7. A stream contains a waste chemical, W, with a concentration of 1 mol/liter. To meet Environmental Protection Agency and state standards, at least 90% of the chemical must be removed by reaction. The chemical decomposes by a second-order reaction with a rate constant of 1.5 liter/(mol hr). The stream flow rate is 100 liter/hour and two available reactors (400 and 2000 liters) have been placed in series (the smaller reactor is placed before the larger one). a. Write the modeling equations for the concentration of the waste chemical. Assume constant volume and constant density. Let Cwl Cw2 F V₁ V₂ k = concentration in reactor 1, mol/liter = concentration in reactor 2, mol/liter = volumetric flow rate, liter/hr = liquid volume in reactor 1, liters == = liquid volume in reactor 2, liters second-order rate constant, liter/(mol hr) b. Show that the steady-state concentrations are 0.33333 mol/liter (reactor 1) and 0.09005 mol/liter (reactor 2), so the specification is met. (Hint: You need to solve quadratic equations to obtain the concentrations.) c. Linearize at steady state and develop the state space model (analytical) of the form where x = Ax + Bu x = Cwl - Cwls Cw2-Cw2s u = - F-F Cwin - Cwins d. Show that the A and B matrices are -1.25 0 0.001667 0.25] A B: 0.05 -0.32015] 0.0001216 0 (also, show the units associated with each coefficient). e. Assuming that each state is an output, show that the C and D matrices are C [ 1 0 D= 0 00
7. A stream contains a waste chemical, W, with a concentration of 1 mol/liter. To meet Environmental Protection Agency and state standards, at least 90% of the chemical must be removed by reaction. The chemical decomposes by a second-order reaction with a rate constant of 1.5 liter/(mol hr). The stream flow rate is 100 liter/hour and two available reactors (400 and 2000 liters) have been placed in series (the smaller reactor is placed before the larger one). a. Write the modeling equations for the concentration of the waste chemical. Assume constant volume and constant density. Let Cwl Cw2 F V₁ V₂ k = concentration in reactor 1, mol/liter = concentration in reactor 2, mol/liter = volumetric flow rate, liter/hr = liquid volume in reactor 1, liters == = liquid volume in reactor 2, liters second-order rate constant, liter/(mol hr) b. Show that the steady-state concentrations are 0.33333 mol/liter (reactor 1) and 0.09005 mol/liter (reactor 2), so the specification is met. (Hint: You need to solve quadratic equations to obtain the concentrations.) c. Linearize at steady state and develop the state space model (analytical) of the form where x = Ax + Bu x = Cwl - Cwls Cw2-Cw2s u = - F-F Cwin - Cwins d. Show that the A and B matrices are -1.25 0 0.001667 0.25] A B: 0.05 -0.32015] 0.0001216 0 (also, show the units associated with each coefficient). e. Assuming that each state is an output, show that the C and D matrices are C [ 1 0 D= 0 00
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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Please help with a, b, c, d, and e
![7. A stream contains a waste chemical, W, with a concentration of 1 mol/liter. To
meet Environmental Protection Agency and state standards, at least 90% of the
chemical must be removed by reaction. The chemical decomposes by a
second-order reaction with a rate constant of 1.5 liter/(mol hr). The stream
flow rate is 100 liter/hour and two available reactors (400 and 2000 liters) have
been placed in series (the smaller reactor is placed before the larger one).
a. Write the modeling equations for the concentration of the waste chemical.
Assume constant volume and constant density. Let
Cwl
Cw2
F
V₁
V₂
k
= concentration in reactor 1, mol/liter
=
concentration in reactor 2, mol/liter
= volumetric flow rate, liter/hr
= liquid volume in reactor 1, liters
==
=
liquid volume in reactor 2, liters
second-order rate constant, liter/(mol hr)
b. Show that the steady-state concentrations are 0.33333 mol/liter (reactor 1)
and 0.09005 mol/liter (reactor 2), so the specification is met.
(Hint: You need to solve quadratic equations to obtain the concentrations.)
c. Linearize at steady state and develop the state space model (analytical) of
the form
where
x = Ax + Bu
x =
Cwl - Cwls
Cw2-Cw2s
u =
-
F-F
Cwin - Cwins
d. Show that the A and B matrices are
-1.25
0
0.001667 0.25]
A
B:
0.05
-0.32015]
0.0001216 0
(also, show the units associated with each coefficient).
e. Assuming that each state is an output, show that the C and D matrices are
C
[
1
0
D=
0
00](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6fb3429f-a406-4812-82fc-346d6a20f662%2F5cb65435-f49e-45d1-9d27-2cbe84357809%2Fgw676b9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:7. A stream contains a waste chemical, W, with a concentration of 1 mol/liter. To
meet Environmental Protection Agency and state standards, at least 90% of the
chemical must be removed by reaction. The chemical decomposes by a
second-order reaction with a rate constant of 1.5 liter/(mol hr). The stream
flow rate is 100 liter/hour and two available reactors (400 and 2000 liters) have
been placed in series (the smaller reactor is placed before the larger one).
a. Write the modeling equations for the concentration of the waste chemical.
Assume constant volume and constant density. Let
Cwl
Cw2
F
V₁
V₂
k
= concentration in reactor 1, mol/liter
=
concentration in reactor 2, mol/liter
= volumetric flow rate, liter/hr
= liquid volume in reactor 1, liters
==
=
liquid volume in reactor 2, liters
second-order rate constant, liter/(mol hr)
b. Show that the steady-state concentrations are 0.33333 mol/liter (reactor 1)
and 0.09005 mol/liter (reactor 2), so the specification is met.
(Hint: You need to solve quadratic equations to obtain the concentrations.)
c. Linearize at steady state and develop the state space model (analytical) of
the form
where
x = Ax + Bu
x =
Cwl - Cwls
Cw2-Cw2s
u =
-
F-F
Cwin - Cwins
d. Show that the A and B matrices are
-1.25
0
0.001667 0.25]
A
B:
0.05
-0.32015]
0.0001216 0
(also, show the units associated with each coefficient).
e. Assuming that each state is an output, show that the C and D matrices are
C
[
1
0
D=
0
00
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