Exp 4 - Venturi Meter - Lab Report PDF

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Department of Chemical Engineering - An-Najah National University Unit operations Lab I (10626339)

Flow Through a Venturi Meter (Experiment No. 4, Performed on April 20, 2021)

Prepared By: Mahmoud Sallam Mohammad Debes

Group Members: Mahmoud Sallam Mohammad Debes Hussam Saleh Kareema Hamdan Syndia Abu Hardan Hala Badawi

Submitted To: Dr. Amjad El-Qanni Eng. Muath Nassar

May 10, 2021

Abstract

In this experiment, we investigate the Venturi tube, a basic device used to measure to flow rate of a fluid in a pipe. The venturi tube operates by increasing the velocity of fluid through a contraction, then reverting it back to normal by a slow enlargement to the original pipe diameter. We measure the pressure head before and at the contraction to calculate the flow rate using equations based on Bernoulli’s equation and the continuity principle, which will be derived in the theory section. The increase in velocity in the “throat” of the Venturi tube causes a pressure drop that turns out to be dependent on the flow rate of the fluid, so we will use the pressure readings taking from piezometric tubes to infer the flow rate of the fluid through the Venturi meter. The results present a clear dependence between the flow rate and the venturi meter coefficient, as well as the head difference between two sections. We also demonstrate the relationship between the horizontal position of the fluid in the Venturi tube (tap water) and the piezometric head coefficient.

Theory[1] The fluid used in this experiment is water, which has negligible compressibility. We also assume that it is inviscid.

Notation: The upstream section is called Section 1, the throat (smallest section) is Section

2 and any other arbitrary section would be called Section 𝑛.

By Bernoulli’s equation, the total head in the first section would be the same in the sections

that follow, and since there is no variation in elevation (Venturi tube is horizontal), we get: 𝑢1 2 𝑢𝑛 2 𝑢2 2 + ℎ𝑛 + ℎ2 = + ℎ1 = 2𝑔 2𝑔 2𝑔

Where ℎ for any section is measured by the piezometric tubes that are directly connected to the section.

We can use the principle of flow continuity 𝑄 = 𝑢1 𝐴1 = 𝑢2 𝐴2 , where 𝑄 denotes the

volumetric flow rate, to solve Bernoulli’s equation for 𝑢2 with some algebraic manipulation: 𝐴2 2 𝑢2 2 𝐴1 ) + ℎ2 + ℎ1 = 2𝑔 2𝑔

(𝑢2

→

𝑢2 = √

available values. To find the flow rate, we multiply by 𝐴2 :

2𝑔(ℎ1 − ℎ2 ) 1 − (𝐴2⁄𝐴 )2 1

This relationship allows us to calculate the flow velocity at Section 2 using readily 2𝑔(ℎ1 − ℎ2 ) 𝑄 = 𝐴2 √ 1 − ( 𝐴2⁄𝐴 )2 1

(1𝐴)

This is the ideal discharge rate with unrealistic assumptions. In reality, there are head losses constant 𝐶 which we can call the Venturi meter coefficient. Since head is lost, the value of 𝐶 will

when the fluid flows through the contraction that must be accounted for. To do this, we use a surely be less than 1, but not much less because the contraction diameter reduction and its length

are not that significant. We arrive at the final form of our flow rate equation: 𝑄 = 𝐶 ∙ 𝐴2 √

2𝑔(ℎ1 − ℎ2 )

1 − (𝐴2

⁄𝐴 ) 1

(1𝐵)

2

On another note, we seek a dimensionless method to express the change in piezometric piezometric head coefficient 𝐶𝑝ℎ:

head between any two sections. It is convenient to use the velocity head at Section 2 to define a 𝐶𝑝ℎ =

ℎ𝑛 − ℎ1 𝑢2 2 ⁄2𝑔

(2𝐴)

We can use this equation when we read piezometric heads along the length of the Venturi tube to obtain a dimensionless head coefficient that we can use to do various calculations. We can

also manipulate this equation to eliminate the need for the velocity, making the coefficient solely a function of the geometry of the tube. From Bernoulli’s equation: ℎ𝑛 − ℎ1 =

𝑢1 2 2𝑔

𝑢𝑛 2 − 2𝑔

𝑢𝑛 2 ℎ𝑛 − ℎ1 𝑢1 2 ) ) − ( = ( 𝑢2 𝑢2 𝑢2 2 ⁄2𝑔

Now the left-hand side is equal to 𝐶𝑝ℎ and for the right-hand side, we can use the continuity

equation to write the velocity ratios as area ratios: 𝑢1 𝐴2 = 𝑢2 𝐴1

𝑎𝑛𝑑

𝑢𝑛 𝐴2 = 𝑢2 𝐴𝑛

2 2 𝐶𝑝ℎ = (𝐴2⁄ 𝐴1 ) − (𝐴2⁄ 𝐴𝑛 )

of each section with 𝐷2 to arrive at the final form of the piezometric head coefficient equation:

Since the area is directly proportional to the square of the diameter, we can replace the area 𝐶𝑝ℎ = (𝐷2 /𝐷1 )4 − (𝐷2 /𝐷𝑛 )4

read the piezometric head or determine 𝑢2 .

(2𝐵)

We can now calculate the piezometric head coefficient for any section without the need to

Results and Discussion For the first part, we calculated the real flow rate from the common equation (𝑄 = ), 𝑡 𝑉

determining the values of the Venturi meter coefficient for different flow rates gives values that

lie within the expected range (0.92-0.99)[1], Table 1 shows the measurements of (ℎ₁ − ℎ₂) ⁄2 and 𝐶.

1

Table 1: Measurements of (h₁-h₂)1/2 and C.

Q (m³/s) 0.000349

(h₁₁-h₂₂)1/2 (m1/2) 0.384708

C 0.948868

0.000436

0.461519

0.987648

0.000442

0.475395

0.971538

0.000371

0.406202

0.954700

Plotting (ℎ₁ − ℎ₂) ⁄ 2 versus Q to visualize the relationship between them, then to see the 1

relationship between C and Q, we plot C versus Q, figures 1 and 2 shows these relationships.

Figure 1: The relationship between the square root of the head difference and the volumetric flow rate.

0.00046

0.00044

Q m3/s

0.00042 0.0004 0.00038 0.00036 0.00034

0.37

0.39

0.41

0.43

(h₁-h₂)1/2

0.45

(m1/2)

0.47

0.49

Figure 2: The relationship between the Venturi meter constant and the volumetric flow rate.

0.00046

0.00044

Q m3/s

0.00042

0.0004

0.00038

0.00036

0.00034 0.945

0.95

0.955

0.96

0.965

0.97

0.975

0.98

0.985

0.99

C We can see that when the flow rate increases, the values of C and (ℎ₁ − ℎ₂) ⁄2 also 1

increase, which means that the head loss decreases with increasing flow rate and since we the increase the velocity (also this will increase the flow rate), the friction between the water and the when 𝐶 ≈ 1.

walls of the pipe, which causes the head losses, will decrease and can be neglected at some point

equals to 0.0003494 𝑚3 /𝑠 (arbitrary choice). Using Equation (2𝐴) gives values close to the For the second part, we determine the piezometric head coefficient when the flow rate is

expected ideal values from Equation (2𝐵) with a slight error. Table 2 shows our results. Section

Cph ideal

Cph exp.

Absolute Error

A

0.000

0.071

0.071

B

-0.083

0.032

0.115

C

-0.428

-0.315

0.113

D

-0.857

-0.868

0.011

E

-0.681

-0.739

0.058

F

-0.420

-0.469

0.050

G

-0.253

-0.309

0.055

H

-0.145

-0.212

0.067

J

-0.070

-0.129

0.058

K

-0.019

-0.084

0.065

L

0.000

-0.051

0.051

Plotting Cph versus the position[1] , which is given for each section, we have the relationship between them, shown in Figure 3. Figure 3: The relationship between the piezometric head coefficient and the position on the Venturi tube.

0.2

0 -20

0

20

40

60

80

100

120

140

160

Cph

-0.2

-0.4

-0.6

-0.8

-1

x (mm)

Here, we see a steep decline at the beginning as the fluid flows through the contraction and loses piezometric head. As the fluid flows through the more gradual enlargement, the piezometric head coefficient slowly recovers to value close to, but not exactly the same as, the value at the beginning. This is because it has lost head due to friction.

Sample of calculations In first part, we want to determine the value of Venturi meter coefficient, after arranging the final form of flow rate equation for C,

𝐶=

𝐴2

𝑄

2𝑔(ℎ1 − ℎ2 )2 √ 1 − (𝐴2⁄ ) 𝐴1

we know that the volumetric flow rate is given by the volume over time, so we calculate it with 5 L and 14.31 seconds in this equation below,

With

𝐴2

𝐴1

𝑄=

𝑉 5 × 10−3 3 = 3.49 × 10−4 𝑚 ⁄𝑠 = 14.31 𝑡

= 0.379, 𝐴2 = 2.011 × 10−4 𝑚2 , ℎ1 = 248 𝑚𝑚, ℎ2 = 100 𝑚𝑚 and 𝑔 = 9.81 𝑚/𝑠 2

Substituting these values into the above equation gives a value of 0.942 for C. In the second part, we want to calculate the value of the piezometric head coefficient using the equation:

𝐶𝑝ℎ = (𝐷2 /𝐷1 )4 − (𝐷2 /𝐷𝑛 )4

with 𝐷1 = 26 𝑚𝑚, 𝐷2 = 16 𝑚𝑚 and 𝐷𝑛 (𝑓𝑜𝑟 𝑟𝑒𝑓 𝐵) = 23.2 𝑚𝑚

substituting these values, we end up with a value of −0.083. This is the ideal value, which is

also given in the lab manual [1]. To calculate the experimental value of 𝐶𝑝ℎ we use the equation: 𝐶𝑝ℎ =

ℎ𝑛 − ℎ1 𝑢2 2⁄ 2𝑔

To solve this equation, we must find the velocity in section B (the throat) using the simple flow rate equation

with 𝐴2 =

𝜋𝐷2 2 4

the velocity.

𝑢2 =

𝑄 𝐴2

3 = 2.01 × 10−4 𝑚2 and 𝑄 = 3.49 × 10−4 𝑚 ⁄𝑠 , we have the value 1.74 𝑚/𝑠 for

We can now pick a section and calculate the piezometric head coefficient for that section. For the sake of this sample, we take the head measurement for Section B:

𝐶𝑝ℎ =

240 − 235 2 0.84

= 0.0324

Note: there may be some small differences 2(9.81) between the values calculated here and in the results section. This is purely due to rounding errors.

Conclusions All in all, the results came out better than expected from this experiment. This owes to the accuracy of the Venturi meter method, which is tried-and-tested and has been used in the industry for decades. One of the data points in Figure 2 may or may not be accurate, mostly due to human error, and that would change the general understanding of the graph. The relationship between the flow rate and C is therefore uncertain, and would require more trials to find.

Appendix A: Raw Data

References [1] Department of Chemical Engineering. (2013). Unit Operation (I) Laboratory Manual. AnNajah National University....