Bernoulli’s Equation (College Board AP® Physics 1: Algebra-Based)

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Dan Mitchell-Garnett

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Bernoulli’s equation

Conditions for ideal fluid flow

Bernoulli's equation describes ideal fluid flow between two points a pipe
This requires some large assumptions, however, which are:
1. The fluid is incompressible
2. The fluid has no viscosity (there is no friction)
3. The fluid moves smoothly through the pipe (no turbulence)
These assumptions also mean that flow rate is constant
Bernoulli's equation is a good approximation for some real-world scenarios but not others

Energy conservation in fluid flow

In ideal fluid flow no energy is dissipated, as viscosity is assumed to be negligible
This means that, as fluid flows from one point to another, energy is conserved
- The sum of kinetic energy, potential energy and work done at point 1 is equal to that at point 2
By equating these mechanical energies, Bernoulli's equation is reached (the derivation is not required):

$P_{1} + ρ g y_{1} + \frac{1}{2} ρ {v_{1}}^{2} = P_{2} + ρ g y_{2} + \frac{1}{2} ρ {v_{2}}^{2}$

Where:
- $P_{1}$ = pressure at point 1, measured in $Pa$
- $ρ$ = fluid density, measured in $kg / m^{3}$
- $y_{1}$ = height of point 1 above reference level, measured in $m$
- $v_{1}$ = fluid speed at point 1, measured in $m / s$
- $P_{2}$ = pressure at point 2, measured in $Pa$
- $y_{2}$ = height of point 2 above reference level, measured in $m$
- $v_{2}$ = fluid speed at point 2, measured in $m / s$
To compare the heights of each point, a reference height must be arbitrarily chosen, much the same as gravitational potential energy

Bernoulli's equation in a pipe

A tube is shown with point 1 at its left side and point 2 at its right. Point 1 is labelled with P_1 for pressure, speed of v_1 (arrow pointing right) and height y_1. The fluid is labelled with ρ for its density. Point 2 is labelled with P_2 for pressure, speed of v_2 (arrow pointing right) and height y_2. Heights are measured from a horizontal reference level below the pipe. — **The Bernoulli equation describes the conservation of energy in ideal fluid flow.**

Streamline and turbulent flow

A fluid can be considered to flow in streamlines
- These are lines which point in the direction of flow
Streamline (or laminar) flow refers to the fluid moving smoothly through the tube, without friction
- Streamline flow is an ideal model, but fluids moving with slow flow speeds and low viscosity behave close to this
In turbulent flow the streamlines move in different directions and cross over
- This produces swirling vortices and unpredictable flow
- Fluids moving at high speeds or with high viscosity tend to exhibit this behaviour

Worked Example

A fluid is being pumped along a pipe. Location 1 is 12 m above the ground while location 2 is 6 m above.

The pressure at location 1 is half the pressure at location 2.

Complete the Bernoulli bar chart for location 2.

The image now features a bar for P_2 which is 3 divisions high, a bar for ρgy_2 which is half a division high and a bar for 1/2 ρg(v_2)^2 which is one division high

You may assume the ground is perfectly flat.

Answer:

Step 1: Analyze the scenario

A pipe is carrying a fluid from location 1 to location 2
Between these locations:
- The pressure doubles from location 1 to location 2
- The height above ground halves from location 1 to location 2
- The change in speed is unknown
- Energy is conserved

Step 2: Apply Bernoulli's equation

Recall Bernoulli's equation:

$P_{1} + ρ g y_{1} + \frac{1}{2} ρ {v_{1}}^{2} = P_{2} + ρ g y_{2} + \frac{1}{2} ρ {v_{2}}^{2}$

The sum of the three terms at location 1 is equal to the sum of the three terms at location 2
This means the combined height of each bar chart should be equal

Step 3: Apply the specific conditions

The pressure doubles from location 1 to location 2
- The bar for $P_{2}$ must be twice the height of the bar for $P_{1}$
- This is a height of 3 divisions on the y axis
The height above ground halves from location 1 to location 2
- The bar for $ρ g y_{2}$ must be half the height of the bar for $ρ g y_{1}$
- This is a height of 0.5 divisions

Step 4: Complete the second bar chart

The total height of the Bernoulli bars at location 1 is 4.5 divisions
The total height of the bars at location 2 must also be 4.5 divisions
- The height of the $P_{2}$ and $ρ g y_{2}$ bars combined is 3.5 divisions
- The height of the $\frac{1}{2} ρ {v_{2}}^{2}$ bar must be 1 division

Examiner Tip

Bernoulli's equation is essentially a conservation of energy equation with some constants canceled out on either side:

The pressure term refers to mechanical work
The height term refers to gravitational potential energy
The speed term refers to kinetic energy

The sum of these terms is always the same value at any point in an ideal fluid flow.

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