Observing the Decrease in Pressure with Increase in Velocity of a Fluid
1. Introduction
This laboratory activity demonstrates Bernoulli's principle, which is a fundamental concept in fluid dynamics. The principle states that for an inviscid flow of a non-conducting fluid, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.
Daniel Bernoulli published his principle in his book "Hydrodynamica" in 1738. This principle has wide applications in real-life scenarios including aircraft wing design, carburetor function, atomizer sprays, and many other fluid flow situations.
Bernoulli's equation can be expressed as:
$$P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$$
Where:
- $P$ = Pressure
- $\rho$ = Fluid density
- $v$ = Fluid velocity
- $g$ = Acceleration due to gravity
- $h$ = Height from reference point
2. Objective
To experimentally verify Bernoulli's principle by observing the relationship between fluid velocity and pressure in a pipe with varying cross-sectional areas.
By the end of this experiment, students should be able to:
- Understand the inverse relationship between fluid velocity and pressure
- Measure the pressure difference at various points along a pipe
- Calculate fluid velocities using the continuity equation
- Verify Bernoulli's equation experimentally
- Interpret the results and explain practical applications
3. Theoretical Background
Bernoulli's Principle
Bernoulli's principle states that as the velocity of a fluid increases, its pressure decreases, and vice versa. This is a consequence of the conservation of energy in fluid flow.
For a horizontal flow with constant elevation ($h_1 = h_2$), Bernoulli's equation simplifies to:
$$P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2$$
Continuity Equation
For an incompressible fluid flowing through a pipe, the mass flow rate remains constant. This leads to the continuity equation:
$$A_1 v_1 = A_2 v_2$$
Where:
- $A_1$ and $A_2$ = Cross-sectional areas at points 1 and 2
- $v_1$ and $v_2$ = Fluid velocities at points 1 and 2
Combining these two principles, we can predict that when fluid flows from a wider section to a narrower section:
- The velocity increases in the narrower section
- The pressure decreases in the narrower section
Note: This relationship assumes ideal conditions: steady, incompressible, inviscid flow with no heat transfer.
4. Apparatus Required
- Venturi tube or a pipe with varying cross-sectional area
- U-tube manometers for pressure measurement
- Water supply with flow control valve
- Flow meter or measuring cylinder and stopwatch
- Thermometer for measuring water temperature
- Meter scale for measuring dimensions
- Stopwatch
- Graph paper for data representation
- Calculator
Experimental Setup
Fig 1: Schematic diagram of the experimental setup showing the Venturi tube with manometer attachments at different sections.
5. Procedure
- Record the dimensions of the Venturi tube: measure the diameters $D_1$, $D_2$, and $D_3$ at the inlet, throat, and outlet sections.
- Calculate the corresponding cross-sectional areas $A_1$, $A_2$, and $A_3$.
- Connect the manometers to the pressure taps at various sections of the tube.
- Ensure all connections are airtight and manometers are properly zeroed.
- Start the water flow and adjust the control valve to establish a steady flow.
- Measure the volume flow rate $Q$ using the flow meter or by collecting a volume of water over a measured time period.
- Read and record the pressure head differences in the manometers at each section.
- Repeat steps 6-7 for at least five different flow rates by adjusting the control valve.
- Measure the water temperature to determine its density.
- Turn off the water supply and drain the system after completing all measurements.
Safety Precautions:
- Ensure the laboratory floor is kept dry to prevent slipping.
- Check all connections before starting to avoid leaks.
- Handle glassware components carefully.
6. Observations and Calculations
Record the following parameters:
- Diameter at inlet section, $D_1$ = ________ mm
- Diameter at throat section, $D_2$ = ________ mm
- Diameter at outlet section, $D_3$ = ________ mm
- Water temperature, $T$ = ________ °C
- Water density, $\rho$ = ________ kg/m³
Observation Table:
| Trial | Flow Rate (Q) (m³/s) | Pressure Head at Section 1 (h₁) (mm) | Pressure Head at Section 2 (h₂) (mm) | Pressure Head at Section 3 (h₃) (mm) |
|---|---|---|---|---|
| 1 | ||||
| 2 | ||||
| 3 | ||||
| 4 | ||||
| 5 |
Calculations:
-
Calculate the cross-sectional areas:
$$A_i = \frac{\pi}{4} \times D_i^2$$
-
Calculate the velocity at each section using the continuity equation:
$$v_i = \frac{Q}{A_i}$$
-
Convert the pressure head readings to pressure values:
$$P_i = \rho g h_i$$
-
Calculate the dynamic pressure at each section:
$$P_{dyn,i} = \frac{1}{2}\rho v_i^2$$
-
Verify Bernoulli's equation by calculating the total head at each section:
$$H_i = \frac{P_i}{\rho g} + \frac{v_i^2}{2g} + z_i$$
Where $z_i$ is the elevation of section $i$ (which is constant for a horizontal pipe).
Results Table:
| Trial | Section | Area (m²) | Velocity (m/s) | Pressure (Pa) | Dynamic Pressure (Pa) | Total Head (m) |
|---|---|---|---|---|---|---|
| 1 | 1 | |||||
| 2 | ||||||
| 3 |
7. Graphical Analysis
Plot the following graphs:
- Pressure vs. Velocity for each section
- Pressure vs. Cross-sectional area
- Velocity vs. Cross-sectional area
Graph Templates
Fig 2: Template for plotting Pressure vs. Velocity graph.
Expected Results:
The graphs should show:
- An inverse relationship between pressure and velocity
- A direct relationship between pressure and cross-sectional area
- An inverse relationship between velocity and cross-sectional area
8. Discussion Questions
- Explain how your experimental results confirm or contradict Bernoulli's principle.
- Calculate the percentage error between the theoretical and experimental values of the pressure drop.
- What could be the possible sources of error in this experiment?
- How would the results change if a more viscous fluid (like oil) was used instead of water?
- Explain how Bernoulli's principle applies in the following real-world applications:
- Aircraft wing design
- Venturi meter for flow measurement
- Spray atomizer
- Chimney effect in buildings
- Derive the mathematical relationship between pressure difference and velocity difference using Bernoulli's equation.
For the relationship between pressure difference and velocity difference:
$$P_1 - P_2 = \frac{1}{2}\rho (v_2^2 - v_1^2)$$
For a case where $A_1 \gg A_2$, from continuity equation $A_1v_1 = A_2v_2$, we get $v_1 \ll v_2$, so:
$$P_1 - P_2 \approx \frac{1}{2}\rho v_2^2$$
9. Conclusion
Summarize your findings from the experiment and state whether they support Bernoulli's principle. Include:
- The observed relationship between pressure and velocity
- The accuracy of your experimental results compared to theoretical predictions
- Possible improvements to the experimental setup
- Real-world applications of the principle demonstrated in this experiment
Note: A well-written conclusion should reflect on both the quantitative results and the qualitative understanding gained from the experiment.