Euler's pump and turbine equation
The Euler’s pump and turbine equations are most fundamental equations in the field of turbo-machinery. These equations govern the power, efficiencies and other factors that contribute in the design of Turbo-machines thus making them very important. With the help of these equations the head developed by a pump and the head utilised by a turbine can be easily determined. As the name suggests these equations were formulated by Leonhard Euler in the eighteenth century. These equations can be derived from the moment of momentum equation when applied for a pump or a turbine.
Contents
The equations
Euler equation for a pump states that the work done on a fluid per unit mass or specific work is given by: \[ {\Delta } W = U_2 C_{\theta 2} - U_1 C_{\theta 1} \, \] Where U2 is the blade speed at the exit, U1 is the blade speed at the inlet, Cθ1 is the component of velocity mutually perpendicular to both the axis as well as the radius r at the inlet, and Cθ2 is the component of velocity mutually perpendicular to both the axis as well as the radius r at the outlet.
In case of a turbine, the equation simply modifies to
\[ {\Delta } W = U_1 C_{\theta 1} - U_2 C_{\theta 2} \, \]
Energy and momentum principle
Control volume approach can be used on a volume of fluid flowing through the machine.[1] When torque balance is applied to the moment of momentum equation that describes the relationship between mass flow, velocities and impeller radii. If torque and angular velocity are of same sign, work is being done on the fluid (e.g. a compressor). If torque and angular velocity are of opposite sign work is being extracted from the fluid (a turbine).
\[ T = \dot{m} ( v_2 C_{\theta 2} - v_1 C_{\theta 1} ) \, \]
Thus we have,
\[ T = \dot{m} ( {\omega } r_2 C_{\theta 2} - {\omega } r_1 C_{\theta 1} ) \, \]
where ω is the angular velocity of the system and Thus,
\[ {\Delta } W_{per\,unit\,\,mass} = U_2 C_{\theta 2} - U_1 C_{\theta 1} \, \]
The velocity triangle
A velocity triangle paves the way for a better understanding of the relationship between the various velocities. Very often, it is useful to rewrite the Euler's pump and turbine equation in terms of the relative and absolute velocities of the fluid and the velocity of the turbine blades at the inlet and outlet. This helps in analyzing the various static and dynamic components contributing to the head developed or utilized. In the adjacent figure we have:
\[V_1\,\] and \(V_2\,\) are the absolute velocities at the inlet and outlet respectively. \[V_{f 1}\,\] and \(V_{f 2}\,\) are the flow velocities at the inlet and outlet respectively. \[V_{w 1}\,\] and \(V_{w 2}\,\) are the swirl velocities at the inlet and outlet respectively. \[V_{r 1}\,\] and \(V_{r 2}\,\) are the relative velocities at the inlet and outlet respectively. \[U_1\,\] and \(U_2\,\) are the velocities of the blade at the inlet and outlet respectively. \[\alpha \] is the guide vane angle and \(\beta \) is the blade angle.
By applying the Euler’s pump equation here we obtain
\[\Delta W_{per\,unit\,mass}\, =\, U_2\, V_{w2}\, -\,U_1 \, V_{w1}\]
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From the inlet velocity triangle shown in the figure, by applying cosine rule:
\[\, V_{r1}^2\,=\,V_{1}^2\,+\,U_{1}^2\,-\,2\,U_1\,V_1\, \cos\alpha_1 \] \[\, V_{r1}^2\,=\,V_{1}^2\,+\,U_{1}^2\,-\,2\,U_1\, V_{w1}\] \[\, V_1\,U_{w1}\,=\, {1 \over 2} ( V_1^2 + U_1^2 -V_{r1}^2 ) \]
Similarly we have
\[\, V_2\,U_{w2}\,=\, {1 \over 2} ( V_2^2 + U_2^2 -V_{r2}^2 ) \]
Substituting these equations in Eqn. A we obtain[2]
\[ H = {1 \over {2g}} [ ( V_2^2 - V_1^2 ) + ( U_2^2 - U_1^2 ) + ( V_{r1 }^2 - V_{r2 }^2 ) ] \, \]
The term ( V22 - V12) represents the dynamic head. The exit velocity almost assumes a negligible value in many turbines and assumes a significant value in the case of pumps and compressors. This difference in Kinetic Energy causes a static pressure rise in a diffuser.
The term ( U22 - U12) represents the static head due to centrifugal force. This is the result of change in diameter of rotation of the fluid.
The term ( Vr12 - Vr22) represents the static head due to the velocity change through the impeller.
In case of a pelton turbine the static component of the head is zero, hence the equation reduces to
\[ H = {1 \over {2g}} ( V_1^2 - V_2^2 ) \, \]
Usage
Euler’s pump and turbine equations can be used to predict the impact of changing the impeller geometry on the head. Qualitative estimations can be made from the impeller geometry about the performance of the turbine/pump. For the design of an aero-engines and the designing of power plants, the equations assume prime significance. Thus for the design aspect of turbines and pumps, the Euler equations are extremely useful.
See also
References
