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Performance estimation of Tesla turbine applied in small scale Organic Rankine Cycle (ORC) system
Applied Thermal Engineering 110 (2017) 318–326
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
Performance estimation of Tesla turbine applied in small scale Organic Rankine Cycle (ORC) system Jian Song, Chun-wei Gu, Xue-song Li ⇑ Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China
h i g h l i g h t s One-dimensional model of the Tesla turbine is improved and applied in ORC system. Working fluid properties and system operating conditions impact efficiency. The influence of turbine efficiency on ORC system performance is evaluated. Potential of using Tesla turbine in ORC systems is estimated.
a r t i c l e
i n f o
Article history: Received 12 June 2016 Revised 10 August 2016 Accepted 26 August 2016 Available online 27 August 2016 Keywords: ORC Tesla turbine One-dimensional model Performance estimation
a b s t r a c t Organic Rankine Cycle (ORC) system has been proven to be an effective method for the low grade energy utilization. In small scale applications, the Tesla turbine offers an attractive option for the organic expander if an efficient design can be achieved. The Tesla turbine is simple in structure and is easy to be manufactured. This paper improves the one-dimensional model for the Tesla turbine, which adopts a non-dimensional formulation that identifies the dimensionless parameters that dictates the performance features of the turbine. The model is used to predict the efficiency of a Tesla turbine that is applied in a small scale ORC system. The influence of the working fluid properties and the operating conditions on the turbine performance is evaluated. Thermodynamic analysis of the ORC system with different organic working fluids and under various operating conditions is conducted. The simulation results reveal that the ORC system can generate a considerable net power output. Therefore, the Tesla turbine can be regarded as a potential choice to be applied in small scale ORC systems. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Primary energy consumption is enlarging rapidly with the development of the human society. Energy shortage and environmental deterioration are two consequent crucial issues that the developing world has to face. In order to solve these problems, the utilization of low grade heat sources, such as the geothermal energy [1,2], the solar energy [3,4], the biomass energy [5,6] and the waste heat [7,8], is attracting broad attention in recent years. Among all of the existing technologies, the Organic Rankine Cycle (ORC) has been proven to be one of the most effective methods for the low grade energy conversion [9–14]. The axial flow turbine and the radial in-flow turbine are typically selected as the expanders in the ORC system. However, in small scale applications, the traditional organic expanders are not suitable since the flow loss will be considerably large. In addition, the high rotation speed of ⇑ Corresponding author. E-mail address: [email protected] (X.-s. Li). http://dx.doi.org/10.1016/j.applthermaleng.2016.08.168 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.
the traditional turbines under small mass flow rate condition also limits their practical applications. In this case, the Tesla turbine allows a low-cost and reliable design for the organic expander that could be an attractive option for small scale ORC systems. The Tesla turbine was invented by the famous scientist, Nikola Tesla, in 1913 [15]. It is a kind of turbo-machinery that combines a series of flat parallel discs rather than rotating blades. Thus, the Tesla turbine is called the bladeless turbine as well. The discs distribute co-axially along a shaft such that a small gap is formed between any two adjacent discs. This design makes use of the viscous effect of the working fluid which occurs in the boundary layer flow between the rotating discs. The working fluid flows spirally from the outer part to the inner part and transfers the kinetic energy to the discs. Then the working fluid flows out through the holes located between the inner part of the discs and the shaft. The combination of the discs and the shaft is placed inside a shell and a plenum chamber is formed, out of which several nozzles are distributed uniformly to supply the inflow working fluid.
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Nomenclature _ m Q h cp T p W v c r b f Re V A v^ h ^ W X C exp sim N n
mass flow rate, kg/s heat load, kW specific enthalpy, kJ/kg specific heat capacity, kJ/kgK temperature, K pressure, kPa power, kW velocity, m/s sonic speed, m/s radius, mm gap distance, mm friction force factor Reynolds number volume, m3 area, m2 relative velocity, m/s dimensionless relative velocity a dimensionless parameter constant number experiment simulation number of the discs rotation speed, rpm
Greek symbols sw shear stress, N/m2 g efficiency
In the subsequent years after the invention of the Tesla turbine, this novel concept has received enormous attention in both technical and industrial fields. Many analytical and experimental investigations have been conducted to explore the performance of the Tesla turbine. Rice [16] reviewed the principles of the Tesla-type turbomachinery and discussed the problems with nozzles and diffusers. In addition, the analytical methods that had been found useful in modeling and calculating the flow in the rotor and the experimental results obtained by some investigators were described. Couto et al. [17] presented a simple and straightforward technique, using basic fluid mechanics, to estimate the needed number of discs required for a Tesla turbine, compressor or pump. Lemma et al. [18] presented experimental and numerical study to explore the performance characteristics of viscous flow turbines and the results indicated that the adiabatic efficiency of this kind of turbomachinery was around 25%. Lampart et al. [19] presented results of the design analysis of a Tesla bladeless turbine intended for a co-generating micro-power plant of heat capacity 20 kW, which operated in an organic Rankine cycle with a low-boiling medium; the simulation results showed that the best obtained solutions can be competitive as compared with classical small bladed turbines. Enign et al. [20] researched the experimental and theoretical characterization of a multiple-disc fan based on the principle of conservation of angular momentum. The effect of gap width and rotational speed were numerically investigated for both design and off-design volume flow rates. Carey [21] developed a 1D model analysis for flow and momentum transport in the Tesla turbine and evaluated the turbine use in Rankine cycle solar thermal power generation systems. Guha and Sengupta [22] presented a simple theory that described the three-dimensional fields of velocity and pressure in the Tesla disc turbine, which gave the torque and power output that had been verified by comparing the theoretical predictions with recently published experimental results. In this paper, the one-dimensional model for the Tesla turbine is used to predict its performance, which focuses on the flow
u
q l n
nozzle velocity coefficient density, kg/m3 viscosity coefficient, Pas relative radius
Subscripts wf working fluid tot total HS heat source in inlet out outlet pump pump evap evaporator exp expander cond condenser c cooling water net net the thermal T turbine 1 outer circumference of the rotor 2 inner circumference of the rotor Acronyms ORC Organic Rankine Cycle GWP global warming potential ODP ozone depletion potential
characteristic and the momentum transfer in the Tesla turbine. As for a low grade heat source, a small scale ORC system is designed to utilize the energy and the Tesla turbine is applied to generate the power output. The one-dimensional model is used to predict the turbine efficiency. The influence of the working fluid properties and the ORC system operating conditions on the Tesla turbine performance is evaluated. Thermodynamic analysis of the ORC system is conducted to explore the potential of applying the Tesla turbine in such small scale systems. 2. Thermodynamic model of ORC system Fig. 1 shows the schematic diagram of a basic ORC system, which consists of a working fluid pump, an evaporator, an organic expander and a condenser. The liquid organic working fluid from the condenser is firstly pumped into the evaporator, where it is converted into saturated or superheated vapor by the heat source. Next, the organic vapor expands in the expander to produce power. Afterwards, the exhaust organic vapor from the expander is condensed to liquid in the condenser by the cooling water. The thermal process of the ORC system is shown in Fig. 2, which can also be described as follows. Process 1–2 in the working fluid pump is given by
W pump ¼
_ wf ðh2s h1 Þ m
gpump
ð1Þ
where h2s is the isentropic enthalpy of the working fluid after being compressed in the pump, and gpump is the efficiency of the pump. Process 2–4 in the evaporator is given by
_ wf ðh4 h2 Þ ¼ m _ HS cp;HS T HS;in T HS;out Q ev ap ¼ m
ð2Þ
where cp,HS is the average specific heat capacity of the heat source, and THS,in and THS,out are defined as its inlet and outlet temperatures, respectively.
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Fig. 1. Schematic diagram of a basic ORC system.
Fig. 3. Schematic diagram of a Tesla turbine. Fig. 2. T-s diagram of ORC.
Process 4–5 in the organic expander is given by
_ wf ðh4 h5s Þ gT WT ¼ m
ð3Þ
where h5s is the isentropic enthalpy of the exhaust organic vapor at the expander outlet, and gT is the efficiency of the organic expander. Process 5–1 in the condenser is given by
_ c cp;c T c;out T c;in _ wf ðh5 h1 Þ ¼ m Q cond ¼ m
ð4Þ
where cp,c is the average specific heat capacity of the cooling water, Tc,in and Tc,out are its inlet and outlet temperatures, respectively. The net power output of the ORC system is
W net ¼ W T W pump
ð5Þ
The thermal efficiency of the ORC system can be calculated as
gnet ¼
W net W T W pump ¼ Q HS GHS cp;HS T HS;in T HS;out
ð6Þ
3. One-dimensional model of the Tesla turbine 3.1. Model analysis Fig. 3 shows the schematic diagram of a Tesla turbine. The working fluid expands in the inlet nozzles and then flows spirally into the rotor. The viscous effect that occurs in the boundary layers drags the discs to rotate, within which the momentum of the working fluid transfers to kinetic energy of the rotating discs. After-
wards, the working fluid flows out through the hole near the inner part of the discs and the shaft. A one-dimensional model [21] is presented to analysis the flow characteristics in the Tesla turbine. First, the working fluid expands in the nozzles, with enthalpy dropping and velocity increasing:
(
v1 ¼ u v 1 ¼ c;
v r1 ¼
pffiffiffiffiffiffiffiffiffiffiffi 2Dhs ; if
_ m 2p r 1 b q
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v h1 ¼ v 21 v 2r1
if
v1 < c v1 P c
ð7Þ
ð8Þ
ð9Þ
where u is the velocity coefficient of the nozzle, which indicates the flow loss that occurs in the expansion process. Eqs. (8) and (9) show the velocity components in r-direction and h-direction. In the original model, the maximum velocity at the nozzle outlet is assumed to be the sonic speed (if choked in the nozzle). In other words, if the outlet velocity calculated is higher than the sonic speed, v1 will be replaced by the sonic speed. In the classical theory of turbine blade, however, the blade profile after throat allows a supersonic flow at the blade outlet by a series of expansion waves, the Mach number of which can reach up to 1.4 based on experience. Similarly, the nozzle profile of the Tesla turbine can also produce a supersonic flow at the nozzle outlet. Moreover, it is not difficult to design a Laval nozzle for the Tesla turbine, in which the numerical model of Eq. (10) is also available even the Mach number of the flow at the nozzle outlet is higher than 1.4. Therefore, in this paper the model is improved by using the actual speed of the flow at the nozzle outlet.
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v1
pffiffiffiffiffiffiffiffiffiffiffi ¼ u 2Dhs
The flow in the gaps between any two adjacent discs is a very thin fluid film, which is similar to that in the bearing [23,24] and the seal [25,26]. Then, it is assumed to be a periodic symmetric, steady, incompressible laminar flow. In the cylindrical coordinates, the governing continuity Eq. (11) and momentum Eqs. (12)–(14) are listed below. Continuity:
1 @ðrv r Þ 1 @ v h @ v z þ þ ¼0 r @r r @h @z
ð11Þ
r-direction momentum:
vr
@v r v h @v r @ v r v 2h 1 @P þ þ vz ¼ @r r @h @z r q @r " # 2 2 1 @ @v r 1 @ v r @ v r v r 2 @v h þt þ 2 þ 2 2 2 r þ fr r @r r @h2 r @h @r @z r h-direction momentum:
@v h v h @v h @v h v r v h þ þ vz þ @r r @h @z r " # 1 @ @v h 1 @2v h @2v h v h 2 @v r þt r þ 2 2 þ 2 þ fh r @r r @h2 r @h @r @z2 r
dv h 24pr lv^ h v h ¼ _ dr r mb
ð23Þ
Several dimensionless parameters are defined and introduced to make the equations simpler
^ ¼ v^ h W U1
n¼
r r1
X¼
_ 2b m r 1 pr 1 l
ð24Þ
ð25Þ
^ dW 48n 1 ^ W 2 ¼ dn X n
ð26Þ
Using the basic mathematical methods, Eq. (26) can be solved and the result is shown below.
^ ¼ 1 e24nX W n ð13Þ
ð14Þ
ð15Þ
2
X 24n2 e X þC 24
ð27Þ
C is a constant value and it can be determined by the boundary condition at the outer circumference of the rotor.
^ 1 ¼ e24X W
According to the assumptions and the idealizations, the four equations above can be reduced to the followings. Continuity:
1 @ðrv r Þ ¼0 r @r
ð22Þ
Finally, the equation can be simplified and given by
1 @P ¼ qr @h
@v z v h @v z @v z 1 @P þ þ vz ¼ @r r @h @z q @z " # 2 1 @ @v z 1 @ v z @2v z þ fz þt r þ þ 2 r @r r @h2 @r @z2
@v h v r v h 12lv^ h þ ¼ @r r qb2
dv^ h 48r 1 v^ h 2U 1 ¼ 2 dr r1 X r1 r r
z-direction momentum:
vr
vr
Then, Eq. (23) can be transformed to
ð12Þ
vr
Thus, the h-direction momentum equation becomes
ð10Þ
X 24 e X þC 24
¼
v h1 U1
ð28Þ
U1
Therefore, the equation can be indicated as
2 2 ^ ¼ 1 e24nX X e24nX þ W ^ 1 X e24X W n 24 24
ð29Þ
The power output and the efficiency of the Tesla turbine can be calculated through the following equations.
_ ½v h1 U 1 v h2 U 2 ¼ m _ WT ¼ m
h
i ^ 1 þ 1 U2 W ^ 2 þ n2 n2 U 2 W 1
1
r-direction momentum:
@v v 1 @P þ fr vr r ¼ @r r q @r 2 h
ð30Þ ð16Þ
gT ¼
h-direction momentum:
vr
@v h v r v h þ ¼ fh @r r
ð17Þ
z-direction momentum:
1 @P 0¼ q @z
ð18Þ
The model focuses on the momentum transfer in the gap, thus the primary interest is the h-direction momentum, Eq. (17). fh is regarded as the wall friction force on a micro fluid element between the discs and it can be given by
fh ¼
Fh
qV
¼
2sw A qAb
ð19Þ
where sw is the shear stress
sw ¼ f
qv^ 2h 2
¼
24 qv^ 2h 24 qv^ 2h 6lv^ h ¼ ¼ ^ Re 2 qv h 2b=l 2 b
ð20Þ
v^ h is the relative velocity of the working fluid
v^ h ¼ v h U ¼ v h
U1 r r1
ð21Þ
v h1 U1 v h2 U2 Dh
¼
^ 1 þ 1 U2 W ^ 2 þ n2 n2 U 2 W 1
1
Dh
ð31Þ
According to Eqs. (29) and (31), if the geometry size of a Tesla turbine is determined, the efficiency of the Tesla turbine is mainly ^ 1 . Fig. 4 related to these two dimensionless parameters, X and W shows the variation of the turbine efficiency with these two parameters for a specified Tesla turbine. The trends are clear from the results. The highest Tesla turbine efficiency is attained for both low X and low dimensionless inlet tangential velocity differ^1 . ence W 3.2. Validation of the model In Ref. [27], Rice presented some experimental results of a Tesla turbine, which is used to verify the one-dimensional model in this paper. The turbine was stated to have discs with an outer radius of 8.89 cm, an inner radius of 3.35 cm and a gap distance of 1.02 mm. Air was selected as the working fluid in this experimental research. In addition, the author declared that virtually all the pressure drop occurred in the nozzle. Thus, the exit of the nozzle was at atmosphere pressure, which could determine the enthalpy drop in the nozzle and the velocity of the working fluid at the rotor inlet. The comparison of the experimental results and those predicted
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properties of these working fluids are listed in Table 2, which are obtained from REFPROP 9.1. Thermodynamic analysis of the ORC system with different working fluids is conducted. The simulation is carried out by a computer program written by the authors in the FORTRAN environment [28–30]. Several conditions and assumptions are given below: (1) the heat loss and the pressure loss of the pipelines in the ORC system are ignored; (2) the efficiency of the working fluid pump is set as 0.8; (3) the Tesla turbine efficiency is calculated by the onedimensional model; (4) the pinch point temperature differences of the evaporator and the condenser are fixed at 6 K; (5) the superheating degree of the working fluid at the inlet of the Tesla turbine is set as 1 K; (6) the condensation temperature of the working fluid is set as 305 K.
^ 1 for a Tesla turbine with Fig. 4. Prediction of the efficiency variation with X and W n2 = 0.4 and n = 8000 rpm.
by the one-dimensional model is listed in Table 1. The simulation results with Eq. (7) are consistent with the results in [21] because they use the same model. Adopting the improved model with Eq. (10), the results become much better and agree well with the experimental data.
4.2. Results of the ORC system with different working fluids In this section, the outlet temperature of the hot water after heat exchange is fixed at 338.15 K. Thus, the total heat load absorbed by the ORC system is constant, which only reaches about 32 kW. Thus, this ORC system can be categorized as a small scale one. For each kind of the working fluids, the computer program arrives to a proper evaporation temperature through continuous iteration, the heat load above which is utilized for evaporation and below which is used for preheating the working fluid to the bubble point temperature. Thermodynamic results of the ORC system with different working fluids are listed in Table 3, including the evaporation temperature, the evaporation pressure, the mass flow rate of the working
4. Results and discussion 4.1. Heat source condition and working fluid selection The selected heat source is hot water, the mass flow rate and the initial temperature of which are 0.5 t/h and 393.15 K, respectively. According to the temperature level of the heat source, seven organic fluids are selected as the working fluid candidates. The Table 1 Comparison of the experimental and simulation results of the Tesla turbine. Tin (K) 368 368 368
Pin (kPa) 377 377 377
_ (kg/s) m
n (rpm) 6300 8500 9200
0.00194 0.00194 0.00194
g (exp)
X 7.35 7.35 7.35
0.217 0.254 0.258
g in [21] 0.155 0.206 0.222
g (sim)
g (sim)
Eq. (7)
Eq. (10)
0.157 0.207 0.222
0.184 0.243 0.261
Table 2 Properties of the working fluid candidates. Working fluid
Molecular weight (g/mol)
Normal boiling point (K)
Critical temperature (K)
Critical pressure (kPa)
GWP
ODP
R123 R600 R600a R236ea R236fa R245ca R245fa
152.93 58.12 58.12 152.04 152.04 134.05 134.05
301.0 272.7 261.4 279.3 271.7 298.3 288.3
456.8 425.1 407.8 412.4 398.1 477.6 427.2
3661.8 3796.0 3629.0 3502.0 3200.0 3940.7 3651.0
120 20 20 1350 9400 693 950
0.012 0 0 0 0 0 0
Table 3 Thermodynamic results of the ORC system with different working fluids. Working fluid
Evaporation temperature (K)
Evaporation pressure (kPa)
Mass flow rate of the working fluid (kg/s)
Viscosity at the rotor inlet (lPas)
R123 R600 R600a R236ea R236fa R245ca R245fa
343.8 346.5 348.9 349.8 352.1 345.8 346.9
383.8 873.2 1230.0 924.2 1216.9 469.9 672.9
0.17 0.08 0.08 0.17 0.19 0.14 0.15
11.4 7.9 8.0 11.7 11.8 15.9 11.1
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Table 4 Initial parameters for the Tesla turbine. Parameter
Symbol
Value
Nozzle velocity coefficient Outer radius of the discs Inner radius of the discs Gap distance Number of the gaps Rotational speed of the rotor
u r1 r2 b N n
0.9 100 mm 40 mm 1 mm 50 8000 rpm
Fig. 5. Efficiency of the Tesla turbine with different working fluids.
fluid and the viscosity at the rotor inlet. The temperature grade of the heat source is low, which results in low evaporation temperature of the working fluids. The difference among the evaporation temperatures is relatively small. However, the evaporation pressure and the mass flow rate of the working fluid vary much. The viscosity of the working fluid has an influence on the effect between the plate and the working fluid. As for the Tesla turbine, several initial geometry parameters are given according to some previous researches [21,23] and are listed in Table 4. Using the aforementioned one-dimensional model, the efficiencies of the Tesla turbine with each kind of the working fluids are calculated. The results are shown in Fig. 5. The efficiency of the Tesla turbine with R245ca reaches 0.387, the highest among all the working fluids. The efficiencies of the Tesla turbine with R600 and R600a are considerably low, which are only 0.346 and 0.341, respectively. There are non-ignorable efficiency differences among the Tesla turbines with different working fluids, which confirms that the working fluid properties and the thermodynamic parameters have an influence on the turbine performance. The two parameters, the dimensionless parameter X and the ^ 1 of the Tesla dimensionless inlet tangential velocity difference W turbine with different working fluids are calculated and compared, the results of which are shown in Fig. 6. As for X, the Tesla turbine with R236fa is the highest, while those with R600 and R600a are considerably low, which is mainly related to the mass flow rate and the viscosity of the working fluid listed in Table 3. On the contrary, the Tesla turbines with R600 and R600a yield high dimen^ 1 and that with sionless inlet tangential velocity difference W R236fa is the lowest. It can be concluded that a higher evaporation pressure corresponds to a low expansion ratio and lower relative ^ 1. velocity, thus lower W High Tesla turbine efficiency can be attained with low X and low ^ 1 . The efficiency of the Tesla turbine with R236fa is higher than W ^ 1 has a more sigthat with R600 and R600a, which indicates that W nificant influence on the turbine performance than X. Since the ^ 1 , it yields Tesla turbine with R245ca has both moderate X and W the highest efficiency.
^ 1 of the Tesla turbine with different working fluids. Fig. 6. Variations of Re⁄ and W
Fig. 7. Thermal performance of the ORC system with different working fluids.
The net power output and the thermal efficiency of the ORC system are simulated with the calculated Tesla turbine efficiency. The results are shown in Fig. 7. Because the total heat load absorbed is fixed, the thermal performance of the ORC system is directly determined by the turbine efficiency. With R245ca as the working fluid, the turbine efficiency is the highest. Therefore, the ORC system with R245ca yields the highest net power output and system thermal efficiency, which are 1.25 kW and 0.04, respectively. 4.3. Results of ORC under different conditions (ORC system with R245ca) With R245ca as the working fluid, the Tesla turbine efficiency is the highest among all the working fluids. R245ca is selected as the working fluid in this section. The outlet temperature of the heat source is not fixed here to obtain different operating conditions.
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(a) Variation of the outlet temperature of the hot water with the evaporation temperature of the working fluid
(b) Variation of the mass flow rate of R245ca with the evaporation temperature
^ 1 with the evaporation temperature. Fig. 9. Variations of Re⁄ and W
Fig. 10. Variations of the Tesla turbine efficiency with the evaporation temperature.
Fig. 8. Variations of the thermal parameters of the ORC system with R245ca.
The heat load absorbed by the ORC system varies with the outlet temperature of the hot water. For each outlet temperature of the hot water, a proper evaporation temperature of the ORC system can be calculated by the computer program, which is still related to the ratio of the latent heat to the sensible heat of the working fluid. The thermodynamic results of the ORC system with R245ca as the working fluid are shown in Fig. 8. Fig. 8a indicates that a higher outlet temperature of the hot water corresponds to a higher evaporation temperature of the working fluid. It is evident that the total heat load absorbed decreases with the increment of the outlet temperature of the hot water, while the heat load required per unit mass of the working fluid increases as the increasing evaporation temperature. Thus, the mass flow rate of R245ca decreases definitely when the evaporation temperature increases, as shown in Fig. 8b. ^ 1 are still calculated to investigate The variations of X and W their influence on the Tesla turbine efficiency. As shown in Fig. 9, with the increment of the evaporation temperature, X decreases ^ 1 increases. High turbine efficiency can be attained with while W ^ 1 . Thus, it is not easy to directly determine how low X and low W the turbine efficiency varies with the evaporation temperature. The efficiency of the Tesla turbine with R245ca under different conditions is shown in Fig. 10. The turbine efficiency decreases first and then increases with the increment of the evaporation temperature. When the evaporation temperature is lower than 335 K, the ^ 1 dominates in the decreasing turbine effieffect of increasing W ciency. When the evaporation temperature is higher than 335 K, the effect of decreasing X dominates in the increasing turbine efficiency. When the evaporation temperature equals 335 K, the
Fig. 11. Power output of the Tesla turbine under different operating conditions.
lowest turbine efficiency is attained at 0.379. The difference between the turbine efficiencies under 335 K and 370 K evaporation temperature conditions is nearly 0.05, which surely has a significant impact on the ORC system performance. The power outputs of the Tesla turbine under different operating conditions are shown in Fig. 11. It increases first and then decreases with the increment of the evaporation temperature. When the evaporation equals 350 K, the Tesla turbine yields the highest net power output at 1.31 kW. The power output of the Tesla turbine is related to the mass flow rate of the working fluid and the specific power output. A higher evaporation temperature corresponds to a higher power output per unit mass of working fluid and mass flow rate of working fluid directly affects the total power output of the ORC system. As shown in Fig. 12, the mass
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temperature. The system thermal efficiency reaches 0.043 when the evaporation temperature is 350 K. 5. Conclusions
Fig. 12. Working fluid mass flow rate and the specific power output of the Tesla turbine under different operating conditions.
Fig. 13. Variations of the power output of the ORC system with R245ca.
The Tesla turbine offers an attractive option for the expander design in small scale ORC systems due to its simplicity and low capital cost. This paper improves the one-dimensional model for the Tesla turbine, which adopts a non-dimensional formulation that identifies the dimensionless parameters that dictate performance features of the turbine. Although the model embodies several simplifying assumptions, its predictions are found to agree reasonably well with available measured performance data for a Tesla turbine from a previous research work. The model is used to indicate the relationship between the turbine parameters and the turbine efficiency, which shows that the dimensionless parameter X and the dimensionless inlet tangential velocity difference ^ 1 have a significant influence on the turbine performance. High W ^ 1. Tesla turbine efficiency is attained with low X and low W This paper focuses on a small scale ORC system using a Tesla turbine as the expander. For the heat source (hot water, 393.15 K, 0.5 t/h), when the outlet temperature is fixed at 338.15 K, the Tesla turbine with R245ca as the working fluid yields the highest efficiency at 0.366, while the turbine efficiency with R600 and R600a is considerably low, which are only 0.346 and 0.341, respectively. There are non-ignorable efficiency differences among the Tesla turbines with different working fluids, which confirms that the working fluid properties and the thermodynamic parameters have an influence on the Tesla turbine performance. With R245ca as the working fluid, the ORC system yields the highest net power output and system thermal efficiency, which are 1.25 kW and 0.04, respectively. With R245ca as the working fluid and outlet temperature of the heat source not fixed, thermodynamic analysis of the ORC system is conducted under different operating conditions. The Tesla turbine efficiency decreases first and then increases with the increment of the evaporation temperature. The difference between the turbine efficiencies under 335 K and 370 K evaporation temperature conditions is nearly 0.05, which surely has a significant impact on the ORC system performance. When the evaporation equals 350 K, the ORC system yields the highest net power output at 1.27 kW and the corresponding system thermal efficiency reaches 0.043. This reveals that the ORC system with a Tesla turbine can generate a considerable net power output and system thermal efficiency in small scale applications. Therefore, the Tesla turbine can be regarded as a potential choice in similar practical cases. Acknowledgement
Fig. 14. Variations of the thermal efficiency of the ORC system with R245ca.
flow rate of R245ca decreases while the specific power output increases with the increment of the evaporation temperature, which results in the trend of the turbine power output in Fig. 11. There is an optimal evaporation temperature (350 K) for the ORC system to reach the maximum net power output. Fig. 13 indicates the variations of the net power output, the turbine power and the pump power with the evaporation temperature. Since the pump power is considerably small, the net power output of the ORC system keeps the same trend with the turbine power. When the evaporation temperature equals 350 K, the ORC system with R245ca yields the maximum net power output at 1.27 kW. As for the system thermal efficiency, Fig. 14 shows that it keeps increasing with the increment of the evaporation
This research study was supported by the cooperative scientific research project of energy conversion and emission reduction among China-Europe enterprises (No. SQ2013ZOC200005). References [1] V. Zare, A comparative exergoeconomic analysis of different ORC configurations for binary geothermal power plants, Energy Convers. Manage. 105 (2015) 127–138. [2] F. Heberle, D. Brüggemann, Exergy based fluid selection for a geothermal Organic Rankine Cycle for combined heat and power generation, Appl. Therm. Eng. 30 (11) (2010) 1326–1332. [3] G. Pei, J. Li, J. Ji, Analysis of low temperature solar thermal electric generation using regenerative Organic Rankine Cycle, Appl. Therm. Eng. 30 (8) (2010) 998–1004. [4] A.M. Delgado-Torres, L. García-Rodríguez, Analysis and optimization of the low-temperature solar organic Rankine cycle (ORC), Energy Convers. Manage. 51 (12) (2010) 2846–2856.
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Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
Performance estimation of Tesla turbine applied in small scale Organic Rankine Cycle (ORC) system Jian Song, Chun-wei Gu, Xue-song Li ⇑ Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China
h i g h l i g h t s One-dimensional model of the Tesla turbine is improved and applied in ORC system. Working fluid properties and system operating conditions impact efficiency. The influence of turbine efficiency on ORC system performance is evaluated. Potential of using Tesla turbine in ORC systems is estimated.
a r t i c l e
i n f o
Article history: Received 12 June 2016 Revised 10 August 2016 Accepted 26 August 2016 Available online 27 August 2016 Keywords: ORC Tesla turbine One-dimensional model Performance estimation
a b s t r a c t Organic Rankine Cycle (ORC) system has been proven to be an effective method for the low grade energy utilization. In small scale applications, the Tesla turbine offers an attractive option for the organic expander if an efficient design can be achieved. The Tesla turbine is simple in structure and is easy to be manufactured. This paper improves the one-dimensional model for the Tesla turbine, which adopts a non-dimensional formulation that identifies the dimensionless parameters that dictates the performance features of the turbine. The model is used to predict the efficiency of a Tesla turbine that is applied in a small scale ORC system. The influence of the working fluid properties and the operating conditions on the turbine performance is evaluated. Thermodynamic analysis of the ORC system with different organic working fluids and under various operating conditions is conducted. The simulation results reveal that the ORC system can generate a considerable net power output. Therefore, the Tesla turbine can be regarded as a potential choice to be applied in small scale ORC systems. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Primary energy consumption is enlarging rapidly with the development of the human society. Energy shortage and environmental deterioration are two consequent crucial issues that the developing world has to face. In order to solve these problems, the utilization of low grade heat sources, such as the geothermal energy [1,2], the solar energy [3,4], the biomass energy [5,6] and the waste heat [7,8], is attracting broad attention in recent years. Among all of the existing technologies, the Organic Rankine Cycle (ORC) has been proven to be one of the most effective methods for the low grade energy conversion [9–14]. The axial flow turbine and the radial in-flow turbine are typically selected as the expanders in the ORC system. However, in small scale applications, the traditional organic expanders are not suitable since the flow loss will be considerably large. In addition, the high rotation speed of ⇑ Corresponding author. E-mail address: [email protected] (X.-s. Li). http://dx.doi.org/10.1016/j.applthermaleng.2016.08.168 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.
the traditional turbines under small mass flow rate condition also limits their practical applications. In this case, the Tesla turbine allows a low-cost and reliable design for the organic expander that could be an attractive option for small scale ORC systems. The Tesla turbine was invented by the famous scientist, Nikola Tesla, in 1913 [15]. It is a kind of turbo-machinery that combines a series of flat parallel discs rather than rotating blades. Thus, the Tesla turbine is called the bladeless turbine as well. The discs distribute co-axially along a shaft such that a small gap is formed between any two adjacent discs. This design makes use of the viscous effect of the working fluid which occurs in the boundary layer flow between the rotating discs. The working fluid flows spirally from the outer part to the inner part and transfers the kinetic energy to the discs. Then the working fluid flows out through the holes located between the inner part of the discs and the shaft. The combination of the discs and the shaft is placed inside a shell and a plenum chamber is formed, out of which several nozzles are distributed uniformly to supply the inflow working fluid.
J. Song et al. / Applied Thermal Engineering 110 (2017) 318–326
319
Nomenclature _ m Q h cp T p W v c r b f Re V A v^ h ^ W X C exp sim N n
mass flow rate, kg/s heat load, kW specific enthalpy, kJ/kg specific heat capacity, kJ/kgK temperature, K pressure, kPa power, kW velocity, m/s sonic speed, m/s radius, mm gap distance, mm friction force factor Reynolds number volume, m3 area, m2 relative velocity, m/s dimensionless relative velocity a dimensionless parameter constant number experiment simulation number of the discs rotation speed, rpm
Greek symbols sw shear stress, N/m2 g efficiency
In the subsequent years after the invention of the Tesla turbine, this novel concept has received enormous attention in both technical and industrial fields. Many analytical and experimental investigations have been conducted to explore the performance of the Tesla turbine. Rice [16] reviewed the principles of the Tesla-type turbomachinery and discussed the problems with nozzles and diffusers. In addition, the analytical methods that had been found useful in modeling and calculating the flow in the rotor and the experimental results obtained by some investigators were described. Couto et al. [17] presented a simple and straightforward technique, using basic fluid mechanics, to estimate the needed number of discs required for a Tesla turbine, compressor or pump. Lemma et al. [18] presented experimental and numerical study to explore the performance characteristics of viscous flow turbines and the results indicated that the adiabatic efficiency of this kind of turbomachinery was around 25%. Lampart et al. [19] presented results of the design analysis of a Tesla bladeless turbine intended for a co-generating micro-power plant of heat capacity 20 kW, which operated in an organic Rankine cycle with a low-boiling medium; the simulation results showed that the best obtained solutions can be competitive as compared with classical small bladed turbines. Enign et al. [20] researched the experimental and theoretical characterization of a multiple-disc fan based on the principle of conservation of angular momentum. The effect of gap width and rotational speed were numerically investigated for both design and off-design volume flow rates. Carey [21] developed a 1D model analysis for flow and momentum transport in the Tesla turbine and evaluated the turbine use in Rankine cycle solar thermal power generation systems. Guha and Sengupta [22] presented a simple theory that described the three-dimensional fields of velocity and pressure in the Tesla disc turbine, which gave the torque and power output that had been verified by comparing the theoretical predictions with recently published experimental results. In this paper, the one-dimensional model for the Tesla turbine is used to predict its performance, which focuses on the flow
u
q l n
nozzle velocity coefficient density, kg/m3 viscosity coefficient, Pas relative radius
Subscripts wf working fluid tot total HS heat source in inlet out outlet pump pump evap evaporator exp expander cond condenser c cooling water net net the thermal T turbine 1 outer circumference of the rotor 2 inner circumference of the rotor Acronyms ORC Organic Rankine Cycle GWP global warming potential ODP ozone depletion potential
characteristic and the momentum transfer in the Tesla turbine. As for a low grade heat source, a small scale ORC system is designed to utilize the energy and the Tesla turbine is applied to generate the power output. The one-dimensional model is used to predict the turbine efficiency. The influence of the working fluid properties and the ORC system operating conditions on the Tesla turbine performance is evaluated. Thermodynamic analysis of the ORC system is conducted to explore the potential of applying the Tesla turbine in such small scale systems. 2. Thermodynamic model of ORC system Fig. 1 shows the schematic diagram of a basic ORC system, which consists of a working fluid pump, an evaporator, an organic expander and a condenser. The liquid organic working fluid from the condenser is firstly pumped into the evaporator, where it is converted into saturated or superheated vapor by the heat source. Next, the organic vapor expands in the expander to produce power. Afterwards, the exhaust organic vapor from the expander is condensed to liquid in the condenser by the cooling water. The thermal process of the ORC system is shown in Fig. 2, which can also be described as follows. Process 1–2 in the working fluid pump is given by
W pump ¼
_ wf ðh2s h1 Þ m
gpump
ð1Þ
where h2s is the isentropic enthalpy of the working fluid after being compressed in the pump, and gpump is the efficiency of the pump. Process 2–4 in the evaporator is given by
_ wf ðh4 h2 Þ ¼ m _ HS cp;HS T HS;in T HS;out Q ev ap ¼ m
ð2Þ
where cp,HS is the average specific heat capacity of the heat source, and THS,in and THS,out are defined as its inlet and outlet temperatures, respectively.
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Fig. 1. Schematic diagram of a basic ORC system.
Fig. 3. Schematic diagram of a Tesla turbine. Fig. 2. T-s diagram of ORC.
Process 4–5 in the organic expander is given by
_ wf ðh4 h5s Þ gT WT ¼ m
ð3Þ
where h5s is the isentropic enthalpy of the exhaust organic vapor at the expander outlet, and gT is the efficiency of the organic expander. Process 5–1 in the condenser is given by
_ c cp;c T c;out T c;in _ wf ðh5 h1 Þ ¼ m Q cond ¼ m
ð4Þ
where cp,c is the average specific heat capacity of the cooling water, Tc,in and Tc,out are its inlet and outlet temperatures, respectively. The net power output of the ORC system is
W net ¼ W T W pump
ð5Þ
The thermal efficiency of the ORC system can be calculated as
gnet ¼
W net W T W pump ¼ Q HS GHS cp;HS T HS;in T HS;out
ð6Þ
3. One-dimensional model of the Tesla turbine 3.1. Model analysis Fig. 3 shows the schematic diagram of a Tesla turbine. The working fluid expands in the inlet nozzles and then flows spirally into the rotor. The viscous effect that occurs in the boundary layers drags the discs to rotate, within which the momentum of the working fluid transfers to kinetic energy of the rotating discs. After-
wards, the working fluid flows out through the hole near the inner part of the discs and the shaft. A one-dimensional model [21] is presented to analysis the flow characteristics in the Tesla turbine. First, the working fluid expands in the nozzles, with enthalpy dropping and velocity increasing:
(
v1 ¼ u v 1 ¼ c;
v r1 ¼
pffiffiffiffiffiffiffiffiffiffiffi 2Dhs ; if
_ m 2p r 1 b q
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v h1 ¼ v 21 v 2r1
if
v1 < c v1 P c
ð7Þ
ð8Þ
ð9Þ
where u is the velocity coefficient of the nozzle, which indicates the flow loss that occurs in the expansion process. Eqs. (8) and (9) show the velocity components in r-direction and h-direction. In the original model, the maximum velocity at the nozzle outlet is assumed to be the sonic speed (if choked in the nozzle). In other words, if the outlet velocity calculated is higher than the sonic speed, v1 will be replaced by the sonic speed. In the classical theory of turbine blade, however, the blade profile after throat allows a supersonic flow at the blade outlet by a series of expansion waves, the Mach number of which can reach up to 1.4 based on experience. Similarly, the nozzle profile of the Tesla turbine can also produce a supersonic flow at the nozzle outlet. Moreover, it is not difficult to design a Laval nozzle for the Tesla turbine, in which the numerical model of Eq. (10) is also available even the Mach number of the flow at the nozzle outlet is higher than 1.4. Therefore, in this paper the model is improved by using the actual speed of the flow at the nozzle outlet.
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v1
pffiffiffiffiffiffiffiffiffiffiffi ¼ u 2Dhs
The flow in the gaps between any two adjacent discs is a very thin fluid film, which is similar to that in the bearing [23,24] and the seal [25,26]. Then, it is assumed to be a periodic symmetric, steady, incompressible laminar flow. In the cylindrical coordinates, the governing continuity Eq. (11) and momentum Eqs. (12)–(14) are listed below. Continuity:
1 @ðrv r Þ 1 @ v h @ v z þ þ ¼0 r @r r @h @z
ð11Þ
r-direction momentum:
vr
@v r v h @v r @ v r v 2h 1 @P þ þ vz ¼ @r r @h @z r q @r " # 2 2 1 @ @v r 1 @ v r @ v r v r 2 @v h þt þ 2 þ 2 2 2 r þ fr r @r r @h2 r @h @r @z r h-direction momentum:
@v h v h @v h @v h v r v h þ þ vz þ @r r @h @z r " # 1 @ @v h 1 @2v h @2v h v h 2 @v r þt r þ 2 2 þ 2 þ fh r @r r @h2 r @h @r @z2 r
dv h 24pr lv^ h v h ¼ _ dr r mb
ð23Þ
Several dimensionless parameters are defined and introduced to make the equations simpler
^ ¼ v^ h W U1
n¼
r r1
X¼
_ 2b m r 1 pr 1 l
ð24Þ
ð25Þ
^ dW 48n 1 ^ W 2 ¼ dn X n
ð26Þ
Using the basic mathematical methods, Eq. (26) can be solved and the result is shown below.
^ ¼ 1 e24nX W n ð13Þ
ð14Þ
ð15Þ
2
X 24n2 e X þC 24
ð27Þ
C is a constant value and it can be determined by the boundary condition at the outer circumference of the rotor.
^ 1 ¼ e24X W
According to the assumptions and the idealizations, the four equations above can be reduced to the followings. Continuity:
1 @ðrv r Þ ¼0 r @r
ð22Þ
Finally, the equation can be simplified and given by
1 @P ¼ qr @h
@v z v h @v z @v z 1 @P þ þ vz ¼ @r r @h @z q @z " # 2 1 @ @v z 1 @ v z @2v z þ fz þt r þ þ 2 r @r r @h2 @r @z2
@v h v r v h 12lv^ h þ ¼ @r r qb2
dv^ h 48r 1 v^ h 2U 1 ¼ 2 dr r1 X r1 r r
z-direction momentum:
vr
vr
Then, Eq. (23) can be transformed to
ð12Þ
vr
Thus, the h-direction momentum equation becomes
ð10Þ
X 24 e X þC 24
¼
v h1 U1
ð28Þ
U1
Therefore, the equation can be indicated as
2 2 ^ ¼ 1 e24nX X e24nX þ W ^ 1 X e24X W n 24 24
ð29Þ
The power output and the efficiency of the Tesla turbine can be calculated through the following equations.
_ ½v h1 U 1 v h2 U 2 ¼ m _ WT ¼ m
h
i ^ 1 þ 1 U2 W ^ 2 þ n2 n2 U 2 W 1
1
r-direction momentum:
@v v 1 @P þ fr vr r ¼ @r r q @r 2 h
ð30Þ ð16Þ
gT ¼
h-direction momentum:
vr
@v h v r v h þ ¼ fh @r r
ð17Þ
z-direction momentum:
1 @P 0¼ q @z
ð18Þ
The model focuses on the momentum transfer in the gap, thus the primary interest is the h-direction momentum, Eq. (17). fh is regarded as the wall friction force on a micro fluid element between the discs and it can be given by
fh ¼
Fh
qV
¼
2sw A qAb
ð19Þ
where sw is the shear stress
sw ¼ f
qv^ 2h 2
¼
24 qv^ 2h 24 qv^ 2h 6lv^ h ¼ ¼ ^ Re 2 qv h 2b=l 2 b
ð20Þ
v^ h is the relative velocity of the working fluid
v^ h ¼ v h U ¼ v h
U1 r r1
ð21Þ
v h1 U1 v h2 U2 Dh
¼
^ 1 þ 1 U2 W ^ 2 þ n2 n2 U 2 W 1
1
Dh
ð31Þ
According to Eqs. (29) and (31), if the geometry size of a Tesla turbine is determined, the efficiency of the Tesla turbine is mainly ^ 1 . Fig. 4 related to these two dimensionless parameters, X and W shows the variation of the turbine efficiency with these two parameters for a specified Tesla turbine. The trends are clear from the results. The highest Tesla turbine efficiency is attained for both low X and low dimensionless inlet tangential velocity differ^1 . ence W 3.2. Validation of the model In Ref. [27], Rice presented some experimental results of a Tesla turbine, which is used to verify the one-dimensional model in this paper. The turbine was stated to have discs with an outer radius of 8.89 cm, an inner radius of 3.35 cm and a gap distance of 1.02 mm. Air was selected as the working fluid in this experimental research. In addition, the author declared that virtually all the pressure drop occurred in the nozzle. Thus, the exit of the nozzle was at atmosphere pressure, which could determine the enthalpy drop in the nozzle and the velocity of the working fluid at the rotor inlet. The comparison of the experimental results and those predicted
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properties of these working fluids are listed in Table 2, which are obtained from REFPROP 9.1. Thermodynamic analysis of the ORC system with different working fluids is conducted. The simulation is carried out by a computer program written by the authors in the FORTRAN environment [28–30]. Several conditions and assumptions are given below: (1) the heat loss and the pressure loss of the pipelines in the ORC system are ignored; (2) the efficiency of the working fluid pump is set as 0.8; (3) the Tesla turbine efficiency is calculated by the onedimensional model; (4) the pinch point temperature differences of the evaporator and the condenser are fixed at 6 K; (5) the superheating degree of the working fluid at the inlet of the Tesla turbine is set as 1 K; (6) the condensation temperature of the working fluid is set as 305 K.
^ 1 for a Tesla turbine with Fig. 4. Prediction of the efficiency variation with X and W n2 = 0.4 and n = 8000 rpm.
by the one-dimensional model is listed in Table 1. The simulation results with Eq. (7) are consistent with the results in [21] because they use the same model. Adopting the improved model with Eq. (10), the results become much better and agree well with the experimental data.
4.2. Results of the ORC system with different working fluids In this section, the outlet temperature of the hot water after heat exchange is fixed at 338.15 K. Thus, the total heat load absorbed by the ORC system is constant, which only reaches about 32 kW. Thus, this ORC system can be categorized as a small scale one. For each kind of the working fluids, the computer program arrives to a proper evaporation temperature through continuous iteration, the heat load above which is utilized for evaporation and below which is used for preheating the working fluid to the bubble point temperature. Thermodynamic results of the ORC system with different working fluids are listed in Table 3, including the evaporation temperature, the evaporation pressure, the mass flow rate of the working
4. Results and discussion 4.1. Heat source condition and working fluid selection The selected heat source is hot water, the mass flow rate and the initial temperature of which are 0.5 t/h and 393.15 K, respectively. According to the temperature level of the heat source, seven organic fluids are selected as the working fluid candidates. The Table 1 Comparison of the experimental and simulation results of the Tesla turbine. Tin (K) 368 368 368
Pin (kPa) 377 377 377
_ (kg/s) m
n (rpm) 6300 8500 9200
0.00194 0.00194 0.00194
g (exp)
X 7.35 7.35 7.35
0.217 0.254 0.258
g in [21] 0.155 0.206 0.222
g (sim)
g (sim)
Eq. (7)
Eq. (10)
0.157 0.207 0.222
0.184 0.243 0.261
Table 2 Properties of the working fluid candidates. Working fluid
Molecular weight (g/mol)
Normal boiling point (K)
Critical temperature (K)
Critical pressure (kPa)
GWP
ODP
R123 R600 R600a R236ea R236fa R245ca R245fa
152.93 58.12 58.12 152.04 152.04 134.05 134.05
301.0 272.7 261.4 279.3 271.7 298.3 288.3
456.8 425.1 407.8 412.4 398.1 477.6 427.2
3661.8 3796.0 3629.0 3502.0 3200.0 3940.7 3651.0
120 20 20 1350 9400 693 950
0.012 0 0 0 0 0 0
Table 3 Thermodynamic results of the ORC system with different working fluids. Working fluid
Evaporation temperature (K)
Evaporation pressure (kPa)
Mass flow rate of the working fluid (kg/s)
Viscosity at the rotor inlet (lPas)
R123 R600 R600a R236ea R236fa R245ca R245fa
343.8 346.5 348.9 349.8 352.1 345.8 346.9
383.8 873.2 1230.0 924.2 1216.9 469.9 672.9
0.17 0.08 0.08 0.17 0.19 0.14 0.15
11.4 7.9 8.0 11.7 11.8 15.9 11.1
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323
Table 4 Initial parameters for the Tesla turbine. Parameter
Symbol
Value
Nozzle velocity coefficient Outer radius of the discs Inner radius of the discs Gap distance Number of the gaps Rotational speed of the rotor
u r1 r2 b N n
0.9 100 mm 40 mm 1 mm 50 8000 rpm
Fig. 5. Efficiency of the Tesla turbine with different working fluids.
fluid and the viscosity at the rotor inlet. The temperature grade of the heat source is low, which results in low evaporation temperature of the working fluids. The difference among the evaporation temperatures is relatively small. However, the evaporation pressure and the mass flow rate of the working fluid vary much. The viscosity of the working fluid has an influence on the effect between the plate and the working fluid. As for the Tesla turbine, several initial geometry parameters are given according to some previous researches [21,23] and are listed in Table 4. Using the aforementioned one-dimensional model, the efficiencies of the Tesla turbine with each kind of the working fluids are calculated. The results are shown in Fig. 5. The efficiency of the Tesla turbine with R245ca reaches 0.387, the highest among all the working fluids. The efficiencies of the Tesla turbine with R600 and R600a are considerably low, which are only 0.346 and 0.341, respectively. There are non-ignorable efficiency differences among the Tesla turbines with different working fluids, which confirms that the working fluid properties and the thermodynamic parameters have an influence on the turbine performance. The two parameters, the dimensionless parameter X and the ^ 1 of the Tesla dimensionless inlet tangential velocity difference W turbine with different working fluids are calculated and compared, the results of which are shown in Fig. 6. As for X, the Tesla turbine with R236fa is the highest, while those with R600 and R600a are considerably low, which is mainly related to the mass flow rate and the viscosity of the working fluid listed in Table 3. On the contrary, the Tesla turbines with R600 and R600a yield high dimen^ 1 and that with sionless inlet tangential velocity difference W R236fa is the lowest. It can be concluded that a higher evaporation pressure corresponds to a low expansion ratio and lower relative ^ 1. velocity, thus lower W High Tesla turbine efficiency can be attained with low X and low ^ 1 . The efficiency of the Tesla turbine with R236fa is higher than W ^ 1 has a more sigthat with R600 and R600a, which indicates that W nificant influence on the turbine performance than X. Since the ^ 1 , it yields Tesla turbine with R245ca has both moderate X and W the highest efficiency.
^ 1 of the Tesla turbine with different working fluids. Fig. 6. Variations of Re⁄ and W
Fig. 7. Thermal performance of the ORC system with different working fluids.
The net power output and the thermal efficiency of the ORC system are simulated with the calculated Tesla turbine efficiency. The results are shown in Fig. 7. Because the total heat load absorbed is fixed, the thermal performance of the ORC system is directly determined by the turbine efficiency. With R245ca as the working fluid, the turbine efficiency is the highest. Therefore, the ORC system with R245ca yields the highest net power output and system thermal efficiency, which are 1.25 kW and 0.04, respectively. 4.3. Results of ORC under different conditions (ORC system with R245ca) With R245ca as the working fluid, the Tesla turbine efficiency is the highest among all the working fluids. R245ca is selected as the working fluid in this section. The outlet temperature of the heat source is not fixed here to obtain different operating conditions.
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(a) Variation of the outlet temperature of the hot water with the evaporation temperature of the working fluid
(b) Variation of the mass flow rate of R245ca with the evaporation temperature
^ 1 with the evaporation temperature. Fig. 9. Variations of Re⁄ and W
Fig. 10. Variations of the Tesla turbine efficiency with the evaporation temperature.
Fig. 8. Variations of the thermal parameters of the ORC system with R245ca.
The heat load absorbed by the ORC system varies with the outlet temperature of the hot water. For each outlet temperature of the hot water, a proper evaporation temperature of the ORC system can be calculated by the computer program, which is still related to the ratio of the latent heat to the sensible heat of the working fluid. The thermodynamic results of the ORC system with R245ca as the working fluid are shown in Fig. 8. Fig. 8a indicates that a higher outlet temperature of the hot water corresponds to a higher evaporation temperature of the working fluid. It is evident that the total heat load absorbed decreases with the increment of the outlet temperature of the hot water, while the heat load required per unit mass of the working fluid increases as the increasing evaporation temperature. Thus, the mass flow rate of R245ca decreases definitely when the evaporation temperature increases, as shown in Fig. 8b. ^ 1 are still calculated to investigate The variations of X and W their influence on the Tesla turbine efficiency. As shown in Fig. 9, with the increment of the evaporation temperature, X decreases ^ 1 increases. High turbine efficiency can be attained with while W ^ 1 . Thus, it is not easy to directly determine how low X and low W the turbine efficiency varies with the evaporation temperature. The efficiency of the Tesla turbine with R245ca under different conditions is shown in Fig. 10. The turbine efficiency decreases first and then increases with the increment of the evaporation temperature. When the evaporation temperature is lower than 335 K, the ^ 1 dominates in the decreasing turbine effieffect of increasing W ciency. When the evaporation temperature is higher than 335 K, the effect of decreasing X dominates in the increasing turbine efficiency. When the evaporation temperature equals 335 K, the
Fig. 11. Power output of the Tesla turbine under different operating conditions.
lowest turbine efficiency is attained at 0.379. The difference between the turbine efficiencies under 335 K and 370 K evaporation temperature conditions is nearly 0.05, which surely has a significant impact on the ORC system performance. The power outputs of the Tesla turbine under different operating conditions are shown in Fig. 11. It increases first and then decreases with the increment of the evaporation temperature. When the evaporation equals 350 K, the Tesla turbine yields the highest net power output at 1.31 kW. The power output of the Tesla turbine is related to the mass flow rate of the working fluid and the specific power output. A higher evaporation temperature corresponds to a higher power output per unit mass of working fluid and mass flow rate of working fluid directly affects the total power output of the ORC system. As shown in Fig. 12, the mass
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temperature. The system thermal efficiency reaches 0.043 when the evaporation temperature is 350 K. 5. Conclusions
Fig. 12. Working fluid mass flow rate and the specific power output of the Tesla turbine under different operating conditions.
Fig. 13. Variations of the power output of the ORC system with R245ca.
The Tesla turbine offers an attractive option for the expander design in small scale ORC systems due to its simplicity and low capital cost. This paper improves the one-dimensional model for the Tesla turbine, which adopts a non-dimensional formulation that identifies the dimensionless parameters that dictate performance features of the turbine. Although the model embodies several simplifying assumptions, its predictions are found to agree reasonably well with available measured performance data for a Tesla turbine from a previous research work. The model is used to indicate the relationship between the turbine parameters and the turbine efficiency, which shows that the dimensionless parameter X and the dimensionless inlet tangential velocity difference ^ 1 have a significant influence on the turbine performance. High W ^ 1. Tesla turbine efficiency is attained with low X and low W This paper focuses on a small scale ORC system using a Tesla turbine as the expander. For the heat source (hot water, 393.15 K, 0.5 t/h), when the outlet temperature is fixed at 338.15 K, the Tesla turbine with R245ca as the working fluid yields the highest efficiency at 0.366, while the turbine efficiency with R600 and R600a is considerably low, which are only 0.346 and 0.341, respectively. There are non-ignorable efficiency differences among the Tesla turbines with different working fluids, which confirms that the working fluid properties and the thermodynamic parameters have an influence on the Tesla turbine performance. With R245ca as the working fluid, the ORC system yields the highest net power output and system thermal efficiency, which are 1.25 kW and 0.04, respectively. With R245ca as the working fluid and outlet temperature of the heat source not fixed, thermodynamic analysis of the ORC system is conducted under different operating conditions. The Tesla turbine efficiency decreases first and then increases with the increment of the evaporation temperature. The difference between the turbine efficiencies under 335 K and 370 K evaporation temperature conditions is nearly 0.05, which surely has a significant impact on the ORC system performance. When the evaporation equals 350 K, the ORC system yields the highest net power output at 1.27 kW and the corresponding system thermal efficiency reaches 0.043. This reveals that the ORC system with a Tesla turbine can generate a considerable net power output and system thermal efficiency in small scale applications. Therefore, the Tesla turbine can be regarded as a potential choice in similar practical cases. Acknowledgement
Fig. 14. Variations of the thermal efficiency of the ORC system with R245ca.
flow rate of R245ca decreases while the specific power output increases with the increment of the evaporation temperature, which results in the trend of the turbine power output in Fig. 11. There is an optimal evaporation temperature (350 K) for the ORC system to reach the maximum net power output. Fig. 13 indicates the variations of the net power output, the turbine power and the pump power with the evaporation temperature. Since the pump power is considerably small, the net power output of the ORC system keeps the same trend with the turbine power. When the evaporation temperature equals 350 K, the ORC system with R245ca yields the maximum net power output at 1.27 kW. As for the system thermal efficiency, Fig. 14 shows that it keeps increasing with the increment of the evaporation
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