DescriptionEML 3303 – Homework 3

Due: 4/10/23 @ 11:59PM

Please, show work for all problems. Otherwise no partial credit.

Problem 1 (2 points)

We need to design a rotameter under the following conditions. We are given a tapered circular

tube with dimensions shown in the figure below:

0.35 m

Fluid is water @ 25 C.

Density = 1000 kg/m3

D=?

Flow

direction

0.7 m

Drag coefficient (CD) = 0.7

PVC density = 1380 kg/m3

0.15 m

The floating object that indicates flow rate is a PVC sphere. Find the range of diameters (D) for

this sphere that would work with the tube provided above for a flow rate of 3.2×10-3 m3/s. (The

sphere has to always float somewhere within the 0.7 m of the tube).

(Sol: D = [0.1354 – 0.3463] m)

Problem 2 ( 1 points)

We want to measure the temperature of an industrial oven with a J-type thermocouple. We

place the measuring junction in the oven and the reference junction is left outside at room

temperature (22 °C). If the voltage output of the thermocouple is 57.062 mV, what is the

temperature of the oven? See table for J-type thermocouple posted on Canvas.

(Sol: ~1004 °C )

Problem 3 ( 2 points)

A thermistor has a resistance of 250 kW at 0 °C. The thermistor is introduced in a voltage

divider circuit to measure temperature. At a temperature of 80 °C the voltage output is 2.25 V.

The reference resistor (R1) in the voltage divider is R1 = 100 kW and the input voltage is 5 V.

a) Calculate the thermistor material constant b.

b) Now that you have b, we place the thermistor in a water bath at an unknown

temperature. Using the same reference resistor R1, the voltage output from the divider

circuit is 3.5 V. What is the temperature of the water bath?

c) Indicate step by step how you would calculate the uncertainty of this temperature if the

resolution of the voltmeter is 0.01 V and its accuracy is 2% of 10V.

(Sol: a) 862.06 K b) 618.3 K )

Problem 4 (2 points)

We need to calibrate a diaphragm for a pressure transducer. For this, we subject the diaphragm

to some known pressure differentials (DP) and we measure the deflection at the center (yc)

with a micrometer. These measurements are shown on the table below.

yc (µm)

DP (MPa)

0

0

79.8

5

159.2

10

240.1

15

320.2

20

a) Find the best linear fit to the measured data in terms of DP as a function of yc. That is, DP in

the y-axis and yc in the x-axis. This will be your calibration equation/curve. No need to convert

units.

b) Find the uncertainty of this fit for a 95% probability.

c) If the micrometer has a resolution of 2 µm and the pressure regulator used to set the

pressure differentials has an accuracy of 0.2% of the FSO, find the total uncertainty of the

calibration equation. The FSO for the regulator is 50 MPa.

d) We are now ready to use this pressure transducer. I connect it to a refrigeration pipe and at

one particular moment I see the deflection is 19 µm. I measured such deflection with an optical

sensor that has an accuracy of 0.8% of the reading. What is the pressure in MPa at that

moment and what is the uncertainty of this pressure?

(Sol: a) DP = 0.0175 + 0.0624yc; b) 0.0369; c) 0.1235; d) 1.2 ± 0.1239 MPa)

Problem 5 (2 points)

Consider the differential pressure flow meter shown in the figure below. It is connected to a

pressure transducer to measure the pressure difference between points 1 and 2. The pressure

transducer has a diaphragm made of aluminum (Bulk modulus = 90 GPa, poisson’s ratio = 0.33).

The radius of the diaphragm is 8 mm and its thickness is 0.2 mm.

The flow meter is made for a pipe diameter of 0.25 m (d1) and has an obstruction of diameter

0.18 m (do). The flow coefficient, Ko, is given by: ! = 640 ∙ “#$.&’ . It is meant to measure

water flow (density of 1000 kg/m3, kinematic viscosity of 1.081×10-6 m2/s). Find the maximum

flow rate we could measure with this setup.

Diaphragm pressure

transducer

P1

2r

P2

P1

Q

d1

P2

d0

Q

(Sol: ~ 0.07186 m3/s)

Problem 6 (1 point)

We want to measure the speed of an aircraft using a Pitot tube. In order to measure the

difference between static and stagnation pressure at the Pitot tube, we use a diaphragm. What

we can measure from the diaphragm is voltage. We then use the following calibration curve to

get pressure: DP = 3877×V + 5, where V is voltage in [V] and DP is pressure difference in [Pa].

With the aircraft in flight, we measure a voltage of 4.3 V on the diaphragm. Calculate the speed

of the aircraft given an air density of 1.225 kg/m3.

Also, if the diaphragm is made of aluminum (Bulk modulus 69 GPa, density 2700 kg/m3,

poisson’s ratio 0.3), its diameter is 15 mm and its thickness is 0.2 mm, what is the maximum

aircraft speed it could measure?

(Sol: a) 165 m/s. b) 333.6 m/s )

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