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Q : In a single-stage turbine designed on free vortex theory the following particulars are given : Mass flow through turbine – 18 kg/s; Inlet total head temperature – 1000 K; Inlet total head pressure – 4 kgf/cm2. The gas leaves the stage in an axial direction. Assuming that all the nozzle loss amounts to 4 per cent of the isentropic heat drop across the nozzle, find the total throat area of the nozzle. Assume a mean specific heat at constant pressure of 0·274 throughout the cycle.
m =18 kg/s
Tt = 1000 K
Pt = 4 kgf/cm2
R = 29.27 kgf m/kg°K
γ = 1.4
g = 9.81 m/s2
J = 427 kgf.m/kcal
Nozzle throat area, A = m/(ρU∗ )…(1)
Perfect gas equation, ρ∗ = P∗/(RT∗ )…(2)
P∗/Pt = (2/(γ+1))γ/(γ-1)
T∗/Tt = (P∗/Pt )(γ-1)/γ = 2/(γ+1)
P∗ = 4×(2/(1.4+1))^(1.4/(1.4-1)) = 2.168 kgf/cm2
T∗=1000×2/(1.4+1) = 858 K
ρ∗ = (2.168×104)/(29.27×858) = 0.863 m3/kg
U∗ =√(2gJcp (Tt-T∗ ))
= √(2×9.81×427×0.274(1000-858) )
= 572 m/s
A = 18/(0.863×572) = 364 cm2
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