Upload
jose-yovany-galindo-diaz
View
214
Download
0
Embed Size (px)
Citation preview
7/26/2019 An Integral Perturbation Model of Flow and Momentum Transport in Rotating
1/12
An integral perturbation model of flow and momentum transport in rotatingmicrochannels with smooth or microstructured wall surfaces
Vince D. Romanina) and Van P. Careyb)
Department of Mechanical Engineering, University of California, Berkeley 94720-1740, USA
(Received 18 October 2010; accepted 7 July 2011; published online 17 August 2011)
This paper summarizes the development of an integral perturbation solution of the equations
governing flow momentum transport and energy conversion in microchannels between disks of
multiple-disk drag turbines such as Tesla turbines. Analysis of this type of flow problem is a key
element in optimal design of Tesla drag-type turbines for geothermal or solar alternative energy
technologies. In multiple-disk turbines, high speed flow enters tangentially at the outer radius of
cylindrical microchannels formed by closely spaced parallel disks, spiraling through the channel to
an exhaust at a small radius, or at the center of the disk. Previous investigations have generally
developed models based on simplifying idealizations of the flow in these circumstances. Here,
beginning with the momentum and continuity equations for incompressible and steady flow in
cylindrical coordinates, an integral solution scheme is developed that leads to a dimensionless
perturbation series solution that retains the full complement of momentum and viscous effects to
consistent levels of approximation in the series solution. This more rigorous approach indicates all
dimensionless parameters that affect flow and transport and allows a direct assessment of the relative
importance of viscous, pressure, and momentum effects in different directions in the flow. Theresulting lowest-order equations are solved explicitly and higher order terms in the series solutions
are determined numerically. Enhancement of rotor drag in this type of turbine enhances energy
conversion efficiency. We also developed a modified version of the integral perturbation analysis that
incorporates the effects of enhanced drag due to surface microstructuring. Results of the model
analysis for smooth disk walls are shown to agree well with experimental performance data for a
prototype Tesla turbine and predictions of performance models developed in earlier investigations.
Model predictions indicate that enhancement of disk drag by strategic microstructuring of the disk
surfaces can significantly increase turbine efficiency. Exploratory calculations with the model
indicate that turbine efficiencies exceeding 75% can be achieved by designing for optimal ranges of
the governing dimensionless parameters. VC 2011 American Institute of Physics.
[doi:10.1063/1.3624599]
I. INTRODUCTION
Because multiple disk drag-type turbines, like the Tesla
turbine,1 are generally simple to manufacture and robust,
they are now being reconsidered as an expander option for
renewable energy applications such as solar Rankine com-
bined heat and power systems, and geothermal power sys-
tems. To achieve optimized designs for applications of this
type, models of the flow, momentum transport, and energy
conversion in such devices are needed that are accurate
and that illuminate the parametric trends in expander
performance.
Multiple-disk Tesla-type drag turbines rely on a mecha-nism of energy transfer that is fundamentally different from
most typical airfoil-bladed turbines or positive-displacement
expanders. A complete understanding of turbine operation
requires an analytical treatment of the unique fluid mechan-
ics processes that effect energy transfer from the fluid to the
rotor. A schematic of the Tesla turbine can be found in
Figure 1. The turbine rotor consists of several flat, parallel
disks mounted on a shaft with a small gap between each
disk; these gaps form the cylindrical microchannels through
which flow will be analyzed. Exhaust holes on each disk are
placed as close to the center shaft as possible. Flow from the
nozzle enters the cylindrical microchannels at an outer radius
ro where roDH (DH is the hydraulic diameter of themicrochannels). The flow enters the channels at a high speed
and a direction nearly tangential to the outer circumference
of the disks and exits through an exhaust port at a much
smaller inner radiusri. Energy is transferred from the fluid to
the rotor via the shear force at the microchannel walls. As
the spiraling fluid loses energy, the angular momentum drops
causing the fluid to drop in radius until it reaches the exhaust
port atri. This process is shown in Figure2.
Several authors have studied Tesla turbines in order to
gain insight into their operation. In the 1960s, Rice2 and
Breiter and Pohlhausen3 conducted extensive analysis and
testing of Tesla turbines. However, Rice did not directly com-
pare experimental data to analytical results, and lacked an an-
alytical treatment of the friction factor. Breiteret al.provided
a preliminary analysis of pumps only and used a numerical so-
lution of the energy and momentum equations. Hoya4 and
Guha5 extensively tested sub-sonic and super-sonic nozzles
a)Electronic mail: [email protected].
b)Electronic mail: [email protected].
1070-6631/2011/23(8)/082003/11/$30.00 VC 2011 American Institute of Physics23, 082003-1
PHYSICS OF FLUIDS23, 082003 (2011)
http://dx.doi.org/10.1063/1.3624599http://dx.doi.org/10.1063/1.3624599http://-/?-http://-/?-http://dx.doi.org/10.1063/1.3624599http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://dx.doi.org/10.1063/1.3624599http://-/?-http://-/?-http://dx.doi.org/10.1063/1.3624599http://dx.doi.org/10.1063/1.36245997/26/2019 An Integral Perturbation Model of Flow and Momentum Transport in Rotating
2/12
with Tesla turbines, however, their analysis was focused on
experimental results and not an analytical treatment of the
fluid mechanics that drive turbine performance. Carey6 pro-
posed an analytical treatment that allowed for a closed-form
solution of the fluid mechanics equations in the flow in the
rotor; however, Careys model analysis invoked several ideal-izations that neglected viscous transport in the radial and axial
directions, and treated tangential viscous effects using a fric-
tion factor approach.
While previous investigations of momentum transport in
Tesla turbines described above have generally developed
models based on idealizations of the flow in these circum-
stances, here, an integral perturbation analysis framework
was explored as a means of providing a more rigorous fluid
mechanical treatment of the flow that also quantifies the
effects of relevant dimensionless parameters on perform-
ance. Beginning with the momentum and continuity equa-
tions for incompressible and steady flow in cylindrical
coordinates, an integral solution scheme was developed that
leads to a dimensionless perturbation series solution that
retains the full complement of momentum and viscous
effects to consistent levels of approximation in the series so-
lution. This more rigorous approach directly indicates the
dimensionless parameters that affect flow and transport and
allows a direct assessment of the relative importance of pres-
sure, viscous, and momentum effects in different directions
in the flow.
The performance analysis in the previous investiga-
tions described above suggest that enhancement of rotor
drag in this type of turbine generally enhances energy con-
version efficiency. Information obtained in recent funda-mental studies indicates that laminar flow drag can be
strongly enhanced by strategic microstructuring of the wall
surfaces in microchannels.79 The conventional Moody dia-
gram shows that for most channels, surface roughness has
no effect on the friction factor for laminar flow in a duct.
However, in micro-scale channels several physical near-
surface effects can begin to become significant compared to
the forces in the bulk flow. First, the Moody diagram only
considers surface roughnesses up to 0.05, which is small
enough not to have meaningful flow constriction effects. In
microchannels, manufacturing techniques may often lead to
surface roughnesses that comprise a larger fraction of the
flow diameter. When the reduced flow area becomes small
enough to affect flow velocity, the corresponding increase
in wall sheer can become significant. Secondly, the size,
shape, and frequency of surface roughness features can
cause small areas of recirculation, downstream wakes, and
other effects which may also impact the wall shear in ways
that become increasingly important in smaller size chan-
nels, as the energy of the perturbations become relevant
compared to the energy of the bulk flow.
In 2005, Kandlikar et al.7 modified the traditional
Moody diagram to account for surfaces with a relativeroughness higher than 0.05, arguing that above this value
flow constriction becomes important. Kandlikar proposes
that the constricted diameter be simplified to be
DcfDt 2e, wheree is the roughness height,Dtis the basediameter, and Dcfis the constricted diameter. Using this for-
mulation, the Moody diagram can be re-constructed to
account for the constricted diameter, and can thus be used
for channels with relative roughness larger than 5%. Kandli-
kar conducts experiments which match closely with this pre-
diction and significantly closer than the prediction of the
classical Moody diagram. Kandlikar, however, only conducts
experiments on one type of roughness element, and does not
analyze the effect of the size, shape, and distribution ofroughness elements, although he does propose a new set of
parameters that could be used to further characterize the
roughness patterns in microchannels.
Croce et al.8 used a computational approach to model
conical roughness elements and their effect on flow through
microchannels. Like Kandlikar, he also reports a shift in the
friction factor due to surface roughness and compares the
results of his computational analysis to the equations pro-
posed by several authors for the constricted hydraulic diame-
ter for two different roughness element periodicities. While
the results of his analysis match Kandlikars equation
(Dcf
Dt
2e) within 2% for one case, for a higher periodic-
ity Kandlikars approximation deviates from numerical
results by 10%. This example, and others discussed in Cro-
ces paper, begins to outline how roughness properties other
than height can effect a shift in the flow Poiseuille number.
Gamratet al.9 provides a detailed summary of previous
studies reporting Poiseuille number increases with surface
roughness. He then develops a semi-empirical model using
both experimental data and numerical results to predict the
influence of surface roughness on the Poiseuille number.
Gamrats analysis, to the best of the authors knowledge, is
the most thorough attempt to predict the effects of surface
roughness on the Poiseuille number of laminar flow in
microchannels.FIG. 2. (Color online) Schematic of flow through a Tesla turbine
microchannel.
FIG. 1. Schematic of a Tesla turbine.
082003-2 V. D. Romanin and V. P. Carey Phys. Fluids 23, 082003 (2011)
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-7/26/2019 An Integral Perturbation Model of Flow and Momentum Transport in Rotating
3/12
There appears to have been no prior efforts to model
and quantitatively predict the impact of this type of drag
enhancement on turbine performance. The integral perturba-
tion analysis can be modified to incorporate the effects of
enhanced drag due to surface microstructuring. The goal of
this analysis is to model surface roughness effects on mo-
mentum transport in drag-type turbines in the most general
way; therefore, surface roughness is modeled as an increase
in Poiseuille number, as reported by Croce and Gamrat. The
development of the integral perturbation analysis and evalua-
tion of its predictions are described in the following sections.
II. ANALYSIS
An analysis will now be outlined that describes first the
flow through the nozzle of the turbine and then the flow
through the microchannels of the turbine, while incorporat-
ing a treatment of microstructured walls. The resulting equa-
tions for velocity and pressure can be used to solve for the
efficiency of the turbine. The closed form solution of the
fluid mechanics equations allows a parametric exploration of
trends in turbine operation.
A. Treatment of the nozzle delivery of flow to the rotor
Before considering the flow in the rotor, a method for
predicting the flow exiting the nozzle in Figure 1 must be
considered. For the purposes of this analysis, the tangential
gas velocity (vh) at the outer radius of the rotor (ro) is taken
to be uniform around the circumference of the rotor and
equal to the nozzle exit velocity determined from one dimen-
sional compressible flow theory.
In expanders of the type considered here, the flow
through the nozzle is often choked. This was the case in ex-
pander tests conducted by Rice,2 who reported that virtuallyall the pressure drop in the device is in the nozzle and little
pressure drop occurs in the flow through the rotor. The pres-
sure ratio Po=Pntacross the nozzle for choked flow must beat the critical pressure ratio (Pt=Pnt)crit at the nozzle inlettemperature. For a perfect gas, this is computed as
Pt
Pnt
crit
2c1
c=c1; (1)
((Pt=Pnt)critis about 0.528 for air at 350 K (Ref. 10)).If the nozzle exit velocity is the sonic speed at the
nozzle throat, it can be computed for a perfect gas as
vo;catffiffiffiffiffiffiffiffifficRTt
p ; (2)
where Tt the nozzle throat temperature for choked flow, is
given by
TtTntPt=Pntc1=ccrit : (3)
For isentropic flow through the nozzle, the energy equa-
tion dictates that the exit velocity would be
vo;i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi2cpTnt 1 Po=Pntc1=c
h ir (4)
and the isentropic efficiency of the nozzle is defined as
gnoz v
2o=2
v2o;i=2
: (5)
It follows from the above relations that for a perfect gas
flowing through nozzles with efficiency gnoz, the tangential
velocity of gas into the rotor at r
ro, taken to be equal to
the nozzle exit velocity, is given by
vh rro vo ffiffiffiffiffiffiffiffignoz
p vo; i; (6)
where vo, i is computed using Eq. (4), and for choked flow
the nozzle efficiency is given by
gnoz cRPt=Pntc1=ccrit
2cp 1 Po=Pntc1=ch i : (7)
Treating the gas flow as an ideal gas with nominally
constant specific heat, Eq. (6) provides the means of deter-mining the rotor gas inlet tangential velocity (vh)rro given
the specified flow conditions for the nozzle.
B. Analysis of the momentum transport in the rotor
For steady incompressible laminar flow in microchan-
nels between the turbine rotor disks, the governing equations
for the flow are
Continuity:
r v0: (8)
Momentum:
v rv rPq
r2vf: (9)
Treatment of the flow as incompressible is justified by
the observation of Rice2 that minimal pressure drop occurs
in the rotor under typical operating conditions for this type
of expander. For this analysis, the following idealizations are
adopted:
(1) The flow is taken to be steady, laminar, and two-dimen-
sional: vz 0 and the z-direction momentum equationhas a trivial solution.
(2) The flow field is taken to be radially symmetric. Theinlet flow at the rotor outer edge is uniform, resulting in
a flow field that is the same at any angle h. All h deriva-
tives of flow quantities are, therefore, zero.
(3) Body force effects are taken to be zero.
(4) Entrance and exit effects are not considered. Only flow
between adjacent rotating disks is modeled.
With the idealizations noted above, the governing Eqs.
(8)and(9), in cylindrical coordinates, reduce to
continuity:
1
r
@rvr@r
0; (10)
082003-3 An integral perturbation model of flow Phys. Fluids 23, 082003 (2011)
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-7/26/2019 An Integral Perturbation Model of Flow and Momentum Transport in Rotating
4/12
r-direction momentum:
vr
@vr@r
v2h
r 1
q
@P
@r
1r
@
@r r
@vr@r
@
2vr
@z2 vr
r2
;
(11)
h-direction momentum:
vr
@vh@r
vrvhr 1
r
@
@r r
@vh@r
@
2vh
@z2 vh
r2
; (12)
z-direction momentum:
0 1q
@P
@z
: (13)
Equation(13)dictates that the pressure is uniform across
the channel at any (r,h) location. For the variations of the ra-
dial and tangential velocities, the following solution forms
are postulated:
vr vrr/z; (14)vhvhr/z Ur; (15)
where
/z n1n
1 2z
b
n (16)
andvrand vh are mean velocities defined as
vrr 1b
b=2
b=2
vrdz; (17)
vhr 1b
b=2b=2
vhUdz; (18)
whereb is the gap distance between disks.
For laminar flow in the tangential direction, the wall
shear is related to the difference between the mean local gas
tangential velocity and the rotor surface tangential velocity
vhvhU through the friction factor definition,
swfqv2h
2 : (19)
For a Newtonian fluid, it follows that
f sqv2h=2
l@vhU=@zzb=zqv2h
: (20)
For the purposes of this analysis, the tangential shear
interaction of the flow with the disk surface is postulated to
be equivalent to that for laminar Poiseuille flow between
parallel plates,
f PoRec
; (21)
where Rec
is the Reynolds number defined as
RecqvhDHl
; (22)
DH 2b; (23)
and Po is a numerical constant usually referred to as the Pois-euille number. For flow between smooth flat plates, the well-
known laminar flow solution predicts Po 24. For flowbetween flat plates with roughened surfaces, experiments79
indicate that a value other than 24 better matches pressure loss
data. We, therefore, define an enhancement numberFPoas
FPoPo=24; (24)which quantifies the enhancement of shear drag that may
result from disk surface geometry modifications. Note that
Eqs.(19)(23)dictate that for the postulated vhform(15),
n
1
Po=8
3FPo: (25)
It follows that:
for laminar flow over smooth walls: n 2, Po 24,FPo 1,
for laminar flow over walls with drag enhancing rough-
ness:n > 2, Po > 24,FPo> 1.
The variation of the velocity profile with n is shown in
Fig.3.
C. Radial velocity solution from the continuityequation
Substituting Eq.(14)into Eq. (10) and integrating withrespect toryields
rvr rvr/constantCr: (26)
Integrating Eq. (16) across the channel and using the
fact that b=2b=2
/dz2b=2
0
/dzb (27)
yields
b=2
b=2
rvrdz b=2
b=2
rvr/dzrvrbbCr C0r: (28)
FIG. 3. (Color online) Variation of velocity profile withn.
082003-4 V. D. Romanin and V. P. Carey Phys. Fluids 23, 082003 (2011)
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-7/26/2019 An Integral Perturbation Model of Flow and Momentum Transport in Rotating
5/12
Mass conservation requires that
2proqb=2b=2
vrdz 2proqvrrob=2b=2
/dz
2proqvrrob _mc; (29)
where _mc is the mass flow rate per channel between rotors.
Combining Eqs. (28)and(29) yields the following solutionfor the radial velocity:
vr rovror ; (30)
where
vro _mc2proqb
: (31)
D. Solution of the tangential and radial momentumequations
The next step is to substitute the postulated solutions
from Eqs. (14) and (15) into the tangential and radial mo-
mentum equations (12) and (11), integrate each term across
the microchannel, and use Eq.(27)together with the results,
b=2b=2
/2dz2b=2
0
/2dz2n12n1 b; (32)
b=2b=2
d2/
dz2
dz2
b=20
d2/
dz2
dz
4n1b
; (33)
doing so and introducing the dimensionless variables
nr=ro; (34)Wvh=Uo vhU=Uo; (35)
^P PPo=qU2o=2; (36)Vrovro=Uo; (37)e2b=ro; (38)
Rem DH=ro _mcDH
2problDH _mc
pr2ol ; (39)
converts Eqs.(11)and(12)to the forms,
@^P
@n ^P0 4n12n1n3 V
2ro W2n2
4 W2n32n1 V
2ro
Remn; (40)
2n1n1
2n12n1
e2
Rem
n W00 2n1
2n1e2
Rem1
W0
1 2n12n1
e2
Rem
1
n82n1n
Rem
W;
(41)
where W0dW=dn and W00d2 W=dn2. Solution of theseequations requires boundary conditions on the dimensionless
relative velocity and the dimensionless pressure ( W and ^P).
Here, it is assumed that the gas tangential velocity and the
disk rotational speed are specified, so Wat the outer radius
of the disk is specified. It follows that
atn
1; W
1
Wo: (42)
In addition, from the definition of ^P, it follows that
^P1 0: (43)Equations (42)(43) provide boundary conditions for solu-
tion of the dimensionless tangential momentum Equation
(41)and the radial momentum Equation (40), which predicts
the radial pressure distribution.
Sincee 2b=rois much less than 1 in the systems of inter-est here, we postulate a series expansion solution of the form,
W W0e W1e2W2 ; (44)^P ^P0e ^P1e2 ^P2 : (45)
Substituting results in Eqs.(46)(54).
O(e0):
6FPo13FPo
W00 1
n86FPo1 n
Rem
W0; (46)
^P00 12FPo
6FPo 11
n3 V2ro W20n2 4 W0 2n V2roFPo 96Remn ;
(47)
atn1 : W0 W0;ro ; ^P00; (48)
O(e1):
0 W01 1
n86FPo1 n
Rem
W1; (49)
^P01 12FPo
6FPo11
n2 W0 W14 W1; (50)
atn1 : W10; ^P10; (51)O(e2):
W02 1
n82n1n
Rem
W2
6FPo16FPo
nW000Rem
W00
Rem W0
Remn
; (52)
^P02 12FPo
6FPo11
n2 W0 W24 W2; (53)
atn1 : W20; ^P20: (54)In solving Eqs.(46)(54), the dimensionless parameters
in the equations,
082003-5 An integral perturbation model of flow Phys. Fluids 23, 082003 (2011)
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-7/26/2019 An Integral Perturbation Model of Flow and Momentum Transport in Rotating
6/12
niri=ro; (55)
W0ro W0:rovh;roUo
Uo; (56)
RemDH _mc
pr2ol ; (57)
Vro
vro=Uo; (58)
e2b=ro (59)are dictated by the choices for the following physical
parameters:
ri,ro: the inner and outer radii of the disks b: the gap between the disks, from which we can compute
DH 2b _mc: the mass flow rate per channel between rotors x Uo=ro: the angular rotation rate vh;ro : the mean tangential velocity at the inlet edge of the
rotor Po=Pnt: pressure ratio Tnt: nozzle upstream total temperature.
Also, for choked nozzle flow, the tangential velocity at
the rotor inlet will equal the sonic velocity (vh, ro a). Thedefinitions ofMo and Wrequire that the choices forMo and
W0; ro satisfy
MoUo=ffiffiffiffiffiffiffiffifficRTt
p Pt=Pnt
c1=2ccrit
W0;ro1: (60)
Solving Eqs. (46), (49), and (52) with boundary conditions
(48), (51), and (54) gives the following solutions:
W0 W0;ro Rem24FPo
efnnef1
Rem24FPo
; (61)
W10; (62)
W2efn
n
n1
nefngndn; (63)
where
gn 6FPo16FPo
W0=n W00n W000Rem
; (64)
fn 46FPo1n2
Rem: (65)
Andn* is a dummy variable of integration.
With this result, the energy efficiency of the rotor and of
the turbine, respectively, can be computed using
grm vh;oUovh;iUivh;oUo
; (66)
gi vh;oUovh;iUi
Dhisen; (67)
which rearrange to
grm1 Winini
Wo1; (68)
Wi Wnniri=ro ; (69)
gi Wo1 Winini
c1M2o
1 P
iPnt
c1=c" # : (70)
The baseline case for comparison with rough wall solu-
tions is that for a smooth wall, orFPo 1. The solution forW0 (Eq.(61)) reduces to
W0 W0;roRem
24
e
20Rem
n21
n Re
m
24 : (71)
The pressure distribution can be found numerically or
analytically by integrating equations(47),(50), and(53).
E. Higher order (e1, e2) terms
In order to evaluate the significance of the higher order
solutions of W and ^P, results are plotted for two different
operating conditions in Figures4 (velocity) and5 (pressure).
Under both scenarios, the 2nd order terms ( W2 and ^P2) are
shown to be the same order of magnitude as the 0th order
terms ( W0 and ^P0). Velocity plots for W0 and for
W0e W1e2 W2 fall nearly directly on top of each other(similarly for ^P). For values of e as high as 1=10, muchlarger than are found in most systems of interest, both e2 W2and e2^P2 are less than 0.1% of the value of the 0th order
FIG. 4. (Color online) Comparison of
velocity plots for 0th order and 2nd
order velocity solutions. In both (a) and
(b), the plots forW0 (solid line) and for
W0e W1e2 W2 (dashed line) arenearly coincident. The dotted-dash line
( W2) is shown to be the same order of
magnitude as W0, thus making it negligi-
ble when multiplied by e2. (a) Case 1:
W0 2, Rem10, ni 0.2, Vro 0.05,e 1=20; choked flow and (b) case 2:W0 1:1, Rem5, ni 0.2,Vro 0.05,e 1/20; choked flow.
082003-6 V. D. Romanin and V. P. Carey Phys. Fluids 23, 082003 (2011)
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-7/26/2019 An Integral Perturbation Model of Flow and Momentum Transport in Rotating
7/12
term for the two cases shown. Note that Eq.(62)along with
Eqs. (50) and(51) show that W1 ^P10. Henceforth, wecan neglect the 1st and 2nd order terms, and only the 0th
order terms ( W0 and ^P0) will be considered.
III. COMPARISON OF SMOOTH WALL CASE WITHEARLIER MODEL PREDICTIONS
The W0 solution corresponds closely with the solution
developed by Carey,6 only differing by numerical constants.
A comparison of results with the model from Careys earlier
model is shown in Figure 6. Carey6 made several assump-
tions, including ignoring radial pressure effects, treating the
flow as inviscid with a body force representation of drag, and
ignoring z-derivatives of velocity. In the present analysis,
initial assumptions were more conservative and terms were
removed based on the arguments of the perturbation analysis.
The similarities in the results of this analysis with that of
Carey verify that the assumptions made were valid. Addi-tionally, Careys analysis was compared extensively with
experimental data in Romanin and Carey,11 so a close corre-
lation between the two approaches is encouraging.
A comparison with previous experimental data11 can be
seen in Table I. The agreement between test data and the
model predicted efficiency is reasonable considering the
uncertainty of the test data, and is similar to the accuracy of
the earlier model developed by Carey.6 However, due to the
small range of nondimensional parameters explored in the
experimental data, a more extensive comparison of test data
with the model may generate additional insights.
Figure7 shows a 3D plot of turbine efficiency as a func-
tion of Rem and W0;ro for the operating parameters in the firstfour lines of Table I. The data points are overlaid on top of
the surface plot, which shows how the data compares to the
predictions of efficiency. The figure shows that the analytical
model correctly predicts that decreasing W0;ro will increaseefficiency, and suggests that decreasing both Rem and W0;rocan dramatically improve performance.
A 3D plot of turbine efficiency with typical operating
parameters, and over ideal ranges of Rem and W0;ro , is shownin Figure8. At very low values of Rem and W0;ro , the analysisshows that very high turbine efficiencies can be achieved.
Practical issues arise when generating power in microchan-
nels such as these at very low values ofW0;ro and Rem.
Reducing W0;ro requires the rotor to be spinning at speeds
very close to the air inlet speeds. This is difficult to achieve
because it requires very low rotor torque and high speeds,
which may require high gear ratios to achieve in some appli-
cations. Also, lower Reynolds numbers require very smalldisk spacings (b) and larger disk radii (ro).
IV. MODELING OF FLOW VELOCITY WITHROUGHENED OR MICROSTRUCTURED SURFACES(FPO>1)
Now that the perturbation analysis has resulted in equa-
tions that define the operating conditions and efficiency of
the turbine as a function ofFPo, we can analyze the effect of
surface roughness on turbine performance. Developing a
direct correlation between surface roughness and FPo is a
FIG. 5. (Color online) Comparison of
pressure plots for 0th order and 2nd
order velocity solutions. In both (a) and
(b), the plots for ^P0 (solid line) and for^P0e ^P1e2^P2 (dashed line) are nearlycoincident. The dotted-dash line ( W2) is
shown to be the same order of magni-
tude as W0, thus making it negligible
when multiplied by e2. (a) Case 1:W0 2, Rem10, ni 0.2, Vro 0.05,e 1=20; choked flow and (b) case 2:W0 1:1, Rem5, ni 0.2,Vro 0.05,e 1=20; choked flow.
FIG. 6. (Color online) Comparison of
solutions from the perturbation method
and the model developed by Carey.6 (a)
Case 1: W02, Rem10, ni 0.2,Vro 0.05; choked flow. The analysispredicts a turbine isentropic efficiency of
gi 26.1% while the analysis by Carey6predicts gi 27.0% (b) case 2:W01:1, Rem5, ni 0.2,Vro 0.05;choked flow. The analysis predicts a tur-
bine isentropic efficiency ofgi 42.3%while the analysis by Carey6 predicts
gi 42.5%.
082003-7 An integral perturbation model of flow Phys. Fluids 23, 082003 (2011)
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-7/26/2019 An Integral Perturbation Model of Flow and Momentum Transport in Rotating
8/12
detailed process that involves characterizing specific geo-
metric properties of the roughness features and is beyond the
scope of this analysis. Here, we will only discuss the effects
of increasing FPo. Kandlikar8 reported values forFPoas high
as 3.5 for roughened surfaces in microchannels, so values up
toFPo 3.5 will be considered.
A. Discussion of the velocity and pressure fields
Figure9 shows that the velocity profile is significantly
altered by using a roughened surface. Equation (70) shows
that the exit velocity Wi should be minimized to increase ef-
ficiency and indeed the efficiency does increase with FPo.
FPo 2 results in a turbine isentropic efficiency ofgi 45.1%, compared to an efficiency of gi 42.3% for
FPo 1.Figure10 shows the dimensionless pressure ^Pas a func-
tion ofn for several values ofFPo. The figure shows that the
dimensionless pressure decreases with increasing FPo. This
can be attributed to the competing effects of centripetal forceand radial pressure. Increasing surface roughness decreases
the velocity and, therefore, the centripetal force is decreased.
The required pressure field to balance the centripetal force
on the fluid is, therefore, also decreased. It is important to
note that this does not contradict the conventional knowledge
that the pressure drop increases along the direction of the
flow as the surface roughness is increased. The pressure drop
described here is in the radial direction, while the fluid flow
has both a radial and circumferential component.
B. Performance enhancement due to mictrostructuredsurfaces
Over the entire range of values for FPo discussed by
Croce,8 Figure11 shows that efficiency increases a total of
3.8 percentage points, which amounts to a 9.2% improve-
ment in performance over a smooth wall.
Figure12 shows a surface plot of efficiency as a func-
tion of two non-dimensional parameters, Rem and W0;ro . It isshown that increasing surface roughness can yield especially
significant performance improvements for higher Reynolds
numbersrather than lower. Similar trends to those reported
by Carey6 and Romanin11 can be seen in Figure12; it is clear
that high efficiency turbine designs should strive for Reyn-
olds numbers and dimensionless inlet velocity differences to
be as small as possible. It is also shown that penalties due to
TABLE I. Comparison of analysis with experimental data from Romanin
and Carey.11
_mcg/s
x
rad/s Rem W0;rogi;exp(%)
gi;model(%)
1.64 450 47.5 18.2 3.1 2.3
1.66 784 48.1 10.0 4.9 3.6
1.65 953 47.8 8.1 5.9 4.4
1.65 1110 47.8 6.8 6.8 5.12.37 708 68.6 11.2 3.0 2.3
2.40 1110 69.5 6.8 4.3 3.3
2.38 1370 68.9 5.3 5.3 4.1
2.38 1590 68.9 4.4 6.0 4.8
FIG. 7. (Color online) A plot of experimental data from the first four lines
of Table I with a surface plot of efficiency (gi) from Eq. (71) (FPo 1(smooth wall),c
1.4 (air),n
i 0.45,P
i=P
nt0.4; choked flow).
FIG. 8. (Color online) A plot of efficiency (gi) as a function of dimension-
less tangential velocity difference at the inlet ( W0; ro ) and modified Reynolds
number (Rem) for typical operating conditions: FPo 1 (smooth wall),c 1.4 (air),ni 0.2,Pi=Pnt 0.5; choked flow.
FIG. 9. (Color online) Velocity vs. n for several values ofFPo. W01:1,Re
m5, n
i 0.2; choked flow.
082003-8 V. D. Romanin and V. P. Carey Phys. Fluids 23, 082003 (2011)
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-7/26/2019 An Integral Perturbation Model of Flow and Momentum Transport in Rotating
9/12
higher values of Rem and W0;ro are less dramatic for rough-ened surfaces.
By taking advantage of microstructured surfaces, largerdisk gaps, and smaller disks can be used while limiting penal-
ties to efficiency. For example, the nondimensional turbine pa-
rameters outlined in case 2 (see Figure 4(b)) can be used to
deduce the physical parameters in the right hand side of Eqs.
(55)(59). Using this set of physical parameters, the Poiseuille
number can be doubled (FPo 2), and the radius can bedecreased while keeping other parameters constant until the
efficiency is equivalent to that achieved by the parameters
from case 2 (Figure4(b)). This process results in a turbine ra-
dius ofro 18.6 cm, down from ro 34.7 cm in the smoothwall case. In other words, doubling the Poiseuille number, in
this case, allowed for a 46% reduction in turbine size with
equivalent performance. Similar trade-offs with other physicalparameters can be explored, allowing greater flexibility in
high-efficiency turbine design. The values outlined in this
example are not universal, however, as Figure 13 shows that
performance increases due to roughened surfaces vary with
non-dimensional parameters (e.g., performance increases are
less significant at low modified Reynolds (Rem) numbers).
V. STREAMLINE VISUALIZATION
The model theory developed here also provides the
means to determine the trajectory of streamlines in the rotor
using the h and r direction velocity components. Starting at
anyh location a the rotor inlet (n r=ro 1), over time, thefluid traces an (r, h) path through the channel between adja-
cent disks that is determined by integrating the differential
relations,
rdh vhdt; (72)dr vrdt: (73)
Combining the above equations yields the following dif-
ferential equation that can be integrated to determine the de-
pendence ofhwithralong the streamline,
dh
dr
st
vhvrr
: (74)
Note that since the velocities are functions only ofr, the
entire right side of the above equation is a function ofr. In
terms of the dimensionless variables described above, the
streamline differential Equation(74)can be converted to the
form,
dh
dn
st
nW
Vro; (75)
whereVrois the ratio of radial gas velocity to rotor tangential
velocity at the outer edge of the rotor,
Vro vroUo
_mc2probqoUo
: (76)
FIG. 10. (Color online) Dimensionless pressure ( ^P) vs.n for several values
ofFPo. W01:1, Rem 5,ni 0.2,Vro 0.05; choked flow.FIG. 11. (Color online) Efficiency (gi) vs.FPoforni 0.2 and choked flow.
FIG. 12. (Color online) A 3D surface
plot of efficiency (gi) as a function of the
inlet dimensionless tangential velocity
difference ( W0;ro ) and Reynolds number
(Rem) for typical operating parameters(c 1.4 (air), ni 0.2, Pi=Pnt 0.5;choked flow). (a) FPo 1 and (b)
FPo 2.
082003-9 An integral perturbation model of flow Phys. Fluids 23, 082003 (2011)
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-7/26/2019 An Integral Perturbation Model of Flow and Momentum Transport in Rotating
10/12
Rotor streamlines determined by integrating Equation
(75) for Wo3:0, (DH=ro)Rem 5.0, Vro 0.05, andni 0.2 are shown in Figure 14. Flow along one streamlineenters the rotor at h 0, whereas the other streamlinebegins at h 180. The model can be used to predict howthe inward spiral path of the flow changes as the governing
parameters are altered. Figure14shows streamlines from the
roughened surfaces have a larger radial component than
those generated with a smooth surface. An analytical method
for predicting streamlines can be useful in designing com-
plex disk surface geometries that consider flow direction,
such as surface contours or airfoils.
VI. CONCLUSIONS
It has been shown that the use of an integral perturbation
analysis scheme allows construction of a series expansion so-
lution of the governing equations for rotating microchannel
flow between the rotor disks of a Tesla-type drag turbine. Two
idealizations in the model may limit its accuracy. One is the
postulated tangential velocity profile used to facilitate the inte-
gral analysis. The other is the idealization of the inlet flow as
being uniform over the outer perimeter of the disk. Real tur-
bines of this type have a discrete number of nozzles that
deliver inlet flow at specific locations. The gap between the
outer edge of the rotor disk and the housing generally allows
the flow to distribute itself somewhat over the perimeter.
Thus, the idealization of uniform inlet flow may be a good
one if the flow is delivered by several nozzles around the pe-
rimeter. Clearly, however, the model developed here is
expected to be most accurate under conditions, where the
postulated tangential velocity profile and the idealization of
uniform inlet velocity are consistent with expected actual con-
ditions in the flow.
Although the model may be somewhat limited by the
idealizations described above, it has several very useful
advantages. One is that it provides a rigorous approach that
retains the full complement of momentum and viscous
effects to consistent levels of approximation in the series
solution. Another is that by constructing the solution in
dimensionless form, the analysis directly indicates all the
dimensionless parameters that dictate the flow and trans-
port, and, in terms of these dimensionless parameters, it
provides a direct assessment of the relative importance of
viscous, pressure, and momentum effects in different direc-tions in the flow. Our analysis also indicated that closed
form equations can be obtained for the lowest order contri-
bution to the series expansion solution, and the higher
order term contributions are very small for conditions of
practical interest. This provides simple mathematical rela-
tions that can be used to compute the flow field velocity
components and the efficiency of the turbine, to very good
accuracy, from values of the dimensionless parameters for
the design of interest. In addition, it has been demonstrated
here that this solution formulation facilitates modeling of
enhanced rotor drag due to rotor surface microstructuring.
The type of drag turbine of interest here is one of very few
instances in which enhancement of drag is advantageous influid machinery. We have demonstrated that by parameter-
izing the roughness in terms of the surface Poiseuille num-
ber ratio (FPo), the model analysis developed here can be
used to predict the enhancing effect of rotor surface micro-
structuring on turbine performance for a wide variety of
surface microstructure geometries.
Predictions of the model analysis have been shown
to agree well with available experimental drag turbine
performance data. However, the available data are lim-
ited to conditions corresponding to low isentropic
FIG. 13. (Color online) A 3D surface plot of the percent increase in efficiency
resulting from increasing FPo from 1 to 2 gi;FPo2gi;FPo1=gi;FPo1
as a
function of the inlet dimensionless tangential velocity difference ( W0;ro ) and
Reynolds number (Rem) for typical operating parameters (FPo 1 andFPo 2,c 1.4 (air),ni 0.2,Pi=Pnt 0.5, choked flow).
FIG. 14. (Color online) Streamlines for
W01:1, Rem5, ni 0.2, andVro 0.05 (a)FPo 1 and (b)FPo 2.
082003-10 V. D. Romanin and V. P. Carey Phys. Fluids 23, 082003 (2011)
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-7/26/2019 An Integral Perturbation Model of Flow and Momentum Transport in Rotating
11/12
efficiency. A particularly interesting prediction of the
model developed here is that low Reynolds numbers and
high rotor speeds result in the highest turbine isentropic
efficiencies. Specifically, for modified Reynolds numbers
(Rem) less than 1.2 and dimensionless inlet velocity dif-ference ( W0) less than 1.2 (orMo> 0.41 for choked flow),efficiencies up to and exceeding 80% can be achieved. In
addition to low Reynolds numbers and high rotor speeds,
roughened or microstructured surfaces can provide effi-
ciency benefits that can further improve turbine perform-
ance. Surface roughness was shown to improve turbine
efficiency by 9.2% in one example case.
The results of this investigation clearly indicate a path
of design changes that can significantly improve the energy
efficiency performance of Tesla-type disk-rotor drag tur-
bines. The trends that indicate this path are supported by
available experimental data. However, because the ranges of
experimental data are limited to low efficiency conditions,
we were not able to validate the model predictions into the
range of conditions predicted to produce high efficiencies.
Comparison of the predictions of the model presented herewith new performance data for Tesla-type drag turbines at
lower Reynolds numbers (Rem) and inlet velocity difference( W0) conditions are needed to fully explore the accuracy of
the model predictions. This model, nevertheless, offers a use-
ful means to explore parametric trends in designs of Tesla-
type drag turbines, and it can be useful in comparisons with
predictions of more detailed computational fluid dynamics
models of the flow in these types of turbines.
ACKNOWLEDGMENTS
Support for this research by the UC Center for Informa-
tion Technology Research in the Interest of Society (CIT-
RIS) is gratefully acknowledged.
1N. Tesla, Turbine, U.S. Patent No. 1,061,206 (May 1913).
2W. Rice, An analytical and experimental investigation of multiple disk
turbines, J. Eng. Power87, 29 (1965).
3M. C. Breiter and K. Pohlhausen, Laminar flow between two parallelrotating disks, Tech. Rep. ARL 62-318 (Aeronautical Research Laborato-
ries, Wright-patterson Air Force Base, Ohio, 1962).4G. P. Hoya and A. Guha, The design of a test rig and study of the per-
formance and efficiency of a tesla disc turbine, Proc. Inst. Mech. Eng.,
Part A 223, 451 (2009).5
A. Guha and B. Smiley, Experiment and analysis for an improved design
of the inlet and nozzle in tesla disc turbines,Proc. Inst. Mech. Eng., Part
A224, 261 (2010).6V. Carey, Assessment of tesla turbine performance for small scale ran-
kine combined heat and power systems,J. Eng. Gas Turbines Power132,
122301 (2010).7S. G. Kandlikar, D. Schmitt, A. L. Carrano, and J. B. Taylor, Characterization
of surface roughness effects on pressure drop in single-phase flow in
minichannels,Phys. Fluids 17, 100606 (2005).8
G. Croce, P. Dagaro, and C. Nonino, Three-dimensional roughness
effect on microchannel heat transfer and pressure drop, Int. J. Heat MassTransfer50, 5249 (2007).
9G. Gamrat, M. Favro-Marinet, S. Le Person, R. Baviere, and F. Ayela,
An experimental study and modelling of roughness effects on laminar
flow in microchannels,J. Fluid Mech.594, 399 (2008).10
B. R. Munson, D. F. Young, and T. H. Okiishi, Fundamentals of Fluid
Mechanics, 5th ed. (Wiley, New York, 2006).11V. Romanin and V. Carey, Strategies for performance enhancement of
tesla turbines for combined heat and power applications, ASME 2010
Energy Sustainability Conference Proceedings, Phoenix, Arizona, USA, 2,
5764 (ASME, New York, 2010).
082003-11 An integral perturbation model of flow Phys. Fluids 23, 082003 (2011)
http://dx.doi.org/10.1243/09576509JPE664http://dx.doi.org/10.1243/09576509JPE664http://dx.doi.org/10.1243/09576509JPE818http://dx.doi.org/10.1243/09576509JPE818http://dx.doi.org/10.1115/1.4001356http://dx.doi.org/10.1063/1.1896985http://dx.doi.org/10.1016/j.ijheatmasstransfer.2007.06.021http://dx.doi.org/10.1016/j.ijheatmasstransfer.2007.06.021http://dx.doi.org/10.1017/S0022112007009111http://dx.doi.org/10.1017/S0022112007009111http://dx.doi.org/10.1016/j.ijheatmasstransfer.2007.06.021http://dx.doi.org/10.1016/j.ijheatmasstransfer.2007.06.021http://dx.doi.org/10.1063/1.1896985http://dx.doi.org/10.1115/1.4001356http://dx.doi.org/10.1243/09576509JPE818http://dx.doi.org/10.1243/09576509JPE818http://dx.doi.org/10.1243/09576509JPE664http://dx.doi.org/10.1243/09576509JPE6647/26/2019 An Integral Perturbation Model of Flow and Momentum Transport in Rotating
12/12
Physics of Fluids is copyrighted by the American Institute of Physics (AIP). Redistribution of journal material
is subject to the AIP online journal license and/or AIP copyright. For more information, see
http://ojps.aip.org/phf/phfcr.jsp