7.1 Transient flow in a paddle-stirred reactor

(a) The problem

The productivity of the reactor depends greatly on the effectiveness of the micro-mixing; but until the arrival of the Multi-Fluid Model, there was no practicable way of predicting this.

The prediction problem is complicated by the facts that the geometry is three-dimensional; and unsteady analysis is needed in order to represent properly the effects of the stirring paddle and of the rotation-impeding fixed baffles.

In the following extract from a recent study, PHOENICS has been used for the simulating such a reactor; and the use of the multi-fluid model has revealed the importance of being able to simulate the micro-mixing process.

The reactor is one for which the US Dow Corporation has reported experimental data (only hydrodynamic data, alas!) which have been used in "benchmark studies".

The geometry and computational domain are shown below.

The impeller speed is 500 rpm, the dynamic laminar viscosity is 1.0cP and the water density is 1000 kg/m

The computational grid is divided into two parts, namely an inner part which rotates at the same speed as the impeller, and an outer part which is at rest.

the geometry

The total number of cells is 31365 (45 vertical, 41 radial and 17 circumferential).

the grid

__________|.|__________ The sketch illustrates the apparatus and | ||| | the initial state of the two liquids. | upper |.| | | liquid ||| | They are both at rest, and are separated | |.| acid | by a horizontal interface |..........|||..........| | lower |.| alkali | The paddle is supposed to be suddenly set | liquid ||| | in motion. | --------- | |paddle ///////// | The computational task is to predict both | . | the MACRO-mixing, represented by the ------------|------------ subsequent distributions of time-average .< axis of pressure, velocity and concentration, and rotation the MICRO-mixing, ie the extent to which the The paddle-stirred two liquids are mixed together at any point. reactor

There is no need, simply because MFM is available, to refrain entirely from using conventional models.

In the present study, therefore, the k-epsilon model was used as the source of the length-scale, effective-viscosity and micro-mixing information.

This allowed the power of the MFM to be concentrated on the process which it alone can simulate, namely micro-mixing and the subsequent chemical reaction.

An eleven-fluid model was employed, with mass-fraction of material from the top half of the reactor (ie the acid) as the population- distinguishing attribute.

This was probably not sufficient to permit achievement of grid- independent of solutions. However, one of the merits of MFM is that population-grid-refinement studies are easily performed.

The simulation procedure therefore generates a very large volume of data, of which only a tiny fraction can be presented here.

What will first be shown is a series of PDFs. They all pertain to:

- a single instant of time, namely that at which the paddle has revolved ten times;
- a single vertical position, namely about one-third of the height from the base; and
- a single radial line.

Specifically, FPDs will be shown for six radial locations, starting near the axis and moving outward to about one third of the radius.

Near the axis, most of the fluid is still alkaline

Farther from the axis, more-acidic fluids are found

This tendency increases with increase of radius

Also the shape of the PDF changes

The shape-change continues

This is the last PDF to be shown

The acid and the alkali are supposed to react chemically in accordance with the classic prescription:

acid + base -> salt + water

Of course, no chemical reaction can occur in the unmixed fluids, which (to use the parent-offspring analogy) are the Adam and Eve of the whole population. It is only their descendants, with both acid and alkali "in their blood", who can produce any salt.

This is why the prediction of the yield of salt necessitates knowledge of the concentrations of the offspring fluids.

The next picture shows the salt concentrations after 10 paddle rotations which the multi-fluid model has predicted.

These concentrations are the averages for all eleven fluids.

The salt concentrations predicted by the multi-fluid model

They are greatest in the region which has been most vigorously stirred by the paddle, which tends to thrust fluid downwards and outwards.

If no account is taken of the existence of the fluctuations, as is perforce the usual case, only the mean value of the acid/base ratio is available.

If the salt-concentration distribution is calculated from this, as the next picture will show, the values will be different from before.

Indeed it can be expected that they will be larger, because the implication of using a single-fluid model (which is what neglect of fluctuations amounts to) is that micro-mixing is perfect.

This is indeed what is revealed by the calculations.

The salt concentrations predicted by the single-fluid model

They are larger than the multi- fluid values, because micro- mixing is presumed (wrongly) to be perfect.

The following conclusions appear to be justified by the studies conducted so far:-

- It is possible, and not very expensive, to compute the probability-density functions describing the extent of micro- mixing of initially-separated materials in a stirred tank.
- The 11 fluids used in the calculations were probably too few; but any number can be used (100 is common) so that (fluid-) grid- independence can be tested.
- When a chemical reaction can take place between the materials, the yield can be computed accurately only by way of a multi- fluid model.
- Even without further refinement, MFM is probably more reliable than any presumed-pdf method, because it predicts instead of presuming.

(a) The problem and its solution

Guessing this is dangerous; but MFM can compute it.

The following pictures result from exercising another case from the
PHOENICS library. The smoke-kinetics formula is highly idealised, so
as not to obscure the main message: **
MFM can be used in practice now.**

The pictures show the distributions of smoke concentrations at the outlet cross-section based on:-

- conventional single-fluid model;
- a five-fluid model;
- a ten-fluid model; and
- a twenty-fluid model.

Smoke predicted by 1-fluid model

Smoke predicted by 5-fluid model

Smoke predicted by 10-fluid model

Smoke predicted by 20-fluid model

Computer times are, in any case, not very large, that for 20 fluids being only three times that for a single fluid.

However, when 100 fluids (say) do prove to be necessary on grounds of accuracy, there are many available means of reducing the computer times.

For example, there is no need to employ the same number in all parts of the field; instead, the number can be varied according to the local behaviour of the solution.

Once, indeed, that it is recognised that MFM entails nothing more than discretizing dependent variables in the same way as is routine for independent ones (space and time), the well-known techniques of grid-adaptation become available.

(a) Why turbo-machinery designers need MFM

Axial-flow compressors and turbines, as used in aircraft propulsion and in ground- (or sea-) level power production, are characterised by the rapid passing of one blade row behind another.

The slower-moving boundary-layer fluid from the upstream row becomes a "wake" of slower-moving fluid fragments, which are distributed across the entrance plane of the downstream row.

The turbulent mixture which passes from row to row through a turbo- machine is therefore best represented as a population of fluids, with (say) axial velocity as their distinguishing characteristic.

MFM, provides a practicable means of calculating the population distribution and its influence on the mean flow, ie the "losses", which Kolmogorov-type models are failing to predict correctly.

The lowest member of the MFM family is the two-fluid model (Spalding, 1987), with which some recent studies have been made.

There follow two pictures which show how the time-mean velocity distribution of a blade row differs according to whether a two- fluid or (as is customary) a single-fluid model is presumed.

The differences are qualitatively similar; but the small quantitative differences are what count when blade-row losses are to be computed.

If two-fluid calculations can already provide meaningful guidance to turbo-machinery designers, much more can be expected from the full MFM treatment.

Unfortunately, most turbo-machinery researchers still follow each other down the approach-1 tunnel, with no light at the end! To switch some effort to MFM would appear timely.

Fig 7.3-1 Radial-velocity contours at outlet ; 1 fluid

Fig 7.3-2 Radial-velocity contours at outlet ; 2 fluids