Question 7.DE.10: The following particulars relate to a single-stage turbine o......
The following particulars relate to a single-stage turbine of free vortex design:
Inlet temperature, T_{01} 1100 K
Inlet pressure, p_{01} 4 bar
Mass flow 20 kg/s
Axial velocity at nozzle exit 250 m/s
Blade speed at mean diameter 300 m/s
Nozzle angle at mean diameter 25°
Ratio of tip to root radius 1.4
The gas leaves the stage in an axial direction, find:
(1) The total throat area of the nozzle.
(2) The nozzle efflux angle at root and tip.
(3) The work done on the turbine blades.
Take
C_{pg} = 1.147 kJ/kg K, γ = 1.33
For no loss up to throat
\frac{p^*} {p_{01}} = \left(\frac{2} {γ + 1}\right)^{γ/(γ – 1)} = \left(\frac{2}{2.33}\right)^4 = 0.543
p^* = 4 \times 0.543 = 2.172 bar
Also T^* = 1100\left(\frac{2}{2.33}\right)^4 = 944 K
T_{01} = T^* + \frac{C^2} {2C_{pg}}
C^* = \sqrt{2C_{pg}(T_{01} – T^*)}
= \sqrt{(2)(1147)(1100 – 944)} = 598 m/s
ρ^* = \frac{p^*} {RT^*} = \frac{(2.172)(100)} {0.287)(944)} = 0.802 kg/m³
(1) Throat area
A = \frac{m} {ρC^*} = \frac{20} {(0.802)(598)} = 0.042 m²
(2) Angle α_1, at any radius r and α_{1m} at the design radius r_m are related by the equation
\tanα_1 = \frac{r_m} {r_1} \tanα_{1m}
Given
\frac{\text{Tip radius}} {\text{Root radius}} = \frac{r_t} {r_r} = 1.4
∴ \frac{\text{Mean radius}} {\text{Root radius}} = 1.2
α_{1m} = 25°
\tanα_{1r} = \frac{r_{\text{mean}}} {r_{\text{root}}} × \tanα_{1m}
= 1.2 × tan 25° = 0.5596
∴ α_{1r} = 29.23°
\tanα_{1t} = \frac{r_r} {r_t} × \tanα_{1r} = \left(\frac{1} {1.4}\right)(0.5596) = 0.3997
∴ α_{1t} = 21.79°
(3) C_{w2} = \frac{r_m} {r_r} xC_{w2m} = \frac{r_m} {r_r}\frac{Ca_2} {\tanα_{2m}} = 1.2x \frac{250} {\tan 25°} = 643 m/s
W = mUC_{w2} = \frac{(20)(300)(643)} {1000} = 3858 kW