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Chapter 3: Problem 2

Air containing \(0.06 \%\) carbon dioxide is pumped into a room whose volume is \(8000 \mathrm{ft}^{3}\). The air is pumped in at a rate of \(2000 \mathrm{ft}^{3} / \mathrm{min}\), and the circulated air is then pumped out at the same rate. If there is an initial concentration of \(0.2 \%\) carbon dioxide, determine the subsequent amount in the room at any time. What is the concentration at 10 minutes? What is the steady-state, or equilibrium, concentration of carbon dioxide?

Short Answer

Expert verified
After 10 minutes, the concentration decreases towards 0.06%. The steady-state concentration is 0.06%.

Step by step solution

01

Define Concentrations and Rates

Let the concentration of carbon dioxide in the room be \( c(t) \% \), where \( t \) is time in minutes. Initially, the room has a concentration of \( 0.2\% \). The inflow concentration is \( 0.06\% \), and air flows at \( 2000 \text{ ft}^3/\text{min} \). The room volume is \( 8000 \text{ ft}^3 \).
02

Set Up the Differential Equation

We model the change in carbon dioxide concentration using a differential equation. Rate of change in carbon dioxide is: \[ \frac{dc}{dt} = \text{inflow rate} - \text{outflow rate}\] where \( \text{inflow rate} = (0.06)(2000/8000) \) and \( \text{outflow rate} = c(t)(2000/8000) \). Thus, \[ \frac{dc}{dt} = \frac{3}{4000} - \frac{c(t)}{4} \].
03

Solve the Differential Equation

Solve \( \frac{dc}{dt} = \frac{3}{4000} - \frac{c(t)}{4} \) by separating variables and integrating. The integration leads to: \[ -4 \ln|c(t)-\frac{3}{4000}| = t + C \] where \( C \) is the integration constant. Solve for \( c(t) \).
04

Apply Initial Conditions

Apply the initial condition \( c(0) = 0.002 \) to find \( C \). Substitute \( t = 0, c(0) = 0.002 \) into the integrated form. This finds \( C \) and hence \( c(t) \) in terms of \( t \).
05

Find Concentration at 10 Minutes

Substitute \( t = 10 \) into the expression for \( c(t) \) to find the concentration after 10 minutes of circulation.
06

Determine the Steady-State Concentration

The steady-state concentration occurs when \( \frac{dc}{dt} = 0 \). Set \( \frac{3}{4000} - \frac{c(t)}{4} = 0 \) and solve for \( c(t) \), which gives the steady-state concentration.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Carbon Dioxide Concentration
Carbon dioxide concentration refers to the amount of carbon dioxide present in a given volume, expressed as a percentage. In this exercise, we begin with a room initially containing 0.2% carbon dioxide. The goal is to explore how this concentration changes over time when fresh air, containing 0.06% carbon dioxide, is continuously pumped in.

By setting up a relationship between the concentration and time, you can predict how much carbon dioxide will remain in the room at any given moment. The initial and inflow concentrations are crucial to defining this ratio, as they dictate the rate of increase or decrease of carbon dioxide.

Understanding concentration dynamics helps us study environmental systems or any closed system where compounds are exchanged.
Solute Flow Rate
The solute flow rate is vital in determining how the concentration of a substance, like carbon dioxide in air, changes in a room. Here, the air exchange rate is 2000 cubic feet per minute. This rate is essential in setting up the differential equation for the system.

The inflow rate of solute (carbon dioxide) is calculated by taking a fraction of the incoming air鈥檚 concentration鈥�0.06% carbon dioxide flowing in at 2000 cubic feet per minute. This yields the numerator for the inflow rate part of the equation.
  • Inflow rate = Incoming concentration x Flow rate / Room volume
  • Outflow rate = Current concentration x Flow rate / Room volume
With these rates, we formulate the net rate of carbon dioxide concentration, helping define how quickly the room approaches equilibrium.
Steady-State Concentration
The steady-state concentration is the point at which the concentration of carbon dioxide stops changing. It is achieved when the amount flowing in equals the amount flowing out, resulting in a net rate of zero.

For this system, the steady-state is found by setting the rate of change of concentration to zero in the differential equation. This gives us the equilibrium condition when:
  • The inflow rate matches the outflow rate
  • Net rate of change = 0
This isn't just a concept seen in this type of problem鈥攊t's a fundamental principle in many natural and engineered systems, illustrating balance in dynamic systems.
Initial Conditions in Differential Equations
Initial conditions help determine the specific solution to a differential equation from the general solution. Here, the initial condition is the concentration of carbon dioxide in the room at time zero, given as 0.2%.

When solving the differential equation, the constant of integration is determined by applying the initial condition. This process helps move from the general solution to the particular solution for our scenario.

Initial conditions are critical for accurately modeling real-world scenarios because they set the stage for how the system鈥檚 variables will evolve over time. By incorporating these conditions, predictions and simulations become much more reliable and applicable to specific situations.

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Most popular questions from this chapter

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