System Analysis

4.2.1 The Inductance Pipe

The fluid inductance pipe is essentially a short length of pipe joining the outlet of the pump to the junction before the E-OWV. The flow through the inductance pipe may be described by considering the forces across its length. Assuming the pipe to be frictionless, the force across the pipe may be represented by Eqn. 4-3, where the

53 acceleration term in the equation may be expressed in terms of its change in velocity with relation to time.

Eqn. 4-3

Considering that and also that the mass of fluid flowing through the pipe may be expressed using the mass-density equation, , then Eqn. 4-3 may be replaced with Eqn. 4-4.

Eqn. 4-4

Within this equation the inductance of the pipe, L, has in previous works [64] been shown in the form outlined by Eqn. 4-5. By substituting into Eqn. 4-4 and rearranging, a secondary formula for the inductance in the pipe can be achieved:

Eqn. 4-5

Eqn. 4-6

This is analogous to the electrical equation for inductance in a circuit, again defined and highlighted by Short [64] and given by Eqn. 4-7:

Eqn. 4-7

Such a relationship is useful when determining the appropriate length to use in the inductance pipe as it allows the inductance value to be fixed for specific values of valve opening times, which are discussed in more detail later in this section.

54 Alternatively Eqn. 4-6 may be defined and proven by using Euler’s equation, assuming that the flow through the pipe is frictionless and one-dimensional [105].

Eqn. 4-8

As is defined in Section 4.3 of this chapter, the change in height across the ECV and induced flow components of the piping system may be neglected. As such,

Eqn. 4-9 is produced, which may be rearranged and written in the form described by

Eqn. 4-10. Eqn. 4-9 Eqn. 4-10 where:

From inspection of Eqn. 4-10 it is clear that by taking the term in the equation to represent the pipe length traversed by the fluid over time, , the equation may be rearranged to produce the same form as Eqn. 4-6. By using the same inductance formula as before, Eqn. 4-4 is again obtained. This equation will become more useful in defining the relationship between the flows across different points in the linear analysis performed in Chapter 5.

4.2.2 The Capacitance Chamber

The capacitance chamber is a component which houses a means of providing a fluid softness, , or degree of compressibility. This could take the form of an

55 arrangement of compressible material housed within the chamber, a compressible fluid cavity or even a mechanical compressibility. In the case of the experimental rig described in Chapter 8, a set of compressible rubber ovoids within a rigid chamber was selected. Regardless of the means, this ‘accumulator’ acts as a fluid capacitance which allows pumping energy to be stored within the system, much in the same way as an electrical capacitor stores energy within an electrical circuit. Using this electrical analogy it is possible to relate the capacitance of the fluid system to the inductance through the natural frequency of the system:

Eqn. 4-11

where the inductance value, , is as described in Eqn. 4-5 and the capacitance, , can be described as the inverse of hydraulic stiffness, or ‘softness’, of the fluid accumulator,

.

Further equations to describe the relationship between pressure and flow in the capacitance chamber have been previously developed and used by Short [64]. These equations make use of Eqn. 4-12, where it is assumed that if the capacitance chamber is always completely flooded with water such that and a linear relationship may be used to describe how the volume of the chamber changes with respect pressure (NB the change in volume is proportional to the change in pressure, due to the fact that as the pressure in the capacitance chamber increases, the ‘fluid

softness’ in the chamber will be compressed and hence the volume of water in the

chamber will increase also. For this reason, the index value, , is used to ensure that the hydraulic stiffness, , remains a constant).

56

Eqn. 4-12

The volume of water in the chamber may be shown to be equal to the integral over time of the flow into the chamber [64], such that:

Eqn. 4-13

Further, it may be shown that this equation is equivalent to Eqn. 4-14 through rearrangement.

Eqn. 4-14

Thus, the pressure in the capacitance chamber is equal to the integral of the flow through the chamber, with respect to time, multiplied by the hydraulic stiffness of the material or mechanisms used within the chamber. Much like the relationship demonstrated previously in Eqn. 4-6, Eqn. 4-14 is of much use in relating the pressure and flow between various points through the system, after the OWV section. This is covered in the linear analysis presented in Chapter 5.

4.1.3 The Electronic Control Valve (ECV)

The ECV is an electronically operated one-way valve which is capable of high frequency operation. It is used to create oscillations in the flow through the pumping system so as to make use of the induced flow subsystems and control the effective static pressure head acting on the centrifugal pump – allowing it to continuously operate near to its BEP (analogous to the operating of the waste-water valve from the hydraulic ram pumps discussed in Chapter 3, or the electronic switch in a boost- converter). The valve is operated by a control circuit and feeds directly back into the borehole water source at times when the valve is open, acting to ‘short circuit’ the flow

57 and hence briefly reduce the effective pressure head acting on the pump. At periods when the valve is closed, the water is allowed to pass freely through the rest of the system to the discharge pipe and pump outlet. Through varying the duty and frequency ratio at which this valve operates, it should be theoretically possible to sustain an effective operating head of the centrifugal pump close to its the BEP.