E. coli lives and grows on mannitol with the following kinetics $$r_{\mathrm{C}}=\frac{1.2 C_{\mathrm{A}} C_{\mathrm{C}}}{C_{\mathrm{A}}+2}, C_{\mathrm{A}}=\mathrm{gm} \operatorname{mannitol} / \mathrm{m}^{3}, \quad(\mathrm{C} \mathrm{A})=0.1 \mathrm{gm} \text { cells/gm mannitol }$$ Find the outlet concentration of cells from the reactor when \(1 \mathrm{m}^{3} / \mathrm{hr}\) of mannitol solution \(\left(C_{\mathrm{A} 0}=6 \mathrm{gm} / \mathrm{m}^{3}\right)\) is fed directly to a mixed flow reactor of volume $$V_{m}=1 \mathrm{m}^{3}$$.

Understanding the growth kinetics of E. coli on mannitol is pivotal for controlling and optimizing bioprocesses in a reactor. The growth rate of these bacteria can be heavily influenced by the availability of nutrients, in this particular case, mannitol. The kinetics equation provided, \( r_C = \frac{1.2 C_A C_C}{C_A + 2} \), encapsulates this relationship; where \( r_C \) is the growth rate, \( C_C \) is the concentration of cells, and \( C_A \) is the concentration of the nutrient, mannitol.

The equation implies that the growth rate has a dependency not only on the nutrient concentration but also on the existing cell concentration. Understanding such kinetics is crucial for predicting how \( E. coli \) populations will expand under certain conditions, making it a fundamental concept in biochemical engineering and reactor design.

The Continuous Stirred-Tank Reactor, commonly known as CSTR, is a type of reactor widely used in industrial production and various chemical processes. In a CSTR, reactants are continuously fed into the reactor, where they are thoroughly mixed to maintain uniform composition throughout. The contents of the reactor are also continuously taken out, usually at the same flow rate to ensure a constant reactor volume.

One of the key features of a CSTR is that it operates at steady state with respect to mass and energy balance. This allows for relatively easy modeling and control, which is why it is often employed in processes that require a steady output, such as the growth of microorganisms like \( E. coli \). For students and practitioners alike, grasping the operation and significance of a CSTR is essential for applications in chemical reaction engineering.

The concept of a steady-state mass balance is integral in understanding how continuous systems like reactors reach an equilibrium. In a steady state, the amounts of substances entering and leaving the system are equal, resulting in no net accumulation over time. This principle can be succinctly expressed by the equation: Input - Output + Generation - Consumption = Accumulation, which, at steady state, reduces to Input = Output since the accumulation term drops out.

For the purpose of CSTR analysis, this balance enables engineers to determine critical design and operating parameters. It ensures that the material being processed reaches a specific desired state before exiting the reactor. This mass balance equation forms the backbone for most calculations regarding reactor design, process control, and optimization in chemical reaction engineering.

Calculating the outlet concentration of reactants or products from a reactor is a fundamental task in process engineering, informing us about the efficiency and effectiveness of the reaction taking place. For the example of \( E. coli \) growth in a CSTR, the outlet concentration of cells helps determine if the need to maintain or enhance nutrient supply or alter reactor conditions is warranted. Using the steady-state mass balance and the reaction kinetics equation provided, we can solve for the outlet concentration \( C_{C_{out}} \).

By rearranging terms and substituting known values, including the reactor volume (\( V_m \) = 1 m³) and flow rate (\( Q \) = 1 m³/hr), students can calculate the outlet concentration. This calculation is crucial for not only academic exercises but also for real-world applications where optimizing product yield and quality are essential.