Manual
FLUID Laboratory MECHANICS
ME 211
C. M. Megaridis and W. J. Minkowycz Mechanical and Industrial Engineering University of Illinois at Chicago August 2003
EXPERIMENT #3
ONE DIMENSIONAL BERNOULLI EQUATION: FLOW METERING DEVICE Objective In this experiment, variations of static pressure and fluid flow velocity in a Venturi tube are investigated. The main goal of the experiment is to utilize the Venturi meter to measure liquid flow rate, and provide comparisons between experiment and theory via the Bernoulli equation. Background Direct Measurement of Flow Rate: By collecting the liquid in a weighing device for a given length of time t, the volume rate of flow can be determined from W Qexp = [ m 3 / s] (1) gt where W = weight of liquid collected during time t g = acceleration of gravity = density of liquid Typically, several measurements are taken to determine the value of Qexp. Theoretical Prediction: Refer to Fig. 1 for the definitions of the symbols used below. Let us apply Bernoullis equation between the inlet (1) and any other station (n) along the center streamline. n can take any value from 2 to 11 (see Fig. 1). Then p1 + 1 1 V1 2 = p n + Vn 2 2 2 ( 2)
The mass conservation equation between stations (1) and (n) gives V1 A1=Vn An=Q From Fig. 1 p1= gh1 and pn= ghn (3) (4)
Combining equations (2), (3) and (4) we obtain 2 g (h1 hn ) (5) A1 2 ( ) 1 An Equation (5) was obtained using the Bernoulli equation, therefore with the underlying assumption that the flow is inviscid (ideal). Thus, use of Eq. (5) to calculate the fluid flow rate produces a value that may not match the one measured in the experiment (Qexp). The theoretical value is denoted by Qideal and is given by V1 =
1
Qideal = V1 A1 = 2 g (h1 hn )
A1 A1 2 An 2 1
( 6)
In general, Equation (5) can be rearranged as
Qexp=C Qideal h1 hn
(7)
(8) = 2 1 V1 2 / 2 g An As does Eq. (5), the above equation also applies for inviscid flow. If the flow were indeed ideal, then a plot of (h1 hn)/(V12/2g) versus A21/A2n 1 would reveal a straight line with a slope of 45. But viscous effects are expected to cause deviations from the straight-line behavior.
A1 2
Experiment outline 1. Review apparatus description and overall procedure. 2. Measure, calculate, or obtain the quantities listed in the attached Tables. 3. Plot the experimental data as described below. 4. Discuss the results of the experiment in a brief and precise manner. 5. Follow the lab report write-up format given in this manual. Apparatus 1. Water pumping table with measuring device and timing mechanism used to find the mass or volume flow rate. 2. Venturi tube manufactured in clear plastic material with known area variation (Fig. 1 and Table 1) and manometer taps for measuring the pressure distribution. Venturi pressure and flow is controlled by supply and outlet valves as shown in Fig. 2. Back pressure for the manometer is controlled by an air bleed valve on the manifold. The whole assembly of Venturi meter, manometer tubes, scale and manifold is supported on a base mounted on adjustable feet to level the equipment.
Experimental Procedure 2
The manometer scale behind the tubes must be leveled first. This is done by opening both the control valve downstream of the meter and the bench supply valve, so as to allow water to flow for a few seconds and clear any air pockets from the supply system. Gradually close the control valve, so that the meter is subjected to a gradually increasing pressure, which will cause water to pass up the tubes, thereby compressing the air contained in the manifold. When the water levels have risen to a convenient height, the bench valve is also gradually closed, so that, as both valves are finally shut off, the meter is left containing static water under moderate pressure. adjusting The screws at the base are then set so that the water elevations in the tubes read the same value when the scale is viewed from the front, and the tubes are reasonably vertical when viewed from the end. The first reading is taken at the maximum available value of h1 h4 that is, with h1 close to the top of the scale and h4 close to the bottom. This condition can be achieved by gradually opening both the bench valve and the control valve. Similar readings are taken for intermediate flow as well as minimum flow. The rate of flow is measured by collection of the fluid in the weighing tank, and while this is in progress, values of h1 and h4 are read from the manometer scale. Record the elevation reading in all tubes (n =1, 11) for the maximum and minimum flow rates.
Data Analysis 1. For station n = 4, fill in Table 2 using the area functions given in Table 1. Then plot the measured values of (h1-h4)1/2 versus the flow rate Q . According to Eq. (6), which is valid for inviscid flow, Qideal is proportional to (h1 hn)1/2. Consequently, the (h1-h4)1/2 versus Qideal curve would be a straight line through the origin in this plot. On the other hand, the points in that plot represented by the pairs of (h1-h4)1/2 and Qexp may not lie on a straight line due to viscous effects. Plot the ratio C=Qexp/Qideal versus Qexp. This ratio quantifies the agreement between Qexp and Qideal. It is important to note that the value of C is not constant as Q varies. Why does the value of C appear to get closer to unity as Q (or likewise, velocity) gets larger? For the maximum and minimum flow rates, fill in Table 3. Using the area functions given in Table 1 and the velocity value V1=Qexp/A1, plot the quantity 2g (h1 hn)/V12 versus the area factor (A12/An21) for both ideal flow (show as a continuous curve), as well as the actual flow (show as points). Comment on the agreement between the two sets.
2.
3.
Questions for Further Discussion 1. Are the assumptions of one-dimensionality and non-viscous flow equally restrictive? 3
2.
Why are stations #1 and 4 best suited for the calculation of Q and V1? What would be the effect of using any other pair on the accuracy of the result? Ideally, h11 should be equal to h1 (Fig. 1). Are these two quantities equal in th...