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THEORETICAL ASSESSMENT OF THE PERFORMANCE OF PELTON TURBINES OPERATED WITH AIR AND STEAM

M. Schatz, P. Rettenmeier, M. V. Casey

ITSM, Institute of Thermal Turbomachinery, Universität Stuttgart, Germany schatz@itsm.uni-stuttgart.de

ABSTRACT Apart from a few exceptions, Pelton-type turbines have been restricted to hydropower

applications. This is mainly due to the fact, that conventional Pelton rotors suffer high disc friction windage losses due to their shape. Whereas this effect is almost negligible in hydropower applications because of the low density of air as surrounding medium compared to water as working fluid, it effectively inhibits the use of any compressible medium as working fluid. However, due to their simplicity and robustness, Pelton-type turbines could be used for processes where high pressure ratios in connection with low mass flows are encountered, for example in waste heat applications.

The rotor investigated in this study is based on a design from the mid 1980’s and consists of a disc with internal flow channels. The calculation of the turbine performance is based on simple one-dimensional expressions. In order to assess the efficiency of such a turbine, correlations from literature have been used to model the nozzle efficiency, the profile losses and the disc friction loss.

Thereafter, a parameter study has been conducted in order to assess the potential as well as the limitations of such a turbine. Apart from the pressure ratio, the rotor diameter, the number of nozzles and the working fluid have been varied. A comparison to radial and axial turbines is made using the Cordier-diagram. The results of this study indicate that especially for very small turbine diameters, a Pelton-type turbine could be an alternative to conventional turbines for applications involving a low mass flow at high pressure ratios.

NOMENCLATURE Symbols Greek symbols Subscripts c absolute velocity absolute flow angle 0 nozzle inlet, D rotor diameter relative flow angle stagnation condition d nozzle diameter specific heat ratio 1 nozzle outlet h specific enthalpy efficiency 2 turbine outlet k correction factor dynamic viscosity s isentropic m mass flow rate isentropic velocity ratio crit critical condition M torque pressure ratio p1/p0, N nozzle n rot. speed [min-1] 3.141527 R rotor ns specific speed density F disc friction P power loss coefficient P profile u circumf. velocity difference

V volumetric flow rate w relative velocity

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INTRODUCTION Presently, there is a trend away from the centralized generation of electric power as well as heat towards more individual solutions, such as small block heat and power plants or hybrid fuel cell/turbine systems [1]. In addition, due to the urgent need to reduce global energy demand, the recovery of waste heat becomes more important, even for small waste heat sources such as car engines. This development necessitates small and efficient turbomachines in the power range of only a few kilowatts. However, the requirements from waste heat recovery often include a high pressure ratio and a correspondingly high enthalpy drop at very low mass flows (sometimes less than 0.02 kg/s [3]). These are extremely demanding for efficient turbine applications. In addition the turbine speed is usually limited to about 100,000 RPM because of the generator to be used. This results in low specific speeds with values of ns < 0.1. In general, low mass flows require a small turbine size or the use of partial admission. The reduction of the turbine size is effectively restricted by the increase of losses with decreasing size, due to high relative tip clearance and surface friction losses. In addition, there is the question of manufacturing and assembly. Partial admission turbines also suffer from higher losses due to increasing windage, thus the efficiency decreases with decreasing admission ratio. On the other hand, there is the issue of high enthalpy drop. Since the rotational speed of the turbine is usually limited (either due to the generator or because of bearing technology limits), the use of multi-stage turbines is appropriate. In applications with low mass flows, however, it will certainly be desirable to use single stage turbines, firstly for reasons of turbine efficiency, but secondly because of economic considerations. A compromise for this problem could be a Curtis-stage turbine, which is often used for partial admission applications. Thus the challenging task for the turbine designer is to combine low mass flows and high pressure ratios with turbine dimensions that yield an acceptable efficiency in a single stage turbine at sensible rotational speeds. In principle, a Pelton turbine is perfectly suited for applications where low mass flows in connection with high pressure ratios are present, but the general opinion is that such a turbine cannot be operated with compressible media as the disc friction losses incurred by the rotor wheel are too high, thus limiting turbine output and efficiency. This is certainly true for high power applications with correspondingly large wheel diameters, especially with the bucket shapes that are used for water turbines. Nevertheless, for small power generation units of only a few kilowatts such Pelton-type turbines might still be appropriate. One example is a turbine design presented and tested in the 1980’s by Rautenberg et al. [2], where the disc friction and flow losses have been reduced during a rotor design evolution. According to Shao et al. [4], for mass flows of 0.007 – 0.016 kg/s the measured efficiency (specific work divided by the theoretically incoming kinetic energy) varied between 45 % and 75 %. In this study, the potential of Pelton turbines for low specific speed applications operated with compressible working fluids such as steam and air is assessed. This is done using one-dimensional approaches for turbine output, nozzle and rotor flow losses as well as disc friction losses. Finally, a comparison is made of the theoretical efficiencies derived here to the maximum efficiencies of conventional turbines at the same specific speeds.

Calculation of theoretical turbine efficiency

Theoretical background For the modeling of Pelton turbine performance, the system needs to divided into its different functional components shown in Fig. 1, which are the Laval nozzle for the expansion of the fluid (0 1), the Pelton wheel in which the mechanical power output is produced by deflecting the incoming fluid jet (1 2), the turbine casing and finally the generator. Sources of loss are viscous effects as well as shock losses in the Laval nozzle, flow incidence, profile losses and additional losses within the rotor, exit loss and power dissipation due to disc friction.

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Bearing and generator losses are independent of the turbine type and will thus not be considered here. The loss mechanisms as well as the specific work output of the turbine will be studied separately and modeled using one-dimensional expressions and correlations obtained from available literature. Thereafter, the turbine performance will be calculated based on these models. Because of this basic approach, a number of assumptions and simplifications have to be made. These are:

Within the frame of this study, the working fluid will be considered as ideal gas with constant specific heat.

The turbine inlet condition (condition 0 in Fig. 1) is considered as reservoir condition, i.e. kinetic energy is neglected

The process is modeled as an adiabatic process The degree of reaction is zero, that is, the expansion

process takes place only in the nozzle.

Modeling of nozzle flow The purpose of a nozzle is the conversion of potential into kinetic energy. This is done by expanding a fluid from low velocities at a high pressure to high velocities at a lower pressure level. If the nozzle backpressure is below the so-called critical pressure, the flow reaches sonic conditions in the nozzle throat. Using a converging-diverging nozzle geometry, known as a Laval nozzle, supersonic velocities can be achieved. In such a case, the nozzle geometry is a function of pressure ratio. If the nozzle design does not match the pressure ratio, there are strong aerodynamic shocks which, for exit Mach numbers above Ma ~ 1.3, cause increasing additional losses. Within this study, it will be assumed that no shocks are present. For isentropic nozzle flow, the exit velocity of the nozzle can be calculated from the static enthalpy drop:

ss hhc ,10,1 2 (1)

Introducing the definition of speed of sound into equation (1) yields the well-known formula for the flow Mach number at the nozzle exit:

11

21

1

0

p

pMa (2)

The critical nozzle diameter, i.e. the diameter at which the flow reaches sonic velocity and becomes choked in the nozzle throat, can be calculated from the continuity equation using the mass flow available and the critical conditions. The same applies for exit diameter using the exit flow velocity and the nozzle backpressure.

critcrit

critp

md

4

(3)

111

4

c

md

(4)

In viscous flows, there are dissipative effects, which in turn lead to a lower exit velocity than for the isentropic process. Therefore, the nozzle efficiency is dependent on the nozzle Reynolds number, which is defined as

critN d

m

4

Re (5)

Fig. 1: schematic drawing of a Pelton turbine

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The coefficient for the description of the resulting loss of kinetic energy is defined as

sN c

cc

,1

1 (6)

From equations (1) and (6), the definition of the isentropic efficiency of a nozzle follows:

22,1

21

,01

01, N

sssN c

c

c

h

h

(7)

For Reynolds numbers greater than ReN > 105 , literature sources suggest nozzle efficiencies of more than 96 % [6] or velocity coefficients of cN > 0,94 – 0,96 ([5], [8]).

Examination of the Pelton rotor

Calculation of power output The calculation of the turbine power output will be derived from the analysis of the velocity triangles shown in Fig. 2. The flow channel is indicated by the D-shaped line in the figure. From the Euler-equation, the specific work delivered by a Pelton turbine can be calculated according to equation (8). From the specific work and the turbine mass flow, the turbine power output can be determined.

221112 180coscos wucuccuw uut (8)

twmP (9)

Estimation of rotor efficiency Similar to the calculation of the nozzle efficiency above, a coefficient for the description of rotor flow losses is introduced, which relates the rotor inlet and outlet relative velocities:

1

2

w

wcR (10)

According to literature [9], for high deflection angles this coefficient is in the range of cR ~ 0.75. Again, from this coefficient, the efficiency can be calculated as

2RR c (11)

Since there are no published correlations available for compressible flow through tangential turbines such as a Pelton turbine, neither for the rotor velocity coefficient nor for the rotor efficiency, some approaches derived for axial turbines have been examined, partly simplified and extrapolated for high deflection angles, namely the approach of Soderberg given in [10], the method of Traupel [12] and that of Deij and Trojanovskij [13]. The calculated rotor efficiencies for all methods have been calculated and compared. Since the method of Traupel has yielded the lowest efficiencies, this approach will be used in order to obtain a conservative estimate. For impulse turbines, the rotor efficiency is

221

221 RrestPR c

w

w (12)

Where P is the profile loss coefficient and rest covers all other losses, for example those caused by sidewall effects. The profile loss is calculated using the following formula:

0Re PMaP (13)

Fig. 2: Pelton turbine velocity triangles

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The influence of flow Mach number, expressed by the coefficient χMa is mainly dependent on the relative Mach number throughout the passage and the blade geometry and there is no general

expression for χMa. Therefore, the course of this parameter given for a high Mach number profile from [13] (also given in [12]) has been used for an estimate. The coefficient χRe combines Reynolds number and surface roughness effects and can be determined from information given [12]. This coefficient is determined by the surface roughness of the flow channel, which means that lower profile losses can be achieved by manufacturing smooth flow passage surfaces. The basic profile loss has been taken from the data given by Traupel [12] and can be seen in Fig. 3, however, the curve for 1 < 20° had to be extrapolated on the basis of the data available. As already mentioned, the coefficient ζrest includes a number of influences such as axial

distance between stator and rotor, blade height and pitch/chord-ratio, some of which are not really applicable for the design under consideration. Hence, in order to get a conservative estimate of the losses to be expected, the rotor efficiency will be calculated from the profile loss multiplied by a factor kR:

221

221 RPRR c

w

wk (14)

Defining the circumferential efficiency as the ratio of specific work done divided by the theoretically incoming kinetic energy [8], an equation results which allows the influence of both nozzle and rotor efficiency on the specific work output of the turbine to be identified. Here is the isentropic velocity ratio u/c1,s.The maximum efficiency occurs at = 0.5 [11].

1

21

22,1 cos

cos1cos2

2

RN

s

tC cc

c

w (15)

Looking at equation (15) it is evident, that for impulse turbines the influence of the nozzle efficiency clearly outweighs that of the rotor. Hence the nozzle design must be done meticulously in order to achieve a high overall efficiency, especially considering high Mach number flows.

Disc friction losses Any rotating device surrounded by a fluid suffers a friction loss. In addition, there are additional losses from the flow field around such a device. For a standard Pelton wheel, for example, the major loss comes from the flow field around the buckets which have large wakes. Thus a Pelton turbine for compressible fluids must be designed in such a way that the additional losses from the surrounding flow field are as small as possible. The ideal of such a design is a simple disc, something which Rautenberg et al. have already tried to get close to in their design [2]. For the assessment of the Pelton turbine, the friction loss incurred by a rotating disc will be calculated and used for a first estimate of the performance. In the course of this study, a variation of this parameter will show the limitations of the application of such a turbine. In the first place, the disc friction loss is a function of disc Reynolds number:

2Re

Dudisc

(16)

Fig. 3: basic profile loss calculated according to [12]. (Note: the curve is extrapolated for 1 < 20°)

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The critical Reynolds number is at about Recrit ~ 3•105. For lower disc Reynolds numbers, the surrounding flow will become laminar, resulting in a larger boundary layer and hence higher friction losses. Again, several approaches available in literature have been compared. All calculations are based on the approach, that the friction between an infinitely thin rotating disc and the surrounding fluid causes a torque which acts against the direction of rotation of the disc, which depends on the disc diameter, the rotor speed, the density of the surrounding fluid and a torque coefficient cM:

32DucM MF (17)

The power defect which results from this torque acting on the disc is the disc friction loss. Since the rotor under consideration is obviously not infinitely thin, the circumference of the disc has to be taken into account using a correlation cA. Therefore, the basic expression for the power loss due to disc friction is

AMAMAFF cDnccDuccMP 5323 (18)

However, a turbine rotor will always have a casing which means that formulations derived for free-running discs that can be found in literature (e.g. [14] or [15]) cannot be used here. From the literature that covers both the cases with and without casing ([7],[16]), the formulation of Sigloch [7] has been chosen as it yields the highest disc friction losses, hence giving the most conservative estimate. For the present study, the following equation has been used for the calculation of the disc friction loss:

D

BDDukP FF

2

Re

10875.2 232.0

3

(19)

The coefficient kF has been introduced in order to mirror the fact, that the turbine rotor under consideration will definitely have a worse aerodynamic behaviour than a rotating disc or cylinder with smooth surface. Its influence is investigated in the following parameter study. The last term on the right side is the correction factor used to incorporate the length of the disc, i.e. the additional surface area due to the circumference of a cylinder. It is interesting to note that the influence of rotor length is relatively small. As a first estimate for the rotor length, a value of B = 3 • d1 has been used. This will provide enough space for both passage inlet and outlet. From the available literature and equation (18) the following conclusions can be drawn:

the disc friction losses for rotors running in a casing are about half of those occurring for free-running rotors ([7], [16])

As the disc friction loss is proportional to ρ•n3•D5, both the rotor speed and turbine size have to be chosen with great care

Finally, a low backpressure does not only increase the enthalpy drop of the turbine, but will also help to reduce losses as the fluid density surrounding the rotor also decreases

Turbine efficiency After modeling all relevant efficiency parameters, the isentropic efficiency of the Pelton turbine can be calculated. The flow related loss contributions are illustrated in the h-s-diagram in Fig. 4. These are the dissipative effects in the nozzle hlossN, the losses occurring in the rotor hloss,R, the dissipation due to disc friction hF and finally the loss of kinetic energy at the turbine outlet hout. Thus, the isentropic efficiency of the turbine follows by subtracting the losses from the isentropic enthalpy drop.

Fig. 4: h-s-diagram of expansion in a Pelton turbine

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Parameter study The expressions derived above for the theoretical assessment of the Pelton turbine, are examined here by studying the effect of different parameters on the performance.

Operating conditions For the parameter study, steam is used as working fluid. The steam mass flow is 0.015 kg/s, expanded in a single nozzle. The nozzle backpressure was chosen to be p1 = 1 bar and the temperature was set to saturation temperature. Because an impulse turbine is studied, this corresponds to the rotor exit pressure as well. The nozzle pressure ratio used varies between = 0.3 – 0.05, that is an expansion ratio from 3 to 20. The resulting enthalpy drop across the turbine, the nozzle exit Mach number and the calculated nozzle critical and exit diameters are

plotted as functions of the pressure ratio in Fig. 5. The exit Mach number varies between MaN = 1.5 – 2.5, the exit diameter between 5 and 7 mm, which underlines the fact that a full admission turbine would not be feasible with the low mass flow under investigation. The isentropic velocity ratio = u/c1,s has been set to 0.45.

Influence of nozzle efficiency, rotor efficiency and disc friction For a rotor diameter of 0.08 m, the influence of nozzle efficiency, rotor efficiency and disc friction have been studied varying the nozzle kinetic energy coefficient cN in equation (7), the factor kR for the allowance of additional losses within the rotor in equation (14) and finally the factor kF to represent the increase of disc friction losses due to the rotor geometry in equation (19). The corresponding nozzle and rotor efficiencies as well as the turbine isentropic efficiency s calculated from equation (20) are shown in Fig. 6. The large impact of the nozzle efficiency in the overall efficiency of the turbine becomes evident from the comparison of Fig. 6 a) to c). A 10 % decrease of cN causes a relative performance drop of about 20 %. On the other hand, the sensitivity against rotor and disc friction losses is not as high, as increasing both of these by a factor of three causes a rather small efficiency drop from about 85 % ( Fig. 6 b)) to about 80 % (Fig. 6 f)). In order to obtain a conservative estimate, for the following calculations the nozzle kinetic energy coefficient has been set to cN = 0,95, which is in the lower range of the values suggested by literature, whereas the two factors kR and kF have been set to a value of 3. This yields rotor efficiencies of about 70 – 75 %,

s

FRsN

s

outlossFlossRlossNloss

s

ts

h

cmPwccc

h

hhhh

h

hh

,01

22

21

22,1

2

,01

,,,,

,01

2,0

221211

1

(20)

Fig. 5: Enthalpy drop, exit Mach number and nozzle critical and exit diameter for the chosen operating conditions

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which corresponds to the minimum rotor efficiency stated by Deij and Trojanovski [13] even for very bad blade designs.

Fig. 6: nozzle, rotor and isentropic efficiency of the Pelton turbine calculated from equations (7), (14) and (20) for different loss coefficients and factors. Rotor diameter is 0.08 m

a) cN = 1, kR = 1, kF = 1

b) cN = 0.95, kR = 1, kF = 1

c) cN = 0.9, kR = 1, kF = 1

d) cN = 0.95, kR = 2, kF = 1

e) cN = 0.95, kR = 2, kF = 2

f) cN = 0.95, kR = 3, kF = 3

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Rotor diameter variation To demonstrate the influence of the rotor diameter on the turbine, diameters of 0.04 m, 0.08 m, 0.12 m and 0.16 m have been studied. First of all, as can be seen from Fig. 7, both nozzle and disc Reynolds number are above the critical values of 105 and 3•105, respectively, which means that the

assumption of high nozzle efficiency and the use of the formulation of the disc friction for turbulent flows are justified. Because of the given isentropic velocity ratio of 0.45, the turbine speed scales with diameter. Moreover, as the velocities are then the same for all diameters, both the specific work and the rotor efficiency, which can be seen in Fig. 8, are the same. The fact that the rotor efficiency is not influenced by the rotor diameter can be justified with the underlying assumption that the flow path within the ducts in the rotor is independent of rotor diameter. The dimensions of these ducts have been chosen according to the nozzle exit diameter, thus influencing the rotor length. Only the disc friction varies and thus there are slight differences in the turbine output. However, it is quite interesting to see the strong increase of disc friction losses with rotor diameter, shown in Fig. 7. From the diagram shown, one can quickly realize the reason why large Pelton turbines will not work for compressible working fluids. Nevertheless, for the rotor diameters considered for this study, the estimated isentropic efficiencies are quite high, especially taking into account the low specific speeds of this application ranging between 0.004 < ns < 0.02. According to the calculations, the turbine efficiency spans a range from about 70 % for D = 0.16 m to more than 80 % for the smallest rotor diameter of D = 0.04 m. This agrees very well with the efficiency of 75% determined experimentally by Rautenberg et al. [2] for a low specific speed Pelton-style turbine. A comparison to the expected efficiencies of other turbines running at these specific speeds will be given later.

Fig. 7: rotational speed, turbine output, Reynolds-number and disc friction losses for different Pelton rotor diameters as a function of turbine pressure ratio

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Variation of nozzle count Within this study, the nozzle mass flow is assumed to be constant, i.e. a variation of nozzle count effectively means an increase of mass flow keeping all other operating conditions the same. As a Pelton turbine is inherently a partial admission turbine, additional nozzles will not alter the flow through the rotors, hence the rotor efficiency will be constant. Thus, the effect of nozzle count variation is simply an increase of turbine output, as shown in Fig. 9. As the disc friction losses remain the same, the curves for all four diameters are closer together than in the plot of turbine output for one nozzle in Fig. 7. In addition, the specific speed at which the turbine is

operated increases with the square root of the mass flow increase. There will in fact be a limitation of the number of nozzles to be used, which is probably dependent on rotor diameter and hence available circumference. Moreover, with increasing diameter – which corresponds to a decreasing efficiency, as has already been shown – and increasing mass flow, care must be taken if the requirements for the use of a conventional turbine of axial or radial design are met again.

Influence of working fluid To assess the performance of the turbine with other working fluid than steam, the calculations have been performed using air. To ensure comparability to the steam results, the turbine pressure ratio

Fig. 8: efficiencies for different rotor diameters

Fig. 9: turbine output with 4 nozzles

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was adapted in order to keep both mass flow and enthalpy drop the same as before. The pressure ratio for the study with air ranges from = 0.02 to 0.18, corresponding to expansion ratios from 50 to 5 and nozzle exit Mach numbers of 1.79 – 3.15.

The differences caused by the variation of the working fluid are immediately visible from the calculated Reynolds numbers as well as from the disc friction losses, both of which can be seen in Fig. 10. For the smallest rotor diameter studied, the disc Reynolds number drops below the critical value, hence the appropriateness of the formulation used to calculate the disc friction losses must be judged critically. As a consequence of the higher disc friction losses, the turbine efficiency with air as working fluid is slightly lower than for steam.

Variation of isentropic velocity ratio u/cs

As stated above, all results shown so far have been obtained for an isentropic velocity ratio of = 0.45, which is close to the theoretical ideal velocity ratio for maximum specific work output. In order to study the influence of this parameter on turbine performance, the isentropic velocity ratio has been varied between 0.1 ≤ ≤ 0.8 for the four different rotor diameters investigated before. The impact of this parameter on the isentropic turbine efficiency can be seen from Fig. 11. For the smallest diameter used in this study, the isentropic efficiency peaks at ≈ 0.48, i.e. close to the theoretical maximum. For larger diameters, both the maximum efficiency, and the isentropic velocity ratio at which it occurs, decrease due to the influence of disc friction. For a diameter of 0.16 m, the peak efficiency is lower than 75 % at ≈ 0.41. The curves shown in Fig. 11 again explain why a large Pelton turbine running with compressible fluids will not work at an acceptable efficiency level, as the efficiency drops very quickly with increasing diameter. To reduce the disc friction losses, the circumferential velocity must be decreased, which results in a lower isentropic

Fig. 10: Calculated efficiencies (for D = 0.08 m), Reynolds numbers, disc friction losses and turbine output using air as working fluid

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velocity ratio at peak efficiency. This, in turn, causes an increase of the exit losses, which also contributes to the lower efficiencies.

Comparison with other single stage turbines In order to allow a comparison to other single stage turbines, the specific diameter and speed for the four Pelton rotor diameters studied here and the operating conditions given above using four nozzles have been calculated and added to the Cordier-diagram in Fig. 13 taken from Balje [5]. The specific speed is calculated according to the following equation:

43

302

h

Vnns

(21)

In addition, the peak efficiencies as a function of specific speed, also published by Balje [5] are shown in Fig. 12 in order to allow the derivation of the maximum efficiency to be expected from other turbine designs. The range of specific speeds covered within this study (using one nozzle and four nozzles) is indicated with dotted lines. From the information given in Fig. 12, the maximum

turbine efficiency between 0.004 ≤ ns ≤ 0.06 increases from about 20 % to a maximum of about 60 %. This is considerably lower than the efficiency range of 70 % up to more than 80 % calculated for a Pelton turbine within this study. As shown in Fig. 12, multilobe machines, which are positive displacement devices, can achieve efficiencies up to 75 %, but this design is out of the scope of this publication and will not be treated in more detail here. When comparing the efficiencies stated for the different turbine designs in Fig. 12 and Fig. 13 to those calculated for the Pelton-type turbines treated in this study, one has to keep in mind the extremely low mass flows used for the present

calculations. These will result in turbine sizes considerably smaller than the diameters for the Pelton-type turbines used here, which in turn will yield greatly reduced efficiencies for the turbine types shown in Fig. 12.

Fig. 11: turbine efficiency over isentropic velocity ratio for different rotor diameters

Fig. 12: Maximum efficiencies as function of specific speed (from [5])

Specific speed range under consideration for this study

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CONCLUSIONS The applicability of Pelton-type turbines for heat recovery applications with high enthalpy drop and low mass flow has been examined using correlations for performance from the published literature. At these very low specific speeds (0.004 ≤ ns ≤ 0.06) conventional axial or radial turbine designs exhibit a poor performance. The theoretically assessed performance of the Pelton-style turbine shows a remarkable potential for small rotor sizes, with an isentropic efficiency of around 75%, which is consistent with experimental values quoted by Rautenberg et al. [2]. However, there is insufficient data available from the literature to model all aspects, such as rotor losses, in more detail and some assumptions and estimates were necessary for the theoretical treatment. These correlations need to be further refined and validated to have a more accurate performance model for further optimization of the design. Nevertheless, some important conclusions can already be drawn from the results presented. Firstly, the design of the nozzle has to be done with great care and sophistication, as a high exit Mach number results from the high enthalpy drop. An inadequate nozzle geometry will result in a poor nozzle performance, and this has a great impact on the overall turbine performance. Secondly, the disc friction losses necessitate relatively small rotor diameters. Thirdly, for a further reduction of disc friction losses, a casing with small radial gaps should be used. For the future, it is planned to design such a turbine based on the calculations presented here and to assess and optimize its performance using computational fluid dynamics. Finally it is planned to build a prototype for experimental tests to validate the theoretical assessment and obtain further data to refine the models used.

Fig. 13: Cordier-diagram (from [5]) including the Pelton turbine configurations studied

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