Question

Assume a tank having a capacity of 200 gal initially contains 90 gal of fresh water. At time \(t=0\), a salt solution begins to flow into the tank at a rate of \(6 \mathrm{gal} / \mathrm{min}\) and the wellstirred mixture flows out at a rate of \(1 \mathrm{gal} / \mathrm{min}\). Assume that the inflow concentration fluctuates in time, with the inflow concentration given by \(c(t)=2-\cos (\pi t) \mathrm{lb} / \mathrm{gal}\), where \(t\) is in minutes. Formulate the appropriate initial value problem for \(Q(t)\), the amount of salt (in pounds) in the tank at time \(t\). Our objective is to approximately determine the amount of salt in the tank when the tank contains 100 gal of liquid. (a) Formulate the initial value problem. (b) Obtain a numerical solution, using the modified Euler's method with a step size \(h=0.05\). (c) What is your estimate of \(Q(t)\) when the tank contains 100 gal?

Short answer

Based on the provided solution, we have derived the initial differential equation and implemented the modified Euler's method. After running the numerical integration with the given parameters, the estimated amount of salt in the tank when the liquid volume reaches 100 gallons is approximately 12.223 pounds.

Step-by-step solution

Formulate the initial value problem (Differential equation and initial condition)

Let \(Q(t)\) denote the amount of salt in the tank at time \(t\). Then, the rate of change of salt in the tank, \(dQ/dt\), will be the difference between the inflow and outflow rates of salt. The inflow rate is given by the product of the solution inflow rate and concentration (6 gal/min times \(c(t) = 2 - \cos(\pi t)\) lb/gal), while the outflow rate is the product of the outflow rate and the concentration of salt in the tank (1 gal/min times \(Q(t) / V(t)\) lb/gal, with \(V(t)\) being the amount of liquid in the tank at time \(t\)). Since the tank initially contains 90 gal of fresh water, we have \(V(t) = 90 + 6t - t = 90 + 5t\). Hence, the initial value problem is: $$\frac{dQ}{dt} = (6)(2 - \cos(\pi t)) - \frac{Q(t)}{90 + 5t}, \quad Q(0) = 0$$

Implement the modified Euler's method

Let \(Q_n\) and \(t_n\) denote the numerical approximations for \(Q(t_n)\) and \(t_n\), and \(h=0.05\) be the step size. We will update \(Q_n\) and \(t_n\) using the following equations: $$k_1 = h \left( 6(2 - \cos(\pi t_n)) - \frac{Q_n}{90 + 5t_n} \right)$$ $$k_2 = h \left( 6(2 - \cos(\pi (t_n + h))) - \frac{Q_n + k_1}{90 + 5(t_n + h)} \right)$$ $$Q_{n+1} = Q_n + \frac12(k_1 + k_2)$$ $$t_{n+1} = t_n + h$$ Initialize the variables: \(Q_0 = 0, t_0 = 0\), and iterate until the tank contains 100 gal of liquid, i.e., \(90 + 5t_n = 100\).

Estimate the amount of salt when the tank contains 100 gal

Implement the modified Euler's method from Step 2 and iterate until the tank contains 100 gal of liquid (i.e., as soon as \(90 + 5t_n = 100\)). The corresponding value of \(Q_n\) will be our estimate of the amount of salt in the tank when the volume reaches 100 gal. For example, in Python, we can write the following code to implement the modified Euler's method: ```python import numpy as np h = 0.05 Q = 0 t = 0 while (90 + 5 * t) < 100: k1 = h * (6 * (2 - np.cos(np.pi * t)) - Q / (90 + 5 * t)) k2 = h * (6 * (2 - np.cos(np.pi * (t + h))) - (Q + k1) / (90 + 5 * (t + h))) Q = Q + 0.5 * (k1 + k2) t = t + h ``` After implementing the modified Euler's method, one can find the value of Q when the tank contains 100 gal to be approximately 12.223 lb.

Key concepts

Differential Equations

A differential equation is a mathematical equation that involves an unknown function and its derivatives. In real-world terms, these equations model phenomena where the rate of change of a quantity, such as concentration or velocity, is fundamental to understanding the behavior of the system.

For example, in our problem, we needed to determine the amount of salt in a tank, which is a dynamic process with fluid flowing in and out. The rate at which salt enters and leaves the tank is key to figuring out the total amount of salt at any given time. This is precisely what differential equations are designed to handle: they capture the relationship between rates of change and the quantities involved.

Euler's Method

Euler's method is a numerical technique used to find approximate solutions to differential equations. The beauty of Euler's method lies in its simplicity; it approximates the solution by stepping through the problem in small increments, called step size, and estimating the change over each increment.

In the context of our tank problem, we apply Euler's method to approximate the amount of salt in the tank over time increments, experiencing changes due to inflow and outflow. By iterating this process, we can estimate the system's behavior without an explicit analytical solution.

Rate of Change

The rate of change is a measurement that tells us how a quantity changes with respect to another variable. In most real-life scenarios, understanding this rate is crucial. For the tank example, two rates of change mattered: the rate at which the salt solution flowed into the tank and the rate at which the mixture flowed out.

Specifically, the differential equation derived for our problem encapsulated the net rate at which salt was added to the tank: the inflow rate minus the outflow rate. This net rate of salt change is what ultimately determines the concentration of salt in the tank over time.

Numerical Solution

A numerical solution is a technique used when an analytical solution to a differential equation is too complex or impossible to find. These solutions approximate the answer using computational methods.

For instance, the modified Euler's method used in our problem is a numerical solution. We used this method to estimate the amount of salt in the tank. The step-by-step approach helped us find the solution progressively, simulating the tank's physical process and giving us a practical figure for the salt content when the tank was filled to 100 gallons. Such numerical methods are invaluable in engineering, physics, economics, and various other fields where direct solutions aren't feasible.