Chapter 10, Problem 10.8P

Water is added at varying rates to a 300-liter holding tank. When a valve in a discharge line is opened, water flows out at a rate proportional to the height and hence to the volume V of water in the tank. The flow of water into the tank is slowly increased and the level rises in consequence, until at a steady input rate of 60.0 L/min the level just reaches the top but does not spill over. The input rate is then abruptly decreased to 40.0 L/min.

Write the equation that relates the discharge rate, ${\stackrel{\dot{}}{V}}_{out}\left(L/\min \right)$

1. , to the volume of water in the tank, V(L), and use it to calculate the steady-state volume when the input rate is 40 L/min.

Write a differential balance on the water in the tank for the period from the moment the input rate is decreased (t = 0) to the attainment of steady state (t ? 8), expressing it in the form dV/dt =? ? ?. Provide an initial condition.

Without integrating the equation, use it to confirm the steady-state value of V calculated in Part (a) and then to predict the shape you would anticipate for a plot of V versus t. Explain your reasoning.

Separate variables and integrate the balance equation to derive an expression for V(t). Calculate the time in minutes required for the volume to decrease to within 1 % of its steady-state value.