Download ME 322: Instrumentation Lecture 6

ME 322: Instrumentation
Lecture 12
February 12, 2016
Professor Miles Greiner
Flow rate devices, variable area, non-linear transfer
function, standards, iterative method
(Needed a few more minutes to complete example)
Announcement/Reminders
• HW 4 due now (please staple)
• Monday – Holiday
• Wednesday – HW 5 due
– Review for Midterm
• Friday, Midterm
Regional Science Olympiad
• Tests middle and high school teams on various science
topics and engineering abilities
• Will be held 8 am to 4 pm Saturday, March 5th 2016
– On campus: SEM, PE and DMS
• ME 322 students who participate in observing and
judging the events for at least two hours (as reported)
will earn 1% extra credit.
• To sign up, contact Rebecca Fisher, [email protected],
(775) 682-7741
– by Wednesday, February 24
• Details
– You cannot get extra-credit in two courses for the same
work.
– If you sign-up but don’t show-up you will loose 1%!
Fluid Flow Rates
VA
𝑉 𝑟
A
dA
V, r
𝑚=
𝑑𝑚 =
𝐴
𝜌𝑉𝑑𝐴
𝐴
• Within a conduit cross section or “area region”
– Pipe, open ditch or channel, 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, Volume/time]
– Q=
𝐴
• 𝑄𝐴 =
𝑉𝑑𝐴 = 𝑉𝐴 𝐴 (How to measure average 𝑄𝐴 over time Δ𝑡?)
Δ𝑉
;
Δ𝑡
Average speed 𝑉𝐴 =
𝑄
𝐴
• Mass Flow Rate, 𝑚 [kg/s, lbm/min, mass/time]
– 𝑚=
𝐴
• 𝑚𝐴 =
𝜌𝑉𝑑𝐴 = rAQ (How to measure average 𝑚𝐴 over time Δ𝑡?)
Δ𝑚
; Average
Δ𝑡
Density: 𝜌𝐴 =
𝑚
𝑄
• Speed: VA [m/s] = 𝑄/𝐴 = 𝑚/ArA
Many Flow Rate Measurement Devices
Rotameters (variable area)
Turbine
Laminar Flow
Vortex (Lab 11)
Coriolis
• Each relies on different phenomena
• When choosing, consider
– Cost, Stability of calibration, Imprecision, Dynamic response, Flow
resistance
– The measurement device can affect the quantity being measured
Variable-Area Meters
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 variable-area meters
– But most expensive
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 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
2
1
z2
z1
• Mass Conservation:
– 𝑚1 = r1A1V1 = 𝑚2 = 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
2
Momentum
Conservation:
Bernoulli
1
z2
z1
• Incompressible, inviscid, steady
•
𝑉1 2
2
+
𝑃1
𝜌
+ 𝑔𝑧1 =
𝑉2 2
2
+
𝑃2
𝜌
+ 𝑔𝑧2 𝜌
𝜌
• 𝑃1 + 𝜌𝑔𝑧1 − 𝑃2 + 𝜌𝑔𝑧2 = 𝑉2 2 − 𝑉1 2
2
• A differential pressure gage at z = 0 will measure
– Δ𝑃 = 𝑃1 + 𝜌𝑔𝑧1 − 𝑃2 + 𝜌𝑔𝑧2 = LHS (Reading)
– Lines must be filled with same fluid as flowing in pipe
• Δ𝑃 =
Reading
𝜌𝑉22
2
1−
2
𝑉1
𝑉2 2
=
𝜌𝑉22
2
1−
𝛽 2 )2
(𝑉2
𝑉2 2
=
𝜌𝑉22
2
– Transfer Function (Reading versus Measurand)
1 − b4 =
Measurand
2
𝜌 𝑄 𝐴2
2
1 − b4
Ideal (inviscid) Transfer Function
∆𝑃 [𝑃𝑎]
𝜕∆𝑃
𝜕𝑄
wDP
∆𝑃
wQ
• Δ𝑃 = 1
𝜌
4
−b
𝑄2
2
2𝐴2
– Sensitivity (slope)
𝑄
𝑚3
Q[ ]
s
: Non-linear (like Pitot probe)
𝜕∆𝑃
𝜕𝑄
– Input resolution 𝑤𝑄 =
than at small values
= 1−
b4
𝜕∆𝑃
𝑤∆𝑃 /
𝜕𝑄
𝜌
𝑄
𝐴22
increases with 𝑄
is smaller (better) at large 𝑄
• Better for measuring large 𝑄’s than for small ones
How to use the gage?
• Invert the transfer function: Δ𝑃 = 1 −
• Get: 𝑄 = 𝐶𝐴2
2Δ𝑃
𝜌 1−β4
=
𝐶(pd2/4)
1−β4
b4
𝜌
2
𝑄
2𝐴22
2Δ𝑃
𝜌
• C = Discharge Coefficient
– Effect of viscosity inside tubes is not always negligible
– C = fn(ReD, b = d/D, exact geometry and port locations)
– 𝑅𝑒𝐷 =
𝑉1 𝐷𝜌
𝜇
=
𝑚
𝜋
𝜌 4 𝐷2
𝐷𝜌
𝜇
=
4𝑚
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
106 𝛽
𝑅𝑒𝐷
0.5
(see restrictions in Text)
• Orifice: page 349, Eqn. 10.13
– C = 0.5959 +
0.0312b2.1 -
91.71𝛽 2.5
8
0.184b +
0.75
𝑅𝑒𝐷
(0.3 < b < 0.7)
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 = 𝐶𝐴2
2Δ𝑃
𝜌 1−β4
=
𝐶(pd2/4)
1−β4
2Δ𝑃
𝜌
• b = d/D
• C = 0.5959 + 0.0312b2.1 - 0.184b8+
• 𝑅𝑒𝐷 =
𝑉1 𝐷𝜌
𝜇
=
𝑚
𝜋
𝜌 4 𝐷2
𝐷𝜌
𝜇
=
4𝑚
𝜋𝐷𝜇
91.71𝛽 2.5
0.75
𝑅𝑒𝐷
=
4𝜌𝑄
𝜋𝐷𝜇
(0.3 < b < 0.7)
Water Properties
• Be careful reading headings and units