Bernoulli Principle Demonstrator Lab #2

Chase Hilderbrand

Joanna Nicholson

Eddwie Perez

November 6, 2015

Professor: Dr. Danvers Johnson

CWR3201C

INTRODUCTION

The Bernoulli Principle Demonstration Lab was conducted in order to determine flow rate, a

velocity distribution in the venturi tube at six pressure ports, determine the pressure distribution at

these same points and to calculate a flowrate coefficient (k) and pressure difference between two

specified ports. The lab is also an exploration of Bernoulli’s Law. The venturi tube is used as a

means of measuring the flow through it. The pressure distribution can be measured from the

column heights corresponding to each of the measurement points in the venturi tube. The results of

each column height reading will show whether or not the column heights increase or decrease

when the pressure difference increases. If the total head is constant, the assumption is that the

stagnation pressure should remain constant.

The venturi tube is used for flow rate measurement because of the assumption that there is

less pressure loss during measurement as compared to a hole or nozzle. As well, assumptions are

being made that the fluid is inviscid, of constant density, the flow is steady and that fluid motion is

governed by pressure and gravity forces only.

THEORY

A single volume flow rate was established using:

𝑸 =𝑽𝒐𝒍

𝑻𝒊𝒎𝒆 (1)

The dynamic pressure head can be calculated from the measurement of the static pressure head (hs)

and total head (ht) by taking a reading at each of the columns. Dynamic head, hd can then be

calculated using the equation:

𝒉𝒅 = 𝒉𝒕 − 𝒉𝒔 (2)

The Bernoulli equation in its form for constant head, includes pressures at a given point (P1,P2),

that can be rearranged in terms of static pressure heads (H1, H2) which includes pressures 𝑃1, 𝑃2

(Equation 3) can be rearranged in terms of static pressure heads 𝐻1, 𝐻2 (Equation 4) when

accounting for friction losses and conversion of pressures,

𝑷𝟏

𝝆+

𝑽𝟏𝟐

𝟐=

𝑷𝟐

𝝆+

𝑽𝟐𝟐

𝟐 (3) 𝒉𝟏 +

𝑽𝟏𝟐

𝟐𝒈= 𝒉𝟐 +

𝑽𝟐𝟐

𝟐𝒈+ 𝒉𝒗 (4)

The demonstration device was assumed to be a closed system, and therefore conservation of flow

applies. The continuity equation is below, where A= Area of cross section at a given point.

𝑨𝟏𝑽𝟏 = 𝑨𝟐𝑽𝟐 = 𝑽 (5)

For the velocity profile in the venturi tube, a reference velocity (Vi) can be derived from the

geometry of the tube where vi=A1/Ai . Then, theoretical velocities, Vcalc can therefore be calculated at

each point on the venturi tube.

Vcalc =Q

Ai (6)

The dynamic pressure found from equation 2 and gravity can then be used to calculate the experimental

velocity, 𝑊𝑚𝑒𝑎𝑠..

𝑽𝒎𝒆𝒂𝒔. = √𝟐𝒈𝒉𝒅 (7)

By knowing the volumetric flow rate found in equation 6 and the pressure loss between the largest

and smallest diameter of the tube, the flow rate factor, k, can be found. The pressure loss, ∆P,

between the largest and smallest diameters in the tube is used to measure the flow rate.

𝑸 = 𝒌√∆𝑷

𝑲 =𝑽

√∆𝑷 (8)

EXPERIMENTAL PROCEDURES

To accurately conduct this experiment we needed several pieces of equipment. We needed pure

laboratory grade distilled water as to not contaminate the inside of the venturi tube. A stop watch

was also used to record the amount of time that it took to fill the venture tube in order to calculate

the flow rate.

The closed system was flushed and air bubbles were allowed to dissipate from each of the six

columns. The water level was also allowed to settle so that there could be height differences which

would affect the pressure in the venture tube at different locations. As water flowed from the inlet

and through the venturi tube in the demonstrator, the static and total pressures were measured at

six pressure points along the venturi tube. A probe measured total head and reading was taken at

each corresponding column with a pressure gage, this was also done to measure any sort of

stagnation point. When the height steadied on the six columns, the change in the height was

recorded along with the reading from the piezometer.

RESULTS AND DISCUSSION

A predetermined volume of water was collected and the number of seconds it took was noted.

Using, Q=V/t, a reference flow rate was established and presented below in Table 1.

Table 1. Single Flow Rate:

Volume (cm3) 920

Average Time (s) 5.75

Flow Rate (L/s) 0.16

The overall pressure was also measured via a probe; it was moved along the venturi tube at various

locations was read from the second single column containing water and a pressure gauge. When the static

head and total settled into an approximately steady height, the reading was taken. The results of the various

head heights, an experimental velocity and theoretical velocity are tabulated below in Table 2.

Table 2. Column Heights, Experimental and Theoretical Velocities

For Q= 0.16 L/s

Points along venturi tube 1 2 3 4 5 6

Static Head (mm) 230 215 52 151 175 182

Total Head (mm) 240 232 228 190 178 168

Dynamic Head (mm) 10 17 176 39 3 -14

Area (mm2) 338.6 233.5 84.6 170.2 255.2 338.6

Experimental Velocity (mm/s) 442.94 577.53 1858.26 874.75 242.61 #NUM!

Theoretical Velocity (mm/s) 472.53 685.22 1891.25 940.07 626.96 472.53

The reading at the sixth point may have been faulty because the dynamic head resulted in a negative

value. As such, an experimental velocity could not be calculated due to the square root in equation

7. The experimental values for velocity are relatively close to one another at points 1, 2, 3 and 4, yet

vary greatly at point 5. This could be as a result of not taking a reading at a steady height in the tube

located above point 5.

A graphical representation of the experimental and calculated velocities is below. The differences

between the two are as a result of the measurements taken.

Figure 1. Comparison of Theoretical and Experimental Velocities

The flow rate factor (K) of a venturi tube can also be determined simply by knowing the flow rate

(Q) and the difference in area (Δp) of the largest point, 1, and the smallest, 3, diameters in the tube.

This value represents the loss in the pipe.

𝑸 = 𝑲 ∙ √∆𝒑 𝐾 =𝑄

√∆𝑝=

0.16𝐿

𝑠

√338.6𝑚𝑚2−84.6𝑚𝑚2= 0.010

𝐿

𝑠∙𝑚𝑚

The pressure distribution reflects the pressure changes are various points and are represented

graphically below in Figure 2. The graph below shows that Equation 2, 𝒉𝒅 = 𝒉𝒕 − 𝒉𝒔 holds true.

Figure 2. Pressure Distribution

0

50

100

150

200

250

1 2 3 4 5 6

Co

lum

n H

eigh

t in

mm

Points On the Venturi Tube

Pressure Distribution

Static Head (mm)

Total Head (mm)

Dynamic Head (mm)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

1 2 3 4 5 6

Ve

loci

ty in

mm

/s

Points Along Venturi Tube

Theoretical Velocity vs Experimental Velocity

TheoreticalVelocity(mm/s)

ExperimentalVelocity(mm/s)

CONCLUSIONS

It is most evident in Figure 2 that the theoretical and experimental velocities do not vary greatly

between differ slightly between points one and two. As the areas decreased, the velocities

increased. As seen in Figure 1, the total head decreases at each stagnation point and did not

remain constant, therefore the stagnation pressure could not remain constant. The static and

dynamic heads nearly mirror each other.

REFERENCES

Gunt Hamburg. (2005) “HM150.07 Bernoulli’s Principle Demonstrator” Gunt Hamburg

Laboratory Manual.

Munson,Young, Okiishi, Huebsch. (2009) Fundamentals of Fluid Mechanics, Sixth Edition, John

Wiley & Sons, Hoboken, NJ