138 CHEMICAL ENGINEERING
139 CHEMICAL ENGINEERING
Continuous-Stirred Tank Reactors in Series
With a first-order reaction A → R, no change in volume.
τN-reactors = Nτindividual k , N
C
C 1 where
AN A
N 0
1
= >e o - H
N = number of CSTRs (equal volume) in series, and CAN = concentration of A leaving the Nth CSTR.
Two Irreversible Reactions in Parallel desired
undesired / / /
/
/
D
A D
A U
r dc dt k
dc dt k dc dt k
d d
D
D
instantaneous fractional yield of
overall fractional yield of
where and are measured at the outlet of the flow reactor.
overall selectivity to
k
k
A A D A
x
U A
y
D D D A
x
U U U A
y
D
D A
D
D A A
A D
DU
D U
D
U
f f
f f
f f
0
C k C
r C
r C
Y
C C
Y
N N N
N N
S
N N
"
"
- = - = +
= =
= =
=
=
-=
=
-=
=
^
^
_
a h
h
i
k
Two First-Order Irreversible Reactions in Series
1 ln
1 1
1
A D U
r dC /dt k dC /dt k dC /dt k
D
/
Yield and selectivity definitions are identical to those for two irreversible reactions in parallel. The optimal yield of in a PFR is
at time
The optimal yield of D in a CSTR is
at time
max log
,max
1 2 2
max
k k
A A D A
D D D A U D
U U U D
A D,max
U D
U D
U D
A D
U D
D U
mean 0
0
D U
U U D
C
r C k C
r C
C C
k k
k k k
k k
C C
k k
k k
k k k
" "
=− =
= = −
= =
=
= =
−
= +
=
−
x
x
e
_ _
` o `
i
i j
j
9 C
Reactions of Shifting Order
ln ln
r k C
k C
C
C k C C k t
C C
C C
k C C
k t
A 1
A A
A A
A A
A A
A A
A A
2 1
2 1
2 1
o
o
o o
o
- =
+
+ - =
- = - +
-e _
`
o i
j
This form of the rate equation is used for elementary enzyme-catalyzed reactions and for elementary surfaced-catalyzed reactions.
Batch Reactor, Variable Volume
If the volume of the reacting mass varies with the conversion (such as a variable-volume batch reactor) according to
V= VXA=0^1+ fA AXh (i.e., at constant pressure), where
V
V V
V
A V
X
X X
A X
A A
A 0
1 0
0
=
-f = D
=
= =
=
then at any time
C C
X X 1
1
A A
A A A
= 0 +
-< f F
and
/
t= -CA0 0#XAdXA 8^1+ fA AXh^-rAhB For a first-order irreversible reaction,
ln ln
kt X
V
1 A 1 V
A XA 0
= - - = - - f D
^ h c = m
Flow Reactors, Steady State
Space-time τ is defined as the reactor volume divided by the inlet volumetric feed rate. Space-velocity SV is the reciprocal of space-time, SV = 1/τ.
Plug-Flow Reactor (PFR) F ,
C V C
r
dX where
A
A PFR
A
A X A 0
0
0 0
= = A
x
-# ^ h FA0 = moles of A fed per unit time.
Continuous-Stirred Tank Reactor (CSTR) For a constant volume, well-mixed CSTR
C F ,
V
r
X where
A A
CSTR A A
0 = 0 =
-x
– rA is evaluated at exit stream conditions.
140 CHEMICAL ENGINEERING
MASS TRANSFER Diffusion
Molecular Diffusion N P RT
p N N D p
N x N N CD z
x Gas:
Liquid:
A A
A B m A
A A A B m A
2
2 2
= +
-= +
-2z _
_ i
i where,
Ni = molar flux of component i P = pressure
pi = partial pressure of component i Dm = mass diffusivity
R = universal gas constant T = temperature
z = length
Unidirectional Diffusion of a Gas A Through a Second Stagnant Gas B (Nb= 0)
N R
T p D P
z z
p p
2 1
2 1
A
B lm
m A A
= # −−
_
` i
j
in which (pB)lm is the log mean of pB2 and pB1
ln p
p p
p p
BM lm
B B
B B
1 2
2 1
=
-_ i c
m
Ni = diffusive flux [mole/(time × area)] of component i through area A, in z direction
Dm = mass diffusivity
pI = partial pressure of species I C = concentration (mole/volume) (z2 – z1)= diffusion flow path length Equimolar Counter-Diffusion (Gases) (NB = –NA)
N D RT p p z
N D C C z
A m A A
A m A A
1 2
1 2
= #
-=
-D D
] _ ^
^
g i h
h
8 B
Convection
Two-Film Theory (for Equimolar Counter-Diffusion) NA = k'G (pAG – pAi)
= k'L (CAi – CAL) = K'G (pAG – pA*) = K'L (CA* – CAL) where,
NA = molar flux of component A k'G = gas phase mass transfer coefficient k'L = liquid phase mass transfer coefficient K'G = overall gas phase mass transfer coefficient K'L = overall liquid phase mass transfer coefficient
pAG = partial pressure in component A in the bulk gas phase pAi = partial pressure at component A at the gas-liquid
interface
CAi = concentration (mole/volume) of component A in the liquid phase at the gas-liquid interface
CAL = concentration of component A in the bulk liquid phase pA* = partial pressure of component A in equilibrium with CAL CA*= concentration of component A in equilibrium with the
bulk gas vapor composition of A Overall Coefficients
1/K'G = 1/k'G + H/k'L 1/K'L = 1/Hk'G + 1/k'L
H = Henry's Law constant where pA* = H CAL and CA* = pAG/H Dimensionless Group Equation (Sherwood)
For the turbulent flow inside a tube the Sherwood number D .
k D DV
0 023 D Sh
.
m m
m
0 8 1 3
= = nt
tn
d n c m d n
where,
D = inside diameter Dm = diffusion coefficient V = average velocity in the tube ρ = fluid density
µ = fluid viscosity
km = mass transfer coefficient
141 CHEMICAL ENGINEERING
Continuous Distillation (Binary System) Constant molal overflow is assumed.
Equilibrium stages numbered from top.
Overall Material Balances Total Material:
F = D + B Component A:
FzF = DxD + BxB Operating Lines
Rectifying section Total Material:
Vn+1 = Ln + D Component A:
Vn+1 yn+1 = Lnxn + DxD
yn+1 = [Ln /(Ln + D)] xn + DxD /(Ln + D) Stripping section
Total Material:
Lm = Vm+1 + B Component A:
Lmxm = Vm+1ym+1 + BxB
ym+1 = [Lm /(Lm – B)] xm – BxB /(Lm – B) Reflux ratio
Ratio of reflux to overhead product RD = LR/D = (VR – D)/D
Minimum reflux ratio is defined as that value which results in an infinite number of contact stages. For a binary system the equation of the operating line is
y R
R x
R x
1 1
min min
min
= + + D+
Feed condition line
slope = q/(q – 1), where
q molar heat of vaporization
heat to convert one mol of feed to saturated vapor
=
SATURATED LIQUID q = 1
SUBCOOLED LIQUID q > 1
y = x LIQUID
+ VAPOR 0 <
q < 1
SATURATED VAPOR q = 0
SUPERHEA TED VAPOR q < 0
q-LINE SLOPES
FEEDCOMPOSITION
Distillation Definitions:
α = relative volatility
B = molar bottoms-product rate D = molar overhead-product rate F = molar feed rate
L = molar liquid downflow rate RD = ratio of reflux to overhead product V = molar vapor upflow rate
W = total moles in still pot
x = mole fraction of the more volatile component in the liquid phase
y = mole fraction of the more volatile component in the vapor phase
Subscripts:
B = bottoms product D = overhead product F = feed
m = any plate in stripping section of column m+1= plate below plate m
n = any plate in rectifying section of column n+1 = plate below plate n
o = original charge in still pot Flash (or equilibrium) Distillation Component material balance:
FzF = yV + xL Overall material balance:
F = V + L
Differential (Simple or Rayleigh) Distillation ln W
W
ydxx
o x
x
= o
-c m #
When the relative volatility α is constant, y = αx/[1 + (α – 1) x]
can be substituted to give
ln ln ln
W W
x x
x x
x x 1
1
1 1
1 1
o o
o o
= -
-- +
-c m ^a h > _^ hiH ; E
For binary system following Raoult's Law α = (y/x)a /(y/x)b = pa /pb, where pi = partial pressure of component i.
142 CHEMICAL ENGINEERING
Murphree plate efficiency
/ ,
EME=_yn- yn+1i `y*n- yn+1j where
yn = concentration of vapor above equilibrium stage n yn+1 = concentration of vapor entering from equilibrium stage
below n
y*n = concentration of vapor in equilibrium with liquid leaving equilibrium stage n
R
Absorption (packed columns) Continuous Contact Columns
Z = NTUG • HTUG = NTUL • HTUL = NEQ • HETP Z = column height
NTUG = number of transfer units (gas phase) NTUL = number of transfer units (liquid phase) NEQ = number of equilibrium stages
HTUG = height of transfer unit (gas phase) HTUL = height of transfer unit (liquid phase) HETP = height equivalent to theoretical plate (stage)
HTU K a
G HTU
K a
G L
G
L L
= l = l
G = gas phase mass velocity (mass or moles/flow area • time)
L = liquid phase mass velocity (mass or moles/flow area • time)
KlG = overall gas phase mass transfer coefficient (mass or moles/mass transfer area • time) KlL = overall liquid phase mass transfer coefficient
(mass or moles/mass transfer area • time) a = mass transfer area/volume of column (length–1)
NTU
y y
dy NTU
x x
dx
* *
G y
y
L x
x
1 2
1 2
= - =
` j _ - i
# #
y = gas phase solute mole fraction x = liquid phase solute mole fraction y* = K • x, where K = equilibrium constant x* = y/K, where K = equilibrium constant y2, x2 = mole fractions at the lean end of column y1, x1 = mole fractions at the rich end of column
For dilute solutions (constant G/L and constant K value for entire column):
ln NTU
y y y y
y y
y y
y y
*
1 2
2 2
*
1 1
*
1 1
*
2 2
* G
LM
LM
y y
y y
= −
−
− =
−
−
− − −
) J
L KKKK KKK
`
`
b b
N
P OOOO OOO j
j
l l
For a chemically reacting system—absorbed solute reacts in the liquid phase—the preceding relation simplifies to:
ln
NTU y
y
G 2
= d n1
143 CHEMICAL ENGINEERING
Rate of Transfer as a Function of Gradients at the Wall Momentum Transfer:
dy
dv f V D
L p
8 4
w
w f
2
= - = - =
-x n t D
c m b lc m
Heat Transfer
A
Q k
dy dT
w w
o =
-d n c m
Mass Transfer in Dilute Solutions A
N D
dy dc
w m m
w
b l = - d n
Rate of Transfer in Terms of Coefficients Momentum Transfer
f V
w 8
2
x = t
Heat Transfer A
Q h T
w= D
d no
Mass Transfer A
N k c
w= mD m
b l
Use of Friction Factor (f ) to Predict Heat-Transfer and Mass Transfer Coefficients (Turbulent Flow)
Heat Transfer
Pr
j f
8 RePr
Nu
H
= b l 2 3=
Mass Transfer
j f
8 ReSc
Sh Sc
M
= b l 2 3=
TRANSPORT PHENOMENA-MOMENTUM, HEAT, AND MASS TRANSFER ANALOGY
For the equations which apply to turbulent flow in circular tubes, the following definitions apply:
Nu = Nusselt Number k
;hDE Pr = Prandtl Number (cpµ/k) Re = Reynolds Number (DVρ/µ) Sc = Schmidt Number [µ/(ρDm)]
Sh = Sherwood Number (kmD/Dm) St = Stanton Number [h/(cpG)]
cm = concentration (mol/m3)
cp = heat capacity of fluid [J/(kg•K)]
D = tube inside diameter (m) Dm = diffusion coefficient (m2/s)
(dcm/dy)w = concentration gradient at the wall (mol/m4) (dT/dy)w = temperature gradient at the wall (K/m) (dv/dy)w = velocity gradient at the wall (s–1) f = Moody friction factor
G = mass velocity [kg/(m2•s)]
h = heat-transfer coefficient at the wall [W/(m2•K)]
k = thermal conductivity of fluid [W/(m•K)]
km = mass-transfer coefficient (m/s)
L = length over which pressure drop occurs (m)
(N/A)w = inward mass-transfer flux at the wall [mol/(m2•s)]
Q Ao w
_ i = inward heat-transfer flux at the wall (W/m2) y = distance measured from inner wall toward centerline
(m)
∆cm = concentration difference between wall and bulk fluid (mol/m3)
∆T = temperature difference between wall and bulk fluid (K) µ = absolute dynamic viscosity (N•s/m2)
τw = shear stress (momentum flux) at the tube wall (N/m2) Definitions already introduced also apply.
144 CHEMICAL ENGINEERING
COST ESTIMATION Cost Indexes
Cost indexes are used to update historical cost data to the present. If a purchase cost is available for an item of equipment in year M, the equivalent current cost would be found by:
$ M
Current Cost in year M
Index in year Current Index
=_ id n
Capital Cost Estimation
Solid processing 4.7
Solid-fluid processing 5.0
4.0 4.3
Type of plant Fixed capital investment Total capital investment
Fluid processing 5.0 6.0
Lang factors
From Green, Don W., and Robert H. Perry, Perry’s Chemical Engineers’ Handbook, 8th ed., McGraw-Hill, 2008.
Adapted from M. S. Peters, K. D. Timmerhaus, and R. West, Plant Design and Economics for Chemical Engineers, 5th ed., McGraw-Hill, 2004.
e g n a R t
n e n o p m o C
Direct costs
Purchased equipment-delivered (including fabricated equipment and process machinery such as pumps and compressors)
100 7 4 – 9 3 n
o i t a l l a t s n i t n e m p i u q e -d e s a h c r u P
8 1 – 9 )
d e l l a t s n i ( s l o r t n o c d n a n o i t a t n e m u r t s n I
6 6 – 6 1 )
d e l l a t s n i ( g n i p i P
1 1 – 0 1 )
d e l l a t s n i ( l a c i r t c e l E
9 2 – 8 1 )
s e c i v r e s g n i d u l c n i ( s g n i d l i u B
3 1 – 0 1 s
t n e m e v o r p m i d r a Y
0 7 – 0 4 )
d e l l a t s n i ( s e i t i l i c a f e c i v r e S
6 )
d e r i u q e r s i e s a h c r u p f i ( d n a L
6 4 3 – 4 6 2 t
s o c t n a l p t c e r i d l a t o T
Indirect costs
3 3 – 2 3 n
o i s i v r e p u s d n a g n i r e e n i g n E
1 4 – 4 3 s
e s n e p x e n o i t c u r t s n o C
Total direct and indirect plant costs 336–420 Contractor's fee (about 5% of direct and
indirect plant costs) 17–21
Contingency (about 10% of direct and
indirect plant costs) 36–42
3 8 4 – 7 8 3 t
n e m t s e v n i l a t i p a c -d e x i F
Working capital (about 15% of total
capital investment) 68–86
9 6 5 – 5 5 4 t
n e m t s e v n i l a t i p a c l a t o T
145 CHEMICAL ENGINEERING
Scaling of Equipment Costs
The cost of Unit A at one capacity related to the cost of a similar Unit B with X times the capacity of Unit A is approximately Xn times the cost of Unit B.
Cost of Unit A Cost of Unit B Capacity of Unit B Capacity of Unit A n
= e o
Typical Exponents (n) for Equipment Cost vs. Capacity
t n e n o p x E e
g n a r e z i S t
n e m p i u q E
Dryer, drum, single vacuum 0.76
Dryer, drum, single atmospheric 0.40
l a g u f i r t n e c , n a
F 0.44
Fan, centrifugal 1.17
Heat exchanger, shell and tube, floating head, c.s. 0.60 Heat exchanger, shell and tube, fixed sheet, c.s. 0.44 Motor, squirrel cage, induction, 440 volts,
explosion proof
0.69
Motor, squirrel cage, induction, 440 volts, explosion proof
0.99
. s . c , p u c e l b b u b , y a r T
. s . c , e v e i s , y a r T
10 to 102 ft2 10 to 102 ft2 103 to 104 ft3/min 2 × 104 to 7 × 104 ft3/min 100 to 400 ft2
100 to 400 ft2 5 to 20 hp 20 to 200 hp 3- to 10-ft diameter
3- to 10-ft diameter 0.86 1.20
146 CIVIL ENGINEERING