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

2. 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.

3. 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

4. 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

5. 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

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 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)

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

9. Pressure Drop, Inclined in gravitational field, gz2 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

10. Momentum Conservation: Bernoulliz2 z1
Incompressible, inviscid, steady
𝑉 𝑃 1 𝜌 +𝑔 𝑧 1 = 𝑉 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 − 𝑉 𝑉 = 𝜌 𝑉 − ( 𝑉 2 𝛽 2 ) 2 𝑉 = 𝜌 𝑉 −b4 = 𝜌 𝑄 𝐴 −b4
Transfer Function (Reading versus Measurand)
Measurand
Reading

11. Ideal (inviscid) Transfer Function
∆𝑃 [𝑃𝑎
wDP
𝜕∆𝑃 𝜕𝑄 ∆𝑃 wQ
Q [ 𝑚 3 s ] 𝑄
Δ𝑃= 1−b4 𝜌 2 𝐴 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

12. How to use the gage?
Invert the transfer function: Δ𝑃= 1−b4 𝜌 2 𝐴 𝑄 2
Get: 𝑄=𝐶 𝐴 Δ𝑃 𝜌 1−β4 = 𝐶(pd2/4) 1−β Δ𝑃 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 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…

13. Discharge Coefficient Data from Text
Nozzle: page 344, Eqn
C = – 𝛽 𝑅𝑒 𝐷 (see restrictions in Text)
Orifice: page 349, Eqn
C = b b 𝛽 𝑅𝑒 𝐷 (0.3 < b < 0.7)

14. Water Properties
Be careful reading headings and units

15. 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

16. Solution Equations 𝑄=𝐶 𝐴 2 2Δ𝑃 𝜌 1−β4 = 𝐶(pd2/4) 1−β4 2Δ𝑃 𝜌 b = d/D
C = b b 𝛽 𝑅𝑒 𝐷 (0.3 < b < 0.7)
𝑅𝑒 𝐷 = 𝑉 1 𝐷𝜌 𝜇 = 𝑚 𝜌 𝜋 4 𝐷 𝐷𝜌 𝜇 = 4 𝑚 𝜋𝐷𝜇 = 4𝜌𝑄 𝜋𝐷𝜇