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MultiCraft

International Journal of Engineering, Science and Technology

Vol. 2, No. 1, 2010, pp. 175-191

INTERNATIONAL

JOURNAL OF

ENGINEERING,

SCIENCE AND

TECHNOLOGY

www.ijest-ng.com

© 2010 MultiCraft Limited. All rights reserved

A numerical model for the design of a mixed flow cryogenic turbine

Subrata Kr. Ghosh

Department of ME & MME, Indian School of Mines, Dhanbad, Jharkhand, INDIA

E-mail: subratarec@yahoo.co.in

Abstract

Present day cryogenic gas turbines are in more popular as they meet the growing need for low pressure cycles. This calls for

improved methods of turbine wheel design. The present study is aimed at the design of the turbine wheel of mixed flow

impellers with radial entry and axial discharge. In this paper, a computer code in detail has been developed for designing such

turbine wheel. To determine the principal dimensions of the turbine wheel, optimum operating speed has been taken from design

charts based on Similarity principles. The algorithm developed, allows any arbitrary combination of fluid properties, inlet

conditions and expansion ratio, since the fluid properties are properly taken care of in the relevant equations. The computational

process is illustrated with an example. The main dimensions, thermodynamic properties at different states, velocity and angles at

entry and exit of the turbine wheel were worked out. The work may help the researchers for further design and development of

cryogenic turboexpander depending on their operating parameters.

Keywords: Cryogenic, turboexpander, mixed flow turbine, flow angle, flow velocity

1. Introduction

Compressor, heat exchanger, expansion turbine, and vacuum vessel are the main components to establish any cryogenic liquid

plant. To establish the set up, different companies are indigenously available in India for compressor, heat exchanger and vacuum

vessel. But the turboexpander is not readily available in market. As the technology is not yet developed indigenously, we are

forced to import the whole liquid plant. A simple method sufficient for the design of a high efficiency expansion turbine is outlined

by Kun (1987) and Kun et al. (1985). A study was initiated in 1979 to survey operating plants and generates the cost factors

relating to turbine by Kun & Sentz (1985). They are also sometimes referred to as design parameters, since the shape dictates the

type of design to be selected. Corresponding approximately to the optimum efficiency a cryogenic expander may be designed with

selected specific speed and specific diameter. Sixsmith et al. (1988) in collaboration with Goddard Space Flight Centre of NASA,

developed miniature turbines for Brayton Cycle cryocoolers. Another programme at IIT Kharagpur developed a turboexpander

unit by using aerostatic thrust and journal bearings which had a working speed up to 80,000 rpm. The detailed summary of

technical features of the cryogenic turboexpander developed in various laboratories has been given in the PhD dissertation of

Ghosh (2002). The detailed design parameters for a 90° inward radial flow turbine is shown in the PhD dissertation of Ghosh.

Bruce (Bruce, 1998) described the aerodynamic and structural analysis for the complete design of turbo-machinery rotor. The

major elements of the turboexpander are briefly discussed in the paper published by Ghosh et al. (2005). Baines (2002) has shown

that mixed flow turbine concepts can achieve stage loadings that are about 20% greater than those of a conventional radial turbine,

without any increase in blade speed and maintaining structural integrity. Descombes (2003) has high-lighted the variability of the

local distribution of velocities and pressures within the rotor and gives a real image of energy transformation for nearly ideal zero

incidence conditions at the rotor inlet. Abidat (2006) has taken care in the design of the volute essential component of a radial and

mixed flow turbines and interest was focused on the influence of the volute inlet flow conditions on its performance. A method of

computing blade profiles has been worked out by Hasselgruber (1958), which has been employed by Kun & Sentz (1985) and by

Balje (1970, 1981). Effect of some of the turbine operating and design parameters on the flow path and its curvature have been

analyzed and presented by Ghosh et al. (2009). The computational procedure developed describes the three-dimensional contours

of the blades for the turbine wheel.

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176

2. Fluid parameters and layout of components

The fluid specifications have been dictated by the requirements of a small refrigerator producing less than 1 KW of refrigeration.

Turbine efficiency of 75% has been assumed following the experience of the workers (Beasley et al., 1965; Yang et al., 1990;

Denton, 1996; Jekat, 1957). The inlet temperature has been specified rather arbitrarily, chosen in such a way that even with ideal

(isentropic) expansion; the exit state should not fall in the two-phase region. The basic input parameters for the system are given in

Table 1. Figure 1 shows the longitudinal section of a typical cryogenic turboexpander displaying the layout of the components

within the system. The major components of the turboexpander are shown in Table 2.

Table 1: Basic input parameters for the cryogenic expansion turbine system

Working fluid : Nitrogen

Turbine inlet temperature : 122 K

Turbine inlet pressure : 6.0 bar

Discharge pressure

Throughput

Expected efficiency

: 1.5 bar

: 67.5 nm3/hr

: 75%

Table 2: Basic unit of a turboexpander assembly

Diffuser

Journal bearings Thrust bearings

Turbine wheel

Brake compressor

Nozzle Shaft

Housing

Figure 1: Longitudinal section of the expansion turbine displaying the layout of the components

3. Design of turbine wheel

The design of turbine wheel has been done following the method outlined by Balje (1981) and Kun & Sentz (1985), which are

based on the well known “similarity principles”. The similarity laws state that for given Reynolds number, Mach number and

Specific heat ratio of the working fluid, to achieve optimized geometry for maximum efficiency, two dimensionless parameters:

specific speed and specific diameter uniquely determine the major dimensions of the wheel and its inlet and exit velocity triangles.

Specific speed (

Q

n

Δ

1

32

Q

s n ) and specific diameter (

s d ) are defined as:

Specific speed

()4

3

3

3

s in

s

h

−

(

=

ω

(1)

Specific diameter

)

3

4

hD

d

s in

s

−

Δ×

=

(2)

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177

In the definition of

s n and

s d the volumetric flow rate

3

Q is that at the exit of the turbine wheel. Kun and Sentz (1985),

however suggest two empirical factors

QkQ

1

/k

ex

ρρ =

h

=Δ

−

3

The factors

rise in temperature and density in the diffuser as shown in Figure 2. Following the suggestion of Kun and Sentz (1985),

03. 1

. The factor

ex

Q/Q3

stage, where as

3

Q and

3

ρ

condition of the turbine wheel can be calculated from equations (3) and (4) respectively. If the guess value is correct, then

ρ should give a turbine exit velocity

3

C that satisfies the velocity triangle as described in equation (13); otherwise the iteration

process is repeated with a new guess value of

Q determines turbine exit velocity uniquely. The thermodynamic

relations for reversible isentropic flow in the diffuser are,

2

3

033

hh

−=

1 k and

2 k for evaluating

3

Q and

s

h3, which define

s n and

s d .

ex3=

and

(

in

h

02

(3)

(4)

(5)

13

)

exs

h

sin

k

−

1 k and

2

k account for the difference between the states ‘3’ and ‘ex’ caused by pressure recovery and consequent

2

k value is

1 k represents the ratio

are unknown. By taking a guess value of

, which is also equal to

3 ex/ρρ

. The value of

ex

Q and

ex

ρ

are known at this

ρ ) at the exit

1 k , the volume flow rate (

3

Q ) and the density (

3

3

Q and

3

1 k . The value of

3

ex

hh

003=

,

ex

ss =

3

and

2

C

Figure 2: State points of turboexpander

Using the property tables, the value of

initial values of

3

ρ is within the prescribed limit, the iteration is converges. Since the change in entropy in the diffuser is small

compared to the total entropy change, assumption of isentropic flow will lead to very little error. The estimation

deviate appreciably, if the expansion of fluid from ‘in’ to ‘ex’ is non-isentropic. With this assumption, the value of

to be 1.11, starting with the initial guess value of 1.02. A flow chart for determining the value of

For estimating the thermodynamic properties at different states along the flow passage, the software package ALLPROPS 4.2

available from the University of Idaho, Moscow (Lemmon et al., 1995) is used. Table 3 represents the thermodynamic states at the

inlet of the nozzle and the exit of the diffuser according to input specifications. The exit state has two different columns, one is

isentropic expansion and other is with isentropic efficiency of 75%.

Using data from Table 3,

.

10 26.23

−

×===

ex

ρ

86. 5

3

===

k

−

×==

ex

QkQ

3

ρ can be estimated from

3

s and

3

h . When the difference between the calculated and

1 k does not

1 k is estimated

1 k is described in the Figure 9.

3

3

1097. 3

86 . 5

−

×

tr

ex

m

Q

m3/s

27. 5

11. 1

1

ex

ρ

ρ

kg/m3

(6)

3

13

1042 . 4

m3/s

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Ghosh / International Journal of Engineering, Science and Technology, Vol. 2, No.1, 2010, pp. 175-191

178

()

39861107 .38 03. 1

3

,, 0

h

23

=××=−=Δ

−

sexinsin

hkh

J/kg

From Balje (1981) the peak efficiency of a radial inflow turbine corresponds to the values of:

54 . 0

=

s n

and

4 . 3

=

s

d

Substituting these values in equations (1) and (2) respectively, yields

Rotational speed

22910

=

ω

rad/s = 2,18,775 r/min,

Wheel diameter

mmD0 . 16

.

Power produced

()

hhmP

ex0in

=−=

&

Tip speed

22

= DU

ω

Spouting velocity

h2C

in0

Δ=

U

(7)

2=

()

KW

9 . 0h

m/s

h

183

η

=

m

&

2

exs 0in

=−

28 ./

20.278

exs

=

−

m/s and (9)

Velocity ratio

66. 0

0

2=

C

According to Whitfield and Baines (1990), the velocity ratio

0

2C

U

in a radial inflow turbine generally remains within 0.66 and

0.70. The ratio of exit tip diameter to inlet diameter should be limited to a maximum value of 0.70 (Dixon, 1978; Rohlik, 1968) to

avoid excessive shroud curvature. Corresponding to the peak efficiency point (Kun and Sentz, 1985):

676 . 0DD

2

tip

8 .10

=

tip

D

mm

Table 3: Thermodynamic states at inlet and exit of prototype turbine

Inlet isentropic exit state (ex,s)

Pressure (bar) 6 1.50

Temperature (K) 122 81.72

Density (kg/m3) 17.78 6.55

Enthalpy (kJ/kg) 119.14 80.44

Entropy (kJ/kg.K) 5.339 5.339

Balje (1981) prescribes values for the hub ratio

tip

D/

=λ

against

recommendation for radial flow machines. In axial flow and large radial flow turbines, a small hub ratio would lead to large blade

height, with associated machining difficulties and vibration problems. But in a small radial flow machine, a lower hub ratio can be

adopted without any serious difficulty and with the benefit of a larger cross section and lower fluid velocity. According to Rohlik

(1968), the exit hub to tip diameter ratio should maintained above a value of 0.4 to avoid excessive hub blade blockage and energy

loss. Kun and Sentz (1985) have taken a hub ratio of 0.35 citing mechanical considerations.

425./

==

tiphub

DD

λ

6 . 4

=

hub

D

mm

There are different approaches for choosing the number of blades, the most common method is based on the concept of ‘slip’, as

applied to centrifugal compressors (Rohlik, 1968; Stewart and Glassman, 1973; Jadeja et al., 1985). Denton (1996) has given same

guidance on the choice of number of blades by ensuring that the flow is not stagnant on the pressure surface. For small turbines,

the hub circumference at exit and diameter of milling cutters available determine the number of blades. In this design the number

of blades (Ztr) are chosen to be 10, and the thickness of the blades to be 0.6 mm throughout.

From geometrical considerations:

(

mean

β

sin24

==

ξ

(10)

Actual exit state (ex)

1.50

89.93

5.86

90.11

5.452

hub D

s n and

s d for axial flow turbines, but makes no specific

(11)

)

()

hubtiptr tr

hubtip

DDtZ

DDA

π

22

3

−

−−=

(12)

(8)

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Ghosh / International Journal of Engineering, Science and Technology, Vol. 2, No.1, 2010, pp. 175-191

179

()

(

(

sin

)

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

−−==

mean

β

)

hub

hubtiptr

2

tr

hubtip

DDtZ

DDCCAQ

π

4

22

3333

(13.a)

()

3

22

33

24

W

DDtZ

DDCQ

tip trtr

hubtip

×

−

−−=

π

(13.b)

2

U

2

C

2

W

θ

θ

3

C

3

W

3

U

Direction of rotation

Z

r

β

(a) Inlet velocity triangle in the r-θ plane (b) Exit velocity triangle in the θ -z plane.

Figure 3: Inlet and exit velocity triangles of the turbine wheel

From the velocity triangle in Figure 3

()

hubtipmean

mean

β

DD

C

+

U

C

==

ω

3

, 3

3

4

tan

(14)

For a given value of

mean relative velocity angle

U

3

Q as given by equation (6), equations (13) and (14) are solved simultaneously for exhaust velocity

β

, giving:

3

C and

mean

°=

=

=

1 .

6 . 45

/90

/ 2 .88

m

3

3

mean

β

mean

sC

sm

(15)

In summary, the major dimensions for our prototype turbine have been computed as follows:

Rotational speed: N = 22910 rad/s = 218,775 rpm

Wheel diameter:

2

D = 16.0 mm

Eye tip diameter:

tip

D

= 10.8 mm

Eye hub diameter:

hub

D

= 4.6 mm

Number of blades:

tr

Z = 10

Thickness of blades tr t

4. Design of diffuser

For the purpose of design, the diffuser can be seen as an assembly of three separate sections operating in series – a converging

section or shroud, a short parallel section and finally the diverging section. The converging portion of the diffuser acts as a casing

to the turbine. The straight portion of the diffuser helps in reducing the non-uniformity of flow, and in the diverging section, the

pressure recovery takes place.

The geometrical specifications of the diffuser have chosen somewhat arbitrarily. Diameter of diffuser inlet is equal to diameter of

the turbine inlet. Diameter of throat of diffuser is dependent on the shroud clearance. The recommended clearance is 2% of the exit

(16)

= 0.6 mm