ME 322: InstrumentationLecture 20

March 6, 2015

Professor Miles Greiner

myDAQ A/D converter, temperature uncertainty, First-order, centered numerical differentiation and random

errors

Announcements/Reminders• HW 7 due now

– Did the computers and software in the ECC work the way they are supposed to?

• HW 8 Due next Friday – Then Spring Break!

• Please complete the Lab Preparation Problems and fully participate in each lab– For the final you will repeat one of the last three

labs, including performing the measurements, and writing Excel, LabVIEW and PowerPoint, solo.

A/D Converter Characteristics• Full-scale range VRL ≤ V ≤ VRU

– FS = VRU - VRL

– For myDAQ the user can chose between two ranges• ±10 V, ±2 V (FS = 4 or 20 V)

• Number of Bits N – Resolves full-scale range into 2N sub-ranges– Smallest voltage change a conditioner can detect:

• DV = FS/2N

– For myDAQ, N = 16, 216 = 65,536• ±10 V scale: DV = 0.000,310 V = 0.305 mV = 305 mV• ±2 V scale: DV = 0.000,061 V = 0.061 mV = 61 mV

• Sampling Rate fS = 1/TS

– For myDAQ, (fS)MAX = 200,000 Hz, TS = 5 msec

Example• For a ±10 Volt, N = 2 bit A/D converter, what digitized

voltages will it report for -∞<V<+∞ ?– M = 2N = __ sub-ranges

• Break input range into __ steps

• IOUT can be 0, 1, 2, 3

– Step size

– How do we interpret IOUT (VDigitized) and what is its uncertainty?

-15 -10 -5 0 5 10 150

1

2

3

VIN [volts]

IOU

T

A/D ConverterTransfer Function

Input Resolution Error• The reported voltage is the center of the digitization sub-range in which the measured voltage is found to reside.

– So the maximum error is half the sub-range size.• Inside the FS voltage range• At edge or outside of FS range

– To avoid this, estimate the range of voltage that must be measured before conducting an experiment, and choose appropriate A/D converter and/or signal conditioners.

• The IRE is the uncertainty caused by the digitization process

Summary of myDAQ Uncertainties

• What are these?– AA: Maximum error of the voltage measurement reported by

the manufacturer for all voltage levels• At different temperatures

– MSVE: Maximum error measured at V = 0V for one device– IRE: Random error due to digitization process

• Which best characterizes voltage uncertainty?

Scale Absolute

Accurcacy 23°C

Absolute Accurcacy 18-28°C

Measurd Shorted

Voltage Error

Input Resolution Error (IRE)

±10 Volts 22.8 mV 38.9 mV 2.4 mV 0.15 mV±2 Volts 4.9 mV 8.6 mV 0.9 mv 0.03 mV

0.1% FS 0.2% FS 0.01 -0.02% FS 0.0008% FS

Lab 7 Boiling Water Temperature in Reno

• Water temperature uncertainty• Standard TC wire uncertainty

– Larger of 2.2°C or 0.75% of measurement– Note: 0.0075 x 293°C = 2.2°C

– For T < 293°C: wTC = 2.2°C; For T > 293°C: wTC = 0.0075*T

• For ±10 Volts, measured shorted voltage uncertainty MSVU = 0.0024V– For TC signal conditioner SSC = 0.025 V/°C

– wTsc = MSVU/SSC = 0.0024V/(0.025 V/°C) = 0.096°C

• 2.202°C ~ 2.2°C

A/D Converters can be used to measure a long series of very rapidly changing voltage

• Great for measuring a voltage signal – How voltage or measurand changes with time– Would be very difficult using a regular voltmeter

• Allows determination of– Rates of Change and – Spectral (Frequency) Content

• The voltage and time associated with each measurement has some error– It is associated with the centers of the voltage sub-range and

sampling time. – Additional systematic and random errors as well

• What can go wrong?

Example

• A small thermocouple at initial temperature T i is placed in boiling water at temperature TB

• Its measured temperature versus time T(t) is shown • What caused the temperature to change?

– What do you expect the time-dependent heat-transfer rate to the thermocouple [joules/sec = watts] to look like qualitatively?

– How can we determine it quantitatively?

T(t)

Ti

TB

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

1st Law of Thermodynamics

• How to estimate a time-derivative from a table of T versus t data?– is the sampling time step [sec] (TS)

• First-order numerical differentiation– Centered differencing

– is the differentiation time step [sec]

• , m = integer (1, 2, or ?)• Will we get the same result for different values

of m?– What is the best value for m? (1, 10, 20, ?)

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

U �̇�

Sample Data• Lab 9 Transient Thermocouple Measurements

– Download sample data– http://

wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%2009%20TransientTCResponse/LabIndex.htm

• Plot T vs t for t < 2 sec• Show how to evaluate and plot first-order centered

derivatives with different differentiation time steps– Plot dT/dt vs t for m = 1, 10, 50

• Slow T vs t – for 0.95s < t < 1.05s and 25°C < T < 45°C– How do random errors affect “local” and “time-averaged”

slopes?

Effect of Random Noise on Differentiation

• Measured voltage has Real and Noise components – VM = VR+VN

– • For small is large and random• Want

– Want to be large enough to avoid random error but small enough to capture real events

– If wV is mostly IRE, then decreases as FS gets smaller and N increases

RF, IRE, other random errors, does not increase

with

Common Temperature Measurement Errors

• Even for steady temperatures• Lead wires act like a fin, cooling a hot the surface

compared to the case when the sensor is not there• The temperature of a sensor on a post will be

between the fluid and duct surface temperature

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(Tgas – TS) = Ase(TS

4 -TW4)

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

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

• Measurement Error = Tgas – TS = (se/h)(TS4 -TW

4)

QConv=Ah(Tgas– TS)

TS

QRad=Ase(TS4 -TW

4)

Tgas

TW

Sensorh, TS, A, e

Problem 9.39 (p. 335)

• Calculate the actual temperature of exhaust gas from a diesel engine in a pipe, if the measuring thermocouple reads 500°C and the exhaust pipe is 350°C. The emissivity of the thermocouple is 0.7 and the convection heat-transfer coefficient of the flow over the thermocouple is 200W/m2-C.

• ID: Steady or Unsteady?• What if there is uncertainty in emissivity?

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)

• Sensor temperature at tip, • Dimensionless Tip Temperature Error from conduction

– , (want this to be small)– Decreases as decreases

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

T∞

h xLA, P, k

T0

TS

Example

• A 1-cm-long, 1-mm-diameter stainless steel support (k = 20 W/mK) is mounted inside a pipe whose temperature is 200°C. The heat transfer coefficient between gas in the pipe and the support is 100 W/m2K, and a sensor at the end of the support reads 350°C. What is the gas temperature? Assume esensor = 0

• Steady or unsteady• Radiation or Conduction errors

Solution

• Sensor temperature: •

• What is given and what must be found?

• What if esensor = 0.2?

t = 0 t

T

Ti

TB

Example

A/D N= 2 ±10V

Interpret:

Input Range (v)

Iout Vout,D Max Error (V)

-∞ to -5 0 -7.5 ∞ -5 to 0 1 -2.5 ± 2.5V 0 to 5 2 2.5 ± 2.5 V 5 to ∞ 3 7.5 ∞