Lecture 1.2Blade Element Theory

1

• Consider the axial and radial flow turbomachines, schematicallyshown in Figs. 1.2.1 and 1.2.2. Let the number of blades besmall.

• The passages between the blades is considered as the flowdomain. The control surfaces are shown by dotted lines in thefigures.

.

Fig. 1.2.1 Axial flow Turbomachine Fig. 1.2.2 Radial flow Turbomachine2

Analogy with Isolated Airfoil

• Drawing an analogy between the control surfaces around bladesof the turbomachine and the control surface of an isolatedaerofoil of Fig. 1.2.3, gives

3

1

2

21

2

1Γ

Γ

Γ Γ

,

,

,

sc

sc

s

B

Ks

D

C

V ds

V ds

V ds V ds

Fig 1.2.3. Analogy with an isolated airfoil

• In Lecture 1.1, it is noted that the circulation around the body (Γ)is independent of the size and shape of the contour, for the givenflow conditions of constant energy.

• Apply the above analogy to an isolated element of aturbomachinery blade as shown in Fig. 1.2.4.

Fig. 1.2.4. Isolated blade of a turbomachine 4

• The blade may be moving with a linear speed of U. Then, Cu2and Cu1 are the tangential components of the flow velocity at theoutlet and inlet, respectively.

• The lengths AB and CD represent the blade pitches (S1 and S2)respectively at inlet and outlet.

• AD and BC are the lines dividing the flow passages betweenadjacent blades.

• The circulation is given by the line integral around ABCD.

• Call (ΓABCD) as Γb, circulation around the isolated blade element5

• The circulation around ABCD may by evaluated by summing theindividual circulations comprising the circuit, such that

(1.2.1)

• Here the circulation is positive anticlockwise.

But and ,

• Therefore, (1.2.2)

2 2 2 B

uA

V ds C s 1 1 1 D

uC

V ds C s C D

B A

V dl V dl

ABCD 2 1Γ B C D A

A B C D

V ds V dl V ds V dlV ds

ABCD 2 2 1 1 Γ b u us C s C

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• Referring to Fig. 1.2.5., Consider the circulation around theblade elements. The circulation about the part AEF contour is

where denotes the line integral from E to A by way of F

• The circulation about the contour ABDE is

• The circulation about the contour BCD is

1Γ F

E A

S s sAEF A EV ds V ds V ds

FA

E

2ΓB D E A

S s s s sABDE A B D EV ds V ds V ds V ds V ds

3Γ C

D B

S s sBCD B DV ds V ds V ds

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Fig. 1.2.5 Circulation about several blade elements8

• Thus

• Considering that , these integrals in the

foregoing expressions will cancel out. It is therefore

• It is seen from Fig. 1.2.5 that the remaining integrations inthe last expression just cover the outer contour ABCDEF.Their sum is therefore equal to the line integral of Vs ds alongthis contour, which is the circulation Γo.

• It follows that (1.2.3)

andE A D B

A E B D

1 2 3Γ Γ Γ F C

A B E D

s s s sE A D BV ds V ds V ds V ds

1 2 3Γ Γ Γ Γo SABCDEFV ds

1 2 3Γ Γ Γ F

C

E A B D E

s s s s sA E A B DA D B

s s sE B D

V ds V ds V ds V ds V ds

V ds V ds V ds

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• In other words, the circulation about a contour is equal to the(algebraic) sum of the circulations around all parts of the regioninside that contour. The direction of circulation determines itssign in Eq. (1.2.3), which can easily be demonstrated byreversing in the foregoing derivation the direction of one of thepartial circulations and thereby the sign of the line integrals ofwhich it is composed.

• The parts of the contour considered do not need to border oneach other, as assumed in the derivation, because, according toEq. (1.2.3) and Fig. 1.2.5, the circulation about the contour AEFpreviously considered, if the shaded area between these twocontours does not contain a deflecting body and has a flow ofconstant energy.

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• Consider a cylindrical section AB through an axial flow runneras shown in Fig. 1.2.6. The circulation around every bladeelement appearing in this section shall be designated by Γb, sothat, according to Eq. (1.2.2), the total circulation of thedeveloped section is

Γo = nbΓb (1.2.4)where nb is the number of blades

• Substitution into Eq. (1.2.2) gives the circulation for the wholeimpeller as

• But, and

• Therefore,

2 2 1 1Γ ( )b b u un s C s C

.

1 12bn s r 2 22bn s r

2 2 1 1Γ 2 ( )b u ur C rC 11

Fig.1.2.6 Circulation around a number of isolated blades

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• The expression specific work for a turbomachine is given by

This equation is also know as Euler’s turbomachinery equation.• Here, U1 = r1ω and U2 = r2ω

• Thus, 2b b

spnW

2 2 1 1( )sp u uW U C UC

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Frames of Reference

• Referring once again to Figs. 1.2.1 and 1.2.2, it is important tonote that, in turbomachinery, fluid flows through stationary aswell as rotating parts of the machine.

• More often, the flow switches from stationary to rotating partsand vice versa.

• The governing equations to describe such flows are thereforewritten by fixing the coordinate system on a rotating frame ofreference.

• It is therefore important to understand the difference in definingthe fluid motion in the stationary as well as the moving (rotating)frames of reference with respect to a fixed inertial frame.

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• Fig. 1.2.7 shows the general Cartesian coordinate system fixed inspace (inertial reference frame, I) and a system (A) acceleratingwith respect to it. The system is translating with a velocityand rotating with an angular velocity .

• In the following chapter, the description of fluid motion withrespect to stationary frame of reference is first discussed. Thedescription with rotating frame will be taken up later.

v t

t

15Fig 1.2.7 Inertial and rotating reference frames

Summary of Lecture 1.2

The concepts of circulation around a contour, extended to anumber of closed contours surrounding a row of blades, is usedto derived the expression for specific work of a turbomachine.The concepts of inertial and rotational frames of reference areintroduced.

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END OF LECTURE 1.2