an integral perturbation model of flow and momentum transport in rotating

7/26/2019 An Integral Perturbation Model of Flow and Momentum Transport in Rotating

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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)
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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)
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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)

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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.

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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,

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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.

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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


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.


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%.

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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.

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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.

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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.

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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).

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an integral perturbation model of flow and momentum transport in rotating

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