Mei C., Zhou J., Peng X. Simulation and Optimization of Furnaces and Kilns for Nonferrous Metallurgical Engineering
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9 Multiple Modeling of the Single-ended Radiant Tubes
quite different: This is the fundamental difference that we have to overcome for
model dimension reducing. Note that in this case the driving force of the mixing
process is the momentum transfer. The dimension reducing should be then based
on a momentum similarity instead of a geometry similarity.
Fig. 9.4 3D flow field of SER tube
(a) I=2; (b)I=9; (c)K=22; (d)K=26
Fig. 9.5 The SIMPLE scheme of 3D computation
Feng Mei
Fig. 9.6
The radial flow velocities along the circumference directions
Defining
#
V as the mean velocity of 95% of the total mass flow as showing in
Eq. 9.1 and defining V
0
as the initial velocity at the burner mouth(J=13), we may use
#
V /V
0
to represent the attenuation of the radial injection before mixing. If the model
is reduced from 3D to 2D,
#
V will be found diminished comparing to that in the
3D case at the height close to the mixing zone. This is because in 2D modeling the
same mass flow is forced to spread equally in the circumferential direction, which
results in weaker mixing effect. A possible solution to retain the mixing intensity is
to enlarge the diameter of the burner head in the geometry model so that the distance
between the injection mouth and the mixing interface is reduced, therefore the
injection flow is more concentrated and
#
V /V
0
is less diminished.
#
0.95
0.95
1
d
S
VVs
S
=
∫
(9.1)
The above 3D analysis of cold flow helped us to set up the guideline of
simplifying the model from 3D to 2D. Note that the transforming of the 3D
geometry to 2D geometry cannot be fully based on the 3D simulation because the
cold flow is still quite much different from the combusting flow. The V
#
/V
0
ratio is
not a constant either (varying from 0.22 to 0.436 over the range operation
condition). In practice the diameter of the 2D burner geometry was actually
determined by compromising all these factors based on information obtained from
numerical experiments using both the 3D and 2D models.
The second objective of 3D modeling was to move forward the boundary
conditions to the line where the fuel and air enter into the combustion chamber.
This was to avoid modeling the separating flow inside the burner head. The results
of 3D simulation indicated that the ratio of the axial flow rate to the radial flow
rate is quite close to a constant around 0.56
± 0.02. This information enables us to
estimate the axial and radial flow rates according to the total incoming flow rate.
9 Multiple Modeling of the Single-ended Radiant Tubes
The geometry model used for the 2D simulation is illustrated in Fig. 9.7.
Fig. 9.7 The geometry model for the 2D simulation
9.3
2D Modeling of the SER Tube
In this section, we talk about how to assess and validate all of the necessary
models to be used for the involved thermal physical processes. Specific thoughts
on the selecting of these models have been presented as well.
9.3.1
Selecting the turbulence model
The turbulence models we were evaluating were the high Reynolds number
k-
ε model (or called the standard k-ε model), the low Reynolds number (LRN)
k-
ε model, the RSM(refer to Section 2.1.6) and the RNG(refer to Section
2.1.5) model. From the foregoing discussion, we understood that a
turbulence model suitable for this application on the one hand should reliably
predict low Reynolds number flow and transition processes, and on the other
hand should be robust in solving complex and highly coupling problems.
Form the concern of prediction accuracy, the LRN model and the RNG model
should be superior to the standard k-ε model. In terms of numerical
robustness, however, the standard k-ε model should be the best choice,
followed by the RNG model, then the RSM and the LRN model.
Compromising the needs from both sides, we ranked the RNG model as the
Feng Mei
most appropriate one. The RSM seems to be less attractive because this
model not necessarily results in higher accuracy under varying density
condition but it is surely more difficult to converge in the present case.
To ensure we made the right choice performances of different models should
still be compared against actual experimental data. First we compared the LRN
model and the standard RNG model. The experimental data employed here were
the measurements reported by C.T. Bowman and L.S.Cohen (Bowman and Cohen,
1975) regarding straight-through radiant tubes. These measurements had also been
used by Viskanta et al. in their studies of 2D radiant tube model (Ramaurthy et al.,
1994,1995) that employed Nagano-Hishida LRN model. Here we compared the
predictions of Viskanta et al. with those predicted by the standard RNG model
developed by the commercial CFD package Fluent 4.0. The combustion process
was modeled by both £-PDF and double-α-PDF models (Borghi,1998;Libby and
Williams, 1980; Kreinin and Kafiren,1980;Chan and Kumar,1990;O’Brien,1980). The
DTRM model was employed for radiation. The nonequilibrium assumption wall
function was applied to predict the near-wall phenomena.
The predicted temperature and velocity fields are shown in Fig. 9.8. The
comparison indicated that the RNG model and the Nagano-Hishida LRN model
perform quite similarly even though the RNG model seemed to be slightly better.
The predictions in the starting section were found remarkably different from the
measurements, which could be explained by errors caused by the measuring
devices. The LRN model converged to 1E-2 level whereas the RNG model could
go down to 1E-3. Understandably, more serious convergence problem might
happen with this LRN model in computing the SER tube that is more complicated
developed geometry and physical processes.
The performances of two RNG models namely the RNG model developed by
Fluent 4.0 (here-after named as Fluent-RNG model) and the standand RNG model
(see section 2.1.5 for detail), were also compared. Eq.9.2.is the estimation of the
Fluent-RNG model for the effective viscosity (turbulent viscosity + laminar
viscosity). The Fluent-RNG model estimated the Prandtl numbers of the scalars by
the turbulent viscosity. These treatments indeed faithfully reflected the original
idea of the Re-normalization Group process. This model, however, was found
difficult to reach convergence under complex flow in numerical experiments.
Fig.9.9 compares the predicted temperatures along the SER tube at two Reynolds
numbers. The high Reynolds number k-
ε
model has basically failed to predict the
detail of the temperature profile, the performance of the Fluent-RNG model was
found to deteriorate at lower Reynolds number. The standard RNG model (refer to
Section 2.1.5) demonstrated satisfying prediction in both cases.
2
eff
1
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+≈
ε
μμ
μ
k
v
C
(9.2)
9 Multiple Modeling of the Single-ended Radiant Tubes
Fig. 9.8
A comparison of the simulated radial velocity fields (a)and the simulated radial
temperature fields (b) with the measurements in reference(Bilger,1987)
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Fig. 9.9 A comparison on performance of the different RNG models
(In the figure, D is equivalent diameter of the outlet of SER)
9.3.2
Selecting the combustion model
A combustion model actually consists of two submodels, namely a chemical
reaction model that estimates the reaction rate and a mass transport model that
governs the diffusion and convection of the reaction components. The combustion
model should properly reflect the interaction between the chemistry and fluid
dgnamics. This interaction is usually measured by the deteriorate Damkholer number:
t
a
C
D
τ
τ
= (9.3)
where
C
τ
represents the chemical time scale and
t
τ
represents the turbulent
time scale.
There is large number of chemical reactions models which either assume
infinitively large reaction rate or defined reaction rate. In the infinitively large
reaction rate class find we the one-step reaction models (mixed-is-burned) and
9 Multiple Modeling of the Single-ended Radiant Tubes
multiple-step reaction models. The later take into account intermediate products
and equilibrium effects. There are also a large variety of models with defined
reaction rates, among which the most frequently used one is the so-called
Arrhenius model(Kreinin and Kafiren,1980).
Regarding the mass transfer process, the eddy break-up (EBU) model is among
the earliest ones but the probability density function models are more widely
applied with new model development continuously coming out(O’Brien,1980).
A complete combustion model is basically an assembly of a chemical reaction
model and a mass transport model. Even though the EBU model is usually
coupled with the Arrhenius reaction rate law model, theoretically other reaction
models can also be “plugged-in”. For example, the mixed-is-burned model or the
locational equilibrium model can be jointly used with the presumed PDF model
such as the double-
¢-PDF model or the£-PDF model which make up various
quick reaction combustion models for diffusion flames. Meanwhile the Arrhenius
reaction rate law model or other defined reaction rate models can be assembled
with the presumed PDF models to make up premixed combustion models.
In the present case the partially premixed combustion mode was used and
the partial premix ratio (defined as the volume percentage of the premix air
against air equivalent) varied from zero to 20%. Under different premix ratio,
the combustion process might shift from pure diffusion mode to premixed
mode. The fuel consisted of chiefly methane with minor ethane, propane and
butane. Apparently this was not a typical fast reaction such as the one between
hydrogen and. oxygen. As a matter of fact, many investigations reported
remarkable error of prediction with instant locational equilibrium assumption
for methane combustion due to the fact that the actual reaction rates were too
far away from the required “fast chemistry” assumption. Therefore the best
approach should be the definite reaction rate model plus a species components
transport model that should be applicable to both diffusion and premixed
flames.
The above analysis allowed us to make proper assessment of the available
combustion models in Fluent 4.0. The EBU (eddy break-up) model was proved,
by numerical experiments, to be a failure for predicting methane combustion. Note
also that Fluent 4.0 did not provide any partially premixed combustion model.
Under this situation a so-called “pseudo-partial-premix-PDF
ā(or 4P’s) model was
developed by adapting the available functions in Fluent 4.0 without carrying out
expensive model developing or programming work. This 4P’s model enabled a
very economical approach with fairly satisfactory simulation performance (Mei,
1999).
As shown in Fig. 9.7, the 4P’s model clustered the mesh points on the inlet of
the partial premixed stream into a number of artificial sub-jets that were defined
either as pure natural gas jet or pure air jet. The total mass flow rates of the radial
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and axial jets were determined by the results of the 3D model and the jet velocities
were calculated based on the opening area of the burner mouths. The computation
was carried out using the Fluent
β-PDF model with instant locational equilibrium
assumption. The idea was to approach the premixing effect using very intensive
turbulent diffusion process in order to cover the “blind point” of the commercial
CFD package. For premixed combustion, the 4P’s approach forced to slower down
the reaction rates that were overestimated by the instant locational equilibrium
assumption.
The weakness of the 4P’s approach was obvious. It was a ad hoc empirical
treatment that could hardly be applied in wide range of application. Numerical
experiments proved, however, that the 4P’s model were quite robust and the
simulations were not the sensitive to the mesh points cluttering scheme or the
velocity boundaries. Understandably, the flow velocities would be rapidly soaring
high once the heat release started and the gases volume expanded. As a result, the
turbulent effect downstream the inlet boundaries became the overwhelming factor
dominating the diffusion process. Fig. 9.10 compares the predictions of the Fluent
EBU model and the 4P’s approach. The later showed better simulation to the
relaminarized combustion at low Reynolds number flow, in which the flame
length is sensitive to the to operation conditions.
Fig. 9.10 A comparison of temperature fields predicted by EBU and the 4P’s approach
For carrying out the 2D simulation, the radiation model, the gases emissivity
model and the wall functions had also been selected. These models are
summarized together with aforementioned models in Table 9.1. Note that the
standard wall function was selected instead of the nonequilibrium wall
function was selected even though the latter normally gives higher accuracy.
9 Multiple Modeling of the Single-ended Radiant Tubes
This decision was made to trade-off local accuracy with overall accuracy. The
nonequilibrium wall function required that the near-wall mesh points must be
arranged in the fully-developed turbulent region. Due to the relaminarization,
the location of the turbulence region varied remarkably as a function of the
flow rate, the premix ratio and the excess air ratio. This fact made it difficult to
decide where and how to arrange the near-wall mesh points.
Table 9.1 2D model structure of SER tube
Process Model
Turbulence Standard RNG k-ε model
Near wall flow and heat transfer Standard wall function
Combustion 4P’s approach
Radiation DTRM
Discrete scheme Quick for velocities, power-law for the others
Mesh
Axis-symmetric finite difference 290h49
9.3.3
Results and analysis of the 2D simulation
Fig. 9.11 to Fig.9.13 demonstrate the predicted temperatures on the inner tube and
outer tube under various premix ratios, excess air ratios and fuel flow rates as
listed in Table 9.2. The dash-lines represent measurements.
Table 9.2 Practical operational conditions and computational cases of SER tube
Test code Fuel supply / kW Premix ratio / % Excess air ration / %
1* 16.3 55.0 11.0
1 16.3 25.1 11.0
2 16.3 14.4 11.0
2* 17.23 0.0 22.7
3* 17.23 20.0 60.0
3 17.23 20.0 22.7
4 17.23 20.0 5.4
4* 17.23 20.0 0.0
5 11.52 22.6 26.1
6 24.8 15.9 8.9
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Fig. 9.11
A comparison of tube wall temperatures simulated with the measurements at
different premix ratio (the numbers of curves are related in Table 9.2)
(a)For inner tube; (b)For outer tube
The premix ratio was found to be the major factor determining the overall fuel
consuming rate of the flames. The temperatures on the inlet side of the tubes were
a good reflection of the different fuel consuming rate. Fig. 9.11(a), (b) compared
the temperatures along the outer and inner tubes. The temperatures demonstrated a
wave-like pattern at large premix ratio, indicating that the fuel had been consumed
rapidly before the flow reaching the dead end of the SER tube. By contrast, the
temperatures rose steadily at lower premix ratio, indicating a moderate but
continuous heat release along the tubes. Unfortunately the model failed to predict
the overheating at the dead ends (Fig. 9.11(a)), which was probably due to the
DTRM radiation model that underestimated the radiation contributed by the
directional emissivity effect on metal surface.
Fig. 9.12
A comparison of predicted tube wall temperatures against
the measurements at different excess air ratio
(a)For outer tube; (b)For inner tube
The impact of excess air ratio was found relatively moderate but far-reaching.
More access air might drag down combustion temperature at the initial stage but