An Integral Perturbation Model Of Flow And Momentum Transport In Rotating 503x72

PHYSICS OF FLUIDS 23, 082003 (2011)

An integral perturbation model of flow and momentum transport in rotating microchannels 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. The resulting lowest-order equations are solved explicitly and higher order 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 deg for optimal ranges of C 2011 American Institute of Physics. the governing dimensionless parameters. V [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 combined heat and power systems, and geothermal power systems. 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 mechanism 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 mechanics 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 a)

Electronic mail: [email protected].

b)

1070-6631/2011/23(8)/082003/11/$30.00

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 ro DH (DH is the hydraulic diameter of the microchannels). 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 radius ri. 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 at ri. This process is shown in Figure 2. 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 compare experimental data to analytical results, and lacked an analytical treatment of the friction factor. Breiter et al. provided a preliminary analysis of pumps only and used a numerical solution of the energy and momentum equations. Hoya4 and Guha5 extensively tested sub-sonic and super-sonic nozzles

23, 082003-1

C 2011 American Institute of Physics V

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FIG. 1. Schematic of a Tesla turbine.

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 proposed an analytical treatment that allowed for a closed-form solution of the fluid mechanics equations in the flow in the rotor; however, Carey’s model analysis invoked several idealizations that neglected viscous transport in the radial and axial directions, and treated tangential viscous effects using a friction 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 circumstances, 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 performance. Beginning with the momentum and continuity equations 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 solution. This more rigorous approach directly indicates the dimensionless parameters that affect flow and transport and allows a direct assessment of the relative importance of pressure, viscous, and momentum effects in different directions in the flow. The performance analysis in the previous investigations described above suggest that enhancement of rotor drag in this type of turbine generally enhances energy conversion efficiency. Information obtained in recent fundamental studies indicates that laminar flow drag can be strongly enhanced by strategic microstructuring of the wall

FIG. 2. (Color online) Schematic of flow through a Tesla turbine microchannel.

Phys. Fluids 23, 082003 (2011)

surfaces in microchannels.7–9 The conventional Moody diagram 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 nearsurface 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 channels, 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 for surfaces with a relative roughness higher than 0.05, arguing that above this value flow constriction becomes important. Kandlikar proposes that the constricted diameter be simplified to be Dcf ¼ Dt 2e, where e is the roughness height, Dt is the base diameter, and Dcf is the constricted diameter. Using this formulation, the Moody diagram can be re-constructed to for the constricted diameter, and can thus be used for channels with relative roughness larger than 5%. Kandlikar conducts experiments which match closely with this prediction 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 of roughness 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 proposed by several authors for the constricted hydraulic diameter for two different roughness element periodicities. While the results of his analysis match Kandlikar’s equation (Dcf ¼ Dt 2e) within 2% for one case, for a higher periodicity Kandlikar’s approximation deviates from numerical results by 10%. This example, and others discussed in Croce’s paper, begins to outline how roughness properties other than height can effect a shift in the flow Poiseuille number. Gamrat et 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. Gamrat’s analysis, to the best of the author’s knowledge, is the most thorough attempt to predict the effects of surface roughness on the Poiseuille number of laminar flow in microchannels.

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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 perturbation 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 momentum 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 evaluation of its predictions are described in the following sections.

and the isentropic efficiency of the nozzle is defined as gnoz ¼

v2o =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 ¼ ro, taken to be equal to the nozzle exit velocity, is given by pffiffiffiffiffiffiffiffi ðvh Þr¼ro ¼ vo ¼ gnoz vo; i ; (6) where vo, i is computed using Eq. (4), and for choked flow the nozzle efficiency is given by

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 incorporating a treatment of microstructured walls. The resulting equations 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.

ðc1Þ=c

gnoz ¼

cRðPt =Pnt Þcrit h i: 2 1 ðPo =Pnt Þðc1Þ=c

(7)

Treating the gas flow as an ideal gas with nominally constant specific heat, Eq. (6) provides the means of determining the rotor gas inlet tangential velocity (vh)r¼ro given the specified flow conditions for the nozzle.

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 dimensional compressible flow theory. In expanders of the type considered here, the flow through the nozzle is often choked. This was the case in expander tests conducted by Rice,2 who reported that virtually all the pressure drop in the device is in the nozzle and little pressure drop occurs in the flow through the rotor. The pressure ratio Po=Pnt across the nozzle for choked flow must be at the critical pressure ratio (Pt=Pnt)crit at the nozzle inlet temperature. For a perfect gas, this is computed as

Pt Pnt



¼ crit

2 cþ1

c=ðc1Þ ;

(1)

((Pt=Pnt)crit is 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 pffiffiffiffiffiffiffiffiffi (2) vo;c ¼ at ¼ cRTt ; where Tt the nozzle throat temperature for choked flow, is given by ðc1Þ=c

Tt ¼ Tnt ðPt =Pnt Þcrit

:

(3)

For isentropic flow through the nozzle, the energy equation dictates that the exit velocity would be vo;i

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi ðc1Þ=c ¼ 2 Tnt 1 ðPo =Pnt Þ

(4)

B. Analysis of the momentum transport in the rotor

For steady incompressible laminar flow in microchannels between the turbine rotor disks, the governing equations for the flow are Continuity: r v ¼ 0:

(8)

Momentum: v rv ¼

rP þ r2 v þ f: q

(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-dimensional: vz ¼ 0 and the z-direction momentum equation has a trivial solution. (2) The flow field is taken to be radially symmetric. The inlet flow at the rotor outer edge is uniform, resulting in a flow field that is the same at any angle h. All h derivatives 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 @ðrvr Þ ¼ 0; r @r

(10)

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r-direction momentum: @vr v2h 1 @P ¼ vr @r r q @r 1@ @vr @ 2 vr vr þ 2 2 ; r þ @r @z r r @r h-direction momentum: @vh vr vh 1@ @vh @ 2 vh vh vr þ ¼ þ 2 2 ; r @r r @r @z r r @r

(11)

(12)

z-direction momentum: 0¼

1 @P : q @z

Equation (13) dictates that the pressure is uniform across the channel at any (r, h) location. For the variations of the radial and tangential velocities, the following solution forms are postulated: vr ¼ vr ðrÞ/ðzÞ;

(14)

vh ¼ v^h ðrÞ/ðzÞ þ UðrÞ;

(15)

where

/ðzÞ ¼

n nþ1 2z 1 n b

v^h ðrÞ ¼

1 b

1 b

ð b=2 vr dz;

(17)

ðvh UÞdz;

(18)

b=2

sw ¼ f

q^ v2h : 2

l½@ðvh UÞ=@zz¼b=z s ¼ : 2 q^ v2h q^ vh =2

(19)

where Rec is the Reynolds number defined as

(23)

FPo ¼ Po=24;

(24)

ðn þ 1Þ ¼ Po=8 ¼ 3FPo :

(25)

It follows that:

•

for laminar flow over smooth walls: n ¼ 2, Po ¼ 24, FPo ¼ 1, for laminar flow over walls with drag enhancing roughness: 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 continuity equation

Substituting Eq. (14) into Eq. (10) and integrating with respect to r yields (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, Po ; Rec

(22)

and Po is a numerical constant usually referred to as the Poiseuille number. For flow between smooth flat plates, the wellknown laminar flow solution predicts Po ¼ 24. For flow between flat plates with roughened surfaces, experiments7–9 indicate that a value other than 24 better matches pressure loss data. We, therefore, define an enhancement number FPo as

•

For a Newtonian fluid, it follows that

f ¼

DH ¼ 2b;

b=2

ð b=2

q^ v h DH ; l

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 vh form (15),

where b 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 ðv^h ¼ vh U Þ through the friction factor definition,

f ¼

Rec ¼

(16)

and vr and v^h are mean velocities defined as vr ðrÞ ¼

FIG. 3. (Color online) Variation of velocity profile with n.

(13)

vr / ¼ constant ¼ Cr : rvr ¼ r

(26)

Integrating Eq. (16) across the channel and using the fact that ð b=2 ð b=2 /dz ¼ 2 /dz ¼ b (27) b=2

0

yields (21)

ð b=2 b=2

rvr dz ¼

ð b=2 b=2

r vr /dz ¼ r vr b ¼ bCr ¼ C0r :

(28)

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Mass conservation requires that 2pro q

ð b=2

vr dz ¼ 2pro q vr ðro Þ

b=2

ð b=2 /dz b=2

vr ðro Þb ¼ m_ c ; ¼ 2pro q

(29)

where m_ c is the mass flow rate per channel between rotors. Combining Eqs. (28) and (29) yields the following solution for the radial velocity: ro vro ; (30) vr ¼ r where vro ¼

m_ c : 2pro qb

(31)

D. Solution of the tangential and radial momentum equations

2 ^ 0 ¼ dW=dn ^ ^ 00 ¼ d 2 W=dn ^ where W and W . Solution of these equations requires boundary conditions on the dimensionless ^ and P). ^ relative velocity and the dimensionless pressure (W Here, it is assumed that the gas tangential velocity and the ^ at the outer radius disk rotational speed are specified, so W of the disk is specified. It follows that

at n ¼ 1;

2

/ dz ¼ 2

b=2

ð b=2 0

2ðn þ 1Þ b; / dz ¼ 2n þ 1 2

(32)

ð b=2 2 ð b=2 2 d / d / dz ¼ 2 dz 2 dz dz2 b=2 0 4ðn þ 1Þ ; ¼ b

(33)

^ ¼ 0: Pð1Þ

^ ¼ v^h =Uo ¼ ð vh UÞ=Uo ; W

(35)

P^ ¼ ðP Po Þ=ðqUo2 =2Þ;

(36)

Vro ¼ vro =Uo ;

(37)

e ¼ 2b=ro ;

(38)

Rem ¼ ðDH =ro Þ

DH m_ c m_ c DH ¼ 2 ; 2pro bl pro l

(44)

P^ ¼ P^0 þ eP^1 þ e2 P^2 þ :

(45)

Substituting results in Eqs. (46)–(54). O(e0): 6FPo 1 ^ 0 þ 1 8ð6FPo 1Þ n W ^ 0; ¼W 0 3FPo n Rem

(46)

12FPo 1 2 ^ 2 n2 þ 4W ^0 þ 2n þ V 2 FPo 96 ; P^00 ¼ V þW 0 ro 6FPo 1 n3 ro Rem n (47) P^0 ¼ 0;

(48)

O(e1): 1 n 0 ^ ^1 ; 8ð6FPo 1Þ W 0 ¼ W1 þ n Rem

(49)

12FPo 1 ^ ^ ^1 ; 2W0 W1 þ 4W P^01 ¼ 6FPo 1 n

(50)

^1 ¼ 0; at n ¼ 1 : W

(39)

P^1 ¼ 0;

(51)

O(e2):

converts Eqs. (11) and (12) to the forms,

2 Vro ; Rem n

1 8ð2n þ 1Þn ^ W2 n Rem 00 ^ ^0 ^0 6FPo 1 nW W W 0 0 ; ¼ þ Rem Rem Rem n 6FPo

^20 W

@ P^ ^0 4ðn þ 1Þ 2 ^ 2 n2 ¼P ¼ Vro þ W 3 @n ð2n þ 1Þn ^ þ 2n þ 32ðn þ 1Þ þ 4W

^ 1 þ e2 W ^¼W ^ 0 þ eW ^2 þ ; W

^0 ¼ W ^0;ro ; at n ¼ 1 : W (34)

(43)

Equations (42)–(43) provide boundary conditions for solution of the dimensionless tangential momentum Equation (41) and the radial momentum Equation (40), which predicts the radial pressure distribution. Since e ¼ 2b=ro is much less than 1 in the systems of interest here, we postulate a series expansion solution of the form,

doing so and introducing the dimensionless variables n ¼ r=ro ;

(42)

^ it follows that In addition, from the definition of P,

The next step is to substitute the postulated solutions from Eqs. (14) and (15) into the tangential and radial momentum equations (12) and (11), integrate each term across the microchannel, and use Eq. (27) together with the results, ð b=2

^ ^o : Wð1Þ ¼W

(40)

2n þ 1 2n þ 1 e2 2n þ 1 e2 00 ^ ^0 nW þ ¼ þ1 W nþ1 2ðn þ 1Þ Rem 2ðn þ 1Þ Rem 2n þ 1 e2 1 8ð2n þ 1Þn ^ þ 1 W; 2ðn þ 1Þ Rem n Rem (41)



12FPo 1 ^ ^ ^2 ; P^02 ¼ 2W0 W2 þ 4W 6FPo 1 n at n ¼ 1 :

^2 ¼ 0; P^2 ¼ 0: W

(52)

(53) (54)

In solving Eqs. (46)–(54), the dimensionless parameters in the equations,

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ni ¼ ri =ro ; ^0 Þ ¼ W ^0:ro ¼ vh;ro Uo ; ðW ro Uo

Phys. Fluids 23, 082003 (2011)

(55) f ðnÞ ¼ (56) (57)

Vro ¼ vro =Uo ;

(58)

e ¼ 2b=ro

(59)

grm ¼

are dictated by the choices for the following physical parameters:

gi ¼

• • • • • • •

ri, ro: the inner and outer radii of the disks b: the gap between the disks, from which we can compute DH ¼ 2b m_ c : the mass flow rate per channel between rotors x ¼ 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 ¼ a). The ^ require that the choices for Mo and definitions of Mo and W ^0; ro satisfy W pffiffiffiffiffiffiffiffiffi ðPt =Pnt Þðc1Þ=2c crit : Mo Uo = cRTt ¼ ^0;ro þ 1 W

(60)

Solving Eqs. (46), (49), and (52) with boundary conditions (48), (51), and (54) gives the following solutions: ^0 ¼ W



f ðnÞ Rem e Rem ^ W0;ro þ ; 24FPo nef ð1Þ 24FPo

^1 ¼ 0; W f ðnÞ ð n ^2 ¼ e n ef ðn Þ gðn Þdn ; W n 1

(61) (62) (63)

where gðnÞ ¼

^00 nW ^000 ^0 =n W 6FPo 1 W ; Rem 6FPo

(64)

(65)

And n* is a dummy variable of integration. With this result, the energy efficiency of the rotor and of the turbine, respectively, can be computed using

DH m_ c ; pro2 l

Rem ¼

4ð6FPo 1Þn2 : Rem

vh;o Uo vh;i Ui ; vh;o Uo

(66)

vh;o Uo vh;i Ui ; Dhisen

(67)

which rearrange to grm ¼ 1

^i þ ni Þni ðW ; ^o þ 1 W

(68)

^n¼n ¼r =r ; ^i ¼ W W i o i

(69)

^o þ 1Þ ðW ^i þ ni Þni ðc 1ÞM2 ðW o " : gi ¼ ðc1Þ=c # Pi 1 Pnt

(70)



The baseline case for comparison with rough wall solutions is that for a smooth wall, or FPo ¼ 1. The solution for W^0 (Eq. (61)) reduces to ^0 ¼ W



20 ðn2 1Þ Rem eRem Re ^ þ m: W0;ro 24 n 24

(71)

The pressure distribution can be found numerically or analytically by integrating equations (47), (50), and (53). E. Higher order (e1, e2)

In order to evaluate the significance of the higher order ^ and P, ^ results are plotted for two different solutions of W operating conditions in Figures 4 (velocity) and 5 (pressure). ^2 and P^2 ) are Under both scenarios, the 2nd order (W shown to be the same order of magnitude as the 0th order ^0 and for ^0 and P^0 ). Velocity plots for W (W ^ 1 þ e2 W ^ 0 þ eW ^2 fall nearly directly on top of each other W ^ For values of e as high as 1=10, much (similarly for P). ^2 larger than are found in most systems of interest, both e2 W and e2 P^2 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 ^0 (solid line) and for (b), the plots for W ^1 þ e2 W ^2 (dashed line) are ^0 þ eW W nearly coincident. The dotted-dash line ^2 ) is shown to be the same order of (W ^0 , thus making it negligimagnitude as W ble when multiplied by e2. (a) Case 1: ^0 ¼ 2, Re ¼ 10, ni ¼ 0.2, Vro ¼ 0.05, W m e ¼ 1=20; choked flow and (b) case 2: ^0 ¼ 1:1, Rem ¼ 5, ni ¼ 0.2, Vro ¼ 0.05, W e ¼ 1/20; choked flow.

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FIG. 5. (Color online) Comparison of pressure plots for 0th order and 2nd order velocity solutions. In both (a) and (b), the plots for P^0 (solid line) and for P^0 þ eP^1 þ e2 P^2 (dashed line) are nearly ^2 ) is coincident. The dotted-dash line (W shown to be the same order of magni^0 , thus making it negligible tude as W when multiplied by e2. (a) Case 1: ^0 ¼ 2, Re ¼ 10, ni ¼ 0.2, Vro ¼ 0.05, W m e ¼ 1=20; choked flow and (b) case 2: ^0 ¼ 1:1, Re ¼ 5, ni ¼ 0.2, Vro ¼ 0.05, W m e ¼ 1=20; choked flow.

term for the two cases shown. Note that Eq. (62) along with ^1 ¼ P^1 ¼ 0. Henceforth, we Eqs. (50) and (51) show that W can neglect the 1st and 2nd order , and only the 0th ^0 and P^0 ) will be considered. order (W III. COMPARISON OF SMOOTH WALL CASE WITH EARLIER MODEL PREDICTIONS

^0 solution corresponds closely with the solution The W developed by Carey,6 only differing by numerical constants. A comparison of results with the model from Carey’s earlier model is shown in Figure 6. Carey6 made several assumptions, 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 were removed based on the arguments of the perturbation analysis. The similarities in the results of this analysis with that of Carey that the assumptions made were valid. Additionally, Carey’s analysis was compared extensively with experimental data in Romanin and Carey,11 so a close correlation 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.

Figure 7 shows a 3D plot of turbine efficiency as a func^0;ro for the operating parameters in the first tion of Rem and W four 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 ^0;ro will increase model correctly predicts that decreasing W ^0;ro efficiency, and suggests that decreasing both Rem and W can dramatically improve performance. A 3D plot of turbine efficiency with typical operating ^0;ro , is shown parameters, and over ideal ranges of Rem and W ^0;ro , the analysis in Figure 8. At very low values of Rem and W shows that very high turbine efficiencies can be achieved. Practical issues arise when generating power in microchan^0;ro and Rem . nels such as these at very low values of W ^0;ro requires the rotor to be spinning at speeds Reducing W 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 applications. Also, lower Reynolds numbers require very small disk spacings (b) and larger disk radii (ro). IV. MODELING OF FLOW VELOCITY WITH ROUGHENED OR MICROSTRUCTURED SURFACES (FPO > 1)

Now that the perturbation analysis has resulted in equations that define the operating conditions and efficiency of the turbine as a function of FPo, we can analyze the effect of surface roughness on turbine performance. Developing a direct correlation between surface roughness and FPo is a

FIG. 6. (Color online) Comparison of solutions from the perturbation method and the model developed by Carey.6 (a) ^0 ¼ 2, Rem ¼ 10, ni ¼ 0.2, Case 1: W Vro ¼ 0.05; choked flow. The analysis predicts a turbine isentropic efficiency of gi ¼ 26.1% while the analysis by Carey6 predicts gi ¼ 27.0% (b) case 2: ^0 ¼ 1:1, Rem ¼ 5, ni ¼ 0.2, Vro ¼ 0.05; W choked flow. The analysis predicts a turbine isentropic efficiency of gi ¼ 42.3% while the analysis by Carey6 predicts gi ¼ 42.5%.

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TABLE I. Comparison of analysis with experimental data from Romanin and Carey.11 m_ c g/s

x rad/s

Rem

^0;ro W

gi;exp (%)

gi;model (%)

1.64 1.66 1.65 1.65 2.37 2.40 2.38 2.38

450 784 953 1110 708 1110 1370 1590

47.5 48.1 47.8 47.8 68.6 69.5 68.9 68.9

18.2 10.0 8.1 6.8 11.2 6.8 5.3 4.4

3.1 4.9 5.9 6.8 3.0 4.3 5.3 6.0

2.3 3.6 4.4 5.1 2.3 3.3 4.1 4.8

detailed process that involves characterizing specific geometric 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 for FPo as high as 3.5 for roughened surfaces in microchannels, so values up to FPo ¼ 3.5 will be considered. A. Discussion of the velocity and pressure fields

Figure 9 shows that the velocity profile is significantly altered by using a roughened surface. Equation (70) shows ^i should be minimized to increase efthat the exit velocity W ficiency and indeed the efficiency does increase with FPo. FPo ¼ 2 results in a turbine isentropic efficiency of gi ¼ 45.1%, compared to an efficiency of gi ¼ 42.3% for FPo ¼ 1. Figure 10 shows the dimensionless pressure P^ as a function of n for several values of FPo. The figure shows that the dimensionless pressure decreases with increasing FPo. This can be attributed to the competing effects of centripetal force and 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

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 ¼ 1 (smooth wall), c ¼ 1.4 (air), ni ¼ 0.45, Pi=Pnt ¼ 0.4; choked flow).

FIG. 8. (Color online) A plot of efficiency (gi) as a function of dimension^0; ro ) and modified Reynolds less tangential velocity difference at the inlet (W number (Rem ) for typical operating conditions: FPo ¼ 1 (smooth wall), c ¼ 1.4 (air), ni ¼ 0.2, Pi=Pnt ¼ 0.5; choked flow.

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 mictrostructured surfaces

Over the entire range of values for FPo discussed by Croce,8 Figure 11 shows that efficiency increases a total of 3.8 percentage points, which amounts to a 9.2% improvement in performance over a smooth wall. Figure 12 shows a surface plot of efficiency as a func^0;ro . It is tion of two non-dimensional parameters, Rem and W shown that increasing surface roughness can yield especially significant performance improvements for higher Reynolds numbers rather than lower. Similar trends to those reported by Carey6 and Romanin11 can be seen in Figure 12; it is clear that high efficiency turbine designs should strive for Reynolds numbers and dimensionless inlet velocity differences to be as small as possible. It is also shown that penalties due to

^0 ¼ 1:1, FIG. 9. (Color online) Velocity vs. n for several values of FPo. W Rem ¼ 5, ni ¼ 0.2; choked flow.

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Phys. Fluids 23, 082003 (2011)

^ vs. n for several values FIG. 10. (Color online) Dimensionless pressure (P) ^0 ¼ 1:1, Re ¼ 5, ni ¼ 0.2, Vro ¼ 0.05; choked flow. of FPo. W m

FIG. 11. (Color online) Efficiency (gi) vs. FPo for ni ¼ 0.2 and choked flow.

^0;ro are less dramatic for roughhigher values of Rem and W ened surfaces. By taking advantage of microstructured surfaces, larger disk gaps, and smaller disks can be used while limiting penalties to efficiency. For example, the nondimensional turbine parameters 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 ¼ 2), and the radius can be decreased while keeping other parameters constant until the efficiency is equivalent to that achieved by the parameters from case 2 (Figure 4(b)). This process results in a turbine radius of ro ¼ 18.6 cm, down from ro ¼ 34.7 cm in the smooth wall 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 physical parameters 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).

any h location a the rotor inlet (n ¼ r=ro ¼ 1), over time, the fluid traces an (r, h) path through the channel between adjacent disks that is determined by integrating the differential relations,

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

rdh ¼ vh dt;

(72)

dr ¼ vr dt:

(73)

Combining the above equations yields the following differential equation that can be integrated to determine the dependence of h with r along the streamline, dh vh : (74) ¼ dr st vr r Note that since the velocities are functions only of r, the entire right side of the above equation is a function of r. In of the dimensionless variables described above, the streamline differential Equation (74) can be converted to the form, ^ dh nþW ¼ ; (75) dn st Vro where Vro is the ratio of radial gas velocity to rotor tangential velocity at the outer edge of the rotor, Vro ¼

vro m_ c ¼ : Uo 2pro bqo Uo

(76)

FIG. 12. (Color online) A 3D surface plot of efficiency (gi) as a function of the inlet dimensionless tangential velocity ^0;ro ) and Reynolds number difference (W (Rem ) for typical operating parameters (c ¼ 1.4 (air), ni ¼ 0.2, Pi=Pnt ¼ 0.5; choked flow). (a) FPo ¼ 1 and (b) FPo ¼ 2.

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FIG. 13. (Color online) A 3D surface plot of the percent increase in efficiency

resulting from increasing FPo from 1 to 2 ðgi;FPo ¼2 gi;FPo ¼1 Þ=gi;FPo ¼1 as a ^ function of the inlet dimensionless tangential velocity difference (W0;ro ) and Reynolds number (Rem ) for typical operating parameters (FPo ¼ 1 and FPo ¼ 2, c ¼ 1.4 (air), ni ¼ 0.2, Pi=Pnt ¼ 0.5, choked flow).

Rotor streamlines determined by integrating Equation ^o ¼ 3:0, (DH=ro)Rem ¼ 5.0, Vro ¼ 0.05, and (75) for W ni ¼ 0.2 are shown in Figure 14. Flow along one streamline enters the rotor at h ¼ 0 , whereas the other streamline begins at h ¼ 180 . The model can be used to predict how the inward spiral path of the flow changes as the governing parameters are altered. Figure 14 shows 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 deg complex 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 solution 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 integral analysis. The other is the idealization of the inlet flow as being uniform over the outer perimeter of the disk. Real turbines of this type have a discrete number of nozzles that

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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 perimeter. 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 conditions 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 transport, and, in of these dimensionless parameters, it provides a direct assessment of the relative importance of viscous, pressure, and momentum effects in different directions in the flow. Our analysis also indicated that closed form equations can be obtained for the lowest order contribution to the series expansion solution, and the higher order term contributions are very small for conditions of practical interest. This provides simple mathematical relations 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 in fluid machinery. We have demonstrated that by parameterizing the roughness in of the surface Poiseuille number ratio (FPo), the model analysis developed here can be used to predict the enhancing effect of rotor surface microstructuring 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 limited to conditions corresponding to low isentropic

FIG. 14. (Color online) Streamlines for ^0 ¼ 1:1, Rem ¼ 5, ni ¼ 0.2, and W Vro ¼ 0.05 (a) FPo ¼ 1 and (b) FPo ¼ 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^0 ) less than 1.2 (or Mo > 0.41 for choked flow), ference ( W 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 efficiency benefits that can further improve turbine performance. 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 turbines. The trends that indicate this path are ed 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 here with new performance data for Tesla-type drag turbines at lower Reynolds numbers (Rem ) and inlet velocity difference ^0 ) conditions are needed to fully explore the accuracy of (W the model predictions. This model, nevertheless, offers a useful means to explore parametric trends in designs of Teslatype 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.

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ACKNOWLEDGMENTS

for this research by the UC Center for Information Technology Research in the Interest of Society (CITRIS) is gratefully acknowledged. 1

N. Tesla, “Turbine,” U.S. Patent No. 1,061,206 (May 1913). W. Rice, “An analytical and experimental investigation of multiple disk turbines,” J. Eng. Power 87, 29 (1965). 3 M. C. Breiter and K. Pohlhausen, “Laminar flow between two parallel rotating disks,” Tech. Rep. ARL 62-318 (Aeronautical Research Laboratories, Wright-patterson Air Force Base, Ohio, 1962). 4 G. P. Hoya and A. Guha, “The design of a test rig and study of the performance 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 A 224, 261 (2010). 6 V. Carey, “Assessment of tesla turbine performance for small scale rankine combined heat and power systems,” J. Eng. Gas Turbines Power 132, 122301 (2010). 7 S. 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. D’agaro, and C. Nonino, “Three-dimensional roughness effect on microchannel heat transfer and pressure drop,” Int. J. Heat Mass Transfer 50, 5249 (2007). 9 G. Gamrat, M. Favro-Marinet, S. Le Person, R. Bavie`re, 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). 11 V. 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, 57–64 (ASME, New York, 2010). 2

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