Question

A tank, containing 1000 gal of liquid, has a brine solution entering at a constant rate of \(2 \mathrm{gal} / \mathrm{min}\). The well-stirred solution leaves the tank at the same rate. The concentration within the tank is monitored and is found to be the function of time specified. In each exercise, determine (a) the amount of salt initially present within the tank. (b) the inflow concentration \(c_{i}(t)\), where \(c_{i}(t)\) denotes the concentration of salt in the brine solution flowing into the tank. $$c(t)=\frac{1}{20}\left(1-e^{-t / 500}\right) \mathrm{lb} / \mathrm{gal}$$

Short answer

Answer: The amount of salt initially present within the tank is 0 lb. The inflow concentration, \(c_{i}(t)\), is given by: $$c_{i}(t) = \frac{1}{5000}e^{-t/500}-\frac{1}{20}(1-e^{-t / 500}) \ \mathrm{lb/gal}$$

Step-by-step solution

Find the initial concentration of salt and calculate the amount of salt initially present

To find the initial concentration of salt in the tank when \(t=0\), we'll plug \(t=0\) into \(c(t)\): $$c(0)=\frac{1}{20}\left(1-e^{-0/500}\right) = \frac{1}{20}(1-e^{0})=\frac{1}{20}(1-1)=0 \ \mathrm{lb / gal}$$ Since the tank has a volume of 1000 gal and the initial concentration is 0 lb/gal, the amount of salt initially present within the tank is: $$\text{Amount of salt} = 1000 \ \mathrm{gal} \times 0 \ \mathrm{lb/gal}=0 \ \mathrm{lb}$$ (a) The amount of salt initially present within the tank is 0 lb.

Acknowledge the constant volume of the tank

Since the brine solution enters and leaves the tank at the same rate (2 gal/min), the volume of the liquid within the tank remains constant at 1000 gal.

Find the derivative of the concentration function with respect to time

To find how the concentration of salt in the tank changes with respect to time, we'll take the derivative of \(c(t)\) with respect to \(t\): $$\frac{dc(t)}{dt} = \frac{d}{dt} \left(\frac{1}{20}\left(1-e^{-t / 500}\right)\right) = -\frac{1}{10000}e^{-t/500}$$

Find the inflow concentration \(c_{i}(t)\)

Since the volume of liquid remains constant, we can use a simple balance equation to find the inflow concentration \(c_{i}(t)\). The balance equation relates the change in salt concentration, inflow concentration, and the rate of flow: $$\frac{dc(t)}{dt} = \frac{c_{i}(t) - c(t)}{V} \times q$$ where \(V=1000 \ \mathrm{gal}\) is the volume of the tank and \(q=2 \ \mathrm{gal/min}\) is the rate of flow. Plugging in the values and solving for \(c_{i}(t)\): $$-\frac{1}{10000}e^{-t/500} = \frac{c_{i}(t) - \frac{1}{20}\left(1-e^{-t / 500}\right)}{1000} \times 2$$ Next, simplify and isolate \(c_{i}(t)\): $$c_{i}(t) = \frac{1}{5000}e^{-t/500}-\frac{1}{20}(1-e^{-t / 500})$$ (b) The inflow concentration \(c_{i}(t)\) is given by: $$c_{i}(t) = \frac{1}{5000}e^{-t/500}-\frac{1}{20}(1-e^{-t / 500}) \ \mathrm{lb/gal}$$

Key concepts

Understanding Initial Salt Concentration

When it comes to mixing solutions in a well-stirred tank, the initial salt concentration is a fundamental starting point for analysis. In our example, the concentration is given by a function of time, and at the very beginning (\(t=0\)), it's natural to ask what this concentration would be. By substituting time zero into the concentration function, we find that the tank starts with no salt at all—this is a significant piece of information because it provides a baseline for understanding how the concentration evolves over time.

From a practical perspective, knowing the initial state of the solution is crucial. It helps in predicting future states and in determining the necessary amount of salt to reach a desired concentration. For students working through such problems, it's important to remember that if the initial condition is not explicitly given, it can often be deduced by evaluating the concentration function at time zero.

Determining Inflow Concentration Over Time

The inflow concentration refers to the amount of salt entering the tank at any given moment. As time progresses, this inflow concentration might change, impacting the overall dynamics of the solution in the tank. In the exercise, we calculate the inflow concentration by employing a balance equation, which accounts for the inflow and outflow rates and the change in concentration within the tank.

For students tackling similar brine problems, focusing on the relationship between the rate of flow and the inflow concentration is essential. The inflow concentration is not static—it has its unique behavior represented by a mathematical expression that changes over time, and understanding this variation is crucial for controlling the salt levels in the tank.

Grasping Brine Solution Dynamics

The dynamics of a brine solution in a well-stirred tank are intriguing. Within our exercise, we're observing a scenario where the liquid enters and exits the tank at a steady rate, which keeps the overall volume consistent. Yet, the concentration of salt can still change because of the difference in concentration between the inflow and the mixture already in the tank.

Understanding brine solution dynamics is all about the interplay between these flows and the resulting concentrations. It combines principles of physical mixing with mathematical expressions that represent them. This balancing act is essential to ensure students appreciate how variables such as flow rate, tank volume, and inflow concentration work together to influence the overall concentration at any time.

Applying Differential Equations

Differential equations are the mathematical backbone of many real-world phenomena, including the brine tank problem. These equations describe how a quantity changes over time and are critical for modeling and solving problems of dynamic systems. The brine tank exercise showcases how the differential equation serves as a tool to express the rate of change of concentration in the tank.

For students venturing into differential equations, it's vital to associate every term in the equation with its physical counterpart in the problem. The derivative of the concentration function tells us how the concentration changes, while the balance equation incorporates the rate of inflow and outflow. By marrying the concept of flow dynamics with differential equations, students gain a powerful approach to predict and control various processes—not just in brine solutions but in countless other applications from population models to heat transfer.