1. Solutions Manual for Elements of Chemical Reaction
Engineering Fourth Edition Brian Vicente Max Nori H. Scott
Fogler
2. 1-1 Solutions for Chapter 1 Mole Balances Synopsis General:
The goal of these problems are to reinforce the definitions and
provide an understanding of the mole balances of the different
types of reactors. It lays the foundation for step 1 of the
algorithm in Chapter 4. P1-1. This problem helps the student
understand the course goals and objectives. P1-2. Part (d) gives
hints on how to solve problems when they get stuck. Encourages
students to get in the habit of writing down what they learned from
each chapter. It also gives tips on problem solving. P1-3. Helps
the student understand critical thinking and creative thinking,
which are two major goals of the course. P1-4. Requires the student
to at least look at the wide and wonderful resources available on
the CD-ROM and the Web. P1-5. The ICMs have been found to be a
great motivation for this material. P1-6. Uses Example 1-1 to
calculate a CSTR volume. It is straight forward and gives the
student an idea of things to come in terms of sizing reactors in
chapter 4. An alternative to P1-15. P1-7. Straight forward
modification of Example 1-1. P1-8. Helps the student review and
member assumption for each design equation. P1-9 and P1-10. The
results of these problems will appear in later chapters. Straight
forward application of chapter 1 principles. P1-11. Straight
forward modification of the mole balance. Assigned for those who
emphasize bioreaction problems. P1-12. Can be assigned to just be
read and not necessarily to be worked. It will give students a
flavor of the top selling chemicals and top chemical companies.
P1-13. Will be useful when the table is completed and the students
can refer back to it in later chapters. Answers to this problem can
be found on Professor Susan Montgomerys equipment module on the
CD-ROM. See P1-17.
3. 1-2 P1-14. Many students like this straight forward problem
because they see how CRE principles can be applied to an everyday
example. It is often assigned as an in- class problem where parts
(a) through (f) are printed out from the web. Part (g) is usually
omitted. P1-15. Shows a bit of things to come in terms of reactor
sizing. Can be rotated from year to year with P1-6. P1-16.
Open-ended problem. P1-17. I always assign this problem so that the
students will learn how to use POLYMATH/MaLab before needing it for
chemical reaction engineering problems. P1-18. Parts (a) and (b)
are open-ended problem. P1-19 and P1-20. Help develop critical
thinking and analysis. CDP1-A Similar to problems 3, 4, 11, and 12.
CDP1-B Points out difference in rate per unit liquid volume and
rate per reactor volume. Summary Assigned Alternates Difficulty
Time (min) P1-1 AA SF 60 P1-2 I SF 30 P1-3 O SF 30 P1-4 O SF 30
P1-5 AA SF 30 P1-6 AA 1-15 SF 15 P1-7 I SF 15 P1-8 S SF 15 P1-9 S
SF 15 P1-10 S SF 15 P1-11 O FSF 15 P1-12 I Read Only SF 30 P1-13 I
SF 1 P1-14 O FSF 30 P1-15 O SF 60 P1-16 S SF 15 P1-17 AA SF 60
P1-18 S SF 30 P1-19 O -- 30 P1-20 O FSF 15
4. 1-3 CDP1-A AA FSF 30 CDP1-B I FSF 30 Assigned = Always
assigned, AA = Always assign one from the group of alternates, O =
Often, I = Infrequently, S = Seldom, G = Graduate level Alternates
In problems that have a dot in conjunction with AA means that one
of the problem, either the problem with a dot or any one of the
alternates are always assigned. Time Approximate time in minutes it
would take a B/B+ student to solve the problem. Difficulty SF =
Straight forward reinforcement of principles (plug and chug) FSF =
Fairly straight forward (requires some manipulation of equations or
an intermediate calculation). IC = Intermediate calculation
required M = More difficult OE = Some parts open-ended.
____________ * Note the letter problems are found on the CD-ROM.
For example A CDP1-A. Summary Table Ch-1 Review of Definitions and
Assumptions 1,5,6,7,8,9 Introduction to the CD-ROM 1,2,3,4 Make a
calculation 6 Open-ended 8,16 P1-1 Individualized solution. P1-2
Individualized solution. P1-3 Individualized solution. P1-4
Individualized solution. P1-5 Solution is in the decoding algorithm
given with the modules.
5. 1-4 P1-6 The general equation for a CSTR is: A AA r FF V ! !
= 0 Here rA is the rate of a first order reaction given by: rA = -
kCA Given : CA = 0.1CA0 , k = 0.23 min-1 , v0 = 10dm3 min-1 , FA =
5.0 mol/hr And we know that FA = CAv0 and FA0 = CA0v0 => CA0 =
FA0/ v0 = 0.5 mol/dm3 Substituting in the above equation we get:
))/5.0(1.0)(min23.0( min)/10)(/5.0(1.0min)/10)(/5.0( 31 3333 00A0
dmmol dmdmmoldmdmmol kC vCvC V A A ! ! = ! = V = 391.3 dm3 P1-7 1
0.23mink ! = From mole balance: dNA dt rA V! Rate law: rA! k CA"
rA! k NA V " Combine: dNA dt k! NA" 0! = 0 t t1 ! " # d 1 k $ %
& ' ( ) * NA0 NA NA 1 NA ! " " # d+ at 0! = , NAO = 100 mol and
! != , NA = (0.01)NAO t = !! " # $$ % & A A N N k 0 ln 1 ( )
min100ln 23.0 1 = t = 20 min
6. 1-5 P1-8 (a) The assumptions made in deriving the design
equation of a batch reactor are: - Closed system: no streams
carrying mass enter or leave the system. - Well mixed, no spatial
variation in system properties - Constant Volume or constant
pressure. P1-8 (b) The assumptions made in deriving the design
equation of CSTR, are: - Steady state. - No spatial variation in
concentration, temperature, or reaction rate throughout the vessel.
P1-8 (c) The assumptions made in deriving the design equation of
PFR are: - Steady state. - No radial variation in properties of the
system. P1-8 (d) The assumptions made in deriving the design
equation of PBR are: - Steady state. - No radial variation in
properties of the system. P1-8 (e) For a reaction, A B -rA is the
number of moles of A reacting (disappearing) per unit time per unit
volume [=] moles/ (dm3 .s). -rA is the rate of disappearance of
species A per unit mass (or area) of catalyst [=] moles/ (time.
mass of catalyst). rA is the rate of formation (generation) of
species A per unit mass (or area) of catalyst [=] moles/ (time.
mass catalyst). -rA is an intensive property, that is, it is a
function of concentration, temperature, pressure, and the type of
catalyst (if any), and is defined at any point (location) within
the system. It is independent of amount. On the other hand, an
extensive property is obtained by summing up the properties of
individual subsystems within the total system; in this sense, -rA
is independent of the extent of the system. P 1-9 Rate of
homogenous reaction rA is defined as the mole of A formed per unit
volume of the reactor per second. It is an Intensive property and
the concentration, temperature and hence the rate varies with
spatial coordinates. ' Ar on the other hand is defined as g mol of
A reacted per gm. of the catalyst per second. Here mass of catalyst
is the basis as this is what is important in catalyst reactions and
not the reactor volume. Applying general mole balance we get: !+"=
dVrFF dt dN jjj j 0 No accumulation and no spatial variation
implies !+"= dVrFF jjj00
7. 1-6 Also rj = b rj` and W = Vb where b is the bulk density
of the bed. => ' 00 ( ) ( )j j j bF F r dV!= " + # Hence the
above equation becomes 0 ' j j j F F W r ! = ! We can also just
apply the general mole balance as ' 0( ) ( ) j j j j dN F F r dW dt
= ! + " Assuming no accumulation and no spatial variation in rate,
we get the same form as above: 0 ' j j j F F W r ! = ! P1-10 Mole
balance on species j is: ! =+" V j jjj dt dN dVrFF 0 0 Let jM =
molecular wt. of species j Then 00 jjj wMF = = mass flow rate of j
into reactor: jjj mMN = = mass of species j in the reactor
Multiplying the mole balance on species j by jM ! =+" V j jjjjjjj
dt dN MdVrMMFMF 0 0 Now jM is constant: ( ) ( ) dt md dt NMd
dVrMMFMF j V jj jjjjjj ==+! "0 0
8. 1-7 dt dm dVrMww j V jjjj =+! "0 0 P1-11 Applying mole
balance to Penicillin: Penicillin is produced in the cells
stationary state (See Chapter 7), so there is no cell growth and
the nutrients are used in making product. Lets do part c first. [In
flowrate (moles/time)] penicillin + [generation rate
(moles/time)]penicillin [Out flowrate (moles/time)] penicillin =
[rate of accumulation (moles/time)]penicillin Fp,in + Gp Fp,out =
dNp dt Fp,in = 0 .(because no penicillin inflow) Gp = . V pr dV!
Therefore, . V pr dV! - Fp,out = dNp dt Assuming steady state for
the rate of production of penicillin in the cells stationary state,
dNp dt = 0 And no variations , ,p in p out p F F V r ! = ! Or, ,p
out p F V r = Similarly, for Corn Steep Liquor with FC = 0 C C C CC
r F r FF V ! = ! ! = 00 Assume RNA concentration does not change in
the stationary state and no RNA is generated or destroyed.
9. 1-8 P1-12 (a) Ranking of 10 most produced chemicals in 1995
and 2002 are listed in table below: Rank 2002 Rank 1995 Chemical 1
1 H2SO4 2 2 N2 3 4 C2H4 4 3 O2 5 9 C3H6 6 - H2 7 6 NH3 8 10 Cl2 9 -
P2O5 10 - C2H2Cl2 Chemicals like H2 , P2O5 , C2H2Cl2 has come in
top 10 chemicals and C3H6 has jumped to rank 5 now then rank 9 in
1995 . P1-12 (b) Ranking of top 10 chemical companies in sales in
year 2003 and 2002: s o u r c e : C h e m i c al and Engineering
News may 17,2004 We have Chevron Phillips which jumped to 5 rank in
2003 from 8th rank in 2002 and Air Products coming to 9th rank in
2003 from 11th in 2003. Chemical sales of each company has
increased compared to year 2002 from 9%(Eastman Chemical)
to28.2%(Chevron Phillips) but Huntsman Corp. has a decrease by
2.9%. P1-12 (c) Sulfuric acid is prime importance in manufacturing.
It is used in some phase of the manufacture of nearly all
industrial products .It is used in production of every other strong
acid. Because of its large number of uses, its the most produced
chemical. Sulfuric acid uses are: It is consumed in production of
fertilizers such as ammonium sulphate (NH4)2SO4 and superphosphate
(Ca(H2PO4)2) , which is formed when rock phosphate is treated with
sulfuric acid. Used as dehydrating agent. Used in manufacturing of
explosives, dyestuffs, other acids, parchment paper, glue ,
purification of petroleum and picking of metals. Used to remove
oxides from iron and steel before galvanizing or electroplating.
Used in non-ferrous metallurgy, in production of rayon and film. as
laboratory reagent and etchant and in storage batteries. It is also
general purpose food additive. 2003 2002 Company Chemical Sales ($
million 2003) 1 1 Dow Chemical 32632 2 2 DuPont 30249 3 3
ExxonMobil 20190 4 4 General Electric 8371 5 8 Chevron Phillips
7018 6 5 Huntsman Corp. 6990 7 6 PPG Industries 6606 8 7 Equistar
Chemicals 6545 9 11 Air Products 6029 10 9 Eastman Chemicals
5800
10. 1-9 P1-12 (d) Annual Production rate of ethylene for year
2002 is 5.21x 1010 lb/year Annual Production rate of benzene for
year 2002 is 1.58 x 1010 lb/year Annual Production rate of ethylene
oxide for year 2002 is 7.6 x109 lb/year P1-12 (e) Because the basic
raw material coal and petroleum for organic chemicals is very
limited and their production is not increasing as production of raw
material for inorganic chemicals. P1-13 Type Characteristics Phases
Usage Advantage Disadvantage Batc h All the reactants fed into the
reactor. During reaction nothing is added or removed. Easy heating
or cooling. 1. Liquid phase 2. Gas phase 3. Liquid Solid 1. Small
scale pdn. 2. Used for lab experimentation. 3. Pharmaceuticals 4.
Fermentation 1. High Conversion per unit volume. 2. Flexibility of
using for multiple reactions. 3. Easy to clean 1. High Operating
cost. 2. Variable product quality. CST R Continuous flow of
reactants and products. Uniform composition throughout. 1. Liquid
phase 2. Gas liquid 3. Solid - liquid 1. Used when agitation
required. 2. Series Configuration possible for different
configuration streams 1. Continuous Operation. 2.Good Temperature
Control 3. Two phase reactions possible. 4.Good Control 5.
Simplicity of construction. 6. Low operating cost 7. Easy to clean
1. Lowest conversion per unit volume. 2. By passing possible with
poor agitation. 3 High power Input reqd. PFR One long reactor or
number of CSTRs in series. No radial variations. Conc. changes
along the length. 1. Primarily gas Phase 1. Large Scale pdn. 2.
Fast reactions 3. Homogenous reactions 4. Heterogeneous reactions
5. Continuous pdn. 1. High conversion per unit volume 2. Easy to
maintain (No moving parts) 3. low operating cost 4. continuous
operation 1. Undesired thermal gradient. 2. Poor temperature
control 3. Shutdown and cleaning expensive.
11. 1-10 PBR Tubular reactor that is packed with solid catalyst
particles. 1. Gas Phase (Solid Catalyst) 2.Gas solid reactions. 1.
Used primarily in the heterogeneous gas phase reaction with solid
catalyst e.g Fischer tropsch synthesis. 1. High conversion per unit
mass of catalyst 2. low operating cost 3. Continuous operation 1.
Undesired thermal gradient. 2. Poor temperature control 3.
Channeling 4. Cleaning expensive. P1-14 Given 210 10*2 ftA = RTSTP
69.491= ftH 2000= 313 10*4 ftV = T = 534.7 R PO = 1atm Rlbmol ftatm
R 3 7302.0= yA = 0.02 3 10 10*04.2 ft lbmol CS ! = C = 4*105 cars
FS = CO in Santa Ana wind FA = CO emission from autos hr ft vA 3
3000= per car at STP P1-14 (a) Total number of lb moles gas in the
system: N = 13 3 3 1 (4 10 ) . 0.73 534.69 . atm ft atm ft R lbmol
R ! ! " # !$ % & ' = 1.025 x 1011 lb mol P1-14 (b) Molar
flowrate of CO into L.A. Basin by cars. STP A ATAA TR PCv yFyF 0
!== cars ft lbmol carhr ft FT 400000 359 13000 3 3 !!= (See
appendix B) FA = 6.685 x 104 lb mol/hr N P0 V R T :=
12. 1-11 P1-14 (c) Wind speed through corridor is v = 15mph W =
20 miles The volumetric flowrate in the corridor is vO = v.W.H =
(15x5280)(20x5280)(2000) ft3 /hr = 1.673 x 1013 ft3 /hr P1-14 (d)
Molar flowrate of CO into basin from Sant Ana wind. FS v0 CS!:= =
1.673 x 1013 ft3 /hr 10 2.04 10! " " lbmol/ft3 FS = 3.412 x 103
lbmol/hr P1-14 (e) Rate of emission of CO by cars + Rate of CO by
Wind - Rate of removal of CO = COdN dt dt dC VCvFF co cooSA =!+
(V=constant, VCN coco = ) P1-14 (f) t = 0 , coOco CC = 0 co coO Ct
co A S o coC dC dt V F F v C = + !" " !! " # $$ % & '+ '+ =
cooSA coOoSA o CvFF CvFF v V t ln P1-14 (g) Time for concentration
to reach 8 ppm. 8 0 3 2.04 10CO lbmol C ft ! = " , 8 3 2.04 10 4 CO
lbmol C ft ! = " From (f), 0 3 4 3 13 8 3 3 3 3 13 4 3 13 8 3 . ln
. 6.7 10 3.4 10 1.673 10 2.04 10 4 ln 1.673 10 6.7 10 3.4 10 1.673
10 0.51 10 A S O CO o A S O CO F F v CV t v F F v C lbmol lbmol ft
lbmol ft hr hr hr ft ft lbmol lbmol ft lbmol hr hr hr hr ft ! ! "
#+ ! = $ % + !& ' " # ( + ( ! ( ( ($ % $ %= $ % ( ( + ( ! ( (
($ % & ' t = 6.92 hr
13. 1-12 P1-14 (h) (1) to = 0 tf = 72 hrs coC = 2.00E-10
lbmol/ft3 a = 3.50E+04 lbmol/hr ov = 1.67E+12 ft3 /hr b = 3.00E+04
lbmol/hr Fs = 341.23 lbmol/hr V = 4.0E+13 ft3 dt dC VCvF t ba co
coos =!+" # $ % & ' + 6 sin ( Now solving this equation using
POLYMATH we get plot between Cco vs t See Polymath program
P1-14-h-1.pol. POLYMATH Results Calculated values of the DEQ
variables Variable initial value minimal value maximal value final
value t 0 0 72 72 C 2.0E-10 2.0E-10 2.134E-08 1.877E-08 v0 1.67E+12
1.67E+12 1.67E+12 1.67E+12 a 3.5E+04 3.5E+04 3.5E+04 3.5E+04 b
3.0E+04 3.0E+04 3.0E+04 3.0E+04 F 341.23 341.23 341.23 341.23 V
4.0E+13 4.0E+13 4.0E+13 4.0E+13 ODE Report (RKF45) Differential
equations as entered by the user [1] d(C)/d(t) =
(a+b*sin(3.14*t/6)+F-v0*C)/V Explicit equations as entered by the
user [1] v0 = 1.67*10^12 [2] a = 35000 [3] b = 30000 [4] F = 341.23
[5] V = 4*10^13 (2) tf = 48 hrs sF = 0 dt dC VCv t ba co coo =!" #
$ % & ' + 6 sin ( Now solving this equation using POLYMATH we
get plot between Cco vs t
14. 1-13 See Polymath program P1-14-h-2.pol. POLYMATH Results
Calculated values of the DEQ variables Variable initial value
minimal value maximal value final value t 0 0 48 48 C 2.0E-10
2.0E-10 1.904E-08 1.693E-08 v0 1.67E+12 1.67E+12 1.67E+12 1.67E+12
a 3.5E+04 3.5E+04 3.5E+04 3.5E+04 b 3.0E+04 3.0E+04 3.0E+04 3.0E+04
V 4.0E+13 4.0E+13 4.0E+13 4.0E+13 ODE Report (RKF45) Differential
equations as entered by the user [1] d(C)/d(t) =
(a+b*sin(3.14*t/6)-v0*C)/V Explicit equations as entered by the
user [1] v0 = 1.67*10^12 [2] a = 35000 [3] b = 30000 [4] V =
4*10^13 (3) Changing a Increasing a reduces the amplitude of
ripples in graph. It reduces the effect of the sine function by
adding to the baseline. Changing b The amplitude of ripples is
directly proportional to b. As b decreases amplitude decreases and
graph becomes smooth. Changing v0 As the value of v0 is increased
the graph changes to a shifted sin-curve. And as v0 is decreased
graph changes to a smooth increasing curve. P1-15 (a) rA = k with k
= 0.05 mol/h dm3 CSTR: The general equation is A AA r FF V ! ! = 0
Here CA = 0.01CA0 , v0 = 10 dm3 /min, FA = 5.0 mol/hr Also we know
that FA = CAv0 and FA0 = CA0v0, CA0 = FA0/ v0 = 0.5 mol/dm3
15. 1-14 Substituting the values in the above equation we get,
05.0 10)5.0(01.010)5.0(00A0 ! = ! = k vCvC V A V = 99 dm3 PFR: The
general equation is kr dV dF A A == , Now FA = CAv0 and FA0 = CA0v0
=> k dV vdCA !=0 Integrating the above equation we get !=! VC C
A dVdC k v A A 0 0 0 => )( 0 0 AA CC k v V != Hence V = 99 dm3
Volume of PFR is same as the volume for a CSTR since the rate is
constant and independent of concentration. P1-15 (b) - rA = kCA
with k = 0.0001 s-1 CSTR: We have already derived that A A r vCvC V
! ! = 00A0 A A kC Cv )01.01(00 ! = k = 0.0001s-1 = 0.0001 x 3600
hr-1 = 0.36 hr-1 )/5.0*01.0)(36.0( )99.0)(/5.0)(/10( 31 33 dmmolhr
dmmolhrdm V ! = => V = 2750 dm3 PFR: From above we already know
that for a PFR AA A kCr dV vdC ==0 Integrating !"=! VC C A A dV C
dC k v A A 0 0 0 V C C k v A A =00 ln Again k = 0.0001s-1 = 0.0001
x 3600 hr-1 = 0.36 hr-1 Substituing the values in above equation we
get V = 127.9 dm3
16. 1-15 P1-15 (c) - rA = kCA 2 with k = 3 dm3 /mol.hr CSTR: !
V = CA0v0 " CAv0 "rA ! = v0CA0(1" 0.01) kCA 2 Substituting all the
values we get ! V = (10dm3 /hr)(0.5mol/dm3 )(0.99) (3dm3
/hr)(0.01*0.5mol/dm3 )2 => V = 66000 dm3 PFR: ! dCAv0 dV = rA =
kCA 2 Integrating ! v0 k dCA CA 2 CA 0 CA " = # dV 0 V " => ! v0
k ( 1 CA " 1 CA0 ) = V => ! V = 10dm3 /hr 3dm3 /mol.hr ( 1
0.01CA0 " 1 CA0 ) = 660 dm3 P1-15 (d) CA = .001CA0 ! t = dN "rAVNA
NA 0 # Constant Volume V=V0 ! t = dCA "rA CA CA 0 # Zero order: ! t
= 1 k CA0 " 0.001CA0[ ] = .999CAo 0.05 = 9.99h First order: ! t = 1
k ln CA0 CA " # $ % & ' = 1 0.001 ln 1 .001 " # $ % & ' =
6908s Second order: ! t = 1 k 1 CA " 1 CA0 # $ % & ' ( = 1 3 1
0.0005 " 1 0.5 # $% & '( = 666h
17. 1-16 P1-16 Individualized Solution P1-17 (a) Initial number
of rabbits, x(0) = 500 Initial number of foxes, y(0) = 200 Number
of days = 500 1 2 dx k x k xy dt = ! .(1) 3 4 dy k xy k y dt = !
..(2) Given, 1 1 2 3 1 4 0.02 0.00004/( ) 0.0004/( ) 0.04 k day k
day foxes k day rabbits k day ! ! = = " = " = See Polymath program
P1-17-a.pol. POLYMATH Results Calculated values of the DEQ
variables Variable initial value minimal value maximal value final
value t 0 0 500 500 x 500 2.9626929 519.40024 4.2199691 y 200
1.1285722 4099.517 117.62928 k1 0.02 0.02 0.02 0.02 k2 4.0E-05
4.0E-05 4.0E-05 4.0E-05 k3 4.0E-04 4.0E-04 4.0E-04 4.0E-04 k4 0.04
0.04 0.04 0.04 ODE Report (RKF45) Differential equations as entered
by the user [1] d(x)/d(t) = (k1*x)-(k2*x*y) [2] d(y)/d(t) =
(k3*x*y)-(k4*y) Explicit equations as entered by the user [1] k1 =
0.02 [2] k2 = 0.00004 [3] k3 = 0.0004 [4] k4 = 0.04
18. 1-17 When, tfinal = 800 and 3 0.00004/( )k day rabbits= !
Plotting rabbits Vs. foxes P1-17 (b) POLYMATH Results See Polymath
program P1-17-b.pol.
19. 1-18 POLYMATH Results NLES Solution Variable Value f(x) Ini
Guess x 2.3850387 2.53E-11 2 y 3.7970279 1.72E-12 2 NLES Report
(safenewt) Nonlinear equations [1] f(x) = x^3*y-4*y^2+3*x-1 = 0 [2]
f(y) = 6*y^2-9*x*y-5 = 0 P1-18 (a) No preheating of the benzene
feed will diminish the rate of reaction and thus lesser conversions
will be achieved. P1-18 (b) An interpolation can be done on the
logarithmic scale to find the desired values from the given data.
Now we can interpolate to the get the cost at 6000 gallons and
15000 gallons Cost of 6000 gal reactor = 1.905 x 105 $ Cost of
15000 gal reactor = 5.623 x 105 $ P1-18 (c) We are given CA is 0.1%
of initial concentration CA = 0.001CA0 Also from Example 1.3, )ln(
00 A A C C k v V = Substituting v0 = 10 dm3 /min and k = 0.23 min-1
we get 3 300dmV = which is three times the volume of reactor used
in Example 1-3. P1-18 (d) Safety of Plant. P1-19 Enrico Fermi
Problem no definite solution P1-20 Enrico Fermi Problem no definite
solution P1-21 Individualized solution. Cost vs Volume of reactor
(log - log plot) 0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 log
volume (gallons)
20. 1-19 CDP1-A (a) How many moles of A are in the reactor
initially? What is the initial concentration of A? If we assume
ideal gas behavior, then calculating the moles of A initially
present in the reactor is quite simple. We insert our variables
into the ideal gas equation: 3 3 (20 )(200 ) 101.33 97.5 1. 8.3145
(500) PV atm dm kPa n moles RT atmkPa dm molK ! " = = =# $ ! " %
& # $ % & Knowing the mole fraction of A (yAo) is 75%, we
multiply the total number of moles (NTo) by the yA: 0.75 97.5
73.1AomolesA N= = ! = The initial concentration of A (CAo) is just
the moles of A divided by the volume: 3 3 73.1 0.37 / 200 Ao Ao
Nmoles moles C moles dm volume V dm = = = = CDP1-A (b) Time (t) for
a 1st order reaction to consume 99% of A. A A dC r dt = Our first
order rate law is: A Ar kC! = mole balance: A A dC kC dt = ! =>
0 A Ao Ct A AC dC k dt C ! =" " ln A Ao C kt C ! " # = $ % & '
, knowing CA=0.01 CAo and our rate constant (k=0.1 min-1 ), we can
solve for the time of the reaction: 1 1 4.61 ln(0.01) 46.1min
0.1min t k ! = ! = = CDP1-A (c) Time for 2nd order reaction to
consume 80% of A and final pressure (P) at T = 127 C. rate law: 2 A
Ar kC! = mole balance: 2A A dC kC dt ! = " => 2 0 A Ao Ct A AC
dC k dt C ! =" " => 1 1 A Ao kt C C ! = ! + We can solve for the
time in terms of our rate constant (k = 0.7) and our initial
concentration (CAo): 5 1 Ao Ao kt C C ! = ! +
21. 1-20 ( )( )3 3 4 4 15.4min. 0.7 / min 0.37 /Ao t kC dm mol
mol dm = = = To determine the pressure of the reactor following
this reaction, we will again use the ideal gas law. First, we
determine the number of moles in the reactor: 0.25 0.2T I A B C To
Ao B CN N N N N N N N N= + + + = + + + 0.8B C AoN N N= = (0.25)
(0.2 0.8 0.8) 0.25(97.6) (1.8)(73.2) 156.1T To AoN N N moles= + + +
= + = 3 3 (156.1 ) 0.082 (500 ) 32 200 T dm atm mole K molKN RT P
atm V dm ! " # $ % &= = = CDP1-B Given: Liquid phase reaction
in a foam reactor, A !B Consider a differential element, V! of the
reactor: By material balance ( ) (1 )A A A AF F F r e V! + " = ! !
" Where, (1 )e V! " = fraction of reactor element which is liquid.
or: (1 )A AF r e V! = ! ! " (1 )A A dF r e dV = ! Must relate (
)Ar! to AF , where, AF is the total (gas +liquid) molar flow rate
of A. Ar! =rate of reaction (g mol A per cubic cm. of liquid per
sec.); e = volume fraction of gas; AF = molar flow rate of A (g
mol/sec.); V = volume of reactor
22. 2-1 Solutions for Chapter 2 - Conversion and Reactor Sizing
Synopsis General: The overall goal of these problems is to help the
student realize that if they have rA=f(X) they can design or size a
large number of reaction systems. It sets the stage for the
algorithm developed in Chapter 4. P2-1. This problem will keep
students thinking about writing down what they learned every
chapter. P2-2. This forces the students to determine their learning
style so they can better use the resources in the text and on the
CDROM and the web. P2-3. ICMs have been found to motivate the
students learning. P2-4. Introduces one of the new concepts of the
4th edition whereby the students play with the example problems
before going on to other solutions. P2-5. This is a reasonably
challenging problem that reinforces Levenspiels plots. P2-6. Novel
application of Levenspiel plots from an article by Professor Alice
Gast at Massachusetts Institute of Technology in CEE. P2-7.
Straight forward problem alternative to problems 8, 9, and 12.
P2-8. To be used in those courses emphasizing bio reaction
engineering. P2-9. The answer gives ridiculously large reactor
volume. The point is to encourage the student to question their
numerical answers. P2-10. Helps the students get a feel of real
reactor sizes. P2-11. Great motivating problem. Students remember
this problem long after the course is over. P2-12. Alternative
problem to P2-7 and P2-9. CDP2-A Similar to 2-9 CDP2-B Good problem
to get groups started working together (e.g. cooperative
learning.
23. 2-2 CDP2-C Similar to problems 2-8, 2-9, 2-12. CDP2-D
Similar to problems 2-8, 2-9, 2-12. Summary Assigned Alternates
Difficulty Time (min) P2-1 O 15 P2-2 A 30 P2-3 A 30 P2-4 O 75 P2-5
O M 75 P2-6 S M 60 P2-7 AA 8,9,12 FSF 45 P2-8 S FSF 45 P2-9 AA
7,9,12 SF 45 P2-10 S SF 15 P2-11 AA SF 1 P2-12 AA 7,8,9 SF 60
CDP2-A O 9,B,C,D FSF 5 CDP2-B O 9,B,C,D FSF 30 CDP2-C O 9,B,C,D FSF
30 CDP2-D O 9,B,C,D FSF 45 Assigned = Always assigned, AA = Always
assign one from the group of alternates, O = Often, I =
Infrequently, S = Seldom, G = Graduate level Alternates In problems
that have a dot in conjunction with AA means that one of the
problems, either the problem with a dot or any one of the
alternates are always assigned. Time Approximate time in minutes it
would take a B/B+ student to solve the problem. Difficulty SF =
Straight forward reinforcement of principles (plug and chug) FSF =
Fairly straight forward (requires some manipulation of equations or
an intermediate calculation). IC = Intermediate calculation
required M = More difficult OE = Some parts open-ended.
____________ * Note the letter problems are found on the CD-ROM.
For example A CDP1-A.
24. 2-3 Summary Table Ch-2 Straight forward 1,2,3,4,10 Fairly
straight forward 7,9,12 More difficult 5,6,8 Open-ended 6
Comprehensive 4,5,6,7,8,9,12 Critical thinking P2-9 P2-1
Individualized solution. P2-2 Individualized solution. P2-3
Solution is in the decoding algorithm given with the modules. P2-4
(a) Example 2-1 through 2-3 If flow rate FAO is cut in half. v1 =
v/2 , F1= FAO/2 and CAO will remain same. Therefore, volume of CSTR
in example 2-3, 2.34.6 2 1 2 1 01 1 == ! = ! = A A A r XF r XF V If
the flow rate is doubled, F2 = 2FAO and CAO will remain same,
Volume of CSTR in example 2-3, V2 = F2X/-rA = 12.8 m3 P2-4 (b)
Example 2-5 Now, FAO = 0.4/2 = 0.2 mol/s,
25. 2-4 New Table: Divide each term A A r F ! 0 in Table 2-3 by
2. X 0 0.1 0.2 0.4 0.6 0.7 0.8 [FAO/-rA](m3 ) 0.445 0.545 0.665
1.025 1.77 2.53 4 Reactor 1 Reactor 2 V1 = 0.82m3 V2 = 3.2 m3 V =
(FAO/-rA)X ( )182.0 1 0 X r F XA A !! " # $$ % & ' = ( )22.3 2
0 X r F XA A !! " # $$ % & ' = By trial and error we get: X1 =
0.546 and X2 = 0.8 Overall conversion X0verall = (1/2)X1 + (1/2)X2
= (0.546+0.8)/2 = 0.673 P2-4 (c) Example 2-6 Now, FAO = 0.4/2 = 0.2
mol/s, 0201 202101 0 AA AA FF XFXF X + + = New Table: Divide each
term A A r F ! 0 in Table 2-3 by 2. X 0 0.1 0.2 0.4 0.6 0.7 0.8
[FAO/-rA](m3 ) 0.445 0.545 0.665 1.025 1.77 2.53 4 V1 = 0.551m3 V2
= 1.614 m3 1 0 X AO A dX V F r = !" Plot FA0/-rA versus conversion.
Estimate outlet conversions by computing the integral of the
plotted function.
26. 2-5 X1 = 0.603 for V1 = 0.551m3 X2 = 0.89 for V2 = 1.614m3
Overall conversion X0 = (1/2)X1 + (1/2)X2 = (0.603+0.89)/2 = 0.746
P2-4 (d) Example 2-7 (1) For PFR, 0.2 2 0 3 0.222 AO A F V dX r m !
" = # $ %& ' = ( For first CSTR, X2 = 0.6, 3 1.32AOF m rA = ! ,
V1 = 32 1( ) .528AOF X X m rA ! = ! For second CSTR, X3 = 0.65, 3
2.0AOF m rA = ! , V3 = ( )3 2 3 0.1AOF X X m rA ! = ! (2) First
CSTR remains unchanged For PFR: 0.5 0.2 AO A F V dX r ! " = # $
%& ' ( , Using the Levenspiel Plot VPFR = 0.22
27. 2-6 For CSTR, VCSTR2 = ( )3 2 3 0.3AOF X X m rA ! = ! (3)
The worst arrangement is to put the PFR first, followed by the
larger CSTR and finally the smaller CSTR. Conversion Original
Reactor Volumes Worst Arrangement X1 = 0.20 V1 = 0.188 (CSTR) V1 =
0.23 (PFR) X2 = 0.60 V2 = 0.38 (PFR) V2 = 0.53 (CSTR) X3 = 0.65 V3
= 0.10 (CSTR) V3 = 0.10 (CSTR) For PFR, X1 = 0.2 1 1 0 X AO A F V
dX r ! " = # $ %& ' ( Using trapezoidal rule, XO = 0.1, X1 =
0.1 ( ) ( ) ( ) [ ] 1 1 1 3 3 0.2 1.28 0.98 2 0.23 O O X X V f X f
X rA m m ! = +" #$ %! = + = For CSTR, For X2 = 0.6, 3 1.32AOF m rA
= ! , V2 = ( )2 1 AO A F X X r ! ! = 1.32(0.6 0.2) = 0.53 m3 For
2nd CSTR, For X3 = 0.65, 3 2AOF m rA = ! , V3 = 0.1 m3 P2-4 (e)
Example 2-8 hrs5=! v0 = 1dm3 /min = 60dm3 /hr CA = 2.5 mol/dm3 X =
0.8 For CSTR, 0v V =! V = 300dm3
28. 2-7 (1) hrdmmol hrdmmol XC r A A 3 30 /4.0 / 5 8.05.2 = !
==" # (2) V = 300dm3 (3)CA = CA0(1-X) = 0.5 mol/dm3 P2-5 X 0 0.1
0.2 0.4 0.6 0.7 0.8 FAO/-rA (m3 ) 0.89 1.08 1.33 2.05 3.54 5.06 8.0
V = 1.0 m3 P2-5 (a) Two CSTRs in series For first CSTR, V =
(FAo/-rAX1) X1 => X1 = 0.44 For second CSTR, V = (FAo/-rAX2) (X2
X1) => X2 = 0.67 P2-5 (b) Two PFRs in series 1 2 10 X X Ao Ao A
AX F F V dX dX r r ! " ! " = +# $ # $ % %& ' & ' ( ( By
extrapolating and solving, we get X1 = 0.50 X2 = 0.74 P2-5 (c) Two
CSTRs in parallel with the feed, FAO, divided equally between two
reactors. FANEW/-rAX1 = 0.5FAO/-rAX1 V = (0.5FAO/-rAX1) X1 Solving
we get, Xout = 0.60 P2-5 (d) Two PFRs in parallel with the feed
equally divided between the two reactors. FANEW/-rAX1 =
0.5FAO/-rAX1 By extrapolating and solving as part (b), we get Xout
= 0.74
29. 2-8 P2-5 (e) A CSTR and a PFR are in parallel with flow
equally divided Since the flow is divided equally between the two
reactors, the overall conversion is the average of the CSTR
conversion (part C) and the PFR conversion (part D) Xo = (0.60 +
0.74) / 2 = 0.67 P2-5 (f) A PFR followed by a CSTR, XPFR = 0.50
(using part(b)) V = (FAo/-rA-XCSTR) (XCSTR XPFR) Solving we get,
XCSTR = 0.70 P2-5 (g) A CSTR followed by a PFR, XCSTR = 0.44 (using
part(a)) ! " = PFR CSTR X X A AO dX r F V By extrapolating and
solving, we get XPFR = 0.72 P2-5 (h) A 1 m3 PFR followed by two 0.5
m3 CSTRs, For PFR, XPFR = 0.50 (using part(b)) CSTR1: V =
(FAo/-rA-XCSTR) (XCSTR XPFR) = 0.5 m3 XCSTR = 0.63 CSTR2: V =
(FAo/-rA-XCSTR2) (XCSTR2 XCSTR1) = 0.5 m3 XCSTR2 = 0.72 P2-6 (a)
Individualized Solution P2-6 (b) 1) In order to find the age of the
baby hippo, we need to know the volume of the stomach. The
metabolic rate, -rA, is the same for mother and baby, so if the
baby hippo eats one half of what the mother eats then Fao (baby) =
Fao (mother). The Levenspiel Plot is drawn for the baby hippo
below.
30. 2-9 31.36 *0.34 0.23 2 baby A FaoX V m r = = = ! Since the
volume of the stomach is proportional to the age of the baby hippo,
and the volume of the babys stomach is half of an adult, then the
baby hippo is half the age of a full grown hippo. 4.5 2.25 2 years
Age years= = 2) If Vmax and mao are both one half of the mothers
then 0 2 2 1 2 motherA Ao AM AM m m r r ! " # $ % &= ' ' and
since max 2 A AM M A v C r K C ! = + then max 2 2 1 12 2baby mother
A AM AM M A v C r r K C ! = = ! +
31. 2-10 2 2 2 1 2 1 2 Ao Ao Ao AM AM motherbaby AM mother mm m
r rr ! " ! " # $ ! " = =# $ # $ # $# $% %& '# $& ' % &
' 2 Ao AM m r! will be identical for both the baby and mother.
Assuming that like the stomach the intestine volume is proportional
to age then the volume of the intestine would be 0.75 m3 and the
final conversion would be 0.40 P2-6 (c) Vstomach = 0.2 m3 From the
web module we see that if a polynomial is fit to the autocatalytic
reaction we get: 0 1 A AM m r! = 127X4 - 172.36X3 + 100.18X2 -
28.354X + 4.499 And since Vstomach = 0 1 A AM m r! X, solve V=
127X5 - 172.36X4 + 100.18X3 - 28.354X2 + 4.499X = 0.2 m3 Xstomach =
.067. For the intestine: The Levenspiel plot for the intestine is
shown below. The outlet conversion is 0.178 Since the hippo needs
30% conversion to survive but only achieves 17.8%, the hippo cannot
survive.
32. 2-11 P2-6 (d) PFR CSTR PFR: Outlet conversion of PFR =
0.111 CSTR: We must solve V = 0.46 = (X-0.111)(127X4 - 172.36X3 +
100.18X2 - 28.354X + 4.499) X=0.42 Since the hippo gets a
conversion over 30% it will survive. P2-7 Exothermic reaction: A B
+ C X r(mol/dm3 .min) 1/- r(dm3 .min/mol) 0 1 1 0.20 1.67 0.6 0.40
5 0.2 0.45 5 0.2 0.50 5 0.2 0.60 5 0.2 0.80 1.25 0.8 0.90 0.91 1.1
P2-7 (a) To solve this problem, first plot 1/-rA vs. X from the
chart above. Second, use mole balance as given below. CSTR:
33. 2-12 Mole balance: ( )( ) ( )min./5 4.0min/300F V 3 A0
dmmol mol r X A CSTR = ! = => =>VCSTR = 24 dm3 PFR: Mole
balance: ! " = X A APFR r dX FV 0 0 = 300(area under the curve)
VPFR = 72 dm3 P2-7 (b) For a feed stream that enters the reaction
with a previous conversion of 0.40 and leaves at any conversion up
to 0.60, the volumes of the PFR and CSTR will be identical because
of the rate is constant over this conversion range. ! ! " = " = " =
6. 4. 6. 4. 0 6. 4. 00 X r F dX r F dX r F V A A A A A A PFR P2-7
(c) VCSTR = 105 dm3 Mole balance: A CSTR r X ! = A0F V moldm mol dm
r X A min/35.0 min/300 105 3 3 == ! Use trial and error to find
maximum conversion. At X = 0.70, 1/-rA = 0.5, and X/-rA = 0.35
dm3.min/mol Maximum conversion = 0.70
34. 2-13 P2-7 (d) From part (a) we know that X1 = 0.40. Use
trial and error to find X2. Mole balance: ( ) 2 120 XA A r XXF V !
! = Rearranging, we get 008.0 40.0 0 2 2 == ! ! AXA F V r X At X2 =
0.64, 008.0 40.0 2 2 = ! ! XAr X Conversion = 0.64 P2-7 (e) From
part (a), we know that X1 = 0.40. Use trial and error to find X2.
Mole balance: !! " = " == 22 40.040.0 0 30072 X A X A APFR r dX r
dX FV At X2 = 0.908, V = 300 x (area under the curve) => V =
300(0.24) = 72dm3 Conversion = 0.908. P2-7 (f) See Polymath program
P2-7-f.pol.
35. 2-14 P2-8 (a) S S r XF V ! = 0 FS0 = 1000 g/hr At a
conversion of 40% g hrdm rS 3 15.0 1 = ! Therefore ( ) 3
60)40.0)(1000(15.0 dmV == P2-8 (b) At a conversion of 80%, g hrdm
rS 3 8.0 1 = ! FS0 = 1000 g/hr Therefore ( ) 3 640)80.0)(1000(8.0
dmV == P2-8 (c) ! " = X S SPFR r dX FV 0 0 From the plot of 1/-rS
Calculate the area under the curve such that the area is equal to
V/FS0 = 80 / 1000 = 0.08 X = 12% For the 80 dm3 CSTR, S S r XF dmV
! == 03 80 X/-rs = 0.08. From guess and check we get X = 55%
36. 2-15 P2-8 (d) To achieve 80% conversion with a CSTR
followed by a CSTR, the optimum arrangement is to have a CSTR with
a volume to achieve a conversion of about 45%, or the conversion
that corresponds to the minimum value of 1/-rs. Next is a PFR with
the necessary volume to achieve the 80% conversion following the
CSTR. This arrangement has the smallest reactor volume to achieve
80% conversion For two CSTRs in series, the optimum arrangement
would still include a CSTR with the volume to achieve a conversion
of about 45%, or the conversion that corresponds to the minimum
value of 1/-rs,first. A second CSTR with a volume sufficient to
reach 80% would follow the first CSTR. P2-8 (e) SM CS s CK CkC r +
=! and [ ] 001.01.0 0 +!= SSC CCC [ ]( ) SM SSS s CK CCkC r + +! =!
001.01.0 0 [ ]( )001.01.0 1 0 +! + =! SSS SM s CCkC CK r Let us
first consider when CS is small. CS0 is a constant and if we group
together the constants and simplify then CskCk CK r S SM s 2 2 1 1
+ + =! since CS < KM CskCk K r S M s 2 2 1 1 + =! which is
consistent with the shape of the graph when X is large (if CS is
small X is large and as CS grows X decreases). Now consider when CS
is large (X is small) As CS gets larger CC approaches 0: [ ]
001.01.0 0 +!= SSC CCC and 0SS CC ! If SM CS s CK CkC r + =! then
CS SM s CkC CK r + =! 1 As CS grows larger, CS >> KM
37. 2-16 And CCS S s kCCkC C r 11 ==! And since CC is becoming
very small and approaching 0 at X = 0, 1/-rs should be increasing
with CS (or decreasing X). This is what is observed at small values
of X. At intermediate levels of CS and X, these driving forces are
competing and why the curve of 1/-rS has a minimum. P9-2
Irreversible gas phase reaction 2A + B 2C See Polymath program
P2-9.pol. P2-9 (a) PFR volume necessary to achieve 50% conversion
Mole Balance ! " = 2 1 )( 0 X X A A r dX FV Volume = Geometric area
under the curve of (FA0/-rA) vs X) ( )5.01000005.0400000 2 1 !+" #
$ % & ' !!=V V = 150000 m3 One of the points of this problem is
for the students to recognize 150,000 m 3 is ridiculously reactor
volume. P2-9 (b) CSTR Volume to achieve 50% conversion Mole Balance
)( 0 A A r XF V ! = 1000005.0 !=V V = 50000m3 One of the points of
this problem is for the students to recognize 150,000 m 3 is
ridiculously reactor volume.
38. 2-17 P2-9 (c) Volume of second CSTR added in series to
achieve 80% conversion )( )( 120 2 A A r XXF V ! ! =
)5.08.0(5000002 !"=V V2 = 150000m3 Ridiculous volume P2-9 (d)
Volume of PFR added in series to first CSTR to achieve 80%
conversion )3.0100000()3.0400000 2 1 ( !+!!=PFRV VPFR = 90000m3
Ridiculous volume P2-9 (e) For CSTR, V = 60000 m3 (CSTR) Ridiculous
volume Mole Balance XX r XF V A A )500000800000(60000 )( 0 +!= ! =
X = 0.463 For PFR, V = 60000 m3 (PFR) Mole balance ! " = X A A r dX
FV 0 0 )( dXX X )100000800000(60000 0 ! +"= X = 0.134
39. 2-18 P2-9 (f) Real rates would not give that shape. The
reactor volumes are absurdly large. P2-10 Problem 2-10 involves
estimating the volume of three reactors from a picture. The door on
the side of the building was used as a reference. It was assumed to
be 8 ft high. The following estimates were made: CSTR h = 56ft d =
9 ft V = r2 h = (4.5 ft)2 (56 ft) = 3562 ft3 = 100,865 L PFR Length
of one segment = 23 ft Length of entire reactor = (23 ft)(12)(11) =
3036 ft D = 1 ft V = r2 h = (0.5 ft)2 (3036 ft) = 2384 ft3 = 67,507
L Answers will vary slightly for each individual. P2-11 No solution
necessary. P2-12 (a) The smallest amount of catalyst necessary to
achieve 80 % conversion in a CSTR and PBR connected in series and
containing equal amounts of catalyst can be calculated from the
figure below.
40. 2-19 The lightly shaded area on the left denotes the CSTR
while the darker shaded area denotes the PBR. This figure shows
that the smallest amount of catalyst is used when the CSTR is
upstream of the PBR. See Polymath program P2-12.pol. P2-12 (b)
Calculate the necessary amount of catalyst to reach 80 % conversion
using a single CSTR by determining the area of the shaded region in
the figure below. The area of the rectangle is approximately 23.2
kg of catalyst. P2-12 (c) The CSTR catalyst weight necessary to
achieve 40 % conversion can be obtained by calculating the area of
the shaded rectangle shown in the figure below. The area of the
rectangle is approximately 7.6 kg of catalyst.
41. 2-20 P2-12 (d) The catalyst weight necessary to achieve 80
% conversion in a PBR is found by calculating the area of the
shaded region in the figure below. The necessary catalyst weight is
approximately 22 kg. P2-12 (e) The amount of catalyst necessary to
achieve 40 % conversion in a single PBR can be found from
calculating the area of the shaded region in the graph below. The
necessary catalyst weight is approximately 13 kg.
42. 2-21 P2-12 (f) P2-12 (g) For different (-rA) vs. (X)
curves, reactors should be arranged so that the smallest amount of
catalyst is needed to give the maximum conversion. One useful
heuristic is that for curves with a negative slope, it is generally
better to use a CSTR. Similarly, when a curve has a positive slope,
it is generally better to use a PBR. CDP2-A (a) Over what range of
conversions are the plug-flow reactor and CSTR volumes identical?
We first plot the inverse of the reaction rate versus conversion.
Mole balance equations for a CSTR and a PFR: CSTR: A A r XF V ! = 0
PFR: ! " = X Ar dX V 0 Until the conversion (X) reaches 0.5, the
reaction rate is independent of conversion and the reactor volumes
will be identical.
43. 2-22 i.e. CSTR A A A A A PFR V r XF dX r F r dX V = ! =" !
=" ! = 0 5.0 0 0 5.0 0 CDP2-A (b) What conversion will be achieved
in a CSTR that has a volume of 90 L? For now, we will assume that
conversion (X) will be less that 0.5. CSTR mole balance: A A A A r
XCv r XF V ! = ! = 000 13 3 8 3 3 3 00 103 . 1032005 09.0 ! "= """
= ! = mol sm m mol s m m r Cv V X A A CDP2-A (c) This problem will
be divided into two parts, as seen below: The PFR volume required
in reaching X=0.5 (reaction rate is independent of conversion).
311000 1 105.1 m r XCv r XF V A A A A != " = " = The PFR volume
required to go from X=0.5 to X=0.7 (reaction rate depends on
conversion).
44. 2-23 Finally, we add V2 to V1 and get: Vtot = V1 + V2 = 2.3
x1011 m3 CDP2-A (d) What CSTR reactor volume is required if
effluent from the plug-flow reactor in part (c) is fed to a CSTR to
raise the conversion to 90 % We notice that the new inverse of the
reaction rate (1/-rA) is 7*108. We insert this new value into our
CSTR mole balance equation: 311000 104.1 m r XCv r XF V A A A A
CSTR != " # = " # = CDP2-A (e) If the reaction is carried out in a
constant-pressure batch reactor in which pure A is fed to the
reactor, what length of time is necessary to achieve 40%
conversion? Since there is no flow into or out of the system, mole
balance can be written as: Mole Balance: dt dN Vr A A =
Stoichiometry: )1(0 XNN AA !=
45. 2-24 Combine: dt dX NVr AA 0= From the stoichiometry of the
reaction we know that V = Vo(1+eX) and e is 1. We insert this into
our mole balance equation and solve for time (t): dt dX X N V r A A
=+! )1( 0 0 ! +" =! X A A t Xr dX Cdt 0 0 0 )1( After integration,
we have: )1ln( 1 0 XC r t A A + ! = Inserting the values for our
variables: t = 2.02 x 1010 s That is 640 years. CDP2-A (f) Plot the
rate of reaction and conversion as a function of PFR volume. The
following graph plots the reaction rate (-rA) versus the PFR
volume: Below is a plot of conversion versus the PFR volume. Notice
how the relation is linear until the conversion exceeds 50%.
46. 2-25 The volume required for 99% conversion exceeds 4*1011
m3 . CDP2-A (g) Critique the answers to this problem. The rate of
reaction for this problem is extremely small, and the flow rate is
quite large. To obtain the desired conversion, it would require a
reactor of geological proportions (a CSTR or PFR approximately the
size of the Los Angeles Basin), or as we saw in the case of the
batch reactor, a very long time. CDP2-B Individualized solution
CDP2-C (a) For an intermediate conversion of 0.3, Figure below
shows that a PFR yields the smallest volume, since for the PFR we
use the area under the curve. A minimum volume is also achieved by
following the PFR with a CSTR. In this case the area considered
would be the rectangle bounded by X =0.3 and X = 0.7 with a height
equal to the CA0/-rA value at X = 0.7, which is less than the area
under the curve.
47. 2-26 CDP2-C (b) CDP2-C (c) CDP2-C (d) For the PFR,
56. 3-1 Solutions for Chapter 3 Rate Law and Stoichiometry P3-1
Individualized solution. P3-2 (a) Example 3-1 0 0.001 0.002 0.003
0.004 0.005 0.006 0.007 0.008 310 315 320 325 330 335 T (K) k(1/s)
For E = 60kJ/mol For E1 = 240kJ/mol ! k =1.32 "1016 exp #60000J RT
$ % & ' ( ) ! k1=1.32 "1016 exp #240000J RT $ % & ' ( ) E =
240 kj/mol 0 5E-23 1E-22 1.5E-22 2E-22 2.5E-22 3E-22 3.5E-22
0.00295 0.003 0.00305 0.0031 0.00315 0.0032 0.00325 1/T (1/K)
k(1/s) E = 60 kj/mol 0 1000000 2000000 3000000 4000000 5000000
6000000 0.00295 0.003 0.00305 0.0031 0.00315 0.0032 0.00325 1/T
(1/K) k(1/s) P3-2 (b) Example 3-2 Yes, water is already considered
inert. P3-2 (c) Example 3-3 The solution to the example at a
conversion of 20% would remain unchanged. For 90 % conversions of
the caustic soda, the final concentration of glyceryl sterate is 0
instead of a negative concentration. Therefore 90 % of caustic soda
is possible.
57. 3-2 P3-2 (d) Example 3-4 ! A + b a B " c a C + d a D So,
the minimum value of B = b/a = ! 1/3 1 = 0.33 P3-2 (e) Example 3-5
For the concentration of N2 to be constant, the volume of reactor
must be constant. V = VO. Plot: 2 1 0.5(1 0.14 ) (1 )(0.54 0.5 )A X
r X X ! = ! ! ! 1/(-ra) vs X 0 20 40 60 80 100 120 140 160 180 0
0.2 0.4 0.6 0.8 1 1.2 X 1/(-ra) The rate of reaction decreases
drastically with increase in conversion at higher conversions. P3-2
(f) Example 3-6 For a given conversion, concentration of B is lower
in flow reactor than a constant volume batch reactor. Therefore the
reverse reaction decreases. CT0 = constant and inerts are varied. !
N2O4 " 2NO2 A 2B Equilibrium rate constant is given by: ! KC = CB,e
2 CA,e Stoichiometry: ! " = yA0# = yA0(2 $1) = yA0 Constant volume
Batch: ! CA = NA0(1" X) V0 = CA0(1" X) and ! CB = 2NA0X V0 = 2CA0X
Plug flow reactor: ! CA = FA0(1" X) v0(1+ #X) = CA0(1" X) (1+ #X)
and ! CB = 2FA0X v0(1+ "X) = 2CA0X (1+ "X)
58. 3-3 ! CAO = yAOPO RTO = yAO 0.07176( )mol/dm3 Combining:
For constant volume batch: ! KC = CB,e 2 CA,e = 4CAo 2 X2 CAO (1"
X) ! " Xe = KC (1# Xe) 4CA0 For flow reactor: )1)(1( 4 22 , 2 , XXC
XC C C K AO Ao eA eB C !+" == 04 )1)(1( A eeC e C XXK X !+" =# See
Polymath program P3-2-f.pol. POLYMATH Results NLES Report
(safenewt) Nonlinear equations [1] f(Xeb) = Xeb -
(kc*(1-Xeb)/(4*Cao))^0.5 = 0 [2] f(Xef) = Xef -
(kc*(1-Xef)*(1+eps*Xef)/(4*Cao))^0.5 = 0 Explicit equations [1] yao
= 1 [2] kc = 0.1 [3] Cao = 0.07174*yao [4] eps = yao 0 0.1 0.2 0.3
0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 y inerts
EquilibriumConversion Batch Flow
59. 3-4 Yinert Yao Xeb Xef 0 1 0.44 0.508 0.1 0.9 0.458 0.5217
0.2 0.8 0.4777 0.537 0.3 0.7 0.5 0.5547 0.4 0.6 0.525 0.576 0.5 0.5
0.556 0.601 0.6 0.4 0.5944 0.633 0.7 0.3 0.6435 0.6743 0.8 0.2 0.71
0.732 0.9 0.1 0.8112 0.8212 0.95 0.05 0.887 0.89 0.956 0.044 0.893
0.896 P3-2 (g) No solution will be given P3-2 (h) A + 2 1 B 2 1 C
Rate law: ! "rA = kACA 2 CB and ! kA = 25 1 s dm3 mol " # $ % &
' 2 ! rA "1 = rB "1/2 = rC 1/2 ! " 25CA 2 CB = kBCA 2 CB 1/2 = kCCA
2 CB #1/2 ! kC = kB =12.5 1 s dm3 mol " # $ % & ' 2 P3-2 (i) !
A + 3B " 2C Rate law: ! "rA = kACACB at low temperatures. At
equilibrium, ! KC = CC,e CA,e 1/2 CB,e 3/2 ! ! CA 1/2 CB 3/2 " CC
KC = 0 At equilibrium, -rA = 0 , so we can suggest that ! "rA = kA
CA 1/2 CB 1/2 " CC KC # $ % & ' ( But at t = 0, CC = 0 So the
rate law is not valid at t = 0. Next guess: ! KC 2 = CC,e 2
CA,eCB,e 3 , or ! CACB " CC 2 KC 2 CB 2 = 0 ! ! "rA = kA CACB " CC
2 KC 2 CB 2 # $ %% & ' (( which satisfies both the initial
conditions and equilibrium rate law.
60. 3-5 Hence ! "rA = kA CACB " CC 2 KC 2 CB 2 # $ %% & '
(( is the required rate law. P3-3 Solution is in the decoding
algorithm available separately from the author. P3-4 (a) Note: This
problem can have many solutions as data fitting can be done in many
ways. Using Arrhenius Equation For Fire flies: T(in K) 1/T Flashes/
min ln(flashe s/min) 294 0.003401 9 2.197 298 0.003356 12.16 2.498
303 0.003300 16.2 2.785 Plotting ln(flashes/min) vs 1/T, we get a
straight line. See Polymath program P3-4-fireflies.pol. For
Crickets: T(in K) 1/T x10 3 chrips/ min ln(chirps/ min) 287.2 3.482
80 4.382 293.3 3.409 126 4.836 300 3.333 200 5.298 Plotting
ln(chirps/min) Vs 1/T, we get a straight line. Both, Fireflies and
Crickets data follow the Arrhenius Model. ln y = A + B/T , and have
the same activation energy. See Polymath program P3-4-crickets.pol.
P3-4 (b) For Honeybee: T(in K) 1/T x10 3 V(cm/s) ln(V) 298 3.356
0.7 -0.357 303 3.300 1.8 0.588 308 3.247 3 1.098 Plotting ln(V) Vs
1/T, almost straight line. ln(V) = 44.6 1.33E4/T At T = 40o C(313K)
V = 6.4cm/s At T = -5o C(268K) V = 0.005cm/s(But bee would not be
alive at this temperature) See Polymath program P3-4-bees.pol.
61. 3-6 P3-4 (c) For ants: T(in K) 1/T x10 3 V(cm/s) ln(V) 283
3.53 0.5 -0.69 293 3.41 2 0.69 303 3.30 3.4 1.22 311 3.21 6.5 1.87
Plotting ln(V) Vs 1/T, almost straight line. See Polymath program
P3-4-ants.pol. So activity of bees, ants, crickets and fireflies
follow Arrhenius model. So activity increases with an increase in
temperature. Activation energies for fireflies and crickets are
almost the same. Insect Activation Energy Cricket 52150 Firefly
54800 Ant 95570 Honeybee 141800 P3-4 (d) There is a limit to
temperature for which data for any one of he insect can be
extrapolate. Data which would be helpful is the maximum and the
minimum temperature that these insects can endure before death.
Therefore, even if extrapolation gives us a value that looks
reasonable, at certain temperature it could be useless. P3-5 There
are two competing effects that bring about the maximum in the
corrosion rate: Temperature and HCN-H2SO4 concentration. The
corrosion rate increases with increasing temperature and increasing
concentration of HCN-H2SO4 complex. The temperature increases as we
go from top to bottom of the column and consequently the rate of
corrosion should increase. However, the HCN concentrations (and the
HCN-H2SO4 complex) decrease as we go from top to bottom of the
column. There is virtually no HCN in the bottom of the column.
These two opposing factors results in the maximum of the corrosion
rate somewhere around the middle of the column. P3-6 Antidote did
not dissolve from glass at low temperatures. P3-7 (a) If a reaction
rate doubles for an increase in 10C, at T = T1 let k = k1 and at T
= T2 = T1+10, let k = k2 = 2k1. Then with k = Ae-E/RT in general,
1/ 1 E RT k Ae! = and 2/ 2 E RT k Ae! = , or
62. 3-7 2 1 1 1 2 1 E R T Tk e k ! " # #$ % & ' = or ( ) 2
2 1 1 1 2 1 22 1 ln ln 1 1 k k k kE T TR TTT T ! " ! " # $ # $ %
& % &= ' = ' '! " '# $ % & Therefore: ( )( ) ( ) ( ) (
)( ) 2 1 1 1 11 2 1 ln 10 ln 2 10 10 k T T T Tk E R R T T ! " +# $
+% &= = ' ( )1 1 10 10 ln 2 E T T R + = which can be
approximated by 0.5 10 ln 2 E T R = P3-7 (b) Equation 3-18 is E RT
k Ae ! = From the data, at T1 = 0C, 1/ 1 E RT k Ae! = , and at T2 =
100C, 2/ 2 E RT k Ae! = Dividing gives 2 1 1 1 2 1 E R T Tk e k ! "
# #$ % & ' = , or ( ) 2 1 1 2 2 1 2 1 2 1 ln ln 1 1 k R k RTT k
E T T k T T ! " # $ ! "% &= ' = ' # $ '! " % &'# $ % &
[ ][ ]1.99 273 373 .050 ln 7960 100 .001 cal K K mol K cal E K mol
! " # $ % &' (= =) * + , ( ) 1 3 1 1 1 7960 10 min exp 2100min
1.99 273 E RT cal molA k e cal K mol K ! ! ! " # $ % $ %= = = $
%& ' $ %( ) * +, - P3-7 (c) Individualized solution
63. 3-8 P3-8 When the components inside air bag are ignited,
following reactions take place, 2NaN3 2Na + 3N2 ..(1) 10Na + 2KNO3
K2O + 5Na2O + N2 (2) K2O + Na2O + SiO2 alkaline silicate glass
...(3) ( ) ( ) ( ) ( )43215 rxnrxnrxnrxn =++! NaN3 + 0.2KNO3 +
0.1SiO2 0.4Na2O + 1.6N2 + complex/10(4) Stoichiometric table:
Species Symbol Initial Change Final NaN3 A NA -NAX NA(1-X) KNO3 B
NA B! -0.2X NA NA( B! - 0.2X) SiO2 C NA C! -0.1XNA NA( C! - 0.1X)
Na2O D 0 0.4XNA 0.4XNA N2 E 0 1.6XNA 1.6XNA Given weight of NaN3 =
150g Mwt of NaN3 = 65 Therefore, no. of moles of NaN3 = 2.3 1 moles
of NaN3 requires 0.2 mole of KNO3 => Moles of B, KNO3 = 0.2(2.3)
= 0.46 moles Mwt of KNO3 = 101.1 Therefore, grams of KNO3 required
= 0.46 x 101.1 = 46.5 g 1 moles of NaN3 requires 0.1 mole of SiO2.
Moles of C, SiO2 = 0.1(2.3) = 0.23 moles Mwt of SiO2 = 60.08
Therefore, grams of SiO2 required = 0.23 x 60.08 = 13.8 g Following
proposals are given to handle all the un-detonated air bags in cars
piling up in the junkyards: Store cars in cool, dry, ventilated
areas. Avoid Physical damage of the bag in car. It is stable under
ordinary conditions of storage. Decomposes explosively upon heating
(over 221F or 105 C), shock, concussion, or friction. Conditions to
avoid: heat, flames, ignition sources and incompatibles. P3-9 (a)
From the web module we know that (1 ) dX k x dt = ! and that k is a
function of temperature, but not a linear function. Therefore
doubling the temperature will not necessarily double the reaction
rate, and therefore halve the cooking time. P3-9 (b) When you boil
the potato in water, the heat transfer coefficient is much larger,
but the temperature can only be 100C. When you bake the potato, the
heat transfer coefficient is smaller, but the temperature can be
more than double that of boiling water. P3-9 (c) No solution will
be given
64. 3-9 P3-10 (a) 1) C2H6 C2H4 + H2 Rate law: -rA = 62HCkC 2)
C2H4 + 1/2O2 C2H4O Rate law: -rA = 2/1 0242 CkC HC 3)
(CH3)3COOC(CH3)3 C2H6 + 2CH3COCH3 A B + 2C Rate law: -rA = k[CA
CBCC 2 /KC] 4) n-C4H10 I- C4H10 Rate law: -rA = k[ 104HnCC 104HiCC
/Kc] 5) CH3COOC2H5 + C4H9OH CH3COOC4H9 + C2H5OH A + B C + D Rate
law: -rA = k[CACB CCCD/KC] P3-10 (b) 2A + B C (1) -rA = kCACB 2 (2)
-rA = kCB (3) -rA = k (4) -rA = kCACB -1 P3-10 (c) (1) C2H6 C2H4 +
H2 Rate law: -rA = 62HCkC (2) H2 + Br2 2HBr Rate law: -rHBr = 2 22
2 2/1 1 Br HBr BrH C C k CCk + (3) H2 + I2 2HI Rate law: 222 IHH
CkCr =! P3-11 (a) Liquid phase reaction, O CH2--OH CH2 - CH2 + H2O
CH2--OH A + B C CAO = 1 lbmol/ft3 CBO = 3.47 lbmol/ft3
Stoichiometric Table: Species Symbol Initial Change Remaining
Ethylene oxide A CAO=1 lbmol/ft3 - CAOX CA= CAO(1-X) = (1-X)
lbmol/ft3 Water B CBO= 3.47 lbmol/ft3 , B! =3.47 -CAOX CB = CAO( B!
-X) =(3.47-X) lbmol/ft3 Glycol C 0 CAOX CC = CAOX = X lbmol/ft3
Rate law: -rA = kCACB
65. 3-10 Therefore, -rA = k 2 AOC (1-X)( B! -X) =
k(1-X)(3.47-X) At 300K E = 12500 cal/mol, X = 0.9, k = 0.1dm3
/mol.s =1.6018 slbmolft ./3 CSTR! = AO A C X r! = ( )( ) ( )( ) (
)( ) sec186.2 9.047.39.0116018.1 9.01 2 = !! At 350K, k2 = k
exp((E/R)(1/T-1/T2))= 1.6018exp((12500/1.987)(1/300-1/350)) =
32.034 dm3 /mol.s Therefore, CSTR! = AO A C X r! = ( )( ) ( )( ) (
)( ) sec109.0 9.047.39.011034.32 9.01 2 = !! P3-11 (b) Isothermal,
isobaric gas-phase pyrolysis, C2H6 C2H4 + H2 A B + C Stoichiometric
table: Species symbol Entering Change Leaving C2H6 A FAO -FAOX
FA=FAO(1-X) C2H4 B 0 +FAOX FB=FAOX H2 C 0 +FAOX FC=FAOX FTO=FAO
FT=FAO(1+X) ! = yao ! = 1(1+1-1) = 1 v = vo(1+! X) => v =
vo(1+X) CAO = yAO CTO = yAO P RT = ( )( ) ( ) 3 1 6 0.082 1100 .
atm m atm K K kmol ! " # $ % & = 0.067 kmol/m3 = 0.067 mol/dm3
CA = ( ) ( ) 1(1 ) (1 ) 1 AOA AO O XF XF C v v X X !! = = + +
mol/dm3 CB = ( ) ( ) (1 ) 1 AOB AO O F XF X C v v X X = = + +
mol/dm3 CC = ( ) ( ) (1 ) 1 C AO AO O F F X X C v v X X = = + +
mol/dm3 Rate law: -rA = kCA = kCAO ( ) ( ) 1 1 X X ! + =0.067 k ( )
( ) 1 1 X X ! + If the reaction is carried out in a constant volume
batch reactor, =>( ! = 0) CA = CAO(1-X) mol/dm3 CB = CAO X
mol/dm3 CC = CAO X mol/dm3 P3-11 (c)
66. 3-11 Isothermal, isobaric, catalytic gas phase oxidation,
C2H4 + 1 2 O2 C2H4O A + 1 2 B C Stoichiometric table: Species
Symbol Entering Change Leaving C2H4 A FAO -FAOX FA=FAO(1-X) O2 B
FBO - B! FAOX FB=FAO( B! -X) C2H4O C 0 +FAOX FC=FAOX B! = 1 12 2 AO
BO AO AO FF F F = = 2 3 AO AO AO TO AO BO F F y F F F = = = + 2 1 1
1 0.33 3 2 AOy! " # $ = = % % =& ' ( ) ( ) ( ) 33 62 0.092 3 .
0.082 533 . AO AO TO AO atmP mol C y C y RT dmatm dm K mol K = = =
= ! " # $ % & ( ) ( ) ( ) ( ) ( ) ( ) 1 1 0.092 1 1 1 0.33 1
0.33 AO AOA A O F X C X XF C v v X X X! " " " = = = = + " " ( ) ( )
( ) 0.046 12 1 1 0.33 AO B B B O X F XF C v v X X ! " # $ %& '
%( )= = = + % ( ) ( ) ( ) 0.092 1 1 0.33 C AO C O XF F X C v v X X!
= = = + " If the reaction follow elementary rate law Rate law: 0.5
A A Br kC C! = ! ( ) ( ) ( ) ( ) 0.5 0.092 1 0.046 1 1 0.33 1 0.33
A X X r k X X ! "! "# #$ $$ $ # = % &% & # #$ $$ $' (' (
P3-11 (d) Isothermal, isobaric, catalytic gas phase reaction in a
PBR C6H6 + 2H2 C6H10 A + 2B C Stoichiometric table: Species Symbol
Entering Change Leaving Benzene A FAO -FAOX FA=FAO(1-X) H2 B
FBO=2FAO -2FAOX FB=FAO( B! -2X) C6H10 C 0 FAOX FC=FAOX
67. 3-12 ( ) 2 2 1 3 1 2 1 2 1 3 3 BO AO B AO AO AO AO AO TO AO
BO AO F F F F F F y F F F y ! " # = = = = = = + = = $ $ = $ =AOC (
) 3 3 1 6 1 0.055 / 3 3. 0.082 443.2 . TO AO P atm CAO C y mol dm
RT atm dm K mol K ! " ! " = = = =# $ # $ ! "% & % & # $ %
& ( ) ( ) ( ) ( )1 1 0.055 1 2 21 1 1 3 3 AO AOA A O F X C X XF
C v v X X X ! " " " = = = = + # $ # $ " "% & % & ' ( ' ( (
) ( ) ( )2 0.11 1 21 1 3 AO BB B O F X XF C v v X X ! " # # = = = +
$ % #& ' ( ) ( ) 0.055 2 21 1 1 3 3 C AO AO C O F F X C X X C v
v X X X ! = = = = + " # " # $ $% & % & ' ( ' ( If the
reaction follow elementary rate law. Rate law: ( ) 2 3 3 ' 1 '
0.0007 2 1 3 A A B A r kC C X r k X ! = ! ! = " # !$ % & ' For
a fluidized CSTR: 0 ' A A F X W r = ! ( ) 0 3 3 1 0.0007 2 1 3 AF X
W X k X = ! " # !$ % & ' k = 53 3 min mol kgcat atm at 300K @ T
= 170oC 1 1 1 1 80000 1 1 exp 53exp 8.314 300 443 E k k R T T ! "!
" ! "! " = # = #$ %$ % $ %$ %$ % & '& '& '& ' =
1663000 3 min mol kgcat atm
68. 3-13 FA0 = CA0* v0 v0 = 5 dm3 /min ( ) 0 0 3 3 1 0.0007 2 1
3 AC v X W X k X = ! " # !$ % & ' at X = 0.8 3 104.2 ! "=W kg
of catalyst @ T = 270oC 3 1 1 atmminkgcat mol 90787719 543 1 300 1
314.8 80000 exp53 11 exp !! ="" # $ %% & ' " # $ % & ' (="
" # $ % % & ' "" # $ %% & ' (= TTR E kk FA0 = CA0* v0 v0 =
5 dm3 /min ( ) 0 0 3 3 1 0.0007 2 1 3 AC v X W X k X = ! " # !$ %
& ' at X = 0.8 5 104.4 ! "=W kg of catalyst P3-12 C2H4 + 1 2 O2
C2H4O A + 1 2 B C Stoichiometric table for the given problem will
be as follows Assuming gas phase Species Symbol Entering Change
Leaving C2H4 A FA0 - FA0X FA0(1-X) O2 B FB0 = BFA0 -1/2 FA0X FA0(B
X/2) N2 I FI = IFA0 ------- FA0I C2H4O C 0 FA0X FA0X ! "B = 1 2 FA0
FA0 = 1 2 ! "I = FIO FAO ,FIO = 0.79 0.21 FBO ! " #I = #B 0.79 0.21
=1.88
69. 3-14 ! yA0 = FA0 FT 0 = 0.30, ! " = yA0# = $0.15 ! CA0 =
yA0P RT = 0.041 mol dm3 ! CA = FA0 v = CA0 (1" X) (1+ #X) =
0.041(1" X) 1" 0.15X ! CB = FB v = CA0 ( 1 2 " 1 2 X) 1" 0.15X =
0.020(1" X) 1" 0.15X ! CC = FC v = CA0X 1" 0.15X = 0.041X 1" 0.15X
P3-13 (a) Let A = ONCB C = Nibroanaline B = NH3 D = Ammonium
Chloride A + 2B C + D - A A Br kC C !!" = P3-13 (b) Species
Entering Change Leaving A FA0 - FA0X FA0(1-X) B FB0 = BFA0 =6.6/1.8
FA0 -2 FA0X FB= FA0(B 2X) C 0 FA0X FC= FA0X D 0 FA0X FD=FA0X P3-13
(c) For batch system, CA=NA/V -rA = kNANB/V2 P3-13 (d) - A A Br kC
C= ( ) ( ) ( )0 0 0 0 0 0 1 1 , 1AA A A A A A A A NN N F F F X C X
C C X V V V v v = = = ! = ! = = = ! ( ) ( ) ( )0 0 0 0 0 0 2 2 ,
2AB B B B B A B B A B NN N F F X C X C C X V V V v ! ! != = = " = "
= = " ( )( )2 0 1 2A A Br kC X X!" = " " 0 0 6.6 3.67 1.8 B B A C C
! = = =
70. 3-15 0 3 1.8A kmol C m = ( ) ( )( ) 2 1.8 1 3.67 2Ar k X X!
= ! ! P3-13 (e) 1) At X = 0 and T = 188C = 461 K min 0202.067.38.1
min 0017.0 3 2 3 3 2 00 m kmol m kmol kmol m kCr BAA =! " # $ %
& ='=( 0 3 kmol 0.0202 m min Ar! = 2) At X = 0 and T = 25C =
298K ! ! " # $ $ % & !! " # $$ % & '= TTR E kk O O 11 exp
min. 3 1003.2 298 1 461 1 . 987.1 11273 exp min. 0017.0 6 3 kmol m
Kmol cal mol cal kmol m k ! "= # # # # $ % & & & &
' ( # $ % & ' ( != -rAO = kCAOCBO = 2.41 X 10-5 kmol/m3 min 3)
0 0 1 1 exp E k k R T T ! "# $ = %& '( ) & '* +, - 3 11273
m 1 1 0.0017 exp kmolmin 461 5611.987 cal molk cal K K mol K ! " #
$% & = # ' $( ) # $* + # $, - 3 m 0.0152 kmolmin k = 0 0 0A A
Br kC C! = 3 3 3 m 0.0152 1.8 6.6 kmolmin A kmol kmol r m m ! "! "
# = $ %$ % & '& ' 3 kmol 0.1806 m min Ar! = P3-13 (f) rA =
kCAO 2 (1-X)(B-2X)
71. 3-16 At X = 0.90 and T = 188C = 461K 1) at T = 188 C = 461
K ( ) ( )( ) min 00103.0 9.0267.39.018.1 min. 0017.0 3 2 3 3 m kmol
m kmol kmol m rA = !!" # $ % & ' "" # $ %% & ' =! 2) At X =
0.90 and T = 25C = 298K ( ) ( )( ) min 1023.1 9.0267.39.018.1 min.
1003.2 3 6 2 3 3 6 m kmol m kmol kmol m rA ! ! "= !!# $ % & ' (
## $ % && ' ( "=! 3) At X = 0.90 and T = 288C = 561K ( ) (
)( ) min 0092.0 9.0267.39.018.1 min. 0152.0 3 2 3 3 m kmol m kmol
kmol m rA = !!" # $ % & ' "" # $ %% & ' =! P3-13 (g) FAO =
2 mol/min 1) For CSTR at 25o C -rA min 1023.1 3 6 m kmol! "= ( ) 3
3 3 9.0 60.162 min 1023.1 1.0min/2 1 m m mol mol r XF V XA AO = ! !
= " " = " =M 2)At 288o C, -rA min 0092.0 3 m kmol = ( ) 3 3 9.0
739.21 min 0092.0 1.0min/2 1 m m mol mol r XF V XA AO = ! = " " =
=M P3-14 C6H12O6 + aO2 + bNH3 c(C4.4H7.3N0.86O1.2) + dH2O + eCO2 To
calculate the yields of biomass, you must first balance the
reaction equation by finding the coefficients a, b, c, d, and e.
This can be done with mass balances on each element involved in the
reaction. Once all the coefficients are found, you can then
calculate the yield coefficients by simply assuming the
reaction
72. 3-17 proceeds to completion and calculating the ending mass
of the cells. P3-14 (a) Apply mass balance For C 6 = 4.4c + e For O
6 + 2a = 1.2c + d + 2e For N b = 0.86c For H 12 + 3b = 7.3c + 2d
Also for C, 6(2/3) = 4.4c which gives c = 0.909 Next we solve for e
using the other carbon balance 6 = 4.4 (0.909) + e e = 2 We can
solve for b using the nitrogen balance b = 0.86c = 0.86* (0.909) b
= 0.78 Next we use the hydrogen balance to solve for d 12 + 3b =
7.3c + 2d 12 + 3(0.78) = 7.3(0.909) + 2d d = 3.85 Finally we solve
for a using the oxygen balance 6 + 2a = 1.2c + d + 2e 6 + 2a =
1.2(0.909) + 3.85 + 2(2) a = 1.47 P3-14 (b) Assume 1 mole of
glucose (180 g) reacts: Yc/s= mass of cells / mass of glucose =
mass of cells / 180 g mass of cells = c*(molecular weight) = 0.909
mol* (91.34g/mol) mass of cells = 83.12 g Yc/s = 83.12 g / 180 g
Yc/s = 0.46 Yc/o2 = mass of cells / mass of O2 If we assume 1 mole
of glucose reacted, then 1.47 moles of O2 are needed and 83.12 g of
cells are produced. mass of O2 = 1.47 mol * (32 g/mol) mass of O2 =
47.04 g Yc/o2 = 83.12 g /47.04 g Yc/o2 =1.77 P3-15 (a) Isothermal
gas phase reaction.
73. 3-18 2 2 3 1 3 2 2 N H NH+ ! Making H2 as the basis of
calculation: 2 2 3 1 2 3 3 H N NH+ ! 1 2 3 3 A B C+ !
Stoichiometric table: Species Symbol Initial change Leaving H2 A
FAO -FAOX FA=FAO(1-X) N2 B FBO= B! FAO -FAOX/3 FB=FAO( B! -X/3) NH3
C 0 +2FAOX/3 FC=(2/3)FAOX P3-15 (b) 2 1 2 1 3 3 3 2 1 0.5 3 3 AOy !
" ! # $ = % % = %& ' ( ) # $ = = * % = %& ' ( ) ( ) ( ) 3
16.4 0.5 . 0.082 500 . AO atm C atm dm K mol K = ! " # $ % & =
0.2 mol/dm3 ( ) ( ) ( ) ( ) ( ) ( ) 2 3 3 3 1 0.2 1 0.1 / 1 1 3 1
0.22 2 0.1 / 3 1 3 1 3 AO H A AO NH C C X X C C mol dm XX C X X C C
mol dm XX ! ! " " = = = = + # $ "% & ' ( " = = ) = ) = + # $ "%
& ' ( P3-15 (c) kN2 = 40 dm3 /mol.s (1) For Flow system: ( ) (
) 2 2 2 2 31 2 2 31 2 2 2 1 13 40 1 1 3 3 N N N H AO r k C C X X C
X X ! " ! "# = $ % $ % ! " ! "& ' #( )* + * +#, -* + * += &
' & '* + * +# #( ) ( )* + * +, - , -$ % $ %
74. 3-19 ! "rN2 =1.6 1" X( ) 1" X 3 # $ % & ' ( ) * + + + +
, - . . . . 3 2 (2) For batch system, constant volume. ( ) ( ) [ ]
2 2 2 2 2 31 2 2 1 2 32 2 1 2 3 2 40 1 1 3 1.6 1 1 3 N N N H AO N r
k C C X C X X r X ! " ! "# = $ % $ % ! "& ' = # #! "( )* + $ %
, -$ % ! " # = # #* +$ % P3-16 (a) Liquid phase reaction assume
constant volume Rate Law (reversible reaction): C A A B C C r k C C
K ! " # = #$ % & ' Stoichiometry: ( )0 1A AC C X= ! , ( )0 1B
AC C X= ! , 0C AC C X= To find the equilibrium conversion, set -rA
= 0, combine stoichiometry and the rate law, and solve for Xe. A B
C CC C K C= ( ) 22 0 01A e C A eC X K C X! = 2 0 1 2 1 0e e A C X X
C K ! " # + + =$ % & ' 0.80eX = To find the equilibrium
concentrations, substitute the equilibrium conversion into the
stiochiometric relations. ( ) ( )0 3 3 1 2 1 0.80 0.4A A mol mol C
C X dm dm = ! = ! = ( ) ( )0 3 3 1 2 1 0.80 0.4B A mol mol C C X dm
dm = ! = ! = 0 3 3 2 *0.80 1.6A A mol mol C C X dm dm = = = P3-16
(b) Stoichiometry:
75. 3-20 ( )( )0 1 3 1 2Ay! "= = # = and 0C! = ( ) ( ) ( ) ( )
0 0 0 1 1 1 1 2 AA A A N X XN C C V V X X! " " = = = + + ( ) ( ) 0
0 0 3 3 1 1 2 C A C A N N X X C C V V X X! = = = + + Combine and
solve for Xe. ( ) ( ) ( ) 3 0 0 1 3 1 2 1 2 e e C A A e e X X K C C
X X ! "# = $ % + +$ %& ' ( )( ) 2 2 3 01 1 2 27C e e A eK X X C
X! + = 2 3027 4 3 1 0A e e C C X X K ! " # + + + =$ % & '
0.58eX = Equilibrium concentrations: ( ) 0 0 33 0 10 0.305 400
0.082 A P atm mol C RT dmdm atm K mol K = = = ! " # $ % & ( ) (
)( ) 3 1 0.58 0.305 0.059 1 2 0.58 A mol C dm ! = = + ( )( ) ( )( )
3 3 0.58 0.305 0.246 1 2 0.58 C mol C dm = = + P3-16 (c) Same
reaction, rate law, and initial concentration as part (b) gas
phase, batch reaction. Stoichiometry: ( ) ( )0 0 0 1 1AA A A N XN C
C X V V ! = = = ! 0 0 0 3 3C A C A N N X C C X V V = = = Combine
and solve for Xe ( ) ( ) 3 0 01 3C A e A eK C X C X! = 0.39eX =
Equilibrium concentrations
76. 3-21 ( )( ) 3 0.305 1 0.39 0.19A mol C dm = ! = ( )( ) 3
0.305 0.39 0.36C mol C dm = = P3-16 (d) Gas phase reaction in a
constant pressure, batch reactor Rate law (reversible reaction): 3
C A A C C r k C K ! " # = #$ % & ' Stoichiometry: ( )( )0 1 3 1
2Ay! "= = # = and 0C! = ( ) ( ) ( ) ( ) 0 0 0 1 1 1 1 2 AA A A N X
XN C C V V X X! " " = = = + + ( ) ( ) 0 0 0 3 3 1 1 2 C A C A N N X
X C C V V X X! = = = + + Combine and solve for Xe: ( ) 3 0 0 1 3 1
2 1 2 C A e A e e e K C X C X X X ! " # = $ % + +& ' 0.58eX =
Equilibrium concentrations: ( ) ( ) 3 0.305 1 0.58 0.059 1 2 0.58 A
mol C dm ! = = + ( )( ) ( ) 3 3 0.305 0.58 0.246 1 2 0.58 C mol C
dm = = + P3-17 Given: Gas phase reaction A + B 8C in a batch
reactor fitted with a piston such that V = 0.1P0 ( ) 2 3 2 1.0 sec
ft k lbmol = 2 A A Br kC C! = NA0 = NB0 at t = 0 V0 = 0.15 ft3 T =
140C = 600R = Constant
77. 3-22 P3-17 (a) 0 0 0 0 0.5A A A B N y N N = = + 8 1 1 6! =
" " = 0 3Ay! "= = Now ( )0 0 0 1 V P V X T P T != + " # $ % & '
and 0 1 T T = , 0 010P V= , and 10P V= Therefore ( ) 2 010 1 10 V V
X V != + or ( )2 2 0 1V V X!= + [ ]0 1A AN N X= ! [ ]0B A BN N X!=
" 0 0 1B B A N N ! = = [ ] ( ) 332 02 33 3 2 0 1 1 AA B A A B kN
XkN N r kC C V V X! " " = = = + 0 0 0 0 A A y P N V RT ! " = # $ %
& Therefore [ ] ( ) 3 0 0 3 2 1 1 A A Xy P r k RT X! "# $ " = %
& ' ( + [ ] ( ) 3 9 3 3 2 1 5.03*10 sec 1 3 A X lbmol r ft X !
! ! = + P3-17 (b) ( )2 2 0 1V V X!= + ( )2 2 0.2 0.15 1 X!= +
0.259X = 10 3 8.63*10 sec A lbmol r ft ! ! = P3-18 No solution will
be given. P3-19 No solution will be given.
78. 3-23 P3-20 No solution will be given. CDP3-A CDP3-B Polanyi
equation: E = C (-HR) We have to calculate E for the reaction CH3 +
RBr CH3Br + R Given: HR = - 6 kcal/mol From the given data table,
we get 6.8 = C (17.5) and 6.0 = C (20) => C = 12.4 KJ/mol and =
0.32 Using these values, and HR = - 6 kcal/mol, we get E = 10.48
KJ/mol CDP3-C (a) A B Rate law at low temperature: ! "rA = kCA The
rate law t higher temperature must: 1) Satisfy thermodynamics
relationships at equilibrium, and 2) Reduce to irreversible rate
law when the concentration of one or more of the reaction products
is zero. Also, We know, ! KC = CBe CAe Rearranging, we get ! CAe "
CBe KC = 0
79. 3-24 So, lets assume rate law as ! "rA = kA CA " CB KC # $
% & ' ( Also when CB = 0, it satisfies the given rate law.
Hence the proposed rate law is correct. CDP3-C (b) A + 2B 2D Rate
law at low temperature: ! "rA = kCA 1/2 CB Here, ! KC = CDe 2
CAeCBe 2 ! CAeCBe 2 " CDe 2 KC = 0 ! "rA = kA CAeCBe 2 " CDe 2 KC #
$ %% & ' (( But it does not satisfy the irreversible rate law
at low temperatures. Hence it is not correct So, taking square root
of KC ! KC = CDe CAe 1/2 CBe , ! CAe 1/2 CBe " CDe KC = 0 ! "rA =
kA CAe 1/2 CBe " CDe KC # $ %% & ' (( Which satisfies the
irreversible rate law. Hence it is the required rate law. CDP3-C
(c) A + B C + D Irreversible rate law: ! "rA '= kPAPB 1+ KAPA +
KBPB We know, ! KP = PC PD PBPA or ! PBPA " PC PD KP = 0 Hence
assume rate law as: ! "rA = k PAPB " PC PD KP # $ % & ' ( 1+
KAPA + KBPB + KC PC + KDPD Which satisfies both the above mentioned
conditions. CDP3-D
80. 3-25
81. 3-26
82. 3-27 CDP3-E
83. 3-28
84. 3-29 CDP3-F
85. 3-30 CDP3-G
86. 3-31 CDP3-H
87. 3-32
88. 3-33 CDP3-I
89. 3-34
90. 3-35
91. 3-36 CDP3-J Species Symbol Entering Change Leaving SiH4 A
FA0 3FA0 FA0(1X) NH3 B FB0 = BFA0 4FA0X FA0(4X/3) Si3N4 C 0 FA0X
FA0X/3 H2 D 0 12FA0X 4FA0X
95. 4-1 Solutions for Chapter 4 - Isothermal Reactor Design
P4-1 Individualized solution. P4-2 (a) Cooking food (effect of
temperature), removing of stains with bleach (effect of bleach
conc.), dissolution of sugar in coffee or tea. P4-2 (b) Example 4-1
There would be no error! The initial liquid phase concentration
remains the same. P4-2 (c) Example 4-2 For 50% conversion, X = 0.5
and k = 0.311min-1 FAO = 6.137 12.27 / min 0.5 CF lbmol X = = Also,
3 12.27 min AO AO AO F ft v C = = vo = vAo + vBo = 2vAo and also,
vAo = vBo vo= 24.52 ft3 /min Using Mole Balance, 24.52 0.5 78.93 (1
) 0.311(0.5) 592 ov X V k X gal ! = = = " = This is less volume
than Example 4-2 because the rate is higher. P4-2 (d) Example 4-3
For P = 60atm, CAO = 0.0415 lbmol/ft3 ( 198073.0 60 ! == O OAO AO
RT Py C ) Using equation E-4-3.6, for X = 0.8 We see that the only
thing that changes is CA0 and it increases by a factor of 10,
therby decreasing the volume by a factor of 10. ! V " 1 P P4-2 (e)
Example 4-4 New Dp = 3D0/4 Because the flow is turbulent ! "o # 1
Dp "1 = "2 Dp2 Dp1 = 0.0775 1 0.75 = 0.1033
96. 4-2 ( ) 1 2 1 12 2 2 0.103 60 2 1 1 0.24 10o atm ft L ft y
P atm ! " # $ $% &" # % &= ' = ' = '% & % &( ) %
& ( ) Now 0 2 1 !" o o P L# , so too much pressure drop P = 0
and the flow stops. P4-2 (f) Example 4-5 For without pressure drop,
conversion will remain same as example X = 0.82. With Pressure
drop, 1 0037.0 10 ! == kgo new " " For decrease in diameter, o =
20.01244[266.92+12920.8] = 334.75lbf/ft3 = 334.75(lbf/ft3
)1/144(ft2 /in2 )1/14.7(atm/( lbf/in2 )) = 0.15814atm/ft = 52.71
kPa/m For turbulent flow: ! " # 1 Dp and $ # 1 P0 % $2 = $1 P01Dp1
P02Dp2 1 2 4.6(0.949) 1 4.6(0.949) 1 1 2 AO o AO o kC W W v X kC W
W v ! ! " # $% & ' ( = = +" # + $% & ' ( X = 0.8136
virtually the same (2) Optimum diameter would be larger 152.71
0.037 0.0756 25.8 kg! " = # = if 2 1 1 2 P pD D= 1 2 1 1 2 2 p p D
D ! ! != = 2 11 1 1 (0.037)(27.5) 0.0175 2 W W ! !" = " = " = " Now
1-W < 0, too much pressure drop due to higher superficial
velocity.
97. 4-3 P4-2 (g) Example 4-6 For turbulent flow ! " # 1 Dp and
" # 1 P0 011 2 1 1 1 2 02 1 1 1 5 5 P P PD D P ! ! ! ! " # $ %" #"
# " # = = =$ %$ %$ % $ % & '& '& ' $ % & '
Therefore there is no change. P4-2 (h) Exmple 4-7 For pressure
doubled and temperature decrease CTO = 2*Po/RT and T = 688K See
Polymath program P4-2-h.pol. POLYMATH Results Calculated values of
the DEQ variables Variable initial value minimal value maximal
value final value V 0 0 1.0E-04 1.0E-04 Fa 2.26E-04 1.363E-05
2.26E-04 1.363E-05 Fb 0 0 2.124E-04 2.124E-04 Fc 0 0 1.062E-04
1.062E-04 E 2.4E+04 2.4E+04 2.4E+04 2.4E+04 T 688 688 688 688 Cto
0.573773 0.573773 0.573773 0.573773 Ft 2.26E-04 2.26E-04 3.322E-04
3.322E-04 Ca 0.573773 0.0236075 0.573773 0.0236075 k 213.40078
213.40078 213.40078 213.40078 ra -70.254837 -70.254837 -0.1189309
-0.1189309 Fao 2.26E-04 2.26E-04 2.26E-04 2.26E-04 rb 70.254837
0.1189309 70.254837 0.1189309 vo 3.939E-04 3.939E-04 3.939E-04
3.939E-04 rc 35.127419 0.0594654 35.127419 0.0594654 X 0 0
0.9395277 0.9395277 Tau 0 0 0.253044 0.253044 rateA 70.254837
0.1189309 70.254837 0.1189309 ODE Report (RKF45) Differential
equations as entered by the user [1] d(Fa)/d(V) = ra [2] d(Fb)/d(V)
= rb [3] d(Fc)/d(V) = rc
98. 4-4 Explicit equations as entered by the user [1] E = 24000
[2] T = 688 [3] Cto = 2*1641/8.314/T [4] Ft = Fa+Fb+Fc [5] Ca =
Cto*Fa/Ft [6] k = 0.29*exp(E/1.987*(1/500-1/T)) [7] ra = -k*Ca^2
[8] Fao = 0.000226 [9] rb = -ra [10] vo = Fao/Cto [11] rc = -ra/2
[12] X = 1-Fa/Fao [13] Tau = V/vo [14] rateA = -ra P4-2 (i) Example
4-8 Individualized solution. P4-2 (j) Example 4-9 Using trial and
error, we get maximum feed rate of B = 0.0251dm3 /s to keep
concentration of B 0.01mol/dm3 . See Polymath program P4-2-j.pol.
POLYMATH Results Calculated values of the DEQ variables Variable
initial value minimal value maximal value final value t 0 0 500 500
ca 0.05 0.0063485 0.05 0.0063485 cb 0 0 0.009981 0.009981 cc 0 0
0.0078965 0.0078965 cd 0 0 0.0078965 0.0078965 k 0.22 0.22 0.22
0.22 v00 0.0251 0.0251 0.0251 0.0251 cb0 0.025 0.025 0.025 0.025 v0
5 5 5 5 ca0 0.05 0.05 0.05 0.05 rate 0 0 3.91E-05 1.394E-05 v 5 5
17.55 17.55 x 0 0 0.5543321 0.5543321
99. 4-5 Differential equations as entered by the user [1]
d(ca)/d(t) = -k*ca*cb-v00*ca/v [2] d(cb)/d(t) =
-k*ca*cb+v00*(cb0-cb)/v [3] d(cc)/d(t) = k*ca*cb-v00*cc/v [4]
d(cd)/d(t) = k*ca*cb-v00*cd/v Explicit equations as entered by the
user [1] k = .22 [2] v00 = 0.0251 [3] cb0 = 0.025 [4] v0 = 5 [5]
ca0 = 0.05 [6] rate = k*ca*cb [7] v = v0+v00*t [8] x =
(ca0*v0-ca*v)/(ca0*v0) If the concentration of A is tripled the
maximum feed rate becomes 0.064 dm3 /s P4-2 (k through r)
Individualized solution. P4-3 Solution is in the decoding algorithm
given with the modules. P4-4 We have to find the time required to
cook spaghetti in Cuzco, Peru. Location Elevation (km) Pressure (mm
Hg) Boiling Point (C) Time (min) Ann Arbor 0.21 739 99.2 15 Boulder
1.63 625 94.6 17 Cuzco 3.416 493 88.3 ? Assume reaction is zero
order with respect to spaghetti conversion: E ART A dC r k Ae dt !
! = = = ! so that 0 A A E C RT C C Ae t ! " # ! = $ % & ' For
complete conversion (i.e.: well cooked) CA = 0 at time t. Therefore
0 0 E RT A E A RT C tAe C te A ! ! = = 0 1 ln ln lnA b C E k t A R
T ! " = = #$ % & '
100. 4-6 Now, plot the natural log of the cooking time versus
1/Tb and get a linear relationship. Extrapolation to Tb = 88.3C =
361.45 K yields t = 21 minutes. P4-5 (a) ( )( )22 0 1 10 AV kC X X
! =
101. 4-7 2 2 0 0 (1 )AOA A A kC XrdX dV F F !! = = 2 0 01 A A
kC VX X F = ! X=0.85 So, considering the above results, we will
choose a CSTR. P4-5 (b) P4-5 (c) P4-5 (d) 1) CSTR and PFR are
connected in series: 3 3 3 2 2 (200 )(0.07 / .min)(1 / ) (1 ) 10 /
min CSTR dm dm mol mol dm X X mol ! = Solving the quadratic
equation, XCSTR = 0.44 For PFR, 0A A dX F r dV = !
102. 4-8 3 2 (0.07 / / min) (1 )(1 ) 10 / min AOdm mol C X X dX
dV mole ! ! = 3 3 2 3 3 0.44 (0.07 / / min)(1 / ) (800 ) (1 ) 10 /
min X dX dm mol mol dm dm X mole = !" 0.736X = 2) when CSTR and PFR
are connected in parallel, 3 3 3 2 2 (200 )(0.07 / .min)(1 / ) (1 )
5 / min. CSTR dm dm mol mol dm X X mol ! = XCSTR = 0.56 For PFR, 3
3 2 3 2 0 (0.07 / .min)(1 / ) (800 ) (1 ) 5 / min X dX dm mol mol
dm dm X mol = !" XPFR = 0.92 Hence, final conversion X = 0.56 0.92
2 + =0.74 P4-5 (e) To process the same amount of species A, the
batch reactor must handle 3 5 60min 24 2 14400 min dm h mol M hr
day day ! " ! "! " =# $ # $# $ % &% &% & If the
reactants are in the same concentrations as in the flow reactors,
then 3 3 1 14400 14400 mol dm dm V day mol day ! "! " = =# $# $ %
&% & So the batch reactor must be able to process 14400 dm3
every 24 hours. Now we find the time required to reach 90%
conversion. Assume the reaction temperature is 300K. ( )2 0 0 0 1AA
A A kC X Vr VdX dt N N !! = = 0 2 0 1 A R A N X t VkC X = ! , and
since 0 0 A A N C V = 3 0 3 1 1 0.9 2.14 1 0.1 4.2 *1 R A X t hr dm
molkC X mol hr dm = = = ! " # $ % & ' (
103. 4-9 Assume that it takes three hours to fill, empty, and
heat to the reaction temperature. tf = 3 hours ttotal = tR + tf
ttotal = 2.14hours + 3 hours = 5.14 hours. Therefore, we can run 4
batches in a day and the necessary reactor volume is 3 314400 3600
4 dm dm= Referring to Table 1-1 and noting that 3600 dm3 is about
1000 gallons, we see that the price would be approximately $85,000
for the reactor. P4-5 (f) The points of the problem are: 1) To note
the significant differences in processing times at different
temperatures (i.e. compare part (b) and (c)). 2) That the reaction
is so fast at 77C that a batch reactor is not appropriate. One
minute to react and 180 to fill and empty. 3) Not to be confused by
irrelevant information. It does not matter if the reactor is red or
black. P4-6 (a)
104. 4-10 P4-6 (b) To increase conversion, use PFR, higher
temperature, or use better catalyst. P4-6 (c) P4-6 (d)
106. 4-12 P4-7 (a) Elementary gas phase reaction. P4-7 (b) P4-7
(c) For = 0.001dm-3 See Polymath program P4-7-c.pol.
107. 4-13 POLYMATH Results Calculated values of the DEQ
variables Variable initial value minimal value maximal value final
value v 0 0 500 500 x 0 0 0.656431 0.656431 y 1 0.1721111 1
0.1721111 Co 0.3 0.3 0.3 0.3 esp 2 2 2 2 alfa 0.001 0.001 0.001
0.001 C 0.3 0.0077768 0.3 0.0077768 k 0.044 0.044 0.044 0.044 r
-0.0132 -0.0132 -3.422E-04 -3.422E-04 fo 2.5 2.5 2.5 2.5
Differential equations as entered by the user [1] d(x)/d(v) = -r/fo
[2] d(y)/d(v) = -alfa*(1+esp*x)/(2*y) Explicit equations as entered
by the user [1] Co = 0.3 [2] esp = 2 [3] alfa = 0.001 [4] C =
Co*(1-x)*y/(1+esp*x) [5] k = 0.044 [6] r = -k*C [7] fo = 2.5 At V =
500, x = 0.66, y = 0.17 P4-7 (d) Individualized solution P4-7 (e) A
B + 2C
108. 4-14 Using these equations in Polymath we get the volume
to be 290 dm3 . P4-7 (f) Using Polymath to solve the differential
equation gives a volume of 290 dm3 See Polymath program
P4-7-f.pol.
109. 4-15 POLYMATH Results Calculated values of the DEQ
variables Variable initial value minimal value maximal value final
value X 0 0 0.47 0.47 V 0 0 290.23883 290.23883 Kc 0.025 0.025
0.025 0.025 Fao 2.5 2.5 2.5 2.5 Cao 0.3 0.3 0.3 0.3 k 0.044 0.044
0.044 0.044 e 2 2 2 2 ra -0.0132 -0.0132 -9.391E-04 -9.391E-04 ODE
Report (RKF45) Differential equations as entered by the user [1]
d(V)/d(X) = Fao/(-ra) Explicit equations as entered by the user [1]
Kc = .025 [2] Fao = 2.5 [3] Cao = .3 [4] k = .044 [5] e = 2 [6] ra
= -(k*Cao/(1+e*X))*((1-X)-(4*Cao^2*X^3)/((1+e*X)^2*Kc)) PFR with
pressure drop: Alter the Polymath equations from part (c). See
Polymath program P4-7-f-pressure.pol. POLYMATH Results Calculated
values of the DEQ variables Variable initial value minimal value
maximal value final value v 0 0 500 500 x 0 0 0.5077714 0.5077714 y
1 0.3181585 1 0.3181585 Kc 0.025 0.025 0.025 0.025 alfa 0.001 0.001
0.001 0.001 Cao 0.3 0.3 0.3 0.3 k 0.044 0.044 0.044 0.044 esp 2 2 2
2 fo 2.5 2.5 2.5 2.5 r -0.0132 -0.0132 -1.846E-04 -1.846E-04
110. 4-16 ODE Report (STIFF) Differential equations as entered
by the user [1] d(x)/d(v) = -r/fo [2] d(y)/d(v) =
-alfa*(1+esp*x)/(2*y) Explicit equations as entered by the user [1]
Kc = .025 [2] alfa = 0.001 [3] Cao = 0.3 [4] k = 0.044 [5] esp = 2
[6] fo = 2.5 [7] r =
-(k*Cao/(1+esp*x))*(4*Cao^2*x^3/((1+esp*x)^2*Kc)) At V = 500 dm3 X
= 0.507 and y = 0.381 P4-7 (g) Membrane reactor: A B + 2C CA =
COFA/FT CB = COFB/FT CC = COFC/FT FT = FA + FB + FC and -rA = rB =
rC/2 Using polymath, For PFR, See Polymath program P4-7-g.pol.
POLYMATH Results Calculated values of the DEQ variables Variable
initial value minimal value maximal value final value v 0 0 1040
1040 Fa 2.5 1.3231889 2.5 1.3231889 Fb 0 0 0.3635477 0.0684325 Fc 0
0 2.3536223 2.3536223 Kc 0.025 0.025 0.025 0.025 Ft 2.5 2.5
3.7452437 3.7452437 Co 0.3 0.3 0.3 0.3 K 0.044 0.044 0.044 0.044 kc
0.08 0.08 0.08 0.08 ra -0.0132 -0.0132 -3.827E-04 -3.827E-04 X 0 0
0.4707245 0.4707245 Differential equations as entered by the user
[1] d(Fa)/d(v) = ra [2] d(Fb)/d(v) = -ra - kc*Co*Fb/Ft [3]
d(Fc)/d(v) = -2*ra Explicit equations as entered by the user [1] Kc
= 0.025 [2] Ft = Fa+ Fb+ Fc [3] Co = 0.3 [4] K = 0.044 [5] kc = .08
[6] ra = - (K*Co)*(Fa/Ft- Co^2*Fb*Fc^2/(Kc*Ft^2)) [7] X = 1 -
Fa/2.5 Solving for when X = 0.47, we get V = 1040 dm3
111. 4-17 P4-8 (a) The blades makes two equal volumes zones of
500gal each rather than one big mixing zone of 1000gal.
112. 4-18 P4-8 (b) A CSTR is been created at the bend due to
backmixing, so the effective arrangement is a PFR is in series with
a CSTR. A B k = 5 min-1 vo = 5 dm3 /min. Xexpected = 0.632 Xactual
= 0.586 0 1 ln 1 X AO o A Expected F dX v V r k X ! " = = # $ % %#
$& ' ( = 5 1 ln 1.0 5 1 .632 ! " =# $ %& ' Now, For PFR, 1
1 ln 1 PV X ! " = # $ %& ' .1 For CSTR, ( ) ( ) 11( ) 1
actualAO actual C A actual X XF X X V r X !! = = ! ! 2 CSTR zone
created due to backmixing
113. 4-19 Also, 1CP VV V V + = ..3 Solving 1, 2 and 3 by using
polymath, See Polymath program P4-8-b.pol. POLYMATH Results NLES
Solution Variable Value f(x) Ini Guess X1 0.350949 3.148E-10 0 Vc
0.567756 -3.297E-14 1 V 1 X2 0.586 Vp 0.432244 NLES Report
(safenewt) Nonlinear equations [1] f(X1) = ln(1/(1-X1)) -Vp = 0 [2]
f(Vc) = (X2-X1)/(1-X2) - Vc = 0 Explicit equations [1] V = 1 [2] X2
= .586 [3] Vp = V - Vc VC = 0.57 dm3 ; VP = 0.43 dm3 ; X1 = 0.35
P4-8 (c) CAO = 2 mol/dm3 A !B Assuming 1st order reaction, For
CSTR, AO A C X r ! = " -rA = kCA = kCAO(1-X) => 0.4 0.67 1 0.6 X
k X ! = = = " For PFR, ( )0 00 1 X A A dX V F kC X = !" , 0 1 X dX
k X ! = "# => 1 exp( )PFRX k!= " " =1-exp(-0.67) = 0.486 Now
assuming 2nd order reaction, For CSTR, Now, assuming 2nd order
reaction, For CSTR, AO A C X r ! = " 2 A Ar kC! = = ( ) 22 1AOkC X!
=> ( ) 2 2 0.4 1.111 0.61 AO X kC X ! = = = " For PFR, ( ) 2 0 1
1 11 X AO AO dX X kC kC XX ! = = "" #
114. 4-20 => 1 1 1 1 .526 1 2.111AO X kC! = " = " = + So,
while calculating PFR conversion they considered reaction to be 1st
order. But actually it is a second order reaction. P4-8 (d) A graph
between conversion and particle size is as follows: Originally we
are at point A in graph, when particle size is decreased by 15%, we
move to point B, which have same conversion as particle size at A.
But when we decrease the particle size by 20%, we reach at point C,
so a decrease in conversion is noticed. Also when we increase the
particle size from position A, we reach at point D, again there is
a decrease in the conversion. P4-9 A ! B To = 300K KCO (300K)= 3.0
V = 1000gal = 3785.4 dm3 Mole balance: A AO r XF V ! = Rate law: !
" # $ % & '=' C B AOA K C Ckr 2 2 Stoichiometry: ( )XCC AOA !=
1 and XCC BOB = ! ( ) ! " # $ % & '' = C AOO AO K X X X Ck F V
2 2 2 1 ( ) 2 2 2 2 2 1 3785.4 0.4 0.6 0.4 3 CAO O AO X V X KF Z k
C X ! " # #$ % ! "& '= = = #$ % & ' 3 2902.2Z dm= , ! Z = k
ko = exp E R 1 T0 " 1 T # $ % & ' ( ) * + , - .
115. 4-21 Now using: ( ) ( ) ( ) 2 2 2 2 0 1 1 C C Z X Z X V f
X V z zX X X X K K ! " ! " = # = = $% & % & ' ( ' () * ) *
$ $ $ $+ , + , - . - . where 1 1 exp O O k E z k R T T ! "# $ = =
%& '( )& ' * +, - and ! ! " # $ $ % & ' ( ) * + , - . =
TTR H KK O RX COC 11 exp Solving using polymath to get a table of
values of X Vs T. See Polymath program P4-9.pol. POLYMATH Results
NLE Solution Variable Value f(x) Ini Guess X 0.4229453 3.638E-12
0.5 To 300 T 305.5 z 2902.2 V 3785.4 E 1.5E+04 R 2 y 1.5684405 Kco
3 Hrx -2.5E+04 Kc 1.4169064 NLE Report (safenewt) Nonlinear
equations [1] f(X) = (z/y)*X/((1-X)^2 - X^2/Kc) -V = 0 Explicit
equations [1] To = 300 [2] T = 305.5 [3] z = 2902.2 [4] V = 3785.4
[5] E = 15000 [6] R = 2 [7] y = exp(E/R*(1/To-1/T)) [8] Kco = 3 [9]
Hrx = -25000 [10] Kc = Kco*exp(Hrx/R*(1/To-1/T)) 0.36 0.37 0.38
0.39 0.4 0.41 0.42 0.43 295 300 305 310 315 320 Temperature X
116. 4-22 T(in K) X 300 0.40 301 0.4075 303 0.4182 304 0.4213
305 0.4228 305.5 0.4229 305.9 0.4227 307 0.421 310 0.4072 315
0.3635 We get maximum X = 0.4229 at T = 305.5 K. P4-10 (a) For
substrate: Cell: voCC = rgV FSO FS + rsV = 0 (CSO CS)vO = rg VYS/C
= VYS/C max S C M S C C K C ! " # $ +% & P4-10 (b) [ ]SSOSCC
CCYC != / ( ) / 0MAX S SO S O S C C M S C C C v VY C K C ! " # # =$
% +& ' ! ( ) ( )/ 0MAX S SO S O C S SO S M S C C C v VY C C K C
! " # # # =$ % +& ' ( ) ( ) 0.5 30 5 25 0.8 30 0 5 S S S S C C
C C ! "# $ $ # # $ =% & +' ( Solving we get CS = 5.0 g/dm3 or
30 g/dm3 . if CS = CS0 no reaction has occurred so the only valid
answer is CS = 0.5 g/dm3 . P4-10 (c) CC = YC/S(CSO - CS) = 0.8(30
5.0)g/dm3 = 20 g/dm3 . P4-10 (d) vnew = vo/2 = 2.5 dm3 /h Using
equation from above, we get CS = 1.67 g/dm3 and CC = 22.67
g/dm3
117. 4-23 P4-10 (e) Vnew = Vo/3 = 25/3 dm3 Using equation from
above, we get CS = 3.0 g/dm3 and CC = 21.6 g/dm3 P4-10 (f) For
batch reactor: CSO = 30 g/dm3 CCO = 0.1 g/dm3 CC = CC0 + YC/S(CSO -
CS) V = 10 dm3 See Polymath program P4-10-f.pol. POLYMATH Results
Calculated values of the DEQ variables Variable initial value
minimal value maximal value final value t 0 0 15 15 Cs 30 0.0382152
30 0.0382152 Cso 30 30 30 30 Ycs 0.8 0.8 0.8 0.8 Km 5 5 5 5 Umax
0.5 0.5 0.5 0.5 Cco 0.1 0.1 0.1 0.1 Cc 0.1 0.1 24.069428 24.069428
rg 0.0428571 0.0428571 5.4444349 0.0912841 rs -0.0535714 -6.8055436
-0.0535714 -0.1141052 negative_ 0.0535714 0.0535714 6.8055436
0.1141052 ODE Report (RKF45) Differential equations as entered by
the user [1] d(Cs)/d(t) = rs Explicit equations as entered by the
user [1] Cso = 30 [2] Ycs = 0.8 [3] Km = 5 [4] Umax = 0.5 [5] Cco =
0.1 [6] Cc = Cco+Ycs*(Cso-Cs) [7] rg = (Umax*Cs/(Km+Cs))*Cc [8] rs
= -(1/Ycs)*rg [9] negative_rs = -rs
118. 4-24 P4-10 (g) Graphs should look the same as part (f)
since reactor volume is not in the design equations for a constant
volume batch reactor. P4-11 Gaseous reactant in a tubular reactor:
A B 1 0.0015min at 80 F A Ar kC k ! ! = = o 25,000 cal E g mol =
0.90X = 1000B lb M hr = 58A B lb MW MW lbmol = = 1 inch (I.D.)tD =
10L ft= 132 146.7P psig psia= = 260 720T F R= =o o number of
tubestn = 1000 17.21 58 B lb lbmolhrF lb hr lbmol = = 0 17.21 19.1
0.9 B A lbmol F lbmolhrF X hr = = = For a plug flow reactor: 0.92 0
0 4 t t A A n D L dX V F r ! = = "# 1 1 0! = " = 1.0Ay = 0Ay! "= =
( )0 1A Ar kC X! = ! 0 A A P P C RT RT = = ( ) 0.9 0.9 0 0 0 0 0 00
0 1 ln ln10 1 1 0.9 A A A A A A A F F RTdX dX V F F r kC X kC kP !
" = = = =# $ % % %& ' ( ( At T2 = 260F = 720R, with k1 = 0.0015
min-1 at T1 = 80F = 540R, 1 2 1 1 2 1 1 25000 1 1 exp 0.0015exp
53.6min 1.104 540 720 E k k R T T ! " #" # " #" # = ! = ! =$ %$ % $
%$ %$ % & '& '& '& ' 1 1 2 53.6min 3219k hr! ! = =
( ) ( )( ) 3 0 1 19.1 10.73 720 ln10 ln10 3219 146.7 A lbmol ft
psia R hr lbmol RF RT V kP hr psia! " #" # $ %$ % & '& '= =
o o 3 0.72V ft= 2 4 t tn D L V ! =
119. 4-25 ( ) ( ) 3 22 4 0.724 13.2 1 10 12 t t ftV n D L ft ft
! ! = = " # $ % & ' Therefore 14 pipes are necessary. P4-12 A
B/2 AO AO TOC y C= 2 2 2 2 (1 1/ 4) 0.8 2( 1/ 4)(1 1/ 4)ln(1 0.8) (
1/ 4) 0.8 2.9 1 0.8 PFR AO TO AO V ky C F ! = ! ! ! + ! + = !
.(6)
120. 4-26 2 2 2 ' ' ' PFR AO TO AO V ky C F = ( ) 2 2 8 8 2.9
2.58 9 9 PFR AO TO AO V ky C F = = = P4-13 Given: The metal
catalyzed isomerization BA ! , liquid phase reaction 1 B A A eq C r
k C K ! " # = #$ %$ % & ' with Keq = 5.8 For a plug flow
reactor with yA = 1.0, X1 = 0.55 Case 1: an identical plug flow
reactor connected in series with the original reactor. Since yA =
1.0, B = 0. For a liquid phase reaction ( )A A0C C 1 X= ! and B A0C
C X= ( )0 1A A eq X r kC X K ! " # = # #$ %$ % & ' For the
first reactor, ( ) 1 1 1 0 0 0 0 0 1 X X A A A A eq dX dX V F F r X
kC X K = = ! " # ! !$ %$ % & ' ( ( or 1 1 0 1 0 0 0 1 1 ln 1 1
11 11 1 X X A A eq eq eq kC V dX X F K X KK ! "# $ = = % % +&
'( )( )# $ & '* ++ , -% +( )( ) * + .
121. 4-27 ( )0 1 1 0 1 1 ln 1 1 0.853ln .355 0.883 1 1 A A eq
eq kC V X F K K ! "# $ = % % + = % =& '( )( )& '* ++ , -
Take advantage of the fact that two PFRs in series is the same as
one PFR with the volume of the two combined. VF = V1 + V2 = 2V1 and
at VF X = X2 0 0 0 0 1 1 ln 1 1 11 11 1 F F X X A F A eq eq eq kC V
dX X F K X KK ! "# $ = = % % +& '( )( )# $ & '* ++ , -% +(
)( ) * + . 0 0 1 2 0 0 1 1 2 ln 1 1 1 1 A F A A A eq eq kC V kC V X
F F K K ! "# $ = = % % +& '( )( )& '* ++ , - ( )0 1 0 2 2
0.883 1.766A A kC V F = = 2 1 1 1.766 ln 1 1 1 5.81 5.8 X ! "# $ =
% % +& '( ) * +, -+ X2 = 0.74 Case 2: Products from 1st reactor
are separated and pure A is fed to the second reactor, The analysis
for the first reactor is the same as for case 1. 0 1 1 0 1 1 ln 1 1
1 1 A A eq eq kC V X F K K ! "# $ = % % +& '( )( )& '* ++ ,
- By performing a material balance on the separator, FA0,2 =
FA0(1-X1)