MCE380: Measurements and Instrumentation Lab

Transcription

1 MCE380: Measurements and Instrumentation Lab Chapter 8: Flow Measurements Topics: Basic Flow Equations Flow Obstruction Meters Positive Displacement Flowmeters Other Methods Holman, Ch. 7 Cleveland State University Mechanical Engineering Hanz Richter, PhD MCE380 p.1/0 Commonly used liquid and air flow units Gallons per minute: (gpm) = 31 cu.in/min=63.09 cc/min. 1 liter per minute: (lpm) = gallons/min. Since the volume of gases depends on pressure and temperature, we have to be careful when trying to find mass flow rate from volumetric rate. A standard cubic foot (of air) at has been defined at 0 C and 1 atm. It weighs gr= lb. MCE380 p./0

2 Summary of Basic Flow Equations These equations apply to horizontal, frictionless, adiabatic flow. They apply to both compressible and incompressible flows (varying or constant density). Continuity: A 1 ρ 1 u 1 = A ρ u = ṁ Energy Conservation (SI units, g c = 1): h 1 + u 1 = h + u 1 h = e + p ρ is the enthalpy, e is the internal energy. Refer to Fig. 7.4 in Holman. MCE380 p.3/0 Flow Equations for Incompressible Fluids If the fluid is incompressible, ρ 1 = ρ. Substituting h = e + p ρ that e 1 = e (no thermal effects are present): Continuity: and noting A 1 u 1 = A u Energy Conservation: p 1 ρ + u 1 = p ρ + u 1 This is the same as Eq. 7. in Holman. We can solve the two equations simultaneously to obtain the flowrate: Q = A u = A 1 ( A A 1 ) This represents an ideal value for the flow. (p 1 p ) ρ MCE380 p.4/0

3 Discharge Coefficient The actual and ideal flows are related by C, the discharge coefficient. so that Q = C C = Q actual Q ideal A 1 ( A A 1 ) (p 1 p ) ρ C is obtained as a function of Reynolds number from experimental calibration (see Figure 7.9 in Holman). Flowmeter manufacturers will sometimes give empirical formulas for C or alternate empirical coefficients. MCE380 p.5/0 Flow Equations for Ideal Gas In this case ρ 1 ρ. Additional information is obtained from the equation of state p = ρrt Also, remembering that h = C p T, the energy conservation equation is C p T 1 + u 1 = C pt + u For small velocity of approach (u 1 small), the equations give a formula for the mass flow rate: [ (p ṁ = A γ p ) ( ) γ+1 ] γ 1 p γ γ 1 RT 1 p 1 p 1 γ is the ratio of specific heat capacities C p /C v. Remember to use absolute pressures (Pa) and temperatures (K). Look at simplifications in Eq. 7.9 and MCE380 p.6/0

4 Flow Stagnation and Properties Stagnation refers to the process of bringing a fluid stream to zero velocity. The pressure and temperature of the fluid at zero velocity will differ from those upstream of the stagnation point. For incompressible fluids, we use the energy equation with u = 0 and denote the stagnation pressure as p 0 = p. Also, the free-stream velocity and pressure are denoted p = p 1 and u = u. We obtain: p 0 = p + 1 ρu The quantity 1 ρu is known as dynamic pressure. The use of stagnation properties is essential to several flow measurement instruments. MCE380 p.7/0 Compressible Flow Stagnation The velocity of acoustic waves in an ideal gas with temperature T is a = γrt The Mach number is defined as M = u a The stagnation (or total) pressure for ideal gases (assuming isentropic stagnation process) is ( p 0 = p 1 + γ 1 M ) γ γ 1 This is the same as Eq.7.49 in Holman. The stagnation temperature can be obtained from the energy equation as T 0 = T 1 + u C p This is the same as Eq. [b] in Example 7.4 in Holman. Remember that the subscript indicates free-stream conditions. MCE380 p.8/0

5 Choked Flow of an Ideal Gas When the downstream pressure is decreased with all other quantities left constant, the flow rate increases. However, there is a limit. When the ratio of downstream to upstream pressures is less than a critical value, no further increase in flow rate can be obtained. The flow is now choked. The critical pressure ratio is: ) = p 1 crit ( p1 ( ) γ γ 1 γ + 1 Choking has the important consequence that the flowrate no longer depends on p. The maximum flowrate is given by ṁ = A p 1 RT 1 [ γ γ + 1 ( ) ] 1/ γ 1 γ + 1 Important: This equation must be used with stagnation values for T 1 and p 1 MCE380 p.9/0 Obstruction Flowmeters These meters are based on the pressure - flow relations we saw. A pressure drop is created on purpose, so that the flow can be measured. In some cases, a large pressure drop is not desirable. The Venturi minimizes this effect, but is expensive in comparison with the orifice plate. The basic flowrate calculation is based on Eq. 7.4 (liquids) or 7.10 (gases), using the appropriate discharge coefficients, seen next. MCE380 p.10/0

6 Empirical Constants: Incompressible Flow Define the following constants: Velocity of approach factor: M = 1 1 (A /A 1 ) Flow coefficient: K = CM (C is the discharge coefficient) Diameter ratio: β = d D = A /A 1 Incompressible flow in Venturi: Q = CMA ρ p1 p Incompressible flow in Nozzles or Orifices: Q = KA ρ p1 p MCE380 p.11/0 Empirical Constants: Compressible Flow Define the following expansion factors Y : Venturi and nozzle: See Eq Orifice with pressure taps on the flange or vena contracta: See Eq Orifice with pipe taps: See Eq Compressible Flow in Venturi: Compressible Flow in Nozzles and Orifices: ṁ = Y CMA p ρ1 (p 1 p ) ṁ = Y KA p ρ1 (p 1 p ) Note: A value of C can be found from Figs. 7.9, 7.10, 7.11 as a function of Reynolds number. Remember Re = ρud µ where u is velocity, D diameter and µ dynamic viscosity.(n-s/m ). MCE380 p.1/0

7 Example: Incompressible Flow A Venturi is used to measure the flow of water at 0 C. The pipe diameter is 70mm and the throat diameter is 35mm. The pressure differential indicated by the Venturi is 50kPa. Find the flowrate in gpm. Notes: 1. 1 corresponds to upstream pipe, corresponds to throat.. Since the velocity is unknown, you ll have to express the Reynolds number as a function of C and use the graph in a convenient way. 3. The Reynolds number is calculated at the throat. 4. Density of water: kg/m3; Dynamic viscosity: Pa.s MCE380 p.13/0 Example: Compressible Flow Nozzles are sometimes designed to operate under choked conditions. In Example 7.4, the throat diameter is selected so that this occurs at nominal conditions. Observe the use of stagnation conditions. MCE380 p.14/0

8 The Pitot Tube The Pitot tube is a pressure probe designed to indicate the difference between static and stagnation pressures. For incompressible fluids, the Pitot tube will indicate the dynamic pressure 1 ρu. The freestream velocity can be calculated. Look at Example 7.9 in Holman. MCE380 p.15/0 Application of Pitot Tubes MCE380 p.16/0

9 Positive Displacement Flow Metering These devices rely on cavities of known volume (one turn implies a known displaced volume). The lobed-impeller flowmeter can be used for both liquids and gases. MCE380 p.17/0 Drag Effect Flowmeters: Rotameter The rotameter bob will stabilize at an equilibrium position that depends on the velocity of the flow. See Eqs and 7.30a. MCE380 p.18/0

10 Hot-Wire Anemometers We have studied how materials change resistance with temperature. RTDs and thermistors exploit this effect for the measurement of temperature. The temperature, in turn, is affected by the conditions of the surrounding medium. We can extract information about the medium (velocity) by looking at the temperature. A hot-wire anemometer uses a thin wire carrying a current and exposed to the flow whose velocity will be measured. The wire is used as sensing resistor in a leg of a Wheatstone bridge. The following equations apply: q = (a + bu 0.5 )(T w T ) q = i R w = i R 0 (1 + α(t w T 0 )) where q is the heat transfer rate, T w is the wire temperature, R 0 is the wire resistance at reference temperature T 0, T is the free-stream temperature, α is the temperature coefficient of resistance, i is the current and a and b are constants. The required velocity is u. The current i and resistance R w are obtained from the Wheatstone bridge. MCE380 p.19/0 Example Suppose a hot-wire anemometer has been custom-built and needs to be calibrated. Current i and resistance R w will be measured along with true values of speed u. What would be the procedure to follow to obtain a and b for the device? Give an expression for the velocity as a function of R w and i. MCE380 p.0/0