 # 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 ${T}_{\mathrm{\infty }}={1200}^{\circ }C$. and maintains a convection coefficient of $h=250\frac{W}{{m}^{2}}\cdot K$ over the blade.
The blades, which are fabricated from Inconel, $k\approx 20\frac{W}{m}\cdot K$, have a length of L=50 mm. The blade profile has a uniform cross-sectional area of ${A}_{c}=6×{10}^{-4}{m}^{2}$ 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 ${T}_{b}={300}^{\circ }C$
a) If the maximum allowable blade temperature is ${1050}^{\circ }C$ 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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Given:$L=0.05m,{A}_{c}=6x{10}^{-4},P=0.11m,k=20\frac{W}{m}\cdot K,h=250\frac{W}{{m}^{2}}\cdot K,{T}_{b}=573K,{T}_{\mathrm{\infty }}=1473K,T{\left(L\right)}_{maz}=1323K$
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
$\frac{\theta }{{\theta }_{b}}=\frac{\mathrm{cos}h\left(L-x\right)}{\mathrm{cos}h\left(mL\right)}here,\theta =T-{T}_{\mathrm{\infty }}$
$At,x=L,\theta =T\left(L\right)-{T}_{\mathrm{\infty }}$
$\frac{T\left(L\right)-{T}_{\mathrm{\infty }}}{{T}_{b}-{T}_{\mathrm{\infty }}}=\frac{\mathrm{cos}h\left(L-x\right)}{\mathrm{cos}h\left(mL\right)}$
$\frac{T\left(L\right)-{T}_{\mathrm{\infty }}}{{T}_{b}-{T}_{\mathrm{\infty }}}=\frac{1}{\mathrm{cos}h\left(mL\right)}$
Calculate the value of m as follows:
$m=\sqrt{\frac{hP}{k{A}_{c}}}$
$=\sqrt{\frac{250×0.11}{20×6×{10}^{-4}}}$
$m=48.87{m}^{-1}$
Calculate the value of T(L) as folows
$\frac{T\left(L\right)-1473}{573-1473}=\frac{1}{\mathrm{cos}h\left(47.87×0.05\right)}$
$T\left(L\right)-1473=-163$
$T\left(L\right)=1310K$
As $T\left(L\right)=1310K. The cooling scheme is satisfactory
(b)
We have the fin heat transfer rate as
$qf=-qb$
$qf=\sqrt{hPk{A}_{c}{\theta }_{b}}\text{tanh}\left(mL\right)$
$-qb=\sqrt{hPk{A}_{c}{\theta }_{b}}\left({T}_{b}-{T}_{\mathrm{\infty }}\right)\text{tanh}\left(mL\right)$

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(a) With the maximum temperature existing at x = L, Eq. 3.75 yields
$\frac{T\left(L\right)-T\mathrm{\infty }}{{T}_{b}-{T}_{\mathrm{\infty }}}=\frac{1}{\mathrm{cos}hmL}$
$m={\left(h\frac{P}{k}{A}_{c}\right)}^{\frac{1}{2}}={\left(250\frac{W}{{m}^{2}}\cdot K×0.11\frac{m}{20}\frac{W}{m}\cdot K×6×{10}^{-4}{m}^{2}\right)}^{\frac{1}{2}}$
$m=47.87{m}^{-1}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}mL=47.87{m}^{-1}×0.05m=2.39$
From Table $B.1,\mathrm{cos}hmL=5.51$. Hence,
$T\left(L\right)={1200}^{\circ }C+{\left(300-1200\right)}^{\circ }\frac{C}{5.51}={1037}^{\circ }C$
And the operating conditions are acceptable. NS (b)With $M={\left(hPk{A}_{c}\right)}^{\frac{1}{2}}{\theta }_{b}={\left(250Wi{m}^{2}\cdot K×0.11m×20\frac{W}{m}×K×6×{10}^{-4}{m}^{2}\right)}^{\frac{1}{2}}\left(-900°c\right)=-517W$
Eq. 3.76 and Table B.1 yield
$qf=M\text{tanh}mL=-517W\left(0.938\right)=-508W$
Hence,$qb=-qf=508W$

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