ME 322: Instrumentation Lecture 12 February 13, 2015 Professor Miles Greiner Flow rate devices, variable area, non-linear transfer function, standards, iterative method

Announcement/Reminders HW 4 due now (please staple) Monday – Holiday Wednesday – HW 5 due and review for Midterm Friday, Feb. 20, 2016 Midterm How was lab yesterday and the day before? Trouble bonding stain gages to beams? Access to glue? We will purchase one more pressure standard and more glue before next year’s class.

Fluid Flow Rates Within a conduit cross section or “area region” dA V, r VA 𝑉 𝑟 𝑚 = 𝐴 𝑑 𝑚 = 𝐴 𝜌𝑉𝑑𝐴 Within a conduit cross section or “area region” Pipe, open ditch, ventilation duct, river, blood vessel, bronchial tube (flow is not always steady) V and r can vary over the cross section Volume Flow Rate, Q [m3/s, gal/min, cc/hour, Vol/time] Q= 𝐴 𝑉𝑑𝐴 = VAA (How to measure average 𝑄 𝐴 over time Δ𝑡?) 𝑄 𝐴 = Δ𝑉 Δ𝑡 ; Average speed 𝑉 𝐴 = 𝑄 𝐴 Mass Flow Rate, 𝑚 [kg/s, lbm/min, mass/time] 𝑚 = 𝐴 𝜌𝑉𝑑𝐴 = rAQ (How to measure average over time Δ𝑡) 𝑚 𝐴 = Δ𝑚 Δ𝑡 ; Average Density: 𝜌 𝐴 = 𝑚 𝑄 Speed: VA [m/s] = 𝑄/𝐴 = 𝑚 /ArA

Many Flow Rate Measurement Devices Turbine Rotameters (variable area) Laminar Flow Coriolis Vortex (Lab 11) Each relies on different phenomena When choosing, consider Cost, Stability of calibration, Imprecision, Dynamic response, Flow resistance

Variable-Area Meters Three varieties Venturi Tube Nozzle Orifice Plate Three varieties All cause fluid to accelerate and pressure to decrease In the pipe the pressure, diameter and area are denoted: P1, A, D At Throat: P2, a, d (all smaller than pipe values) Diameter Ratio: b = d/D < 1 To use, measure pressure drop between pipe and throat using a pressure transmitter (Reading) Use standard geometries and pressure port locations for consistent results All three restrict the pipe and so reduce flow rate compared to no device

Venturi Tube Insert between pipe sections Convergent Entrance: smoothly accelerates the flow reduces pressure Diverging outlet (diffuser) decelerates the flow gradually, avoiding recirculating zones, and increases (recovers) pressure Reading DP increases as b = d/D decreases Smallest flow restriction of the three But most expensive

Orifice Plate Does not increase pipe length as much as Venturi Vena contracta Does not increase pipe length as much as Venturi Rapid flow convergence forms a very small “vena contracta” through which all the fluid must pass No diffuser: flow “separates” from wall forming a turbulent recirculating zone that causes more drag on the fluid than a long, gradual diffuser Least expensive of the three but has a the largest flow restriction (permanent pressure drop)

Nozzles 1-b2 Permanent pressure drop, cost and size are all between the values for Ventrui tubes and orifice plates.

Pressure Drop, Inclined in gravitational field, g z2 z1 Mass Conservation: 𝑚 1 = r1A1V1 = 𝑚 2 = r2A2V2 where V1 and V2 are average speeds For r1= r2 (incompressible, liquid, low speed gas) V1 = V2(A2 /A1) = V2[(pd2/4) /(pD2/4)] = V2(d/D)2 = V2b2

Momentum Conservation: Bernoulli z2 z1 Incompressible, inviscid, steady 𝑉 1 2 2 + 𝑃 1 𝜌 +𝑔 𝑧 1 = 𝑉 2 2 2 + 𝑃 2 𝜌 +𝑔 𝑧 2 𝜌 𝑃 1 +𝜌𝑔 𝑧 1 − 𝑃 2 +𝜌𝑔 𝑧 2 = 𝜌 2 𝑉 2 2 − 𝑉 1 2 A transmitter at z = 0 will measure Δ𝑃= 𝑃 1 +𝜌𝑔 𝑧 1 − 𝑃 2 +𝜌𝑔 𝑧 2 = LHS (Reading) Lines must be filled with same fluid as flowing in pipe Δ𝑃= 𝜌 𝑉 2 2 2 1− 𝑉 1 2 𝑉 2 2 = 𝜌 𝑉 2 2 2 1− ( 𝑉 2 𝛽 2 ) 2 𝑉 2 2 = 𝜌 𝑉 2 2 2 1−b4 = 𝜌 𝑄 𝐴 2 2 2 1−b4 Transfer Function (Reading versus Measurand) Measurand Reading

Ideal (inviscid) Transfer Function ∆𝑃 [𝑃𝑎 wDP 𝜕∆𝑃 𝜕𝑄 ∆𝑃 wQ Q [ 𝑚 3 s ] 𝑄 Δ𝑃= 1−b4 𝜌 2 𝐴 2 2 𝑄 2 : Non-linear (like Pitot probe) Sensitivity (slope) 𝜕∆𝑃 𝜕𝑄 increases with 𝑄 Input resolution 𝑤 𝑄 = 𝑤 ∆𝑃 / 𝜕∆𝑃 𝜕𝑄 is smaller (better) at large Q than at small values Better for measuring large Q than for small ones

How to use the gage? Invert the transfer function: Δ𝑃= 1−b4 𝜌 2 𝐴 2 2 𝑄 2 Get: 𝑄=𝐶 𝐴 2 2Δ𝑃 𝜌 1−β4 = 𝐶(pd2/4) 1−β4 2Δ𝑃 𝜌 C = Discharge Coefficient Effect of viscosity, not always negligible C = fn(ReD, b = d/D, exact geometry and port locations) 𝑅𝑒 𝐷 = 𝑉 1 𝐷𝜌 𝜇 = 𝑚 𝜌 𝜋 4 𝐷 2 𝐷𝜌 𝜇 = 4 𝑚 𝜋𝐷𝜇 = 4𝜌𝑄 𝜋𝐷𝜇 Problem: Need to know Q to find Q, so iterate Assume C ~ 1, find Q, then Re, then C, then check…

Discharge Coefficient Data from Text Nozzle: page 344, Eqn. 10.10 C = 0.9975 – 0.00653 10 6 𝛽 𝑅𝑒 𝐷 0.5 (see restrictions in Text) Orifice: page 349, Eqn. 10.13 C = 0.5959 + 0.0312b2.1 - 0.184b8+ 91.71 𝛽 2.5 𝑅𝑒 𝐷 0.75 (0.3 < b < 0.7)

Water Properties Be careful reading headings and units

Example: Problem 10.15, page 384 A square-edge orifice meter with corner taps is used to measure water flow in a 25.5-cm-diameter ID pipe. The diameter of the orifice is 15 cm. Calculate the water flow rate if the pressure drop across the orifice is 14 kPa. The water temperature is 10°C. Solution: Identify, then Do ID What type of meter? What fluid? Given pressure drop, find flow rate

Solution Equations 𝑄=𝐶 𝐴 2 2Δ𝑃 𝜌 1−β4 = 𝐶(pd2/4) 1−β4 2Δ𝑃 𝜌 b = d/D C = 0.5959 + 0.0312b2.1 - 0.184b8+ 91.71 𝛽 2.5 𝑅𝑒 𝐷 0.75 (0.3 < b < 0.7) 𝑅𝑒 𝐷 = 𝑉 1 𝐷𝜌 𝜇 = 𝑚 𝜌 𝜋 4 𝐷 2 𝐷𝜌 𝜇 = 4 𝑚 𝜋𝐷𝜇 = 4𝜌𝑄 𝜋𝐷𝜇