Answer:
a) Qe = 0.01963 m^3 / s , mass flow rate m^ = 0.1334 kg/s
b) Inlet cross sectional area = Ai = 0.11217 m^2 , Qi = 0.11217 m^3 / s
Explanation:
Given:-
- The compressor exit conditions are given as follows:
Pressure ( Pe ) = 825 KPa
Temperature ( Te ) = 150°C
Velocity ( Ve ) = 10 m/s
Diameter ( de ) = 5.0 cm
Solution:-
- Define inlet parameters:
Pressure = Pi = 100 KPa
Temperature = Ti = 20.0
Velocity = Vi = 1.0 m/s
Area = Ai
- From definition the volumetric flow rate at outlet ( Qe ) is determined by the following equation:
Qe = Ae*Ve
Where,
Ae: The exit cross sectional area
Ae = π*de^2 / 4
Therefore,
Qe = Ve*π*de^2 / 4
Qe = 10*π*0.05^2 / 4
Qe = 0.01963 m^3 / s
- To determine the mass flow rate ( m^ ) through the compressor we need to determine the density of air at exit using exit conditions.
- We will assume air to be an ideal gas. Thus using the ideal gas state equation we have:
Pe / ρe = R*Te
Where,
Te: The absolute temperature at exit
ρe: The density of air at exit
R: the specific gas constant for air = 0.287 KJ /kg.K
ρe = Pe / (R*Te)
ρe = 825 / (0.287*( 273 + 150 ) )
ρe = 6.79566 kg/m^3
- The mass flow rate ( m^ ) is given:
m^ = ρe*Qe
= ( 6.79566 )*( 0.01963 )
= 0.1334 kg/s
- We will use the "continuity equation " for steady state flow inside the compressor i.e mass flow rate remains constant:
m^ = ρe*Ae*Ve = ρi*Ai*Vi
- Density of air at inlet using inlet conditions. Again, using the ideal gas state equation:
Pi / ρi = R*Ti
Where,
Ti: The absolute temperature at inlet
ρi: The density of air at inlet
R: the specific gas constant for air = 0.287 KJ /kg.K
ρi = Pi / (R*Ti)
ρi = 100 / (0.287*( 273 + 20 ) )
ρi = 1.18918 kg/m^3
Using continuity expression:
Ai = m^ / ρi*Vi
Ai = 0.1334 / 1.18918*1
Ai = 0.11217 m^2
- From definition the volumetric flow rate at inlet ( Qi ) is determined by the following equation:
Qi = Ai*Vi
Where,
Ai: The inlet cross sectional area
Qi = 0.11217*1
Qi = 0.11217 m^3 / s
- The equations that will help us with required plots are:
Inlet cross section area ( Ai )
Ai = m^ / ρi*Vi
Ai = 0.1334 / 1.18918*Vi
Ai ( V ) = 0.11217 / Vi .... Eq 1
Inlet flow rate ( Qi ):
Qi = 0.11217 m^3 / s ... constant Eq 2
