Proceedings of the ASME 2014 12th International Conference on Nanochannels, Microchannels, and Minichannels
ICNMM2014
August 3-7, 2014, Chicago, Illinois, USA
ICNMM2014-21920
Pressure Drop Analysis Using the Homogeneous Model for Open
Microchannel with Manifold (OMM)
Ankit Kalani, Satish G. Kandlikar*
Department of Mechanical Engineering
Rochester Institute of Technology
Rochester, NY, USA
*[email protected]
ABSTRACT
Flow boiling in microchannels has the ability to dissipate
high heat fluxes due to the associated small hydraulic diameter
and latent heat effects. However, flow instabilities and early
critical heat flux have often limited the heat transfer
performance of such systems. In a previous study, the open
microchannel with manifold (OMM) design was introduced to
address these issues. Low pressure drop at high heat flux were
obtained with this configuration. In this work, theoretical
modeling of pressure drop for the OMM geometry with a
uniform and a tapered manifold is undertaken. Applicability of
the homogeneous model is evaluated using seven different
viscosity averaging schemes. Experiments were performed with
two test sections (one plain and one with open microchannels)
and with four different manifolds (one uniform and three
tapered). All experimental data with various configurations
were compared with the different viscosity models. The
viscosity model of Owen et al. predicted the highest value of
pressure drop, while the lowest value was obtained with that of
Dukler et al. All models underpredicted for uniform manifold
with plain and microchannel chips with an average MAE of
50%. For tapered manifolds, plain chip underpredicted, while
good agreement was obtained with microchannel chip for
McAdams et al. and Akers et al.
INTRODUCTION
Flow boiling in microchannels has been extensively
studied over the past two decades. It is considered to be an
attractive cooling technique due to its ability to dissipate high
heat fluxes while maintaining a low wall superheat. In addition
to the excellent thermal performance, low coolant inventory,
small hydraulic diameter and compactness are also added
benefits. However, early critical heat flux (CHF) [1], flow
instability [2] and low heat transfer coefficient [3] have
adversely affected its performance. Artificial nucleation sites
[4], inlet/exit restrictors [5], diverging microchannels [6],
microgap [7,8] and stepped microchannels [9] were some of the
techniques employed by various researchers to counter the
above mentioned disadvantages. A detailed literature review on
various techniques investigated by different researchers to
reduce pressure drop is presented in a previous publication
[10]. In the current work, the literature review is focused on the
predictive pressure drop models and correlations of two-phase
flow in microchannels. Pressure drop modeling can be
classified into three approaches [11], the homogeneous
equilibrium model, separated flow model, and theoretical flow
regime specific models.
The homogeneous model [12] is based on the assumption
of equal liquid and vapor velocities. It is one of the most widely
used models [13-20] and has shown to provide consistently
reasonable (within about 30 to 40 percent) prediction of
pressure drop over a wide range of parameters for flow patterns
that do not have widely differing individual phase velocities.
The separated flow model allows different velocities for the
liquid and vapor phases. The separated flow model based on the
Lockhart–Martinelli method [21] has been widely used. Several
correlations [22-25] have been recommended on the LockhartMartinelli method for microscale applications, where the C
parameter in the original model has been modified according to
the geometry used by the researchers. The theoretical models
[26-29] presented in literature have focused on the annular flow
regime. These models have shown the ability to provide good
agreement with various geometries used by different
researchers.
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In the present work, the homogeneous model was used for
pressure drop prediction. Seven viscosity averaging schemes
were applied in the model. Plain and microchannel chips were
used as the boiling surfaces with distilled water at one
atmospheric pressure. A flow rate of 80 mL/min was
maintained for all test runs. Four manifolds were used, one
uniform and three tapered (200 µm, 400 µm, and 600 µm taper
height across a 10 mm flow length). Pressure drop
performances of all configurations are compared with the
homogeneous model.
meter. The flow rate was controlled through a rotameter and
was set to 80 mL/min for this work. The water then entered an
inline heater. A subcooling of 10°C was maintained at the
entrance to the test section. A throttle valve was present at the
upstream of the test section. The heated water and steam
coming out from the test section was sent back to the reservoir.
NOMENCLATURE
G
ṁ
Ac
f
ρ
um
L
D
Dh
Po
Re
P
Kc
z’
Aca
Ap
Q
W
∆TSub
Cp
vf
vg
x
dx
dz
dA
dz
µl
µv
hfg
q”
dp
dz
Mass flux, kg/s
Mass flow rate, kg/m2s
Cross-sectional area, m2
Fanning frictional factor
Density, kg/m3
Average velocity, m/s
Length, m
Diameter, m
Hydraulic diameter
Poiseuille number
Reynolds number
Wetted perimeter, m
Contraction loss
Single phase length, m
Channel area, m2
Total plenum cross-sectional area, m2
Total heat transferred, W
Channel width, m
Degree of subcooling, °C
Specific heat
Specific volume of liquid, m3/kg
Specific volume of gas, m3/kg
Exit quality
Change of quality w.r.t channel length
Change of c/s area w.r.t channel length
Liquid viscosity, Pa-s
Vapor viscosity, Pa-s
Latent heat of vaporization, kJ/kg
Heat flux, W/m2
Pressure drop, kPa
EXPERIMENTAL SETUP
Figure 1 shows the schematic of the experimental flow
boiling setup. A five gallon pressure canner served as the
reservoir for the distilled water. A hot plate was used to
vigorously degas the water for 8 hrs before the start of each test
run. A Micropump was used to drive the water through the
entire loop. A heat exchanger was placed before the Micropump
so as to reduce the temperature of the water going to the flow
Figure 1. Schematic of the flow boiling test loop
Details of the test section, which included the heater and
the manifold configuration, have been discussed in a previous
publication [10]. Four manifolds with various taper heights
were used in the study. The inlet height remained constant for
all test runs, while the exit height changed depending on the
manifold configuration used. Both the inlet and exit heights
were referenced from the top plane of the microchannels to the
top of the manifold. Table 1 shows the details of the manifold
configuration and the mass fluxes obtained for the
microchannel chip.
Table 1. Manifold configuration and mass fluxes at inlet
and outlet
Taper
Inlet
Exit
Ginlet
Goutlet
Manifold
height height height
2
(kg/m s) (kg/m2s)
(µm)
(µm)
(µm)
Uniform
0
127
127
372
372
Taper A
200
127
327
372
238
Taper B
400
127
527
372
175
Taper C
600
127
727
372
138
The mass flux shown in the above table was calculated
using the following equation:
ṁ
𝐺=
(1)
𝐴𝑐
where the cross-sectional area, Ac, was the actual flow area
calculated as the sum of the total microchannel area and the
manifold gap area. The cross-sectional area at the inlet was the
same for all manifolds, while the cross-sectional area at the exit
for the tapered manifold varied depending on the taper height.
2
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𝑧′ =
PRESSURE DROP ANALYSIS
The differential pressure sensors were located at the inlet
and exit regions of the manifold block as shown in Fig. 3. The
pressure drop calculations consisted of the single phase region
(due to the inlet subcooling) and the two phase region. The
single phase region losses were divided into the inlet losses
(pipe and bends) and channel region which consisted of the
frictional pressure losses and the contraction (entrance) losses.
The following equations were used for the calculation of the
single phase pressure drop:
𝑚̇𝐶𝑝 𝛥𝑇
𝑄𝑊
(6)
where W is the channel width, ∆T is the degree of subcooling,
and Q is the total heat transferred.
The two phase region length z, is given by
𝑧 = 𝐿 − 𝑧′
(7)
where L is the total channel length (10 mm) and z’ is the single
phase region length.
The two-phase pressure drop can be characterized as the
sum of frictional, accelerational, and gravitational components.
The gravitational term was zero since the test section was kept
horizontal. The homogeneous model [10] was used for the twophase pressure drop calculations. For the uniform manifold, the
homogeneous model consisted of the frictional and the
accelerational terms as shown in the equation below.
2𝑓𝑇𝑃 𝐺 2 𝑣𝑓
𝑣𝑓𝑔
𝑑𝑥
[1 + 𝑥 ( )] + 𝐺 2 𝑣𝑓𝑔
𝐷ℎ
𝑣𝑓
𝑑𝑧
𝑑𝑃
−( ) =
𝑑𝑣
𝑑𝑧
𝑔
1 + 𝐺2𝑥 (
)
𝑑𝑝
Figure 2. Schematic of the uniform manifold (not to scale)
𝛥𝑃𝑖𝑛𝑙𝑒𝑡 =
𝑢𝑚 =
2
2𝑓𝜌𝑢𝑚
𝐿
𝐷
𝑚̇
,
𝜌𝐴𝑐
𝑃𝑜 = 𝑓𝑅𝑒
(2)
(3)
where f is the fanning friction factor, um is the mean velocity,
and Po is the Poiseuille number. Equation (4) was used to
calculate the single phase pressure drop in the inlet region (pipe
and duct).
Single phase pressure drop in the channel region was
calculated using the equation below which consisted of the
losses due to bends, contraction, and the frictional component.
2
4𝑓𝑎𝑝𝑝 𝑧′
𝜌𝑢𝑚
𝛥𝑃 =
[𝐾𝑐 +
]
2
𝐷ℎ
𝐷ℎ =
4𝐴𝐶
𝑃
(4)
(5)
where Aca is the channel area, Ap is the total plenum crosssectional area, Kc is the contraction losses coefficient due to
area changes, Dh is the hydraulic diameter, P is the wetted
perimeter and z’ is the single phase length.
The distance (z’) at which the water becomes saturated was
calculated by
(8)
where x is the exit quality, vf is the specific volume of the
liquid, vfg is the difference in the specific volume of saturated
liquid and vapor, G is the mass flux, and fTP is the two phase
friction factor.
For the tapered manifold, to compensate for the increase in
the cross-sectional area due to the taper, an additional area term
was added to the frictional and the accelerational terms in the
numerator. A similar equation to that of the uniform manifold
(Eq. 8) was used with the addition of the below term for
pressure drop calculation.
𝑣𝑓𝑔 𝑑𝐴
−2𝐺 2 𝑣𝑓
[1 + 𝑥 ( )]
𝑣𝑓 𝑑𝑧
𝐴𝑐
𝑑𝑃
−(
)=
𝑑𝑣𝑔
𝑑𝑧 𝑡𝑎𝑝𝑒𝑟,𝑎𝑟𝑒𝑎
1 + 𝐺 2𝑥 (
)
𝑑𝑝
(9)
Where A is the cross-sectional area and dA/dz is the change
in cross-sectional area along the channel length (two phase
region).
The two phase friction factor was calculated using the two
phase viscosity. Various researchers have obtained different
models for the two-phase viscosity; the well-established
relationships in literature for viscosity are shown in the table
below.
Table 2. Various two-phase viscosity models used in the
homogenous model [14 -20]
1
Owens et al. [14]
µ𝑡𝑝 = µ𝑓
2
Cicchitti et al. [15]
µ𝑡𝑝 = 𝑥µ𝑔 + (1 − 𝑥) µ𝑓
3
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3
Lin et al. [16]
4
McAdams et al. [17]
5
Akers et al. [18]
6
Beattie and Whalley
[19]
7
Dukler et al. [20]
𝑓𝑇𝑃 =
µ𝑓 µ𝑔
µ𝑔 + 𝑥 1.4 (µ𝑓 − µ𝑔 )
1
𝑥
1−𝑥
=
+
µ𝑡𝑝
µ𝑔
µ𝑓
µ𝑓
µ𝑡𝑝 =
𝑣𝑔
[(1 − 𝑥) + 𝑥 √ ]
𝑣𝑓
µ𝑡𝑝 = 𝜔µ𝑔 + (1 − 𝜔)(1
+ 2.5𝜔) µ𝑓
𝑥𝑣𝑔
𝜔=
𝑣𝑓 + 𝑥𝑣𝑓𝑔
𝑥𝑣𝑔 µ𝑔 + (1 − 𝑥)𝑣𝑓 µ𝑓
µ𝑡𝑝 =
𝑥𝑣𝑔 + (1 − 𝑥)𝑣𝑓
µ𝑡𝑝 =
𝑃𝑜µ̅
𝐺𝐷ℎ
over a significant portion of the heated length. These two data
points are therefore neglected since subcooled flow boiling
pressure drop is not modeled in the present work. Beyond 100
W/cm2, the pressure drop increases almost linearly with heat
flux. The maximum pressure drop obtained is 158 kPa at a heat
flux of 227 W/cm2. The homogenous model underpredicts the
experimental data across the range of viscosity models. The
absolute error at lower heat fluxes (<100 W/cm2) is greater than
that at higher heat fluxes. The model shows an increasing trend
in pressure drop with heat flux similar to the experimental data
points. Owens et al. viscosity model shows the least MEA
(35%) compared to the other models, While Dukler et al.
viscosity model shows the highest error (62%).
(10)
Similar to the inlet single phase losses, the expansion (exit)
losses were calculated at the exit, and the frictional pressure
drop in the rest of the non-heated region was calculated using
the two-phase friction factor. Both contraction and expansion
loss were obtained through look up tables from Kays and
London.
The exit quality was calculated taking into account the
inlet subcooling as given by:
1 q"
x=
[( ) − Cp ∆T𝑆𝑢𝑏 ]
hfg ṁ
(11)
where hfg is the latent heat of vaporization, Cp is the specific
heat capacity, and ∆TSub is the degree of subcooling.
RESULTS
In this section, the theoretical pressure drop value, obtained
using the homogeneous model with seven different viscosity
models, was compared with the experimental pressure drop
data. Plain and microchannel chips are compared first with a
uniform manifold, and then with all three tapered manifolds
using the tapered manifold pressure drop equations discussed in
the next section. For all the plots, pressure drop was computed
as a function of heat flux. Mean absolute error (MAE) was
calculated using the following equation.
𝑀𝐴𝐸% =
1
𝑁
∑|
∆𝑃
− ∆𝑃
∆𝑃
| × 100
Figure 3. Comparison of experimental data with homogenous
model for uniform manifold and plain chip
Figure 4 shows the comparison of the model with data for
the uniform manifold and the open microchannel chip. The
introduction of open microchannel to the geometry shows a
reduction in the pressure drop compared to the plain chip. A
maximum pressure drop of 62 kPa was recorded at a heat flux
of around 280 W/cm2. Similar to the plain chip, the model
underpredicts the experimental data. The model showed an
increasing trend with increasing heat flux. The average MAE
was found to be 69%. In regards to the viscosity model, Owens
et al showed the lowest absolute error (35%), while Dukler et
al. correlation showed the maximum error (72%).
(12)
Uniform manifold pressure drop comparison
Figure 3 compares the pressure drop data for the uniform
manifold and plain chip with the homogeneous model. For the
plain chip, an initial decrease in the pressure drop is seen which
could be attributed to the presence of subcooled flow boiling
Figure 4. Comparison of experimental data with homogenous
model for uniform manifold and microchannel chip
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Tapered manifold with plain chip
The experimental data of the plain chip with three levels of
tapered manifold were compared with homogeneous model as
seen in Fig. 5. The introduction of the tapered manifold showed
drastic reduction in the pressure drop compared to the uniform
manifold. The tapered manifolds recorded a maximum pressure
drop of 20 kPa and 6 kPa with 600 µm taper height.
Figure 5 (a) shows the 200 µm taper height manifold data
comparison. The model underpredicts pressure drop for all
viscosity models at low heat fluxes. At higher heat fluxes (>
150 W/cm2), five models show good agreement with
experimental data. The model shows an increasing pressure
drop with an increase in heat flux. The viscosity models of
Owens et al., Cicchitti et al. and Lin et al. showed the best
comparison with the data, with an average MAE of less than
10%
For the 400 µm taper height manifold, the model
underpredicted over the entire range of heat flux. Different
viscosity model showed varying trends. The viscosity models
of Beattie and Whalley and Dukler et al. showed a decreasing
trend with increasing heat fluxes.
The 600 µm taper height manifold showed a similar
decreasing trend with increasing heat fluxes as the 400 µm
taper. All viscosity models showed a decreasing trend. The
model shows good comparison with experimental data with low
MAE. At higher heat flux the absolute error was greater due to
the diverging trends of the model and the data.
The homogeneous model includes friction, acceleration
and area change terms for the calculation of pressure drop for
tapered manifolds. The area change term was included to
account for the increase in the cross-sectional area at the exit
due to the taper. As the taper height increases, the area term
becomes more dominant and hence for the 600 µm taper height,
a decreasing trend is observed.
Tapered manifold with microchannel chip
Figure 6 (a) – (c) compares the experimental data with the
homogeneous model for tapered manifolds and the
microchannel chip. The combination of tapered manifold and
microchannels worked best to provide a low pressure drop in
the system. The 600 µm tapered manifold provided the lowest
pressure drop of 3.3 kPa at a heat flux of 282 W/cm2.
The 200 µm tapered manifold with microchannel showed
similar performance to the plain chip (200 µm tapered). The
model underpredicted at low heat fluxes and overpredicted at
higher heat fluxes. Similar to the data points, the model showed
a greater than linear increasing trend. The viscosity models of
Beattie and Whalley (MAE – 5%) and Dukler et al. (MAE –
29%) show the best comparison with the data.
For both, the 400 µm and 600 µm taper height manifolds,
the model underpredicted pressure drop for heat fluxes around
150 W/cm2 and below. At higher heat fluxes the models showed
good agreement with the experimental data. Unlike the 200 µm
taper configuration, Beattie and Whalley (MAE – 58%) and
Dukler et al. (MAE – 50%) showed the maximum error.
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Unlike the case of the plain chips (400 µm and 600 µm
taper), the models with OMM show an increasing pressure drop
trend with increasing heat fluxes. This can be attributed to the
increase in the cross-sectional area provided by the
microchannels. The surface area increase provided by the
microchannels balanced the cross-section area term in the
model due to the tapered manifold.
Viscosity models
Seven viscosity models (shown in Tables 3 and4) for all
configurations were compared with the experimental data.
MAE was calculated using the Eq. 12, while not taking into
account values below 100 W/cm2. For the uniform manifold for
both the plain and microchannel chips, all seven viscosity
models underpredicted the experimental values of pressure
drop. Owens et al. predicted the highest values, hence resulting
in the lowest error. This was expected since the model
considers the two-phase viscosity to be equal to the liquid
viscosity and does not take into account the effect of the exit
quality. While, Dukler et al. predicted the lowest pressure drop
values among the viscosity models.
Table 3. MAE% for plain chip with uniform and tapered
manifold with all viscosity models
Mean absolute error%
(MAE%)
Viscosity models
Plain chip
Uniform
200
400
600
1. Owens et al. [14]
36
10
54
56
2. Cicchitti et al. [15]
37
10
55
57
3. Lin et al. [16]
39
7
57
58
4. McAdams et al. [17]
45
22
62
62
5. Akers et al. [18]
6. Beattie and Whalley
[19]
7. Dukler et al. [20]
48
28
68
65
60
26
79
73
62
49
81
74
Similar to plain and uniform manifold, Owens et al. gave
the lowest MAE% for all tapers (400 and 600 µm, in
particular), while Dukler et al. obtained the highest error.
Furthermore, due to high MAE seen for the 400 and 600 µm
tapers specifically for all viscosity models, the homogenous
model does not provide god prediction with plain and tapered
configurations. Hence, the separated flow model is
recommended for future investigation.
Table 4. MAE% for microchannel chip with uniform and
tapered manifold with all viscosity models
Mean absolute error%
(MAE%)
Viscosity models
Microchannel chip
Uniform
200
400
600
1. Owens et al. [14]
35
101
20
43
2. Cicchitti et al. [15]
36
99
20
42
3. Lin et al. [16]
41
88
17
36
4. McAdams et al. [17]
51
57
23
15
5. Akers et al. [18]
6. Beattie and Whalley
[19]
7. Dukler et al. [20]
56
53
32
10
70
5
58
45
72
29
62
50
For microchannel and tapered manifold configuration,
Owens et al., Cicchitti et al. and Lin et al. showed high MAE
especially for the 200 µm taper. While low MAE% was
obtained with McAdams et al. (MAE – 20%) and Akers et al.
(MAE – 22%) for the 400 and 600 µm tapers. Beattie and
Whalley and Dukler et al. resulted in high MAE of over 50%
for the 400 and 600 µm tapers. Good agreement with
experimental data can be obtained with the models of
McAdams et al. and Akers et al. for the tapered and
microchannel configuration.
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CONCLUSIONS
Pressure drop performance of uniform and tapered
manifolds with plain and microchannel chips were compared
with the homogeneous model. Seven viscosity averaging
equations were applied in the homogeneous model. The
following conclusions are made based on the current work:
[2]
[3]
[4]
1.
2.
3.
4.
5.
6.
7.
8.
The combination of tapered manifold and
microchannels worked best to provide a low pressure
drop in the system. A pressure drop of only 3.3 kPa
was recorded for the 600 µm taper height manifold
with a microchannel chip at a heat flux of 282 W/cm2.
The homogeneous model underpredicted pressure drop
for the uniform manifold with plain and microchannel
chips for all seven viscosities averaging schemes. The
average MAE obtained for the uniform manifold was
50%.
The 200 µm taper manifold with plain chip showed
good agreement with the model at higher heat flux,
while the model underpredicted at lower heat fluxes.
For the 400 µm and 600 µm taper manifolds, the
model underpredicted compared to the data. A
decreasing trend was observed for 600 µm taper
manifold with plain chip. The area term in the model
was identified as the key reason for the trend.
The 200 µm taper manifold and microchannel chip
showed the opposite trend as the plain chip. The model
underpredicted at low heat fluxes and overpredicted at
higher heat fluxes. The viscosity models of Beattie and
Whalley (MAE – 5%) and Dukler et al. (MAE – 29%)
showed the best comparison with the data.
Both 400 µm and 600 µm tapered manifolds showed
an increasing trend with increasing heat flux. Good
agreement with the McAdams et al. and Akers et al.
was obtained with these geometries.
For all configurations, the Owen et al. viscosity model
predicted the highest value of pressure drop, while the
lowest value was obtained with the Dukler et al model.
The maximum exit quality calculated was 0.1 at the
highest heat flux. Further testing of this geometry is
recommended.
For the plain and tapered manifold configuration, the
homogeneous model did not provide a good prediction
with experimental data. Good prediction (McAdams et
al. and Akers et al.) was obtained with microchannel
and tapered manifold.
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
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