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Is height and pressure proportional ? TBR PG 97 problem 21
- Thread starter dahmsom
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you can narrow the equation to p1 + pgh2 = p2 + pgh2. You can use your knowledge of pressure/cross-sectional area relationship to figure out that as you proceed from left to right the water pressure is increasing. so from p1 --> p2, p is increasing. Therefore on side (2) one of the units would have to decrease in pgh to keep both sides of the equation equal. You can't change the density or the effect of gravity, therefore the h (height) has to be decreased. Is this what you were asking? By the way, just in case some people hadn't realized this; the p (pressure) in fluid dynamics are based on the forces exerted on the OUTER walls, not the force of the fluid moving forward.
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The height of the water in each column will be a result of the Pressure of the fluid on the outer walls near that column. So, the water level will be lowest in the column that has the lowest Pressure.
Bernoulli's eqn, comparing columns 1 and 2:
P1 + density * g * h1 + 1/2 * density * v1^2 = P2 + density * g * h2 + 1/2 * density * v2^2
And continuity equation:
a1 * v1 = a2 * v2
h1 and h2 refer to the height of the fluid above some datum, which I think is negligible in this question. Betweens columns 1 and 2, the cross sectional area is decreasing. From the continuity equation, a1 > a2, so v2 > v1. Since v2 > v1, P1 > P2 to satisfy Bernoulli's equation describing conservation of energy. A lower pressure in column 2 will result in a lower height of water in column 2- this is different from the h2 variable in Bernoulli's equation.
Following this logic, column 4 has the smallest area, so it has the largest velocity of fluid flow (because the same amount of fluid has to get through a smaller area in the same amount of time -> it has to move faster!). With a larger velocity, the pressure will be decreased resulting in the lowest height of water in the column.
Bernoulli's eqn, comparing columns 1 and 2:
P1 + density * g * h1 + 1/2 * density * v1^2 = P2 + density * g * h2 + 1/2 * density * v2^2
And continuity equation:
a1 * v1 = a2 * v2
h1 and h2 refer to the height of the fluid above some datum, which I think is negligible in this question. Betweens columns 1 and 2, the cross sectional area is decreasing. From the continuity equation, a1 > a2, so v2 > v1. Since v2 > v1, P1 > P2 to satisfy Bernoulli's equation describing conservation of energy. A lower pressure in column 2 will result in a lower height of water in column 2- this is different from the h2 variable in Bernoulli's equation.
Following this logic, column 4 has the smallest area, so it has the largest velocity of fluid flow (because the same amount of fluid has to get through a smaller area in the same amount of time -> it has to move faster!). With a larger velocity, the pressure will be decreased resulting in the lowest height of water in the column.
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