Question

Consider a CMFR of volume V that initially contains a mass M of nonreactive contaminant. Beginning at time t=0, contaminant-free fluid flows into the reactor at rate Q, and the fluid flow is balanced by equal outflow. How does the concen- tration vary with time for t>0?

Answer 1

In a Continuous Stirred Tank Reactor (CMFR) with contaminant-free inflow and outflow at a rate Q, the concentration C(t) of the nonreactive contaminant varies over time according to the differential equation:

\[ \frac{dC}{dt} = \frac{Q}{V} \left(C_{\text{in}} - C\right) \]

where \(C_{\text{in}}\) is the concentration of the contaminant in the inflowing fluid. This first-order linear ordinary differential equation describes the change in concentration with respect to time. The solution to this equation is given by:

\[ C(t) = C_{\text{in}} - \left(C_{\text{in}} - C_0\right) \exp\left(-\frac{Qt}{V}\right) \]

Here, \(C_0\) is the initial concentration at time \(t=0\). This equation illustrates an exponential decrease in concentration over time as the contaminant-free fluid continuously dilutes the initially contaminated volume.

Feel free to ask if you have specific values for \(C_{\text{in}}\), \(C_0\), \(Q\), or \(V\), and I can help you further with the calculation.