(a) Interpretation: For a first order system, it is to be shown that the step response for the change in the input of magnitude M can be approximately modeled by the step response of an integratorfor low values of t . Concept introduction: For chemical processes, dynamic models consisting ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances The process models generally include algebraic relationships which commence from thermodynamics , transport phenomena , chemical kinetics, and physical properties of the processes. For a first-order system, the transfer function takes the form: G ( s ) = Y ( s ) U ( s ) = K τ s + 1 Here, K and τ are the system parameters. Step response for the change in input of magnitude M is given by: y ( t ) = K M ( 1 − e − t / τ ) The transfer function for an integrating process is: G ( s ) = Y ( s ) U ( s ) = K s Here, K is the system parameters. Step response for the change in input of magnitude M is given by: y ( t ) = K M t

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Answer 1

Question

Chapter 5, Problem 5.22E

Interpretation Introduction

(a)

Interpretation:

For a first order system, it is to be shown that the step response for the change in the input of magnitude M can be approximately modeled by the step response of an integratorfor low values of t.

Concept introduction:

For chemical processes, dynamic models consisting ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

For a first-order system, the transfer function takes the form:

G(s)=Y(s)U(s)=Kτs+1

Here, K and τ are the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(1−e−t/τ)

The transfer function for an integrating process is:

G(s)=Y(s)U(s)=Ks

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KMt

Interpretation Introduction

(b)

Interpretation:

For t≪τ, the relationship between K1 and K0 is to be derived if the two responses match.

Concept introduction:

For chemical processes, dynamic models consisting of ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

For a first-order system, the transfer function takes the form:

G(s)=Y(s)U(s)=Kτs+1

Here, K and τ are the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(1−e−t/τ)

The transfer function for an integrating process is:

G(s)=Y(s)U(s)=Ks

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KMt

Interpretation Introduction

(c)

Interpretation:

Single step analysis to find K0 and time delay in an integrating process is to be explained.

Concept introduction:

For chemical processes, dynamic models consisting of ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

The transfer function for an integrating process with the time delay of θ is:

G(s)=Y(s)U(s)=Kse−θs

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(t−θ)

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A biodiesel mixture consisting of 60 mol% methyl oleate (MO), 25 mol% methyl linoleate (ML), and 15 mol% methyl palmitate (MP) is held at 373.15 K and 200 MPa. Given PC-SAFT parameters: segment number \( m_i = [5.7, 6.3, 4.8] \), segment diameter \( \sigma_i = [3.95, 3.98, 3.91] \) Å, dispersion energy \( \epsilon_i/k = [260, 270, 250] \) K, and binary interaction parameters \( k_{ij} = 0.01 \), determine the isentropic speed of sound (m/s) using the PC-SAFT Helmholtz energy formulation and the thermodynamic identity\[c^2 = \left( \frac{\partial P}{\partial \rho} \right)_T + \frac{T \left( \frac{\partial P}{\partial T} \right)_\rho^2 }{ \rho^2 c_v },\]assuming the density is precomputed at 200 MPa and \( c_v \) is obtained from ideal mixing of pure-component values.

A steady Williamson nanofluid containing Cu nanoparticles flows over a permeable wedge with wall suction \( V_w = 0.015 \, \text{m/s} \), under a transverse magnetic field \( B_0 = 0.6 \, \text{T} \). The flow obeys the Buongiorno model, with \( D_B = 9 \times 10^{-10} \), \( D_T = 3.5 \times 10^{-8} \), and activation energy \( E_a = 65 \times 10^3 \). Hall and ion-slip effects are included with \( m_e = 0.4 \), \( \beta = 0.15 \). Thermal conductivity varies as \( k(T) = 0.6 (1 + 0.002 (T - 305)) \). Apply velocity and thermal jump conditions with \( \alpha_u = 0.9 \), \( \alpha_T = 0.8 \), \( \lambda = 2 \times 10^{-7} \). Using Keller’s method and similarity variables for wedge parameter \( m = 0.4 \), determine the entropy generation number \( N_s \) at \( x = 0.03 \), where \[N_s = \frac{k(T)}{T_\infty^2} \left( \frac{\partial T}{\partial y} \right)^2 + \frac{\mu}{T_\infty} \left( \frac{\partial u}{\partial y} \right)^2.\]

E. coli was continuously cultured in a continuous stirred tank fermenter with a working volume of 1 L by chemostat. A medium containing 4.0 g/L of glucose as a carbon source was fed to the fermenter at a constant flow rate of 0.5 L/hr, and the glucose concentration in the output stream was 0.20 g/L. The cell yield with respect to glucose was 0.42 g dry cells per gram glucose.

Answer 2

Question

Chapter 5, Problem 5.22E

Interpretation Introduction

(a)

Interpretation:

For a first order system, it is to be shown that the step response for the change in the input of magnitude M can be approximately modeled by the step response of an integratorfor low values of t.

Concept introduction:

For chemical processes, dynamic models consisting ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

For a first-order system, the transfer function takes the form:

G(s)=Y(s)U(s)=Kτs+1

Here, K and τ are the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(1−e−t/τ)

The transfer function for an integrating process is:

G(s)=Y(s)U(s)=Ks

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KMt

Interpretation Introduction

(b)

Interpretation:

For t≪τ, the relationship between K1 and K0 is to be derived if the two responses match.

Concept introduction:

For chemical processes, dynamic models consisting of ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

For a first-order system, the transfer function takes the form:

G(s)=Y(s)U(s)=Kτs+1

Here, K and τ are the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(1−e−t/τ)

The transfer function for an integrating process is:

G(s)=Y(s)U(s)=Ks

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KMt

Interpretation Introduction

(c)

Interpretation:

Single step analysis to find K0 and time delay in an integrating process is to be explained.

Concept introduction:

For chemical processes, dynamic models consisting of ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

The transfer function for an integrating process with the time delay of θ is:

G(s)=Y(s)U(s)=Kse−θs

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(t−θ)

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 3

Question

Chapter 5, Problem 5.22E

Interpretation Introduction

(a)

Interpretation:

For a first order system, it is to be shown that the step response for the change in the input of magnitude M can be approximately modeled by the step response of an integratorfor low values of t.

Concept introduction:

For chemical processes, dynamic models consisting ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

For a first-order system, the transfer function takes the form:

G(s)=Y(s)U(s)=Kτs+1

Here, K and τ are the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(1−e−t/τ)

The transfer function for an integrating process is:

G(s)=Y(s)U(s)=Ks

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KMt

Interpretation Introduction

(b)

Interpretation:

For t≪τ, the relationship between K1 and K0 is to be derived if the two responses match.

Concept introduction:

For chemical processes, dynamic models consisting of ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

For a first-order system, the transfer function takes the form:

G(s)=Y(s)U(s)=Kτs+1

Here, K and τ are the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(1−e−t/τ)

The transfer function for an integrating process is:

G(s)=Y(s)U(s)=Ks

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KMt

Interpretation Introduction

(c)

Interpretation:

Single step analysis to find K0 and time delay in an integrating process is to be explained.

Concept introduction:

For chemical processes, dynamic models consisting of ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

The transfer function for an integrating process with the time delay of θ is:

G(s)=Y(s)U(s)=Kse−θs

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(t−θ)

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 4

Question

Chapter 5, Problem 5.22E

Interpretation Introduction

(a)

Interpretation:

For a first order system, it is to be shown that the step response for the change in the input of magnitude M can be approximately modeled by the step response of an integratorfor low values of t.

Concept introduction:

For chemical processes, dynamic models consisting ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

For a first-order system, the transfer function takes the form:

G(s)=Y(s)U(s)=Kτs+1

Here, K and τ are the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(1−e−t/τ)

The transfer function for an integrating process is:

G(s)=Y(s)U(s)=Ks

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KMt

Interpretation Introduction

(b)

Interpretation:

For t≪τ, the relationship between K1 and K0 is to be derived if the two responses match.

Concept introduction:

For chemical processes, dynamic models consisting of ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

For a first-order system, the transfer function takes the form:

G(s)=Y(s)U(s)=Kτs+1

Here, K and τ are the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(1−e−t/τ)

The transfer function for an integrating process is:

G(s)=Y(s)U(s)=Ks

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KMt

Interpretation Introduction

(c)

Interpretation:

Single step analysis to find K0 and time delay in an integrating process is to be explained.

Concept introduction:

For chemical processes, dynamic models consisting of ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

The transfer function for an integrating process with the time delay of θ is:

G(s)=Y(s)U(s)=Kse−θs

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(t−θ)

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 5

Chapter 5, Problem 5.22E

Interpretation Introduction

(a)

Interpretation:

For a first order system, it is to be shown that the step response for the change in the input of magnitude M can be approximately modeled by the step response of an integratorfor low values of t.

Concept introduction:

For chemical processes, dynamic models consisting ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

For a first-order system, the transfer function takes the form:

G(s)=Y(s)U(s)=Kτs+1

Here, K and τ are the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(1−e−t/τ)

The transfer function for an integrating process is:

G(s)=Y(s)U(s)=Ks

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KMt

Interpretation Introduction

(b)

Interpretation:

For t≪τ, the relationship between K1 and K0 is to be derived if the two responses match.

Concept introduction:

For chemical processes, dynamic models consisting of ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

For a first-order system, the transfer function takes the form:

G(s)=Y(s)U(s)=Kτs+1

Here, K and τ are the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(1−e−t/τ)

The transfer function for an integrating process is:

G(s)=Y(s)U(s)=Ks

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KMt

Interpretation Introduction

(c)

Interpretation:

Single step analysis to find K0 and time delay in an integrating process is to be explained.

Concept introduction:

For chemical processes, dynamic models consisting of ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

The transfer function for an integrating process with the time delay of θ is:

G(s)=Y(s)U(s)=Kse−θs

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(t−θ)

Answer 6

Chapter 5, Problem 5.22E

Interpretation Introduction

(a)

Interpretation:

For a first order system, it is to be shown that the step response for the change in the input of magnitude M can be approximately modeled by the step response of an integratorfor low values of t.

Concept introduction:

For chemical processes, dynamic models consisting ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

For a first-order system, the transfer function takes the form:

G(s)=Y(s)U(s)=Kτs+1

Here, K and τ are the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(1−e−t/τ)

The transfer function for an integrating process is:

G(s)=Y(s)U(s)=Ks

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KMt

Answer 7

Chapter 5, Problem 5.22E

Interpretation Introduction

(a)

Interpretation:

For a first order system, it is to be shown that the step response for the change in the input of magnitude M can be approximately modeled by the step response of an integratorfor low values of t.

Concept introduction:

For chemical processes, dynamic models consisting ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

For a first-order system, the transfer function takes the form:

G(s)=Y(s)U(s)=Kτs+1

Here, K and τ are the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(1−e−t/τ)

The transfer function for an integrating process is:

G(s)=Y(s)U(s)=Ks

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KMt

Answer 8

Interpretation Introduction

(b)

Interpretation:

For t≪τ, the relationship between K1 and K0 is to be derived if the two responses match.

Concept introduction:

For chemical processes, dynamic models consisting of ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

For a first-order system, the transfer function takes the form:

G(s)=Y(s)U(s)=Kτs+1

Here, K and τ are the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(1−e−t/τ)

The transfer function for an integrating process is:

G(s)=Y(s)U(s)=Ks

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KMt

Answer 9

Interpretation Introduction

(b)

Interpretation:

For t≪τ, the relationship between K1 and K0 is to be derived if the two responses match.

Concept introduction:

For chemical processes, dynamic models consisting of ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

For a first-order system, the transfer function takes the form:

G(s)=Y(s)U(s)=Kτs+1

Here, K and τ are the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(1−e−t/τ)

The transfer function for an integrating process is:

G(s)=Y(s)U(s)=Ks

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KMt

Answer 10

Interpretation Introduction

(c)

Interpretation:

Single step analysis to find K0 and time delay in an integrating process is to be explained.

Concept introduction:

For chemical processes, dynamic models consisting of ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

The transfer function for an integrating process with the time delay of θ is:

G(s)=Y(s)U(s)=Kse−θs

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(t−θ)

Answer 11

Interpretation Introduction

(c)

Interpretation:

Single step analysis to find K0 and time delay in an integrating process is to be explained.

Concept introduction:

For chemical processes, dynamic models consisting of ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.

The transfer function for an integrating process with the time delay of θ is:

G(s)=Y(s)U(s)=Kse−θs

Here, K is the system parameters. Step response for the change in input of magnitude M is given by:

y(t)=KM(t−θ)

Answer 12

Expert Solution & Answer

Answer 13

Expert Solution & Answer

Answer 14

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Answer 15

Ch. 5 - Prob. 5.1E Ch. 5 - Prob. 5.2E Ch. 5 - Prob. 5.3E Ch. 5 - Prob. 5.4E Ch. 5 - Prob. 5.5E Ch. 5 - Prob. 5.6E Ch. 5 - Appelpolscher has just left a meeting with Stella...Ch. 5 - Prob. 5.8E Ch. 5 - Prob. 5.9E Ch. 5 - Prob. 5.10E

Solution Summary: The author explains how the step response of a first-order system can be approximated by the model of an integrator.

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Chapter 5 Solutions