Hydraulics or fluid dynamics has the primary influence on the geometry of a rotodynamic
pump stage—of all the engineering disciplines involved in the design of the machine. It is
basic to the energy transfer or pumping process. Staging is also influenced by the other
disciplines, especially in high-energy pumps. The basic energy transfer relationships need
to be thoroughly understood to achieve a credible design and to understand the operation
of these machines. Action of the mechanical input shaft power to effect an increase in the
of energy of the pumpage is governed by the first law of thermodynamics. Realization of
that energy in terms of pump pressure rise or head involves losses and the second law of
thermodynamics.

The First Law of Thermodynamics   Fluid flow, whether liquid or gas, through a centrifugal
pump is essentially adiabatic, heat transfer being negligible in comparison to the
other forms of energy involved in the energy transfer process. (Yet, even if the process were
not adiabatic, the density of a liquid is only weakly dependent on temperature.) Further,
while the delivery of energy to fluid by rotating blades is inherently unsteady (varying
pressure from blade to blade as viewed in an absolute reference frame), the flow across
the boundaries of a control volume surrounding the pump is essentially steady, and the
first law of thermodynamics for the pump can be expressed in the form of the adiabatic
steady-flow energy equation (Eq. 1) as follows:

where

(1)

Here, shaft power P_s is transformed into fluid power, which is the mass flow rate times
the change in the total enthalpy (which includes static enthalpy, velocity energy per unit
mass, and potential energy due to elevation in a gravitational field that produces acceleration
at rate g) from inlet to outlet of the control volume (Figure 1).

When dealing with essentially incompressible liquids, the shaft power is commonly
expressed in terms of “head” and mass flow rate, as in Eq. 2:

(2)

where

(3)

The change in H is called the “head” ΔH of the pump; and, because H (Eq. 3) includes the
velocity headV²/2g and the elevation headZ_e at the point of interest, ΔH is often called the
“total dynamic head.” ΔH is often abbreviated to simply “H” and is the increase in height
of a column of liquid that the pump would create if the static pressure head p/pg and the
velocity head V²/2g were converted without loss into elevation head Z_e at their respective
locations at the inlet to and outlet from the control volume; that is, both upstream and
downstream of the pump.

The Second Law of Thermodynamics:Losses and Efficiency  As can be seen from
Eq. 2, not all of the mechanical input energy per unit mass (that is, the shaft power per
unit of mass flow rate) ends up as useful pump output energy per unit mass gΔH. Rather,
losses produce an internal energy increase Δu (accompanied by a temperature increase)
in addition to that due to any heat transfer into the control volume. This fact is due to the
second law of thermodynamics and is expressed for pumps in Eq 4:

(4)

where

The losses in the pump are quantified by the overall efficiency η, which must be less
than unity and is expressed in Eq. 5:

(5)

It should be pointed out here that real liquids undergo some compression—which is
accompanied by a reversible increase in the temperature ΔT_c of the liquid—called the
“heat of compression.” This portion of the actual total temperature rise ΔT is in addition
to that arising from losses and must therefore be taken into account when determining
efficiency from measurements of the temperature rise of the pumpage.¹ See the discussion
on this subject in Section 2.1.3.

To pinpoint the losses, it is convenient to deal with them in terms of “component
efficiencies.” For the typical shrouded- or closed-impeller pump shown in Figure 2, Eq. 5
can be rewritten as follows:

(6)

Noting that

		(7)
and

one may rewrite Eq. 6 as follows

(8)

where

Approximate formulas for the three component efficiencies of Eq. 8 will be given further
on. Their product yields the overall pump efficiency as defined in Eq. 5, and reflects
the following division of the pump losses:

1. External drags on the rotating element due to i) bearings, ii) seals, and iii) fluid friction
   on the outside surfaces of the impeller shrouds—called “disk friction”; the total
   being P_D = P_S  - P_I. Generally, the major component of P_D is the disk friction, and the
   “mechanical efficiency” is that portion of the shaft power that is delivered to the fluid
   flowing through the impeller passages.
2. Hydraulic losses in the main flow passages of the pump; namely, inlet branch,
   impeller, diffuser or volute, return passages in multistage pumps, and outlet branch.
   The energy loss per unit mass is gΣH_L= g(H_i  - ΔH), the ratio of output head ΔH to
   the input head H_i being the hydraulic efficiency. This is the major focus of the designer
   for typical centrifugal pump geometries (which are associated with normal “specific
   speeds”—to be defined later). The other two component efficiencies are then quite high
   and of relatively little consequence.
3. External leakages totaling Q_L leaking past the impeller and back into the inlet eye.
   This leakage has received its share of the full amount of power P_I = ρg ΔH_i(Q + Q_L)
   delivered to all the fluid (Q + Q_L) passing through the impeller. This leakage power
   is P_L = ρg ΔH_iQ_L, which is lost as this fluid leaks back to the impeller inlet. The
   remaining fluid input power is thus (P_I  - P_L) = ρg ΔHiQ, the ratio of this power to
   the total (P_I) being the volumetric efficiency.

There are exceptions to this convenient model for dividing up pump losses. The main
exception is that if the pump has an open impeller, that is, one without either or both
shrouds, that portion of the total leakage Q_L disappears. The leakage now occurs across the
blade tips and affects the main flow passage hydraulic losses. The volumetric efficiency
is now higher, but the hydraulic efficiency is lower. In that case disk friction is still present,
as the impeller still has to drag fluid along the adjacent stationary wall(s). Another
exception—for closed impellers—is that disk friction is fundamentally an inefficient
pumping action, the fluid being flung radially outward²; and this can result in a slight
increase in pump head if the fluid on the outside of an impeller shroud or disk is pumped
into the main flow downstream of the impeller.

Hydraulics or fluid dynamics has the primary influence on the geometry of a rotodynamic
pump stage—of all the engineering disciplines involved in the design of the machine. It is
basic to the energy transfer or pumping process. Staging is also influenced by the other
disciplines, especially in high-energy pumps. The basic energy transfer relationships need
to be thoroughly understood to achieve a credible design and to understand the operation
of these machines. Action of the mechanical input shaft power to effect an increase in the
of energy of the pumpage is governed by the first law of thermodynamics. Realization of
that energy in terms of pump pressure rise or head involves losses and the second law of
thermodynamics.

The First Law of Thermodynamics   Fluid flow, whether liquid or gas, through a centrifugal
pump is essentially adiabatic, heat transfer being negligible in comparison to the
other forms of energy involved in the energy transfer process. (Yet, even if the process were
not adiabatic, the density of a liquid is only weakly dependent on temperature.) Further,
while the delivery of energy to fluid by rotating blades is inherently unsteady (varying
pressure from blade to blade as viewed in an...