Steady-state concept check: for a system at steady state with respect to substrate concentration, what is the time derivative dCs/dt?

Introduction:
Steady state is a foundational concept in reactor engineering and transport phenomena. It means that macroscopic state variables do not change with time, even though microscopic processes (reactions, flows) continue. Recognizing what steady state implies for time derivatives helps in setting balances and solving design equations.

Given Data / Assumptions:

• We focus on substrate concentration C_s as the variable of interest.
• Conditions are steady in time (startup transients have died out).
• Spatial variations may exist (e.g., gradients), but temporal derivative is zero at each point considered for a steady-state analysis.

Concept / Approach:
By definition, steady state implies d(any state variable)/dt = 0. In a CSTR at steady state, inlet and outlet flows and reaction exactly balance to maintain constant compositions inside the reactor. In plug flow at steady state, compositions vary with position but not with time at a fixed position, so local time derivatives are still zero.

Step-by-Step Solution:
Step 1: Write the general transient species balance including accumulation term.Step 2: Apply the steady-state condition: accumulation term = 0.Step 3: Conclude dC_s/dt = 0 for steady operation.Step 4: Use this to simplify reactor design equations (e.g., algebraic rather than differential forms for CSTRs).

Verification / Alternative check:
Textbook formulations explicitly set time derivatives to zero at steady state, leaving only convective, diffusive, and reaction terms in the balance.

Why Other Options Are Wrong:

• Numerical values (1, >1, <1) do not represent a derivative at steady state.
• Dependence on reactor volume is irrelevant to whether the time derivative vanishes.

Common Pitfalls:
Confusing steady state with equilibrium; at steady state, reactions can proceed with non-zero rates as long as influx equals consumption plus outflow.

Final Answer:
zero