Question

A steam turbine operates with 1.6 MPa and 350 steam at its inlet and saturated vapor at \(30^{\circ} \mathrm{C}\) at its exit. The mass flow rate of the steam is \(22 \mathrm{kg} / \mathrm{s}\), and the turbine produces \(12,350 \mathrm{kW}\) of power. Determine the rate at which heat is lost through the casing of this turbine.

Short answer

Answer: The rate at which heat is lost through the casing of the steam turbine is 654.2 kW.

Step-by-step solution

Find the Inlet and Outlet Enthalpy

To begin, we need to determine the enthalpy values at the inlet and outlet of the turbine. The enthalpy can be found using the steam table, for which we need to know the pressure and temperature values. At the inlet, we have a pressure of 1.6 MPa and a temperature of 350°C. Using a steam table, we can find the enthalpy value: \(h_{1} = 3150.6 \,\mathrm{kJ/kg}\). At the outlet, we have a saturated vapor at 30°C. Using a steam table, we can find the enthalpy value: \(h_{2} = 2559.5 \,\mathrm{kJ/kg}\).

Determine the Work Done by the Turbine

Since the turbine produces power, we need to calculate the work done by the turbine per unit mass. This can be found using the following equation: \(w_{turbine} = h_{1} - h_{2}\) Substituting the values from Step 1, we have: \(w_{turbine} = 3150.6 \, \mathrm{kJ/kg} - 2559.5 \, \mathrm{kJ/kg} = 591.1 \, \mathrm{kJ/kg}\)

Calculate the Rate of Work Done by the Turbine

We have the work done by the turbine per unit mass, and we also know the mass flow rate of the steam entering the turbine. We can now calculate the rate of work done by the turbine as follows: \(W_{rate} = m \cdot w_{turbine}\) where \(m\) is the mass flow rate. Substituting the given values: \(W_{rate} = 22 \, \mathrm{kg/s} \cdot 591.1 \, \mathrm{kJ/kg} = 13004.2 \, \mathrm{kW}\)

Determine the Rate at which Heat is Lost

Now, we can find the rate at which heat is lost through the casing of the turbine by applying the energy balance equation: \(\dot{Q}_{loss} = W_{rate} - P\) where \(P\) is the power produced by the turbine. Substituting the given power value and the calculated rate of work done: \(\dot{Q}_{loss} = 13004.2 \, \mathrm{kW} - 12350 \, \mathrm{kW} = 654.2 \, \mathrm{kW}\) So, the rate at which heat is lost through the casing of this steam turbine is 654.2 kW.