HW 6

Problem 1

A continuous stirred tank reactor reacts chemical species A according to the following ODE:

$$\frac{dA}{dt} = \frac{A_{in}-A}{\tau} - kA^2.$$

Solve this equation using the explicit Euler method. Plot $A(t)$ out to $t=10$ if $A_{in}=1$, $A_0=0$, $\tau=2$, and $k=0.1$. Use $\Delta t=0.1$.
What is the long-time (steady state) value of $A$ from your solution?
In the above equation, set $dA/dt=0$ and solve for $A$. This is the analytic steady state value of $A$. How does it compare to the long-time solution from the ODE? Does this make sense? Why?

Problem 2

A $d_p=1$ mm diameter raindrop falls in still air at 300 K, 1 atm. The initial position and velocity of the drop are both 0. The drop’s velocity and position evolve according to the following 2 ODEs:

$$\frac{dv}{dt} = g - \frac{18\mu}{\rho_p d_p^2}v,$$

$$\frac{dx}{dt} = v.$$

Here, $\mu$ is the viscosity of air, $\rho_p$ is the density of water, and $g$ is the gravitational constant.

Review the notes if needed to get the Explicit Euler equations.
Plot the velocity versus time and the position versus time on two separate plots. Label the axes as usual.
Find the long-time velocity (called the terminal velocity) of the droplet.
Also, find the distance the droplet falls before reaching 90% of this velocity.

Problem 3

A reaction among species $A$, $B$, and $C$ is written as

$$A \rightarrow B \rightarrow C.$$

The first reaction has rate constant $k_1=2$ and the second reaction has rate constant $k_2=1.$

The ODEs for the concentrations of the three species are \begin{align} \frac{dA}{dt} &= -k_1A, \\ \frac{dB}{dt} &= k_1A - k_2B, \\ \frac{dC}{dt} &= k_2B. \end{align}

The initial concentrations are $A_0=1$, $B_0=C_0=0$.

On a single plot, show the concentrations of the three species to a time of 5 seconds. Use $\Delta t=0.1$. The plot should include x and y axis labels and a legend. Don’t include a chart title. Use straight lines connecting data points and no data markers.