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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

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**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

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**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

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**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?

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**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

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**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

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**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 )

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**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

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**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

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**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

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**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

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**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

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**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%)

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**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.

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**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

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**Summary Before Experiment Use hand held barometer to measure PATM TATM**

𝑊 𝑃 𝐴𝑇𝑀 =0.5 𝑘𝑃𝑎 TATM 𝑊 𝑇 𝐴𝑇𝑀 =1°C

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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

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**Consistency Check For a given volume flow rate Q**

VS = Q/A VP = 2VS What area should we use APipe or ATube ?

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

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**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

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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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ME 322: Instrumentation Lecture 39

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