A 200 -gallon tank is half full of distilled water. Starting...

A 200 -gallon tank is half full of distilled water. Starting at time $t=0,$ a solution containing 0.5 pound of concentrate per gallon is admitted to the tank at a rate of 5 gallons per minute, and the well-stirred mixture is withdrawn at a rate of 3 gallons per minute. (a) At what time will the tank be full? (b) At the time the tank is full, how many pounds of concentrate will it contain? (c) Repeat parts (a) and (b), assuming that the solution entering the tank contains 1 pound of concentrate per gallon.

A 200 -gallon tank is half full of distilled water. Starting at time $t=0,$ a solution containing 0.5 pound of concentrate per gallon is admitted to the tank at a rate of 5 gallons per minute, and the well-stirred mixture is withdrawn at a rate of 3 gallons per minute.
(a) At what time will the tank be full?
(b) At the time the tank is full, how many pounds of concentrate will it contain?
(c) Repeat parts (a) and (b), assuming that the solution entering the tank contains 1 pound of concentrate per gallon.

Key Concepts

Inflow and Outflow Analysis

Analyzing the rates at which solution enters and leaves the tank is crucial. In this context, inflows add both volume and solute, while outflows remove them, and understanding these rates allows for the construction of the differential equations that describe the system.

Initial Conditions and Boundary Values

Initial conditions, such as starting volume and concentration, are essential for solving the mixing problem since they provide the necessary information to determine the particular solution of the differential equation. Additionally, boundary conditions, like the maximum tank capacity, are used to calculate specific events in the process.

Dynamic Volume Modeling

In systems where the volume changes over time due to differing inflow and outflow rates, it is important to model the volume dynamically. This approach determines the current volume of the mixture, which affects the concentration of the solute and the overall behavior of the system.

Differential Equations in Mixing Problems

This concept involves using differential equations to model the rate of change of a substance within a system. In mixing problems, the differential equation accounts for the rates at which a solute enters and exits a tank, providing a framework for predicting how the concentration changes over time.

Mass Balance Principle

The mass balance or material balance principle is the foundation for setting up equations in mixing problems. It states that the rate of change of the mass of a solute in the tank equals the mass entering minus the mass leaving the tank, ensuring conservation of mass in the system.

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