Bernoulli’s Theorem LABORATORY MANUAL for MECE2860U-Fluid Mechanics

Bernoulli’s Theorem LABORATORY MANUAL for MECE2860U-Fluid Mechanics Experiment # 3 Bernoulli’s Theorem Demonstration Apparatus LABORATORY MANUAL for MECE2860U-Fluid Mechanics 2 Experiment # 3: Bernoulli’s Theorem Demonstration Apparatus 1.1 Objective The objective of this experiment is to investigate Bernoulli’s law, perform measurements along a venturi tube and determine the flow rate factor (K). 1.2 Introduction and Theoretical Background Bernoulli’s Equation is a very important integral form of the equation of fluid motion. It is one of the most commonly used equations in fluid mechanics. The Bernoulli equation is named in honor of Daniel Bernoulli (1700-1782). Many phenomena regarding the flow of liquids and gases can be analyzed by simply using the Bernoulli equation. However, due to its simplicity, the Bernoulli equation may not provide an accurate enough answer for many situations, but it is a good place to start. It can certainly provide a first estimate of parameter values. Modifications to the Bernoulli equation to incorporate viscous losses, compressibility, and unsteady behavior can be found in other (more complex) calculations in the textbook and Ref. [1]. When viscous effects are incorporated, the resulting equation is called the "energy equation". This experiment utilizes a Venturi tube (as a flow-area varying device) and a Prandtl tube-manometer set up (as a flow measurement device) to demonstrate some of the key concepts of the Bernoulli’s Equation. The Bernoulli’s Equation is a description of the momentum of steady, incompressible, irrotational, and frictionless flow (Figure 1). Figure 1. Steady, incompressible, irritation and frictionless flow A general form of Bernoulli’s Equation can be expressed as: LABORATORY MANUAL for MECE2860U-Fluid Mechanics p + V + gh = p + V + gh2 = const 2 1 2 2 2 1 1 2 1 2 1 ? ? ? ? (1) where p is static pressure, ? is density, V is velocity, g is gravity constant, and h is the height with respect to the reference level (i.e. sea level). The subscripts 1 and 2 denote the stream wise locations of the flow. Equation (1) can be interpreted as: the total energy (sum of static pressure pstat = p , dynamic pressure 2 2 pdyn = 1 ?V and body force pbdy = ?gh ) of a fluid body flowing along the streamline always remains constant. For gas, since the density is low, the body force is practically insignificant. Thus, the term pbdy = ?gh can be ignored and Equation (1) can be simplified to: 2 V const V p 1 2 p 1 2 2 2 2 1 + ? 1 = + ? = (2) This expression is also termed the total pressure pt: 2 2 pt = pstat + pdyn = p + 1 ?V (3) Proper use of the Bernoulli equation requires close attention to the assumptions used in its derivation. To use it correctly we must constantly remember the basic assumptions used in its derivation: (a) viscous effects are assumed negligible, (b) the flow is assumed to be steady, (c) the flow is assumed to be incompressible, (d) the equation is applicable along a streamline. In the derivation of Equation (1), we assume that the flow takes place in a plane (the x–z plane). In general, this equation is valid for both planar and nonplanar (threedimensional) flows, provided it is applied along the streamline. The Bernoulli equation is used to analyze fluid flow along a streamline from a location 1 to a location 2. Most liquids meet the incompressible assumption and many gases can even be treated as incompressible if their density varies only slightly from 1 to 2. The steady flow requirement is usually not too hard to achieve for situations typically analyzed by the Bernoulli equation. Steady flow means that the flow rate (i.e. discharge) does not vary with time. The inviscid fluid requirement implies that the fluid has no viscosity. All fluids have viscosity; however, viscous effects are minimized if travel distances are small. Flow along a 100 km river has significant viscous effects (friction between the channel material and the flowing fluid), but viscous effects along just a 10 m reach of that channel where a sluice gate occurs would be minimal. Sluice gates are typically analyzed with the Bernoulli equation 1.3 Equipment The schematic layout and photos of the Bernouilli’s theorem demonstration on apparatus are shown in Figures 2-4, respectively. Body force Dynamic pressure Static pressure LABORATORY MANUAL for MECE2860U-Fluid Mechanics 4 Figure 2. Layout of Bernoulli’s theorem demonstration apparatus Figure 3. A photo of Bernoulli’s theorem demonstration apparatus 1. Assembly board 2. Water pressure gage 3. Discharge pipe 4. Outlet ball cock 5. Venturi tube with six measurement points 6. Compression gland 7. Probe for measuring overall pressure (can be moved axially) 8. Hose connection, water supply 9. Ball cock at water inlet 10. 6-fold water pressure gage (pressure distribution in venture tube) LABORATORY MANUAL for MECE2860U-Fluid Mechanics 5 Single water pressure gage Venturi tube with six measurement points Figure 4. Various photos of the main components of Bernoulli’s theorem demonstration apparatus Water pressure gage LABORATORY MANUAL for MECE2860U-Fluid Mechanics 6 1.4 Operating Instructions and Procedure The following procedure should be followed during the experiments by taking into account Figure 5. • Arrange the experimental setup on the on the gravimetric hydraulic bench such that the discharge routes the water into the channel. • Make hose connection between the gravimetric hydraulic bench and unit. • Open discharge of the gravimetric hydraulic bench. • Set cap nut (1) of probe compression gland such that slight resistance is felt on moving probe • Open inlet and outlet ball cock. • Close drain valve (2) at bottom of single water pressure gauge. • Switch on pump and slowly open main cock of the gravimetric hydraulic bench. • Open vent valves (3) on water pressure gauges • Close outlet cock carefully until pressure gauges are flushed. • Regulate water level in pressure gauges by simultaneously setting inlet and outlet cock such that neither upper nor lower range limit (4\5) is overshot or undershot. • Record pressures at all measurement points. Then move overall pressure probe to corresponding measurement level and note down overall pressure. • Determine volumetric flow rate. To do so, use stopwatch to establish time t required for raising the level in the volumetric tank of the gravimetric hydraulic bench from 20 to 30 liters. Note that the experimental setup should be arranged absolutely plane to avoid falsification of measurement results (use of spirit level recommended). For taking pressure measurements, the volumetric tank of the gravimetric hydraulic bench must be empty and the outlet cock open, as otherwise the delivery head of the pump will change as the water level in the volumetric tank increases. This results in fluctuating pressure conditions. A constant pump delivery pressure is important with low flow rates to prevent biasing of the measurement results. The zero of the single pressure gauge is 80 mm below that of the 6-fold pressure gauge. Allowance is to be made for this fact when reading the pressure level and performing calculations. Both ball cocks must be reset whenever the flow changes to ensure that the measured pressures are within the display ranges. Figure 5. Experimental procedure steps LABORATORY MANUAL for MECE2860U-Fluid Mechanics 1.5 Calculations For steady, inviscid, incompressible flow the total energy remains constant along a streamline. The concept of “head” was introduced by dividing each term in equation (1) by the specific weight, ?=?g, to give the Bernoulli equation in the following form. z H 2g p V2 + + = ? (=constant on a streamline) (4) Each of the terms in this equation has the units of length (feet or meters) and represents a certain type of head. The Bernoulli equation states that the sum of the pressure head, the velocity head, and the elevation head is constant along a streamline. This constant is called the total head, H. Assuming that z1=z2, Equation (4) can be written as follows (Figure 6): + = 2g V h 2 1 1 L 2 2 2 h 2g V h + + (5) where h1 and h2 are the pressure heads at cross-sectional areas A1 and A2, respectively, while hL is the pressure loss head. Figure 6. Cross-sectional areas of the venturi To conserve mass, the inflow rate must equal the outflow rate. If the inlet is designated as (1) and the outlet as (2), it follows that m?? 1 = m?? 2 (6) Thus, conservation of mass require ?1A1V1 = ?2A2V2 (7a) If the density remains constant, then ?1=?2 and the above becomes the continuity equation for incompressible flow A1V1 = A2V2 or Q1 = Q2 (7b) Condition 1 Condition 2 LABORATORY MANUAL for MECE2860U-Fluid Mechanics 1.5.1 Velocity profile in the venturi tube The venturi tube used has 6 measurement points. Table 1 shows the standardized reference velocity Vsr, while the measurement points along the venturi are illustrated in Figure 7. This parameter is derived from the geometry of the venturi tube and is given by the relation i 1 SR A A V = (8) Table 1. Standardized reference velocities Figure 7. Measurement points along the venturi Calculate the theoretical velocity values (Vcal) at the 6 measuring points of the venturi tube by multiplying the reference velocity values with a starting value. cal SR 1 V = V V (9) with the starting value for calculating the theoretical velocity at a constant flow rate 1 1 A V = Q (10) The dynamic pressure head is calculated as: h dyn = h t -80 mm - hstat (11) where 80 mm is subtracted, as there is a zero-point difference of 80 mm between the pressure gages. Point i Inside diameter Di (mm) Ai (m210-4) Reference velocity VSR (-) 1 28.4 6.33 1.00 2 22.5 3.97 1.59 3 14.0 1.54 4.11 4 17.2 2.32 2.72 5 24.2 4.60 1.37 6 28.4 6.33 1.00 LABORATORY MANUAL for MECE2860U-Fluid Mechanics The measured velocity (Vmeas) is calculated from the dynamic pressure as follows: V (m / s) 2 p (Pa) / (kg / m3 ) meas = ? dyn ? (12a) or V (m / s) 2g(m / s )h dyn (mWC) 2 meas = (12b) Plot the measured and calculated velocity profile along the venture tube at a recorded volumetric flow rate. 1.5.2 Pressure distribution along the venturi tube Plot the values for hdyn, hstat and ht (mmWC) along the venture tube using the values measured and obtained from Equation (11). 1.5.3 Determination of flow rate factor A venturi tube can be used for flow rate measurements. In comparison with orifice or nozzle, there is a far more smaller pressure loss during measurements of flow rate. The pressure loss ?p between largest and smallest diameter of the tube is used as measure for the flow rate (Figure 8): 1 3 Q K p - = ? (13a) Figure 8. Measurement points for the pressure loss (?p1-3) The flow rate factor K is generally made available for the user by the manufacturer of a venturi tube. If the flow rate factor is unknown, it can be determined from the pressure loss ?p1-3 as follows: ( ) p (bar) K1/ s bar Q(l / s) ? 1-3 = (13b) Table 2 shows the pressure loss for various flow rates as well as the flow rate factor K. Read off the pressure loss from the six–tube manometer in mm water column (mmWC) and set in the equation as bar. The flow rate can be used with unit l/s (liter/s). LABORATORY MANUAL for MECE2860U-Fluid Mechanics 10 Table 2. Pressure loss for various flow rates and flow rate factors Q=0.275 l/s Q=0.256 l/s Q=0.166 l/s Measurement points ?p (mmWC) K ) s bar ) ( l ?p (mmWC) ) s bar ) ( l ?p (mmWC) ) s bar ) ( l 1-3 160 2.1 143 2.1 65 2.1 1 mmWC (or mmH2O) = 0.0980665 mbar ˜ 0.1 mbar = 0.0001 bar = 10 Pa 1.6 Worksheet for Experimental Data *1 mmWC (or mmH2O) = 0.0980665 mbar ˜ 0.1 mbar = 0.0001 bar = 10 Pa Volume of water in the tank at the beginning of the experiment (V1)= l Volume of water in the tank after t=60 s (V2) = l ?V = V2 - V1= - = l t = 60 s Q = V/t = /60 = l/s Measurement points (See Figure 7) h1 (mmWC) h2 (mmWC) h3 (mmWC) h4 (mmWC) h5 (mmWC) h6 (mmWC) hstat (mmWC) measured ht (mmWC) measured hdyn (mmWC) Using Equation (11): h dyn = h t -80 mm - hstat Vmeas (m/s) Using Equation (12b): 2g(m/ s )h (mWC) V (m / s) dyn 2 meas = Vcal Using Equations (9)-(10) and Table 1: V1 = Q / A1 ; cal SR 1 V = V V ?p1-3 (See Figure 8) mmWC bar* Flow rate factor (K) Using Equation (13b): ( ) p (bar) K1/ s bar Q(l / s) ? 1-3 = s bar l bar (l / s) p (bar) K Q(l / s) 1 3 = = ? = - LABORATORY MANUAL for MECE2860U-Fluid Mechanics 11 Nomenclature A : Cross-sectional area (m2) g : Gravitational acceleration (m/s2) h : Height with respect to the reference level (m) hL : Pressure head loss (mWC) K : Flow rate factor (1/ s bar ) m?? : Mass flow rate (kg/s) pdyn : Dynamic pressure (bar) pstat : Static pressure (bar) pt : Total pressure (bar) Q : Volumetric flow rate (m3/s) V : Volume (m3) Vcal : Calculated velocity (m/s) Vmeas : Measured velocity (m/s) VSR : Reference velocity (m/s) z : Elevation head (mWC) ?p1-3 : Pressure loss (bar) ? : Density (kg/m3) References 1. Bernoulli Equation Calculator with Applications. Available at: http://www.lmnoeng.com/Flow/bernoulli. htm. 2. Equipment for Engineering Education, Instruction and Operation Manuals, Gunt Hamburg Germany, 1998. 3. Munson, B. A., Young, D. F., and Okiishi, T. H. Fundamentals of Fluid Mechanics. 4th Edition, Wiley, New York, 2002. PLACE THIS ORDER OR A SIMILAR ORDER WITH US TODAY AND GET AN AMAZING DISCOUNT :)