Proceedings of the ASME 2009 International Mechanical Engineering Congress & Exposition
IMECE2009
Proceedings of the 2009 ASME International Mechanical Engineering Congress and
Exposition
November 13-19, Lake Buena Vista, Florida,
USA
IMECE2009
November 13-19, 2009, Lake Buena Vista, Florida, USA
IMECE2009-11039
IMECE2009-11039
MATHEMATICAL MODEL FOR FLUID FLOW IN ARTIFICIALLY ROUGHENED
MICROCHANNELS
Rebecca N. Wagner*, Satish G. Kandlikar
Department of Mechanical Engineering
Rochester Institute of Technology, Rochester, NY, USA
*[email protected]
to model numerically and their associated amplitude and spatial
parameters are more predictable.
In this study, laminar flow through rectangular channels
with structured roughness on the walls is investigated. The
present work deals solely with deliberately designed and
machined two-dimensional surface roughness for which a
periodic pattern can be easily distinguished. The primary
geometric characteristics describing the microscale striations
are the roughness element height h and spacing, or pitch, λ. In
addition to these variables, the slope and alignment of
roughness peaks are of interest.
Theoretical modeling presented here will be referred to as
the wall function method. For comparison with conventional
theory, friction factors for each rough surface are calculated
based on the pressure drop evaluated from this method. The
conventional equations used are the Kakaç correlation for
smooth-walled rectangular ducts and the Fanning friction
factor.
ABSTRACT
Two dimensional lubrication approximation is applied to
the analysis of fluid flow in a rectangular microchannel with
structured roughness. Each of two major walls is composed of
designed microscale transverse ribs, modeled by twodimensional functions. A pressure drop correlation is formed
as a direct function of these surface equations and the resultant
velocity profile is incorporated into boundary layer analysis.
Friction factors are calculated based on the obtained pressure
drop values for comparison with conventional theory. The
method presented here is consistent with conventional
correlations in the case of hydraulically smooth channels, and
is a stronger function of structured roughness geometry than
the existing methods.
INTRODUCTION
All machined surfaces possess natural roughness which
results from machining processes. This random roughness can
influence flow characteristics in internal flow. However, it is
typically smaller than the boundary layer thickness. Thus it is
assumed to have negligible influence in the laminar flow
regime, although it will influence turbulent flow if its height is
larger than the viscous boundary layer [1].
The natural or random roughness due to machining
processes is difficult to control and characterize. Typically, the
average roughness, Ra, or the root-mean-square roughness, Rq,
is used to assess the approximate height of the surface
asperities. These average parameters do not fully account for
the geometry for the surface, such as spacing, slope, and
alignment of peaks, which may be the controlling factors in
hydraulic performance of the surfaces.
Structured roughness has greater repeatability in
manufacturing than the random roughness attributed to the
machining processes themselves. In addition to being simpler
to recreate, two-dimensional periodic surfaces are also simpler
NOMENCLATURE
a
b
bcf
beff
Dh
e
FdRa
Fp
f
f(x)
g
h
h(x)
k
P
Q
1
Channel height
Root channel separation or width
Constricted channel separation
Effective channel seapartion
Hydraulic diameter
Roughness height, used in Moody diagram
Average floor profile
Average floor profile, alternative name
Fanning or Kakaç friction factor, as stated in text
Lower wall function
Gravity
Roughness height, wall function method
Upper wall function
Roughness height, as defined by Von Mises [2]
Pressure
Volumetric flow rate
Copyright © 2009 by ASME
r
Ra
Re
Rq
Rp
u
v
w
Pipe radius
Average roughness
Reynolds number
Root-mean-square roughness
Maximum peak height from mean line
x-component of velocity
y-component of velocity
z-component of velocity
Greek
α
δ
εFp
λ
μ
ρ
Channel aspect ratio, b/a
Boundary layer thickness
Roughness height parameter [3]
Roughness pitch
Dynamic viscosity
Density
flows, Jiménez [13] related flow transition to the ratio of
roughness height k to boundary layer thickness δ, emphasizing
the importance of the interference of the roughness with the
boundary layer. He indicates that flows with high ratios of δ/k
will best be described as flows over obstacles and are also very
dependent on roughness geometry.
Coleman et al. [14] similarly assessed the effect of various
roughness pitch-to-height ratios, λ/h, identifying a
“transitional” roughness occurring for λ/h = 5. Values of λ/h <
5 indicate closely spaced ribs, referred to as d-type roughness,
and λ/h > 5 indicate isolated roughness elements, or k-type
roughness. In both extremes the roughness effect is expected
to diminish.
Brackbill [15] performed exhaustive experiments on
sawtooth roughness in rectangular microchannels, assessing the
effects of relative roughness greater than 5% to show that there
is significant effect in the laminar regime. In the same study,
the constricted parameter derived by Kandlikar was shown to
be effective in predicting flow characteristics.
In general, definite departure from conventional laminar
theory has been identified in many cases. The majority of
recent work however is experimental, resulting in empirical
relations, scaling factors, etc. These correlations are not
universally applicable across the full range of mini- and
microchannels, roughness types, and fluids. Similarly, recent
theoretical and numerical works result in scaling models or
correction factors.
LITERATURE REVIEW
Since Darcy [4] identified the dependence of fluid flow on
diameter, surface type, and slope of pipes in the 1800s, much
effort has been put into experimental and theoretical
assessment of the effect of roughness on internal flows. In the
early 1900’s, Von Mises [2] identified the significance of
relative roughness, the ratio of roughness size to pipe radius,
k/r. Hopf [5] and Fromm [6] in 1923 classified surface types
based on roughness aspect ratio and identified friction factor
and Reynolds number as dependent on surface type,
differentiated by the pitch-to-height ratio, λ/h. Nikuradse [7]
performed exhaustive experiments assessing a complete range
of Reynolds numbers for various k/r values maintaining
geometric similitude in pipe geometries. Colebrook [8]
developed the general (and well-known) formula for friction
factor at high Reynolds numbers, with which Moody [9]
provided a convenient means for estimating friction factors
based on pipe diameter, roughness size, and Reynolds number.
The Moody diagram allows for prediction of fluid
behavior in circular macroscale pipes, but indicates that
roughness has no effect on friction factor in the laminar flow
regime. The diagram also is limited to relative roughness
values of 5%, or e/D of 0.05. Experiments performed since the
1980s indicate that as the pipe diameter decreases there is
significant deviation from conventional theory in both laminar
and turbulent regimes [10]. Wu and Little [11] recognized
higher friction factors and early transition to turbulence of gas
flow in microscale channels. Peng et al. [12] experimented
with rectangular channels having hydraulic diameters ranging
from 0.133 to 0.343 mm, and identified pipe geometry,
particularly channel aspect ratio, as having the most critical
effect on flow characteristics.
Interest in microscale phenomena has increased in the
recent past due to its usefulness for passive enhancement of
microfluidic devices.
Kandlikar et al. [3] proposed a
constricted parameter with which they modified the Moody
diagram to account for higher relative roughness and thus
smaller hydraulic diameter channels. In his review on turbulent
OBJECTIVE
The primary objective of the current work is to develop
and validate a theoretical model that provides a foundation for
understanding the effects of structured roughness on fluid flow
at the microscale level in internal laminar flow. The ratio of
roughness pitch to height, λ/h, will be assessed as well as the
average amplitude parameters to determine their relevance to
flow performance. The theoretical model is to be checked
against both experimental data and numerical simulation.
THEORETICAL ANALYSIS
The Navier-Stokes (N-S) equations form the foundation
for the theoretical treatment of the microchannel problem
presented here. The internal two-dimensional flow is assumed
to be steady and incompressible. In addition, flow is assumed
to be fully developed, so that inlet and outlet effects may be
neglected. For simplification, temperature and thus fluid
properties are assumed constant. This is valid for the geometry
at hand, for which prior experimental data shows a negligible
temperature difference along the computational length.
Channel orientation is as displayed in Figure 1. Bulk flow
is in the x-direction, from left to right. Two-dimensional,
transverse rib roughness is in the x-y plane. Gravity is acting in
the y-direction, perpendicular to the page. Wall roughness is
represented by the functions f(x) and h(x) for the lower and
upper walls, respectively. These physical boundaries are used
as the limits of integration in the z-direction.
2
Copyright © 2009 by ASME
Equation 3 represents the bulk flow velocity profile in the
absence of inertia. In laminar flow, if the channel separation b
is significantly less than the length L, then the flow is
approximately parabolic. The bulk profile shown in equation 3
is parabolic in z.
For relatively high Reynolds numbers, flow can be divided
into a bulk region of inviscid flow unaffected by viscosity, and
a region close to the wall where viscosity is significant (the
boundary layer). By performing boundary layer analysis, we
may test the limits of the lubrication approximation. This is
achieved through integration of the continuity equation across
the channel separation, applying Leibniz Rule, and utilizing
equation 3 as the initial guess for velocity. The resultant
pressure–flow relation is given in equation 4.
z = h(x)
z = f(x)
−
12 μQ
∂P
=−
3
∂x
a (h( x) − f ( x ) )
The full N-S equations require the use of computational
fluid dynamics. However, when the slope of the trajectory of
fluid points is small, the lubrication approximation may be
applied for simplification. Essentially, the z-component of
velocity, w, is assumed to be significantly less than velocity in
the x-direction, u. This approximation is often applied to flow
fields in which the fluid is forced to move between two closely
spaced surfaces. For smaller Reynolds numbers, as with
laminar flow, the convective inertial effects may also be
neglected.
Thus, the simplified N-S equations are as follows:
(1.a)
∂P
= ρg
∂y
(1.b)
∂P
=0
∂z
12μQ
1
∂x
∫
a 0 (h( x) − f ( x) )3
L
P0 − PL =
1
∫ (h( x) − f ( x))
(1.c)
(3)
∂x =
0
L
3
beff
(7)
If f(x) and h(x) are constant values, i.e. the channel walls
are hydraulically smooth, this expression is valid and the
effective separation is:
Integrating the x- and y-components of the simplified N-S
equations (1.a and 1.b) and applying the no-slip boundary
conditions yields the following:
1 ∂P
(z − h( x) )(z − f ( x) )
u=
2 μ ∂x
(6)
From here, it is possible to input the exact wall functions,
f(x) and h(x). However, in prior works by Brackbill [15] and
Brackbill and Kandlikar [16, 17] an effective channel
separation was used for evaluation of friction factor. It is
possible to incorporate said parameter by defining the effective
channel width as follows:
3
(2)
(5)
Integrating the differential equation along a specified
length results in pressure-drop as a function of flow rate and
roughness geometry, as seen in equation 6.
L
P = ρgy + p (x )
(4)
Solving for differential pressure:
Figure 1. Representative Channel and Roughness Geometry;
400μm root separation, 97μm roughness height, 405μm pitch.
∂P
∂ 2u
=μ 2
∂x
∂z
1 ∂P
(h( x) − f ( x) )3 = Q
a
12 μ ∂x
beff = h ( x ) − f ( x )
(8)
The pressure-flow relation from equation 6 then becomes:
P0 − PL =
Equation 2 allows for an understanding of the effect of
gravity on flow conditions. It may be used to evaluate changes
in the flow field in the y-direction, which will be covered in
later analyses, but will not be discussed here.
3
12μQ L
3
a beff
(9)
Copyright © 2009 by ASME
Kandlikar et al. [3] set forth
surface of significant random
roughness. This new roughness
function of the average parameters
surface analysis.
For comparison with existing correlations, the following
equation for friction factor in smooth ducts, based solely on
channel aspect ratio, from Kakaç et al. [18], was evaluated:
f =
24
(1 − 1.3553α + 1.9467α 2 − 1.7012α 3
Re
+ 0.9564α 4 − 0.2537α 5 )
(10)
a method for assessing a
roughness or structured
factor (equation 12) is a
typically obtained through
ε Fp = R p + FdRa
= Max - Ra + FdRa
In addition, the Fanning friction factor was evaluated using
both the constricted parameter method and the wall function
method via the following equation:
f =
Dh ΔP
2 ρu 2 L
(12)
Here, Max represents the highest peak obtained from one
evaluation length of the profile. The floor profile, FdRa, also
called Fp, is the average of all points below the mean line, Ra.
These additional values for the 815μm surface are also shown
in Figure 2.
The usefulness of the new roughness parameter εFp is in
developing a constricted channel separation to use for beff in
equation 9. This constricted separation is calculated as follows:
(11)
Equation 9 is valid for macroscale channels, or channels of
low relative roughness, if one takes the effective separation beff
to be the root dimension of the channel, neglecting the peaks
projecting into the flow. In Figure 1, this value would be
400μm. However, for micro- and minichannels, or channels in
which roughness is relatively high, as in Figure 1, the channel
separation must be given careful consideration.
bcf = b − 2ε Fp
(13)
where b is the root separation, excluding roughness elements,
between rough walls. In Figure 1, the root separation is
400μm. Figure 3 shows the same channel with the separation
based on the constricted parameter as well as the average
roughness. It can be seen that using the average roughness
does not entirely account for the peaks. It was shown by
Young et al. [19] that for surfaces of natural or uniform
roughness, the Ra value for vastly different surfaces may be the
same, whereas the constricted parameter provides a more
representative value for the height of asperities. In a
microscale channel, the distance between the Ra lines is
significantly larger than the constricted separation, as much as
one third of the channel separation.
CHANNEL SEPARATION
Averaging amplitude parameters such as the average
roughness, Ra, and root-mean-square roughness, Rq, are
frequently used to describe roughness of machined surfaces.
Simple average parameters, however, are insufficient for
structured two-dimensional roughness [19].
A sample of transverse rib roughness, machined via ball
end mill on CNC, and its associated Ra and Rq values are
shown in Figure 2. The average lines lie along the middle of
the height of the profile. Utilizing these standard parameters
would neglect the effect of significant peaks, which may
project through the boundary layer.
Figure 3. Representative Channel and Roughness Geometry;
400μm root separation, 97μm roughness height, 405μm pitch,
Average Roughness and Constricted Parameter εFp
Figure 2. Profilometer Data for 815μm Pitch Structured
Roughness Profile; Average Roughness Parameters and
Constricted Parameter εFp
4
Copyright © 2009 by ASME
The opposing wall, represented by h(x), is simply the
negative of f(x). These equations may then be input directly
into equation 6 to evaluate the theoretical pressure drop with
the use of Gaussian quadrature to approximate the complex
integral. For the available milled surfaces and accompanying
experimental flow data, however, the error associated with this
curve-fitting technique is non-negligible.
This constricted parameter was used by Kandlikar et al. [3]
to modify the Moody diagram by incorporating it into
constricted hydraulic diameter as follows:
D h ,cf =
4 Acf
Pcf
=
2 abcf
a + bcf
(14)
This constricted diameter is used to calculate a constricted
Reynolds number, and may be used in further theoretical
analysis.
The constricted parameter is effective in accounting for
natural or random roughness, as well as structured roughness.
However, for structured two-dimensional roughness, as
considered in this work, it becomes necessary to consider the
height and pitch values as well. This is achieved by
maintaining the wall roughness as a function of x.
For comparison with experimental data, curve fitting was
used to approximate the roughness profiles used in experiments
by Brackbill [15]. This is achieved by examining a given
roughness profile and formulating a sinusoidal function that fits
the curvature. Leaving all coefficients as variables, the least
sum of squares method is used to obtain a best fit function.
FURTHER THEORETICAL ANALYSIS
For higher Reynolds numbers, the inertial term in the xcomponent of the N-S equation may be non-negligible. This
further analysis allows us to understand the impact of the
lubrication approximation itself without the added complication
of unknown velocity profile assumptions. Also, it allows for
examination of the velocity profile from the boundary layer
solution and the behavior of the inertial term as the velocity
increases and hydraulic diameter decreases.
Putting the differential pressure function (equation 5) into
the velocity profile (equation 3):
u=
6Q
(z − h( x))(z − f ( x) )
3
a(h( x) − f ( x) )
(16)
In equation 16, velocity is strictly a function of flow rate
and wall geometry. The profile is parabolic and the equation
satisfies both the boundary conditions and the continuity
equation. We can now examine the velocity profile from the
boundary layer solution and the behavior of the inertial term as
the Reynolds number increases.
Without neglecting the inertial term, the x-component of
N-S equation is:
−
∂P
∂ 2u
∂u ⎞
⎛ ∂u
+ μ 2 = ρ⎜ u + w ⎟
∂x
∂z
∂z ⎠
⎝ ∂x
(17)
Through the use of u-substitution and Leibniz Rule, this
differential equation simplifies to:
−
Figure 4. Curve Fit of 405μm Pitch Structured Roughness Profile
Figure 4 gives an example of a curve fit to regular periodic
roughness with pitch of 405μm. This surface, machined in the
same method as that of Figure 2, forms the lower wall of a
channel as seen in Figure 1. The wall function depicted above
takes the following form:
⎛π
f ( x ) = h cos p ⎜
⎝λ
⎞ b
x⎟ −
⎠ 2
∂P
(h( x) − f ( x)) + μ ∂u
∂z
∂x
h( x)
=ρ
f ( x)
∂
∂x
h( x)
∫ u ∂z
2
(18)
f ( x)
If the velocity profile is known, this equation may be
solved for a pressure–flow relation that includes the inertial
term. Therefore, we use the velocity profile developed
previously (equation 16) as an initial guess, yielding the
following pressure-drop equation:
(15)
⎛ ∂h ∂f ⎞
−
L
L
12 μQ
1
6 ρQ 2 ⎜ ∂x ∂x ⎟
⎟∂x
⎜
x
ΔP = −
∂
+
a ∫0 (h − f )3
5a 2 ∫0 ⎜ (h − f )3 ⎟
⎟
⎜
⎠
⎝
where h is the roughness height, λ the pitch, and b the root
channel separation. The εFp value for the surface shown in
Figure 4, as obtained from profilometer data, is 99.71μm, while
the height h obtained from curve fitting is 97μm.
(19)
Notice that the first term of this equation is the same as the
previous analysis (equation 6). The inertial integral is not
5
Copyright © 2009 by ASME
function method. When compared with the Kakaç equation, a
percent error of approximately 1% was found for all values of
Reynolds numbers and channel separations.
Prior to assessing flow in the presence of structured
roughness, the roughness geometry itself was analyzed. For
consistency, the height of roughness elements was maintained
at 50μm and pitch was varied such that λ/h ranged from 2 to
12. A summary of the roughness values is provided in Table 1.
As the pitch increases, average amplitude values decrease
while the constricted parameter, εFp, increases. This is due to
the decrease in the density of asperities on the surface,
indicating that the constricted parameter is more sensitive to
these extreme peaks than the average parameters, Ra and Rq.
trivial to solve, however, and complications arise when using
Gaussian quadrature, as was used in the initial analysis. To
give a preliminary indication of the usefulness of this analysis,
its limits with respect to roughness and flow performance may
be assessed.
If inertia were negligible, this formulation would limit
back to the lubrication approximation equation (equation 6).
Similarly, in the smooth wall case, f(x) and h(x) are constants,
the derivatives of which are zero. The inertial integral will then
yield a negligible constant. This indicates that the formulation
is consistent with the boundary layer analysis for the case of
hydraulically smooth walls.
RESULTS
Table 1. Comparison of Roughness Values for Wall Functions
Initial validation was performed for the hydraulically
smooth channel case. Experimental data from Brackbill [15]
was used for comparison. Flow rate was varied to cover the
range of Reynolds numbers from 487 to 2322. Channel
separation values were chosen as 200, 300, and 500μm for
comparison with experimental data. For theoretical analysis,
the wall functions were taken to be constant and pressure drop
was evaluated via equations 6 and 9. Friction factors were then
calculated using equations 10 and 11. Figure 5 shows the
comparison between the Kakaç equation and Brackbill’s
experimental data for a hydraulically smooth channel. The
Kakaç line shown is for the 200μm separation, as there is
minimal difference in aspect ratio for these channels, thus the
theoretical friction factor lines are very closely spaced. For
each of the channel separations, percent error between theory
and experimental data is less than 3% at all Reynolds numbers.
λ/h
Ra
Rq
FdRa
Rp
εFp
2
25.0
30.6
9.4
25.0
34.4
3
18.5
25.9
4.2
31.5
35.7
5
11.5
20.2
1.3
38.5
39.9
8
7.0
15.8
0.4
43.0
43.4
10
6.3
14.9
0.3
43.7
44.0
12
4.8
13.1
0.2
45.2
45.4
Theoretical analysis of flow in the presence of twodimensional roughness was performed on idealized surfaces by
incorporating the pressure drop from equations 6 and 9 into the
Fanning friction factor equation. The format of equation 15
was used as the framework for the wall functions, where h was
held at a constant value of 50μm, λ was varied according to the
ratios set forth in Table 1, and separation was varied to cover
the range of relative roughness εFp/Dh from 3-15%
(unconstricted). That is, the root separation values modeled
were 150μm, and 200-600μm in 100μm increments.
Another ratio worth noting is the roughness height to
channel separation, h/b. Similar to relative roughness, it is
more specific to rectangular channels of low aspect ratio in that
it is indicative of the extent to which the roughness peaks
intrude into the bulk flow. For example, an h/b ratio of 0.25
indicates that the roughness height is one fourth of the channel
separation. The constricted parameter, though sensitive to
changes in pitch, is based on amplitude averages and is by
definition never greater than or equal to the exact roughness
height. For this reason, the ratio of roughness height to root
channel separation is most useful in comparing channels with
the same roughness profile but different separations.
Evaluating the Kakaç equation with the constricted aspect
ratio proved to have little effect on the friction factor, as
compared with the unconstricted Kakaç equation, because
small changes in separation result in minute changes in the
aspect ratio. The percent error between the constricted and
unconstricted forms of the equation was found to be
consistently at about 0.2% for all Reynolds numbers and all
values of εFp/Dh. This error decreases marginally as channel
separation increases. For this reason, the unconstricted Kakaç
friction factor was used for comparison with the wall function
and constricted parameter forms of the Fanning friction factor.
Figure 5. Validation of Experimental Setup with Smooth Channel
The constricted parameter bcf of a hydraulically smooth
channel (εFp = 0) is equivalent to the root separation. Thus,
incorporating the constricted separation into the Fanning
friction factor equation yields zero percent error when
compared with the friction factor found through the wall
6
Copyright © 2009 by ASME
Figure 6. Comparison of Wall Function Method with Constricted
Flow Method for b = 150μm and λ/h = 3, 5, 8, and 10
Figure 7. Comparison of Wall Function Method with Constricted
Flow Method for λ/h = 5 and b = 150, 200, 300, and 500μm
Overall, the constricted Fanning friction factor compares
well with the Kakaç friction factor in all cases involving
transverse rib roughness, though it consistently predicts
marginally lower values. For all h/b, or εFp/Dh, and λ/h values,
the difference between these friction factors remains less than
2%. The difference increases to 3% as λ/h increases to 10 and
h/b decreases to 0.1. The pitch-to-height ratio has minimal
effect on the constricted flow method as that method is a strong
function of the average amplitude parameters, and not the
roughness pitch.
For a constant channel separation, as λ/h increases, the
wall function method converges to the constricted flow friction
factor, or Kakaç friction factor for smooth ducts. An example
of this convergence for channel separation of 150μm is shown
in Figure 6. Higher values of λ/h indicate lower roughness
effect on fluid flow, in that the isolated peaks become flow
obstructions rather than periodic roughness, eventually limiting
to the hydraulically smooth case.
Similarly, for any constant value of λ/h, as the channel
separation increases, the wall function method again converges
to the constricted friction factor, as seen in Figure 7 and 8.
This is due to the fact that the microscale roughness ceases to
affect the bulk flow when the channel size increases to the
miniscale range [13]. In these cases, aspect ratio is increasing
with the increase of separation from 150μm to 600μm, but the
corresponding decrease in the Kakaç friction factor is 1.2%.
The same decrease is seen in the constricted friction factor
when channel separation increases, though the constricted flow
method predicts a lower friction factor.
The wall function method compares well with the
constricted parameter method when the channel size is
relatively large, i.e. in the minichannel range, and when εFp/Dh
is less than 5%. The percent difference between the two
correlations outside of this extreme is in excess of 10%.
Figure 8. Comparison of Wall Function Method with Constricted
Flow Method for λ/h = 3 and b = 150, 200, 300, and 500μm
Figures 7 and 8 show the increase in theoretical friction
factor from the wall function method as the channel size
decreases, for λ/h of 5 and 3, respectively. In both figures, the
“constricted” line corresponds to a separation of 150μm.
Higher separation values result in reduced friction factors, the
difference being 3.7% between the largest and smallest
separations modeled. Thus, for clarity, only the highest
constricted friction factor obtained from this method is
displayed in the figures. The wall function method sees a
difference greater than 130% for friction factors between the
150μm and 600μm separations.
In general, the wall function method shows that for larger
hydraulic diameters and larger λ/h ratios, i.e. low relative
7
Copyright © 2009 by ASME
roughness (εFp/Dh < 5%) the friction factor deviates little from
the constricted flow and smooth channel correlation friction
factors. As the channel separation decreases, the roughness
height becomes on the order of the channel separation,
resulting in a significant increase in roughness effect. For large
roughness pitch values, this effect is not evident until the ratio
of roughness height to channel separation h/b is at least 0.25.
For lower values of λ/h, the difference between the wall
function method and the constricted flow method is significant
for h/b ≈ 0.08.
REFERENCES
[1]
[2]
[3]
[4]
CONCLUSIONS
Theoretical treatment of laminar flow in rectangular miniand microchannels was assessed via the constricted flow
method and wall function method. Hydraulic diameter ranged
from 296.4μm to 1143.7μm, the corresponding constricted
hydraulic diameter range being 123.2μm to 1018.1μm. The
range of relative roughness assessed was 3% to 15%.
Roughness pitch-to-height ratio ranged from 2 to 12.
The use of the constricted flow method is effective for
random or uniform roughness in that it allows for assessment of
the bulk flow, neglecting the boundary layer. Comparison with
the Kakaç friction factor shows a difference of less than 3% in
all cases of flow rate and channel separation for structured
roughness. It should be noted that using the constricted
hydraulic diameter results in significant differences in relative
roughness values but negligible variance in friction factors.
Both the constricted flow method and wall function method are
comparable with experimental data for laminar flow in smooth
channels.
The wall function method obtained through lubrication
approximation is effective in incorporating the pitch-to-height
ratio of transverse rib roughness into the flow equations. The
resulting friction factors are comparable with the constricted
flow method in the hydraulically smooth minichannel range for
laminar flow, as well as the rough cases when the hydraulic
diameter is large or relative roughness is low. In the case of
high relative roughness, the wall function method is a stronger
function of roughness geometry than the constricted method.
In order to assess the usefulness of the lubrication
approximation further, experimental data will need to be
obtained. A test set is currently being constructed for this
purpose. The ideal surfaces addressed in this work will be
generated via wire EDM and tested experimentally. For
comparison with the wall function and constricted flow
methods, the surfaces will be evaluated with precision
metrology equipment for curve fitting and subsequent analysis.
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
ACKNOWLEDGEMENTS
This material is based upon work supported by the
National Science Foundation under Award No. CBET0829038.
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.
[18]
[19]
8
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