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Created: 2024/12/14 15:25

Experiment No.: 01

Experiment Name: Proof of Bernoulli's Equation

Objective

The objective of this experiment is to,

1. Plot static head, velocity head, elevation head and total head (both measured and theoretical) against pressure measuring points (length) on the graph paper.
2. Determine the total head loss ($H_L$) for the inflow conditions.

Scope

The scope of Bernoulli's equation and this experiment includes:

1. Demonstrating the conservation of energy in steady, incompressible, and non-viscous fluid flow.
2. Construction of flow measuring devices such as venturimeter, flow nozzle, orifice meter and pitot tube.
3. Analysis of practical flow problems under a sluice gate, free liquid jet, radial flow and free vortex motion.

Datasheet and Graph

Parameter	V1	V2	V3	V4	V5	V6
d	0.028	0.023	0.014	0.017	0.024	0.028
A	6.33E-04	3.98E-04	1.54E-04	2.32E-04	4.60E-04	6.33E-04
H_stat	0.235	0.230	0.160	0.190	0.215	0.220
H_total	0.317	0.317	0.317	0.317	0.317	0.317
H_datum	0.080	0.080	0.080	0.080	0.080	0.080
H_dyn	0.002	0.007	0.077	0.047	0.022	0.017
V_mes	0.223	0.356	0.919	0.609	0.308	0.223
V_calc	0.198	0.370	1.228	0.960	0.657	0.577
H_dyn_mes	0.003	0.006	0.043	0.019	0.005	0.003
H_total_mes	0.318	0.316	0.283	0.289	0.300	0.303

Calculation

Measured velocity head is calculated by the formula:

$$H_{dyn\mathrm{\_}mes}=\frac{v_{{mes}^2}}{2\times g}=\frac{1}{2\times 9.8}\times v_{{mes}^2}=0.051v_{{mes}^2}$$ Hence, the measured velocity head at:

• Point 1 is $0.051\times {0.223}^2=0.003\text{[PARSE ERROR: Undefined("Command("space")")]}m$
• Point 2 is $0.051\times {0.356}^2=0.006\text{[PARSE ERROR: Undefined("Command("space")")]}m$
• Point 3 is $0.051\times {0.919}^2=0.043\text{[PARSE ERROR: Undefined("Command("space")")]}m$
• Point 4 is $0.051\times {0.609}^2=0.019\text{[PARSE ERROR: Undefined("Command("space")")]}m$
• Point 5 is $0.051\times {0.308}^2=0.005\text{[PARSE ERROR: Undefined("Command("space")")]}m$
• Point 6 is $0.051\times {0.223}^2=0.003\text{[PARSE ERROR: Undefined("Command("space")")]}m$

Consequently, the measured total head is calculated by the formula:

$$H_{total\mathrm{\_}mes}=H_{stat}+H_{dyn\mathrm{\_}mes}+H_{datum}$$

Hence, the measured total head at:

• Point 1 is $0.235+0.003+0.08=0.318\text{[PARSE ERROR: Undefined("Command("space")")]}m$
• Point 2 is $0.230+0.006+0.08=0.316\text{[PARSE ERROR: Undefined("Command("space")")]}m$
• Point 3 is $0.160+0.043+0.08=0.283\text{[PARSE ERROR: Undefined("Command("space")")]}m$
• Point 4 is $0.190+0.019+0.08=0.289\text{[PARSE ERROR: Undefined("Command("space")")]}m$
• Point 5 is $0.215+0.005+0.08=0.300\text{[PARSE ERROR: Undefined("Command("space")")]}m$
• Point 6 is $0.220+0.003+0.08=0.303\text{[PARSE ERROR: Undefined("Command("space")")]}m$

For real fluids, Bernoulli's equation is modified to:

$$H_{total}=H_{total\mathrm{\_}mes}+H_L$$ Applying the value of $H_{total\mathrm{\_}mes}$ for point 6, we can get,

$$0.317\text{[PARSE ERROR: Undefined("Command("space")")]}m=0.303\text{[PARSE ERROR: Undefined("Command("space")")]}m+H_L$$ or,

$$H_L=0.317-0.303=0.014\text{[PARSE ERROR: Undefined("Command("space")")]}m$$

Discussion

The experiment aimed to verify Bernoulli's Equation and concluded that the total head does indeed tend to remain constant across every section of the fluid flow. There were some head losses due to the fluid flow not being frictionless and incompressible. The head loss was determined to be 0.014 m.

Possible experimental errors:

1. Errors in reading static pressure from manometer due to parallax and inconsistent water level. A better approach might've been to take a photo of the whole system and take readings from the photo.
2. Potential miscalibrations of the Bernoulli's apparatus.

Difference in total head values at first and last section: The measured total head values at first and last section are different due to head loss ($H_L$), caused by:

1. Frictional losses: Real fluids are not frictionless or perfectly incompressible as opposed to the Bernoulli's Theorem assumption.
2. Other losses: Energy loss through heat and sound further reduced total energy and therefore total head at the downstream section.

Nature of curves:

1. Velocity head curve: Increases at higher velocity and therefore narrow flow passage which is correctly reflected by the experiment — reduced diameter $d$ produced increased velocity head $H_{dyn\mathrm{\_}mes}$.
2. Pressure head curve: Increases in wider section which is correctly reflected by the experiment — increased diameter $d$ produced increased pressure head $H_{stat}$.
3. Total head curve: Ideally constant, but in practice, it gradually decreases due to energy losses as reflected by this experiment. However, it is noticeable that the total head curve seemed to have recovered some energy losses towards the end section, it could be due to measurement errors at the mid sections.

Assignments

1. Assumptions underlying the Bernoulli's energy equation:

• The fluid is incompressible.
• The flow is steady — fluid properties do not change with time at any point.
• The fluid is non-viscous.
• The flow is along a streamline, with no energy added or removed by pumps, turbines, or heat exchange.

2. Modifications of Eqn (1):

(a) When frictional head loss ($h_f$) is to be considered:

The equation becomes:

$$\frac{P_1}{\gamma }+\frac{V_{1^2}}{2g}=\frac{P_2}{\gamma }+\frac{V_{2^2}}{2g}+h_f$$

Here, $h_f$ accounts for energy loss due to friction.

(b) When the conduit is not horizontal:

The equation becomes:

$$\frac{P_1}{\gamma }+\frac{V_{1^2}}{2g}+z_1=\frac{P_2}{\gamma }+\frac{V_{2^2}}{2g}+z_2$$

Here, $z$ accounts for the potential energy difference due to elevation changes.

3. Difference between energy line and energy grade line:

• Energy Line (EL): Represents the total head, including pressure head, velocity head, and elevation head. It shows the maximum energy available at a point in the system.
• Energy Grade Line (EGL): Lies below the energy line by the magnitude of the velocity head. It represents the sum of the pressure head and elevation head, excluding velocity head.