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THE CITY COLLEGE OF NEW YORK

CITY UNIVERSITY OF NEW YORK

SECOND TERM DESIGN PROJECT

SPRING 2014

Name: ISRAEL MIRANDA

Title: SIMPLIFIED APPROACH TO PRELIMINARY IMPULSE STAGE

Course Title: Thermo-Fluid Systems Analysis and Design

Course No: ME 430

Section No: DD

Instructor: Prof. Rishi S. Raj

Teacher Assistant: Saman Reshadi

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Abstract

The overall efficiency of thermodynamic systems is derived from different

mechanisms; from the fuel used the furnace to the materials used on every physical

component. One essential mechanism that contributes to the efficiency and the cost of

power plant is the design of the blades in the turbine (high or low pressure). This text is

organized on a simplified impulse stage. The turbine is assume to operate at 3600 RPM

for a frequency of 60 Hz. The change in enthalpy is establish as 30 Btu/lbm. The exting

velocity from the stator is chosen to be 13°. Important factors in the design such as mean

radius was found to be 1.268 ft., the specific work: 𝑤 = 27.35 𝐵𝑡𝑢/𝑙𝑏𝑚 and the rotor

length, 𝐿2 = 1.35 𝑖𝑛.

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Table of Contents

Nomenclature ………………………………………………………………………….…4

Theory ………………………………………………………………...………….….…. 5

Calculations …………………………………………………………..…………………. 8

Discussion and Conclusion ………………………………………..……………….….15

References …………………………………………………………..……………….….16

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Nomenclature

Δh Total heat is converted to work in a stage Btu/lbm

𝑔𝑐 Gravitational constant ft/s2

P Power plant power produced kWh

ηt Turbine efficiency %

v Specific volume ft3/lbm

n Angular frequency RPM

ṁ Mass flow rate lbm/hr

Δhnet Total heat in Btu/lbm

V2 Velocity observed by object outside the turbine, leading edge ft/s

U/V2 Ratio of Circumferential velocity of Observed velocity N/A

U Circumferential velocity ft/s

𝑟𝑚 Median rotor radius ft

α2 Angle between V2and the horizontal, leading edge rad

β2 Angle between W2 and the horizontal, leading edge rad

W2 Velocity observed by object inside the turbine, leading edge ft/s

α3 Angle between V2and the horizontal, trailing edge rad

V3 Velocity observed by object outside the turbine, trailing edge ft/s

w Actual specific work produced Btu/lbm

η Efficiency of the rotor %

dm Mean diameter of rotor ft

C Chord length of blade in

C/s Ratio of Chord over spacing N/A

s Spacing between blades in

N Number of blades #

b Axial chord in

A Area of cross section of rotor ft2

𝑟𝑛/𝑟𝑡 Ration of the radius of hub over the radius of tip N/A

𝑟𝑛 Radius of hub ft

𝑟𝑡 Radius of tip ft

𝑙𝑅 Rotor blade length ft

ψ Loading factor N/A

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Theory

Generally, a turbine is classified as a turbomachine that transfers energy through the

shaft as a result of hot steam flow rotates the rotor. Hence, a turbine main task is the electricity

production found in steam turbine power plants, gas turbine power plants, hydro-electric power

plant, and wind turbines. The internal turbine process can be subdivided as:

1. Fluid flowing in the axial direction towards the turbine blades.

2. Stator blades switch the flow’s direction to line it up with the turbine blades.

3. Turbine blades change the fluid flow back in the axial direction again with the

purpose of rotating the shaft.

The above Figure illustrates the some major mechanical components that can be found

a general turbine. A further visual analysis can be implemented in the first set of stator-rotor

stage known as the cascade view, Figure 3. The stator is the blade that is generally in a fixed

position whereas the rotor rotates with the shaft. The task is to design these two type of turbines

blades

Figure 1 Aircraft Turbine Explode View

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The impulse stage is considered on a basis of zero degree reaction. The reason for this is because the reaction stage is defined to have enthalpy that is distributed to the rotor over the total enthalpy available to change to work. The impulse stage is generally the first stage in a turbine that allows the power plant to process enormous amounts of heat that is eventually converted to shaft work. Absolute velocity, is then defined as the velocity of an object relative to the earth. For this particular stage, the two different blades move at separate velocities relative to each other. Calculation for the separate velocities is determined using the formula

𝑉 − 𝑈 = 𝑊

Where

V: is determined as the absolute velocity as observed from the outside

U: is the absolute velocity of the moving blades

W: is the relative velocity as observed from the rotor.

The steam that originates from the boiler and enters the turbine at a velocity of 𝑣𝑜, in which the kinetic energy can be determined as

𝐾𝐸 = 𝑣𝑜

2

2𝑔𝑐= Δℎ

Using this equation, we can determine the inlet velocity by rearranging the equation to

𝑣𝑜 = √2𝑔𝑐Δℎ

Figure 3 Internal Turbine Process at the Impulse Stage Figure 2 Cascade View for the Impulse Stage of a simple Turbine

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While the steam is in transition between the stator and the rotor blades; it will experience a change due to the turbine blades. As the steam continues through the

system, the fluid velocity will go from 𝑣2 to 𝑣3. This change can be seen through the equation

𝐹 = �̇�

𝑔𝑐(𝑣2𝑐𝑜𝑠𝛼2 − (−𝑣3𝑐𝑜𝑠𝛼3))

Knowing that power is equal to 𝐹𝑥𝑈, the power equation comes to

𝑤 = 𝑚�̇�

𝑔𝑐(𝑣2𝑐𝑜𝑠𝛼2 − (−𝑣3𝑐𝑜𝑠𝛼3))

The ratio of the energy that the rotor receives with the kinetic energy which the fluid can potentially give the rotor, defines the measure of the efficiency of a system. This is define as :

𝜂 =energy that the rotor recievs

𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑=

𝑤

(𝑉0

2

2𝑔𝑐)

A few more parameters are needed to finalize the overall design of the stator blade and the rotor blade. One of these parameters is the distance between the stator blades. To satisfy the equation it is necessary to calculate the spacing with:

𝑠 =0.85 ∗ 𝑏

4 tan(𝑎) ∗ 𝑐𝑜𝑠2(𝑎)

To obtain the number of blades needed for the turbine: 𝑁 =2𝜋𝑟

𝑠

Figure 4 Simplify Schematic of inlet turbine showing the space between the blades

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Calculations

1. Initial parameters established from the previous design project are:

Power plant power output = 200,000 KW Inlet pressure = 1800 psi Inlet temperature = 1000oF Reheat inlet temperature = 1000oF Maximum moisture level = 13%

Mass Flow rate, ṁ = 324.48 𝑙𝑏𝑚

𝑠

Enthalpy drop upon exiting the impulse stage, ∆h = 30 𝑏𝑡𝑢

𝑙𝑏𝑚

Assumed rotor spin, N = 3600 RPM

Assuming 𝛼2 = 13o

To begin designing the impulse stage the following step are taken:

2. The velocity of the fluid is calculated by applying the known drop in enthalpy along with the kinetic energy equation:

𝑉02

2𝑔𝑐= ∆ℎ

𝑉2 = 𝑉0 = √2𝑔𝑐∆ℎ

𝑉0 = √2(32.2𝑓𝑡

𝑠2)(778

𝑙𝑏𝑚 ∗ 𝑓𝑡

𝐵𝑡𝑢)(30

𝐵𝑡𝑢

𝑙𝑏𝑚)

𝑽𝟎 = 𝟏𝟐𝟐𝟔. 𝟎𝒇𝒕

𝒔

3. Assuming 𝑈

𝑉𝑜= 0.39 we calculate:

𝑈 = 𝑉𝑜 ∗𝑈

𝑉𝑜

𝑈 = (1226.01 𝑓𝑡

𝑠)(0.39)

𝑼 = 𝟒𝟕𝟖. 𝟏𝟒 𝒇𝒕

𝒔

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4. Assuming the turbine is rotating at 3600 rpm; we implement previous values to obtain the average stage diameter and radius.

𝑈 = 2𝜋𝑟𝑚𝑁

60

𝑟𝑚 = 60 ∗ 𝑈

2𝜋𝑁

𝑟𝑚 = 60(478.14

𝑓𝑡𝑠 )

2𝜋(3600𝑟𝑝𝑚)

𝒓𝒎 = 1.268 ft

5. Solving for the angles using the velocity triangles, we can implement the

assumed values of 𝛼2 = 13. To obtain the velocity at that angle we calculate:

Figure 5 Velocity Triangles for in coming and out coming Steam Flow for the First Stage

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𝑊2 = √𝑉22 − 2𝑈𝑉2𝐶𝑜𝑠𝛼2 + 𝑈2

𝑾𝟐 = 𝟕𝟔𝟕. 𝟔𝟗 𝒇𝒕

𝒔

𝛽2 = 𝑡𝑎𝑛−1[𝑉2𝑠𝑖𝑛𝛼2

𝑉2𝑐𝑜𝑠𝛼2 − 𝑈]

𝛽2 = 𝑡𝑎𝑛−1[(1226.01

𝑓𝑡𝑠 )(𝑠𝑖𝑛13)

(1226.01 𝑓𝑡𝑠 ) (𝑐𝑜𝑠13) − (478.14

𝑓𝑡𝑠 )

𝜷𝟐 = 𝟐𝟏. 𝟎𝟓 o

6. Knowing 𝑊3 = 𝑊2 (Ideal Case) and that 𝛽2 = 𝛽3

7. By assuming that 𝛽3 = 𝛽2 and that 𝑊3 = 𝑊2 we can calculate the velocity and angle. Implementing this along with the geometry diagram, we can find 𝑉3 and 𝛼3.

𝛼3 = tan−1(𝑉2𝑆𝑖𝑛𝛼2)

𝑉2𝑐𝑜𝑠𝛼2 − 2𝑈

𝛼3 = 𝑡𝑎𝑛−1(1226.1

𝑓𝑡𝑠 )(𝑠𝑖𝑛13)

(1226.0𝑓𝑡𝑠 ) (𝑐𝑜𝑠13) − 2(478.14

𝑓𝑡𝑠 )

𝜶𝟑 = 𝟒𝟗. 𝟏𝟕 o

𝑉3 = √𝑉22 − 4𝑈𝑉2𝐶𝑜𝑠𝛼2 + 4𝑈2

𝑉3 = √(1226.01𝑓𝑡

𝑠)

2

− 4 (478.14𝑓𝑡

𝑠) (1226.01

𝑓𝑡

𝑠) (𝐶𝑜𝑠13) + 4(478.14

𝑓𝑡

𝑠)2

𝑽𝟑 = 𝟑𝟔𝟒. 𝟒𝟖 𝒇𝒕

𝒔

8. By considering the impulse of the fluid acting on the blades, we can calculate the work produced by the rotor:

∆𝑤 =𝑈

𝑔𝑐

(𝑊2𝑐𝑜𝑠𝛽2 + 𝑊3𝑐𝑜𝑠𝛽3) = 2𝑊2𝑐𝑜𝑠𝛽2

𝑈

𝑔𝑐

∆𝑤 =2(767.69𝑓𝑡

𝑠)(cos 21.05)

(478.14𝑓𝑡

𝑠)

(32.2𝑓𝑡

𝑠2)(778𝑙𝑏𝑚∗𝑓𝑡

𝐵𝑡𝑢)

∆𝒘 = 𝟐𝟕. 𝟑𝟓 𝑩𝒕𝒖

𝒍𝒃𝒎

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9. For the Efficiency:

𝜂 =∆𝑤

(𝑉0

2

2𝑔𝑐)

=27.35

𝐵𝑡𝑢𝑙𝑏𝑚

((1226.0

𝑓𝑡𝑠 )2

2 ∗ 32.2𝑓𝑡𝑠2 ∗ 778

𝑙𝑏𝑚 ∗ 𝑓𝑡𝐵𝑡𝑢

)

𝜂 = 0.9116

𝜼 = 𝟗𝟏. 𝟏𝟔%

10. For the lengths finding the specific volume at 𝑃𝑖 = 1800 𝑝𝑠𝑖 and 𝑇1=1000 oF:

𝑣1 = 0.45719 𝑓𝑡3

𝑙𝑏𝑚

𝐿1 =ṁ𝑣1

𝜋𝑑𝑉2𝑠𝑖𝑛 ∝2=

ṁ𝑣1

2𝜋𝑟𝑚𝑉2 𝑠𝑖𝑛 ∝2

𝐿1 = ( 324.48

𝑙𝑏𝑚𝑠 )(0.45719

𝑓𝑡3

𝑙𝑏𝑚)

2𝜋(1.2683 𝑓𝑡)(1226.01𝑓𝑡𝑠 )(𝑠𝑖𝑛13)

𝑳𝟏 = 𝟎. 𝟎𝟔𝟕𝟓 𝒇𝒕= 0.81 in

Figure 6 Simplify Schematic for Impulse Stage for Stator-Rotor Combination

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11. To compute the length of the rotor, one must identify the specific volume when

the flow enter the rotor blade. First we find the first enthalpy value of 1490.8

Btu/lbm from the pressure and temperature values provided in the first project,

and traced it to the left until we hit the temperature curve of 1000 F. Once we find

that point, we trace down keeping the entropy constant because in a turbine

while the steam expands, the entropy remains constant.

Figure 7 Mollier Graph use to Obtained Temperature and Pressure at the Rotor

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Secondly, the fraction heat drop is 20 Btu/lbm, we find a new enthalpy value of

1470.8 Btu/lbm and trace it until it hits that vertical line. The point where they

intersect is the new pressure and temperature. From the graph, the following

values are found.

𝑇2 = 900℉ 𝑅

𝑃2 = 1000 𝑝𝑠𝑖

From the superheated tables, a second specific volume is found to be:

𝑣2 = 0.76136 𝑚3/𝑘𝑔

Therefore, the length of the rotor turns out to be:

𝐿2 =ṁ𝑣2

𝜋𝑑𝑉2𝑠𝑖𝑛 ∝2=

ṁ𝑣2

2𝜋𝑟𝑚𝑉2 𝑠𝑖𝑛 ∝2

𝐿2 = ( 324.48

𝑙𝑏𝑚𝑠 )(0.76136

𝑓𝑡3

𝑙𝑏𝑚)

2𝜋(1.2683 𝑓𝑡)(1226.01𝑓𝑡𝑠 )(𝑠𝑖𝑛13)

𝑳𝟐 = 𝟎. 𝟏𝟏𝟐𝟒 𝒇𝒕= 1.35 in

12. It is possible to determine the number of blades needed for the rotors to produce

the power needed. To accomplish this it is necessary to find the blade spacing.

From the Zweifel relation:

2𝑠

𝑏(𝑡𝑎𝑛𝛼1 + 𝑡𝑎𝑛𝛼2)𝑐𝑜𝑠2𝛼2 = 0.85

Assuming b=0.5 in.

s = 0.85(0.5 𝑖𝑛)

2(2𝑡𝑎𝑛49.17)𝑐𝑜𝑠213

s= 𝟎. 𝟑𝟏𝟕 in.

Knowing Ns=2𝜋𝑟

𝑁 =2𝜋𝑟

𝑠

N= 301.75

Therefore, the number of blades needed are approximately 302 blades.

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With the number of blades known, the radius between the shaft and the blades can be

determined, but first the area is to be found using the formula

𝐴 = 2𝜋𝑟𝑚 ∗ 𝐿2 = 2𝜋 ∗ 1.2683 ∗ 0.1124 = 0.8958 𝑓𝑡2

The area allows the determination of the radii, 𝑟𝑡 𝑎𝑛𝑑 𝑟ℎ, with 𝑟ℎ

𝑟𝑡= 0.3

𝑟ℎ = √𝐴

𝜋 (1

0.32 − 1)= √

0.8958

𝜋 (1

0.32 − 1)= 0.1679 𝑓𝑡 = 2.015 𝑖𝑛

𝑟𝑡 = 𝑟ℎ

0.3= 0.5598 𝑓𝑡 = 6.717 𝑖𝑛

13. Calculating the loading factor leads to:

𝜓 =∆𝑤 ∗ 𝑔𝑐

𝑈2

𝜓 =(27.35

𝐵𝑡𝑢𝑙𝑏𝑚

)(32.2𝑓𝑡𝑠2)(778

𝑙𝑏𝑚 ∗ 𝑓𝑡𝐵𝑡𝑢 )

(478.14𝑓𝑡𝑠 )2

𝝍 = 2.997

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An iteration procedure was used to calculate all parameters above with different angles

(α₂). The major variables such as the loading factor and the efficiency are plotted in the

following graph. The left axis shows the loading factor and the right axis the efficiency of

the impulse stage for a range of 10 < 𝛼2 < 15.

Figure 8 Loading Factors and Impulse Stage Efficiencies according the Velocity Angles

Discussion and Conclusion

One can state that for the first stage impulse design the necessary velocity for the

exit nozzle must be close to 𝟏𝟐𝟐𝟔 ft/s. The exit nozzle’s velocity is the pillar for several

calculations and is a necessary part for the knowledge of the design of turbine blades.

Although further calculations, like the velocity of the rotor blade, are based this value; the

loading factor is the most crucial number that must not be greater than 3 to ensure that

turbine will work without putting the entire power plant in jeopardy. Equally important is

the efficiency of the impulse stage, Figure 8 exposes an important detail when design the

first stage. The efficiency and the loading factor really depend on the orientation of the

incoming velocity vector to the rotor blade. That is the velocity vector exiting the stator

determines the efficiency of the system and also whether the system will work properly

due the value of the loading factor. In this case, the adequate angle is 13°; which produces

an efficiency of 91.16% and a loading factor of 2.997.

89

89.5

90

90.5

91

91.5

92

92.5

93

2.94

2.96

2.98

3

3.02

3.04

3.06

10 11 12 13 14 15

Imp

uls

e St

age

Effi

cien

cy,

η (

%)

Load

ing

Fact

or

,ψ

Velocity Angle,α₂(Degrees)

Loading Factor and Efficiency vs Velocity Angle α₂

Loading Factor

Impulse StageEfficiency

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References

[1]. Cengel, Yunus A., and Boles, Michael A. Thermodynamics An Engineering Approach,

Vapor and combined Power Cycles,5th edition.

[2]. Raj, Rishi S. Thermo-Fluid Systems Analysis and Design, Forms of Rankine Cycle for

Steam Turbine Power Plants and Block Diagrams, 3rd edition.

[3]. Ingram, G. (2009). Concepts in Turbomachinery. Ventus Publishing Aps.