Question

Two 500 gallon tanks \(T_{1}\) and \(T_{2}\) initially contain 100 gallons each of salt solution. A solution with 2 pounds of salt per gallon is pumped into \(T_{1}\) from an external source at \(6 \mathrm{gal} / \mathrm{min},\) and a solution with 1 pound of salt per gallon is pumped into \(T_{2}\) from an external source at \(5 \mathrm{gal} / \mathrm{min} .\) The solution from \(T_{1}\) is pumped into \(T_{2}\) at \(2 \mathrm{gal} / \mathrm{min},\) and the solution from \(T_{2}\) is pumped into \(T_{1}\) at \(1 \mathrm{gal} / \mathrm{min}\). Both tanks are drained at \(3 \mathrm{gal} / \mathrm{min}\). 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 \(Q_{2}\) that's valid until a tank is about to overflow. Assume that both mixtures are well stirred.

Short answer

The differential equations that describe the salt amounts in tanks \(T_1\) and \(T_2\) are: $$\frac{dQ_1(t)}{dt} = 12 - \frac{5Q_1(t) - Q_2(t)}{100+t}$$ $$\frac{dQ_2(t)}{dt} = 5 - \frac{4Q_2(t) - 2Q_1(t)}{100+t}$$

Step-by-step solution

Determine the net flow rates for each tank

We need to find the net flow rates for each tank, taking into account the flows between the tanks and the external drains. For \(T_1\), we have: - Inflow from the external source: \(6\) gal/min - Inflow from \(T_2\): \(1\) gal/min - Outflow to \(T_2\): \(2\) gal/min - Outflow to drain: \(3\) gal/min For \(T_2\), we have: - Inflow from the external source: \(5\) gal/min - Inflow from \(T_1\): \(2\) gal/min - Outflow to \(T_1\): \(1\) gal/min - Outflow to drain: \(3\) gal/min The net inflow rate for \(T_1\) is \(6 + 1 - 2 - 3 = 2\) gal/min, and the net inflow rate for \(T_2\) is \(5 + 2 - 1 - 3 = 3\) gal/min.

Determine the salt balance for each tank

Now, we need to find the inflow and outflow of salt for each tank. We'll use the salt solution concentration and the net flow rates calculated in step 1. For \(T_1\): - Salt inflow from the external source: \(6\) gal/min \(\times 2\) lb/gal \(= 12\) lb/min - Salt inflow from \(T_2\): \(1\) gal/min \(\times \frac{Q_2(t)}{100+t}\) lb/gal \(= \frac{Q_2(t)}{100+t}\) lb/min - Salt outflow to \(T_2\): \(2\) gal/min \(\times \frac{Q_1(t)}{100+t}\) lb/gal \(= \frac{2Q_1(t)}{100+t}\) lb/min - Salt outflow to drain: \(3\) gal/min \(\times \frac{Q_1(t)}{100+t}\) lb/gal \(= \frac{3Q_1(t)}{100+t}\) lb/min For \(T_2\): - Salt inflow from the external source: \(5\) gal/min \(\times 1\) lb/gal \(= 5\) lb/min - Salt inflow from \(T_1\): \(2\) gal/min \(\times \frac{Q_1(t)}{100+t}\) lb/gal \(= \frac{2Q_1(t)}{100+t}\) lb/min - Salt outflow to \(T_1\): \(1\) gal/min \(\times \frac{Q_2(t)}{100+t}\) lb/gal \(= \frac{Q_2(t)}{100+t}\) lb/min - Salt outflow to drain: \(3\) gal/min \(\times \frac{Q_2(t)}{100+t}\) lb/gal \(= \frac{3Q_2(t)}{100+t}\) lb/min

Derive the system of differential equations

Using the salt balance for each tank, we can now write the system of differential equations for \(Q_1(t)\) and \(Q_2(t)\). The change in the amount of salt in each tank is the difference between the total inflow and outflow of salt. For \(T_1\): $$\frac{dQ_1(t)}{dt} = 12 + \frac{Q_2(t)}{100+t} - \frac{2Q_1(t)}{100+t} - \frac{3Q_1(t)}{100+t}$$ For \(T_2\): $$\frac{dQ_2(t)}{dt} = 5 + \frac{2Q_1(t)}{100+t} - \frac{Q_2(t)}{100+t} - \frac{3Q_2(t)}{100+t}$$ Now, we have the system of differential equations for \(Q_1(t)\) and \(Q_2(t)\): $$\frac{dQ_1(t)}{dt} = 12 - \frac{5Q_1(t) - Q_2(t)}{100+t}$$ $$\frac{dQ_2(t)}{dt} = 5 - \frac{4Q_2(t) - 2Q_1(t)}{100+t}$$ This system of differential equations is valid until a tank is about to overflow.

Key concepts

Mathematical Modeling

Mathematical modeling is the process of creating representations of real-world systems through the language of mathematics. This often involves writing down equations that describe the behavior of a system or a process. In the context of the given exercise, mathematical modeling helps us to construct a system of differential equations that accurately represents the dynamics within two interconnected tanks containing salt solutions.

When we begin to model the tanks, we need to consider variables like the volume of solution in each tank, the concentration of salt, and the rates at which the solution is added or removed. By using mathematical modeling, we can derive equations that express how these variables change over time. The models often rely on the assumption of perfect mixing, which is to say the concentration of salt is identical throughout each tank at any given moment, enabling us to use algebraic equations to represent the change in salt concentration.

In our specific problem, the interconnected tanks and the management of fluid flows are analogous to many real-world situations such as chemical reactions in reactors or resource distribution in ecosystems. This makes such exercises valuable not only for understanding the principles of mathematical modeling but also for applying them to practical scenarios in engineering, environmental science, and beyond.

Salt Balance Problems

Salt balance problems form an important category of mathematical problems that arise in environmental engineering, chemistry, and various industrial processes. They involve tracking and predicting the concentration of a substance, such as salt, within a volume of liquid as it is subject to inflows and outflows.

In the given problem, we're asked to focus on two tanks with different salt concentration inflows and interconnected flows. To analyze this, we set up a 'balance' for each tank - accounting for all the salt entering and leaving the system. We consider the concentration of the inflowing solutions, the rate of flow between the tanks, and the draining rates.

The goal is to derive equations that will allow us to calculate the amount of salt in each tank at any given time. This is done by understanding that the rate of change of salt in each tank is a function of the difference between the input of salt and its output. With proper information about these rates and using the principles of conservation of mass, we can set up a differential equation to solve the problem.

These types of problems challenge students to think critically about process dynamics and apply their knowledge of calculus, ultimately leading to a deeper understanding of physical systems and their behaviors.

Rate of Change

The rate of change is a foundational concept in calculus, and it refers to how a quantity varies over time. In the context of differential equations, it is an expression of how a variable, such as the quantity of salt in a tank, changes in response to other variables or processes.

In our tank system, the rate of change of the salt amount is dictated by the flow rates of solution into and out of the tanks, which themselves change over time. To model this, differential equations are used, which are tools that relate functions to their derivatives - rates of change. In essence, by writing a differential equation, we're looking to find a function that predicts the future state of a system based solely on its current state and how its state is changing.

Through these exercises, students learn not just how to perform mathematical operations but also to interpret the results: a high rate of inflow or outflow correlates to faster changes in concentration, while a low rate suggests a more gradual change. By mastering the calculus of rates of change, students equip themselves with the skills necessary to tackle a wide array of problems in science, engineering, and beyond where understanding and predicting dynamic systems are essential.