Experimental investigation and performance prediction of a cryogenic turboexpander using artificial intelligence techniques

Applied Thermal Engineering 162 (2019) 114273

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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

Experimental investigation and performance prediction of a cryogenic turboexpander using artificial intelligence techniques

T

⁎

Manoj Kumara, , Debashis Pandab, Suraj K. Beherab, Ranjit K. Sahoob a b

Mechanical Engineering Department, NIT Rourkela, 769008, India Mechanical Engineering Department, NIT, Rourkela 769008, India

H I GH L IG H T S

methodology of a cryogenic radial turbine is proposed. • Design design parameters are validated with available results. • Turbine optimization and performance prediction, ANN and ANFIS models are developed. • For • Experimental analysis is performed for performance measurement.

A R T I C LE I N FO

A B S T R A C T

Keywords: Radial turbine design Turboexpander Cryogenics ANN ANFIS

As a major component of cryogenic turboexpander, the design and performance estimation of a radial inflow turbine determines the effectiveness of the system. To explore the performance, this paper focuses on to investigate the effect of mass flow rate and operating temperature on isentropic efficiency, temperature drop, enthalpy drop, pressure variation, and power output of a cryogenic turboexpander. Firstly, the mean-line design of a radial inflow turbine is conducted by considering different loss models. Sobol sensitivity analysis is performed to identify the major geometrical parameters which have a significant effect on the performance of the turbine. Based on the geometrical data sets, an ANN and ANFIS models are developed to predict the ranges in which maximum efficiency of the turbine is obtained with minimum losses. The designed turbine is validated with available data in the literature. Secondly, an experimental set-up with extended measuring points for data collection is developed to investigate the performance of a turboexpander at cryogenic temperature. A detailed experimental analysis is carried out to compare the temperature drop, isentropic efficiency, and power output of the turboexpander for mass flow rate in the range of 0.03–0.08 kg/s and the inlet temperature of 130, 140, and 150 K. It is noticed that the highest temperature drop is obtained for the inlet temperature of 150 K. Thirdly, based on the experimental data, an ANN and ANFIS model is developed to predict the optimal range in which the turboexpander have maximum isentropic efficiency and temperature drop. The results deduce some valuable experimental data and also accumulate the design methodology of radial inflow turbine for cryogenic applications.

1. Introduction

component which plays major role in defining the efficiency of the system. The concept of using a turbine as an expander is proposed by Lord Rayleigh in 1898. In 1930, Linde Pvt. Ltd. commercialized the application of turbine in gas liquefaction unit. In 1964, a radial turbine has used an expander in helium refrigerator system which produced 73 W refrigerating power and achieves 3 K temperature at Rutherford helium bubble chamber [41]. Generally, the efficiency of the small radial turbine is relatively low as compared to large gas turbines. Researchers suggest that the radial

The increasing demand for cryogenic fluids and modernization of cryogenic industries need efficient turboexpander which is used in various applications like superconductivity, space appliances, turbochargers, petrochemical industries, biomedical and chemical instruments, etc. The turboexpander consists of different components like nozzle, turbine, diffuser, brake compressor, aerodynamic journal, and thrust bearings, etc. Out of which, the turbine is the most important

⁎

Corresponding author. Tel.: +91 7750825846. E-mail addresses: [email protected] (M. Kumar), [email protected] (S.K. Behera).

https://doi.org/10.1016/j.applthermaleng.2019.114273 Received 12 March 2019; Received in revised form 16 August 2019; Accepted 19 August 2019 Available online 27 August 2019 1359-4311/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature

Greek symbols

Variables

ψ1 ω β α εx εr ηts ηtt ωn ϕ ψ χ σ Θ ηis ρm ν Ω

Cos vs P rp ṁ Lt Z t b R2 Rh Rs Kg E ns M d U Zr C W bt Dt Cmt Cθt ORC RTD ANFIS ANN GA GRNN RGRNN PFHX MLP RBFN

spouting velocity (m/s) blade speed ratio power (kW) pressure ratio mass flow rate (kg/s) total losses number of blades blade thickness (mm) blade height (mm) turbine inlet radius (mm) hub radius (mm) shroud radius (mm) geometrical coefficient Young’s modulus specific speed Mach number turbine diameter (mm) blade speed (m/s) turbine axial length (mm) absolute velocity (m/s) relative velocity (m/s) nozzle height (mm) nozzle throat circle diameter (mm) nozzle throat velocity (Meridional component) (mm) nozzle throat velocity (Tangential component) (mm) Organic Rankine cycle radial turbine design adaptive neuro-fuzzy inference system artificial neural network genetic algorithm general regression neural network rotated general regression neural network plate-fin heat exchanger multilayer perceptron network radial basis function network

blade inlet blockage coefficient rotational speed of turbine (RPM ) absolute flow angle (°) , Premise parameter in ANFIS absolute velocity angle (°) , Premise parameter in ANFIS axial clearance (mm) radial clearance (mm) total to static efficiency total to total efficiency natural frequency of rotor (rad/s) flow coefficient stage head coefficient absolute meridional velocity smoothing parameter rotor meridional velocity coefficient isentropic efficiency material density (kg/m3) Poisson’s ratio degree of reaction

Subscripts I Cl TEL

incidence loss clearance loss trailing edge loss

Acronyms BL 2 3 h s o i o

blade loading loss turbine inlet turbine outlet hub shroud stagnation properties turboexpander inlet turboexpander outlet

loss are assumed to be negligible during the rotor, stator, and diffuser design process. However, the internal losses are considered. The blade profile loss is occurred because of boundary layer formation and viscosity effect at the surface of the blade. The growth of the boundary layer depends on the aerodynamic profile of the turbine blade which causes the boundary layer separation [13]. It can be reduced by decreasing the blade thickness or optimizing the blade geometry. Tako et al. [47] propose that the maximum blade thickness near the leading edge delay the flow separation, turbine stall, and secondary losses. The secondary losses are obtained due to the end wall boundary layer, tip leakage, vortices, and boundary layer separation near the suction side. The flow is reversed due to the adverse pressure gradient whereas the vortices are generated because of the mixing of the secondary flow and the main flow. It occurs due to aerodynamic losses and the main reason for

inflow turbine can be extensively used due to its better aerodynamic performance and relatively low manufacturing cost. It operates at different ranges of power, mass flow rate and rotational speeds with better ruggedness and efficiency [8]. The high-pressure ratio, temperature drop, better aerodynamic design, and high rotational speed emphasize the interest of researchers for their rapid development [29]. Turbine efficiency is one of the most important parameters on which the effectiveness of the system depends. Since the turboexpander is a high power consuming devices, any moderate improvement in turbine efficiency leads to substantial improvement in system effectiveness [30]. Wu [52] suggest that the aerodynamic performance of turbomachines depend on the mean-line design methodology of stator and rotor. Hongli et al. [24] perform the experiments to study the performance of a turboexpander and its components which is used in the Brayton cycle based air refrigerator. Ghosh et al. [18,19] perform an analytical and experimental study of a high-speed cryogenic turboexpander. Some available literature for experimental and numerical analysis of cryogenic turboexpanders are mentioned in Table 1. In turboexpander, there are internal and external losses. Generally, the loss prediction correlations are developed for the large-scale turbine, but in case of small or microturbines, the aerodynamic losses become most significant. Therefore, accurate techniques (for conventional loss models) are required, which can model these kinds of losses in the best possible manner. Additionally, external losses, like bearing

Table 1 Radial inflow turbine design and testing summary.

2

Authors

Field

Pressure ratio

Fluid

Ino et al. [25] Ghosh [20,18] Balaji [10] Sam et al. [44] Li et al. [35]

Cryogenics (Experimental) Cryogenics (Experimental) Cryogenics (Experimental) Cryogenics (Numerical) Cryogenics (Numerical)

7.0 − 12.0 1.8 − 6.0 2.0 − 5.8 1.5 5.53

Helium Air Air Helium Helium

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drop, and power output of the turboexpander are analyzed and discussed in details. The data obtained from the experimental analysis are scattered and unevenly distributed. Consequently, a significant amount of data sets are obtained, which needs to be accurately predicted to save time and expenses. In this regard, the data sets are accumulated to develop the ANN and ANFIS model. The developed ANN and ANFIS structures are incorporated for performance prediction. The results show a good agreement between the experimental data and the predicted results obtained using ANFIS and ANN. It also shows that the proposed model predicts the isentropic efficiency, temperature drop, and power output efficiently. The paper is systematized as follows: The design of radial inflow turbine, description of inlet and outlet blade angles, and Sobol sensitivity analysis are summarized in Section 2. ANN and ANFIS methodologies and range of design variables are discussed in Section 3. The experimental set-up, different components and procedures are discussed in Section 4. Experimental and predicted results (ANN and ANFIS) are summarized in Section 5. Finally, Section 6 concludes the results.

diminishing the performance of the turbine [1,12]. Several researchers report the design of the turbine by considering the different loss correlations [15,6,11,48]. It is beneficial to improve the aerodynamic performance of the turbine blade profile. For this purpose, the nonaxisymmetric end wall profile is used to diminish the secondary flows [22]. The aero-thermal mathematical model of a cryogenic turbine is vital for design and control procedure. Generally, it is based on the conservation of mass, energy, and momentum equations. Although to predict the performance characteristics like dimensions, mass flow rate, isentropic efficiency, etc., nonlinear algebraic equations have opted. Peculiarly, the component parameters defined the design information, so the selection of generic functions is censorious. It can affect the overall performance of the turboexpander. Suhrmann et al. [46] develop a correlation for loss model which improves the design methodology of small-scale radial turbines for the rotor inlet diameter is less than 40 mm. Moustapha et al. [40,39] propose the design methodology for preliminary and mean-line performance analysis by considering the incidence losses, based on which the turbine geometry can be optimized. Whitfield and Baines [50] proposed the mean-line methodology for calculation of different losses and performance prediction of the turbomachines. The usage of ANN and ANFIS are demanding because of its unique features like reliable, fast response and computationally less expensive. ANN has excellent prediction ability because it does not require the iterative solution [26,17,28,16]. Several researchers report the usage of ANN and ANFIS models to predict the performance of gas turbines. Some of them apply this methodology in turbomachinery as well as to model the gas turbines and compressor characteristics [53,14,5,37]. Moraal and Kolmanovsky [38] suggest that ANN modeling is superior to curve fitting approach if it is trained properly, but it is not appropriate to model the compressor flow rate as a function of its pressure ratio and rotational speed. Kong et al. [32,31] relates the mass flow rate, pressure ratio, and isentropic efficiency using third order polynomial equation and its coefficients are obtained through GA and neural network. Moraal and Kolmanovsky [38] study the comparison between neural networks and curve fitting approach. Ghorbanian and Gholamrezaei [16] investigate the performance of a compressor using different ANN techniques like GRNN, RGRNN, RBFN, and MLP. Detailed literature review reveals that there are few works available regarding the design of cryogenic radial inflow turbine, none of them consider the loss models in the preliminary design process. Also, some works are available to predict the performance of radial inflow turbine using artificial intelligence technique. But none of them develop such models for the prediction of performance characteristics of a cryogenic turboexpander. Apart from this, very few experimental works available which investigate the performance of a turboexpander at cryogenic temperature. This paper attempts to clinch this literature gap by explaining the experimental methodology and developing an ANN and ANFIS models to investigate the performance and predicting the optimal parameters of a cryogenic turboexpander. In this paper, the preliminary design of a cryogenic radial turbine is carried out including different loss models. Sobol sensitivity analysis is performed to determine the most effective design variables which have a significant effect on the performance of the turbine. Based on the design variable datasets, an ANN and ANFIS models are developed which can predict the optimal range of input parameters in which the highest total to static efficiency is obtained with minimum losses. The obtained turbine parameters are further validated with the available mean-line design results in the literature. Based on this, the radial turbine and other components of the turboexpander are manufactured. After that, the experimental test rig is developed to investigate the performance of the turboexpander at different inlet temperature and mass flow rate. The investigation is carried out using air as a working fluid at three inlet temperatures (130 K, 140 K, and 150 K) and mass flow rate in the range of 0.03–0.08 kg/s. The effect of an increase in mass flow rate on rotational speed, isentropic efficiency, temperature

2. Design of radial turbine 2.1. Design procedure The mean-line design of a radial inflow turbine takes maximum effort and resources. The dimension of the turbine significantly depends on the properties of the working fluid. In cryogenic, the ultimate aim is to decrease the temperature for which a higher pressure ratio is required inside the turbine. It can be obtained by the continuous variation of radius from the inlet to the outlet which is responsible for the higher pressure ratio as compared to axial turbines [49]. These criteria are fulfilled through a radial expansion turbine which is relatively smaller in size and very useful in small-scale applications. The mean-line design procedure of radial turbine is presented in Fig. 1. It depends on two parameters like flow coefficient and static head coefficient.

• Flow coefficient

Fig. 1. Flowchart for mean-line design of a radial turbine. 3

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Cm3 C = m2 U2 χU2

ϕ=

2.1.1. Incidence loss This loss is based on the principle of conversion of kinetic energy into internal energy of the fluid. It increased the entropy and formulated as follows:

(1)

• Static head coefficient ψ=

C C ▵h 0 = θ 2 − λ θ3 U2 U2 U22

2 ⎡ W2 cos (β2 − β2, opt ) ⎤ ΔhI = ⎢ ⎥ 2 ⎦ ⎣

(2)

(11)

where β2, opt is the incidence angle which is defined as: where χ represents the absolute meridional velocity at the inlet and outlet of the rotor and λ is the outlet to inlet radius ratio of the turbine. The first step is to guess an initial value of total to static efficiency which is important to calculate the spouting velocity.

▵Hideal = C0s =

H02 − H03 ηts

cotβ2, opt =

Z (1 −

2 2 ⎧ L sinβ ⎫ r ⎛ W2 + W3 ⎟⎞ ⎤ Δhpassage = kp ⎛ h ⎞ + 0.68 ⎡1 − t ⎤ × b 3 × ⎡ ⎜ ⎥ ⎢ 3m ⎬ ⎢ ⎨ ⎝ Dh ⎠ r2 ⎥ 2 ⎣ ⎦ ⎠⎦ ⎣⎝ C ⎭ ⎩

(4)

⎜

where ▵Hideal is total to static ideal enthalpy drop, H0 is total enthalpy, ηts is total to static efficiency and C0s is the spouting velocity. Assuming there is no swirl at the outlet of the turbine, the inlet blade speed is obtained from the velocity ratio. The absolute rotor velocity is obtained as:

Cu2 =

(6)

where vs is blade speed ratio. Rohlik [43] proposed several equations for the analytical design of radial turbine to obtain maximum total to static efficiency. The initial value of inlet flow angle is calculated based on these equations. Whitfield and Baines [51] suggested that the number of blades of small radial turbines can be optimized. It may also reduce the flow reversal without affecting the total to static efficiency of the turbine. For this turbine design, the number of blades is optimized in the range of 11–17. The number of blades is calculated as:

Z=

π (110 − α2) tan (α2) 30

ΔhCl =

t3 = 0.02r2

(9)

U2 Z (k x εx Cx + kr εr Cr + k xr εx Cx εr Cr ) 8π

Δprel = p3rel − p2rel =

ρ3 W32 ⎡ Zt ⎤ ⎥ 2 ⎢ ⎣ πrtcosβ3 ⎦

2

Cm3 b = 1 + 5⎛ 2 ⎞ Cm2 ⎝ r2 ⎠ ⎜

(15)

This pressure loss is converted into loss coefficient formulated as:

where t2 and t3 are the blade thickness at the rotor inlet and outlet respectively. The outlet velocity triangle is deduced as follows:

Θ=

(14)

2.1.4. Trailing edge loss This loss has to be modeled by assuming that the relative pressure drop at the rotor exit is proportional to the relative kinetic energy.

After calculating the number of blades, the blade inlet angle β2 and absolute flow angle α2 are determined. The trailing edge thickness is smaller than the leading edge thickness to avoid the trailing edge losses. By considering these effects, Aungier [3] proposed the correlation for blade thickness variation which is as follows: (8)

(13)

where k x and kr are the discharge coefficients in axial and radial directions respectively, k xr is the cross coupling coefficient. In this case, the value of k x , kr , and k xr are 0.40, 0.75, and − 0.3 respectively [4].

(7)

t2 = 0.04r2

⎟

2.1.3. Rotor clearance loss It is assumed that shear flow exists in the tip clearance. The velocity is varied linearly in this region approaching from zero in the vicinity of the casing to surface velocity on the blade. Therefore, axial (εx ) and radial (εr ) clearance of the rotor has major effects on rotor clearance loss which is formulated as follows:

U2 ηts 2vs2

(12)

where Lh , Dh , and C are mean passage hydraulic length, diameter and rotor blade chord. kp is a constant which accounts the secondary losses. In this case, its value is taken as 0.2 .

(5)

U2 = vs C0s

1.98 ) Z

2.1.2. Passage loss The passage loss is a function of mean kinetic energy. It is formulated as follows:

(3)

2▵Hideal

−1.98cot (α2)

⎟

(10)

where b2 and r2 are the height and radius of the turbine blade respectively. It is significant because these variables play an important role in the optimization of the blade profile. The geometry of the rotor is presented in Fig. 2. The next step is to consider the different loss model in the design process. Various losses like incidence loss, tip clearance loss, friction loss, etc. are calculated which affects the efficiency significantly [46,39]. Based on these loss models, the total to static efficiency of the turbine is calculated which is used to update the initial guess. The losses are formulated are as follows [21]:

Fig. 2. Rotor geometry details. 4

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ΔhTEL =

Δprel 2 × p3rel γM32rel

U2 =

(16)

where k1 is a constant and Q is mass flow rate.

(17)

U2 − Cθ2 =

The total losses are formulated as:

∑ h Lt = ΔhI + Δhpassage + ΔhCl + ΔhTEL + ΔhBL

Qc =

(20)

where ρm is the density of the blade material, K g is geometrical coefficient (0.36). The mode shape and natural frequency parameters are also considered in the design process. It is calculated as follows [7]:

Et 3 12ρm (1 − υ2)

(24)

nπd 22 120k1

(25)

The flow rate (Q) is directly proportional to circumferential velocity component and inlet relative flow angle which is independent of the rotational speed. Fig. 3 shows β2 > 90° when Q > Qc , while β2 < 90° when U2 > Cθ2 . The critical flow rate (0.1 kg/s) is calculated based on the obtained parameters which are d2 = 24.90 mm, ω = 140, 000 RPM. To study the effect of blade inlet angle on the performance of the backward curved blades, the effect of six different rated flow rates on inlet and outlet blade angles are compared which are mentioned in Table 2.The flow velocity (radial) Cm2 at the inlet of the impeller is computed as:

(19)

In the design process of an efficient radial turbine, it is necessary to consider the different stresses which occur due to blade loading. Marscher [36] suggest the estimation of the elastic stresses which is obtained due to blade loading are as follows:

6.94 2πb32

nπd 22 − 120k nπd2 2k − = 60 d2 60d2

If Cθ2 = U2 , the critical flow rate (Qc ) is obtained when β2 = 90°.

(18)

The total to static efficiency is updated with the addition of various losses as follows:

ωn =

(23)

k = k1 Q

ZL r2

σr = K g ρm U22

2k d2

Cθ2 =

(W2 − U2)2

h0 ηts = h 0 + ∑ h Lt

(22)

The circumferential velocity component at the inlet of the turbine is calculated as:

2.1.5. Blade loading loss Blade loading losses are a kind of secondary loss obtained due to boundary layer separation. The curvature of the blade profile is also responsible for this type of loss. It is formulated as follows [42]:

ΔhBL = 2

πd2 ω 60

Cm2 =

Qr πd2 b2 ψ1

(26)

where b2 is the width of impeller inlet and ψ1 is blade inlet blockage coefficient which is represented as:

(21)

ψ1 = 1 −

ωn , b32 ,

E , t , ν are the natural frequency of rotor, blade height at where the outlet of the rotor, Young’s modulus, mean blade thickness, and Poisson’s ratio respectively. The properties of aluminum alloy (Al 6082) is used to estimate the stress and frequency.

εr Z πd2

(27)

where εr is the circumferential thickness of the blade leading edge. 2.3. Blade outlet angle

2.2. Blade inlet angle

The outlet flow angle β3 is computed as:

Fig. 3 represents the velocity triangle of the radial turbine which is based on the methodology proposed by Aungier [3]. The impeller diameter and rotational speed (ω) are constant therefore blade velocity at the inlet is also constant:

tan (β3) =

(U3 − Cθ3) Cm3

The meridional velocity at the outlet is computed as:

Fig. 3. Velocity triangles at (a) Inlet (b) Outlet of the rotor. 5

(28)

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Table 2 Blade angle at different flow rates. β2 0(Inlet blade angle)

m (kg / s )

− − − − − −

0.03 0.04 0.05 0.06 0.07 0.08

51.50 52.15 54.36 55.12 57.13 59.70

β3 0 (Outlet blade angle)

− 79.95 − 81.68 − 83.41 − 85.72 − 86.2 − 87.82

Fig. 6. Neuron sensitivity test.

Table 3 Details of ANN training process. Parameter

Value

Network structure Number of hidden layers Training algorithm Error function Number of epochs Hidden neuron range

MLP 1 Back-propagation MAE 1000 10 with step size one

Fig. 4. Sensitivity analysis of different geometric variables.

Cm3 =

Qr A3

Table 4 ANFIS network design.

(29)

where A3 is the outlet cross-sectional area. The selection of hub to rotor inlet diameter ratio (λ ) is an important criterion to avoid hub blade blockage and kinetic energy loss. Kun and Sentz [33] suggest a hub ratio of 0.35 to reduce the hub blade blockage without considering the kinetic energy loss at the turbine exit. Similarly, Rohlik [43] have chosen a hub ratio of 0.50 to reduce kinetic energy loss which has less effect on the hub blade blockage. For current design the average value of λ (0.425) is taken to consider both effects.

D λ = hub d2

MFs type of input parameters

Gauss, Tri, Trap, Gbell

MF type of output parameters Number of MFs Optimization method Error tolerance Number of epochs

Linear 4 Hybrid 0.0001 200

Cmt =

m πDt ρt bt

(31)

where Dt is nozzle throat circle diameter and Cmt the meridian component of the nozzle throat velocity, bt is the height of the nozzle. It is usually smaller than the turbine inlet blade height. This allows some margin for expansion in annular space and also accommodating the axial misalignment.

(30)

2.4. Design of nozzle The convergent type nozzle is used which provide the subsonic velocity at the nozzle exit. It is designed by considering the mass balance equation:

bt = 0.8 × b2

(32)

Dt = 1.068 × d2

(33)

Fig. 5. Structure of the designed MLP network for ANN. 6

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Fig. 7. The basic structure of ANFIS.

3. ANN and ANFIS methodology

The velocity at the exit of the throat consists of two components, Cmt and Cθt . The meridian component is perpendicular to the nozzle throat circle diameter, which determines the mass flow rate whereas the other component Cθt is tangential to the throat.

Cθt =

Cθ2 d2 Dt

3.1. Artificial neural network (ANN) The ANN is an adaptive system used as a simulation and prediction tool, motivated by the biological neural network of the human brain. It consists of interconnecting process units called neurons with linear and nonlinear transfer functions. The interconnecting units are the adaptive parameters of the ANN. Generally, ANN is a statistical data modeling tool which can structure the nonlinear functions into a set of input and output parameters during training. The feed-forward multi-layers perceptron (MLP) network opts during training with the back-propagation algorithm. The MLP consists of an input layer, a hidden layer, and an output layer. There are no restrictions on the selection of hidden layers, but it is well justified that at least one layer of a hidden neuron is sufficient to approximate any continuous functions. The present model has four inputs for the geometrical parameters (Rs / R2 , Rh / R2 , vs , and Z) and three inputs for experimental parameters (m , rp , and RPM). Fig. 5 represents a schematic of an MLP. Each unit in the network is connected with every node in the very next layer. The input parameters are given to the network; it can be multiplied with the weightage of the hidden neurons. The weighted input values are summed up and transformed when it passes through the hidden neurons. The output of the hidden layer is the input of the next layer, and the process is continued in a forward direction. Finally, the model has been developed using three transfer functions logarithmic sigmoid (Logsig), pure linear (Purelin), and tangent sigmoid (Tansig) [23]. Levenberg-Marquardt algorithm (Trainlm) has opted for the training function, and backpropagation method is selected as a network type. The training is performed with 1000 epoch. The functional relation is expressed as follows:

(34)

2.5. Sobol sensitivity analysis The performance of radial turbine depends on some geometrical design variables. It is complicated to identify the effect of these variables. Through the sensitivity analysis, the dependability of such variables on the performance of radial turbine can be evaluated. Sobol method is a variance decomposition based global sensitivity analysis method commonly used for nonlinear systems. In this study, Sobol method is utilized to evaluate the effectiveness of such design variables on the total to static efficiency of the turbine. The variance decomposition, the total output variance to input constants are derived as follows:

Var (Y ) =

∑ Di (Y ) + ∑ Dij (Y ) + …+D12… d (Y )

(35)

where Var (Y ) is the total variance of total loss (output), Di (Y ) is the first-order variance of input constants Xi and Dij (Y ) is the second-order variance of Xi and Xj which are formulated as:

Var (Y ) = E (Yi2) − E 2 (Yi )

(36)

Di (Y ) = Var [E (Y Xi )]

(37)

k

ym = g

Thus, the first-order sensitivity index is written as follows:

Si =

Di (Y ) Var (Y )

n

⎛ ⎞⎞ ⎛ w2 φ w1 x ⎜ ∑ mj ⎜∑ ji i⎟ ⎟ ⎠⎠ ⎝ i=0 ⎝ j=0

(39)

where n is the inputs, k is the hidden neurons and m is the outputs. w1ji is a weight connects the input i to the hidden node j, and w1j0 is the bias for the hidden unit j, corresponds to the fixed input value x 0 2 equals to 1. Similarly, wmj stands for a weight connects the hidden unit j 2 to the output m, and wm0 is the bias for the output m, corresponds to the fixed input value 1. φ (.) and g (.) represent the transfer function of hidden and output neurons respectively. The training process is carried on to minimize the error function. The error function is defined as the

(38)

Fig. 4 represents the sensitivity index of six geometrical design variables. It is find out that the variation of total to static efficiency are higher for vs , Z , Rh / R2 , and Rs / R2 as compared to ϕ and ψ . Therefore, the optimal range of these parameters (vs , Z , Rh / R2 , and Rs / R2 ) need to accurately predicted for better performance of the turbine. 7

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The efficacy of the ANN is analyzed using root mean square error (RMSE ) and R defined as:

Table 5 Comparisons of variation of output parameters with different geometrical inputs

n

∑

(a) Geometrical input and output parameters Geometrical input parameters Z

Rh / R2

Rs / R2

RMSE =

Output parameters

η (%)

vs

P (kW)

Lt

R= 0.98 0.9 0.86 0.85 0.86 0.9 0.9 0.85 0.8 0.94

0.425 0.45 0.45 0.28 0.22 0.25 0.25 0.26 0.26 0.26

10 11 12 13 12 11 13 12 10 13

0.62 0.62 0.68 0.72 0.73 0.74 0.70 0.69 0.66 0.65

0.802 0.821 0.823 0.836 0.778 0.769 0.802 0.803 0.784 0.816

0.22 0.188 0.178 0.188 0.241 0.256 0.181 0.165 0.187 0.154

0.608 1.33 3.191 0.465 1.345 1.289 1.78 1.72 2.1 2.81

R2 error

PURELIN

Lt

η

Lt

η

Lt

0.802 0.820 0.823 0.820 0.784 0.794 0.804 0.806 0.788 0.811 0.9131

0.218 0.186 0.179 0.188 0.241 0.221 0.180 0.169 0.185 0.158 0.9352

0.802 0.817 0.825 0.817 0.798 0.79 0.813 0.805 0.784 0.814 0.9417

0.201 0.203 0.188 0.173 0.187 0.197 0.170 0.177 0.209 0.163 0.9634

0.805 0.817 0.825 0.808 0.799 0.792 0.801 0.808 0.787 0.809 0.9266

0.221 0.188 0.168 0.187 0.241 0.204 0.176 0.172 0.194 0.158 0.9479

Training Errors

(×10−3) Testing Error

TRI

TRAP

Lt

η

Lt

η

Lt

η

Lt

0.802 0.821 0.823 0.836 0.778 0.78 0.802 0.804 0.784 0.816 0.193

0.219 0.193 0.174 0.162 0.171 0.18 0.162 0.171 0.195 0.162 0.012

0.802 0.821 0.823 0.836 0.778 0.78 0.802 0.803 0.784 0.816 0.230

0.22 0.188 0.178 0.163 0.173 0.183 0.163 0.173 0.195 0.163 0.019

0.802 0.821 0.823 0.836 0.778 0.78 0.802 0.803 0.784 0.816 0.196

0.22 0.186 0.18 0.164 0.24 0.187 0.164 0.183 0.20 0.155 0.020

0.802 0.821 0.823 0.823 0.778 0.78 0.802 0.803 0.784 0.816 0.198

0.205 0.191 0.182 0.165 0.182 0.191 0.165 0.182 0.205 0.165 0.022

0.736

0.721

0.954

0.793

0.872

0.737

0.894

0.743

1 n

yi − yi∗

(42)

• The input (80%) and output (20%) data are divided into training and testing. • A fuzzy model is generated in ANFIS editor, and data training is carried out.

Different membership functions (MFs) are determined to establish an ANFIS network. Theoretically, the increase of MFs can decrease the error until it reaches its minimum value after that further increase of MFs deduce higher errors and computational time. Therefore, to determine the optimum number of MFs, the error occurred from each network is checked by varying the number of MFs till the errors stop to decrease and that MFs is the optimum one [45]. There are various membership functions available, but for the present case, Gauss, Tri, Trap, and Gbell MFs are employed for input parameters and compare the obtained errors. For the output parameters, a linear MF has opted. The additional information of ANFIS MFs and network structure are provided in Table 4 and Fig. 7. The output of first order T-S FIS with IF-THEN fuzzy inference rules

n

∑

(yi − y¯i )2

3.2.1. Structure of adaptive neuro-fuzzy inference system ANFIS combines the advantages of learning ability of artificial neural network (ANNs) and modeling capability of fuzzy logic [9]. The step-by-step details for the development of the model are as follows:

difference between the estimated outputs (ym ) , and the measured out2 puts by varying the value of w1ji and wmj . This type of training is called supervised training. The errors are evaluated in the output layer and then propagated in a backward path in the network. In a similar way, the corresponding weights are updated, and the process is repeated until the errors are reduced to an acceptable level. This type of algorithm is known as back-propagation. Once the training 2 is over, the weights (w1ji and wmj ) are stable, and the network is organized to predict the output based on the input data. In this work, mean absolute error (MAE ) is used which is defined as:

MAE =

n

∑

The fuzzy system is a computing framework originated from fuzzy set theory, if-then rules, and fuzzy reasoning. When the system combines with the neural network, it advantages the assimilating features of the neural network and has an implementation equal to the fuzzy inference model [27]. The neural networks are a low-level mathematical structure that executes with the data set, while fuzzy logic deals with higher-level reasoning. But, fuzzy systems can not learn itself and adjusted accordingly. Therefore, a combination of neural network (advantages of robustness, learning, and training) and fuzzy system which is functionally similar to a fuzzy inference model (for interpretability).

GBELL

η

(41)

3.2. Adaptive neuro fuzzy inference system (ANFIS)

(c) Predicted ANFIS results for different MFs Gauss

n

3.1.1. ANN model development and neuron sensitivity test The model develops using design and experimental variables for the prediction of total to static efficiency, total losses, isentropic efficiency, temperature drop and power output. Initially, the input and output datasets are filtered from fallacious data. It is done by manual annotation of the plot of input and output data by considering its range. This step is essential for data-based modeling systems, as it significantly affects the model accuracy. After that, the data sets are divided into three parts, namely, training, testing, and validation data sets containing 70% , 20%, and 10% respectively. The testing and validation data sets are not used during the training process. The hyperbolic tangent (TFs) is opted for hidden as well as the output layer. Fig. 6 represents the neuron sensitivity test for total to static efficiency. It shows that 10 number of hidden neurons is sufficient for the training process. Therefore, ten neurons have opted in the hidden layer and one in the output layer for the remainder of the solution. The additional details of ANN training process are mentioned in Table 3.

TANSIG

η

i=1

where y¯i is the mean value.

(b) Predicted ANN results for different transfer functions LOGSIG

1 n−1

(yi − yi∗ )2

(40)

where yi is the provided value, yi∗ is the predicted one, and n is the provided values. 8

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(a) Total to static ef¿ciency

(b) Total losses

Fig. 8. Geometrical and Predicted data fits for ANN.

Fig. 9. Effect of (a) Rs R2 and Rh / R2 (b) vs and Z on the Total to Static efficiency.

Fig. 10. Effect of (a) Rs / R2 and Rh / R2 (b) vs and Z on total losses.

polynomial) which is determined during the training process. ANFIS structure contains five layers of neurons. Each neuron performs as a processing element with an activation function to generate the output. The different layers are illustrated as follows:

are expressed as follows:

• Rule-1: If x = A

i

and y = Bj then

fi = pi x n + qi yn + ri

(43)

Layer 1: Fuzzy layer: The first layer is termed as fuzzification layer or MFs layer. All the input variables are allocated identical fuzzy labels based on MFs in the layer. The nodes in this layer are adaptive and can be modified during the learning process. The Gauss MF is used which is expressed as

where x and y are non-fuzzy input variables, fi is the output in the fuzzy region which is specified by the fuzzy rules. Ai and Bj are fuzzy variables of the MFs. pi , qi , and ri are the design parameters (first order 9

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Table 6 Comparison of present result with results available in literature. (a) Input parameters range Input parameters

Range

Fluid T02 (K) P02 (bar) Mass flow rate (kg/s) Rotational speed (rpm)

Nitrogen 90 − 180 2.0 − 10.0 0.01 − 0.09 60, 000 − 150, 000

Major design and geometrical specification of the experimental turbine and nozzle (Comparison between present results with Ghosh et al. [21]) Present turbine specification

Ghosh et al.

Relative error (%)

24.90 15.60 2.111 1.005 143.38 121.88 0.862 0.237 0.46 79.20

25.19 15.75 2.00 0.996 130.28 124.42 0.88 0.241 0.42 78.20

0.01 0.01 0.06 0.008 0.09 0.02 0.02 0.017 0.08 0.01

Turbine inlet diameter (d2) (mm) Turbine outlet diameter (d3) (mm) Blade height at turbine inlet (b2) (mm) Blade height at turbine outlet (b3) (mm) Inlet blade speed (U2) (m/s) Outlet blade speed (U3) (m/s) Stage head coefficient (ψ) Flow coefficient (ϕ) Degree of reaction (Ω) Total to static efficiency (ηts ) Total to total efficiency (ηtt )

80.20

79.51

0.01

β2 (Degree)

− 54.98 − 85.04

− 69.78 − 85.74

0.21

0.78 0.25 0.54 0.226 4.52

0.85 0.31 0.47 0.242 2.83

0.08 0.194 0.13 0.06 0.37

β3 (Degree) M2 M3 Specific speed (ns ) Total losses (Lt ) Power (kW)

Layer 4: Defuzzification Layer: In this layer, an output is obtained. The nodes of this layer are adaptive. The output of the nodes is the product of the normalized firing strength and a first-order polynomial function. This model is also termed as 0th order ANFIS. The parameters are referred as follows:

Table 7 Details of major components of turboexpander. Parameters

Values (mm)

Turbine diameter at inlet (mm) Turbine diameter at outlet (mm) Number of blades (Z ) Pitch circle diameter of nozzle (mm) Outlet diameter of nozzle Number of nozzles Shaft diameter (mm)

24.90 15.60 13 40 26 19 16.00

wi fi = wi (pi x + qi y + ri ) i = 1, ….,n

Output = f =

−(x − αi )2 ⎫ 2(βi )2 ⎬ ⎭

∑i wi fi ∑i wi

=

w1 f1 + w2 f2 , i = 1, …, n w1 + w 2

(48)

3.3. Performance prediction of radial inflow turbine from geometrical parameters

(45)

The ANN and ANFIS networks are trained with geometrical data used for designing the turbine. There are two ANFIS networks for geometrical datasets. ANFIS-1- prediction of the total to static efficiency, ANFIS-2 - prediction of total losses. The samples for validation of actual output and predicted one from ANN and ANFIS are summarized in Table 5. It shows that Logsig model has the least error in ANN model and Gauss MF in ANFIS model. Therefore, regression curves and decision surfaces of these models are

Layer 3: Normalisation Layer: In this layer, the corresponding ratio of the firing strength is generated. The nodes are fixed. The layer is defined as:

wi , i = 1, …, n w1 + w 2

=

3.2.2. ANFIS base Estimator The system is trained for 200 epochs and error tolerance is set to be 0.0001. It is noticed that the RMSE of the Gaussian model becomes steady after 30 epochs of the training data. After completion of the training process, ANFIS generates 256 IF-THEN fuzzy rules for geometrical variables and 81 rules for experimental variables for the representation of the inputs and the outputs.

Layer 2: Product Layer: In this layer, the incoming signals are combined. Each node are fixed and labeled ∏ which indicates that it can multiply the incoming signals and produce the firing strength of a rule. It is defined as:

w =

∑ wi fi

(44)

where αi , and βi are termed as premise parameters. The main objective of ANFIS is to minimize the network and actual output for a conferred pairs of training data (input-output) with premise (αi, βi ) and consequent (pi , qi , ri ) parameters.

wi = μ Ai (x ) × μ Bj (y ), i = 1, …, n

(47)

Layer 5: Output Layer: In this layer, only one node ∑ is labeled. It calculates the outputs and summed up all the incoming signals as follows:

follows:

μ Ai (x ) = exp ⎧ ⎨ ⎩

0.01

(46) 10

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Fig. 11. Schematic diagram of experimental set-up.

Fig. 12. Experimental set-up (1) PFHX (2) Hot end turboexpander (3) Cold end turboexpander (4) ADAM (5) Pressure transducer (6) Oscillloscope (7) Rotameter (8) High-pressure reservoir (9) PFHX inlet line (10) Turbine inlet line (11) Turbine outlet line.

of 0.78 − 0.88 and Rh / R2 is in the range of 0.22 − 0.24 and 0.27 − 0.28. For designing the turbine, the value of Rs / R2 and Rh / R2 is taken to be 0.88 and 0.28 respectively. Also, the value of vs and Z is taken to be 0.70 and 13 respectively. The turbine is designed based on the predicted optimal range of geometrical variables from artificial intelligence method and the aforementioned design methodology in MATLAB® 2017b environment. The obtained results are compared with available data in the literature which is mentioned in Table 6. It depicts that the preliminary design procedure is reliable. Some parameters like M3, ns , and P has higher variation. The specific speed (ns ) is selected from Balje ns − ds diagram

presented. Fig. 8 illustrates the regression of predicted data with actual output data. The final decision surfaces, obtained from ANFIS training represent all the information compactly (Figs. 9 and 10)). Fig. 9 illustrates the effect of Rs / R2 , Rh / R2 , vs , and Z on the total to static efficiency. It depicts that the total to static efficiency of the turbine is higher when the value of Rs / R2 and Rh / R2 in the range of 0.88 − 0.92 and 0.25 − 0.27 respectively whereas Z must be in the range of 12 − 15. The velocity ratio (vs ) has relatively less effect on the efficiency prediction, but losses are minimum if the ratio is maintained in the range of 0.66 − 0.71 (Fig. 10 (b)). Similarly, total loss is minimum when Rs / R2 is in the range 11

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Fig. 13. (a) Turboexpander assembly (b) Three-dimensional drawing (c) Other components.

SIGMA BSD 72), a filter, moisture absorber, and an air purifier unit. The filter consists of three layer filtration system which removes the solid dust particles, water, and oil particles to provide clean air. It is important because the impurities can cause the failure of aerodynamic bearing units and also diminishing the performance of the turbine. The compressed air passes through the Freon precooler. Its main aim is to provide dry, subcooled, and dehumidifying the air. After that, it passes through a heat exchanger equipped with a liquid nitrogen coolant to decrease its temperature and make it suitable for the experimental work. The compressed air after cooling from the heat exchanger is accumulated in the reservoir. The heat exchanger used in this study is a plate-fin type heat exchanger [2]. In this way, the high pressure and low temperature (Case 1 (150 K), Case 2 (140 K), and Case 3 (130 K)) air are supplied to the turboexpander. The turbine supply line (cold end side) is insulated with perlite powder (Grade 45) and nitrile rubber polyethylene pipe. The PFHX and high-pressure reservoir are also insulated with perlite powder and covered with thermocol insulation sheet. In this way, the adiabatic

for better efficiency of the turbine. Due to this, the power output increases which has a significant effect on the refrigerating capacity of the turbine. Similarly, the turbine exit Mach number (M3) decreases due to which kinetic energy loss decreases which ultimately reduce the trailing edge loss. The turbine and nozzle specifications are mentioned in Table 7. 4. Experimental set-up and procedure 4.1. Experimental components The test rig consists of eight components which are: 1) a compressed air facility (outdoor), 2) an air filter (Outdoor), 3) an air purifier (outdoor), 4) cooler (outdoor), 5) a PFHX equipped with liquid nitrogen (Outdoor), 6) a reservoir (indoor), 7) a heat exchanger (indoor), and 8) a turboexpander set-up (indoor). All the components are described herewith. The compressed air facility includes a screw compressor (KAESER 12

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temperature sensor, and rotameter at different test points. The specifications of measuring components are as follows: The pressure gauge (Swagelok) is used at different test points having 0 − 25 bar range and ± 0.2% accuracy level. The pressure transducer (Endevco) having 0 − 34.5 bar range is used to obtain the turbine tip pressure. It produces a very small effect on efficiency calculation. The temperature of the fluid at the inlet and outlet of PFHX and turboexpander are measured with resistance thermometer (RTD) using ADAM for better accuracy. The flow meter is used to measure the mass flow rate at test points; its range is 1–30 m3/h with the accuracy of ± 2.5%. The rotational speed of the turbine is measured by placing the speed sensors at the space provided in the bearing housing. The accuracy of eddy current speed sensor is ± 0.005%. Figs. 11 and 12 illustrates the flow process schematic diagram of the experimental set-up. The important components of the turboexpander is a convergent nozzle, radial turbine, brake compressor, aerodynamic bearings, etc. Fig. 13 illustrates the drawing and images of the different components which is used in the turboexpander assembly. The experimental platform is developed to investigate the performance of the turboexpander at cryogenic temperature. It is a reverse Brayton cycle based air turboexpander system whose primary aim to decrease the temperature. The turbine blade profile is designed to rotate at rotational speed in the range of 60, 000 − 120, 000 rpm (approximately 40–80% of the designed rotational speed). The power generated by the turbine is dissipated using brake compressor which is

Fig. 14. Variation of inlet pressure with mass flow rate.

condition is achieved. Additionally, there is no need to insulate the bearing supply line and brake compressor inlet and outlet line. The turboexpander test rig includes measuring devices such as pressure gauge, ADAM, pressure transducers, speed sensors, cryogenic

Fig. 15. Variation of (a) Pressure drop (b) Rotational speed (c) Isentropic efficiency with mass flow rate (These graphs are presented for case 1). 13

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Fig. 16. Effect of rotational speed of the turbine on (a) Temperature drop (b) Enthalpy drop.

Fig. 17. Effect of rotational speed of the turbine on (a) Isentropic efficiency (b) Power output.

supply the high-pressure and low-temperature air to the turboexpander unit. The mass flow rate is varying in the range of 0.01–0.08 kg/s due to which the inlet pressure of the turboexpander varies in the range of 3.5–7.8 bar. The second step is to record the rotational speed of the turbine for different mass flow rate. The third step is to note the outlet pressure and temperature of the turboexpander which determines the pressure ratio and temperature drop. Based on obtained pressure and temperature, the corresponding thermodynamic properties of the working fluids are obtained using ALLPROPS [34].

mounted on the same shaft. The test rig also consists of a flange in which turboexpander assembly is placed, and it is mounted on a table. The shaft and brake compressor is made of chromium-vanadium steel whereas other the stationary parts are made of stainless steel. The turbine is made of aluminum alloy (Al 6082) . The flange is fabricated using stainless steel having an external diameter and height are 160 mm and 157.50 mm respectively. The turboexpander set-up is placed inside the flange and makes it leak proof with the help of O-ring (Fig. 13)). The rotor of the turboexpander is connected with a turbine wheel at one end and a brake compressor at the other end. The rotor is supported by aerodynamic gas foil journal and thrust bearings to overcome the axial thrust on the shaft during the operation.

5.1.1. Effects of mass flow rate of working fluid Fig. 14 represents the inlet pressure variation with an increase in the mass flow rate of the working fluid. The turboexpander inlet pressure increases with an increase in mass flow rate. For approximately similar mass flow rate, the inlet pressure is marginally higher at greater temperatures. It also increases the rotational speed of the turbine because of the pressure energy of the fluid is converted into kinetic energy. During the energy transformation process, the enthalpy of the fluid decreases which results in temperature drop (Fig. 16). Fig. 15 illustrates the variation of pressure drop, rotational speed, and isentropic efficiency of the turboexpander with mass flow rate. The isentropic efficiency, rotational speed, and pressure drop increase with an increase in mass flow rate. The maximum isentropic efficiency (82.1%) of the turboexpander is obtained at a mass flow rate of 0.078 kg/

5. Results and discussion The results and discussion section consists of two parts. The first part presents the experimental results. The second part illustrates the predicted results of ANN and ANFIS. 5.1. Experimental results Experimental tests are conducted to estimate the performance of the turboexpander operating at a range of conditions. The first step is to 14

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Table 8 Comparisons of variation of output parameters with different experimental data inputs. (a) Experimental input and output parameters Experimental input parameters

Output parameters

m (kg/s)

rp

ω(*105)

η

Tdrop

P (kW)

0.02 0.04 0.08 0.01 0.03 0.03 0.04 0.04 0.05 0.06

1.9 2.5 3.8 1.7 2.5 2.6 2.8 2.7 2.9 3.2

0.62 0.87 1.91 0.71 0.73 0.78 0.86 0.85 0.98 1.08

0.651 0.754 0.823 0.656 0.734 0.714 0.751 0.803 0.784 0.816

13.71 19.21 31.82 15.76 15.36 14.81 19.66 17.61 21.76 24.11

1.91 2.36 5.31 2.16 2.08 2.11 2.41 2.46 2.94 3.61

Predicted ANN results for different transfer functions LOGSIG

R2 error

PURELIN

TANSIG

η

Tdrop

P

η

Tdrop

P

η

Tdrop

P

0.644 0.691 0.833 0.671 0.742 0.699 0.741 0.811 0.774 0.831 0.865

13.91 18.61 30.74 14.96 16.68 15.21 18.96 17.82 22.01 23.86 0.911

1.86 2.61 5.26 2.11 2.01 2.24 2.37 2.39 2.86 3.74 0.929

0.621 0.683 0.831 0.657 0.732 0.693 0.737 0.819 0.781 0.827 0.890

13.76 18.57 30.61 14.84 16.76 15.52 19.11 18.02 22.14 23.72 0.926

1.79 2.57 5.34 2.16 2.08 2.31 2.34 2.41 2.74 3.86 0.947

0.619 0.667 0.822 0.672 0.712 0.689 0.739 0.821 0.779 0.834 0.871

13.64 18.72 30.81 14.96 16.24 15.61 19.88 18.56 22.79 23.81 0.916

1.65 2.24 5.15 2.24 2.21 2.66 2.54 2.47 2.87 3.97 0.939

Predicted ANFIS results for different MFs GAUSS

Training Error (×10−3) Testing Error

TRI

TRAP

GBELL

η

Tdrop

P

η

Tdrop

P

η

Tdrop

P

η

Tdrop

P

0.649 0.756 0.827 0.646 0.714 0.719 0.747 0.823 0.781 0.821 0.115

13.69 19.26 31.78 15.71 15.24 14.75 19.68 17.57 21.81 24.21 0.013

1.89 2.38 5.30 2.17 2.06 2.13 2.47 2.39 2.79 3.66 0.096

0.656 0.776 0.834 0.627 0.775 0.686 0.731 0.822 0.747 0.827 0.370

14.22 18.89 31.16 14.84 16.91 15.66 19.02 16.52 22.32 23.79 0.019

1.79 2.57 5.19 2.08 2.26 2.51 2.48 2.41 2.88 3.61 0.134

0.591 0.681 0.849 0.633 0.714 0.728 0.789 0.833 0.792 0.831 0.210

13.49 18.66 30.89 14.43 16.67 15.14 19.08 18.21 21.66 23.22 0.016

1.72 2.61 5.17 2.23 2.11 2.46 2.54 2.49 2.77 3.89 0.255

0.602 0.621 0.823 0.636 0.778 0.769 0.802 0.803 0.784 0.816 0.172

13.74 18.79 30.86 14.31 16.42 16.88 18.43 18.71 21.62 23.79 0.020

1.61 2.33 5.19 2.47 2.35 2.29 2.78 2.73 2.62 2.86 0.103

0.630

0.661

0.529

0.782

0.758

0.661

0.762

0.728

0.683

0.663

0.736

0.613

The power output (P ) of the turboexpander is calculated as:

s and a pressure ratio of 3.85.

P = ṁ (hi − ho)

5.1.2. Effect of rotational speed of the turbine Fig. 16 illustrates the effect of the rotational speed of the turbine on the temperature and enthalpy drops. The rotational speed of the turbine is varied by changing the mass flow rate of the working fluid. The turbine rotational speed increases from 60, 000 rpm to 120,000 rpm approximately by varying the mass flow rate in the range of 0.01–0.08 kg/s. The increase in rotational speed causes higher work done by the fluid. This is responsible for the higher temperature drop. It is also observed that the change in temperature and enthalpy at higher inlet temperature (case 1) is more significant than that of at relatively lower temperature for constant rotational speed. It happens due to the working fluid approaches towards its boiling point, and at these temperatures, a relatively higher pressure ratio is required to achieve the same amount of temperature drop. The maximum temperature and enthalpy drops are 30 K and 34.50 kJ/kg respectively at a rotational speed of 119,600 rpm for the fluid inlet temperature of 150 K (case 1).

(49)

where ṁ is the mass flow rate. Fig. 17 illustrates the effect of rotational speed on the isentropic efficiency and power output of the turboexpander at different inlet temperatures. Initially, isentropic efficiency increases with an increase in rotational speed. For case 1 and 2, the maximum isentropic efficiency is obtained at 80% of the designed speed of the turbine whereas for case 3, it may be increased, but there are some design and material restrictions of the turbine. Therefore, an optimum rotational speed for each case is found by varying the mass flow rate. It is an important parameter because at this speed the turbine produces maximum expansion ratio which is directly related to the temperature drop. It is also noticed that the power output of the turboexpander increases at a moderate rate up to 90, 000 rpm, after that, it increases at a higher rate because of the increase in enthalpy drop of the working fluid at the higher mass flow rate and rotational speed. 15

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(a) Isentropic ef¿ciency

(b) Temperature drop

(c) Power output Fig. 18. Experimental and Predicted data fits for ANN.

Fig. 19. Effect of (a) m and RPM and (b) rp and RPM on the isentropic efficiency.

networks are developed for the experimental datasets. ANFIS-1- prediction of isentropic efficiency, ANFIS-2 - prediction of temperature drop, ANFIS- 3 - prediction of power output. The samples of actual and predicted data from ANN and ANFIS using different transfer functions

5.2. ANN and ANFIS results The ANN and ANFIS networks are trained with the experimental data extracted from a set of experiments. There are three ANFIS 16

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Fig. 20. Effect of (a) m and RPM and (b) RPM and rp on the temperature drop.

Fig. 21. Effect of (a) m and RPM and (b) rp and RPM on the power output.

and MFs are mentioned in Table 8.

parameters with different input data.

5.2.1. Performance prediction of turboexpander from experimental variables The experimental datasets are used to predict the best operating ranges in which turboexpander have maximum isentropic efficiency, maximum temperature drop and a reasonable amount of power output. Fig. 18 illustrates the regression of predicted data with actual experimental output data using ANN. There are three input variables (mass flow rate (m) , pressure ratio (rp) , and rotational speed (rpm) ) are used for the prediction of isentropic efficiency, temperature drop and power output. Since the training errors are minimum for Gauss MFs as mentioned in Table 8, therefore the final decision surfaces of the ANFIS networks are based on it as represented in Figs. 19–21. These ANFIS surfaces are obtained after the training process which represents the optimal operating ranges in which the turboexpander have maximum efficiency. Fig. 19 represents the effect of m , RPM , and rp on isentropic efficiency of the turboexpander. It is observed that the maximum isentropic efficiency of the turboexpander is obtained for m , RPM , and rp in the range of 0.055 − 0.07 kg/s, 1.05 − 1.15 × 105 RPM, and 3.3 − 3.9 respectively. The predicted mass flow rate and rotational speed for maximum isentropic efficiency are in the range of 0.05 − 0.07 kg/s and 0.9 × 105 − 1.1 × 105 respectively. The temperature drop and power output of the turboexpander is maximum in this range (Figs. 20(a), 21(a)). It is also observed that the pressure ratio should be in the range of 3.3 − 4.0 to achieve maximum efficiency, power output, and temperature drop. Therefore, a decision has to be taken in which the efficiency and temperature drop of the turboexpander should be acceptable. In this case, the maximum temperature drop (29.46 K) obtained with an isentropic efficiency of 82.1% for the mass flow rate of 0.07 kg/ s, and at a pressure ratio of 3.8. In general, it can be concluded that the developed ANFIS networks provide an optimal prediction of output

6. Conclusions In this study, the mean-line design methodology of a cryogenic radial inflow turbine and experimental set-up are developed to investigate the performance of a cryogenic turboexpander. Different loss models have opted during the design of a radial turbine which is beneficial to increase the performance of the turboexpander. Sobol sensitivity analysis suggests that Rs / R2 , Rh / R2 , Z , and vs are the most effective parameters to determine the performance of the turbine. The optimal range of these geometrical variables is predicted using an artificial intelligence technique. The experimental results are presented to examine the isentropic efficiency, temperature drop, enthalpy drop, and power output of the turboexpander at different mass flow rate, rotational speed, and pressure ratio. The maximum temperature and enthalpy drop are 30 K and 34.5 kJ/kg respectively which is obtained for case 1 at a rotational speed of 119,600 rpm (80% of the designed rotational speed) and the mass flow rate of 0.078 kg/s. Based on these datasets, ANN and ANFIS models are developed which predict the optimal range of design variables in which the highest total to static efficiency of the turbine is obtained with minimum losses. The similar models are developed for the datasets obtained from the experimental investigation. The maximum isentropic efficiency, temperature drop, and power output of a turboexpander are obtained for different mass flow rate, pressure ratio, and rotational speed. The current investigation can be used to design and development of a cryogenic turboexpander for the liquefaction of gases. Acknowledgments The authors sincerely thank to the Board of Research in Nuclear Sciences (BRNS ) (Project number: 39/23/2015-BRNS/39001), Ministry 17

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of Human Resource Development (MHRD) , Government of India, and National Institute of Technology, Rourkela for providing the financial support. The authors present immense gratitude to Prof. Amitesh Kumar (Assistant professor, IIT, Varanasi) who supports and continuously discuss with us on this regard. We are also grateful to the senior technicians of our institute Mr. Somnath Das and Mr. Sudhananda Pani for delivering valuable suggestions during commissioning of the experimental set-up and helping us in the experiments.

[26]

[27] [28]

[29]

Appendix A. Supplementary material [30]

Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.applthermaleng.2019. 114273.

[31]

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