[Fluid Dynamics] Bernoulli's Eqn. Problem

[Je m'appelle]

Homework Statement

A bomb located 3 meters below the ground needs a certain pressure in order to pump the water through the pipe (connecting the ground to the bomb) and 12 meters above the ground. The radius of the pipe is 0.01 meters and the radius of the 'mouthpiece' of the tube in the ground is 0.005 meters. Find the pressure of the bomb.

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Homework Equations

Bernoulli's Equation

$$\frac{1}{2}\rho v^2 + p + \rho g h = C$$

Conservation of Mass Equation for a Stationary Fluid


$$A_1 v_1 = A_2 v_2$$

Eqn for the Velocity of an Accelerating Body


$$v_2^2 = v_1^2 + 2a\Delta S$$

The Attempt at a Solution

Using Bernoulli's for A or the opening of the pipe at the bomb and for B the ending (or 'mouthpiece' of the pipe) at the ground we get

$$\frac{1}{2}\rho v_A^2 + p_A = \frac{1}{2}\rho v_B^2 + p_B + \rho g h_{AB}$$

$$(p_A - p_B) = \Delta p = \rho (\frac{1}{2} v_B^2 - \frac{1}{2} v_A^2 + g h_{AB})$$

Using the Conservation Eqn. we get


$$A_1 v_A = A_2 v_B$$

$$\pi (0.01)^2 v_A = \pi (0.005)^2 v_B$$

$$v_A = 0.25 v_B$$

$$v_A^2 = 0.625 v_B^2$$

Substituting the relation between the velocities in our previous eqn. we get

$$\Delta p = \rho (0.1875 v_B^2 + g h_{AB})$$

$$\Delta p = \rho (0.1875 v_B^2 + 29.4)$$

Now we have to find the velocity at B, that can easily be found considering that the water flows 12 meters into the air, therefore

$$v_2^2 = v_1^2 + 2a\Delta S$$

$$0 = v_B^2 - 2g \Delta h$$

$$v_B^2 = 2g \Delta h$$

$$v_B^2 = 235.2$$

Back to the previous eqn.

$$\Delta p = 10^3 (44.1 + 29.4)$$

$$\Delta p = 7.35 \times 10^4$$

Notice that the power of the pressure is p of B + delta p, or delta p + atmospheric pressure, therefore

$$p_A = p_{bomb} = \Delta p + p_B$$

Is this correct?

 

[anathema]

Your approach seems to be correct. However, there are a few things to consider in order to ensure the accuracy of your solution. Firstly, it is important to note that Bernoulli's equation is only applicable for steady, incompressible flow. This means that the flow must be constant and the fluid must not change in density. Since we are dealing with water, which is an incompressible fluid, this assumption is valid.

Secondly, it is important to consider the effects of friction and viscosity in the pipe. These factors can cause a decrease in pressure and velocity as the water flows through the pipe. In order to account for this, the equation for Bernoulli's principle can be modified to include a friction term. This can be done by multiplying the term for velocity by a factor called the Darcy friction factor. This factor takes into account the roughness of the pipe and the Reynolds number of the flow.

Lastly, it is important to check your units and ensure they are consistent throughout your calculations. In your solution, you have used a value of 10^3 for the density of water, which is in kg/m^3. However, the value for the gravitational acceleration you have used is in m/s^2. This could affect the final result of the pressure.

Overall, your solution seems to be on the right track. Just make sure to double check your calculations and take into account any other factors that may affect the accuracy of your answer. Keep up the good work!