This paper presents a new incompressible single-phase model for ESP's head performance. Sachdeva (1988) and Cooper (1966) developed models for ESP channels [1, 2] and for inducers [3], respectively. The model presented in this paper is based on one-dimensional approximation along an ESP channel. The new derived pressure ODE (Ordinary Differential Equation) for frictionless incompressible flow is consistent with the pump Euler equation. New models for pump frictional and shock losses have been proposed. Finally, a comparison between the predicted pump performance and the pump performances from Affinity Law for different rotational speeds is presented. The single-phase model can predict ESP performance under different fluid viscosities and also is the basis of gas-liquid model for ESP's head performance.


This paper presents the new single-phase model developed for the prediction of an ESP's performance. The correct ESP head performance is critical for the appropriate design, simulation and troubleshooting of an ESP installation. The model consists of the mass and momentum equations, based on the streamline approach or one-dimensional assumption. In the momentum equations, the calculation of the friction factor proposed by Sachdeva, is improved by incorporating the channel curvature, channel rotation, and channel cross-section effects. A new shock loss model including rotational speeds has been proposed. The new single-phase model is capable of predicting the pump performance for different rotational speeds and for different viscosities.

Literature Review

Sachdeva (1988, 1994) derived the frictionless pressure ODE under incompressible single-phase flow as follows,

  • Equation 1

where p is pressure, ? is angular velocity, r is radius along the channel, ?l is liquid density, and Vr is the radial component of absolute velocity.

Sachdeva's previous equation needs to be extended for any blade angle in an ESP.

The frictionless pressure ODE given by Cooper (1966) for an inducer [3],

  • Equation 2

where W is the relative velocity.

Different investigators such as Sachdeva and Cooper superimposed frictional losses into their frictionless pressure ODE.

The friction factor in Sachdeva's frictional loss model considers the effects of curvature for the diffuser and both curvature and rotational speed for the impeller.

Sachdeva's approach for friction factor calculation is very important to model pump performance. His approach assumed smooth surface and turbulent flow regime inside ESP channels.

Mass Balance Equation

The derivation of the one-dimensional mass balance equation [4] yields the following expression along the channel in an impeller or diffuser.

  • Equation 3

where ß is the geometric blade angle as shown in Fig.1, s is the streamline coordinate, which is the distance between the entrance to any location along the channel, and t is time. The streamline in this one-dimensional model is at the center of the channel and has the same shape as its two blades.

For steady-state incompressible liquid flow along the ESP channel, the relative velocity W can be expressed as,

  • Equation 4

where Ql is the liquid flow rate and H is the channel height.

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