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Chapter 3: Problem 104

A spherical tank of diameter \(D\) has a drain hole of diameter \(d\) at its bottori. A vent at the top of the tank maintains atmospheric pressure at the liquid surface within the tank. The flow is cuasisteady and inviscid and the tank is full of water initially. Determine the water depth as a function of time, \(h=h(t),\) and plot graphs of \(h(t)\) for tank diameters of \(1,5,10,\) and \(20 \mathrm{ft}\) if \(d=1\) in.

Short Answer

Expert verified
The water depth as a function of time, \(h(t)\), is expressed as \(h(t) = [\frac{D}{2} - \frac{gt^2 (D/2)^2}{4(d/2)^2}]^2\). From the graphs plotted using this equation for different values of D, it can be seen that the water level decreases over time, at rates that depend on the diameter of the tank.

Step by step solution

01

Define the Formulas Needed

The formula needed to solve the problem is the Torricelli's law, \(v = \sqrt{2 g h}\). The flow can be determined through the drain hole with area \(A_h = \pi (d/2)^2\), and the volume being drained from the tank per unit time can be expressed using this formula: \(\mathrm{d}V/\mathrm{d}t = A_h v\). To find the change in depth over time, use this formula: \(\mathrm{d}V/\mathrm{d}t = A_t \mathrm{d}h/\mathrm{d}t\), where \(A_t = \pi (D/2)^2\).
02

Equate the two volume flow rates

Equate the two expressions for \(\mathrm{d}V/\mathrm{d}t\) to give \(\pi (d/2)^2 \sqrt{2 g h} = \pi (D/2)^2 \mathrm{d}h/\mathrm{d}t\). This equation provides a relationship between h and t.
03

Solve for \(h(t)\)

The above differential equation can be rearranged and integrated on both sides to give \(h(t)\). The technique used here is called 'separation of variables', which necessitates the rearrangement of the differential equation so that all terms involving h are on one side and all terms involving t are on the other. For simplicity, ignore the constants at first: \(\mathrm{d}h/\sqrt{h} = \mathrm{d}t\). Integrating both sides will provide the equation for \(h(t)\) in the form \(2\sqrt{h} = t + C\), where C is the constant of integration.
04

Find C

To find C, consider the initial condition, which is \(h(0) = D/2\). By substituting \(h = D/2\) and \(t = 0\), we can solve for C. Once C is found, we can substitute it back to find the function \(h(t)\).
05

Plot the Graph of h(t) for different D values

Now, substitute the different values of D (1, 5, 10, and 20 ft) into the function \(h(t)\). Use a suitable graphing tool to plot the depth of water against time for each case to visualize how the water level in the tank decreases with time for different diameters of the tank.

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