Problem 10 - Advanced Engineering Mathematics

Question

Suppose that a large mixing tank initially holds 300 gallons of water in which 50 pounds of salt has been dissolved. Another brine solution is pumped into the tank at a rate of \(3 \mathrm{gal} / \mathrm{min}\), and when the solution is well stirred, it is pumped out at a slower rate of \(2 \mathrm{gal} / \mathrm{min}\). If the concentration of the solution entering is \(2 \mathrm{lb} / \mathrm{gal}\), determine a differential equation for the amount \(A(t)\) of salt in the tank at time \(t\).

Step-by-Step Solution

Answer

The differential equation is \( \frac{dA}{dt} = 6 - \frac{2A}{300 + t} \).

Step 1: Understand the Problem

We're given a tank initially containing 300 gallons of water with 50 pounds of salt dissolved. A brine solution with a concentration of 2 lb/gal is pumped into the tank at 3 gal/min, while the mixed solution is pumped out at 2 gal/min. We need to find a differential equation for the amount of salt \( A(t) \) at time \( t \).

Step 2: Calculate Volume Over Time

Initially, the tank has 300 gallons of water. Since we're adding at 3 gal/min and removing at 2 gal/min, there is a net increase of 1 gal/min. Therefore, the volume \( V(t) \) at time \( t \) is \( V(t) = 300 + t \).

Step 3: Calculate Salt Input Rate

The brine solution entering is 2 lb/gal at a rate of 3 gal/min. Thus, the rate of salt being added to the tank is \( 3 \times 2 = 6 \) lb/min.

Step 4: Calculate Salt Output Rate

The salt leaves the tank along with the solution at 2 gal/min. Since the concentration of salt in the tank is \( \frac{A(t)}{V(t)} \) lb/gal, the rate of salt going out is \( 2 \times \frac{A(t)}{V(t)} = \frac{2A(t)}{300 + t} \) lb/min.

Step 5: Define the Differential Equation

The change in the amount of salt \( A(t) \) in the tank is the difference between the rate of salt entering and leaving:\[ \frac{dA}{dt} = 6 - \frac{2A}{300 + t} \]

Key Concepts

Mixing Problems

Mixing problems are a fascinating application of differential equations in which we analyze how different substances, like salt and water, mix over time in a tank or reservoir. This involves setting up a system where solutions are continuously stirred, and substances are either added or removed. Because of the ongoing in-and-out process, mixing problems require us to calculate the concentration of substances at different times.

In solving these problems, you'll often need to determine:

• The initial conditions: How much of each substance is present initially?


• The rate at which different solutions are entering and leaving the system.
• The concentration of substances in incoming/outgoing solutions.

By understanding these factors, you can create a model, typically a differential equation, enabling you to predict future concentrations and the behavior of the mixture over time.

Rate of Change

Rate of change is a fundamental concept in calculus used to describe how one quantity changes in relation to another. In the context of mixing problems, the rate of change helps determine how the concentration of a substance, like salt, varies as solutions are added or removed.

When dealing with mixing problems, we often need to calculate two main rates:

• Inflow rate: How quickly a particular substance is being added to the tank.
• Outflow rate: How quickly the mixed solution, containing the substance, is being removed.

To formulate a differential equation for the rate of change, we compute these rates separately:
1. **Salt input rate:** The amount of substance entering per time unit. In this context, it is the flow rate of the brine solution times the concentration of salt in the incoming stream.
2. **Salt output rate:** The product of the outflow rate and the concentration of the substance in the tank at any given time. The overall rate of change in the substance's amount inside the tank is then the difference between the input rate and the output rate. This concept is key to developing the differential equation needed to model the mixing process.

Brine Solution

A brine solution is a mixture of salt and water, commonly used in various industrial applications. Understanding how brine behaves when mixed is vital for predicting concentrations and ensuring processes run smoothly.
The brine solution entering a tank often has a defined concentration, expressed in terms like pounds of salt per gallon of water. In a typical mixing problem, this solution is continuously added to a system, where the concentration affects how fast the salt's content changes.

Key considerations include:

• **Concentration:** Initial salinity of the brine solution being added. Higher salinity means more salt content impacts the system rapidly.
• **Flow Rates:** How quickly the brine enters and mixes within the tank. This affects the overall equilibrium achieved after mixing.
• **Stability:** Ensuring the mixing process achieves uniform distribution, especially crucial for systems managed over longer periods.

In summary, analyzing the behavior of a brine solution within a mixing tank allows us to compute differential equations that model the system's dynamics and output. This fundamental understanding is crucial in controlling and predicting the efficacy of the mixing process.