44 Practice Exercises
Multiple Choice Questions
Exercise: Separable Differential Equations
Which of the following is not a separable differential equation?
a) [latex]\frac{t^{2}}{C_{A}} = 4*C_{A}\frac{dC_{A}}{dt}[/latex]
b) [latex]\frac{t^{2}}{T} = 3*T \frac{dT}{dt}[/latex]
c) [latex]\frac{dM_{A}}{dt} + e^{M_{A} + t} = 0[/latex]
d) [latex]\frac{-2t + T^{2}}{2*T*t} = \frac{dT}{dt}[/latex]
Solution
d) [latex]\frac{-2t + T^{2}}{2*T*t} = \frac{dT}{dt}[/latex]
If you try to seperate this equation, you get: [latex]2*t + T^{2} + 2*T*t*\frac{dT}{dt} = 0[/latex]. There is no way to isolate the T and t variables further.
How to separate the other choices:
a) \begin{align*}
\frac{t^{2}}{C_{A}}& = 4*C_{A}\frac{dC_{A}}{dt}\\
t^{2}dt&= 4*C_{A}^2\,dC_{A}
\end{align*}
b) \begin{align*}
\frac{t^{2}}{T} &= 3*T \frac{dT}{dt}\\
t^{2}dt &= 3*T^2dT
\end{align*}
c)\begin{align*}
\frac{dM_{A}}{dt} + e^{M_{A} + t} &= 0\\
\frac{dM_{A}}{dt} + e^{M_{A}}· e^{t} &= 0\\
e^{M_{A}}· e^{t} &= -\frac{dM_{A}}{dt} \\
e^{t}dt &=-\frac{1}{e^{M_{A}}}dM_{A}
\end{align*}
Exercise: Process Control Variables
Consider a mixing tank with 2 inlet streams and 1 outlet stream. Stream 1 is feeding the tank with a solution with a concentration of species A. We can change the flowrate of stream 1 using a valve. Stream 2 is feeding the tank with a solution with a concentration of species B, but we do not have control over stream 2’s flowrate or concentration (species B concentration fluctuates and is decided by an upstream process). We measure the concentration of the outlet stream and adjust our system accordingly. Which of the following variables is the disturbance variable?
a) Stream 1 flowrate
b) Stream 2 flowrate
c) Outlet stream flowrate
d) Outlet stream concentration
Solution
b) Stream 2 flowrate
The flowrate of stream 2 is the best answer here because it is an input variable that is affecting our controlled variable (outlet stream concentration), but we do not have control over it. We must adjust our manipulated variable (stream 1 flowrate) to achieve our desired outlet flowrate or concentration.
Exercise: Unsteady-state Systems
Which of the following is not a steady-state system?
a) CSTR (continuous stirred-tank reactor) with constant flow in and out of all species
b) A filtration system that has fluid entering and exiting the system at constant volumetric flowrates
c) A bathtub filling up with water at the same rate as the drain empties the water
d) Heat exchanger in a process that maintains the process fluid at 50°C
Solution
b) A filtration system that has fluid entering and exiting the system at the same rate
This system will accumulate a build-up of the filtered product. Although the flowrates entering and exiting the system are constant, the concentration will change over time due to the filter present.
Exercise: Methods to Decrease Reactor Temperature
We have the following reactor with an endothermic reaction taking place inside the reactor. There is a heating jacket outside of the r