CE 319F, Elementary Fluid Mechanics Momentum LaboratoryName Date Lab time AIR FLOW AROUND A BEND1) Determine the volumetric flow rate and the average velocity in the bend. To apply the momentum equation for steady flow through the bend, it is necessary to know the inflow and the outflow velocities. Since size of the bend is constant, the cross sectional average velocity is constant. The acceleration comes primarily from changing the direction of the velocity, not from changing its magnitude. The flow rate can be determined by using the contracting nozzle above the elbow as a flow meter. One piezometer (#13 on the manometer board) is attached to the plenum chamber (pc,Fig. 1) and another one (#14) is attached at the downstream end of the contraction (in). Since the pressure change is very small, the air may be assumed to be incompressible. Then, continuity gives pcinpcininpcVAAVAVAV )()(Bernoulli’s equation may be applied along a streamline from pc to in since the effects of shear stress are negligble in this short distance with rapid acceleration of the flow. Thus,    inpcinpcinpcinininininpcinpcpcinininpcpcpczzppVAAVpzVAApzVpzVpz212222222222222Fig. 1 - Contracting Nozzle forMeasuring Q    2212pcininpcinpcnAAzzppVFor this apparatus, 22inpcAA  and    inpcinpczzpp . Thus,  nipcnppV2Procedure:a) Read the piezometers for pc and in and then convert those readings into a pressure difference.h14 = hin = _________ cmh13 = hpc = _________ cmppc - pin = (hpc - hin)/100 cm/m = 9810(hin - hpc)/100 = _________ Pab) Read the air temperature and atmospheric pressure, and then calculate the air density.Taair = _________ oC = _________ K-1-pcinQpaair = mm Hg  Hg/(1000 mm/m) = _______________ mm Hg  (13.559810)/1000 = _______________ PaRair = 287 J/(kg K)3kg/m _______________ airaaairRTpc) Calculate the velocity at the inlet cross section, and then calculate the volumetric flow rate. For this cross section, the area is rectangular with a width (B) of 100 mm or 0.1 m and a depth (D) of 50 mm or 0.05 m. m/s _______________ 22inpcinppVQ = (VA)in= _______________ (0.050.10) = _______________ m3/s2) Determine the pressures around the bend and calculate the z (vertical) component of the net pressure force. The air flow through a 90o bend develops pressure distributions as illustrated in Fig. 2. Thereis also a pressure at the inflow section (in). The pressure at the outlet is atmospheric. The resultant force from these distributed pressures is the force that accomplishes the Eulerian convective acceleration to move the flow around the bend. These pressures can be measured with the piezometers attached to the piezometer taps shown in Fig. 3. Tubes go from piezometer taps to the tops of the tube on the multiple-tube, air-water manometer board. The bottoms of all of the manometer tubes are connected to the same constant head reservoir. When the piezometer tap has a high pressure, as on the outside of the bend, the water level in the manometer tube is pushed down. A low pressure pulls the water level up from its equilibrium position. In Fig. 3, Aa, Ac, Ad, and Af are the areas for the flat sections at the entrance and exit of the bend. Ab and Ae are the areas of the curved sections. Using the bend shown in Fig. 3 as the control volume, using the coordinate system in Fig. 3, and assuming that the shear forces are negligible because of the short flow lengths, integration of the pressure around the control surface gives Rx as 2/022/022/022/02)(cos)(cos))((cos))((cos)(cos)(cosdpBrBLpdpBrBLpdBrpBLpdBrpBLpdAppdAdAppdAReiboeiboedbapiidooaiidooaAAAAxwhere dA on the circular arcs is Brd since rd is the arc length and the subscript p indicates the force that is determined from pressure measurements. This equation assumes that all of the pressures on the outside of the bend are positive while all on the inside of the bend are negative, so -2-that the x component of the force on each subarea is positive (in the positive x direction based on the coordinate system). Piezometer 2 gives approximately the average pressure on the straight vertical sections. Similarly, piezometer 8 gives approximately the average pressure on the straight horizontalsections. Thus, Rz is BLpdpBrBLpdpBrBLpdBrpBLpdBrppdAdAppdAdApRfiicoofiicooAAAAzieobieobfecbp82/082/082/082/0)(sin)(sin))((sin))((sin)(sin)(sinFig. 2 - Pressure Distributions around Bend-3-zxAaAb Ad AcAe AfLaLdLcLf = 22.5o 2o3o4o5o6o7o8opiezometer number2i3i4i5i6i7i8iAin Qpositive pressurenegative pressureR = resultant pressure forceinoutRRzRxzxFig. 3 - Pressure Distributions around BendThe integrals on the curved surfaces can be evaluated numerically using the pressures measured with the piezometers. For example, for Rz, the integrand is p(sin), which is to be integrated with respect to . The pressure will be measured at five know  values. In general, p(sin) is a function of , as illustrated for five measured values in Fig. 4. The integral is the area under this curve. Since the shape of the curve between the measurement points is not known, the curve will be approximated by a series of straight line segments and the area found by a trapezoidal approximation(Fig. 5). Then, Fig. 4 - Actual IntegrandFig. 5 - Straight Line Approximation for the Integrand              677766566655455544344433/20)(sin)(sin21)(sin)(sin21)(sin)(sin21)(sin)(sin21)(sinppppppppdpFor constant  (/8 for this integration),7766554433/20)(sin21)(sin)(sin)(sin)(sin21)(sinpppppdpThis trapezoidal approximation to the integral will be