Question

Residence time distribution (RTD) function E(t) satisfies:

• A. \( \int_0^{\infty} E(t)dt = 0 \)
• B. \( \int_0^{\infty} E(t)dt = 1 \)
• C. \( \int_0^{\infty} tE(t)dt = 0 \)
• D. \( E(t) = F(t) \)

Hint:

The area under any valid \( E(t) \) curve is always equal to 1.
If you are given experimental tracer data, always normalize it by dividing the concentration readings by the total area under the curve to convert it to a standard \( E(t) \) function.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the mathematical normalization condition satisfied by the Residence Time Distribution (RTD) function, \( E(t) \).
This is a fundamental concept in non-ideal reactor analysis.

Step 2: Key Formula or Approach:
The RTD function \( E(t) \), also known as the exit age distribution function, describes quantitatively how much time different fluid elements spend inside a reactor.
The term \( E(t) \, dt \) represents the fraction of fluid leaving the reactor that has spent a time between \( t \) and \( t + dt \) inside the vessel.

Step 3: Detailed Explanation:

• Normalization Condition: Since all fluid elements leaving the reactor must have spent some amount of time between \( t = 0 \) and \( t = \infty \) inside the vessel, the sum of all these fractions must equal 1.
Mathematically, this corresponds to the integral of the probability density function over all time:
\[ \int_0^{\infty} E(t) \, dt = 1 \]
This is the normalization condition for the RTD curve.

• Mean Residence Time (\( t_m \)): The first moment of the \( E(t) \) curve represents the mean residence time of the fluid:
\[ t_m = \int_0^{\infty} t \cdot E(t) \, dt \]
Therefore, option (C) is incorrect because this integral is equal to the mean residence time, not zero.

• Relation to F-curve: The \( F(t) \) function is the cumulative distribution function, related to the \( E(t) \) function by integration:
\[ F(t) = \int_0^t E(t) \, dt \quad \implies \quad E(t) = \frac{dF(t)}{dt} \]
Thus, \( E(t) \neq F(t) \), making option (D) incorrect.


Step 4: Final Answer:
The RTD function \( E(t) \) satisfies the normalization integral \( \int_0^{\infty} E(t)dt = 1 \).