Additional detail and derivations of the equations shown in the lecture can be found in the text book.

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Flow is the movement of particles through a reference cross section. In order to measure flow, we therefore need to define a cross section and then observe or measure the movement of particles through that cross section. In this lecture we will describe techniques to measure flow. The variables we want to measure will be the flow velocity, the pressure in the flow and the temperature.

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It is important to remind yourself of the origin of pressure. It is the molecular motion of the fluid molecules bouncing against a fixed surface. When we measure pressure, we measure the average kinetic energy imparted by the molecules against the boundary. If we measure the pressure against a vacuum, we refer to it as the absolute pressure, which is always positive.

But often we measure pressure against the atmospheric pressure or another reference pressure. In that cause we refer to gage pressure, which is the difference between these two pressure sources. Therefore gage pressure can be negative, which simply means that the measured pressure is lower than the reference pressure when measured on an absolute scale.

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There are many types of pressure sensors based on different mechanisms. All of them make use of the fact that the pressure induces a load or deformation on the sensor thus changing its properties. We have a capacitance sensor in the lab (shown with two ports for high and low pressure). As the pressure changes, the separation between the two conducting plates is changed, leading to a change in capacitance, which can be related to the pressure difference between the high and low ports.

The pressure sensor used in the lab converts the differential pressure (high – low) into a voltage proportional to the pressure. This voltage is then digitized by the DAQ system.

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The first conservation law that we can apply to flow is the conservation of mass. Specifically, for a duct in the absence of ports in the duct, in a steady flow all the mass going into the duct has to come out. Otherwise, it would have to accumulate in the duct, which of course would not be a steady condition.

The mass flow rate is therefore constant. It can be determined from the integral across a duct cross section of the flow density times the flow velocity. If there is no appreciable density change in the flow (incompressible flow or flow with low Mach number), we can simplify the conversation principle to volume flow conservation, which is the product of the average flow velocity and cross sectional area.

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In our experiment we plan to determine the volume flow. This requires calculating the integral, which can be approximated by a sum over the flow velocities measured at multiple locations times their associated areas. The strategy that we will use in the lab is to measure the flow velocity at specific locations so that the area associated with that location is the same for each measurement point. The table in the slide shows, where to measure the flow velocity in the duct so that the area for each “slice” is the same. Given those specific measurement locations, the average flow velocity is simply the average of the measured velocities at the measurementflow velocity is simply the average of the measured velocities at the measurement points.

In the lab we will measure the flow velocity at 12 locations, 6 referenced from the top of the duct and 6 referenced from the bottom. What would you expect the flow profile to look like? Would you expect the flow velocity of measurement points symmetric with respect to the center of the duct to be different?symmetric with respect to the center of the duct to be different?

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The Bernoulli Equation is another conservation law, the conservation of momentum of the flow. It is derived based on a steady state flow of an ideal fluid and incompressible flow (or compressible flow with low Mach Number).

It is important to understand the implications of the Bernoulli equation. Since the sum of the potential (pressure and elevation) and kinetic energy (velocity) of the flow is constant along a stream-line, it can be used to predict the change in pressure or flow velocity at a location 1 from the known conditions at location 2. This is exploited in measurement devices.

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The Pitot Static (P-S) tube is a measurement device that was developed by Henri Pitot in the 18th century to determine the flow velocity in a fluid. The physical principle of how it works is based on the Bernoulli Equation. The idea is to slow down the flow to stagnation (zero velocity) at the front of the tube. Because of the conservation of momentum, the pressure at the tip of the tube (point 2) rises to the stagnation pressure, which can be measured through the hollow tube. A secondary tube surrounding the first tube measures the pressure in the flow as it flows past the tube This pressure is called the static pressure of the flow and is the same as thetube. This pressure is called the static pressure of the flow and is the same as the pressure upstream of the flow (point 1) (Why is this true? The flow velocity going past the tube unimpeded is the same flow velocity as upstream so the pressure has to be the same as well)

Since two variables are measured (p2, the stagnation pressure and p1, the static pressure) Bernoulli’s equation can be solved for v1 the flow velocity upstream Itpressure), Bernoulli s equation can be solved for v1, the flow velocity upstream. It is a function of the pressure difference and the fluid density.

One common application of P-S tubes is to measure airspeed in aircrafts.

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Because of the assumption of stagnation of the airflow, the alignment of the tube into the flow is important. Errors occur, when the tube is misaligned or the underlying assumptions of the Bernoulli equation no longer hold (incompressibility, inviscid).

The slide shows an example of the P-S tube used in the lab. Two hoses are shown coming off the tube, the top hose measures the stagnation pressure, the 2nd one the static pressure.

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Bernoulli’s equation can also be used to estimate the flow rate based on the pressure change as a flow squeezes through an obstruction in a duct. The flow velocity and hence the flow rate is related to the change in pressure before and after the obstruction and therefore a measurement of the pressure drop across the obstruction can be used to determine the overall flow rate. This is the idea behind an obstruction flow meter as shown on the next slide.

Pay close attention to the flow shown in the slide. The flow profile is symmetric upstream and the flow velocity is zero at the boundaries. It speeds up noticably as it squeezes through the orifice. This change in speed causes the static pressure to drop. This can be observed by measuring the static pressure just upstream and downstream of the orifice. There is also a backwash region, where flow is reversed.

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For an obstruction flow meter the flow rate can be determined from the equation shown above. It depends on the cross section area of the orifice the fluid is flowing through, the pressure drop across the orifice and an orifice flow coefficient, which is a function of the duct and orifice geometry and the Reynolds number. The flow coefficient has been determined experimentally for multiple duct geometries and can be obtained from a lookup table (see the slide).

Note that in order to lookup the flow coefficient, the Reynolds number needs to be known but it depends on the flow velocity. Therefore an iteration is required, first a Reynolds number is assumed, then the flow velocity is calculated and the Reynolds number is corrected and the calculations are rechecked. One or two iterations are usually sufficient.

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The static pressure drop in the flow of a duct will continuously drop due to friction losses in the flow. Also there are nonrecoverable “form” losses due to obstructions such as the orifice (backwash,…). These losses are not predicted by the Bernoulli equation, as it assumes conservation of momentum (no frictional losses).

However, the losses can be observed by measuring the static pressure along the duct. In order to determine the frictional losses away from the obstruction, the pressure drop per unit duct length can be measured upstream or downstream from the obstruction. Using the equation given above the wall friction coefficent can then be determined. Students will be asked to calculate the wall friction coefficient from static pressure drop data measured in the lab and compare it to text book values.

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