Flow-rate measurement

The instantaneous flow-rate of a fluid, volumetric or mass flow-rate, can be determined by measuring the fluid’s local velocity at different distances from the pipe’s axis in order to establish a medium velocity for that section. This operation can be done by means presented earlier. Numerous methods allow a simplification of the operation based on the effects implied by the medium velocity measured at a given section. These means are called flow meters. The flow-rate is calculated by integrating the instantaneous flow-rate in a established period of time and some types of flow meters, called counter , are integrating the instantaneous using one of its components.

Metoda reducerii sectiunii

Devices with a constant flow section.

This method is the oldest one used for measuring the flow-rate of a given fluid. At a given point along the pipe a smaller section is placed in order to create a pressure drop. For a compressible fluid, in a isentropic process from section 1 to section 2 taking in consideration a pressure drop from p1 to p2 ,the volumetric flow-rate Is calculated using the following expression:

Qv=α∗A∗√ 2k∗p1k−1∗p1

ρ1∗√ ( p2 p1)

2/ k−(p2 p1)k+1 /k

1−β4 (p2 p1)2/ k =α∗ε∗A∗√ 2ρ1∗( p1−p2)

(11.34)

,where α represents the discharge coefficient; A represents the area of the given section; β the pipes diameter divided to the section’s diameter, β=d/D; ε expansion factor; By experimental activity it can be shown that α is influenced by the design and by Re number for the flowing state. And so the flow rate can be calculated by measuring the pressure drop p1- p2 (e.g. using a differential manometer with on a U tube) and the fluid’s density ( by measuring the pressure and the temperature) only if α is given. For incompressible fluids ε =1.

From the expression (11.34) it can be shown that the flow rate has a nonlinear dependence by the pressure drop, Qv→¿. Due to the fact that the measurement methods of the pressure drop offer a good precision, bigger than 10% of the whole scale, the minimum flow-rate that can be measured represents 30% from the maximum flow-rate.

In order to obtain a precise data the flowing state has to be a steady flowing state. It can be shown that for a non-steady flowing state by using the expression (11.34), for an average pressure drop p1−p2 , the flow-rate is overvalued.

The most common construction used to measure the flow rate is the orifice plate. The design and the method of mounting the pressure intake ports is standardized (fig. 11.31). For these design type of the orifice plate α , the discharge coefficient, is strictly dependent by the Re number and by β, the diameter’s ratio(fig 11.32).

In order to create a contraction point for measuring the flowing rate nozzles and Venturi tubes can be used(fig. 11.34). By comparing with orifice plate method is shown that the nozzles and the Venturi tubes offer a smaller pressure loss(fig. 11.35), but the flow rate coefficients have a greater value.

The rotameter. In opposite with the previously methods the rotameter the working principle implies a constant pressure drop and a variable flowing area. The flow-rate is measured by observing the float’s position in a glass tube in which the fluid is passing.

The float’s dead state is establish using 1) the gravity force of the float, G p= ρp∗g∗V p and the pressure force of the stream, F p=A f∗p2 2) pressure force of the stream, F p=A f∗p2 and the friction force of

the stream applied on the float F f=A I∗k∗wn. The pressure under the float is p1 and above the float is

p2, A f is the frontal area of the float and A I is the lateral area of the float, w-the flow’s velocity in the annular section, V p- the float’s volume, ρ I the density of the float’s material. Based on the continuity equation and the Bernoulli equation, and assuming the pressure drop p1−p2 constant the flow rate expression is:

Q=α1∗K∗A(h)(11.35) where α 1=α∗√g¿¿¿ (11.36) and K=√ 2∗g∗V p∗¿( ρv−ρ)

ρ∗A f¿ (11.37)

In the expressions above ρ is the fluid’s density, α is the discharge coefficient, -the float’s length, A is the section area trough which the fluid passes the float which is different trough the tube. The term α 1∗K is constant for a type of fluid if the Re number has a sufficient value. By making the tube in order to have different values for A at different heights , h, a linear expression can be obtained which connects the flow rate, Q and h. The machine is calibrated differently for different types of fluids. The measuring interval is 10:1, with a 1-2% error at the bottom of the scale and 0.5-0.9% at the upper end of the scale.

11.22 Electromagnetic flow meter

In a conductive liquid which flows through a magnetic field a electromotive force is induced in correspondence with the next expression:

U=B∗D∗w∗10−8∗V (11.38), where B is the magnetic induction, D- the length of the conductor, w-the conductor’s velocity which is direct with the medium flow velocity. The positive ions and the

negative ones are separated at opposite ends of the jet, and so the potential distribution from the fig11.37 is obtained.

The flow meters that are build based on these principle have a pipe from a non-conductive material and non-magnetic also, with two electrodes placed close to the tube’s wall where the difference of potencial is maximum. The magnetic field is in general alternative. The measured volumetric flow-rate is not influenced by the fluid’s density or by its viscosity and the velocity distribution trough the section has to be symmetric. The electric wiring of the electromagnetic flow meter has a reaction servo-system which allows