1. ME 322: Instrumentation Lecture 15
February 23, 2015
Professor Miles Greiner
Relating speed and flowrate, Lab 6 equipment, Presso flow coefficient, linear sum uncertainty

2. Announcements/Reminders
HW 6 due Friday
Lab today (Lab 5) but not the rest of this week
Career Fair, Thursday, February 26, 2015
Internships
Prepare for permanent employment next year

3. Pipe Speed and Volume Flow Rate
Centerline speed increases in the entrance region
In fully-developed flow, speed profile V(r) is
Parabolic in laminar flow (Re <~2000), and develops slowly
Flatter in Turbulent flow (Re > 104), and develops rapidly

4. Speed and Flow Rate Consistency
Does the centerline speed increase with flow rate?
Is there a unique centerline speed for every volume flow rate and every location?
What does this relationship dependent on?
For a given volume flow rate, what is the range of centerline speeds in which we expect VC to be?

5. Possible Centerline Speeds
At the pipe entrance and for fully-developed turbulent flow, the velocity profile is relatively flat compared to fully-developed laminar flow
VC >~ VSlug = Q/A
For fully-developed laminar flow, we expect the velocity profile to be parabolic
𝑉 𝑉 𝑃 = 𝑟 0 2 − 𝑟 2 𝑟 0 2 , where
𝑟 0 is the pipe inner radius, and
𝑉 𝑃 is the centerline velocity (for a parabolic profile).
Relationship between speed and volume flow rate
𝑄= 𝐴 𝑉𝑑𝐴 = 0 𝑟 0 𝑉 𝑝 𝑟 𝑟 0 2 − 𝑟 2 2𝜋𝑟𝑑𝑟
In HW find that VP = 2VSlug
It’s reasonable to expect (VSlug = Q/A) < VC < (VP = 2VSlug)
In Lab 6 measure Q and VC in a small wind tunnel

6. Lab 6 Air Volume Flow Rate and Centerline Speed in a Wind Tunnel
Plexiglas Tube and Schedule-40 Pipe have different diameters
Control flow rate using a variable-speed blower
Cover blower exit for very low speeds
For a range of flow rates, measure
Volume flow Q rate using a Presso Venturi Tube (in pipe)
Centerline speed VC using a Pitot-Static Tube (in Plexiglas tube)
For both measure pressures difference using calibrated transmitters/digital multimeters
Both VC and Q increase with blower flow rate
Check to see if VS < VC < VP

7. Venturi Tube Inverted transfer function: 𝑄= 𝐶 𝐴 2 1− 𝛽 4 2∆𝑃 𝜌
Need 𝛽= 𝑑 𝐷 , 𝐴 2 = 𝜋 4 𝑑 2 (throat), 𝐶=𝑓 𝑅𝑒 𝐷 = 4𝜌𝑄 𝜋𝐷𝜇
These are all characteristics of the venturi.
But 𝐶=𝑓 𝑅𝑒 𝐷 is based on knowing d and D.
Presso Formulation:
𝑄= 𝐴 1 𝐴 2 𝐴 1 𝐶 1− 𝛽 ∆𝑃 𝜌 = 𝐴 1 𝐶 𝛽 − 𝛽 ∆𝑃 𝜌 = 𝜋 4 𝐷 2 𝐾 Presso 2∆𝑃 𝜌
𝐾 Presso = 𝐶 𝐷 𝛽 − 𝛽 4 =𝑓𝑛 𝑅𝑒 𝐷 : Given by manufacturer
Only need D (pipe) and KPresso (not ~1, but don’t need to find 𝛽 or 𝐴 2 )

8. In Lab 6 use a Presso Venturi Tube
In Lab 6 use 2-inch schedule 40 Pipe, ID = inch
Presso Data Sheet – Page 10, Venturi # 38
𝐾 𝑝 = ± 2% ( 𝑝 𝑝 =?; b = , but don’t need to this)
Valid for 54,000 < 𝑅𝑒 < 137,000 (ReD or Red?)
𝑄= 𝜋 4 𝐷 𝑃𝑖𝑝𝑒 2 𝐾 𝑝 2∆𝑃 𝜌
Easier to use than 𝑄= 𝐶 𝐴 − 𝛽 ∆𝑃 𝜌

9. How to find VC and Q (and uncertainties)?
Pitot-Static Probe
𝑉 𝑐 =𝐶 2 𝑃 𝑃 𝜌 Air (power product?)
Presso Venturi Tube
𝑄= 𝜋 4 𝐷 𝑝𝑖𝑝𝑒 2 𝐾 𝑝𝑟𝑒𝑠𝑠𝑜 2 𝑃 𝑣 𝜌 𝐴𝑖𝑟 (power product?)
Both need air-density
𝜌 𝐴𝑖𝑟 = 𝑃 𝑆𝑡𝑎𝑡 𝑅 𝐴𝑖𝑟 𝑇 (power product?)
RAir = kPa-m3/kg-K
Need to measure
Pressure differences PP, PV, and PStat
Air Temperature, T

10. Instrument Schematic To measure PATM and TATM
Variable Speed
Blower
Plexiglas
Tube
Pitot-Static Probe VC
Venturi Tube Q
Barometer
PATM
TATM
Pipe
DPipe
DTube PV - +
Static
40 in WC
Total PP PG IV -
Atm + - +
3 in WC IP IG
40 in WC
To measure PATM and TATM
Use hand-held digital-barometer
Is PStat <, = or > than PATM?
Use 40-in-WC transmitter to find Gage Pressure PG = PATM – PS
PS = PATM - PG
To measure PP
Use 3-in-WC transmitter
To measure PV
Use 40-in-WC transmitter

11. Inlet Pressure and Temperature
Fisher Scientific™ Traceable™ Hand-Held Digital Barometer
Barometric pressure, PATM
Uncertainty: 𝑤 𝑃 𝐴𝑇𝑀 = 5 mbar = 0.5 kPa = 500 Pa (95%?)
Units: 1 bar = 105 Pa; so 1 mbar = 100 Pa = 0.1 kPa
Atmospheric Temperature, TATM
Assume: 𝑤 𝑇 = 1°C (95%?)
T[K] = T[°C]
Assume tunnel and atmospheric temperatures are the same

12. Pressure Transmitter Uncertainty
𝑃= 𝜌 𝑊 𝑔ℎ= 𝜌 𝑊 𝑔(𝐹𝑆) 𝐼−4 𝑚𝐴 16 𝑚𝐴
𝜌 𝑊 = kg/m3, g = 9.82 m/s2
FS = (3 or 40 inch) 2.54 𝑐𝑚 1 𝑖𝑛𝑐ℎ 1 𝑚 100 𝑐𝑚 = 𝑜𝑟 𝑚
Manufacturer stated uncertainty: 0.25% Full Scale
(68%?)
For FS = 3 inch WC
PFS = rWghFS =
(998.7 kg/m3)(9.81 m/s2) (3 inch) 2.54 𝑐𝑚 1 𝑖𝑛𝑐ℎ 1 𝑚 100 𝑐𝑚 = Pa
wP = PFS = 1.9 Pa
For FS = 40 inch WC
(998.7 kg/m3)(9.81 m/s2) (40 inch) 2.54 𝑐𝑚 1 𝑖𝑛𝑐ℎ 1 𝑚 100 𝑐𝑚 = 9954 Pa
wP = PFS = 25 Pa

13. Static Pressure PStat = PATM – PG Inputs
Use for 𝜌 𝐴𝑖𝑟 = 𝑃 𝑆𝑡𝑎𝑡 𝑅 𝐴𝑖𝑟 𝑇 , RAir = kPa-m3/kg-K
Want kPa
Inputs
PATM
Measure using barometer
𝑤 𝑃 𝐴𝑇𝑀 = 500 Pa = 0.5 kPa (68%)
PGAGE
Measure using 40 inch WC gage
𝑤 𝑃 𝐺𝐴𝐺𝐸 = 25 Pa = kPa (68%)

14. Static Pressure Uncertainty
PStat = PATM – PG (power product?)
Need to use general formula for likely uncertainty:
𝑤 𝑃 𝑆𝑡𝑎𝑡 2 = 𝑖= 𝛿 𝑃 𝑆𝑡𝑎𝑡 𝛿 𝑥 𝑖 𝑤 𝑖 2
= 𝛿 𝑃 𝑆𝑡𝑎𝑡 𝛿 𝑃 𝐴𝑇𝑀 𝑤 𝑃 𝐴𝑇𝑀 𝛿 𝑃 𝑆𝑡𝑎𝑡 𝛿 𝑃 𝐺 𝑤 𝑃 𝐺 2
= 1 𝑤 𝑃 𝐴𝑇𝑀 −1 𝑤 𝑃 𝐺 2
= kPa − kPa 2
𝑊 𝑃 𝑆𝑡𝑎𝑡 = 𝑘𝑃𝑎
Square of absolute uncertainty in result is sum of squares of absolute uncertainty in inputs times coefficient.

15. General Expression Likely Error of “Linear Sums”
𝑅=𝑎𝑋+𝑏𝑌+𝑐𝑍+…+= 𝑐 𝑖 𝑋 𝑖
𝑤 𝑅 2 = 𝜕𝑅 𝜕 𝑋 𝑖 𝑤 𝑖 2 = 𝜕𝑅 𝜕𝑋 𝑤 𝑋 𝜕𝑅 𝜕𝑌 𝑤 𝑌 𝜕𝑅 𝜕𝑍 𝑤 𝑍 2 +…
𝑤 𝑅 2 = 𝑎 𝑤 𝑋 𝑏 𝑤 𝑌 𝑐 𝑤 𝑍 2 +…
𝑤 𝑅 2 = 𝑐 𝑖 𝑤 𝑖 2

16. Summary Before Experiment Use hand held barometer to measure PATM TATM
𝑊 𝑃 𝐴𝑇𝑀 =0.5 𝑘𝑃𝑎
TATM
𝑊 𝑇 𝐴𝑇𝑀 =1°C

17. During Experiment
For each blower setting find the value and uncertainty of the
Static Pressure, PStat = Work on Board (WOB)
𝑊 𝑃 𝑆𝑡𝑎𝑡 = WOB
𝑊 𝑃 𝑆𝑡𝑎𝑡 = WOB
Air density 𝜌 𝐴𝑖𝑟 = WOB
𝑊 𝜌 𝐴𝑖𝑟 𝜌 𝐴𝑖𝑟 2 = WOB
Centerline speed 𝑉 𝑐 = WOB
𝑊 𝑉 𝑐 𝑉 𝑐 2 = WOB
Volume flow rate 𝑄= WOB
𝑊 𝑄 𝑄 2 = WOB

18. Consistency Check For a given volume flow rate Q
VS = Q/A
VP = 2VS
What area should we use
APipe or ATube ?

19. Measured Results
Determine speed and flow rate uncertainty for a range of blower speeds

21. Wind Tunnel Schematic Variable Speed Blower Plexiglas Tube
Pitot-Static
Probe, VC
Venturi
Tube, Q
Pipe
DPipe
DTube PV - +
Static
40 in WC
Total PP PG IV - -
Atm + +
3 in WC IP IG
40 in WC

22. During Experiment
For each blower setting find the value and uncertainty of the
Static Pressure, PStat = PATM – Pgage
𝑊 𝑃 𝑆𝑡𝑎𝑡 2 = 1 𝑊 𝑃 𝐴𝑇𝑀 −1 𝑊 𝑃 𝐺 2 = kPa − kPa 2
𝑊 𝑃 𝑆𝑡𝑎𝑡 = 𝑘𝑃𝑎
Air density 𝜌 𝐴𝑖𝑟 = 𝑃 𝑆𝑡𝑎𝑡 𝑅 𝐴𝑖𝑟 𝑇 𝐴𝑖𝑟
𝑊 𝜌 𝐴𝑖𝑟 𝜌 𝐴𝑖𝑟 2 = 1 𝑊 𝑃 𝑆𝑡𝑎𝑡 𝑃 𝑆𝑡𝑎𝑡 −1 𝑊 𝑇 𝐴𝑇𝑀 𝑇 𝐴𝑇𝑀 2
Centerline speed 𝑉 𝑐 =𝐶 2 𝑃 𝑝 𝜌 Air
𝑊 𝑉 𝑐 𝑉 𝑐 2 = 1 𝑊 𝐶 𝐶 𝑊 𝑃 𝑝 𝑃 𝑝 − 𝑊 𝜌 𝐴𝑖𝑟 𝜌 𝐴𝑖𝑟 2
Volume flow rate 𝑄= 𝜋 4 𝐷 𝑝𝑖𝑝𝑒 2 𝐾 𝑝𝑟𝑒𝑠𝑠𝑜 2 𝑃 𝑣 𝜌 𝐴𝑖𝑟
𝑊 𝑄 𝑄 2 = 2 𝑊 𝐷 𝑝𝑖𝑝𝑒 𝐷 𝑝𝑖𝑝𝑒 𝑊 𝐾 𝑝𝑟𝑒𝑠𝑠𝑜 𝐾 𝑝𝑟𝑒𝑠𝑠𝑜 𝑊 𝜌 𝑣 𝜌 𝑣 − 𝑊 𝜌 𝐴𝑖𝑟 𝜌 𝐴𝑖𝑟 2