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Lect - 7

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Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay

Lect - 7

Three DimensionalFlow Analysis in

Axial Compressor

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Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay

Lect - 7

Let us assume thata small elementinside the rotatingblade passagerepresents the fluidflow inside the

rotor, such that theanalysis of thestatus of thiselement may

wholly representthe status of thewhole flow insidethe rotor passage

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Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay

Lect - 7

It may be recalled

that this element isalso executing apath through thecurved diffusing

passage betweenthe rotor blades.

w2

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Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay

Lect - 7

Simple three dimensional flow analysis:

Initial assumptions

1)Radial movement of the flow is governed by theradial equilibrium of forces

2) Radial movements occur within the bladepassage only and not outside it

3) Flow analysis involves balancing the radial forceexerted by the blade rotation

4) Gravitational forces can be neglected

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Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay

Lect - 7

Consider this fluid elementof unit axial lengthsubtended by an angle d, of thickness dr, along

which the pressure variation is from p to p+dp.

Subscript w refers

to tangential / whirlcomponent of theflow

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Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay

Lect - 7

Resolving all the aerodynamic forces, acting onthis element, in the radial direction,

we get,

(p+dp)(r+dr).d

.1 p.r.1.d

2(p+dp/2).dr.(d

/2).1

= . dr. r. Cw2

/ r

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Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay

Lect - 7

Neglecting the second order terms (products

of small terms e.g. dp.dr etc) the equationreduces to

2wdp1 1= .C rdr

This is called the

Sim p l e Ra d i a l Eq u i l i b r i u m Eq u a t i o n

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Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay

Lect - 7

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Invoke the laws of fluid and thermo-dynamics

1) H = h + C2/2 = cpT + (Ca2 + Cw

2)

p

- 12) c

p.T =

p= c

3) Isentropic Law

From Equation of state

Energy Eqn

Where, H is total enthalpy, h is static enthalpypressure p, density , are the fluid properties

and cp and are the thermal properties of air

at the operating condition

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Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay

Lect - 7

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2dC dCdH 1 dp p dpa w= C +C + . -wadr dr dr -1 dr dr

substituting for cp from Eqn(2) and then

differentiating the eqn (1) w.r.t. r ,

we get

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Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay

Lect - 7

Differentiating the eqn 3 (isentropic law) we get

d dp=

dr .p dr

Substituting this in the new energy equationwe get

dC dCdH 1 dpa w= C +C +wadr dr dr dr

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Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay

Lect - 7

Now invoking the simple radial equilibrium equation

developed earlier in the energy equation

2dC dC CdH a w w= C +C +wadr dr dr r

2w

dp1 1= .C rdr

We get

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Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay

Lect - 7

At entry to the compressor, except near the huband the casing, enthalpy H (r) = constant.

Using the condition of uniform work distributionalong the blade length ( i.e. radially constant) we

can saydH

= 0dr

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Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay

Lect - 7

Thus, the energy equation would be written as,2

a w ww

dC dC C+C + = 0

dr dr r

Now, if Ca = constant at all radii, then thefirst term is zero and the above equationreduces to

2

dC Cw wC = -w dr r

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Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay

Lect - 7

dC drw = -dr r

This yields, on integration

Therefore, the equation becomes

Cw . r = constant.

This condition is commonly knownas the Fr e e V o r t e x Law

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Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay

Lect - 7

The term Free Vortex essentially denotes that

the strength of the vortex (or lift per unit length)created by each airfoil section used from the rootto the tip of the blade remains constant

Lift , L = .V.

where, is the density,V is the inlet velocity, and is the strength of circulation

It, therefore, means that at the trailing edge of

the blade the trailing vortex sheet has constantstrength from the root to the tip of the blade

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Prof Bhaskar Roy Prof A M Pradeep Department of Aerospace IIT Bombay

Lect - 7

Next Class ---

Free Vortex Design Lawand

Other Blade design laws