Problem 28 - Advanced Engineering Mathematics

Question

In Example 5 the size of the tank containing the salt mixture was not given. Suppose, as in the discussion following Example 5, that the rate at which brine is pumped into the tank is \(3 \mathrm{gal} / \mathrm{min}\) but that the well-stirred solution is pumped out at a rate of \(2 \mathrm{gal} / \mathrm{min}\). It stands to reason that since brine is accumulating in the tank at the rate of \(1 \mathrm{gal} / \mathrm{min}\), any finite tank must eventually overflow. Now suppose that the tank has an open top and has a total capacity of 400 gallons. (a) When will the tank overflow? (b) What will be the number of pounds of salt in the tank at the instant it overflows? (c) Assume that although the tank is overflowing, the brine solution continues to be pumped in at a rate of \(3 \mathrm{gal} / \mathrm{min}\) and the well- stirred solution continues to be pumped out at a rate of \(2 \mathrm{gal} / \mathrm{min}\). Devise a method for determining the number of pounds of salt in the tank at \(t=150 \mathrm{~min}\). (d) Determine the number of pounds of salt in the tank as \(t \rightarrow \infty\). Does your answer agree with your intuition? (e) Use a graphing utility to plot the graph \(A(t)\) on the interval \([0,500)\).

Step-by-Step Solution

Answer

(a) 400 min, (b) 1200c_0 lbs, (c) Solve \(A(150)\), (d) 600c_0 lbs, (e) Use a plot utility.

Step 1: Determine Overflow Time

First, we need to calculate when the tank will overflow. Since brine is accumulating at a rate of \(1\) gallon per minute, we need to figure out how long it takes to fill the entire \(400\) gallon tank. Using the formula \(\text{time} = \frac{\text{total volume}}{\text{rate of accumulation}}\), we have \(\text{time} = \frac{400}{1} = 400\) minutes. Thus, the tank will overflow after \(400\) minutes.

Step 2: Calculate Salt Content at Overflow

Assume the brine initially has a concentration of \(c_0\) pounds per gallon. The salt concentration changes as the brine flows in and out. Since brine is entering at 3 gal/min with initial salt concentration \(c_0\), and exiting at 2 gal/min, the change in the amount of salt at any moment is expressed as \(\frac{dA}{dt} = 3c_0 - \frac{2}{V}(A)\), where \(V = 400 + (t - 400)\). Solving this equation tells us that at overflow (t = 400), the approximate calculation gives \(A(400) = 3 \times 400 \times c_0 = 1200 c_0\). Thus, the number of pounds of salt is 1200 times the initial concentration per gallon.

Step 3: Determine Salt at t=150 min

At time \(t = 150\), the tank has not yet overflowed. At this point, the tank's volume \(V\) is \(150 + C\) where \(C\) is the initial volume before entering brine (assume 0 since not stated). Solving \(\frac{dA}{dt} = 3c_0 - \frac{2}{V}(A)\) for this time interval with the same concentration shows that solutions develop, leading to \(A(150) = 3 \times 150 \times c_0\).

Step 4: Determine Salt as t Approaches Infinity

As time approaches infinity, the brine still entering holds its concentration. However, with equilibrium between inflow and outflow rates, the salt pounds stabilize when \(\frac{dA}{dt} = 0\). Solving the equilibrium \(3c_0 = \frac{2}{400}(A)\) reveals a solution, leading to \(A = 600c_0\). Thus, the amount of salt stabilizes at \(600c_0\) in equilibrium.

Step 5: Plot A(t) with Graphing Utility

To plot \(A(t)\), use a graphing utility, plotting from \([0, 500]\) minutes. Start the graph at \(A(0) = 0\) and observe changes. The graph will initially rise, then stabilize as the tank's volume approaches overflow and continues after equilibrium.

Key Concepts

Tank Overflow Problem

The tank overflow problem is a common scenario in differential equations exercises, where you analyze how a tank filling up with a liquid surpasses its capacity. Here, we have a tank that starts with no initial brine but is filled with brine at a rate of 3 gallons per minute. However, since the well-stirred mixture is simultaneously being pumped out at 2 gallons per minute, the net accumulation is 1 gallon per minute.
Given the tank capacity of 400 gallons, it is initially empty and will eventually fill up to the brim then overflow. To find out when this happens, you divide the tank’s total volume by the net rate of filling. This results in: \[ \text{overflow time} = \frac{400\, \text{gallons}}{1\, \text{gallon/min}} = 400\, \text{minutes} \]
Thus, after 400 minutes, the tank is full and begins to overflow. Understanding this process of gradual filling and point of overflow helps students draw analogous lessons in dynamic systems and capacity analysis.

Salt Mixture Concentration

Understanding the concentration of salt in a salt mixture presents another valuable learning aspect. Initially, the concentration of the salt in the brine being pumped into the tank is constant, denoted as \( c_0 \), the pounds per gallon.
The change in the amount of salt, \( A(t) \), at any time \( t \) is described by a differential equation reflecting both inflow and outflow of salt:\[ \frac{dA}{dt} = 3c_0 - \frac{2}{V}(A) \]
Here, \( V \) is the volume of the mixture in the tank, dynamically increasing as time passes. By solving this equation, you calculate the salt amount at overflow (400 minutes) as 1200 times the initial concentration: \( A(400) = 1200c_0 \).
This step illustrates how the dynamics of flow rates, concentration, and time all influence the solution mix, emphasizing the importance of differential equations in real-life scenarios.

Rate of Change

The rate of change in this exercise is a crucial part of comprehending how variables evolve over time within a system. In the context of the tank overflow problem, the rate of change of both water volume and salt concentration is governed by differential equations.
The inflow of brine is 3 gallons per minute, while the outflow is 2 gallons per minute, leading to a net change of 1 gallon per minute. This accumulates until overflow occurs.
Additionally, for the salt concentration, we have:\[ \frac{dA}{dt} = 3c_0 - \frac{2}{V}(A) \] This depicts how the rate of change of the salt amount is impacted by both entering and exiting streams, where the balance eventually governs the system to reach equilibrium.
Understanding these rates helps students comprehend how systems behave under varying conditions and can be broadly applicable in various scientific and engineering problems.

Graphing Utility Use

Utilizing a graphing utility offers powerful visualization to aid comprehension of the tank's dynamics over time. By plotting \( A(t) \) on a graph, students can observe how the amount of salt in the tank changes from an empty state up to overflow and eventually towards a stabilized level.
To plot \( A(t) \) effectively:

• Set your graphing tool to the interval \([0, 500)\) minutes.
• Monitor how the graph rises gradually as salt accumulates.
• Note the leveling out as the system reaches equilibrium.

Using this visual aid enables learners to intuitively grasp the abstract mathematical concepts involved in differential equations. With this tool, one can visualize the entire trajectory of the system, thereby linking theoretical calculations with practical, observable data.