ROTULACIÓN DE LA CARPETA

On most process plants, in order to minimise cost, pressure loss, erosion, particle deposition and thermal losses through interconnecting ducts or flues, the carrier gas velocity, at full normal plant output, is typically in the order of 13–20 m s−1. In order to reduce this duct velocity to around 1.5 m s−1 in the precipitation field, some type of diffuser is required having an expansion ratio in the order of 10 or more dependent on the design precipitator gas velocity and type of particle to be collected. The design of this transition section, where kinetic energy is converted to potential energy, usually as a pressure drop, is critical to achieve a uniform flow profile through the precipitation field if optimum performance is to be obtained. With many installations, cost and space restrictions mean that sharp angled diffusers are used to decelerate the gas velocity and distribute the flow through the precipitator. There are two main types of diffuser used in practice; one is to use a fully splittered transition section, based on an ‘egg-box’ principle to spread the gas followed by a smoothing screen further downstream across the face of thefield. This approach is illustrated in Figure 3.9.

The other type of approach is to use a series of perforated screens, a technique developed from wind tunnel diffusers using grids or nets. If the transition is short, having steep angles, the flow will separate from the duct walls creating a central high velocity jet, which hits the perforated plate and spreads sideways, resulting in a recirculating gas flow adjacent to the walls. To minimise this effect it is normal to have a series of perforated plates having different porosities within the transition section to obtain the degree of flow smoothing required. Although both approaches can be found in practice, the advantage of the split- tered transition is a lower pressure loss of some 0.5 kPa on the overall system.

For either system to be fully effective, in terms of ultimate performance of the precipitator, it is important that the gas upstream of the transition is uniform in

terms of flow, temperature and dust distribution, any disturbance in the flow leading to non-uniformity of velocity within the precipitation field.

Although ideally, the gas velocity profile through the precipitation field should be uniform, in practice, because of boundary layer drag and areas above and beneath the collector system where the gas can expand, it has been generally agreed throughout the precipitation industry that an acceptable distribution is one where the ‘root mean square’ deviation is no more than 15 per cent [6].

Although most precipitator suppliers have adopted this standard, there has been a recent trend to consider a ‘skewed’ distribution [7], particularly in the vertical plane, for units having a high carbon carryover or very high particulate loadings. In these instances, some precipitators have had the gas distribution modified such as to increase the gas velocity in the lower part of the inlet field to be some 30 per cent higher than the top of the field, while in the outlet field the converse flow profile is adopted, as indicated in Figure 3.10.

The theory of this form of distribution is that at the inlet, with higher mass loadings or readily entrainable dusts, the bulk of the collected material has the least distance to fall after release from the collectors, which should reduce re- entrainment into the downstream sections of the precipitator. At the outlet, by having the high flow towards the roof of the precipitator, any dust released from the collectors will pass through a lower axial velocity area, which again should reduce any incipient re-entrainment.

Regardless of whether a standard or skewed approach is considered, it is important that the gas distribution in the horizontal plane is maintained as uniform as possible, otherwise the performance will be adversely affected. A further practical consideration is the avoidance/minimisation of gas looping (bypassing) over or under the actual precipitation field. This bypass results from the slightly higher pressure drop across the field because of the internals. To minimise the risk of this bypassing impacting on overall performance, small ‘kicker’ baffles are installed at the top and bottom of the field to break the gas Figure 3.9 Inlet splittering to obtain uniform gas flow

from the walls and divert the flow into the field area. Although this is a palliative solution, since gas will still tend to loop, plant performance measurements have produced emissions of less than 1 mg Nm−3 on some high efficiency installations. To achieve an acceptable gas distribution it is normal practice to carry out testing, either in the field to confirm results obtained from large scale models of the system (∼1/10 full scale) or data based on computational fluid dynamics (CFD) approaches. Although correcting gas distribution in the field is possible, time constraints and the fact that a single splitter plate on large installations can weigh some 500 kg, make this approach impracticable except for the smallest installations.

3.7.1 Correction by model testing

In the case of large scale modelling, it is important that geometrical, kinematic and dynamic similarity is achieved between the model and full scale plant. The physics of flow is described by three Navier–Stokes equations, one per coordin- ate in space and the equation of continuity. By making the equations dimension- less, in terms of mass, length and time, it can be shown that the solution is universal, providing the numbers appearing in the equations are equal in the model and full scale. These include the Reynolds number, Re, the Mach number

Ma and the Euler number Eu. (Note that the Euler number is automatically fulfilled as it depends on the Reynolds number.)

Figure 3.10 Skewed distribution profile

The Reynolds number, Re, is the ratio between the flow inertia and friction

Re= vd/η. (3.4)

The Mach number, Ma, is the ratio between the gas flow velocity and the speed of sound

Ma= v/a = v/(kRT)½_{, (3.5)}

where v is the gas velocity (in m s−1), d is the characteristic length (in m), η is the kinematic viscosity, a is the speed of sound (in m s−1), k is the adiabatic constant,

R is the universal gas constant and T is the absolute temperature.

When using a 1/10 scale model of a power station cold side precipitator, under the same temperature and gas conditions, ‘exact similarity’ would involve run- ning the model with a field velocity some ten times higher than the full scale plant. In practice, because the gas composition, and hence viscosity and tem- perature, are very different, plus reducing the number of collector plates in the model effectively to increase the Reynolds number, the velocity in the model will reduce to around six times the actual for similarity. This velocity, however, may be too high in some duct configurations since the gas will be approaching com- pressibility, which will impact on the flow characteristics. To investigate this, by operating the model at several different but lower velocities, the significance of operating at lower Reynolds numbers on the precipitator gas distribution can be established.

Generally by operating at a model velocity of around twice that of the full scale plant, acceptable correlation with the field measurements can be obtained. During the construction of both model and full scale units it is imperative that the positioning of all splitters, baffles and any ‘kicker’ plates to minimise bypass are faithfully reproduced if correlation is to be achieved.

While the inlet transition determines the distribution profile through the pre- cipitator inlet fields, a steep angled outlet transition can impact on the flow in the outlet field and it is normal for splitter plates to be included at the outlet to maintain control of the gas flow.

3.7.2 Computational fluid dynamic approach

In recent years the development of powerful and fast computers has meant that solving the differential equations describing fluid motion is now possible. Physic- ally the conservation of mass and Newton’s second law are applied to the fluid, expressed mathematically as the equation of continuity plus the Navier–Stokes equation, in two or three dimensions. Solving these equations is a discipline defined as direct numerical simulation, or generally computational fluid dynamics. Even with today’s fastest supercomputers, it would be impossible to divide the calculation domain into small enough parts to describe all the details in the flow field. Solving the equations directly is a discipline called direct numerical simula- tion, which is still restricted to very limited Reynolds numbers and small geometries.

With the smallest eddies, for example as in the Kolmogorov length scale

冢

u3

冣

, where v is the kinematic viscosity ≈40 × 10−6 _m2 _s−1_{,δ is a typical}

shear layer thickness ≈10−1 _{m and U is the bulk velocity ≈1 m s}−1_{, the length scale}

is of order 0.3 mm. This means that a precipitator for a power plant installation having approximate dimensions of 15 × 15 × 20 m3 _{should have a mesh number}

of around one hundred million millions, far beyond the capacity of most com- puters to tackle direct numerical simulation.

Instead, the turbulent variables are taken as average values plus fluctuating parts; for example, u= U + u′, where u is the instantaneous x velocity, U is the time average velocity and u′ the time-dependent fluctuation (time average of

u′ = 0).

Introducing these variables into the Navier–Stokes equation and time averaging, leads to:

ρdUav/dt= ρg − ∇ρav+ ∇τij, (3.6)

where the stress tensor

τij= μ(dui/dxj+ duj/dxi)− ρ(ui′ uj′)av. (3.7)

Thefirst term is the laminar stress, and the second is the turbulent stress. The turbulent part of the stress tensor is either found by solving the so-called Reynolds stress equations or simply by expressing it using average flow values and the so-called Boussinesq assumption

τi= μi/dU/dy,

where μi is the eddy (or turbulent) viscosity, U is an average velocity and y is a

coordinate perpendicular to the vector U.

The value of μi is determined by using either Prandtl’s mixing length theory,

that is ut≡ pl2|dU/dy| (where l is the so-called mixing length), or by using the ‘k–ε’

model:μt≡ Cμk2/ε, where Cμ is a constant or coefficient, k is the turbulent kinetic

energy, k=½(u′2_{+ v′}2_{+ w′}2_{), and ε is the turbulent dissipation, ε = k}3/2_/L d,

where Ld is a length scale for the dissipating eddies.

The more complex the turbulence model is, the more differential equations must be introduced, which can only be solved by closing the system by introducing algebraic expressions based upon empiricism.

The most common methods today are based on transformation of the differ- ential equations for conservation of energy, mass and momentum to difference equations, which can be solved by integration after applying the given boundary conditions. Originally, in two dimensions, the equations were transformed and solved using stream-function values Ψ(ρu = dΨ/dy, ρv = dΨ/dx) and vorticity w (wz= dv/dx − du/dx), thus eliminating pressure as a variable. Later, with more

efficient computers, the equations were formulated and solved in the primitive variables, pressure and velocity, in three dimensions, thus avoiding the problem of defining the exact boundary conditions for vorticity having steep gradients close to the walls. As velocity gradients are also steep at the walls, special

attention must be paid to the description of analytical near-wall variations, the so-called wall functions.

The difference equations are often solved in Cartesian or orthogonal grids, equidistant or almost equidistant, using ‘upwind differences’ and taking into consideration the local direction of flow. One problem with the grids is the appearance of ‘numerical diffusion’, which is a maximum if the flow vector has a 45° angle with the grid axes. In recent years, advanced grids have been developed, e.g. ‘adaptive grids’, where the mesh size changes according to the gradients of the variables, decreasing in size where gradients are steep. Another approach is the use of ‘multigrids’, where calculation shifts between a fine and a coarse mesh, thus smoothing out short wave and long wave variations.

The difference equations are normally solved indirectly (integrated) by iter- ation, sometimes several hundreds of iterations, until numerical stability is achieved by successive under or over relaxation approaches. The normal cri- terion for having found the final solution is one of mass continuity, and the calculation stops when the maximum residue is less than, say, 10−4. The ‘solver procedure’, being a chapter in itself outside the scope of this book, typically uses various numerical principles, continually updating the values in the domain as soon as new values have been found. A number of fast and efficient solvers are now commercially available and are in common usage for CFD work.

With more complicated geometries, such as precipitators including compli- cated transition pieces, guide vanes and screens, grid designing is difficult. A simple method is to use a parallelepipedal domain and block out all the outer elements until an approximate contour is achieved, leaving the geometric domain boundary as a step surface. This means that there is a limit to how precise the solution close to the walls can be, even though the internal flow is only slightly influenced. In fact, it is a grid generator, which is the main issue for the operator. The easier the grid generation, the more different configura- tions can be calculated in a reasonable time, presuming, of course, that the solver is effective and fast. Less than 2 h per new contour and less than 1 h per modification would be considered acceptable.

The real progress in mesh generation is a mesh that can be fitted without any consideration of the interfacing, at least as seen from the point of view of the user. Up to now, mesh structures have had to fit where ducts and transition pieces, or transition pieces and precipitator housing, meet, but new improved grid systems are making it possible to have an essentially polar mesh of a circu- lar configuration, corresponding with a rectangular mesh of a box, without any concern about mesh fitting at the plane where they meet. This block type of approach is well suited for duct and precipitator analysis with respect to both gas distribution and pressure drop determination. Figure 3.11 shows this type of mesh from the code Star-CD [8]. Other forms of program are commercially available from, for example, Fluent Inc. [9].

The more advanced the mesh, the more variables to be treated and the stricter the convergence criteria, the more calculation time is needed to find a correct solution. The need for space in the memory and on the disc is another ‘eye of a

needle’, the cell number increasing roughly with the product of the number of variables and the reciprocal of the mesh size to the third power. While details such as guide vanes, kicker plates and ladder vanes have to be modelled separately, there is a possibility to simulate perforated screens using some sort of porosity model such as a Darcy porosity.

Screens with evenly distributed guide vanes might be modelled as a whole, but this calls for special routines, not normally commercially available. In the equa- tions the so-called ‘source’ terms can be modified in order to reflect the influence of the gas distribution screens [10]. Screens can also be modelled by ‘blocking’ out cells, but this procedure demands a very fine mesh with many nodes. The flow before a perforated screen is often recirculating, unless small screen guide vanes are used. Such flows can be identified using CFD, revealing velocity vectors parallel to the screen, making it impossible to improve the distribution by modi- fying the screen by increasing or decreasing the open area alone. Programs that cannot treat small thin oblique surfaces in an effective and easy way should not be used. As in the use of porosity for simulating a perforated screen, it is possible to design a subroutine for a screen combined with distributed guide vanes and use it as a black box when the calculation is operated in the screen domain, which would reduce the need for a complex mesh with a large number of grid points.