The mechanism of the transfer of shaft torque (or power) to the fluid flowing within the
impeller is fundamentally dynamic; that is, it is connected with changes in fluid velocity.
This requires the introduction of Newton’s second law, which when combined with the first
law of thermodynamics, yields Euler’s Pump Equation. Fluid velocities at inlet and exit of
the impeller are fundamental to this development. Fluid flowing along the blades of an
impeller rotating at angular velocity Ω and viewed in the rotating reference frame of that

impeller has relative velocity W. Vectorially adding W to impeller blade speed U = Ωr
yields the absolute velocity V, as shown in the velocity diagrams of Figure 3.

Newton’s Second Law for Moments of Forces and Euler’s Pump Equation   Relating
impeller torque T to fluid angular momentum per unit mass rV_θ is the convenient way
of applying Newton’s second law to centrifugal pumps. This is stated as follows for the
control volume V that contains the pump impeller:

(12)

where ΣT = T_S- T_D is the summation of torques acting on the impeller; namely, the net
torque T_I acting on the fluid flowing through it. The volume integral (first term on the
right side) of Eq. 12 is the unsteady term, which is zero for steady operation. It comes into
play during changing or transient conditions, such as start-up and shutdown; that is, when
the angular momentum per unit volume ρrV_θ is changing with time within the impeller
control volume V.

The surface integral (second term on the right hand side) of Eq. 12 is the one that
pump designers and users are mainly concerned with. Its integration over the exterior
surface S of the control volume V is effectively accomplished for most impellers by combining
one-dimensional results from inlet to outlet on each of several stream surfaces—
imagined to be nested surfaces of revolution bounded by the hub and shroud stream surfaces
(indicated in Figure 2). Insight into the significance of this term can be gained by
taking the mean value of the integrand in terms of the velocities on a representative
stream surface; that is, essentially the surface of revolution lying at an appropriate mean
location between hub and shroud. Each of the two velocity diagrams of Figure 3 lies in a
plane tangent to this mean stream surface. For flow through an impeller, the torque delivered
to the fluid is therefore given by the following relationship involving these average
quantities:

(13)

or × Ω:

(14)

Eq. 13 says that the torque is equal to the mass flow rate times the change of angular
momentum per unit mass Δ(rV_θ). This becomes the “power” statement of Eq. 14 when both
sides are multiplied by Ω. Following the statement of the second law of thermodynamics
in Eq. 4, we now can similarly say that gΔH must be less than the power input to the fluid
per unit of mass flow rate, namely Δ(UV_θ) from Eq. 14. So, we now arrive at Euler’s Pump
Equation—expressed three different ways as follows:

or

(15c)

The inequality (Eq. 15a) is quantified by Eq. 15b, which follows in view of Eq. 7. Eq. 15c
then follows from the definition of hydraulic efficiency (Eq. 10). Euler’s Pump Equation
makes one of the most profound statements in the field of engineering, because it determines
the major geometrical features of the design of a rotodynamic machine. By reversing
the inequality in Eq. 15a, the same principle applies to turbines; hence, the more
encompassing title, “Euler’s Pump and Turbine Equation.”

So, to design or analyze a pump, one needs to a) obtain the velocity diagrams that will
produce the ideal head at the design flow rate and b) determine how the shape of these
diagrams affects the hydraulic efficiency η_HY, so as to obtain the desired pump stage head.
Step (a) for a given pump is a simple one-dimensional exercise that utilizes the principles
of continuity and kinematics (Eqs. 16 and 17) to construct the velocity diagrams for a given
total impeller volume flow rate Q and pump rotative speed (Ω or N):

Continuity:

(16)

where W=V_m/sin βf

Kinematics:

(17)

Step (b) is in essence the evaluation of the hydraulic losses ΣH_L in Eq. 10, which
depend mainly on the relative and absolute velocities, the associated flow passage dimensions,
and incidence angles. Eq. 15c then gives the head that the pump stage will generate.
Performing steps (a) and (b) at several other flow rates at the same speed enables one
to develop the pump performance characteristics.

The mechanism of the transfer of shaft torque (or power) to the fluid flowing within the
impeller is fundamentally dynamic; that is, it is connected with changes in fluid velocity.
This requires the introduction of Newton’s second law, which when combined with the first
law of thermodynamics, yields Euler’s Pump Equation. Fluid velocities at inlet and exit of
the impeller are fundamental to this development. Fluid flowing along the blades of an
impeller rotating at angular velocity Ω and viewed in the rotating reference frame of that

impeller has relative velocity W. Vectorially adding W to impeller blade speed U = Ωr
yields the absolute velocity V, as shown in the velocity diagrams of Figure 3.

Newton’s Second Law for Moments of Forces and Euler’s Pump Equation   Relating
impeller torque T to fluid angular momentum per unit mass rV_θ is the convenient way
of applying Newton’s second law to centrifugal pumps. This is stated as follows for the
control volume V that contains the pump impeller:

(12)

where ΣT =