
U
m
¼
U
l
U
lm
¼
f
ð
g
Þ
;
g
¼
y
d
ð
15
Þ
and the function is given by
f
(
η
)=1
−
η
3/2
[10].
A
U
A
y
¼
A
U
A
g
A
g
A
y
¼
U
lm
d
f
V
ð
g
Þ ð
16
Þ
Therefore
−
u
i
u
j
P
A
U
i
A
x
j
¼
l
2
U
3lm
d
3
½
f
V
ð
g
Þ
3
¼
b
2
U
3lm
d
½
f
V
ð
g
Þ
3
ð
17
Þ
U
lm
¼
Ax
−
2
=
3
;
d
¼
Bx
1
=
3
ð
18
Þ
The dimensionless length scale
β
=
l
/
δ
is the ratio of the turbulent length scale for a half width of the wake, which is afunction of the streamwise distance. To calculate the average value of the source of the velocity gradient, this term isintegrated for the wake length and the wake width
Z
l
w
0
Z
d
0
278
l
2
U
3lm
d
3
g
3
=
2
y
d
y
d
x
¼
Z
l
w
0
2728
b
5
l
3
U
3lm
d
4
d
x
¼
487
A
3
B
b
2
l
−
2
=
3w
ð
19
Þ
Now, the generation of a turbulent kinetic energy of a wake per unit volume is
l
2
U
3lm
d
3
½
f
V
ð
g
Þ
3
¼
1207
A
3
b
2
B l
−
7
=
3w
ð
20
Þ
Where, as Yarin and Hetsroni proposed [11], the length of a wake is
l
w
¼
d
p
X
3
q
p
q
f
c
!
1
=
3
2435
ð
21
Þ
In Eq. (21),
Ω
is the order of unity determined by the experiments. The third term which means a redistributionof the kinetic energy can be neglected because it is small enough near the center of the pipe flow [12]. The last termis the dissipation rate of the kinetic energy and this term can be modeled with the kinetic energy and the
Fig. 3. Comparisons of the non-dimensional pressure drop with the experimental data.157
S. Kim et al. / International Communications in Heat and Mass Transfer 34 (2007) 153
–
161

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