Question

A mixing tank initially contains 4500 kg of liquid water. The tank is fitted with two inlet pipes: one delivering hot water at a volumetric flow rate of 1.5×10−3 m3/s and one delivering cold water at a mass flow rate of 2.2 kg/s. In addition, the tank has an outlet pipe with a diameter of 0.2 m through which water travels at 0.5 m/s. a. How long does it take for the tank to empty assuming 1-D flow and constant mass flow rate in each of the pipes. Note: density of water is 1000 kg/m3. b. What is the necessary outlet flow velocity to ensure steady state flow?

Answer 1

To determine the time it takes for the tank to empty, we calculate the flow rate of water through the outlet pipe and divide the initial volume by the total flow rate. An outlet flow velocity is necessary to ensure steady-state flow, which can be calculated using the known values for flow rate and cross-sectional area.

Step-by-step explanation:

To determine the time it takes for the tank to empty, we need to calculate the flow rate of water through the outlet pipe. The flow rate is given by the formula Q = Av, where Q is the volumetric flow rate, A is the cross-sectional area of the pipe, and v is the velocity of the water. We can calculate the cross-sectional area of the outlet pipe using the formula A = πr^2, where r is the radius of the pipe. In this case, the radius is half of the diameter (0.2 m / 2 = 0.1 m).

We can substitute the given values into the formulas to find the flow rate:

1. The flow rate through the outlet pipe is Q = (π × 0.1^2) × 0.5 m/s.
2. Next, we need to determine the total flow rate into the tank. The volumetric flow rate of the hot water inlet pipe is given (1.5 × 10^-3 m^3/s), and the volumetric flow rate of the cold water inlet pipe can be calculated using the mass flow rate and the density of water (2.2 kg/s ÷ 1000 kg/m^3).
3. Finally, we can calculate the time it takes for the tank to empty by dividing the initial volume of water in the tank by the total flow rate.

An outlet flow velocity is necessary to ensure steady-state flow. In steady-state flow, the flow rate of water into the tank is equal to the flow rate of water out of the tank. By rearranging the formula Q = Av, we can solve for the velocity v, using the known values for Q and A.