Bernouilli and Torricelli Lab Sheet

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Southern Adventist University *

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Mechanical Engineering

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Apr 3, 2024

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BERNOULLI AND TORRICELLI PRINCIPLES Objective: To experimentally verify Torricelli’s principle using hydrodynamic. Equipment: Transparent plastic bucket with a hole at one of the bottom corners, ruler, graduated cylinder, stopwatch (on your phone), water supply (tap water). Preliminary work: 1. Write Bernoulli’s principle and give the meaning of each symbols. The total mechanical energy of the moving fluid comprising the gravitational potential energy of elevation, the energy associated with the fluid pressure and the kinetic energy of the fluid motion, remains constant. Bernoulli’s equation formula is a relation between pressure, kinetic energy, and gravitational potential energy of a fluid in a container. The formula for Bernoulli’s principle is given as follows: Where p is the pressure exerted by the fluid, v is the velocity of the fluid, ρ is the density of the fluid and h is the height of the container, g the gravitational force. Bernoulli’s equation gives great insight into the balance between pressure, velocity and elevation 2. If a bucket without lid is filled with water, what would be the pressure at the surface of the water that is in contact with the atmosphere? The pressure at the surface of the water that is in contact with the atmosphere is equal to the pressure of the atmosphere. (Patm) 3. If H is the height of the water in the bucket, give the expression of the pression at the bottom of the bucket in function of H, g, and ρ water . First consider the very simple situation where the fluid is static—that is, v1=v2=0 Bernoulli’s equation in that case is: P1+ ρ gh1= P2 + ρ gh2 We can further simplify the equation by setting h2 = 0 (Any height can be chosen for a reference height of zero, as is often done for other situations involving gravitational force, making all other heights relative.) In this case, we get P2 = p1 + ρgh1 4. Assume that there is an opening (hole) at the bottom of the bucket and the water is jetting out through the opening. (a) what would be the pressure of the water at the opening area? (b) If the opening is very small compared to the basis area of the bucket, how would be the downward speed of the water molecules at the upper surface compared to the exit speed at the hole?

a) Assuming that the opening area is exposed to the atmosphere, the pressure at the opening area is equal to the pressure of the atmosphere. 14.7 psi = 1 atm b) The downward speed of the water molecules at the upper surface will be really slow because of the small opening at the bottom that can be consider as zero or negligible. 5. Torricelli’s law is a special case of Bernoulli’s principle. What are the conditions to be fulfilled for Torricelli’s principle? What does Torricelli’s law say? Torricelli's law states that the speed of efflux, v of a fluid through a sharp-edged hole at the bottom of a tank filled to a depth h is the same as the speed that a body (in this case a drop of water) would acquire in falling freely from a height h. And assimilates the one for free fall whether the jet is at an angle or not. This problem is solved using Bernoulli's equation, gh +ρPatm=2v2+ρPatm ⟹ v=2gh This implies that the time to collect the same amount of water through the hole varies inversely with the square root of the height of water, a relationship that you have to derive in your calculations—in another words, show how 1/h = C*t 2 where C is a constant supports Torricelli’s law. 6. The volume flow rate Q is equal to the volume that leaves the hole per unit of time. How is it then related to the exit speed and the opening area? The flow rate will be proportional to the volume. The opening area the smaller it is, the exit speed will be also small, and in the bigger case will be the opposite vice versa. 7. Using your answers in #3, 5, and 6, find the relationship between the pressure at the bottom of the bucket and the time elapsed to collect a given volume of water. The pressure at the bottom of the bucket is greater. Q= (pi. Pr^4)/ 8ln Poiseuille equation Increase velocity and decrease in pressure – Bernoulli principle ACTIVITIES: If one fills the bucket with water until a certain height and then release the finger that clogged the bottom corner, the water would flow according to Torricelli’s principle. Torricelli’s law, principle, or theorem describes the exit speed of a non-viscous fluid when the opening area is very small compared to the upper area of the water that is in contact with the atmosphere. In that case, the exit speed is proportional to the square root of the height of the water (or depth of the hole) and assimilates the one for free fall whether the jet is at an angle or not. This implies that the time to collect the same amount of water through the hole varies inversely with the square root of the height of water, a relationship that you have to derive in your calculations—in another words, show how 1/h = C*t 2 where C is a constant supports Torricelli’s

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