2.2 Detailed Model of the Tubular Reactor
2.2.4 Validation
2.2.4.2 Influence of Discretization
Θ [−]
z/L [−]
0 0.01 0.02 0.03 0.04 0.05 0.06
0.7 0.75 0.8 0.85 0.9 0.95 1
w M/w r [−]
z/L [−]
0 0.02 0.04 0.06
1.7 1.75 1.8 1.85 1.9 1.95 2
w P/w r [−]
z/L [−]
0 0.02 0.04 0.06
0 0.05 0.1 0.15 0.2 0.25
1000 fixed 100 fixed 60 fixed 60 adaptive
Figure 2.12: Temperature profiles and the mass fractions of monomer and polymer at steady state conditions using different discretization schemes and grid
points
unrealistic high. A model of the tubular reactor, with its 29 partial differential and algebraic equations for each of the 16 modules would result in a system of 464· 103 ordinary differential and algebraic equations. Such a system is far too large for standard computers systems available right now. So, in a next step, the influence of the selected discretization method on the simulation result of one module is discussed.
2.2. DETAILEDMODEL OF THETUBULARREACTOR
is used for this comparison, i.e. the energy balance equation for the wall is included.
Moreover, the coolant is operated counter-current wise. For this configuration no simulation data from Luposim T is available, hence the detailed dynamical model with a high resolution fixed grid serves as reference.
The following configurations are compared to each other,
• high resolutionfixedgrid (1000 grid nodes),
• intermediate resolutionfixedgrid (100 grid nodes),
• low resolutionfixedgrid (60 grid nodes) and
• low resolutionadaptivegrid (60 grid nodes).
In Fig. 2.12, process variables, such as dimensionless temperature2 and weight fractions of monomer wM and LDPEwP are depicted along the spatial coordinate.
Both, wM and wP are made dimensionless using a reference weight fraction wr. Again, only one module i.e. one cooling jacket, is considered here. The reference profile, which is the result of the simulation using the high resolution fixed grid is drawn in all three graphs with a solid blue line. The dash-dotted red line represents the intermediate resolution grid, the dotted red line the low resolution fixed and the dashed black line the low resolution adaptive grid. As one can observe, the number of grid points of either fixed or moving grid does not influence the profile of those process variables significantly. Only in regions of the tubular reactor where larger temperature gradients occur, some slight deviations between the four discretization schemes are visible.
But for such issues as optimization or control, also product properties play an important role. These properties can be expressed in terms of the number average,
Pn= µ1P
µ0P (2.88)
variance
σ2= µ0Pµ2P−(µ1P)2
(µ0P)2 (2.89)
and skewness
γ = µ3P
σ2. (2.90)
They are depicted in the left column of Fig. 2.13. Number average means here an aver-age chain length, variance is the averaver-aged squared deviation from the arithmetic mean.
The standard deviation is defined as the square root of the variance. Both standard de-viation and variance describe the spread of a distribution, as does the polydispersity defined in (2.57). To describe, whether a distribution is asymmetric, the skewness can be used. If the distribution is normal, then the skewness calculated with central-ized moments is zero, if the right tail is longer than the left one, then the skewness is negative, otherwise it will be positive. In Fig. 2.13 right column shows the sec-ond moments of the living primary and secsec-ondary and the dead polymer chain-length distribution. Again, the solid line represents the grid with 1000 fixed grid points, the dash-dotted line the fixed grid with 60 nodes and the dashed line the adaptive grid with the same number of grid nodes.
From this picture it is obvious, that there exist significant differences between the high and the low resolution grids in general. But if one compares these differences in magnitude, the adaptive low resolution grid performs considerably better than the fixed one. E.g. the error in the variance of the distribution is approx. 25% using the fixed grid, whereas with the adaptive grid it is 10% less. The reason for these large deviations in the properties of the distributions can be seen in the right column of Fig. 2.13. Even though the zeroth moment of the living polymer chain length distribution is used in the monitor function M(z,x)(Eq. (2.73)), the number of fixed grid points is too small to resolve the formation of living polymer precisely. Hence these deviations sum up in higher moments of living polymer and naturally the error is transferred to the moments of the dead polymer chain length distribution.
In Tab. 2.1 the simulation times with the different discretization schemes are com-pared. The fastest calculation time, utilizing a low-resolution fixed equidistant grid is chosen as the reference time (tdi sc,r e f =42.42s on a standard PC equipped with an AMD Athlon™ 64 Processor 3500+ and 1G B RAM). In order to make the results of this comparison transferable to other computer systems, not the absolute values of the simulation time but multiples of the fastest solution are shown. In the upper row, the
2.2. DETAILEDMODEL OF THETUBULARREACTOR
P n [−]
z/L [−]
0 0.02 0.04 0.06
400 600 800 1000
σ2 [−]
z/L [−]
0 0.02 0.04 0.06
0 500 1000
γ [−]
z/L [−]
0 0.02 0.04 0.06
0 5000 10000 15000
µ 2P [−]
z/L [−]
0 0.02 0.04 0.06
0 500 1000
µ 2R [−]
z/L [−]
0 0.02 0.04 0.06
0 2 4 6
8 1000 fixed
100 fixed 60 fixed 60 adaptive
µ 2R,sec [−]
z/L [−]
0 0.02 0.04 0.06
0 0.01 0.02 0.03 0.04
Figure 2.13: Temperature profiles and the mass fractions of monomer and polymer at steady state conditions using different discretization schemes and grid
points
results for the validation of one module are tabulated (see Sec. 2.2.4.1). There, the model without energy balance for the wall has been used and that the coolant is as-sumed to flow co-current wise. It is interesting to see, that the fixed grid with 100 grid nodes and the low-resolution grid both have similar simulation times, even though the
Wall energy Coolant flux Fixed Adaptive
balance direction 1000 100 60 60
– co-current 83.3 1.9 1.0 2.0
4 counter-current 110.7 4.0 1.8 12.6
Table 2.1:Comparison of simulation times ttdi sc
di sc,r e f on a standard with an AMD
Athlon™ 64 Processor 3500+ and 1G B RAM.
number of state variables is lower by a factor of 1.6. Moreover, the high-resolution scheme, which is larger in size by a factor of 16.7, takes 83.3 times longer to be solved.
In general, the effort of solving equation systems numerically increases exponentially with system size. Additionally, also the type of equations and back-mixing effects increase simulation times. This can be seen in the bottom row of the table, which tabulates the results of a simulation with the energy balance equation for the wall and with the counter-current flow. Both effects contribute to an almost doubled simulation time for the fixed low-resolution grid. The fixed intermediate-resolution grid scales with the same factor, but the adaptive grid now takes three times as long as the fixed grid with the same error. Even though, the simulation is much faster, it is not possible to use the fixed grid with 100 grid nodes, since this adds up to 29·16·100=46000 equations, which has not been possible to solve on a standard PC. Here, the adaptive
0 0.02 0.04 0.06
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
z/L [−]
τ [−]
0.76 0.78 0.8
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
z/L [−]
τ [−]
Figure 2.14: Movement of grid nodes during the startup of the tubular reactor