Question

Steam enters the first-stage turbine shown in Fig. P4.50 at 40 bar and 500℃ with a volumetric flow rate of 90 m³/min. Steam exits the turbine at 20 bar and 400℃. The steam is then reheated at constant pressure to 500℃ before entering the second-stage turbine. Steam leaves the second stage as saturated vapor at 0.6 bar. For operation at steady state, and ignoring stray heat transfer and kinetic and potential energy effects, determine the

(a) mass flow rate of the steam, in kg/h.
(b) total power produced by the two stages of the turbine, in kW.
(c) rate of heat transfer

Answer 1

Concepts and reason

Volume flow rate:

The volume flow rate of a fluid is defined as the volume of a fluid that passes through a given cross-sectional area per unit of time.

Units: The S.I. unit of volume of flow rate in m3/s.

Mass flow rate:

The mass flow rate of a fluid is defined as the mass of a fluid that passes through a given cross sectional area. It is also defined as the rate of movement of liquid mass through a unit area. This mass flow rate depends on the density of the fluid, velocity of the flowing fluid and also the area of cross section.

Units: The S.I. unit of mass of flow rate is kg/s.

Compounding of steam turbines:

In Compounding of steam turbines the energy from steam is extracted in a number of stages to reduce the wastage of steam and also to protect the turbine structure from centrifugal forces caused by the high speeds of the rotor.

Fundamentals

Mass flow rate:

The volume rate of a fluid is denoted by Q which is defined as,

Here, the volume of the fluid is and the time for which the fluid flows is t.

But, the volume of a body is the product of cross-sectional area A and thickness d.

Hence,

The term is the thickness of the fluid per unit time which is nothing but the speed of the fluid. So, we can replace the term with the velocity of the fluid V.

$Q=AxV $

The mass flow rate of a fluid is denoted by which is defined as,

Here, the mass of the fluid is m, and the time for which a particular mass of fluid flows is t.

But, the mass of a body is the product of volume v and density P.

Hence,

The term is known as the volume rate Q of a fluid.

But, $Q=AxV $

Hence,

But the density of a fluid is equal to the reciprocal of the specific volume. Then,

Hence,

Work output from a two stage turbine:

Consider a steam turbine in which the steam is expanded in two stages. The T-s diagram of the turbine is shown below:

Here, the steam expansion in the first stage of the turbine from state (1) to  state(2) and then it is heated in the reheater from state (2) to state (2’) and again the steam is expanded from state (2’) to state (3).

The work output of the two stage turbine is given by,

The rate of heat transfer to the steam flowing through the reheater is given by,

Draw the T-s diagram of the cycle as shown below.

Here, process 1-2 is isentropic process in first stage turbine,

Process 2-3 is constant pressure heat addition in reheater,

Process 3-4 is isentropic process in second stage turbine.

Calculate the mass flow rate of steam by using the equation,

                                                              …… (1)

Here, the volumetric flow rate at state 1 is (AV)1 and the specific volume of steam at state 1 is v1.

Obtain specific volume at pressureand temperature from superheated water tables,

Now substitute in equation (1)

Calculate the work output from the two stages of the turbine by using the equation,

                                                                                 …… (2)

Here, the enthalpy of steam at respective points is h.

Obtain enthalpies at states 1, 2, 3 and 4 from “superheated water tables” as shown below:

At pressureand temperature,

Atand,

Atand,

At state 4 the steam is saturated vapor, obtain specific enthalpy of saturated vapor atfrom saturated water tables

Substitute the above values in equation (2).

Calculate the rate of heat transfer to the steam flowing through the reheater by using the equation: