ME 322: Instrumentation Lecture 12

1. ME 322: InstrumentationLecture 12 February 14, 2014 Professor Miles Greiner

2. Announcement/Reminders • HW 4 due now (please staple) • Monday – Holiday • Wednesday – HW 5 due and review for Midterm • Friday, Feb. 21, 2014 Midterm • How was lab this week? • Any problems or confusion?

3. Fluid Flow Rates • Within a conduit cross section or “area region” • Pipe, open channel, river, blood vessel (not steady) • V and r can vary over cross section • Mass Flow Rate, [kg/s, lbm/min, mass/time] • = rAQ • Average Density: rA[kg/m3] • Volume Flow Rate, Q [m3/s, gal/min, cc/hour, Vol/time] • Q = = VAA • Averages • Density: rA = • Speed: VA[m/s] = Q/A = A V, r dA

4. Many Flow Rate Measurement Devices • Each relies on different phenomena • When choosing, consider • cost, stability of calibration, imprecision, dynamic response, flow resistance Turbine Rotameters (variable area) Laminar Flow Coriolis Vortex (Lab 11)

5. Variable-Area Meters Nozzle Venturi Tube Orifice Plate • Three varieties • All cause fluid to accelerate and pressure to decrease • In Pipe pressure, diameter and area, 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 flow (compared to no device), and so reduce flow rate

6. 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

7. Orifice Plate 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 flow • No diffuser: • flow “separates” from wall forming a turbulent recirculating zone that causes more drag on the fluid than a long, gradual diffuser • Lest expensive of the three but has a the largest permanent pressure drop

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

9. Flow restriction, Inclined in gravitational field, g • Mass Conservation: • r1A1V1 = r2A2V2 • where V1 and V2 are average speeds • For r1r2 • V1= V2(A2 /A1) = V2[(pd2/4)/(pD2/4)] = V2(d/D)2 = V2b2 z2 z1

10. Momentum Conservation: Bernoulli • Incompressible, inviscid, steady • A transmitter at z = 0 will measure • = LHS (Reading) • Lines must be filled with same fluid as flowing in pipe • Transfer Function z2 z1 Measurand Reading

11. Ideal (inviscid) Transfer Function wDP • : Non-linear • Input resolution is smaller (better) at large Q than at small values • Better for measuring large Q than for small ones wQ Q

12. Invert Transfer Function • Invert the transfer function: • Get: • C = Discharge Coefficient • Effect of viscosity, not always negligible • C = fn(ReD, b = d/D, exact geometry and port locations) • Problem: Need to know Q to find Q, so iterate • Assume C ~ 1, find Q, then Re, then CD and check…

13. Discharge Coefficient Data from Text • Nozzle: page 344, Eqn. 10.10 • C = 0.9975 – 0.00653 (see restrictions in Text) • Orifice: page 349, Eqn. 10.13 • C = 0.5959 + 0.0312b2.1 - 0.184b8+ (0.3 < b < 0.7)

14. Water Properties • Be careful reading headings and units