ME 322: InstrumentationLecture 27

Midterm Review

March 1, 2014Professor Miles Greiner

Announcements/Reminders• HW 9 is due now• This week: Lab 9 Transient Temperature Response• Next week: Open-ended extra-credit Lab 9.1

– Proposal due Friday, 4/3/15 (if you want to participate)– 1%-of-grade extra-credit for active participation

• Midterm II, Wednesday, April 1, 2015– Marissa Tsugawa will hold a review: See WebCampus

Midterm II• Handout last year’s exam

– Neither Marissa nor I will work those problems– These specific problems will not be on this year’s exam

• Open book + bookmarks + 1 pages of notes• ~4 problems, with parts• Focus on materials not covers on Midterm I• Study

– HW, Lab Calculations, Notes, Text reading– If you missed a lecture you may want to talk to student’s who attended,

since some information is not on the lecture slides• Units, significant figures

– Especially on statistical-analysis and propagation-of-uncertainty problems• Know how to used your calculator• Special needs: See Dr. Fayed and me today (to confirm)

– Do not come to class if you are schedule to go to the DRC

Fluid Speed and Uncertainty

V

PSPS

PT > PS PT > PS

• Pitot and Pitot-Static Probes– (power product?)

• C accounts for viscous effects, which are small– Assume C = 1 unless told otherwise

• Need to determine pressure difference and fluid density

Fluid Density and Uncertainty• Ideal Gases

– (power product?) • R = Gas Constant = RU/MM,

– Ru = Universal Gas Constant = 8.314 kJ/kmol K– MM = Molar Mass of the gas– RAir = 0.2870 kPa-m3/kg-K

• T[K] = T[°C] + 273.15, Gas Absolute Temperature • P, Gas static pressure

– Can incorporate into speed calculation• (power product?)

• Liquids– (Tables)

Water Properties

• Be careful reading headings and units

Pressure Transmitter Measurement

– = 998.7 kg/m3, g = 9.81 m/s2

– FS = (3, 40 or ? inch)• Stated uncertainty: 0.25% (or ?) of Full Scale

– Certainty level = ? (need to be told on test)– For FS = 3 inch WC

• PFS = rWghFS = (998.7 kg/m3)(9.81 m/s2) (3 inch) = 746.6 Pa• wP = 0.0025 PFS = 1.9 Pa

– For FS = 40 inch WC• PFS = rWghFS = (998.7 kg/m3)(9.81 m/s2) (40 inch) = 9954 Pa• wP = 0.0025 PFS = 25 Pa

Static Pressure• PStat = PATM – PG (power product or linear sum?)

– Uncertainty: • For general linear sums

–

Volume FlowRate

• Variable Area Meter (venturi): – Need , (throat), (iterate)– This expression needs pipe and throat dimensions

• Presso Formulation: – = – : Given by manufacturer

• Don’t need A2 or b

Discharge Coefficient Data from Text

• Nozzle: page 344, Eqn. 10.10– C = 0.9975 – 0.00653 (see restrictions in Text)

• Orifice: page 349, Eqn. 10.13– C = 0.5959 + 0.0312b2.1 - 0.184b8+ (0.3 < b < 0.7)

Centerline-Speed/Volume-Flow-Rate Consistency

• Estimated centerline-speeds for a given volume flow rate Q– Slug Flow: VS = Q/A– Parabolic Speed Profile: VP = 2VS

Temperature Measurements

TT

TT

TS

• Thermocouple, metal pair AB

• from page 300 (bookmark)• Standard Uncertainty, certainty level = ? (need to be told)

– 2.2°C for T < 314°C– 0.7% of reading for T > 314°C

𝑉 𝑜𝑢𝑡

+¿

−

• Not quite linear• Different sensitivities

(slopes)

Transfer Function (Type-J-TC/DRE–TC-J TC)

• For TS < 400C– (linear)

• ; = 500– Inverted transfer function: TS = (40°C/V)*VSC

• Conditioner Provides– Reference Junction Compensation (not sensitive to TT)– Amplification (Allows normal DVM or computer acquisition to be used) – Low Pass Filtration (Rejects high frequency RF noise) – Linearization (Easy to convert voltage to temperature)– Galvanic Isolation (TC can be used in water environments)

ReadingVSC [V]

Measurand, T [°C]00

400

10? Out of

rangeTransferFunction

𝑆𝑆𝐶=𝜕𝑉 𝑆𝐶

𝜕𝑇

A/D Converter Characteristics• Sampling Rate fS [samples/second]

– Sampling time DtS = 1/fS [seconds/sample]• Full-scale range VRL ≤ V ≤ VRU

– FS = VRU - VRL

• Number of Bits N – Converter resolves full-scale range into 2N sub-ranges– Smallest voltage change that can be detected: FS/2N

• Input Resolution Error, IRS– Random error due to digitization process

• Inside full-scale range: • Outside range: ∞

• Absolute Voltage Accuracy, AVA– Larger than IRS, Includes calibration and other errors

Numerical Differentiation of Discretely Sampled Signals

• First-order Centered Differencing

• is the differentiation time step [sec]– , – is the sampling time – m = integer (1, 2, or ?)

• What is the best value for m (1, 10, 20, ?)– Compromise between responsiveness and

sensitivity to random errors

t [sec] T [oC]0 20.599

0.001 20.3870.002 20.6460.003 20.3160.004 20.9050.005 20.5280.006 20.7160.007 20.8580.008 20.6930.009 20.9050.01 20.6690.011 20.8110.012 20.8110.013 20.7160.014 20.2460.015 20.6460.016 20.3870.017 20.3870.018 20.6930.019 20.222

Fourier Transform of Discretely Sampled Signal

• Any function V(t), over interval 0 < t < T1, may be decomposed into an infinite sum of sine and cosine waves

– ,

• Discrete frequencies: , n = 0, 1, 2, … ∞ (integers) (not continuous) – Only admits modes for which an integer number of oscillations span the total sampling time T1.

• The root-mean-square (RMS) coefficient for each mode quantifies its total energy content for a given frequency (from sine and cosine waves)

• LabVIEW find versus numerically– When processing, need to add these frequencies: 0, , ,

0 t T1

Vn = 0 n = 1 n = 2

sine

cosine

Examples (ME 322 Labs)

• Converts signals from time-domain to frequency-domain (spectral energy content)

Function Generator100 Hz sine wave

Unsteady Speed AirDownstream froma Cylinder in Cross

Flow

Time Domain Frequency Domain

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 2 4 6 8 10

Time t [sec]

Dim

ensi

nole

ss A

ccel

erat

ion,

g t1 = 1.14 sec, a1 = 0.314 g

t2 = 5.88 sec, a2 = 0.152 g

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 10 20 30 40 50 60f [Hz]

a rm

s [g'

s]Damped VibratingCantilever Beam

Upper, Lower, and Resolution Frequencies

• If a signal is sampled at a rate of fS for a total time of T1 , the highest and lowest finite frequencies that can be accurately detected are: – (f1 = 1/T1) < f < (fN = fS/2)

• The frequency resolution – Smallest frequency change that can be detected – f1 = 1/T1 (same as minimum frequency)

• To reduce lowest frequency (and increase frequency resolution), increase total sampling time T1

• To observe higher frequencies, increase the sampling rate fS.

How to predict indicated (or Alias) Frequency?

• fa = fm if fs > 2fm • Otherwise using folding chart on page 106 (bookmark)• fN = fs/2 is the maximum frequency that can be accurately observed using

sampling frequency fs.

Maximum frequency that canbe accurately measured usingsampling frequency fS .

TC Response to Temperature Step Change

• At time t = t0 a thermocouple at TI is put into a fluid at TF. – Error: E = TF – T

• Theory for a lumped (uniform temperature) TC predicts:– Dimensionless Error: – (spherical thermocouple)

T

tt = t0

TI

TF

Error = E = TF – T ≠ 0

T(t)

TI

TF

Environment Temperature

Initial Error EI = TF – TI

𝜌 ,𝑐 ,𝐷

h

Slower TCFaster

To find heat transfer coeff. h from T vs t Data

• If given T versus t data in the exponential decay period• Calculate and for each time• Find the least-squares coefficients a and b of

– Calculate (power product?), ?• Assume uncertainty in b is small compared to other components• Find and for TC from appendix

t [sec] T [oC] qBoil ln(qBoil)

TC Response to Sinusoidally-Varying Temp

• Environment Temp: • TC Temp:

– TC will have same mean temperature and frequency ()– TC temperature amplitude will be attenuated and delayed

• Minimal if , where , otherwise:

T

tD

High Temperature (combustion) Gas Measurements

• Radiation heat transfer is important and can cause errors• Convection heat transfer to the sensor equals radiation heat transfer

from the sensor– Q = Ah(T∞ – TS) = Ase(TS

4 –T04)

• s = Stefan-Boltzmann constant = 5.67x10-8W/m2K4

• e = Sensor emissivity (surface property ≤ 1)• T[K] = T[C] + 273.15

• Correction: = T∞ – TS = (se/h)(TS4 –T0

4)– How does uncertainty in e and h affect ?– Tgas= TS +

QConv=Ah(Tgas– TS)

TS

QRad=Ase(TS4 -TW

4)

T∞

T0

Sensorh, TS, A, e

Conduction through Support (Fin Configuration)

• Sensor temperature TS will be between those of the fluid T∞ and duct surface T0

– Support: cross sectional area A, parameter length P, conductivity k– Convection heat transfer coefficient between gas and support h

• Fin Temperature Profile (from conduction heat transfer analysis):– – (dimensionless length)

• Dimensionless Tip Temperature Error from conduction– , (want this to be small); – ; Adjunstment – Decreases as

• L, h and P increase, k and A decrease

• If both conduction and radiation corrections are required then – +

T∞

h xLA, P, k

T0

TS