Question

Tanks \(T_{1}\) and \(T_{2}\) contain 50 gallons and 100 gallons of salt solutions, respectively. A solution with 2 pounds of salt per gallon is pumped into \(T_{1}\) from an external source at \(1 \mathrm{gal} / \mathrm{min},\) and a solution with 3 pounds of salt per gallon is pumped into \(T_{2}\) from an external source at \(2 \mathrm{gal} / \mathrm{min} .\) The solution from \(T_{1}\) is pumped into \(T_{2}\) at \(3 \mathrm{gal} / \mathrm{min},\) and the solution from \(T_{2}\) is pumped into \(T_{1}\) at \(4 \mathrm{gal} / \mathrm{min} . T_{1}\) is drained at \(2 \mathrm{gal} / \mathrm{min}\) and \(T_{2}\) is drained at \(1 \mathrm{gal} / \mathrm{min}\). Let \(Q_{1}(t)\) and \(Q_{2}(t)\) be the number of pounds of salt in \(T_{1}\) and \(T_{2}\), respectively, at time \(t>0\). Derive a system of differential equations for \(Q_{1}\) and \(Q_{2}\). Assume that both mixtures are well stirred.

Short answer

Question: Express the number of pounds of salt in each tank, \(Q_1(t)\) and \(Q_2(t)\), with respect to time using a system of differential equations. Answer: The system of differential equations describing the amount of salt in each tank at any given time \(t\) is: \(\frac{dQ_1}{dt} = 2 + \frac{Q_2(t)}{100} \cdot 4 - \left(\frac{Q_1(t)}{50} \cdot 3 + \frac{Q_1(t)}{50} \cdot 2\right)\) \(\frac{dQ_2}{dt} = 6 + \frac{Q_1(t)}{50} \cdot 3 - \left(\frac{Q_2(t)}{100} \cdot 4 + \frac{Q_2(t)}{100} \cdot 1\right)\)

Step-by-step solution

Analyze Tank T1

At any given moment \(t\), let's analyze the salt amounts and the flow rates in Tank T1. We have: - Incoming salt solution at a rate of 1 gal/min with 2 pounds of salt per gallon from an external source. - Outgoing salt solution towards Tank T2 at a rate of 3 gal/min. - Incoming salt solution from Tank T2 at a rate of 4 gal/min. - Outgoing salt solution being drained at a rate of 2 gal/min. Total inflow rate of salt solution in Tank T1: \((1 + 4) \ \mathrm{gal/min} = 5 \ \mathrm{gal/min}\) Total outflow rate of salt solution in Tank T1: \((3 + 2) \ \mathrm{gal/min} = 5 \ \mathrm{gal/min}\) Now let's consider the flow of salt specifically. We know that \(Q_1(t)\) is the amount of salt in Tank T1 at any time \(t\). The rate of salt inflow from the external source is \(1 \ \mathrm{gal/min} \cdot 2 \ \mathrm{lb/gal} = 2 \ \mathrm{lb/min}\). The rate of salt inflow from Tank T2 can be expressed as \(\frac{Q_2(t)}{100} \cdot 4 \ \mathrm{lb/min}\), assuming that the mixture in T2 is uniform. The rate of salt being pumped out from T1 to T2 is \(\frac{Q_1(t)}{50} \cdot 3 \ \mathrm{lb/min}\), and the rate of salt being drained is \(\frac{Q_1(t)}{50} \cdot 2 \ \mathrm{lb/min}\).

Set up the differential equation for Tank T1

Considering the inflows and outflows of salt in Tank T1, we can write the differential equation for \(Q_1(t)\) as follows: \(\frac{dQ_1}{dt} = \text{(rate of salt entering)} - \text{(rate of salt exiting)}\) \(\frac{dQ_1}{dt} = 2 + \frac{Q_2(t)}{100} \cdot 4 - \left(\frac{Q_1(t)}{50} \cdot 3 + \frac{Q_1(t)}{50} \cdot 2\right)\)

Analyze Tank T2

Similarly, let's analyze the salt amounts and the flow rates in Tank T2: - Incoming salt solution at a rate of 2 gal/min with 3 pounds of salt per gallon from an external source. - Outgoing salt solution towards Tank T1 at a rate of 4 gal/min. - Incoming salt solution from Tank T1 at a rate of 3 gal/min. - Outgoing salt solution being drained at a rate of 1 gal/min. Total inflow rate of salt solution in Tank T2: \((2 + 3) \ \mathrm{gal/min} = 5 \ \mathrm{gal/min}\) Total outflow rate of salt solution in Tank T2: \((4 + 1) \ \mathrm{gal/min} = 5 \ \mathrm{gal/min}\) The rate of salt inflow from the external source is \(2 \ \mathrm{gal/min} \cdot 3 \ \mathrm{lb/gal} = 6 \ \mathrm{lb/min}\). The rate of salt inflow from Tank T1 can be expressed as \(\frac{Q_1(t)}{50} \cdot 3 \ \mathrm{lb/min}\). The rate of salt being pumped out from T2 to T1 is \(\frac{Q_2(t)}{100} \cdot 4 \ \mathrm{lb/min}\), and the rate of salt being drained is \(\frac{Q_2(t)}{100} \cdot 1 \ \mathrm{lb/min}\).

Set up the differential equation for Tank T2

Considering the inflows and outflows of salt in Tank T2, we can write the differential equation for \(Q_2(t)\) as follows: \(\frac{dQ_2}{dt} = \text{(rate of salt entering)} - \text{(rate of salt exiting)}\) \(\frac{dQ_2}{dt} = 6 + \frac{Q_1(t)}{50} \cdot 3 - \left(\frac{Q_2(t)}{100} \cdot 4 + \frac{Q_2(t)}{100} \cdot 1\right)\)

Write the system of differential equations

Now that we have the differential equations for both Tank T1 and Tank T2, we can write the system of differential equations for \(Q_1(t)\) and \(Q_2(t)\) as follows: \(\frac{dQ_1}{dt} = 2 + \frac{Q_2(t)}{100} \cdot 4 - \left(\frac{Q_1(t)}{50} \cdot 3 + \frac{Q_1(t)}{50} \cdot 2\right)\) \(\frac{dQ_2}{dt} = 6 + \frac{Q_1(t)}{50} \cdot 3 - \left(\frac{Q_2(t)}{100} \cdot 4 + \frac{Q_2(t)}{100} \cdot 1\right)\) This is the system of differential equations that describes the amount of salt in each tank at any given time \(t\).

Key concepts

Salt Mixing Problem

The 'salt mixing problem' is a classic example used to illustrate the principles of differential equation modeling in the context of real-world applications. This type of problem typically involves two or more tanks containing a mixture of water and salt, where the concentration and amount of salt in each tank changes over time due to the flow of the mixtures between the tanks and external sources.

In our specific exercise, two tanks are interconnected and have varying quantities of salt solution. Salty water is pumped into and out of these tanks at different rates, leading to a dynamic system where the concentration of salt is constantly changing. The central question is to determine the amount of salt in each tank at any given time, using differential equations to represent the rates of change based on the given conditions.

This problem is key for students learning about differential equations because it simulates real-world mixing scenarios, such as industrial mixing processes, environmental engineering scenarios dealing with pollution dispersion in lakes or rivers, or even pharmacokinetics in medical treatments.

Differential Equation Modeling

Differential equation modeling involves creating equations to describe how a particular quantity changes over time. These models are based on the relationship between the rate of change of a variable and the variable itself.

In the context of our salt mixing problem, we model the process by deriving a set of differential equations that describe the rates at which salt enters and leaves each tank. Each term in the differential equation corresponds to a flow of salt solution either into or out of each tank.For example, the rate at which salt enters a tank is a product of the flow rate and the concentration of the incoming solution. Likewise, the rate at which salt exits is a function of the tank's concentration and its outflow rate. These relationships are used to develop a system of differential equations that can predict the amount of salt in each tank at any time, considering the initial conditions of the system.

Rate of Change

The 'rate of change' is a fundamental concept in calculus and differential equations that expresses how a quantity varies over time or another variable. It is essentially the 'speed' at which one quantity changes with respect to another.

In our exercise, the rate of change is crucial as it determines how quickly the amount of salt in each tank is altering as a result of the various inflows and outflows. By examining these rates, we understand the behavior of the salt concentration over time. The differential equations we derive represent these rates mathematically, allowing us to predict future states of the system. Understanding the rate of change helps students to grasp the essence of dynamic processes and is widely applicable across many fields of study in both science and engineering.

Initial Value Problem

An 'initial value problem' in differential equations is a problem in which the solution to a differential equation is to be found given initial conditions—he specific values at a starting point (usually time).

Our salt mixing problem typically employs an initial value problem to determine the amount of salt in each tank at any time. The initial conditions would be the initial amounts of salt present in the tanks before the mixing process starts. By solving the obtained system of differential equations with these initial conditions, we can predict the amount of salt at any subsequent time.

Initial value problems are essential in ensuring that the solutions to differential equations accurately reflect the real-world scenarios they are intended to model. They ensure the solution path is uniquely determined, providing the precise behavior of the system from a specific starting point.