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Table 4.4: Results of influence of shrinkage after 100 s in fixed-bed and 10 s in fluidized-bed
Bed Shrinkage %Conversion %Char % Gas* %Tar
Fixed-bed Yes 82.5 25.6 13.2 61.2
Fixed-bed No 76.1 25.9 13.9 60.2
Fluidized-bed Yes 90.2 22.1 14.0 63.9
Fluidized-bed No 86.3 22.3 14.1 63.6
a typical fixed-bed environment and the respective quantities of a spherical particle of 2 mm radius after 10 seconds in a fluidized-bed environment are summarized.
The temperature evolution of the surface and several internal temperatures are de-picted in Fig. 4.2. The results are from the multi-step ODEs solver, considered as the reference solution. Other methods provide however the same results within the significant figures considered here. As it will be later discussed, the total error of every method is always lower than 0.1%. Tests were performed to ensure that the obtained results were grid independent and convergent. 20 grid points were enough in both conditions. The influence of including or neglecting diffusion and to model convection with QUICK or UDS was not significant.
It can be seen that fixed-bed pyrolysis conditions, with slower heating rates, favour the formation of char (reaction 3 in Tab. 4.2); while fluidized-bed conditions, with a faster conversion, favour the formation of tar (reaction 2 in Tab. 5.4) [31].
In both cases the importance of the correct modelling of the particle due to the significant gradients can be observed. The shrinkage of the particle, that is even sometimes neglected in particle models, has also an important impact both on the conversion time and in the product distribution. When shrinkage is included the conversion proceeds faster, because the temperatures inside the particle are higher.
In the non shrinking particle there are longer residence times, so more tar is cracked decreasing the tar yield and increasing the permanent gas (gas*) yield [38]. This effect becomes clearer in the bigger fixed-bed particles.
The main objective of this section is to find out which is the most suitable solution method for coupling the particle model to a reactor model. The com-putational times needed for each method on a dual core 2.4 GHz AMD Opteron processor are shown on Fig. 4.3 for fixed-bed and fluidized-bed conditions. The codes are written on Fortran 77. The abscissa represents the maximum time step that each solver is allowed to use, that is, the integration interval at which they should provide a solution. It should be taken into account that although in this example the boundary conditions are fixed, they will not be fixed when solving a
4.3 Numerical results 67
Figure 4.2: Temperature profile in -always from higher to lower temperatures - the surface, at a radius equal to 25% and 50% of the particle radius and in the middle of the particle with shrink-age (solid line) and without it (dashed line). Above fixed-bed conditions and below fluidized-bed conditions.
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particle model coupled to a reactor model. The particle model solver should give a solution each time step of the reactor solver. This is simulated by imposing several different maximum time steps to the solvers, although in our example it does not affect the solution because the boundary conditions are constant. Typical reactor solvers of biomass thermo-chemical processes use time steps in the order of millisec-onds, around 1 ms for fixed-beds [147,148] and 0.1 ms for fluidized-beds [149,150]. It should be noted that a particular time step may be higher than the maximum time step possible with the use of the consecutive or iterative method (called from now on internal maximum time step). Therefore these solution methods should be applied in intervals with length of their maximum internal time step until they achieve in their time evolution the particular time step when they should provide the solution.
When there is no limitation to the maximum time step the multi-step and one-step ODEs solvers are the fastest. Therefore they are the most commonly used methods to solve single-particle models. But when the maximum allowable time step decreases to the order of magnitude typical for reactor simulations they be-come much slower. This is due to the need of initializating the ODEs solver each time step. This is the most time consuming task, because a new Jacobian should be calculated each time step, no matter how small it is. In reactor conditions the boundary conditions will be always different, so the Jacobian of the previous time step could not be used. It could be argued that the initialization of the solvers could be done faster taking information of the previous solution of the solver, which would reduce the computational time needed to calculate the new Jacobian. But this would not allow to use the available solvers as some modifications should be done. And based on these results, where the differences in computational time are of several orders of magnitude, it was not considered to try it due to the conclu-sions will remain the same. A similar behaviour as for the ODEs solvers would be expected for other different iterative methods such as SIMPLE-like methods, which are designed to calculate steady state conditions and take advantage of their implicit character to employ large time steps when the time step is not limited. The multi-step ODEs solver CVODE is faster almost always than the one-multi-step ODEs solver LIMEX. CVODE, which employs BDF, as a multi-step method uses information from several previous time steps, information that is not available in the initializa-tion in every time step. As LIMEX is a one-step solver, its initializainitializa-tion is easier and, therefore, becomes faster when small time steps are used. When the time step decreases to 10−4 seconds LIMEX is faster than CVODE (Fig. 4.3).
In the iterative method the maximum internal time step is fixed depending on the Courant convection characteristic time shown on Tab. 4.3. As this method is solving this phenomenon in an explicit way, the internal time step should be lower
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Figure 4.3: Computational time to integrate the particle model over 100 s under fixed-bed condi-tions (top) and over 10 s under fluidized-bed condicondi-tions (bottom) with different solution methods:
Multi-step ODEs solver -CVODE- (circle), one-step ODEs solver -LIMEX- (rhombus), consecutive (triangle) and iterative (square).