Krantz W.B. Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach to Model Building and the Art of Approximation
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FLUID-WALL AEROSOL FLOW REACTOR FOR HYDROGEN 453
We have introduced reference factors for the temperature of both the gas and carbon
particles since neither is naturally referenced to zero. We have also introduced scale
factors for the spatial gradients of the fractional conversion and the temperature in
both the gas and cloud of carbon particles since there is no reason to believe that any
of these will scale with the length of the reactor.
16
We have not introduced a scale
for the axial spatial coordinate since this dependent variable does not enter explicitly
into the describing equations. Scales for the enthalpies, molar flow rates, heat
capacities, heat-transfer coefficients, gas velocity, kinetic parameter, and thermal
conductivity of the gas have also been introduced since these quantities are not
necessarily constant for the fluid-wall aerosol flow reactor. We have not scaled the
fractional conversion since it is dimensionless and bounded of
◦
(1).
Substitute these dimensionless variables into the describing equations and divide
each equation through by the dimensional coefficient of one term (steps 5 and 6):
C
pCs
T
Czs
U
Gs
ρ
C
εσa
C
M
C
T
4
Cs
W
∗
C
C
∗
pC
dT
C
dz
∗
+
W
0
M
H
Cs
k
ns
ρ
C
εσa
C
W
0
C
M
C
T
4
Cs
k
∗
n
H
∗
C
U
∗
G
(1 − X
c
)
n
=−
h
Ps
T
Gs
εσT
4
Cs
h
∗
P
W
∗
C
U
∗
G
T
Cs
T
Gs
T
∗
C
+
T
Cr
T
Cs
−
T
∗
G
+
T
Gr
T
Gs
+
W
∗
C
U
∗
G
!
T
4
W
T
4
Cs
−
T
∗
C
+
T
Cr
T
Cs
4
"
(7.5-13)
W
Ms
C
pMs
W
Hs
C
pHs
W
∗
M
C
∗
pM
+ W
∗
H
C
∗
pH
dT
G
dz
∗
−
W
0
M
H
Ms
k
ns
U
Gs
W
Hs
C
pHs
T
Gzs
H
∗
M
k
∗
n
U
∗
G
(1−X
c
)
n
+
2W
0
M
H
Hs
k
ns
W
Hs
C
pHs
T
Gzs
U
Gs
H
∗
H
k
∗
n
U
∗
G
(1 − X
c
)
n
=
ˆ
W
H
H
Hs
W
Hs
C
pHs
T
Gzs
H
HW
H
Hs
− H
∗
H
+
2πR
i
h
Ws
T
Gs
W
Hs
C
pHs
T
Gzs
h
∗
W
T
W
− T
Gr
T
Gs
− T
∗
G
+
h
Ps
a
C
M
C
W
Cs
T
Gs
U
Gs
ρ
C
W
Hs
C
pHs
T
Gzs
h
∗
P
W
∗
C
U
∗
G
×
T
Cs
T
Gs
T
∗
C
+
T
Cr
T
Cs
−
T
∗
G
+
T
Gr
T
Gs
(7.5-14)
dX
c
dz
∗
=
k
ns
X
czs
U
Gs
k
∗
n
U
∗
G
(1 − X
c
)
n
(7.5-15)
k
∗
n
=
k
n0
k
ns
exp
−
E
RT
s
(T
∗
G
+ T
0
/
T
s
)
=
k
n0
k
ns
exp
−
E
R(T
W
− T
0
)
× exp
E
R(T
W
− T
0
)
(T
∗
G
+ T
0
/
T
s
− 1)
(T
∗
G
+ T
0
/
T
s
)
(7.5-16)
16
The scaling done here differs from the development by Dahl et al. (footnote 15) in that a separate
scale is introduced here for the gradient of the fractional conversion, whereas Dahl et al. assume that
the length scale for this gradient is the length of the reactor; the latter would apply only at very
high wall temperatures for which complete thermal dissociation occurs over the full length of the
reactor.
454 APPLICATIONS IN PROCESS DESIGN
W
∗
M
= (1 − X
c
)
W
0
M
W
Ms
; W
∗
H
=
2X
c
W
0
M
+
ˆ
W
H
z
s
z
∗
W
Hs
; W
∗
C
=
X
c
W
0
M
+ W
0
C
W
Cs
(7.5-17)
H
∗
M
=
C
pMs
T
Gs
H
Ms
T
∗
G
(T
0
−T
Gr
)/T
Gs
C
∗
pM
dT
∗
; H
∗
H
=
C
pHs
T
Gs
H
Hs
T
∗
G
(T
0
−T
Gr
)/T
Gs
C
∗
pH
dT
∗
;
H
∗
C
=
C
pCs
T
Cs
H
Cs
T
∗
C
(T
0
−T
Cr
)/T
Cs
C
∗
pC
dT
∗
(7.5-18)
h
∗
P
=
k
Gs
h
Ps
R
P
k
∗
G
(7.5-19)
T
∗
G
=
T
0
− T
Gr
T
Gs
,T
∗
C
=
T
0
− T
Cr
T
Cs
,X
c
=0atz
∗
= 0 (7.5-20)
We now set appropriate dimensionless groups equal to zero or 1 to determine the
reference and scale factors that will ensure that all the dimensionless variables are
◦
(1) (step 7). Setting the two dimensionless groups equal to zero in the boundary
conditions given by equation (7.5-20) references both the gas and carbon particle
temperatures to zero:
T
0
− T
Gr
T
Gs
= 0 ⇒ T
Gr
= T
0
;
T
0
− T
Cr
T
Cs
= 0 ⇒ T
Cr
= T
0
(7.5-21)
The dimensionless gas and carbon particle temperatures can be bounded to be
◦
(1) by
setting the appropriate dimensionless groups in equation (7.5-14) equal to 1
17
:
T
W
− T
Gr
T
Gs
= 1 ⇒ T
Gs
= T
W
− T
Gr
= T
W
− T
0
;
T
Cs
T
Gs
= 1 ⇒ T
Cs
= T
Gs
= T
W
− T
0
(7.5-22)
The molar flow rates can be bounded to be
◦
(1) by setting the appropriate dimen-
sionless groups in equations (7.5-17) equal to 1:
W
0
M
W
Ms
= 1 ⇒ W
Ms
= W
0
M
2W
0
M
+
ˆ
W
H
L
W
Hs
= 1 ⇒ W
Hs
= 2W
0
M
+
ˆ
W
H
L (7.5-23)
W
0
M
+ W
0
C
W
Cs
= 1 ⇒ W
Cs
= W
0
M
+ W
0
C
17
The scaling here differs from that in the development of Dahl et al. (footnote 15) in that the temperature
reference scale here is T
W
− T
0
, whereas Dahl et al. use T
W
; the former is the appropriate scale for the
temperature gradient since the temperatures range between T
0
and T
W
.
FLUID-WALL AEROSOL FLOW REACTOR FOR HYDROGEN 455
These scale factors for the molar flow rates are the maximum values, which
include both the injected flows and those produced by the thermal decomposition.
18
The scale for the reaction parameter is obtained from the dimensionless group in
equation (7.5-16):
k
n0
k
ns
e
−E/R(T
W
−T
0
)
= 1 ⇒ k
ns
= k
n0
e
−E/R(T
W
−T
0
)
(7.5-24)
The scale for the heat-transfer coefficient from the carbon particles to the gas is
obtained from the dimensionless group in equation (7.5-19):
k
Gs
h
Ps
R
P
= 1 ⇒ h
Ps
=
k
Gs
R
P
(7.5-25)
The dimensionless physical properties and h
W
are bounded to be
◦
(1) by choosing
the scale factors to be characteristic maximum values.
19
Hence, the heat capaci-
ties and thermal conductivities are scaled with their values at T
0
, whereas the
heat-transfer coefficient h
W
is scaled with its value at T
W
. The scale factors for
the enthalpies are obtained by setting the appropriate dimensionless groups in
equations (7.5-18) equal to 1:
C
pMs
T
Gs
H
Ms
= 1 ⇒ H
Ms
= C
pMs
T
Gs
= C
pMs
(T
W
− T
0
)
C
pHs
T
Gs
H
Hs
= 1 ⇒ H
Hs
= C
pHs
T
Gs
= C
pHs
(T
W
− T
0
) (7.5-26)
C
pCs
T
Cs
H
Cs
= 1 ⇒ H
Cs
= C
pCs
T
Cs
= C
pCs
(T
W
− T
0
)
where C
pMs
,C
pHs
,andC
pCs
are the characteristic maximum values of the heat
capacities for the methane, hydrogen, and carbon, respectively. The dimensionless
gas velocity is also bounded to be
◦
(1) by basing it on the maximum hydrogen
flow rate assuming complete thermal conversion of methane
20
:
U
Gs
=
(2W
0
M
+
ˆ
W
H
L)M
H
ρ
H
S
c
(7.5-27)
where ρ
H
and M
H
are the mass density and molecular weight of hydrogen, and
S
c
is the cross-sectional area of the reactor. The scale factor for the temperature
18
The scaling here differs from that in the development of Dahl et al. (footnote 15) in that the scale
for the flow rate of the cloud of carbon particles here is W
0
M
+ W
0
C
, whereas Dahl et al. use W
0
M
;the
former is the appropriate scale since it is the maximum possible value.
19
Note that by choosing maximum values for the physical properties, they are bounded to be
◦
(1),
in contrast to the scaling of the dependent and independent variables that necessarily are bounded to
be
◦
(1).
20
The scaling here differs from that in the development of Dahl et al. (footnote 15) in that the scale
for the gas velocity here includes both the hydrogen produced by thermal decomposition as well as
injected, whereas Dahl et al. do not include the hydrogen injected; the former is the appropriate scale
since it is the maximum possible value.
456 APPLICATIONS IN PROCESS DESIGN
gradient in the cloud of carbon particles is obtained by recognizing that it is heated
by radiation from the hot reactor wall. Hence, we set the dimensionless group
multiplying the first term in equation (7.5-13) equal to one:
C
pCs
T
Czs
U
Gs
ρ
C
εσa
C
M
C
T
4
Cs
= 1 ⇒ T
Czs
=
εσa
C
M
C
ρ
H
S
c
(T
W
− T
0
)
4
(2W
0
M
+
ˆ
W
H
L)M
H
C
pCs
ρ
C
(7.5-28)
To determine the temperature gradient in the gas phase, we need to assess which
term in equation (7.5-14) is the principal source of heat. We assume here that the
heat is provided mainly by the hydrogen gas that is injected through the porous
inner reactor wall. This is a reasonable assumption since considerable hydrogen
must be injected to prevent reaction at the wall; moreover, it is injected at the
maximum possible temperature, T
W
. Hence,
ˆ
W
H
H
Hs
W
Hs
C
pHs
T
Gzs
= 1 ⇒ T
Gzs
=
ˆ
W
H
(T
W
− T
0
)
2W
0
M
+
ˆ
W
H
L
(7.5-29)
The gas-phase temperature gradient would scale differently if one of the other terms
were the principal heat source.
21
Nonetheless, the forgiving nature of scaling will
tell us if we have balanced the wrong terms in any of the describing equations;
that is, an incorrect scaling will be manifest by a dimensionless group that is not
bounded of
◦
(1). Since the two terms in equation (7.5-15) must balance, we set
the dimensionless group in this equation equal to 1 to obtain the scale factor for
the gradient of the fractional conversion:
k
ns
X
czs
U
Gs
= 1 ⇒ X
czs
=
k
ns
U
Gs
=
k
ns
ρ
H
S
c
(2W
0
M
+
ˆ
W
H
L)M
H
(7.5-30)
The scale and reference factors defined by equations (7.5-21) through 7.5-30
result in the following minimum parametric representation of the dimensionless
describing equations:
W
∗
C
C
∗
pC
dT
C
dz
∗
+
1
k
∗
n
H
∗
C
U
∗
G
(1 − X
c
)
n
=−
2
h
∗
P
W
∗
C
U
∗
G
(T
∗
C
− T
∗
G
)
+
W
∗
C
U
∗
G
!
1
1 −
3
4
−
T
∗
C
+
3
1 −
3
4
"
(7.5-31)
(
4
W
∗
M
C
∗
pM
+ W
∗
H
C
∗
pH
)
dT
G
dz
∗
−
5
H
∗
M
k
∗
n
U
∗
G
(1 − X
c
)
n
+
6
H
∗
H
k
∗
n
U
∗
G
(1 − X
c
)
n
= (1 − H
∗
H
) +
7
h
∗
W
(1 − T
∗
G
) +
8
h
∗
P
W
∗
C
U
∗
G
(T
∗
C
− T
∗
G
) (7.5-32)
21
Determining the scale for the temperature gradient in the gas phase based on other assumptions for
the principal heat source is considered in Practice Problems 7.P.19 and 7.P.20.
FLUID-WALL AEROSOL FLOW REACTOR FOR HYDROGEN 457
dX
c
dz
∗
=
k
∗
n
U
∗
G
(1 − X
c
)
n
(7.5-33)
k
∗
n
= e
9
[(T
∗
G
+
3
−1)/(T
∗
G
+
3
)]
(7.5-34)
h
∗
P
= k
∗
G
(7.5-35)
T
∗
G
= 0,T
∗
C
= 0atz
∗
= 0 (7.5-36)
The dimensionless groups appearing in the minimum parametric representation and
their physical significance are defined below:
1
≡
W
0
M
C
pCs
ρ
C
k
ns
εσa
C
W
0
C
M
C
(T
W
− T
0
)
3
∼
heat consumed for carbon production
radiative heat transfer to particles
(7.5-37)
2
≡
k
Gs
R
P
εσ(T
W
− T
0
)
3
∼
heat lost to gas
radiative heat transfer to particles
(7.5-38)
3
≡
T
0
T
W
∼
initial temperature of feed
temperature of reactor wall
(7.5-39)
4
≡
W
0
M
C
pMs
(2W
0
M
+
ˆ
W
H
L)C
pHs
∼
sensible heat consumed by methane
heat supplied by injected hydrogen
(7.5-40)
5
=
W
0
M
C
pMs
k
ns
ρ
H
S
c
(2W
0
M
+
ˆ
W
H
L)
ˆ
W
H
C
pHs
M
H
∼
heat released by reaction of methane
heat supplied by injected hydrogen
(7.5-41)
6
=
2W
0
M
k
ns
ρ
H
S
c
(2W
0
M
+
ˆ
W
H
L)
ˆ
W
H
M
H
∼
heat consumed by production of hydrogen
heat supplied by injected hydrogen
(7.5-42)
7
≡
2πR
i
h
Ws
ˆ
W
H
C
pHs
∼
heat transferred from reactor wall to gas
heat supplied by injected hydrogen
(7.5-43)
8
≡
a
C
S
c
M
C
(W
0
M
+ W
0
C
)ρ
H
k
Gs
(2W
0
M
+
ˆ
W
H
L)
ˆ
W
H
M
H
ρ
C
C
pHs
R
P
∼
heat transferred from particles to gas
heat supplied by injected hydrogen
(7.5-44)
9
≡
E
R(T
W
− T
0
)
∼ dimensionless activation energy (7.5-45)
Now let us consider how these dimensionless describing equations can be
simplified (step 8). Table 7.5-1 summarizes typical system parameters for the fluid-
wall aerosol flow reactor. These system parameters lead to the values of the
dimensionless groups that characterize the fluid-wall aerosol flow reactor summa-
rized in Table 7.5-2. The additional groups
10
through
17
appearing in this table
will be defined when we consider correlating the fractional conversion in terms of
458 APPLICATIONS IN PROCESS DESIGN
TABLE 7.5-1 System Parameters for the Fluid-Wall Aerosol Flow Reactor
Parameter Value Parameter Value
a
C
(m
−1
) 6.0 ×10
4
R
P
(m) 5.0 ×10
−5
C
pCs
(J
/
mol ·K) 8.8 n 4.4
C
pHs
(J
/
mol ·K) 29.0 S
c
(m
2
) 4.6 ×10
−3
C
pMs
(J
/
mol · K) 34.0 T
0
(K) 298
h
Ps
(W
/
m
2
·K) 3.6 ×10
3
T
W
(K) 1200
h
Ws
(W
/
m
2
·K) 18.8 W
0
C
(mol
/
s) 0.12
k
Gs
(W
/
m ·K) 0.18
ˆ
W
H
(mol
/
m · s) 0.11
k
ns
(s
−1
) 0.54 W
0
M
(mol
/
s) 0.057
k
n0
(s
−1
) 6.0 × 10
11
E(J
/
mol) 2.08 × 10
5
L(m) 0.91 ε 1.0
M
C
(kg
/
mol) 0.012 σ(W
/
m
2
· K
4
) 5.67 × 10
−8
M
H
(kg
/
mol) 0.002 ρ
C
(kg
/
m
3
) 2270
R(J
/
mol · K) 8.314
ρ
H
(kg
/
m
3
) 0.080
R
i
(m) 0.038 ρ
M
(kg
/
m
3
) 0.65
TABLE 7.5-2 Dimensionless Groups Characterizing the Fluid-Wall Aerosol
Flow Reactor for Producing Hydrogen from a Methane Feed Stream and the
Process Parameters in Table 7.5-1
Dimensionless Dimensionless
Group Value Group Value
1
0.17
10
180
2
87
11
2.4 ×10
3
3
0.25
12
1.17
4
0.31
13
1.0 ×10
12
5
0.28
14
0.57
6
0.48
15
84
7
1.41
16
3.8 ×10
16
8
54
17
3.8 ×10
−6
9
28
dimensionless groups that isolate particular dimensional quantities of interest. The
magnitudes of the groups
10
through
17
have no relevance to our scaling anal-
ysis since these groups do not represent any ratios of the terms appearing in our
describing equations.
It might appear that we have scaled equation (7.5-31) incorrectly since
2
1, yet all the terms in this equation should be bounded of
◦
(1). Recall that
we balanced the two principal terms in equation (7.5-13) that had to be retained:
the term involving the heat required to increase the temperature of the cloud of
carbon particles and the heat supplied by radiation from the reactor wall. The fact
that
2
1 means that the dependent variable combination that it multiplies must
be very small; that is, T
∗
C
− T
∗
G
1, so that the product
2
(T
∗
C
− T
∗
G
)is
◦
(1). This
means that the gas is essentially in local thermal equilibrium with the cloud of
FLUID-WALL AEROSOL FLOW REACTOR FOR HYDROGEN 459
carbon particles; that is,
2
h
∗
P
W
∗
C
U
∗
G
(T
∗
C
− T
∗
G
)
∼
=
87(T
∗
C
− T
∗
G
)
∼
=
1 ⇒ T
∗
C
− T
∗
G
∼
=
0.011 ⇒ T
∗
G
∼
=
0.99T
∗
C
(7.5-46)
Hence, the fluid-wall aerosol reactor can be modeled assuming that the gas is in
local thermal equilibrium with the carbon particles; that is, it is not necessary to
solve the thermal energy equation for the gas. However, note that this approximation
was based on the system parameters given in Table 7.5-1. For sufficiently large
particles and/or high reactor wall temperatures, the dimensionless group
2
will
become less than 1, implying that the gas temperature is much less than that
of the carbon. The conclusion that the gas and carbon particles are in thermal
equilibrium can also be obtained from the scaled thermal energy balance for the
gas phase since
8
1, which again implies that T
∗
G
∼
=
T
∗
C
. The fact that
2
1
and
8
1, both of which imply that T
∗
G
∼
=
T
∗
C
, confirms that we have chosen
our scales correctly for the process parameters given in Table 7.5-1. If one is
not aware that rapid thermal equilibrium is achieved between the gas and carbon
particles, serious problems can be encountered in solving the describing equations
numerically. For sufficiently large values of
2
there is a thermal boundary layer
at the entrance of the reactor wherein the gas and carbon particles come to thermal
equilibrium whose thickness is much less than the length of the reactor. Hence,
numerical integration employing a step size based on the reactor length will not
resolve the transport processes occurring in this boundary layer.
Let us now consider the temperature gradient in the reactor. This can be esti-
mated from equation (7.5-28). For the parameter values given in Table 7.5-1, we
obtain the following estimate:
T
Gzs
∼
=
T
Czs
=
εσa
C
M
C
ρ
H
S
c
(T
W
− T
0
)
4
(2W
0
M
+
ˆ
W
H
L)M
H
C
pCs
ρ
C
= 1163 K/m (7.5-47)
This estimate for T
Gzs
implies that the cloud of carbon particles and gas reach the
wall temperature before they exit from the downstream end of the reactor; that is,
z =
1200 K − 298 K
1163 K/m
= 0.78 m <L= 0.91 m (7.5-48)
Note that the temperature gradient increases markedly with increasing T
W
,which
implies that the gas and carbon particles will reach T
W
progressively farther
upstream as T
W
increases.
The dimensionless groups
1
,
5
,and
6
are of
◦
(1) and thus do not per-
mit us to conclude that the term they multiply, (1 −X
c
)
n
, is small; that is, we
cannot conclude from the magnitude of these groups that the conversion is nearly
complete. However, our scaling analysis allows us to estimate whether the reactor
is sufficiently long to achieve complete conversion of the methane. An estimate
460 APPLICATIONS IN PROCESS DESIGN
of this can be obtained from the scale factor for the spatial gradient of the frac-
tional conversion given by equation (7.5-30). For the parameter values given in
Table 7.5-1 we obtain the following estimate:
X
czs
=
k
ns
U
Gs
=
k
ns
ρ
H
S
c
(2W
0
M
+
ˆ
W
H
L)M
H
= 0.46 m
−1
(7.5-49)
This estimate for X
czs
implies that complete conversion of the methane is not
attained for the reactor length given in Table 7.5-1; that is,
z =
1 − 0
0.46 m
−1
= 2.2m>L(0.91 m) (7.5-50)
One way to increase the fractional conversion is to increase the reactor wall tem-
perature. The temperature dependence of X
czs
enters through the scale factor for
the reaction parameter k
ns
, which is defined by equation (7.5-24). An estimate of
the fractional conversion can be obtained from equation (7.5-33), which can be
rearranged as follows:
dX
c
dz
∗
=
1
X
czs
dX
c
dz
=
k
∗
n
U
∗
G
(1 − X
c
)
n
⇒
dX
c
dz
= X
czs
k
∗
n
U
∗
G
(1 − X
c
)
n
(7.5-51)
Since both k
∗
n
and U
∗
G
are bounded of
◦
(1), they will be assumed to be constant in
order to integrate equation (7.5-46) to obtain an estimate of X
c
, which is given by
X
c
= 1 − [1 +(n − 1)X
czs
L)]
1
(1−n)
(7.5-52)
The effect of the wall temperature on the fractional conversion can be estimated
by combining equations (7.5-24), (7.5-30), and (7.5-52) to obtain
X
c
= 1 −
1 +
!
(n − 1)
ρ
H
S
c
Lk
0
e
−E/R(T
W
−T
0
)
(2W
0
M
+
ˆ
W
H
L)M
H
"#
1
(1−n)
(7.5-53)
Equation (7.5-53) will provide a reasonable estimate of the conversion only if the
dimensionless group
2
is sufficiently large to ensure that the gas phase is in
thermal equilibrium with the cloud of carbon particles. The magnitude of
2
is
directly proportional to h
Ps
, the characteristic heat-transfer coefficient between the
carbon particles and the gas, which in turn is inversely proportional to the radius of
the carbon particles. Hence, as the radius of the carbon particles increases,
2
will
become sufficiently small so that the gas is no longer in thermal equilibrium with
the cloud of carbon particles. For the parameter values in Table 7.5-1, equation (7.5-
53) indicates that X
c
∼
=
0.23 for T
W
= 1200 K. To achieve a fractional conversion
of X
c
= 0.90 for the parameter values in Table 7.5-1, a reactor wall temperature
of 1530 K would be required.
Let us now use the advanced dimensional analysis concepts for presenting
numerical or experimental data that were developed in Chapter 2. In particular, let
FLUID-WALL AEROSOL FLOW REACTOR FOR HYDROGEN 461
us develop a dimensional analysis correlation for the fractional conversion achieved
in the gas exiting the fluid-wall aerosol flow reactor. This is obtained in principle by
solving the coupled describing equations for X
c
evaluated at z = L. This introduces
L into our dimensional analysis as an independent quantity, whereas in our scaling
analysis it appeared as the product
ˆ
W
H
L. Hence, an additional dimensionless group
is required that we will arbitrarily define as
10
≡
L
2
S
c
(7.5-54)
Therefore, X
c
will be a function of the nine dimensionless groups defined by
equations (7.5-37) through (7.5-45) and equation (7.5-54). However, if
2
1,
the gas will be in thermal equilibrium with the cloud of carbon particles, in which
case the thermal energy equation for the gas need not be considered. This eliminates
groups
2
,
4
,
5
,
6
,
7
,and
8
from the dimensional analysis correlation for
X
c
; that is, for sufficiently large values of
2
, the fractional conversion X
c
will be
a function of only the dimensionless groups
1
,
3
,
9
,and
10
.
Now let us return to the general case for which
2
is not necessarily very large.
Groups
1
through
10
are not particularly convenient for correlating X
c
and
other quantities of interest in assessing the performance of the fluid-wall aerosol
flow reactor. In particular, we would like to isolate the process parameters that can
easily be changed, such as the wall temperature T
W
, the flow rate of carbon particles
injected with the feed W
0
C
, and the radius of the carbon particles R
P
. In isolating
the latter quantity we must be aware that a
C
, the surface area per unit volume of
particles, depends on the particle size. For spherical particles, a
C
= 3
/
R
P
,which
is a reasonable approximation for the small carbon particles involved in the fluid-
wall aerosol flow reactor. Let us first invoke the formalism of steps 9 and 10 in
the scaling approach to dimensional analysis that was outlined in Chapter 2. These
steps allow us to remove redundant quantities that appear in sums or differences in
dimensionless groups. Hence, seven of the nine dimensionless groups that resulted
from our scaling analysis can be replaced by the following redefined groups:
10
=
1
≡
W
0
M
C
pCs
ρ
C
k
0
R
P
eσ W
0
C
M
C
T
3
0
(7.5-55)
11
=
2
≡
k
Gs
R
P
eσ T
3
0
(7.5-56)
12
=
4
≡
C
pMs
C
pHs
(7.5-57)
13
=
5
=
k
0
ρ
H
S
c
ˆ
W
H
M
H
(7.5-58)
14
=
6
=
W
0
M
ˆ
W
H
L
(7.5-59)
462 APPLICATIONS IN PROCESS DESIGN
8
≡
S
c
M
C
ρ
H
k
Gs
ˆ
W
H
M
H
ρ
C
C
pHs
R
2
P
(7.5-60)
15
=
9
≡
E
RT
0
(7.5-61)
where the
i
denotes the result of redefining
i
using the formalism of steps
9 and 10. Note that it was not necessary to redefine groups
3
,
7
,and
10
.In
arriving at equations (7.5-55) through (7.5-61), we have assumed spherical car-
bon particles and therefore have made the substitution a
C
= 3/R
P
.Wehavealso
dropped numerical constants to simplify the dimensionless groups. Although these
numerical constants need to be retained in scaling analysis, they serve no purpose
in dimensional analysis. Applying steps 9 and 10 has isolated T
W
into
3
and W
0
C
into
1
. However, R
P
is still contained in groups
1
,
2
,and
8
. To isolate R
P
into just one dimensionless group, we apply the formalism of step 7 in the scaling
analysis approach to dimensional analysis.
16
=
1
·
2
=
W
0
M
C
pCs
ρ
C
k
0
R
P
eσ W
0
C
M
C
T
3
0
k
Gs
R
P
eσ T
3
0
=
W
0
M
C
pCs
ρ
C
k
0
k
Gs
(eσ )
2
W
0
C
M
C
T
6
0
(7.5-62)
17
=
8
1
(
2
)
2
=
S
c
M
C
ρ
H
k
Gs
ˆ
W
H
M
H
ρ
C
C
pHs
R
2
P
$
R
P
eσ T
3
0
k
Gs
%
2
=
(eσ )
2
S
c
M
C
ρ
H
T
6
0
ˆ
W
H
M
H
ρ
C
C
pHs
k
Gs
(7.5-63)
Hence, X
c
can now be correlated in terms of the ten dimensionless groups
3
,
7
,
10
,
11
,
12
,
13
,
14
,
15
,
16
,and
17
,inwhichT
W
,R
P
,andW
0
C
have been isolated into groups
3
,
11
,and
16
, respectively; that is,
X
c
= f(
3
,
7
,
10
,
11
,
12
,
13
,
14
,
15
,
16
,
17
) (7.5-64)
Note that all the dimensional quantities contained in the original 10 dimensionless
groups are also included in the redefined set of dimensionless groups.
The utility of scaling analysis for simplifying the describing equations and for
correlating the output from a numerical solution is demonstrated in Figure 7.5-2,
which shows a plot of the fractional conversion X
c
as a function of the dimension-
less group
−1
3
with the group
11
as a parameter; the other eight dimensionless
groups are held constant. The solid and dashed lines are the predictions of a numer-
ical solution to the full set of describing equations.
22
The dotted line is the estimate
of the fractional conversion obtained from scaling analysis for the limiting case of
2
1 given by equation (7.5-53). The scaling analysis estimate agrees within
the expected
◦
(1)accuracy with the predictions of the numerical solution. One can
immediately appreciate the utility of isolating the quantities T
W
and R
P
into just one
dimensionless group; that is, Figure 7.5-2 indicates how increasing the wall tem-
perature or the particle size affects the fractional conversion. Larger values of
−1
3
22
J.K.Dahl,A.W.Weimer,andW.B.Krantz,Int. J. Hydrogen Energy, 29, 57 (2004).