Problem 25 - Thomas Calculus

Question

A tank initially contains 100 gal of brine in which 50 lb of salt are dissolved. A brine containing 2 \(\mathrm{lb} / \mathrm{gal}\) of salt runs into the tank at the rate of 5 \(\mathrm{gal} / \mathrm{min}\) . The mixture is kept uniform by stirring and flows out of the tank at the rate of 4 \(\mathrm{gal} / \mathrm{min.}\) a. At what rate (pounds per minute) does salt enter the tank at time \(t ?\) b. What is the volume of brine in the tank at time \(t ?\) c. At what rate (pounds per minute) does salt leave the tank at time \(t ?\) d. Write down and solve the initial value problem describing the mixing process. e. Find the concentration of salt in the tank 25 min after the process starts.

Step-by-Step Solution

Answer

a. 10 lb/min. b. 100 + t gal. c. \( \frac{4S}{100+t} \) lb/min. d. Solve: \( \frac{dS}{dt} = 10 - \frac{4S}{100+t} \). e. Use \( S(t) \) to find concentration at 25 min.

Step 1: Determine Rate of Salt Entering the Tank

The brine entering the tank has a concentration of 2 lb/gal, and it flows in at 5 gal/min. Thus, the rate of salt entering the tank is given by \( 2 \times 5 = 10 \) pounds per minute.

Step 2: Determine Volume of Brine in the Tank

Initially, the tank has 100 gallons. Brine enters at 5 gal/min and leaves at 4 gal/min. The net change in volume per minute is \( 5 - 4 = 1 \) gal/min. Hence, at time \( t \), volume is \( V(t) = 100 + t \) gallons.

Step 3: Determine Rate of Salt Leaving the Tank

Let \( S(t) \) represent the amount of salt in the tank at time \( t \). The concentration of salt in the tank is \( \frac{S(t)}{V(t)} \) lb/gal. Since the outflow is 4 gal/min, the rate of salt leaving is \( 4 \times \frac{S(t)}{100 + t} \) lb/min.

Step 4: Write the Differential Equation

The rate of change of salt in the tank is the rate in minus the rate out, leading to the equation: \[ \frac{dS}{dt} = 10 - \frac{4S}{100 + t}. \]

Step 5: Solve the Differential Equation

Using an integrating factor, \( \mu(t) = (100+t)^4 \), multiply through the differential equation: \[ (100+t)^4 \frac{dS}{dt} + 4(100+t)^3S = 10(100+t)^4. \] Integrate both sides with respect to \( t \) and solve for \( S(t) \). Applying the initial condition \( S(0) = 50 \) allows for determination of constants.

Step 6: Calculate Concentration at t = 25 min

Once \( S(t) \) is determined, calculate \( S(25) \). The concentration is \( \frac{S(25)}{125} \), as at \( t = 25, V(t) = 125 \) gallons.

Key Concepts

Rate of Change

Understanding the rate of change is crucial in analyzing how certain quantities evolve over time. In the context of our exercise, it refers to the change in the amount of salt within the tank. To determine how much salt enters or leaves, we look at how these quantities adjust per minute.

• The salt enters the tank at a constant rate, calculated by multiplying the concentration of incoming brine with its flow rate.
• The salt leaves the tank at a rate dependent on both the outflow rate and the salt concentration within the tank at any given time.

This means that the net change in salt within the tank is impacted by both incoming and outgoing rates. Mathematically, this is expressed using a differential equation, allowing us to model how salt concentration changes over time.

Brine Solution

A brine solution is a mixture primarily composed of salt and water. In industrial and scientific applications, understanding the properties of brine is essential for processes like salt concentration adjustments.

In our exercise, we start with 100 gallons of brine containing 50 pounds of salt. The brine entering the tank has a known salt concentration of 2 lb/gal. By mixing and understanding the properties of brine solutions, we maintain consistent salt movement and concentrations within industrial operations.

The continuous inflow of brine with a known concentration affects the overall salt concentration in the system, making understanding these solutions crucial for solving related problems.

Mixing Process

The mixing process is vital for ensuring uniform distribution of components within a solution. In our case, the process involves continuously stirring the tank to maintain uniformity of the brine's salt concentration.

With a flow of brine entering at 5 gallons per minute and leaving at 4 gallons per minute, the movement ensures thorough mixing. The slight increase in volume (1 gal/min) ensures the system expands but also dilutes the overall concentration if not carefully monitored.

Effective mixing avoids localized concentration changes, which could result in inconsistencies during processes requiring homogenous mixtures. This is key to ensuring accurate predictions of salt concentrations at specific times.

Salt Concentration

Salt concentration is a measure of how much salt is dissolved in the brine. Initially, our brine contains 50 pounds of salt in 100 gallons, and the concentration changes due to the inflow and outflow of brine.

As new brine enters the tank at 2 lb/gal, and with stirring ensuring mixing, the concentration varies with time and flow rates. Calculating the concentration involves determining the amount of salt present at any given time and dividing by the current volume of brine.

The exercise shows that at time \( t \), the concentration can be expressed as \( \frac{S(t)}{V(t)} \), where \( S(t) \) represents the amount of salt and \( V(t) \) the volume of brine.

Analyzing this concentration over time requires solving the differential equation derived earlier, ensuring we understand both immediate and long-term effects on the solution's salinity.