The dynamic response of a stirred-tank bioreactor can be represented by the transfer function C'(s) - 5/(3s+1) C(s) where C' is the exit substrate concentration (mol/liter), and C,' is the feed substrate concentration (mol/liter). a) Derive an expression for c'(t) if c,'(t) is a rectangular pulse with the following characteristics: t<0 CF (t) = {5 0st<2 2st<00 b) What is the maximum value of c'(t)? When does it occur? What is the final value of c'(t)? E) If the initial is c(0) = 1, how long does it take for c(t) to return to a value of 2.10 after it has reached its maximum value?
The dynamic response of a stirred-tank bioreactor can be represented by the transfer function C'(s) - 5/(3s+1) C(s) where C' is the exit substrate concentration (mol/liter), and C,' is the feed substrate concentration (mol/liter). a) Derive an expression for c'(t) if c,'(t) is a rectangular pulse with the following characteristics: t<0 CF (t) = {5 0st<2 2st<00 b) What is the maximum value of c'(t)? When does it occur? What is the final value of c'(t)? E) If the initial is c(0) = 1, how long does it take for c(t) to return to a value of 2.10 after it has reached its maximum value?
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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![**Dynamic Response of a Stirred-Tank Bioreactor**
The dynamic response of a stirred-tank bioreactor can be represented by the transfer function:
\[ \frac{C'(s)}{C'_F(s)} = \frac{5}{3s + 1} \]
where \( C' \) is the exit substrate concentration (mol/liter), and \( C'_F \) is the feed substrate concentration (mol/liter).
**Problem Statements:**
**a) Derivation of an Expression for \( c'(t) \):**
- Given \( c_F(t) \) as a rectangular pulse with the following characteristics:
\[
c_F(t) =
\begin{cases}
2 & t < 0 \\
5 & 0 \leq t < 2 \\
2 & 2 \leq t < \infty
\end{cases}
\]
**b) Maximum Value of \( c'(t) \):**
- What is the maximum value of \( c'(t) \)?
- When does it occur?
- What is the final value of \( c'(t) \)?
**c) Time for \( c(t) \) to Return to 2.10:**
- If the initial \( c(0) = 1 \), how long does it take for \( c(t) \) to return to a value of 2.10 after it has reached its maximum value?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ed819a1-7cc7-4b35-b605-cdaaef8761fd%2F2d540b60-8b52-401e-a153-b5637bf0a215%2Fw2rvkl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Dynamic Response of a Stirred-Tank Bioreactor**
The dynamic response of a stirred-tank bioreactor can be represented by the transfer function:
\[ \frac{C'(s)}{C'_F(s)} = \frac{5}{3s + 1} \]
where \( C' \) is the exit substrate concentration (mol/liter), and \( C'_F \) is the feed substrate concentration (mol/liter).
**Problem Statements:**
**a) Derivation of an Expression for \( c'(t) \):**
- Given \( c_F(t) \) as a rectangular pulse with the following characteristics:
\[
c_F(t) =
\begin{cases}
2 & t < 0 \\
5 & 0 \leq t < 2 \\
2 & 2 \leq t < \infty
\end{cases}
\]
**b) Maximum Value of \( c'(t) \):**
- What is the maximum value of \( c'(t) \)?
- When does it occur?
- What is the final value of \( c'(t) \)?
**c) Time for \( c(t) \) to Return to 2.10:**
- If the initial \( c(0) = 1 \), how long does it take for \( c(t) \) to return to a value of 2.10 after it has reached its maximum value?
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