8 Liquid flow

Liquid flow

Liquid flow explains the flow characteristics in pipe system of a pump installation. The process of calculation and the necessary data for quantifying flow losses are reviewed for Newtonian liquids, suspensions and liquids with solid particles suspended in water.

The processes of flow occurring in nature are usually very complex and difficult to deal with. In a good many technical applications, however, perfectly acceptable results can be obtained from calculations based upon simplified considerations. Some of the concepts and conditions germane to this context are discussed below.

A flow process is steady if all flow parameters, pressure, velocity, etc. at any specific point in the flow field are independent of time. According to this definition practically all flow processes are unsteady. In practice, many flow processes can, with sufficient accuracy, be treated as steady using suitable mean time values. The basic equations presented in later sections are valid for steady flow.

Flow is generally three-dimensional, i.e. the flow parameters vary with all three coordinates defining some point in space. In a good many technical cases of flow the number of dimensions studied may be reduced whilst still maintaining accuracy. A common example is pipe flow, where the parameters are assumed to vary in only one direction, i.e. the longitudinal dimension of the pipe. One-dimensional flow assumes that flow parameters can be described by mean values across the flow through-section. In principle different mean values should be applied when studying continuity, momentum and energy. In the case of pipe flow, the mean velocity in the pipe is defined as the volume flow divided by the cross-sectional area of the pipe (v = Q/A). This mean velocity can as a rule be used with adequate accuracy for most purposes.

An important property of a flowing medium is its density and the changes in density which occur during flow. A gas is compressed, i.e. the density increases, when the pressure somewhere in the flow increases. This type of flow is called compressible. In a liquid density variations are very small under great changes of pressure. Liquid flow can thus, with sufficient accuracy, be treated as incompressible. This is also the case with gases at low flow velocities when the pressure fluctuations are insignificant.

A streamline is a curve to which the velocity vector is tangential at any point. In the case of steady flow, the streamlines remain unchanged with time and describe the path of a liquid particle passing through the flow field. The streamlines through all points on a closed curve in the flow field constitute a flow tube. No mass will pass through the boundary surface of a flow tube. The flow tube is thus reminiscent of an ordinary’ pipe. In the case of a pipe, however, there are strong frictional effects associated with the wall of the pipe which do not necessarily occur in a flow tube.

Figure 8a Streamline and flow tube

The continuity equation

The continuity equation is an expression for the condition that mass is not created or destroyed during a flow process.

Figure 8b Example of one-dimensional flow.

Assuming that the flow is steady, the mass flow ṁ must be of equal magnitude everywhere along the pipe or the flow tube. In the case of one-dimensional flow in figure 8b,

ṁ = ρ_I * v_I * A_I = ρ_II * v_II * A_II      (Equ. 8a)
or for an incompressible liquid flow,
Q =v_I * A_I = v_II * A_II         (Equ. 8b)

where
Q = volume flow (m³/s)
v = Q/A = flow velocity (m/s)
A = cross-sectional area (m²)

When the cross-sectional area in a pipe reduces, then, to the continuity equation, the flow velocity increases. If the area is halved, the velocity doubles and so on.

Figure 8c Branching

Since there is no increase in mass at the point of branching, the mass flow entering will equal the total mass discharging per unit of time. Using the symbols of figure 8c.

Q_I = Q_II + Q_III      (Equ. 8c)

or

v_I * A_I = v_II * A_II + v_III * A_III      (Equ. 8d)

Bernoulli’s equation

Bernoulli’s equation is an equation of motion. It is an extension of Newton’s second law (force = mass x acceleration). Bernoulli’s equation thus applies regardless of whether or not heat is added during the process. Bernoulli’s equation, for steady, one-dimensional and incompressible flow between stations I and II, becomes:

where

p = pressure (Pa)
ρ = density (kg/m³)
v = flow velocity (m/s)
g = acceleration due to gravity 9,806 (m/s²)
h = height above horizontal datum reference (m)
Δ_pf = pressure loss (Pa)

Figure 8d Symbols used in Bernoulli’s equation

The term ρ v²/2 is called the dynamic or kinetic pressure and is sometimes combined with the static pressure p to give the total pressure (stagnation pressure) p_o.

Total pressure = static pressure = dynamic pressure

p_o = p + ρ v²/2 = p + p_dyn       (Equ. 8f)

In the case of a loss-free (Δp_f = 0) and horizontal (h_I = h_II) flow, the total pressure remains unchanged. If the dynamic pressure (the velocity) increases, the static pressure will decrease correspondingly. The velocity in a pipe increases as the area (the diameter) reduces.

Figure 8e Loss-free, horizontal pipe flow.

Total and static pressure are measured in different ways. A pressure tap at right angles to the direction of flow will sense the static pressure. Ahead of a pitot tube the velocity is reduced and the static pressure rises. At the pitot tube opening, the velocity is equal to zero. The pitot tube thus senses the total pressure in the flow.

As a rule, it is easy to measure the total pressure, whereas the static pressure can easily be effected by measurement errors. A Prandtl tube measures both total and static pressure. The differential pressure, the dynamic pressure, ρ v²/2 is obtained directly.

Figure 8f Measuring of static, total and dynamic pressure.

Considering the remaining terms in Bernoulli’s equation, ρ g h corresponds to the static pressure due to a column of liquid of height h. The term Δp_f is the drop in pressure (pressure drop) caused by flow losses due to friction.

In pump technology it is practical to express the Bernoulli equation in terms of head, i.e. meters column of liquid. If all terms in Bernoulli’s equation are divided through by ρg we get:

The various terms are then called:

p/ρg = static head (m)
v²/2g = velocity head (m)
h = height above a horizontal datum (m)
h_f = head loss (Pa)

Since all the terms in Equation 8g refer to height they are easy to illustrate graphically.

Figure 8g Graphic illustration of Bernoulli’s equation.