Problem 8 - Advanced Engineering Mathematics

Question

A model is $$\begin{aligned}&\frac{d x_{1}}{d t}=(4 \mathrm{gal} / \mathrm{min})(0 \mathrm{lb} / \mathrm{gal})-(4 \mathrm{gal} / \mathrm{min})\left(\frac{1}{200} x_{1} \mathrm{lb} / \mathrm{gal}\right)\\\&\begin{array}{l}\frac{d x_{2}}{d t}=(4 \mathrm{gal} / \mathrm{min})\left(\frac{1}{200} x_{1} \mathrm{lb} / \mathrm{gal}\right)-(4 \mathrm{gal} / \mathrm{min})\left(\frac{1}{150} x_{2} \mathrm{lb} / \mathrm{gal}\right) \\\\\frac{d x_{3}}{d t}=(4 \mathrm{gal} / \mathrm{min})\left(\frac{1}{150} x_{2} \mathrm{lb} / \mathrm{gal}\right)-(4 \mathrm{gal} / \mathrm{min})\left(\frac{1}{100} x_{3} \mathrm{lb} / \mathrm{gal}\right)\end{array}\end{aligned}$$ or $$\begin{aligned}\frac{d x_{1}}{d t} &=-\frac{1}{50} x_{1} \\\\\frac{d x_{2}}{d t} &=\frac{1}{50} x_{1}-\frac{2}{75} x_{2} \\\\\frac{d x_{3}}{d t} &=\frac{2}{75} x_{2}-\frac{1}{25} x_{3}\end{aligned}$$ Over a long period of time we would expect \(x_{1}, x_{2},\) and \(x_{3}\) to approach 0 because the entering pure water should flush the salt out of all three tanks.

Step-by-Step Solution

Answer

Over time, \(x_1, x_2,\) and \(x_3\) will all approach zero.

Step 1: Identify the Initial Value Problem (IVP)

The problem describes a system of differential equations governing the change in concentration of salt (\(x_1, x_2, x_3\)) in three tanks over time. Each differential equation describes the rate at which salt leaves or enters a tank due to the flow of water. The set of equations is: \(\frac{d x_{1}}{d t} = -\frac{1}{50} x_{1}\), \(\frac{d x_{2}}{d t} = \frac{1}{50} x_{1} - \frac{2}{75} x_{2}\), and \(\frac{d x_{3}}{d t} = \frac{2}{75} x_{2} - \frac{1}{25} x_{3}\).

Step 2: Analyze the Behavior Over Time

Each equation includes a decay term for its respective salt concentration due to the outflow minus inflow changes. As each tank's inflow is sourced from the previous tank (or pure water for the first tank), and the outflow reduces the concentration, all equations suggest a tendency toward zero concentration as time approaches infinity.

Step 3: Evaluate the Long-Term Solution

Consider the differential form where the coefficient of negative terms are larger than any potential sources (inflows from preceding tanks). Over long periods of time, any initial amount of salt will be washed out by continuous pure water in-flow. Hence, \(x_1, x_2,\) and \(x_3\) tend towards zero.

Key Concepts

Initial Value Problem

An initial value problem (IVP) in the context of differential equations involves finding a function that satisfies a given differential equation and an initial condition. In this problem, we are tasked with determining the salt concentration dynamics in a series of tanks over time, for which an initial concentration value is generally given. Unlike general differential equations, IVPs specify the state (value) of the unknown function at a starting point, providing a foundation to solve the equation consistently.
In our exercise, the IVP consists of differential equations for three connected tanks, and our goal is to analyze how initial salt concentrations, if any, evolve as water flows through these tanks.

Salt Concentration

Salt concentration refers to the amount of salt present per unit volume of solution – typically water in our case. In this system of tanks, each differential equation provides insight into how salt concentration in each tank changes with time given certain inflow and outflow rates.
The core principle here is dilution, where pure water introduced into a tank dilutes the existing salt in it. Consequently, the salt concentration described by variables \(x_1, x_2,\) and \(x_3\), denotes the concentration in each tank respectively. Over time, as more pure water flows in, this concentration tends to decrease, ideally reaching zero if the water keeps flushing through uninterrupted.

System of Equations

The system of equations refers to the set of related differential equations describing the change in different entities — here, salt concentrations in tanks. Each equation models the rate of salt change in one tank, caused by both internal and external processes such as inflow from the previous tank and outflow into the next.
These equations are intertwined, as the output of one affects the input of another, stating:

• \(\frac{d x_{1}}{d t} = -\frac{1}{50} x_{1}\)
• \(\frac{d x_{2}}{d t} = \frac{1}{50} x_{1} - \frac{2}{75} x_{2}\)
• \(\frac{d x_{3}}{d t} = \frac{2}{75} x_{2} - \frac{1}{25} x_{3}\)

This system requires solving each equation while considering its relationship with others to determine the evolution of salt concentration in each tank.

Rate of Change

The rate of change explains how the concentration of salt in each tank changes over time. Represented by the differential \(\frac{d x_{i}}{d t}\), it quantifies the speed at which salt is being added or removed from a tank. This is driven by the inflow from either pure water or from another tank, and an outflow which usually leads to a decrease in concentration.
In this scenario, it signifies how quickly the system reaches a stable state, often zero for all tanks in a constant flush of pure water. Understanding this allows predictions of how long it might take for the salt concentration to dissipate significantly in a realistic setting. This concept is key for dynamic systems where time-variant processes are analyzed.