NE418 spring 2017 HW#3 (Total 60 points), Tuesday March 14 th Due Tuesday, March 21 st by class time

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1 NE48 spring 07 HW#3 (Total 60 points), Tuesday March 4 th Due Tuesday, March st by class time - (0points) Thermocouples measure the temperature as a result of the voltage across the two-wire contact, known as the Seebeck voltage. The relation between the voltage changes as a result of the temperature change is known as the Seebeck coefficient a -. a) If the temperature T= 600 o K, find the value of the resistor R of the OpAmp that produces the same exact output on the readout unit for each of the listed thermocouple types if =kω. TC Type Contact Materials (m/ o K) E Chromel/Constantan 60 J Iron/Constantan 5 T Copper/Constantan 40 K Chrole/Alumel 40 N Nicrosil/Nisil 38 S Platinum 0% Rhodium/Platinum B Platinum 30% Rhodium/Platinum 6% Rhodium 8 R Platinum 3% Rhodium/Platinum = = a T R output T b) If changing the feedback to a capacitor C=0μF, what is output, in m, for each thermocouple if the input is fluctuating at a frequency f=0hz Worked Solutions: oltage equation: = a T Then all voltages are obtained in m for all thermocouple types. a) To have same output for all sensors, then we can select one of the sensors as the reference. As E-type has the largest Seebeck coefficient then we can select it as the reference. We calculate the ratio between all types to that of the E-type, which yields the needed OpAmp amplification then we calculate the value of R. The other option is to

2 select a voltage value to be the standard and amplify all thermocouples voltages to the selected value. Amplification = R / R R = Amplification R R = Amplification R ( kω) ( kω) 0 TC Type Contact Materials (m/ o K) = alpha * T =c *600 (m) Setting E-type as the limit for the voltage Amplification needed R (kω) for same output E Chromel/Constantan J Iron/Constantan T Copper/Constantan K Chrole/Alumel N Nicrosil/Nisil S Platinum 0% Rhodium/Platinum B Platinum 30% Rhodium/Platinum 6% Rhodium R Platinum 3% Rhodium/Platinum Alternative solution: = a T so one3 can decide on the output voltage to be displayed as a measure of temperature, for example choosing an output voltage of 4,000μ to be the standard voltage to display then = a T = a 600 = 4000m from which it indicates α = 40 which is the K-type thermocouple. Then scale the output of each thermocouple to the same 4,000μ, which will determine the ration of amplification TC Type Contact Materials (m/ o K) = alpha * T =c *600 (m) Ratio with respect to 4000 R (kω) for same output E Chromel/Constantan J Iron/Constantan T Copper/Constantan K Chrole/Alumel N Nicrosil/Nisil S Platinum 0% Rhodium/Platinum B Platinum 30% Rhodium/Platinum 6% Rhodium R Platinum 3% Rhodium/Platinum b) Changing feedback to a capacitor will turn the OpAm to integrate the input. Z f ( / jωc) Z f = / jωc, and output = input = input = input Z R jωrc s

3 Then the magnitude is output = ωrc TC Type Contact Materials E Chromel/Constantan 60 J Iron/Constantan 5 T Copper/Constantan 40 K Chrole/Alumel 40 N Nicrosil/Nisil 38 S Platinum 0% Rh di /Pl ti B Platinum 30% Rhodi m/platin m R Platinum 3% Rh di /Pl ti o 8 = alpha * T =c *600 ( ) - (40points, as per distribution shown for each part a through d) The piping system shown below is equipped with pressure and temperature measuring sensors. Temperature is by thermocouples and pressure is by strain gauges. T T T 3 ω C=*3.46* 0*e 3 *0e -6 output = /ω C (m) output (m) P 3 P P D D D L L a) (0 points) If the pressure by P is half of that by P, derive the equation that relates the velocity in the second section (v z ) to the inlet velocity (v z ). Do not neglect the gravitational effect (known as the potential energy per unit volune). b) (5 points) What is the quantity (Q) of the liquid flowing into thend section? c) (5 points) Denoting the viscosity of the fluid as m fluid, write down the equation of the Reynold s number for this fluid as the fluid flows through the nd section d) (5 points) As the pressure sensors are strain gauges, and as the pressure sensor P half of that by P, obtain the value of the resistance of gauge (R g ) with respect to that of gauge (R g ). The batteries supplying the input voltage for the sensors are identical. Resistors in the bridges (in kω) are =, R =6, R 3 =5, =8, R5=4, R 6 =8kΩ. L R g R g R R 3 R 5 R 6 P sensor P sensor

4 e) (5 points) A C-type thermocouple, which has a measuring range up to 50 o C measures the temperature in the 3 rd section. The relation between the temperature and the voltage across the thermocouple leads has the form: T = Determine the Seebeck coefficient of this thermocouple when the voltage is 0m. Worked Solutions a) Bernoulli s equation: P + rv + rgh z = P + rv + rgh z = constant st section nd section P + rv + rgh = P + rv + rgh z z rv = z ( P P ) + rg( h h ) + rvz / P from which v = g z + ( h h ) + vz r b) What is the quantity (Q) of the liquid flowing into thend section? Q = va / P π D nd section = ( ) = z = ( ) nd section + + z Q va v A g h h v ρ c) Denoting the viscosity of the fluid as m fluid, write down the equation of the Reynold s number for this fluid as the fluid flows through the nd section / ρv D ρd P z R = = g e ( h h ) vz µ µ + + ρ fluid fluid d) As the pressure sensors are strain gauges, and as the pressure sensor P half of that by P, obtain the value of the resistance of gauge (R g ) with respect to that of gauge (R g ). The batteries supplying the input voltage for the sensors are identical. Resistors in the bridges (in kω) are =, R =6, R 3 =5, =8, R5=4, R 6 =8kΩ. 4 R g R g R R 3 R 5 R 6 P sensor P sensor

5 R=, R=6, R3 =5, R4=8, R5=4, R6=8 Sensor R3 R = input R3 Rg R R + + Sensor R6 R 5 = input R6 Rg R4 R P = = = P 3 g R6 R5 R + R R + R R3 R R + R R + R 6 g g R6 R R6 + Rg R4 + R5 8+ Rg Rg = = = = R3 R Rg 3 5+ Rg 4 R + R R + R 5+ R g g = = = 8 + R R 5 + R 3 4 g g g g g = = 8+ R 5+ R 8+ R 5+ R g g g g R = = 8+ R 5+ R 5 + R g ( ) ( + Rg ) ( 55 Rg ) g g g 9 5 R + 8 = e) A C-type thermocouple, which has a measuring range up to 50 o C measures the temperature in the 3 rd section. The relation between the temperature and the voltage across the thermocouple leads has the form: T = Determine the Seebeck coefficient of this thermocouple when the voltage is 0m. The Seebeck voltage is given by α T 0 = α ( ) 0 α = = o m / C =, hence = α T = α ( )