Question

Steam enters a turbine at a velocity of 200 m/s. The inlet conditions of the steam are at 4000 kPA and 500°C. The diameter of the inlet pipe is 50 mm. The outlet conditions of the steam are 80 kPa and a quality of 1.0. The diameter of the outlet pipe is 250 mm. Determine the turbine power output in kJ/s assuming the kinetic energy change and potential energy change are both negligible. Calculate the change in kinetic energy to show it is indeed negligible.

Answer 1

Given: Inlet velocity of steam, V₁=200 m/s

Inlet pressure of steam, P₁=4000 kPa

Inlet temperature of steam, T₁=500⁰C

Diameter of pipe at inlet, d₁=50 mm Radius, r₁=25 mm=0.025 m

Outlet pressure of steam, P₂=80 kPa

Quality of steam at outlet, x₂=1

Diameter of pipe at outlet, d₂=250 mm Radius, r₂=125 mm=0.125 m

Assumption: Heat transfer is negligible.

The energy equation for this system, neglecting the kinetic and potential energies change is

where _T is the Turbine power output

is the Mass flow rate of steam

h₁ and h₂ is the Specific enthalpy at inlet and outlet conditions respectively

Mass flow rate of steam, , is obtained using the expression

where ₁ is the Density of steam at inlet conditions,

v₁ is the Specific volume of steam at inlet conditions

V₁ is the Inlet velocity

A₁ is the Cross sectional area of pipe at inlet, A₁=r₁²

r₁ is the Inlet radius of pipe

Refer Properties of Superheated Water Vapor Table.

At P₁=4000 kPa=40 bar and T₁=500⁰C, specific volume, v₁=0.08643 m³/kg and specific enthalpy, h₁=3445.3 kJ/kg.

Refer Properties of Saturated Water-Pressure table to obtain specific volume and specific enthalpy at outlet conditions.

At P₂=80 kPa=0.8 bar, specific volume of saturated vapor, v_g=v₂=2.087 m³/kg and specific enthalpy of saturated vapor, h_g=h₂=2665.8 kJ/kg

Thus the turbine power output is

Thus the turbine power output is 3542.048 kJ/s.

Change in Kinetic energy, KE, is

where V₂ is the Outlet velocity

Outlet velocity, V₂, is obtained using the Continuity equation

where A₂ is the Cross sectional area of pipe at outlet, A₂=r₂²

r₂ is the Radius of outlet pipe

₂ is the Density of steam at outlet conditions,

Thus change in kinetic energy is

(Unit: kg m²/s² is J)

Change in enthalpy, h=(3445.3-2665.3)kJ/kg=779.5 kJ/kg.

The change in kinetic energy is less than 0.1% of the enthalpy change. Hence kinetic energy change is negligible in the case of turbine.