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Review. Old Faithful Geyser in Yellowstone National Park erupts at approximately one-hour intervals, and the height of the water column reaches 40.0 m (Fig. P14.25). (a) Model the rising stream as a series of separate droplets. Analyze the free-fall motion of one of the droplets to determine the speed at which the water leaves the ground. (b) What If? Model the rising stream as an ideal fluid in streamline flow. Use Bernoulli’s equation to determine the speed of the water as it leaves ground level. (c) How does the answer from part (a) compare with the answer from part (b)? (d) What is the pressure (above atmospheric) in the heated underground chamber if its depth is 175 m? Assume the chamber is large compared with the geyser’s vent.
Figure P14.25
(a)
The speed at which the water leaves the ground when it is modeled as separate droplets.
Answer to Problem 14.50P
The speed at which the water leaves the ground when it is modeled as separate droplets is
Explanation of Solution
The height of the water column is
For the free fall, the equation of motion for upward flight is,
Here,
Substitute
Conclusion:
Therefore, the speed at which the water leaves the ground when it is modeled as separate droplets is
(b)
The speed of water as it leaves the ground using Bernoulli’s equation.
Answer to Problem 14.50P
The speed of water as it leaves the ground using Bernoulli’s equation is
Explanation of Solution
Apply Bernoulli’s equation at the ground level and at the top of the column.
Here,
Substitute
Conclusion:
Therefore, the speed of water as it leaves the ground using Bernoulli’s equation is
(c)
The answer obtained in part (a) with the answer obtained in part (b).
Answer to Problem 14.50P
Both the models are consistence with each other as the speed of water leaving the ground is same in both models.
Explanation of Solution
Along a streamline for an ideal fluid, the sum of the pressure, kinetic energy per unit volume and the potential energy per unit volume is same at all the points.
The speed of the water leaving the ground obtained in both part (a) as well as in part (b) is
This shows that the both the models are consistence with each other.
Conclusion:
Therefore, both the models are consistence with each other as the speed of water leaving the ground is same in both models.
(d)
The pressure in the heated underground chamber.
Answer to Problem 14.50P
The pressure in the heated underground chamber is
Explanation of Solution
Apply the Bernoulli’s equation between the chamber and the geyser vent.
Here,
Substitute
Substitute
Conclusion:
Therefore, the pressure in the heated underground chamber is
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