M15.2 Flowmeters

An orifice plate meter is the most common flow measurement device. A constricting orifice is placed between two flanges, as shown in Figure M15-4. As the fluid "speeds up" through the orifice, the pressure drops. The volumetric flow rate is related to the pressure drop across the orifice. As shown in the motivating example, a DP cell is typically used to measure the pressure drop; here, we use a manometer for illustrative purposes.

The equation for flow through an orifice is (McCabe and Smith, 1976)

Equation M15.5

where v_o is the average velocity through the orifice;

b is the ratio of orifice to pipe diameters, D_o/D_p;

g_c is Newton's law gravitational constant, 32.174 ft lb_m/lb_fs²; r is the fluid density;

C_o is the orifice coefficient (typically 0.61); and

P₁,P₂ are the upstream and downstream pressures.

The volumetric flow rate, F, is

Equation M15.6

so the volumetric flow rate [combining Equations (M15.5) and (M15.6)] is

Equation M15.7

which can be written

Equation M15.8

where

Equation M15.9

Notice that C₁ will be a constant for a given system. Notice also that an orifice plate has a nonlinear input-output relationship. Consider the input to be the volumetric flow rate. The output is the differential pressure. After rearranging Equation (M15.8) we find Equation (M15.10) and notice that the pressure drop is a function of the square of the flow rate,

Equation M15.10

This means that the orifice plate gain (pressure drop change/flow rate change) increases with increasing flow. The orifice plate gain is

Equation M15.11

and we see that if the flow rate doubles, then the orifice plate gain doubles. If a controller is tuned to obtain good control when the flow rate is low, there is a chance that the control loop will become unstable when the flow rate is high, because of the increase in overall control loop gain, which is due to the increasing orifice plate gain. A common way of handling this varying gain problem is to use a square root extractor.

Square Root Extractor

Assume that the input to a square root extractor is the pressure drop signal. The output is a 4- to 20-mA signal that is proportional to the square root of the pressure drop. Let P_op represent the signal entering the square root extractor and I represent the signal out of the square root extractor. The input-output relationship is then

Equation M15.12

where C₂ is a constant. The gain of the square root extractor is

Equation M15.13

Notice that the square root extractor gain is inversely proportional to the square root of the pressure drop. The purpose of a square root extractor can be seen by combining Equations (M15.12) and (M15.10) to yield

Equation M15.14

The gain between flow rate and the signal out of the square root extractor is

where . That is, we have a signal that is now directly proportional to the flow rate.