Joint Meeting of the Greek and Italian Sections of The Combustion Institute
Calculations of Nu and Sh in Monoliths
2
A. Di Benedetto, 1F. Donsì, 2F.S. Marra
1
Dipartimento di Ingegneria Chimica - Università Federico II, Naples - ITALY
2
Istituto di Ricerche sulla Combustione - C.N.R., Naples - ITALY
INTRODUCTION
Monolithic catalysts are continuous structures made of a large number of straight parallel and
thin channels, whose walls are coated with catalyst. They are commonly used a wide range of
applications, especially involving the oxidation of hydrocarbons in fast heterogeneous
reactions [1-3]. Accurate modeling of monolithic reactors may be performed by means of
two- and three-dimensional models. Nevertheless, for applications of real time simulations or
kinetic parameters estimation, at the present computing power, non-computationally
expensive one-dimensional models are required.
Notwithstanding the fact that it was shown that the critical point in achieving high accuracy
of 1D models consists of the correct evaluation of mass and thermal fluxes between the bulk
gas phase and the surface [4], heat and mass fluxes in monolithic catalysts are presently
described with correlations for heat and mass transfer coefficients, derived for flow in ducts,
hence neglecting the occurrence of the superficial reaction. Such correlations are based on the
solution of the Graetz problem for constant wall temperature, NuT, and constant wall heat
flux, NuH [5], where NuT and NuH are given as unique functions of the dimensionless
coordinate x* (x* = z / 2 Re Pe). Nevertheless, in the presence of a fast exothermic reaction at
the surface, Nu and Sh numbers may be severely affected by inlet conditions [4], and by the
kinetic parameters of the surface reaction [6, 7].
In our previous work [8, 9] we showed that Nu is significantly enhanced by the perturbation
induced by surface reaction on the radial profiles of temperature, concentration and velocity.
We found that the perturbation on mass and heat fluxes generated by the light-off at different
fuel mass fractions, inlet temperatures and Re numbers, may be uniquely correlated with the
adiabatic temperature of the reacting mixture via a modified Nu number (Nuad), which
anchors the heat transfer efficiency to the light-off position rather than to the geometrical
position (x*) [8, 9].
In the present paper we extend the previous analysis to the effect of the variation of the
kinetic parameters of the superficial oxidation of propane on the heat and mass transfer
efficiency, by means of a two-dimensional dynamic model of a single channel of an adiabatic
monolithic reactor in which coupling between mass, energy and flow-field is solved.
COMPUTATIONAL MODEL
A two-dimensional model has been adopted to simulate the behavior of a circular adiabatic
monolith reactor, of length L = 0.12 m and radius R = 0.00045 m, in which the catalytic
combustion of propane occurs. Internal diffusion resistances and homogeneous reactions are
neglected to focus the study on the effect of the superficial reaction on heat and mass fluxes.
Even though the unsteady balance equations have been solved to ease solution convergence,
only steady state results will be presented in this paper.
The rate of the surface reaction is assumed to be a one step irreversible reaction zeroth order
with respect to oxygen and first order with respect to propane mass fraction as follows:
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Joint Meeting of the Greek and Italian Sections of The Combustion Institute
 ∆E 
 yC H
 RT  3 8
ω y ,C H = k0 exp −
3
8
Therefore ωh = ωy,C3H8 |∆HC3H8|, being ∆HC3H8 the heat of combustion of propane. The values
of the adopted kinetic parameters are k0 = 1·106 mol/m2 s, Ea/Rg = 10900 K [10].
The model equations have been discretized by adopting the control volume approach.
Numerical solutions have been performed by means of the commercial software package
CFD-ACE+ [11]. Simulations were carried out for a developing flow of propane in air,
whose propane mass fraction is 0.015, inlet temperature 600 K, and inlet velocity 36 m/s.
The total number of cells is 19200. A complete simulation takes about 24 h of CPU time on a
AMD 2100+ PC to reach a converged solution starting from non-ignited conditions.
RESULTS
The results reported in Figure 1 refers to the computed Nu (Sh) for Re = 640, as function of
the dimensionless coordinate x*, in comparison with the correlations available in literature
for NuT and NuH [5]. By comparing the calculated Nu (Sh) in the presence of a fast
superficial reaction with the literature Nu numbers, it turns out that the computed Nu (Sh) is
roughly approximated by the correlation curves for developing flow at the inlet of the reactor
(NuH) before ignition occurs, where the reactor behavior may be assimilated to a constant
wall heat flux problem, while it is quite well correlated by NuT after the ignition point, where
wall temperature can be considered quite safely constant and equal to the adiabatic
temperature (T = Tad).
a.
Nu (Sh)
60
45
Nu
H
30
Nu
T
15
T (K)
0
1200
b.
1000
800
600
1e-4
1e-3
1e-2
1e-1
x*
Fig. 1. Computed Nu (a) in a monolithic catalyst with a fast exothermic reaction occurring
at the surface, in comparison with literature values [5]; computed wall and bulk
temperature under the same conditions (b).
Nevertheless we observe that Nu exhibits a significant enhancement, with a maximum in
correspondence of the light-off point (identified by the steep rise of Twall in Fig. 1b) and a
non-monotonic trend, for which there are no literature correlations available.
The increase of efficiency of heat and mass transfer at light-off was related to the heavy
perturbation of the flow field, generated by the ignition of the fast exothermic reaction, with
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Joint Meeting of the Greek and Italian Sections of The Combustion Institute
strong temperature, concentration and velocity gradients. The formation of brand new fluid
dynamic, thermal and concentration boundary layers determine the appearance of a new
entrance effect, accompanied by radial convective fluxes due to the gas expansion in the bulk,
as previously discussed [8, 9].
Effect of the kinetic parameters
The investigation of the effect of a change in activity of the surface reaction on Nu and Sh is
reported in Figure 2, at varying the pre-exponential factor and the activation energy,
according to Table 1.
a.
Ea_9
k0
T
Nu
60
k0*5
H
Nu
Nu (Sh)
200
160
120
80
Ea_10.5
40
k0/2
20
0
1200
Twall (K)
1100
1000
900
800
k0*5
Ea_9
k0
Ea_10.5
k0/2
700
b.
600
0
k0/2
Ea_10.5
v (m/s)
-2
k0
-4
-6
k0*5
-8
1e-5
Ea_9
c.
1e-4
1e-3
1e-2
1e-1
x*
Fig. 2. Effect of the kinetic parameters of the surface reaction, varied as reported in
Table 1, on Nu (Sh) (a), wall temperature (b) and radial velocity profiles evaluated
at r = 0.0004 m (c).
k0 (mol/m2 s)
Ea/Rg (K)
6
1·10
10869
k0
5
5·10
10869
k0/2
6
5·10
10869
k0*5
1·106
9869
Ea_9
6
1·10
11200
Ea_10.5
Tab. 1 List of parameter values of the surface reaction used in the simulations.
In particular, Fig. 2a shows along the dimensionless coordinate x* the Nu (Sh) number, the
wall temperature (2b) and the radial component of velocity (2c) evaluated at r = 0.0004 m, at
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Joint Meeting of the Greek and Italian Sections of The Combustion Institute
different values of the pre-exponential factor and the activation energy.
It is evident that when the rate of the surface reaction is varied, by varying its kinetic
parameters, light-off position is shifted. In particular, in correspondence of a reduction of the
reaction rate with respect to the case of Fig. 1 (k0), either due to a reduction in the preexponential factor (k0/2) or an increase in the activation energy (Ea_10.5) light-off position is
moved downstream, as shown by the wall temperature profiles reported in Fig. 2b. The spikes
of Nu and Sh, anchored to the perturbation generated by the ignition, move consequently.
The heavy perturbation of the flow field is clearly revealed by the radial velocity profiles
(Fig. 2c), which are perturbed in dependence on the kinetic parameters.
The heat release associated to the superficial reaction determines the formation of a second
negative peak of the radial velocity, which may or may not overlap to the negative peak due
to the entrance effects. For k0/2, Ea_10.5 and k0, the two contributions are clearly separated
and distinguishable (Fig. 2c). The peaks due to entrance effects, which are always located
soon after the inlet and extinguish for x* > 10-3, are all similar in shape and dimension, with
only minor differences due to the different rates of occurrence of the surface reactions. On the
contrary, the peaks due to the ignition of the reaction are placed in correspondence of the
light off, and depend strongly on the kinetic parameters in shape and dimension.
For k0*5 and of Ea_9, the ignition of the surface reaction occurs while the entrance effects
are still present. Hence, the perturbation induced by the reaction is completely superimposed
to the entrance effects, as shown by the peak of radial velocity, which is single, but more than
double in size the peak corresponding to the entrance effects for k0. As a consequence, mass
and heat fluxes are significantly enhanced soon after the inlet, and Nu exhibit a single, high
intensity peak (Fig. 2a).
The overall effect of the variation of the kinetic parameters suggests the existence of a strong
link between the rate of the surface reaction (which increases at increasing pre-exponential
factor, and at decreasing activation energy) and the intensity of the perturbation of the flow
field and of heat and mass fluxes.
Indeed, at larger pre-exponential factor (or low activation energy), the time of reaction, and
hence of heat production, is shorter, thus increasing the extent of the local gas expansion,
generating a stronger perturbation of the flow field and then of the radial component of
velocity. In addition, also the radial profiles of temperature and concentration become much
steeper.
Following the shown results, it can be generally stated that heat and mass transport are
affected not only by the amount of heat generated [8] but also by its production rate.
DISCUSSION
From the phenomenology analyzed in the previous section, it results that the variation of the
kinetic parameters of the surface reaction gives rise to a variability in the mass and heat
fluxes, with a significant and non-predictable displacement of calculated Nu and Sh with
respect to the correlations currently available for their prediction (NuT and NuH).
In a previous work [8] the identification of a new driving force, embedding the dependence
on the actual strength of heat production, let the effect of the surface reaction come into
action in a new definition of Nu and Sh numbers, evaluated as a function of the difference
between the adiabatic temperature and gas bulk temperature:
Nu ad =
Shad =
− ∂T ∂r r = R 2 R k w
(Tad − Tb )
− ∂YC3 H 8 ∂r
kb
2R D
w
(Yb − Yeq )
Db
4
r =R
Joint Meeting of the Greek and Italian Sections of The Combustion Institute
The good overlap we obtained plotting Nuad and Shad versus the dimensionless temperature
∆Tb = (Tad – Tb)/∆Tad and concentration ∆Yb = (Yb – Yeq)/(Yin – Yeq) holds for different values
of the adiabatic temperature (at varying inlet concentration and temperature). Indeed, in this
way a single peak remains along Nuad (Shad), in correspondence of light off, while the region
on the left of the peak (LHS branch) corresponds to the region after the ignition (RHS will
define the other branch). The peak due to the entrance effects indeed disappears due to the
chosen driving force (Tad – Tb), as Tw << Tad. It follows that for ∆Tb that goes to 0, Nuad tends
asymptotically to Nu.
This correlation works fairly well at varying the adiabatic temperature rise of the reactor
(inlet concentration and inlet temperature) [8], but it does not take into account the kinetic
parameter of the surface reaction. Nevertheless, the intensity of mass and heat transfer
strongly depends on such kinetic parameters. Hence, in the case herein considered, also the
LHS of the Nuad (Shad) curves collapse when plotted versus ∆Tb, as shown in Fig. 3a.
∆Tb (∆Yb)
0.2
120
100
0.4
0.6
0.8
1.0
a.
k0/2
k0
k0*5
Ea_9
Ea_10.5
b.
120
Nuad
Shad
100
60
80
40
60
20
Nuad (Shad)
Nuad (Shad)
80
40
0
20
0.2
0.4
0.6
0.8
1.0
0
1.2
∆b '
Fig. 3
Nuad (Shad) plotted versus the dimensionless coordinate ∆Tb (∆Yb) (a) and left-handside curves of Nuad and Shad plotted versus the modified dimensionless coordinate∆b′
at varying the kinetic parameters as from Table 1.
A unique correlation of Nuad, and analogously of Shad curves, may be obtained, if Nuad and
Shad are plotted respectively versus a modified dimensionless temperature ∆Tb′
( ∆Tb' = ∆Tb Ba Da b ) and concentration ∆Yb′ ( ∆Yb' = Yb Ba Da b ), where now the kinetic
parameters of the surface reaction are embedded via the dimensionless numbers B (thermal
number) and Da (Damkholer number), defined as follows:
B=
L ω C3 H 8
Ea ∆Tad
, Da =
Rr Tin Tin
u in CCin3 H 8 R
In particular, the LHS of Nuad and Shad curves for the 5 cases herein investigated collapse for
a = 0.07 and b = 0.018, as shown in Fig. 3b. The RHS branches of the Nuad curves (under a
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Joint Meeting of the Greek and Italian Sections of The Combustion Institute
kinetic regime) at varying the kinetic parameters do not overlap, but the need for the correct
prediction of mass and heat fluxes is of less importance than after ignition, when the reactor
is under mass and heat transfer limited regime.
On the contrary, the LHS branch (mass and heat transfer regime) of the Nuad (Shad) curves
when plotted versus ∆Tb′ (∆Yb′) overlap, and can be correlated by the following expression,
where ∆b′ stands for ∆Tb′ in Nuad and for ∆Yb′ in Shad.
Shad ≡ Nu ad = 3.656 + ∆'b ⋅
exp(3.75 ⋅ ∆'b )
(37.5 − 32.4 ⋅ ∆'b )
(1)
The parameters used in the correlation proposed herein are related to the cases analyzed here.
To extend its validity a more extensive analysis of parameters has to be performed.
The equation (1) represents the trend of Nuad in the presence of superficial reaction and it is
anchored to the reaction co-ordinate instead of the axial position in the channel as in the
literature correlation. This fact allows us to take into account the effect of the increase of
transfer efficiency at the light-off location.
CONCLUSION
The occurrence of a fast exothermic surface reaction in a channel where a uniform velocity
profile develops into parabolic modifies the radial profiles of temperature, concentration and
velocity. In particular, the reaction generates a perturbation of the flow field due to the local
gas expansion associated to the heat produced. This results in an enhancement of mass and
heat transfer in the channel to such an extent that the literature correlations for simultaneous
developing flow in channels obtained in the absence of superficial reaction can not be used
anymore for the prediction of Nu and Sh numbers. Indeed, Nu and Sh trends strongly depend
on the kinetics of the surface reaction, which must then be included in their correlation.
The results of the investigation suggest that a suitable correlation for the evaluation of heat
and mass transfer coefficients has to take into account the coupling between light-off position
and flow field. Moreover, they reinforce the assumption that the modified Nu and Sh
numbers (Nuad and Shad) defined in our previous paper, can be suitable for the selective
description of reaction effects on heat and mass transfer, if plotted versus modified
dimensionless temperature ∆Tb′ and concentration Yb′, which include the kinetic parameters.
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