International Journal of Heat and Mass Transfer 57 (2013) 190–201
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ijhmt
A numerical study on the effects of 2d structured sinusoidal elements on fluid
flow and heat transfer at microscale
V.V. Dharaiya, S.G. Kandlikar ⇑
Rochester Institute of Technology, Rochester, NY, USA
a r t i c l e
i n f o
Article history:
Received 18 March 2012
Received in revised form 18 September 2012
Accepted 2 October 2012
Available online 3 November 2012
Keywords:
Numerical simulation
Computational fluid dynamics
Microchannel
Minichannel
Surface roughness
Roughness elements
Heat transfer
Fluid flow
Structured roughness elements
Laminar flow
a b s t r a c t
Better understanding of laminar flow at microscale level is gaining importance with recent interest in
microfluidics devices. The surface roughness has been acknowledged to affect the laminar flow, and this
feature is the focus of the current work to evaluate its potential in heat transfer enhancement. Based on
various roughness characterization schemes, the effect of structured roughness elements on incompressible laminar fluid flow is analyzed and the hydrodynamic and thermal characteristics of minichannels
and microchannels are studied in the presence of roughness elements using CFD software, FLUENT. Structured roughness elements following a sinusoidal pattern are generated on two opposed rectangular channel walls with a variable gap. A detailed study is performed to understand the effects of roughness height,
roughness pitch, and channel separation on pressure drop and heat transfer coefficient. As expected, the
structured roughness elements on channel walls result in pressure drop and heat transfer enhancements
as compared to smooth channels due to the combined effects of area increase and flow modification. The
current numerical scheme is validated with the experimental data and can be used for design and
estimation of transport processes in the presence of roughness features.
Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction
1.1. Previous experimental work on roughness
Microchannel heat sinks are of interest to researchers due to
their ability to dissipate very high heat fluxes. The advantage of
microchannel heat sinks arises from their small passage size and
high surface area to volume ratio which makes it possible to
achieve enhancement in heat transfer with relatively low cooling
fluid flow rate [1]. In spite of having wide experimental data for
pressure drop in literature, the flow behavior over roughness elements still remains an open problem [2]. Wall roughness has been
studied extensively for its effect on pressure drop. However, studies on their effect on heat transfer performance are scarce. A structured roughness on the walls of microchannel or minichannel
passages is a simple technique for enhancing heat transfer and is
explored in this work. Random roughnesses are three-dimensional
and fractal in nature. However, the current work focuses on investigating structured two-dimensional roughness elements in microchannels that are being considered for microscale applications.
Nikuradse [3] conducted an extensive experimental study with
water in circular pipes to predict the effect of surface roughness on
pressure drop in laminar and turbulent regions, varying Re from
600 to 106. Kandlikar [4] performed a critical review on Nikuradse’s [3] experimental data to understand the mechanisms that
affect fluid-wall interactions in rough channels. For narrow channels, experimental data sometimes misleads as it is strongly influenced by a number of competing factors such as surface roughness,
flow modifications and varying experimental uncertainties.
Nikuradse [3] used sand grains deposition to generate roughness structures on inner channel walls with diameters from 2.42
to 9.92 cm. The results predicted that surface roughness on channel walls do not have significant effect on pressure drop in laminar
flow regime. Kandlikar [4] concluded that the overall uncertainty
associated with pressure drop measurement was estimated to be
between 3% and 5% in the turbulent region whereas it was
significantly higher in the laminar region. These errors were
mainly due to large inaccuracies in manometers used to measure
pressure drop in the laminar region and surface geometry
measurements. Kandlikar [4] showed that measurements of channel dimensions, reading accurate flow parameters, and recognizing
⇑ Corresponding author. Address: Mechanical Engineering Department, Rochester
Institute of Technology, Rochester, NY, USA. Tel.: +1 585 475 6728; fax: +1 585 475
7710.
E-mail addresses: [email protected] (V.V. Dharaiya), [email protected]
(S.G. Kandlikar).
0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.10.004
V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) 190–201
191
Nomenclature
a
b
L
Dh
_
m
V
Ac
P
f
k
l
Re
height of rectangular microchannel, (m)
width of rectangular microchannel, (m)
length of rectangular microchannel, (m)
hydraulic diameter, (m)
mass flow rate (kg/s)
velocity (m/s)
cross-sectional area of microchannel, (m2)
perimeter of microchannel (m)
friction factor
thermal conductivity of fluid (W/m K)
dynamic viscosity (N s/m2)
Reynolds number
Roughness parameters
Fp
floor profile line
Ra
average roughness (lm)
Rp
maximum peak height (lm)
FdRa
distance from Fp to average (lm)
experimental uncertainties were the major problems in obtaining
accurate experimental data in late 80s and 90s.
Wu and Little [5,6] performed experiments on etched channels
with hydraulic diameters varying from 45 to 150 microns and
found an early transition from laminar to turbulent regime in
microminiature J-T refrigerators. Their results were obtained for
hydrogen, neon and argon and the friction factor values were found
to be higher compared to the conventional theory. In 2005, Kandlikar et al. [2] presented experimental data for water and air as fluids
in rectangular channels with Dh varying from 325 to 1819 lm. The
pressure drop showed a significant difference compared to conventional channels and also predicted early transition to turbulence
due to the presence of roughness.
Weilin et al. [7] carried out experiments on trapezoidal silicon
microchannels to investigate flow characteristics of water with Dh
ranging from 51 to 169 lm. Experimental results predicted higher
pressure gradient and friction factor values compared to conventional laminar flow theory which may be due to the effects of surface
roughness. Recently, Gamrat et al. [8] presented an experimental
and numerical study to predict the influence of roughness on laminar flow in microchannels. The results showed that the Poiseuille
number increased with relative roughness and was independent of
Reynolds number in laminar regime (Re < 2000).
Brackbill [9], and Brackbill and Kandlikar [10–12] generated a
considerable amount of experimental data to investigate the effects of surface roughness on pressure drop. Structured saw-tooth
roughness was fabricated on channel walls with relative roughness
ranging from 0% to 24%. The results predicted early transition to
turbulence and showed the use of constricted flow parameters that
would cause the data to collapse onto conventional theory line for
laminar flow regime.
1.2. Previous numerical work on roughness
Hu et al. [13] developed a numerical model to simulate rectangular prism shaped rough elements on the surfaces and showed
significant effects of roughness height, spacing and channel height
on velocity distribution and pressure drop. Croce and D’Agaro [14]
and Croce et al. [15] investigated the effects of roughness elements
on heat transfer and pressure drop in microchannels through a finite element CFD code. They modeled roughness elements as a set
of random peak heights and different peak arrangements along the
ideal smooth surface. Their results predicted a significant increase
RR
Sm
relative roughness
mean spacing of irregularities
Subscripts
w
wall
f
fluid
fd
fully developed
avg
average
x
local
m
mean
cf
constricted flow parameter
expt
experiment based calculation
num
numerical based calculation
Greeks
a
k
e
aspect ratio (a/b)
roughness pitch (lm)
roughness height (lm)
in Poiseuille number as a function of Reynolds number. Moreover,
a remarkable effect of surface roughness on pressure drop was observed as well as a weaker effect on the Nusselt number.
Rawool et al. [16] presented a three-dimensional simulation of
flow through serpentine microchannels with roughness elements
in the form of obstructions generated along the channel walls. They
found that the obstruction geometry plays a vital role in predicting
friction factor in microchannels. The effect on friction factor in case
of rectangular and triangular obstructions was higher compared to
the trapizoidal roughness element. The numerical results concluded that the roughness pitch is an important design parameter
and the pressure drop value decreases with an increase in roughness pitch.
Kleinstreuer and Koo [17,18] proposed a new approach to capture relative surface roughness in terms of a porous medium layer
(PML) model. They evaluated the microfluidics variables, such as
roughness layer thickness and porosity, uncertainties in measuring
hydraulic diameters, and inlet Reynolds number as a function of
PML characteristics. In 2008, Stoesser and Nikora [19] numerically
investigated the turbulent open-channel flow over 2D square
roughness for two roughness regimes using Large Eddy Simulation
(LES). They modeled the effects of roughness height, roughness
pitch, and roughness width as the relative contributions on pressure drag and viscous friction.
1.3. Previous experimental data used for current numerical model
validation
In order to study the effects of surface roughness on fluid flow
and friction factor in microchannels, an extensive experimental
data set was generated by Wagner and Kandlikar [20]. Structured
sinusoidal roughness elements were fabricated on opposed rectangular channel walls to predict the effect of roughness pitch and
height on pressure drop along the channel length. The authors observed that as the hydraulic diameter decreased, the experimental
friction factor increased and became more pronounced for tall and
closely spaced roughness elements.
For the current work, the same experimental data set was utilized for comparison and validation of the proposed numerical
model to predict the effects of surface roughness on pressure drop
in microchannels. The test specimens used for experimentation
were measured under a laser confocal microscope to determine
roughness height and roughness pitch accurately. LabView
192
V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) 190–201
software was used for measuring the flow rate and pressure at inlet, outlet and along the flow length of the channels. The data was
further processed and values of friction factors were calculated for
a given set of test matrices.
Similar experiments to predict the effects of structured sinusoidal roughness elements on heat transfer were presented by Lin and
Kandlikar [22]. The results predicted significant heat transfer
enhancement caused by the presence of roughness structures.
Moreover, the surfaces with higher roughness height to hydraulic
diameter ratio (e/Dh) showed large enhancements in both heat
transfer and pressure drop. The experiments showed that the
roughness pitch did not significantly affect the heat transfer in
the range investigated.
The results presented in this current work are applicable to
incompressible liquid flows. In case of gas flows, compressibility,
accommodation coefficient and rarefaction effects as described
by Cao et al. [23] need to be considered.
2. Objectives
As seen from the above literature review, it is difficult to theoretically analyze the effects of surface roughness on channel walls
in microchannels due to the shape of roughness elements and
inhomogeneities of their distribution. The main objective of the
proposed numerical model is to develop a better understanding
of the roughness effects on heat transfer and fluid flow by generating structured sinusoidal roughness elements on opposed rectangular channel walls. Initially, the numerical model is developed
to characterize flow in smooth rectangular microchannels using
CFD software code, FLUENT. The results for fully developed region
are analyzed for smooth channels and validated with the conventional theory [21] and experimental data [20]. Thereafter, numerical simulations are performed for two different roughness
geometries and the results obtained for friction factor and heat
transfer are compared with available experimental data [20,22].
Constricted flow parameters are used to solve the numerical cases
with surface roughness elements. This systematic study on structured roughness elements in microchannels can be extensively
used for design, optimization and estimation of transport characteristics in the presence of different roughness features.
3. Theoretical formulation
Kandlikar et al. [2] characterized the effects of surface roughness
on pressure drop in single phase fluid flow. Based on their experimental results, the relation between critical Reynolds number versus relative roughness (e/Dh,cf) and friction factor vs. constricted
flow hydraulic diameter Dh,cf was observed. Kandlikar et al. [1] proposed that critical Reynolds number decreases with increase in relative roughness. Based on the enhancement of roughness effects on
transport behavior in minichannels and microchannels, a new modified Moody diagram was developed using the constricted flow
parameters over the entire range of Reynolds number. The new
Moody diagram shows the representation of early transition from
laminar regime to turbulent at micro level, as observed by many
researchers.
In representing the roughness effects on microscale, Kandlikar
et al. [2] proposed a new set of roughness parameters. Fig. 1 shows
the new set of parameters. Rp is the maximum height from the
mean line along the profile. Next, Sm is defined as the mean separation of profile irregularities, which correspond to the pitch of
roughness elements in this work. Lastly, FdRa is the distance of
the floor profile (Fp) which lies below the mean line. These values
are established to replace the assumption that a surface needs only
be defined by the average roughness, Ra. The roughness height e, is
Fig. 1. Schematic diagram of roughness parameters.
the sum of FdRa and Rp. These parameters detail the surface profile
in a more in-depth fashion compared to Ra.
The derivation of the constricted parameters is useful in accurately calculating friction factor in high roughness channels [11].
The current work is based on the constricted parameter scheme
of Kandlikar et al. [2] and Brackbill and Kandlikar [10–12]. In the
case of smooth microchannels, a channel has a cross-section of
height a, and width b, whereas for roughness on two opposite sides
of a rectangular microchannel, the parameter acf represents the
constricted channel height. Fig. 2 shows a generic representation
of the constricted geometric parameters.
In order to re-calculate the constricted parameters, acf and
cross-sectional channel area Acf can be defined as follows:
acf ¼ a 2e
ð1Þ
A ¼ ba and Acf ¼ bacf
ð2; 3Þ
Constricted perimeter for rough channels was obtained by substituting acf instead of channel separation a used for smooth channels;
P ¼ 2b þ 2a and Pcf ¼ 2b þ 2acf
ð4; 5Þ
Similarly, hydraulic diameter and constricted hydraulic diameter
were defined as:
Dh ¼
4A
2ðb þ aÞ
and Dhcf ¼
4Acf
2ðb þ acf Þ
ð6; 7Þ
Theoretical friction factors are calculated using the constricted
parameters. For laminar flow in rectangular channels, the theoretical friction factor is predicted by Kakac et al. [21], by Eq. (8). In
smooth channels, the friction factor was obtained by substituting
the conventional aspect ratio as defined in Eq. (9), whereas in case
of microchannels with surface roughness on channel walls, Eq. (10)
was used to define constricted aspect ratio as shown below:
ftheory ¼
a¼
a
b
24
1 1:3553a þ 1:9467a2 1:7012a3
Re
þ 0:9564a4 0:2537a5 b
and acf ¼
acf
b
ð8Þ
ð9; 10Þ
Relative roughness is defined as the ratio of roughness height to
hydraulic diameter of a channel. Eqs. (11) and (12) defines the relative roughness for smooth and rough channels respectively.
Fig. 2. Generic constricted parameters.
V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) 190–201
e
RR ¼
Dh
and RRcf ¼
e
ð11; 12Þ
Dh;cf
Reynolds number and constricted Reynolds number can be calculated as follows:
_ ¼ V_ q
m
Re ¼
_
4m
lP
ð13Þ
and Recf ¼
_
4m
ð14; 15Þ
lPcf
The theoretical pressure drop can be calculated using Eq. (16) which
was used for validating the current numerical model for predicting
effects of fluid flow in smooth rectangular channels. Using the constricted flow parameters, Eq. (17) can be used to find the friction
factor for channels with surface roughness on channel walls. The
pressure drop data obtained from numerical simulations for rough
microchannels will be compared with Eq. (17). This comparison will
validate the usage of current numerical code to accurately predict
the effects of structured roughness elements on fluid flow in minichannels and microchannels.
f ¼
dP qDh A2
_2
dx 2m
and f cf ¼
2
dP qDh;cf Acf
_2
dx 2m
ð16; 17Þ
A finite volume approach was employed to investigate the thermally developing flow regime. The local and average Nusselt numbers are calculated numerically as a function of non-dimensional
axial distance and channel aspect ratio. The heat transfer coefficient and Nusselt number for rectangular channels can be calculated using Eq. (18) below. Also, constricted hydraulic diameter
was used in the equation for calculating constricted fully developed Nusselt number.
h¼
q00
T w;av g T f ;av g
and NuH2 ¼
Dh
hðxÞ
kf
ð18Þ
In Eq. (18), the average fluid temperature along the flow length of
the channel was calculated using the energy balance equation as
follows:
T f ;x ¼
q00 Pheated walls x
þ T f :in
m Cp
ð19Þ
In case of surface roughness generated on all the four sides of a
rectangular microchannel, all the above equations should be modified by using constricted channel width as bcf, instead of b.
193
very large as compared to the channel height to test the numerical
simulations for extremely low aspect ratios. The width and length
of the channel were kept as 12.7 and 114.3 mm respectively for all
cases of smooth and rough channels. The dimensions were determined based on the experimental test matrix, so as to provide
comparison of numerical model with experimental results.
All the cases were numerically simulated using CFD software,
FLUENT. GAMBIT was used as pre-solver software for designing
geometric models, grid generation and boundary definition. Water
enters the rectangular microchannels with a fully developed velocity profile. The first step in CFD modeling was to compare smooth
channel geometries with conventional theory and prior experimental data. Hence, the experimental tested flow rates were selected to be used for inlet flow rates for numerical simulation.
For numerical investigation of smooth microchannels, the
geometries were created for four different channel separations of
100, 400, 550, and 750 lm. The pre-solver software GAMBIT was
used for performing mesh generation and defining inlet and outlet
boundaries. Waters enters the rectangular channel from left face as
shown in Fig. 3.
All smooth rectangular geometries were created with varying
aspect ratios from 0.007 to 0.06 in a similar way. Tet/Hybrid T-grid
scheme was used for generating mesh for all geometries. The mesh
spacing was kept as low as possible to ensure accurate predictability depending upon each generated model. There were approximately three million grid elements in all the roughness
geometries and the processing time for the each simulation was
noted to be around 14–16 h (Intel Core 2 Duo processor). Grid
independence tests were carried out for all the geometries to ensure best mesh spacing for the numerical model.
Furthermore, all the mesh geometries were imported in numerical simulation CFD software, FLUENT. The Reynolds number was
kept at 100 for all the cases in order to confirm laminar flow
through rectangular microchannels. However, Reynolds numbers
used for the numerical validation cases were selected to be the
same as the experiments for better comparison. Pressure-based
solver was used to achieve steady state analysis. The SIMPLE
(Semi-Implicit Method for Pressure-Linked Equations) algorithm
was used for introducing pressure into the continuity equation.
The inlet temperature was kept as room temperature for all the
cases and for the experimental data as well. The flow equations
were solved with a first-order upwind scheme. The convergence
criteria for velocities, continuity and energy equation in each
numerical model were kept as 1E-6.
4. Model description
4.2. Case II: Rough channels
Initially the fluid flow effects in smooth rectangular microchannels with varying aspect ratios from 0.007 to 0.06 were investigated using commercial CFD software, FLUENT. The obtained
results were compared with conventional theory and prior experimental data to provide validation of the current numerical
scheme. Later on, the numerical scheme was extended to predict
the flow characteristics with structured surface roughness elements on channel walls using constricted flow parameters. The
geometries selected for computational analysis were based on
the experimental test matrix. The reason for using long section of
a channel (L = 114 mm) was due to the fact that CFD model was
used to handle wide range of Reynolds number. Moreover, entrance region may be observed over significant lengths at higher
Reynolds number. This paper focuses on fully developed region beyond the entrance region.
4.1. Case I: Smooth channels
Fig. 3 shows a schematic with channel height (a), channel width
(b) and channel length (L). The channel width and length were kept
The numerical model used to predict the pressure drop for
smooth rectangular microchannels was extended to rough channels. In order to investigate structured roughness effects on pressure drop, sinusoidal roughness elements on opposed rectangular
channel walls were generated as shown in Fig. 3. Two roughness
geometries as shown in Fig. 3 were selected from the experimental
test matrix to validate the numerical model. In both the roughness
profiles, the only varying parameter was roughness pitch.
Table 1 shows the channel geometries used for validating the
numerical model. There were four smooth channel geometries
and two geometries with surface roughnesses on opposed rectangular channel walls. The values for constricted flow parameters
were used to calculate pressure drop in rough microchannels. As
seen in Table 1, the usage of constricted parameters makes significant difference in values of aspect ratio and hydraulic diameters
for channels with roughness elements. The controlling parameters
that were used to define the channel width and the channel length
for all the cases in Table 1 were kept as 12.7 mm and approximately 114.3 mm respectively.
194
V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) 190–201
Fig. 3. Meshed geometry for smooth rectangular channel having channel height (a = 400 lm), channel width (b = 12.7 mm), and channel length (L = 114.3 mm). Fig. 3:
Schematic and meshing of rough microchannel created in GAMBIT software having channel separation = 550 lm, roughness pitch = 250 lm, and roughness height = 50 lm.
Table 1
Test matrix used for CFD model validation.
a
acf
Dh (lm)
Dh,cf (lm)
Case I: Smooth channels
100
N/A
N/A
400
550
750
0.008
0.032
0.043
0.059
–
–
–
–
198.4
775.6
1054.3
1416.4
198.4
775.6
1054.3
1416.4
Case II: Rough channels
550
250
50
550
150
50
0.043
0.043
0.035
0.035
1054.3
1054.3
869.2
869.2
a (lm)
k (lm)
e (lm)
In a similar way, all other roughness geometries were generated
as shown in Table 2 to predict the effects of roughness pitch,
roughness height, and channel separation on heat transfer. Tet/Hybrid T-grid scheme of mesh generation and lowest possible mesh
spacing was used for all roughness geometries. For each analysis,
it was found that any further reduction in mesh spacing does not
affect the pressure profiles at varying cross-sections. This confirms
the grid independency for accurately predicting the numerically
simulated results.
5. Results and discussions
As discussed earlier, two roughness geometries were used to
validate the proposed numerical scheme. Fig. 4 represents the outline and solid model of roughness geometry with structured sinusoidal elements generated on opposed channel walls of width (b)
12.7 mm. The two major parameters used to define roughness elements were roughness height and roughness pitch. Fig. 4 represents roughness geometry with channel separation of 550 lm,
roughness height of 50 lm, and roughness pitch of 250 lm. For
creating a geometric model in pre-solver GAMBIT, a single roughness element of 250 lm pitch was duplicated 457 times to match
the channel length of 114.25 mm.
All 457 segments were unified to create a single volume with
roughness elements on opposite walls as depicted in Fig. 4. The inlet
was given on the left face and the flow direction was kept as shown
in Fig. 4. The section of meshed geometry of the model created is
also shown. Tet/Hybrid T-grid meshing was used and conformance
of mesh spacing was done employing a grid independence plot.
Table 1 summarizes the detailed geometric parameters used for
generating smooth and rough rectangular microchannels for validating current numerical model with available experimental data
[20,22].
The different models were generated and meshed using pre-solver software, GAMBIT. Further, the meshed geometries were
numerically simulated with prescribed boundary conditions using
a CFD software tool, FLUENT. The pressure data along the length of
microchannel was extracted from the converged simulated models. Later on, the pressure drop was calculated and once the pressure drop achieves linearity along the length of a channel, a fully
developed region was assumed to have reached. Also, hydrodynamic entrance lengths were calculated for each case and ascertained that the values for fully developed pressure drop were
obtained at a length greater than Lh. The fully developed friction
factors were then calculated from the numerical simulated pressure drop data. All the cases were solved with a Reynolds number
in a range of laminar flow for microchannels.
Four smooth geometries and two rough geometries were selected from experimental data of Wagner and Kandlikar [20] to
compare and validate the current numerical model. The comparison can rate the accuracy of numerical model to predict the effects
of fluid flow in microchannels with and without roughness
Table 2
Test matrix used for numerical simulation of rough channels with varying roughness
height, roughness pitch, and channel separation.
e (lm)
k/e
Varying roughness height, e
550
250
550
250
550
250
550
250
550
250
550
250
10
16
20
25
50
100
25
16
12.5
10
5
2.5
Varying roughness pitch, k
550
150
550
250
550
400
50
50
50
3
5
8
Varying channel separation, a
250
250
550
250
750
250
50
50
50
5
5
5
a (lm)
Fig. 4. Schematic and meshing of rough microchannel created in GAMBIT software
having channel separation = 550 lm, roughness pitch = 250 lm, and roughness
height = 50 lm.
k (lm)
195
V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) 190–201
elements. In case of rough channels, the numerical cases were
solved on the basis of constricted flow parameters and the resultant numerical fully developed laminar friction factor (ffd,cf,num)
was compared with the experimental friction factor (ffd,cf,expt) for
validating current numerical model. Similar study was conducted
to validate the numerical model with available heat transfer experimental data generated by Lin and Kandlikar [22].
5.1. CFD model validation for DP
5.1.1. Case I: Smooth channels
Table 3 shows the results for smooth rectangular microchannels
and minichannels tested for channel separations of 100, 400, 550,
and 750 lm. For each case, pressure drop data along the flow
length obtained from FLUENT simulations were in good agreement
with the measured experimental pressure drop data. This shows
the conformity of the numerical model to accurately predict the
fully developed friction factor in microchannels. Eq. (16) was used
to calculate the numerical friction factor values for smooth channels, where pressure drop value was obtained from FLUENT.
Boundary conditions and geometric parameters were maintained
exactly the same as experimental data [20] to show better comparison. Hence, the Reynolds number for each smooth channel was
varied as seen in Table 3.
The discrepancies in the numerical results for smooth channels
with theory are mainly due to interpolation used in theoretical
predictions for very low aspect ratios ranging from 0.008 to
0.043. Since the theoretical values have been obtained using interpolation for very low aspect ratios, the numerical values are compared with carefully obtained experimental values by Wagner and
Kandlikar [20] for the low aspect ratio range. The numerical data
for smooth channels were in good agreement with the experimental data [20]. For smooth narrow rectangular channels, the percentage deviation of numerical model with experimental data was less
than 2.58%.
5.1.2. Case II: Rough channels
The roughness geometries were numerically simulated using
the same numerical scheme with constricted flow parameters. Table 4 shows the Reynolds number used for each simulation which
was utilized from the experimental data set. Two roughness cases
with varying roughness pitches of 150 and 250 lm were
simulated. Eq. (17) was used to calculate fully developed laminar
friction factor values for both experimental and numerical cases.
The numerically simulated results show good agreement with
the experimental data as seen in Table 4. The percentage deviation
between experimental and numerical work was found to be less
than 2.58% for all the cases simulated. This shows that a numerical
model is always a good tool to predict the fluid flow characteristics
in microchannels.
5.2. CFD model validation for Nusselt number
The current numerical model was also validated for one of the
heat transfer experimental data generated by Lin and Kandlikar
Table 3
Comparison of fully developed laminar friction factor for smooth channels.
Case I: Smooth channels
a (lm)
Re
ftheory
fexpt [20]
fnum
% Errorexptnum
100
400
550
750
784
800
784
785
0.0303
0.0288
0.0289
0.0283
0.0232
0.0331
0.0330
0.0299
0.0238
0.0336
0.03325
0.0304
2.58
1.51
0.76
1.67
Table 4
Comparison of fully developed laminar friction factor for rough channels using
constricted flow parameters.
Case II: Rough channels
a (lm)
a
acf
k (lm)
Re
fcf,expt
fcf,num
% Errorexptnum
550
550
550
550
0.043
0.043
0.043
0.043
0.035
0.035
0.035
0.035
250
250
150
150
80
190
85
190
0.398
0.157
0.345
0.158
0.406
0.162
0.343
0.168
2.01
3.12
0.72
6.33
[22]. The roughness geometry used for validating had a channel
separation of 354 lm, roughness height of 100 lm, and roughness
pitch of 250 lm. This geometry was tested at Reynolds number of
286. The experimental value of fully developed Nusselt number
was found to be 31.4 [22].
The numerically simulated case resulted in fully developed Nusselt number of 30.7 with percentage deviation of 2.25%. The results
in both, experiments as well as numerical model predicted very
high heat transfer enhancement. The results were analyzed by
plotting velocity vectors as shown in Fig. 5 and it was found that
there were several formations of vortex generators. Hence, the current numerical model used for generating heat transfer results in
presence of structured sinusoidal roughness elements in rectangular channels was successfully validated.
5.3. Data analysis for rough channels
As seen earlier, the accuracy of this numerical model to evaluate
the pressure drop and heat transfer coefficient was found to be in
good agreement with the previous experimental results. Thereafter, the current numerical model was extended to predict the effects of various roughness parameters such as roughness height,
roughness pitch and channel separation on heat transfer. All the
geometric parameters were calculated using constricted flow theory for rough channels. The physical properties were calculated
using the average temperature difference between the wall and
fluid obtained from numerical simulations.
Fig. 6 shows the temperature variation of wall and fluid along
the length of the channel for one of the roughness geometries having channel separation of 550 lm, roughness pitch of 250 lm, and
roughness height of 50 lm. Fig. 6 shows that the heat transfer coefficient is very high in the entrance region and progressively becomes constant, i.e., when fully developed laminar flow is
achieved. Based on the temperature difference (DT) in the figure,
heat transfer coefficient can be estimated to calculate the fully
developed Nusselt number. As seen from the figure, the average
wall temperature profile seems to have a larger bandwidth compared to mean bulk temperature along the length of the channel.
Fig. 6 also shows the zoomed view of the same temperature
profile for roughness geometry for the first 10 mm length of the
channel. It can be seen that the wall temperature varies on the basis of roughness profile along the channel. Also, the analysis in a
single roughness element shows that the temperature at the extreme bottom node of the roughness element has the highest temperature whereas the extreme top node has the lowest
temperature. This was due to the fact that the extreme top node
is in closer contact with the bulk fluid flow.
The sinusoidal roughness elements provide flow modifications
which increase heat transfer compared to the smooth channels at
the expense of pressure head. Moreover, the temperature variation
for sinusoidal roughness geometries was observed along the length
as well as the width of the channels. Therefore, it becomes important to properly analyze the average wall temperature for roughness
geometries to estimate correct heat transfer from the system.
196
V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) 190–201
Fig. 5. Velocity vector for geometry with a = 354 lm, k = 250 lm, and e = 250 lm.
Fig. 6. Temperature variation for the roughness geometry having a = 550 lm,
k = 250 lm, and e = 50 lm.
As discussed above, the wall temperature varies along the
length of the microchannel due to flow modifications caused by
the fluid following the path of sinusoidal roughness pattern.
Fig. 7 shows the schematic of roughness profiles for a geometry
having channel separation of 550 lm, roughness pitch of 250 lm,
and roughness height of 50 lm. Fig. 7 also shows the corresponding variation of wall temperature along the length of the microchannel for the same geometry.
Initially, in order to calculate the average wall temperature, several node points were computed along a single roughness element.
Thereafter, the mean of all nodes was calculated to estimate the
average wall temperature. Based on the definition discussed in
the earlier section, the wall temperature was also determined at
a distance of the average maximum profile peak height (Rpm) to
estimate the average wall temperature based on theory. Both values of average wall temperatures tended to match with a maximum deviation of 0.03%.
Therefore, to simplify the calculations, the average wall temperature was determined at the main profile mean line as shown in
Fig. 7. The data points computed for wall temperature in Fig. 7
were in the fully developed laminar region and therefore the
corresponding average wall temperature follows a linear increasing trend as expected. In a similar way, the heat transfer coefficients for all the roughness geometries were estimated based on
the wall temperatures computed at a distance of averaged maximum profile peak height (Rpm).
Fig. 8 shows the flow direction of water along the length of the
microchannel having a width of 12.7 mm. Fig. 9 represents single
roughness element of the same geometry. In the case of the roughness geometries, a constant heat flux H2 boundary condition was
applied on the two opposite channel walls having surface roughness.
The maximum temperature was observed at the corners and
minimum at the center line subjected under H2 boundary condition as seen in Fig. 9. A similar temperature trend was obtained
when the rectangular microchannels were applied with uniform
heat flux H2 boundary conditions as shown by Dharaiya and Kandlikar [24]. Fig. 8 shows the temperature variation along the width
of the rough microchannel at 90 mm axial distance (along the
width highlighted shown in Fig. 8). The average of all nodes along
the width of the microchannel were used in calculating the average
wall temperature.
The average wall temperature obtained from the computational results was calculated using temperature at different nodes
on each heated wall along the length of the microchannel. Initially, several points are computed along the heated walls to estimate average wall temperature. Thereafter, to simplify the
calculations, five nodes are selected on the heated wall which
would be able to predict the average wall temperature with a
maximum error of 0.06%. Eq. (20) below represents the five-node
method used to average the wall temperature profile for heated
walls 1 and 2 respectively. For each wall of the rectangular microchannel under constant wall heat flux, temperature peak was observed at corners and hence the temperature values at each
corner nodes are halved while calculating the average temperature for each wall as represented by Eq. (20) for wall-1 (refer to
[24] for further details):
T1 ¼
½ð0:5 T N1 Þ þ T N2 þ T N3 þ T N4 þ ð0:5 T N5 Þ
4
ð20Þ
where, subscripts N1, N2, . . ., N5 represents the different nodes
which are used to calculate the average wall temperature for different heating configurations.
V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) 190–201
197
Fig. 7. Wall temperature variations along the length of the channel for roughness geometry.
height, roughness pitch, and channel separation on pressure drop
and heat transfer in minichannels and microchannels. As expected,
roughness elements on channel walls resulted in enhancements in
transport behavior compared to smooth channels due to surface
area enhancement and flow modifications.
5.5. Effects of roughness height
Fig. 8. Geometric representation of flow over the roughness geometry.
Fig. 9. Wall temperature variation along the width of the rough channel at 90 mm
(a = 550 lm, k = 250 lm, and e = 50 lm).
5.4. Effects of roughness parameters on heat transfer
In the current work, roughness elements were configured as
structured sinusoidal pattern along the channel walls. Using structured roughness features on surfaces will provide better understanding of different roughness parameters such as roughness
In narrow channels, structured sinusoidal roughness elements
act as small obstructions in the flow which provides mixing of fluid
and aids in heat removal from the system. Fig. 10 shows the temperature profiles along the axial length of a microchannel for three
different roughness geometries with varying roughness heights.
The temperature profiles are shown for the first 25 mm length of
the microchannel as the flow becomes fully developed within that
length and it follows a linear trend thereafter. The channel separation and roughness pitch were kept as 550 and 250 lm respectively for all cases to study the effect of roughness height. The
roughness height was varied from 10 to 100 lm. This study signifies the effects of roughness height on transport processes.
The temperature variation in Fig. 10 was plotted by computing
the wall temperatures at different locations on roughness elements
such as tip, base and average roughness peak height. The heat
transfer coefficient and fully developed laminar Nusselt number
were estimated by considering the temperature profile at the main
profile mean line as discussed earlier. It was observed from the figure that the wall temperature was maximum at the base of the
roughness element and lowest at the tip. Moreover, the temperature profile at a distance of average roughness peak height (Rpm)
lied between the maximum roughness peak height and floor profile mean line. In all the cases, constant heat flux H2 boundary condition was applied on the two opposite roughness walls which was
calculated based on the constricted flow parameters. As seen from
three plots in Fig. 10, in a fully developed laminar flow region, the
temperature difference between the wall and fluid tends to increase with increase in roughness height. This corresponds to the
decrease in the heat transfer coefficient with increment in the
roughness height from 10 to 100 lm.
These results were observed due to the fact that the roughness
height grows taller keeping the channel separation and roughness
pitch constant. Fig. 11 shows the velocity vector for a case showing
highest heat transfer enhancement. The velocity vector’s path
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V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) 190–201
Fig. 10. Temperature variation along the length of channel for roughness geometry to predict the effects of roughness height on heat transfer.
Fig. 11. Velocity vectors for a roughness geometry with a = 550 lm, k = 250 lm, e = 20 lm.
Fig. 12. Velocity streamlines for the roughness geometries – (i) a = 550 lm, k = 250 lm, e = 20 lm; (ii) a = 550 lm, k = 250 lm, e = 100 lm.
predicts smooth flow along the sinusoidal roughened channel
walls. Hence, the enhancement on heat transfer was observed
mainly due to smooth-structured rough geometries studied in this
current work. There was also a certain rise in pressure drop due to
presence of roughness but was very negligible compared to other
geometries such as conical peaks, sharp corners, rectangular prism
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Table 5
Rough Channels: NuH2,fd and fH2,fd for fully developed laminar flow with varying roughness height, roughness pitch, and channel separation.
e (lm)
k/e
b (mm)
Dh (lm)
Dh,cf (lm)
NuH2,fd
NuH2,fd,cf
fH2,fd
fH2,fd,cf
Effects of roughness height, e
550
250
550
250
550
250
550
250
550
250
550
250
10
16
20
25
50
100
25
16
12.5
10
5
2.5
12.7
12.7
12.7
12.7
12.7
12.7
1054.3
1054.3
1054.3
1054.3
1054.3
1054.3
1017.5
995.4
980.6
962.1
869.2
681.2
7.235
17.231
19.332
7.408
8.039
7.678
6.972
16.268
17.981
6.76
6.628
4.961
0.2216
0.2910
0.2968
0.2150
0.2105
0.2810
0.2138
0.2748
0.2760
0.1962
0.1736
0.1816
Effects of roughness pitch, k
550
150
550
250
550
400
50
50
50
3
5
8
12.7
12.7
12.7
1054.3
1054.3
1054.3
869.2
869.2
869.2
7.979
8.039
7.378
6.578
6.628
6.082
0.2216
0.2105
0.2111
0.1827
0.1736
0.1741
Effects of channel separation, a
250
250
50
550
250
50
750
250
50
5
5
5
12.7
12.7
12.7
490.4
1054.3
1416.4
296.5
869.2
1236.7
14.871
8.039
6.163
8.993
6.628
5.381
0.2135
0.2105
0.2115
0.1291
0.1736
0.1847
a (lm)
k (lm)
elements, and others mostly studied in literature. The smooth
structured sinusoidal roughness elements, if properly employed
in geometry can allow better mixing and provide significantly enhanced heat transfer rate with a minimal increase in pressure drop.
Fig. 12 above compares velocity streamlines for two different
geometries with roughness heights of 20 and 100 lm respectively.
The streamlines clearly show no flow-path in a rough geometry
with very high roughness height. Hence, there was not much heat
transfer enhancement seen for the cases with taller roughness elements. Similar theory applies for a geometry having extremely low
roughness height (for a case with roughness height = 10 lm). The
roughness height in this case was too small to produce any flow
modifications to enhance heat transfer rate. As seen in Fig. 12(ii);
the majority of the fluid path tends to flow through the gap between the roughness peaks generated on two opposite channel
walls.
Table 5 shows the effect of roughness height on the fully developed Nusselt number for rough microchannels. The constricted
Nusselt number was calculated based on constant heat flux on
two rough surfaces. The heat transfer coefficient decreases with
the corresponding increase in the structured roughness height.
This was due to the decrease in the magnitude of the effective
hydraulic diameter which was calculated using the constricted
flow parameters as seen in the table below. The function of the
roughness element was to enhance the mixing of the fluid flowing
through the channel. As the roughness element height increases,
its functionality decreases.
5.6. Effects of roughness pitch
To study the effects of roughness pitch on fluid flow and heat
transfer properties, three geometries were selected with varying
Fig. 13. Velocity vectors near roughness elements – (i) a = 550 lm, k = 150 lm, e = 50 lm; (ii) a = 550 lm, k = 250 lm, e = 50 lm; (iii) a = 550 lm, k = 400 lm, e = 50 lm.
200
V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) 190–201
Fig. 14. Temperature variation along the length of channel for roughness geometry to predict the effects of channel separation on heat transfer.
pitch as 150, 250, and 400 lm respectively. Also, all the other
roughness parameters such as channel separation and roughness
height were kept constant. The effect of roughness pitch on fluid
behavior seems to have very less influence of transport phenomena
for selected range of roughness geometries.
Fig. 13 displays the velocity vectors for rough channels with
varying roughness pitch of 150, 250, and 400 lm respectively.
The results show that the flow tends to follow the path of sinusoidal rough elements but does not contribute to provide a high heat
transfer rate. The values of fully developed Nusselt number were
decreasing with increase in magnitude of roughness pitch but
the variation was very small. These results can be attributed to
the fact that the effective hydraulic diameter remained the same
for all the cases as the channel separation and roughness height
were kept constant. Similar results were observed by the experimental heat transfer data generated by Lin and Kandlikar [22] on
sinusoidal rough elements. Table 5 shows the results of fully developed Nusselt number with varying roughness pitch.
5.7. Effects of channel separation
In order to perform numerical simulation to evaluate the effects
of channel separation on heat transfer, the other two roughness
parameters such as roughness pitch and roughness height were
kept constant as 250 and 50 lm respectively. The channel separation was varied from 250 to 750 lm. Fig. 14 shows the temperature
profile for the first 25 mm length of microchannel as the flow becomes fully developed and follows constant slope thereafter.
The values of temperature difference between the wall and fluid
in a fully developed region increased with increase in channel separation as expected. These resulted in lower values of heat transfer
coefficient with corresponding increase in channel separation
assuming the roughness height and roughness pitch as constant.
Table 5 showed the effects of channel separation on fully developed Nusselt number for rough channels.
The above results were observed due to the fact that the heat
transfer coefficients can only be affected by change in channel
separation as all other roughness parameters were kept constant.
The numerical analysis predicts that the tendency of the roughness
elements to affect the fluid flow behavior diminishes with increase
in channel separation due to inactiveness of roughness elements
with higher separation.
6. Conclusions
A numerical model was developed to predict the effects of fluid
flow characteristics in smooth channels and channels with surface
roughness. Smooth rectangular geometries were tested for hydraulic
diameters varying from 0.0078 to 0.06. Fully developed laminar friction factors were calculated from the pressure drop values obtained
from numerical simulated cases. The numerical model was validated
with experimental data and the percentage deviation was less than
2.58% for smooth rectangular channels. Also, the fully developed friction factor values were found higher as compared to conventional
theory for smooth channels. The constricted flow parameters were
further used to simulate cases with structured sinusoidal surface
roughness elements on channel walls. The numerical simulation
for two roughness geometries (k = 150 and 250 lm) showed very
good agreement with experiments as well. The maximum friction
factor percentage deviation for rough channels was found to be less
than 6.33%. The numerical model was also successfully validated
with experiments [22] to predict the effects of heat transfer properties in presence of surface roughness.
The following conclusions were drawn on the roughness
parameters based on the current numerical work:
Channel separation-The Nufd decreases with increase in channel
separation due to the fact that roughness effect diminishes.
Roughness height-The Nufd was significantly high for k/e of 12.5
and 15.625, whereas the enhancement diminishes for lower k/e
ratios. This was observed due to the fact that the constricted
hydraulic diameter decreased with increase in roughness
height. Also, the zone of flow modification beneath the rough
elements became inactive with increase in roughness height.
Roughness pitch-There was no significant effect of roughness
pitch found on the fully developed Nusselt number for the range
selected. Similar results were also observed by experimental
data published by Lin and Kandlikar [22].
Heat transfer enhancement of magnitude as high as 264.8% was
found in one of the roughness geometries with channel separation
of 550 lm, roughness pitch of 250 lm, and roughness height of
20 lm. Moreover, looking at the velocity vectors for different
rough geometries, continuous streamlines were observed in all
geometries. There was no presence of vortices formation behind
the roughness elements which normally increases the effects heat
transfer but also results in high penalty on pressure drop. Therefore the purpose of designing roughness as smooth structured
sinusoidal rather than having sharp corners in roughness elements
significantly increased the potential of Nusselt number with slight
increase in friction factor. The enhancement observed in the current work was purely due to flow modifications and area enrichment due to the presence of roughness elements. As discussed in
literature earlier, researchers have used sharp wedges to increase
effects of heat transfer phenomenon but have also shown high
penalties on friction factor. Moreover other researchers [14–16]
have used random roughness peak arrangements along the ideal
V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) 190–201
smooth surface and predicted a remarkable effect of surface roughness on friction factor compared to weaker one on the heat transfer. The current study focuses on carefully obtaining numerical CFD
data for two-dimensional smooth sinusoidal wall shape that offers
high heat transfer enhancement of 264.8% with penalty on friction
factor of as low as 30.9% for one of the roughness geometry. Therefore, these types of smooth sinusoidal roughness structures can be
used as an important feature in designing and operation of microsystems as it promises to show significant heat transfer enhancement as compared to smooth channels.
Acknowledgments
The current work was carried out in Thermal Analysis, Microfluidics, and Fuel cell Lab (TAlFL) at Rochester Institute of Technology, Rochester, NY, USA. This material is based upon work
supported by the National Science Foundation under Award No.
CBET-0829038. Any opinion, findings, and conclusions or recommendation expressed in this material are those of the authors
and do not necessarily reflect the views of the National Science
Foundation.
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