Turbine blades mounted to a rotating disc in a gas turbine engine are

aspifsGak5u 2021-12-14 Answered

Turbine blades mounted to a rotating disc in a gas turbine engine are exposed to a gas stream that is at T=1200C. and maintains a convection coefficient of h=250Wm2K over the blade. 
The blades, which are fabricated from Inconel, k20WmK, have a length of L=50 mm. The blade profile has a uniform cross-sectional area of Ac=6×104m2 and a perimeter of P=110 mm. 
A proposed blade-cooling scheme, which involves routing air through the supporting disc, is able to maintain the base of each blade at a temperature of Tb=300C
a) If the maximum allowable blade temperature is 1050C and the blade tip may be assumed to be adiabatic, is the proposed cooling scheme satisfactory? 
b) For the proposed cooling scheme, what is the rate at which heat is transferred from each blade to the coolant?

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Expert Answer

psor32
Answered 2021-12-15 Author has 33 answers

Given:L=0.05m,Ac=6x104,P=0.11m,k=20WmK,h=250Wm2K,Tb=573K,T=1473K,T(L)maz=1323K 
Schematic of turbine blade mounted to a rorating disc as follows: 
<img src="19610907981.jpgZSK 
(a) 
Temperature distribution of a fin with an adiabatic tip as follows 
θθb=cosh(Lx)cosh(mL)here,θ=TT 
At,x=L,θ=T(L)T 
T(L)TTbT=cosh(Lx)cosh(mL) 
T(L)TTbT=1cosh(mL) 
Calculate the value of m as follows: 
m=hPkAc 
=250×0.1120×6×104 
m=48.87m1 
Calculate the value of T(L) as folows 
T(L)14735731473=1cosh(47.87×0.05) 
T(L)1473=163 
T(L)=1310K 
As T(L)=1310K<T(L)maz. The cooling scheme is satisfactory 
(b) 
We have the fin heat transfer rate as 
qf=qb 
qf=hPkAcθbtanh(mL) 
qb=hPkAcθb(TbT)tanh(mL) 
=250×0.11×20×6×1064×(5731473)×tanh 

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usaho4w
Answered 2021-12-16 Author has 39 answers
(a) With the maximum temperature existing at x = L, Eq. 3.75 yields
T(L)TTbT=1coshmL
m=(hPkAc)12=(250Wm2K×0.11m20WmK×6×104m2)12
m=47.87m1andmL=47.87m1×0.05m=2.39
From Table B.1,coshmL=5.51. Hence,
T(L)=1200C+(3001200)C5.51=1037C
And the operating conditions are acceptable. NS (b)With M=(hPkAc)12θb=(250Wim2K×0.11m×20Wm×K×6×104m2)12(900°c)=517W
Eq. 3.76 and Table B.1 yield
qf=MtanhmL=517W(0.938)=508W
Hence,qb=qf=508W

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