Lab #3

Experiment #1

Air Flow Rig

Fluid Mechanics Lab

Abstract

The fluid flow rates and pressure drops through various common piping constructs are all subject to Bernoulli’s Equation – energy is conserved (though viscosity and friction losses must be taken into account). Changes in pressure result in a change in velocity when the energy of the fluid is constant.

Introduction

Each of the devices in the air flow rig can be used for velocity and/or flow rate measurement in a pipe, and each one will impart its own level of pressure drop (energy loss) on the fluid.

Figure 1: Air Flow Rig

Velocity measurement using the pitot tube is based upon the fact that when a flow strikes the rounded tip of the tube, a stagnation point is created where velocity is zero. According to Bernoulli’s Equation, all of the velocity head is converted to pressure head – the pressure at this point is the maximum. Holes along the side of the tube measure the “ambient” pressure inside the pipe. The difference between the two pressures represents the velocity head which had been converted to pressure at the stagnation point.

(1) Pitot Tube Velocity

Equation 1 represents the velocity in terms of pressure difference – it is derived from Bernoulli’s Equation. The velocity profile for the pitot tube should appear to be low near the wall of the pipe (due to viscous forces slowing down the flow) and plateau as the tube is moved away from the wall of the pipe (due to turbulent flow the velocity is essentially constant except near the pipe walls).

(2) Volumetric Flow Rate through Orifice

Equation 2 represents flow through an orifice. Like a venturi, an orifice limits the diameter of the pipe and converts pressure head into velocity head. Also like a venturi, this does not happen in the real world without loss. A flow coefficient (C_f) is used to correct for the energy which is converted to heat due to viscous forces as the air flows through the orifice.

Elbows causes pressure drop due to the fact that the fluid must change direction. The fluid’s inertia attempts to resist such changes and causes pressure gradients in the elbow resulting in virtual flow constriction and turbulence which again cause energy to be dissipated. Equation 3 represents the derivation of the loss coefficient for an elbow assuming that the other variables are known. Equation 5 represents the flow rate through an elbow. A loss coefficient, K, is once again used to account for the energy losses.

(3)

Minor Loss Coefficient, used in determining head loss through fittings.

(4)

Rearrangement of Equation 3, determining velocity with a known constant, K.

(5)

Volumetric Flow Rate through 90 degree elbow.

Nozzles once again act as flow constrictions, very similar to a venturi. The pressure drop, difference in cross-sectional area, and a discharge coefficient to correct for loss are used to determine flow rate with a nozzle. The difference in cross-sectional areas of the inlet and outlet (throat) determines the flow rate through the nozzle with a given pressure drop. Equations 6 and 7 describe the mass flow rate and volumetric flow rates through a nozzle, respectively.

(6) Mass flow rate through nozzle

(7) Volumetric flow rate through nozzle

Procedure

Four distinct differential pressures are measured – a pitot tube and pressure drop across an orifice, 90° elbow, and nozzle. A digital manometer is connected between the two measurement points of each fitting, the fan is turned on and air pressure difference measured. (The manometer reports the pressure on each point relative to atmospheric pressure – “zero”).

For the pitot tube, six points at a constant flow rate on the radius of the pipe are measured. (These points are represented by r in Table 1). The velocity will be different at the periphery of the pipe due to viscous forces slowing the flow.

The orifice, elbow, and nozzle pressure drops are all measured sequentially at five different flow rates (A flow rate is set, the pressure drops of each device are measured, then the flow rate is changed and the measurements repeated, for a total of five times). This allows each device to be used as a flow meter and the results to be compared.

Results and Discussion

For all results, the density of air is assumed to be 1.220 kg/m³. All Δp values are pressure losses. The major diameter/area of the pipe is 0.1524 m / 0.018 m². The orifices and nozzles will constrict this down to a level, given above the respective data tables.

Table 1: Pitot tube experimental data and results

Trial	Δp (Pa)	r (m)*	V (m/s)	Q (m3/s)
1	14	0.0762	4.791	0.0874
2	7	0.0699	3.388	0.0618
3	7	0.0572	3.388	0.0618
4	7	0.0445	3.388	0.0618
5	7	0.0318	3.388	0.0618
6	7	0.0191	3.388	0.0618

The radius r is the distance that the pitot tube is inserted into the pipe, therefore the larger number means that the tube is closer to the center. Velocity is found from Equation 1 and volumetric flow rate by multiplying this by the cross-sectional area of the pipe. The results were fairly inconclusive as the pressure difference should have been less on the periphery of the pipe and fairly constant in all other regions. A peak was experienced when the pitot tube was fully inserted, indicating a higher velocity in the center of the pipe. Due to the turbulent flow (Re ≈ 1.9 x 10⁵), this should not have been so defined.

Figure 2: Velocity Profile as measured by pitot tube – higher Radial Distance = closer to center of pipe.

Table 2: Orifice experimental data and results (C_f=0.8, A₀=0.153 m)

Trial	Δp (Pa)	Q (m3/s)
5	6	0.0385
4	7	0.0415
1	14	0.0587
2	34	0.0915
3	145	0.1891

The orifice results were more sensible, with increases in pressure loss representing a marked increase in flow rate. Volumetric flow rate through an orifice is calculated from Equation 2, derived from Bernoulli’s Equation with the coefficient of flow taken into account. The ordering of the trials is based upon the fan speed which was used during that trial – from lowest to highest. This is the case through the elbow and nozzle measurements as well, as the same speed of fan was used for the respective measurements of those devices as well.

Table 3: Elbow experimental data and results (K=0.19)

Trial	Δp (Pa)	V (m/s)	Q (m3/s)
1	0	0.0000	0.0000
2	0	0.0000	0.0000
3	0	0.0000	0.0000
4	7	1.4766	0.0269
5	7	1.4766	0.0269

The velocity and volumetric flow rates through the elbow were calculated using Equations 4 and 5, respectively. The value K of 0.19 is obtained from Crowe, page 337 Table 10.5, with a 90° smooth bend of radius/diameter ratio of 2. An elbow with a tight radius will cause greater pressure drop as fluid is forced to change direction but an elbow with a large radius will incur pressure drop as well due to the long length of pipe. According the Crowe Table 10.5, a r/d ratio of 4 seems to be optimal. The elbow presented a bit of difficulty as some of the trials produced a pressure loss of zero – it may have been below the scale of the manometer or there may have been a leak, but a pressure drop of seven pascals was obtained in the last two trials. A pressure drop of zero technically means there is no flow, but there WAS in fact flow it was not being measured properly. The flow rate resulting from this compared with the orifice measurement is quite a bit lower.

Table 4: Nozzle experimental data and results (C=0.98, A_throat=0.00456 m²)

Trial	Δp (Pa)	Qmass (kg/s)	Q (m3/s)	Percentage of orifice Q value (%)
4	21	0.0330	0.0271	65.20
5	28	0.0382	0.0313	81.32
1	28	0.0382	0.0313	53.23
2	61	0.0563	0.0462	50.42
3	249	0.1138	0.0933	49.33

The results of the nozzle were more in line with those obtained from the orifice, at least compared with the elbow. The percentage of flowrate compared with the flowrate obtained by the orifice method is listed in the last column of Table 4. Nozzle mass flow rate was determined using Equation 6. The fact that the pressure was not exactly atmospheric the density of air is not exactly the 1.22 kg/m³ assumed, but the pressure difference compared with atmospheric was so low that the effect is negligible. The nozzle discharge coefficient was assumed to be 0.98.

Conclusions

Orifices and nozzles can effectively be used to determine flow rate and/or velocity through a closed pipe. Pitot tubes can determine velocity, and if they are within a closed channel (e.g. a pipe), the flow rate can be found, especially where flow is turbulent and velocity is essentially constant throughout the cross-section of the pipe. Elbows, while not an extremely effective flow determining device, can incur significant pressure loss, especially if they are tightly bent or have a very long radius.

References

1. Crowe, Clayton T., Donald F. Elger, Barbara C. Williams, and John A. Roberson. Engineering Fluid Mechanics. 9^th edition. John Wiley & Sons, 2009.