# ME 322: Instrumentation Lecture 15

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

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

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

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?

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

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

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 )

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

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

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

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

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

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%)

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.

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

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

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

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

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

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

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

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