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MIDDLE EAST TECHNICAL UNIVERSITY
DEPARTMENT OF MECHANICAL ENGINEERING
ME 305 FLUID MECHANICS I
EXPERIMENT 2
DETERMINATION OF LINEAR MOMENTUM RATE OF AIR FLOW
PREPARATION: In ME 305 course, you will conduct the experiments, by yourself, with little help or instruction from your lab instructor (the teaching assistants - TAโs - of the course) in the limited lab time allocated to you. Therefore, you must read the experiment manual thoroughly and understand what you are expected to do and how (and why) for each experiment, before coming to the lab. You must use a pen (not a pencil) when recording your data. Data should be recorded in the data tables provided in the โreportโ section at the end of the experiment manual. Following the experiment, you will write down and turn in your lab report in the remaining time of your lab session โ for this, you will have approximately 1 hour ( note that time is limited so you must complete the experiment โ on your own - without delay and proceed to report-writing ). The report for the experiment is included at the end of the experiment manual in this document โ you will prepare your report by detaching it from this document and completing the relevant sections. Up to 3 students in a lab group will prepare and submit a single report, i.e. in a single lab session, two lab reports will be submitted by dividing the lab group into two report groups.
1. Objective
The purpose of this experiment is the calculation of the linear momentum rate of air flowing out of a duct, using the integral formulation of the momentum equation. In the first part of the experiment, the force with which the air flowing out from the duct impinges on a plate is to be measured using a measurement plate (scale). In the second part, the velocity distribution at the exit plane of the duct is to be obtained through Pitot tube measurements and the use of Bernoulli equation. This velocity distribution can be used to obtain the momentum rate of air. From the results of this portion of the experiment, the momentum coefficient will also be calculated.
2. Theoretical Background
2.1. Momentum Equation in Integral Form
According to Newtonโs second law of motion, the net external force acting on a control volume is
equal to the total rate of change of its linear momentum. This can be stated in integral form with the
below equation
โ
๐ด
In Equation 1 , first term on the right hand side represents the local rate of change of linear momentum in the control volume. The second term on the right hand side represents the net linear momentum rate of flow crossing the surfaces of the control volume. Under steady state conditions, first term on the right hand side of equation (1) cancels out and we get โ ๐นโ = โซ ๐๐โโ (๐โโ โ ๐ฬ )๐๐ด ๐ด
2. 2. Fluid Jet Impinging Normally on a Flat Plate
Suppose that a jet of air (leaving a duct) strikes a planar surface (a plate) steadily as illustrated in the below figure. Figure 1. Air stream hitting a planar surface planar surface control volume
section A ๐แถ๐ด
The Bernoulli equation for the steady, incompressible flow of air with a density of ๐๐ may be applied between points X and O along this streamline to yield ๐๐ฅ ๐๐
๐๐ฅ^2
๐๐^2
where ๐๐ฅ and ๐๐ are the static pressures at points X and O, and ๐๐ฅ and ๐๐ are the corresponding velocities at these two sections. In the above equation, the change in the elevation is not considered since points X and O are close to each other. Likewise, frictional effects are negligible, too. Observing that point X is exposed to the atmosphere (pressure is atmospheric at X) and point O is a stagnation point, then ๐๐ = 0 and ๐๐ฅ = ๐๐๐ก๐. The velocity at point X is found as ๐๐ฅ = โ
Neglecting the pressure changes in an air column, for a manometer fluid density of ๐๐, ๐๐ = ๐๐ฆ = ๐๐๐ก๐ + ๐๐๐ โโโฒ (^) ( 5 ) where ๐๐ฆ is the pressure at point Y (shown in Figure 2), โโโฒ^ = โโ sin ๐ผ, โโ is the reading from the inclined manometer, inclined at an angle of ๐ผ, and ๐๐ is the manometer fluid density. Thus, ๐๐ฅ = โ 2 ๐๐๐ โโ sin ๐ผ ๐๐
If a series of Pitot tubes, connected to individual manometers, are positioned at a flow section (such as the exit plane of the duct in Figure 2, i.e. section A in Figure 1), the velocity distribution at that flow section can be obtained using equation ( 6 ). The average velocity ๐ฬ
at this flow section (duct exit) can be obtained by mass-averaging the velocity distribution ๐๐ as ๐ฬ
=
where ๐แถ is the total mass flow rate through the duct across the duct area (total flow area) ๐ด and ๐แถ๐ is the mass flow rate across the ๐ด๐ portion of the total flow area, with โ ๐ด๐ = ๐ด. ๐๐ (the velocity measured by the Pitot tube) is assumed to be the average velocity on ๐ด๐ and the density is constant.
2.4. Momentum Coefficient
The momentum rate at a flow section may change depending on the velocity profile even if the mass flow rate does not change. Consider Figure 3 where two different velocity profiles (one uniform, one parabolic) are shown for the same flow section. The mass flow rates are identical in both profiles. Even though the average velocities are the same in both profiles, the momentum rates of the flows are not the same. (a) Nonuniform velocity profile (b) Uniform velocity profile Figure 3. Nonuniform and uniform velocity profiles of the same mass flow rate at a section The momentum coefficient ๐ฝ is defined for a specific flow profile at a flow section. It indicates the deviation of the momentum rate associated with this velocity profile, from the momentum rate of a uniform flow of the same mass flow rate, and is defined as ๐ฝ =
โซ๐ด ๐๐^2 ๐๐ด
Note that ๐ฝ = 1 for uniform flow and ๐ฝ > 1 for nonuniform flow.
3. Experimental Set-Up
The sketch of the experimental set-up is shown in Figure 4. There are three major components in this set-up. The first is the air duct that discharges air created by the working of a fan (air blower). The fan is not shown in the sketch; however, the duct is connected to the outlet of the fan and by starting the fan an air flow is created in the duct. The second component of the experimental set-up is the measurement plate (scale) and the table on which it lies (Figure 4 .a). This component will be used to measure the force with which the air jet (leaving the duct) hits a horizontal plate. Air is directed through the duct onto the top plate, so that it hits the measurement plate. The load cell under the measurement plate (not visible in the figure) reads the โweightโ on the plate and displays it on the reading screen. The momentum rate of this air flow is to be obtained during this experiment. The third component is the Pitot tube-manometer apparatus that will be used to obtain the air velocity profile at the exit of the duct, in order to obtain the momentum rate (Figure 4 .b). In the first part of the experiment, the measurement plate will be positioned below the duct (Figure 4 .a). Once
๐ด ๐ฬ
= โซ๐ด^ ๐๐๐๐ด
๐๐ด
measurement plate is zeroed so that the measurements to be taken during the flow will only reflect the effect of air momentum rate. c) Move the table that carries the measurement plate under the duct air outlet. Air exit area must be centered with respect to the opening on the top plate. To help you position the table properly, markers have been placed on the floor for the four wheels of the table. Study the positioning of the wheels in Figure 5. Your lab supervisor will help you with the positioning. d) Read and note the value in the reading screen. Donโt forget that this value is in [kg]. There might be fluctuations in the reading. Record an average value as best as you can, on the data sheet. e) Unplug the electric cord and move the table aside, clearing the space below the exit of the duct for the next portion of the experiment. Do not turn off the air blower. Figure 5. Experimental set-up for creating air flow in a duct and for force measurement Part 2 : The photographs of the set-up in the laboratory for this part of the experiment are shown in Figure 6. a) Record the ambient temperature and pressure on the data sheet. Your teaching assistant will direct you to the thermometer & barometer. You will use these values to calculate the density of air using ideal gas law. measurement^ duct plate top plate bottom plate markers reading screen duct Air coming in to the duct from behind the wall Behind the wall air flow fan (to the duct) air flow air flow Reading screen
b) Move the Pitot tube rake under the duct by using markers. Your lab supervisor will help you with the positioning of the rake.
c) Read the inclination angle, ๏ก of the manometer rack from the protractor and record on the data sheet.
d) Record the deflection, ๏ h in each manometer connected to the respective Pitot tube facing the exit
flow, on the data sheet. Note that there are a total of 12 Pitot tubes, numbered as shown in Figure 6. These Pitot tubes are connected to the respective manometers, also numbered as shown. The manometer rack has several manometers open to atmosphere as shown. The deflection you read for the Pitot tubes should be the relative to the deflection in a manometer open to atmosphere. e) Once you have finished recording the deflections, stop the air blower with the help of your teaching assistant. Move aside the Pitot tube rake so that you clear the space below the duct exit. Figure 6. Experimental set-up for obtaining the air velocity profile at the exit of the duct 1 2 3 4 5 6 7 8 9101112 manometer fluid supply Pitot tubes protractor manometer rack ๏ก duct Pitot tube rake manometer rack markers to position the Pitot tube rack plastic tubing manometers open to atmosphere manometers connected to the 12 Pitot tubes to Pitot tubes manometers open to atmosphere
ME 305 FLUID MECHANICS I - EXPERIMENT 2 REPORT DETERMINATION OF LINEAR MOMENTUM RATE OF AIR FLOW Student Name: Student ID Number: Student Course Section: Lab. Group: Date: Lab. Supervisor:
1. Data & Results
Part 2 Ambient temperature (ฮฟC) Ambient pressure (Pa) Gas constant of air (^) 287 J/kgK Air density (kg/m^3 ) Density of alcohol (manometer fluid) (^810) kg/m Manometer inclination, ๏ก (ยบ) โโ (cm of alcohol) ๐๐๐ (m / s) ๐ด๐๐ (m^2 ) ๐แถ๐๐ (linear momentum rate) 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 0. 10 0. 11 0. 12 0. ๏ฅ: Average duct exit velocity (m/s) Momentum Coefficient Part 1 Measurement Plate Reading (kg) Force on the plate (N)
ME 305 โ Experiment 2 Report 2 of 3
2. Calculations
2.1. Force on the measurement plate 2.2. Air density
2. 3. Sample calculation of the velocity measured by Pitot tube 5 2. 4. Sample calculation of the linear momentum rate across the area segment 5 on the duct exit section 2. 5. Average velocity of air flow at the duct exit 2. 6. Momentum coefficient
