III. Theoretical Underpinnings
The total developed head (TDH) of an individual pump is given in equation (i) below. Total developed head is fundamentally a measure of the total output energy of a pump. The pressure term and elevation term are the potential energy components and the velocity term is the kinetic energy component. By adding all three terms, the total energy is obtained.
where P = pressure,
r = density of water,
Z = change in elevation,
V = flow velocity, and
gc = gravitational constant to units conversion.
The relationship between volumetric flow and flow velocity for water in a pipe is given by the basic continuity equation for incompressible fluid flow given in equation (ii).
where Q = volumetric flow (i.e., volume of flow per unit of time),
V = flow velocity, and
A = cross sectional area of the pipe.
Substituting equation (ii) into equation (i) allows us to directly relate TDH to volumetric flow in the following way.
If the TDH remains constant within a particular operating range and the elevation (the "Z" term) does not change, equation (iii) indicates that when pressure goes down, the volumetric flow must increase. Likewise, when the volumetric flow decreases, the pressure must increase. By taking the differential of equation (iii), the exact relationship between pressure and volumetric flow when TDH is a constant can be determined.
(iv) TDH = P1/r + Q2/(A2)(2gc) + Z
0 = [1/r][dP/dQ] + [1/(A2)(gc)][Q]
dP/dQ = - [r/(A2)(gc)][Q] = - [K]Q
where K = a constant = [r/(A2)(gc)]
To relate pump motor energy input to pump output, the following relationship holds.
where E = electrical energy input to the pump motor,
L = losses due to friction and heat in both the pump and motor,
dm/dt = mass flow rate of water through the pump
Of course, equation (iv) is basically a re-statement of the Conservation of Energy Law as it applies to pumps driven by electric motors.
Assuming for a particular operating range that the normal losses from heat and friction in both the pump and the motor are simply a constant percentage of the electrical energy input, then equation (iv) can be re-written as follows.
where h = overall mechanical efficiency of the motor and pump combination.
The electrical energy input itself is composed of several parameters. Typically when a three phase motor is used to drive the pump, those parameters are as follows.
Where Irms = measured root mean squared current value,
Vrms = measured root mean squared voltage value,
1.732 = factor to account for three-phase power, and
PF = power factor of motor.
Substituting equation (vi) into equation (v) gives the following relationship that relates pump output pressure, flow output and electrical consumption of the pump motor.
= [P1/r + Q2/(A2)(2gc) + Z][dm/dt]
When the impeller clearance of a vertical pump is out of its recommended range, it creates energy losses in addition to the normal losses captured in the "h" term. This addition loss of energy is accounted for in equation (viii), which follows, by the addition of the extra term, "Lclearance" on the left hand side of the equation.
where Lclearance = energy losses associated with an out-of-range clearance.
At Cooper Nuclear Station, the vertical pump motors are the synchronous type. They maintain nearly constant RPM’s, and will, within certain limits, maintain constant flow output unless throttling is done.
As indicated by inspection of equation (viii), if the clearance losses in a vertical pump are significant, at a given Total Developed Head, these losses will cause a corresponding loss in the output pressure of the pump. Likewise, if a lift adjustment is made and the clearance losses are minimized (i.e., Lclearance ~ 0), an increase in output pressure will be realized.
Thus, anytime the lift clearance on a particular pump is too tight or too loose, output pressure improvements can be made by resetting the lift so that it is within the recommended range.