Textbook Question
11–16. Initial value problems Solve the following initial value problems.
y'(t) − 3y = 12, y(1) = 4
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13. Intro to Differential Equations
Separable Differential Equations
3:46 minutesProblem 9.5.23a
Textbook Question
23–26. Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis.
a. Write an initial value problem for the mass of the substance.
A 500-L tank is initially filled with pure water. A copper sulfate solution with a concentration of 20 g/L flows into the tank at a rate of 4 L/min. The thoroughly mixed solution is drained from the tank at a rate of 4 L/min.
Verified step by step guidance1
Define the variable: Let \(m(t)\) represent the mass of copper sulfate (in grams) in the tank at time \(t\) (in minutes).
Identify the inflow rate of copper sulfate: The solution enters at 4 L/min with a concentration of 20 g/L, so the mass inflow rate is \(4 \times 20 = 80\) g/min.
Identify the outflow rate of copper sulfate: Since the tank is well mixed, the concentration inside the tank at time \(t\) is \(\frac{m(t)}{500}\) g/L. The outflow rate is 4 L/min, so the mass outflow rate is \(4 \times \frac{m(t)}{500} = \frac{4m(t)}{500}\) g/min.
Write the differential equation expressing the rate of change of mass in the tank: \(\frac{dm}{dt} = \) (mass inflow rate) \(-\) (mass outflow rate), which gives \(\frac{dm}{dt} = 80 - \frac{4m(t)}{500}\).
Specify the initial condition: Since the tank is initially filled with pure water, the initial mass of copper sulfate is zero, so \(m(0) = 0\).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
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3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Formulating Initial Value Problems (IVPs)
An initial value problem involves setting up a differential equation that models the rate of change of a quantity along with an initial condition. In this context, it means expressing how the mass of copper sulfate in the tank changes over time, starting from zero since the tank initially contains pure water.
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Mass Balance in Continuous Stirred Tank Reactors (CSTR)
A mass balance accounts for the mass entering, leaving, and accumulating in the tank. For a stirred tank with inflow and outflow at equal rates, the change in mass equals the mass entering minus the mass leaving, assuming perfect mixing ensures uniform concentration throughout the tank.
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Differential Equations for Concentration and Volume
The problem involves setting up a first-order linear differential equation relating the mass of solute to time, considering constant volume due to equal inflow and outflow rates. Concentration is mass divided by volume, and the inflow concentration and flow rate determine the input term in the equation.
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