- The document discusses the mathematical modeling of a steam turbine unit. It describes how steam turbines work by converting the high pressure and temperature steam into rotational energy via multiple turbine stages. - It presents the equations used to model the steam turbine as a transfer function based on the continuity equation and assumptions about steam flow and density relationships. The turbine torque is also modeled as proportional to the steam flow rate. - Block diagrams show the typical multi-stage configuration of a steam turbine with high, intermediate, and low pressure sections, each modeled as first-order lag systems.

1. Mathematical Modelling Of
Steam Turbine
BY,
TUSHAR VILAS GHAG
M.E MACHINE DESIGN

2. Contents
• Introduction
• Mathematical modelling of steam turbine unit
• Block diagram of steam turbine configuration
2

3. Introduction
• The steam turbine converts stored energy of high pressure and high temperature steam into rotary
energy, which in turn is converted into electrical energy by the generator.
• Each turbine section consists of a set of a moving blades attached to rotor and a set of stationary
vanes in which steam is accelerated to high velocity.
• The kinetic energy of the high velocity steam is converted into shaft torque.
• A large variety of steam turbines have been built, with respect to the capacity, application and
desired performance.
• To increase the thermal efficiency in applications, steam turbines consist of multistage steam
expansion.
3

4. Types Of Steam Turbine
4

5. Rankine Cycle
5

6. Rankine Cycle With
Reheat
6

7. Mathematical Modelling Of Steam Turbine
Unit
• In many cases, the steam turbine models are simplified, many intermediate variables are omitted.
• The transfer function of a steam turbine and the expression for mechanical power developed by a
turbine are based on the continuity equation.
𝑑𝑊
𝑑𝑡
= V
𝑑𝜌
𝑑𝑡
= 𝐹𝑖𝑛 𝑡 − 𝐹𝑜𝑢𝑡(𝑡) ………………. (1)
Where,
W = weight of steam in the turbine (kg)
7

8. Mathematical Modelling Of Steam Turbine
Unit
V = volume of turbine (𝑚3
)
𝜌 = density of steam (kg/𝑚3)
F = steam mass flow rate (kg/s)
t = time (sec)
• Assuming the flow out of the turbine proportional to pressure in the turbine.
𝐹𝑜𝑢𝑡(𝑡) = P
𝐹0
𝑃0
…………………………… (2)
Where,
P = Pressure of steam in the turbine (kPa)
𝑃0 = Rated pressure
𝐹0 = Rated flow out of the turbine
8

9. Mathematical Modelling Of Steam Turbine
Unit
• With constant temperature in the turbine:
𝑑𝜌
𝑑𝑡
=
𝑑𝑃
𝑑𝑡
.
𝜕𝜌
𝜕𝑃
…………………………… (3)
• From equations (1), (2) & (3), result the mathematical model:
𝐹𝑖𝑛 𝑡 − 𝐹𝑜𝑢𝑡 𝑡 = 𝑉
𝑑𝑃
𝑑𝑡
.
𝜕𝜌
𝜕𝑃
= V
𝜕𝜌
𝜕𝑃
.
𝑃0
𝐹0
.
𝑑𝐹𝑜𝑢𝑡
𝑑𝑡
= 𝑇𝑇
𝑑𝐹𝑜𝑢𝑡
𝑑𝑡
……………………(4)
𝑇𝑇
𝑑𝐹𝑜𝑢𝑡
𝑑𝑡
+ 𝐹𝑜𝑢𝑡(𝑡) = 𝐹𝑖𝑛 𝑡 …………………………………. (5)
9

10. Mathematical Modelling Of Steam
Turbine Unit
• After Laplace transform, the transfer function is:
𝐻𝑇(𝑠) =
𝐹𝑜𝑢𝑡(𝑠)
𝐹𝑖𝑛 𝑠
=
1
𝑇𝑇𝑠+1
………………………… (6)
Where,
𝑇𝑇 = V
𝜕𝜌
𝜕𝑃
.
𝑃0
𝐹0
is the time constant (sec)
10

11. Mathematical Modelling Of Steam
Turbine Unit
• The turbine torque is proportional to the steam flow rate:
𝑇𝑚(𝑡) = k . F(t)
Where,
k = constant of proportionality.
11

12. Mathematical Modelling Of Steam
Turbine Unit
𝐻𝐻𝑃(𝑠) =
1
𝑇𝐻𝑃𝑠+1
…………………………… (7)
𝐻𝑅𝐻(𝑠) =
1
𝑇𝑅𝐻𝑠
…………………………….... (8)
𝐻𝐿𝑃(𝑠) =
1
𝑇𝐿𝑃𝑠+1
……………………………. (9)
12

13. Block Diagram Of Steam Turbine
Configuration
Fig. Steam turbine configuration
13

14. Block Diagram Of Steam Turbine
Configuration
14
Fig. Block diagram of steam turbine configuration

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