Agustin, Crystal Mae C. ChE424 Lab

Bigso, Jhullie Ann S. 4ChE-C

Dimaano, Carlito Jr. R.

Micaller, Ian Kenneth D.

A Study of Bernoulli’s Theorem

I. Introduction

In 1738, Swiss mathematician Daniel Bernoulli stated that the total mechanical

energy of the fluid, comprising the energy associated with fluid pressure, the

gravitational potential energy of elevation, and the kinetic energy of fluid motion

remains constant [1]. This theory is supported by the Bernoulli Equation and is valid

in regions of steady, incompressible flow where net frictional forces are negligible,

and flow is along the streamline [1, 2]. The key approximation in the derivation of the

Bernoulli Equation is that viscous effects are negligibly small compared to inertial,

gravitational, and pressure effects.

Pρ+ v2

2+gz=constant (1)

Where, P is pressure; ρ is the density of the flowing fluid; v is the mean velocity

of fluid flow at the cross section; g is the acceleration due to gravity; z is the elevation

head. Note that if the equation is to be used for English units, the gravitational

constant, gc, should be taken into consideration.

Since there is no such thing as “inviscid” (fluids having no viscosity) this

approximation cannot be valid for an entire flow field of practical interest. As a result,

the Bernoulli Equation cannot be applied everywhere in a flow. However, the

approximation is reasonable in certain regions of many practical flows and they are

called inviscid regions of flow. In these regions, fluids are not really inviscid or

frictionless, but rather these are regions where net viscous or frictional forces are

negligibly small compared to other forces acting on fluid particles [2].

The Bernoulli Equation is later derived by Leonhard Euler in 1755; he established

that the value of the constant in Eq. 1 can be evaluated at any point along the

streamline where the pressure, density, velocity, and elevation are known [2]. Hence,

the Bernoulli Equation can also be written between as

P1

ρ+

v12

2+g z1=

P2

ρ+

v22

2+g z2 (2)

The implication of Bernoulli’s Equation is that if the fluid flows horizontally so

that no change in gravitational potential energy occurs, then a decrease in fluid

pressure will result to an increase in fluid velocity. In addition, if the fluid is flowing

through a horizontal pipe of varying cross-sectional area, the fluid speeds up in

constricted areas so that the pressure the fluid exerts is least where the cross section is

smallest. This phenomenon is sometimes called the Venturi effect, after the Italian

scientist G.B. Venturi (1746–1822) [1].

Figure 1 Venturi effect [3]

The Bernoulli Equation can be expressed in units of pressure by multiplying the

equation by the density ρ

P1+ρv1

2

2+ρg z1=P2+ρ

v22

2+ρg z2 (4)

Where, Pn is the static pressure that represents the actual thermodynamic pressure

of the fluid at any point, ρvn2/2 is the dynamic pressure but is not a pressure in a real

sense it is simply a convenient name for the quantity which represents the decrease in

the pressure due to the velocity of the fluid, and ρgzn is the hydrostatic pressure that is

usually neglected when the fluid flow is not elevated [2, 4].

The Bernoulli apparatus consists of a convergent duct, a straight section called the

throat, and a divergent duct back to the original pipe diameter.  The contracting part

gives a convenient method to demonstrate the application of Bernoulli's equation,

provided that there is a very gradual taper. Piezometers are attached to the apparatus

which can be used to measure the piezometric heads of the flow and a stagnation tube

that can be used to measure the total head of the flow at different longitudinal

locations within the tube [5]. In a Bernoulli apparatus, the fluid along the dividing or

“stagnation streamline” slows down and eventually comes to rest without deflection

at the stagnation point [2, 4].

Figure 2 Stagnation streamlines illustration [6]

Taking the stagnation point as point 2, Eq. 4 will become

Pe+ρve

2

2=Po (5)

Where, Pe is the static pressure, ρve2/2 is the dynamic pressure, and Po is the

stagnation pressure. As mentioned earlier, a common method of expressing pressures

is in terms of head (h) of a fluid. This height or head (in units of length) will exert the

same pressure as the pressures it represent [7].

h= Pρg (6)

Equation 5 can now be expressed in terms of the head of the fluid

he+ve

2

2 g=ho (7)

Where, he is the static head of the fluid, ve2/2g is the dynamic or velocity head, and

ho is the stagnation or total head.

The objectives of this study is to determine the validity of the Bernoulli’s

Theorem on the flow of water in a tapering circular duct by comparing the theoretical

total head and the observed total head at a given flow rate, and to determine the range

of validity of the Bernoulli Theorem using water flowing at different flow rates.

II. Method

A. Equipment and Apparatus

The Bernoulli’s Theorem Demonstration Apparatus, a stopwatch and a 1-L beaker were used in the experiment.

B. Experimental Procedure

The Bernoulli apparatus was checked if it is connected to the faucet and that the 14 degree tapered duct was in the flow direction. Water from the faucet was allowed to flow in the apparatus and the flow valve was adjusted. Hand pump was used when there is a finite lowering of levels in the tubes. The manometer level readings in each tube were recorded. The probe manometer levels were also recorded for each tapered portion. These steps were repeated using three different flow rates.

The faucet was closed and the apparatus was drained off. The probe was withdrawn. Figure 3 shows the set up of the experiment.

Figure 3 Set-up of the Experiment

pump

test section (Venturi section)

pressure tappings

hose

manometer board with 8 tubes

14 degree tapered duct

probe

III. Results and Discussion

A. Data and Results

The data gathered from the experiment are presented in Table 1 and Table 2.

Table 1 shows the diameter and area of different cross sections and the probe

distances of each tube in the Bernoulli’s Theorem Demonstration Apparatus. The area

was computed by using the diameter of each tube in the equation, A = ¼ πd2.

Table 1. Diameter and Area of diff. Cross-section of the Tubes and the Probe Distances

Tube number Diameter of diff. Cross-section (mm)

Area of cross-section (mm2)

Probe Distance

(mm)1 10.0000 78.5400 72.00002 10.6000 88.2500 64.90003 11.3000 100.2900 62.00004 12.4000 120.7600 57.50005 14.6000 167.4200 49.60006 25.0000 490.8700 0.0000

Three different flow rates were used in the experiment which correspond to

laminar, transitional and turbulent flow. The flow rates were controlled such that the

Reynold’s number, NRe, would result to a value that falls within the range of each

flow (i.e. laminar flow if NRe ≤ 2100, transitional flow if 2100 < NRe < 4000 and

turbulent flow if NRe ≥ 4000).

A volumetric flow rate of 0.0144 L/s was used in the laminar flow which is the

same for all the tubes in the apparatus. The flow rates 0.0174 L/s and 0.0971 L/s were

used in the transitional and turbulent flow, respectively. The manometer levels and

actual probe manometer levels obtained from each flow are presented in Table 2.

Table 2. Manometer Levels and Probe Manometer Levels from 3 Different Flow Regimes

Tube no.

Laminar Flow Transitional Flow Turbulent FlowML

(mm) PML actual (mm) ML (mm) PML actual (mm) ML

(mm) PML actual (mm)

1 131.0000 134.0000 189.0000 193.0000 95.0000 193.0000

2 133.0000 135.0000 190.0000 193.0000 118.0000 194.00003 134.0000 135.0000 191.0000 193.0000 138.0000 194.00004 135.0000 135.0000 192.0000 193.0000 139.0000 197.00005 135.0000 135.0000 193.0000 194.0000 168.0000 197.00006 135.0000 135.0000 194.0000 194.0000 194.0000 197.0000

B. Treatment of Results

The Reynolds numbers of each tube from the different flows were

computed for the flow regime (shown in Table 6 in the Appendix). The density

and viscosity were calculated from Perry’s Chemical Engineer’s Handbook

Tables 2-32 and 2-313, respectively, at 28.4°C. The fluid velocities, theoretical

velocity head, theoretical head of probe and percent error for each trial are added

up for further analysis and are presented in Table 3 to Table 5.

Table 3. Trial 1 (Laminar and Transitional flow)Tube no.

fluid velocity (m/s)

theoretical velocity head (mm)

theoretical head of probe (mm)

% Error

1 0.1833 1.7133 132.7125 0.972 0.1632 1.3571 134.3571 0.483 0.1436 1.0508 135.0508 0.044 0.1192 0.7247 135.7247 0.535 0.0860 0.3771 135.3771 0.286 0.0293 0.0439 135.0439 0.03

Table 4. Trial 2 (Transitional and Laminar Flow)

Tube no. fluid velocity (m/s)

theoretical velocity head (mm)

theoretical head of probe (mm)

% Error

1 0.2215 2.5016 191.5016 0.782 0.1972 1.9814 191.9814 0.533 0.1735 1.5342 192.5342 0.244 0.1441 1.0582 193.0582 0.035 0.1039 0.5505 193.5505 0.236 0.0354 0.0640 194.0640 0.03

Table 5. Trial 3 (Turbulent Flow)

Tube no. fluid velocity (m/s)

theoretical velocity head (mm)

theoretical head of probe (mm)

% Error

1 1.2363 77.9036 172.9036 11.622 1.1003 61.7035 179.7035 7.963 0.9682 47.7776 185.7776 4.434 0.8041 32.9529 171.9529 14.57

5 0.5800 17.1445 185.1445 6.406 0.1978 1.9944 195.9944 0.51

Sample Calculations:

Preliminary calculations:T=301 .55 K

MW of H2 O=18 . 015 kgkmol

ρ=C1+C2T+C3T2+C 4 T 3

ρ=−13. 851+(0. 64038 )(301. 55)+(−0. 00191)(301 .55 )2+(1 .8211 x 10−6 )(301 . 55)3

ρ=55. 51047503 moldm 3=1000 .0212 kg

m3

μ=exp(C1+C2T

+C 3 ln T+C 4 T C 5)μ=exp[−52.843+3703 . 6

301 . 55+5 . 866 ln 301.55+(−5 . 879 x 10−29)(301 .55 )10]

μ=8 . 478 x 10−4 Pa⋅s

Data CalculationsTrial 1, Tube No. 1

v1=qA

=(0 .0144 L

s )(1 m3

1000 L )(78.54 mm2 )(1m2

10002mm2 )=0. 1833m

s

Continuity Equation : P1+12

ρv12

gc+ρ gz1=P2+

12

ρv22

gc+ρ gz2

Neglect change in Potential Energy . The fluid comes to rest at stagnation point, uo=0.Take points 1-6 as point 1 and the probe as point 2:static pressure+dynamic pressure=stagnation pressure

Pe+12

ρve2

gc=Po

in terms of head (h) :he+hev

=ho

where :he=measured static head

hev= theoretical velocity head=ve

2

2gho= theoretical head of probe

he+hev=hotheoretical

131+(0. 1833m

s)2

(2 )(9 . 81ms2)

(1000 mm1 m )=hotheoretical

hotheoretical=132. 7125 mm

% Error=|hotheoretical

−hoactual|

hotheoretical

x 100=|132 . 7125−134|132 .7125 x100=0. 97 %

C. Analysis of Results

The assumptions in Bernoulli’s Theorem may seem rigorous but the

simplicity of the equation gives major importance into the balance between

pressure, velocity and elevation.

In Table 3 to Table 5, values calculated for the fluid velocities, theoretical

velocity head, theoretical head of probe and percent error for each trial are

presented. The fluid velocity was derived from the diameter of the cross-section

particular in the apparatus and the measured flow rates for each tube of each the

three trials. The theoretical velocity head and head of probe were computed using

Bernoulli’s Equation.

The computed heads in Trials 1 and 2 both having laminar and transitional

flows have almost the same value with that of the total head probe, only having

errors less than 1.00% for each tube. However, in the third trial, the average

percent error for all the tubes resulted to 7.58%, revealing that the actual total

head probe is relatively larger than the theoretical head probe. Due to turbulence,

presence of vortices and eddies may have occurred [7]. As a result, the

consistency of the density of the fluid at different points in the flow system cannot

be assured which violates one of the assumptions of Bernoulli’s Theorem.

Deviations may also be caused by difficulty of reading the manometer

levels because of the graduation of the apparatus’s manometer. Another cause of

the variations may be instrumental error due to leakage that happened in the

apparatus (shown in Figure 5 in the Appendix).

In Table 1, the data for convergence and divergence are presented.

Consider the steady flow of a fluid in a converging duct with constant density

without losses due to friction. The flow satisfies all the assumptions governing the

use of Bernoulli’s equation. The parallel upstream and downstream of the

contraction makes the assumption that over the inlet and outlet areas, the velocity

is constant. When the streamlines are parallel, the pressure across them is

constant, excluding hydrostatic head differences if effect of gravity is disregarded.

In Bernoulli’s and the one-dimensional continuity equation, the theory states that

the fluid velocity will increase with the decrease of pressure of the fluid and area

of the cross-section. The results of the experiment validate this theory because the

velocity is highest at the point where the cross sectional area of the converging

duct is lowest. For a fluid under divergent flow, Bernoulli’s equation implies that

when the area perpendicular to flow is increased, the velocity of the fluid

decreases which is also validated in the results.

In addition, the results show that as the area decreases, the head of the

fluid at each tube and at the probe also decreases; hence, the velocity is inversely

proportional to the head of the fluid.

IV. Conclusion

In the derivation of the Bernoulli Equation, one of the assumptions is that the fluid has constant density throughout the flow. Results show that there are significant errors in the turbulent flow with a mean value of 7.58%. Under this flow condition, eddies caused the fluid to move rapidly in the system which violated the assumption of constant density throughout the fluid. Therefore, Bernoulli’s Theorem only holds true for laminar and transitional flows having mean values of 0.39% and 0.31% errors from the data, respectively. Higher derivations from the theoretical head of the probe observed

using a higher flow rate which results to a turbulent flow lessens the validity of the theorem. Additionally, the relationship between the velocity with the static head and total head as observed is indirectly proportional.

V. Appendix

Additional Tables and Figures from the Experiment:

Table 6 shows the computed values of the Reynolds number and the

corresponding type of flow for each tube in the three trials.

Table 6. Reynolds number and Type of Flow of the 3 TrialsTube no.

Trial 1 Trial 2 Trial 3NRe flow NRe flow NRe flow

1 2162.6608 transitional 2613.2151 transitional 14582.9418 turbulent2 2040.2460 laminar 2465.2973 transitional 13757.4923 turbulent3 1913.8591 laminar 2312.5797 transitional 12905.2583 turbulent4 1744.0813 laminar 2107.4315 transitional 11760.4370 turbulent5 1481.2745 laminar 1789.8734 laminar 9988.3163 turbulent6 865.0643 laminar 1045.2860 laminar 5833.1767 turbulent

Figures 4 to Figure 6 show the probe tapping at point 1, the leakage observed on the

apparatus and the diameter at different portion in the venture section.

Figure 4. Probe at tapping 1 Figure 5. Leak on the test section

Figure 6. The diameter at different portion in the venture section

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