Turbine (P)#

The Turbine (P) block models the dynamics of hydraulic turbines used in hydroelectric power plants and other energy systems. The block allows analyzing the conversion of water flow energy into mechanical energy on the turbine shaft, which is a key stage in electricity generation.

Mass Conservation Law

The block conserves mass such that (1):

(1)#\[\dot m_A+ \dot m_B=0\]

Energy Conservation Law (2):

(2)#\[\phi_A+\phi_B+ P_{fluid}=0\]

where:

\(\phi_A\) - energy flow at the input, [W];

\(\phi_B\) - energy flow at the output, [W].

\(P_{fluid}=\dot m_A \cdot \Delta h_{total}\)

Adjusted Mass Flow Rate (3):

(3)#\[\dot m_A \cdot \sqrt{\frac{T_A}{T_{corr}}}=\dot m_{corr}\cdot \frac{p_A}{p_{corr}}\]

where \(T_{corr}, m_{corr}, p_{corr}\) — adjusted temperatures, mass flow rate, pressure.

Parameter adjustment depends on the selected turbine calculation type:

a) Analytical

Nominal values of temperature, pressure ratios, and isentropic Efficiency are specified. Adjusted variables are found (4), (5), (6), (7):

(4)#\[\dot m_{corr}=\dot m_{nom}\cdot \frac{\sqrt{1-(\frac{P_B}{P_A})^2}}{\sqrt{1-(\frac{1}{Pr_{nom}})^2}}\]

(5)#\[Pr_{corr}=\frac{P_A}{P_{ref}}\]

(6)#\[T_{corr}=T_{ref}\]

(7)#\[\eta_t=\eta_{nom}\]

where:

\(\dot m_{nom}\) - nominal mass flow rate;

\(Pr_{nom}\) - pressure ratio for the analytical calculation mode;

\(Pr_{corr}\) - adjusted pressure ratio;

\(T_{corr}\) - adjusted temperature;

\(\eta_t\) - Efficiency of the isentropic process equals the nominal value \(\eta_{nom}\). Efficiency is used to find the total enthalpy change \(\Delta h_{total}\).

b) Tabular from Pressure Ratios

Adjustment data is interpolated from vectors of pressure ratios, mass flow rates, and isentropic Efficiency from a single pressure ratio parameter.

\(\dot m_{corr}\) – is found from vectors of pressure ratios and mass flow rates (8), (9):

(8)#\[Pr_{corr}=\frac{P_A}{P_{ref}}\]

(9)#\[T_{corr}=T_{ref}\]

\(\eta_t\) of the isentropic process is found from vectors of Efficiency and pressure ratios.

c) Tabular from Pressure Ratios and Shaft Rotation Speed

Adjustment data is interpolated from tables of pressure ratios, mass flow rates, and isentropic Efficiency from two parameters: pressure ratios and shaft rotation speed.

1. Torque (10):#

(10)#\[\tau=\frac{\eta_m \cdot \dot m_A \cdot \Delta h_{total}}{\omega}\]

where:

\(\tau\) - turbine torque;

\(\eta_m\) - mechanical Efficiency;

\(\dot m_A\) - mass flow rate in the turbine;

\(\Delta h_{total}\) - total enthalpy change;

\(\omega\) – turbine angular velocity.

2. Properties#

Property	Default Value	Description	Units of measurement
Type of turbine calculation: Analytical; Tabulated data by pressure ratio	Analytical	-	-
Corrected temperature	288.15	-	К
Corrected pressure	101325	-	Pa
Mechanical efficiency	0.9	-	-
Inlet area port	1	-	m\(^2\)
Outlet area port	1	-	m\(^2\)
Pressure ratio (analytical), available when `Type of turbine calculation` - `Analytical`	2	-	-
Mass flow rate (analytical), available when `Type of turbine calculation` - `Analytical`	48	-	kg/s
Isentropic efficiency (analytical), available when `Type of turbine calculation` - `Analytical`	0.8	-	-
Pressure ratio vector, available when `Type of turbine calculation` - `Tabulated data by pressure ratio`	1.251;1.345;1.442;1.521;1.596;1.692; 1.849;1.972;2.1;2.173;2.259;2.3424;	-	-
Mass flow rate vector, available when `Type of turbine calculation` - `Tabulated data by pressure ratio`	39.439;43.01;45.556;46.801;47.792;48.388; 48.634;48.67;48.689;48.747;48.825;48.912;	-	kg/s
Isentropic efficiency vector, available when `Type of turbine calculation` - `Tabulated data by pressure ratio`	0.559;0.61;0.673;0.741;0.788;0.835; 0.869;0.895;0.908;0.899;0.854;0.751;	-	-

3. Inputs and outputs#

Parameters	Input/Output	Description	Units of measurement	Port type
Input	Input	-	-	Pneumatics
Shaft rotation speed	Input	Mechanics Input	rpm	Rotational motion
Output	Output	-	-	Pneumatics

In order to see the connection capabilities of the block, you need to go to page Description of block ports