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Oct 17-21, 2007

Xian, China

CFD: Progress and

Prospects

by

Brian Spalding,

of CHAM, Ltd

A lecture at the

Asian Symposium ASCHT-2007

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Oct 17-21, 2007

Xian, China

1. Introduction

1.1 Purpose

Computational fluid dynamics started half a century ago. In this

lecture, I review its progress and seek to indicate how it may

profitably develop further.

I direct my words to research

workers seeking problems which it

is possible and beneficial to solve.

I address also engineers,

especially those working in

process industries, whose

designs can be improved if the

indicated developments are carried

out.

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Oct 17-21, 2007

Xian, China1.2 Patterns of analysis

The problems facing applied science are multi-dimensional; and

they can be approached in various ways.

The main dimensions of variation are in:

• time,• space, and

• population (to be explained below).

Variations in time are easiest to handle,

because we all grow older at the same

rate: one day per day.

Variations in space are more complex,

but easy to understand; for some of us

can run faster than others.

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Oct 17-21, 2007

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Populations which are relevant to CFD include those of:

• liquid droplets with differing diameters;

• solid particles with differing velocities;

• gas ‘fragments’ with differing compositions, or temperatures; and

• radiation fluxes with differing directions.

1.2 Patterns of analysis

Variations in population?

Here is a one-dimensional histogram

representing the distribution of the

age of persons for a particular

community at a particular time;

and here is a picture to show that

histograms can be two-dimensional.

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Oct 17-21, 2007

Xian, China

and I shall argue that, in respect of calculation, the methods which

are used for spatial variations can be applied to population

variations also.


I shall further distinguish the three main approaches to non-

uniformity, whether in time, space or population dimensions,

namely:

• neglect,

• presume,

which means in effect, guess, and

• calculate;

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I shall not argue that 'neglect' is always bad, or that 'calculate' is

always best.

Indeed, most successful approaches are hybrid; thus:

• even the most extreme of the calculators neglect something; and• nearly all presume rather than calculate some non-uniformities.

What is necessary is to make wise decisions about

• what to neglect,

• what to presume,

• what to calculate, and

• when to do each.


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Oct 17-21, 2007

Xian, China1.3 The structure of

the lecture

In part 2, I shall explain my 3-dimension ~ 3-

approach classification; and I shall illustrate it by

way of examples from science and engineering.

In part 3, I shall recommend that CFD specialistsshould provide:

• heat-exchanger designers with software based

on less presumption and more calculation;

• chemical-reactor operators with prediction toolswhich calculate the distribution of fluid fragments

in composition space; and

• mechanical engineers with computer codes

which calculate the flow of fluids and the stressesin solids simultaneously.

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Oct 17-21, 2007

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2. Examples of engineering analysis

2.1 Piston engines; space-direction

variations

The steam engine

For this example, the 'neglect' approach is quite

satisfactory, because the variations of steam temperature

and pressure with position in the space above the piston

are small at any instant of time.

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Oct 17-21, 2007

Xian, China


2.1 Piston engines; space-direction

variations

Internal-combustion engines

The 'presume' approach is best,

especially when flame speed or spray

burning rates are based on experimental

observations.

The 'calculate' approach, i.e. conventional CFD, is often

employed; with limited success. Why? Because

it neglects 'population' aspects of:

(1) turbulent combustion and (2) droplet vaporisation.

Here the 'neglect' approach is not

satisfactory,because flames spread

slowly.

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Oct 17-21, 2007

Xian, China

The plane turbulent mixing layer; non-uniformity in space


2.2 Simpler turbulent flows

I start with the simplest of

all turbulent flows; the

plane mixing layer.

The task is to predict the angle of the wedge-shaped layer of

turbulent fluid at the edge of a jet injected into fluid at rest.

.

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Oct 17-21, 2007

Xian, ChinaShape functions and weighting functions

The presumed-profile approach involves:

• Guess the shapes of the velocity and effective-viscosity

profies, e.g. as sloping or horizontal straight lines

• Multiply the differential equations by weighting functions.

• Integrate across the layer analytically.

• Deduce the angle by algebra.

Advantage: quick and easy.

Disadvantage: accuracy is uncertain.

The 'neglect' approach is not applicable here; for non-

uniformity is of the essence.

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Oct 17-21, 2007

Xian, ChinaThe plane turbulent mixing layer;

the Finite-Volume Method

The ‘calculate’ approach

(version of Patankar and myself, 1967):

This is now known as the 'finite-volume' method' (FVM),the

general form of its equations being:

value in the volume = sum for all faces of coefficient * value in

neighbou r volume + sum of additional sources

wherein the coefficients express diffusion and convection.

• presumes only that the velocity profile is ahistogram, with unknown column heights;

• uses weighting functions of 1, i.e. none at all;

• integrates across each histogram interval;

• deduces the unknowns numerically.

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Oct 17-21, 2007

Xian, China

Other steady-state turbulent jets,

wakes, plumes and boundary layers

The FVM was soon applied to these flows

which:

• had already been extensively studied

experimentally, and by presumed-profile

methods;

• are 'parabolic' (i.e. downstream events

do not influence upstream ones);

• therefore permitted solution by 'marching'methods' on memory-scarce computers;

• allowed turbulence models to be tested;

• gave us confidence to extend the FVM to

recirculating, three-dimensional,

unsteady, compressible and chemically-reactin flows

The early days of CFD; a condensed history

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Oct 17-21, 2007

Xian, China2.3 Steady flow around solid bodies

immersed in fluid streams

• aircraft design was based mainly

on a 'neglect' approach, in that thevariations of stagnation pressurewere neglected. The aerodynamic forces on the aircraft were

then computed by way of ideal-fluid theory.

• The effects of viscosity, and indeed turbulence were expressed by

the supposition that the 'displacement thickness' of thin boundarylayers enveloping wings and fuselage made these, in effect, rather

thicker than they truly were.

• The presumption approach was used, however, to calculate

the displacement-thickness distribution; so the whole method can be

characterised as being 'hybrid'.

Streamlined objects

Before CFD,

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Oct 17-21, 2007

Xian, ChinaCurrent practice

Now that CFD exists,

• the calculation' approach is adopted for the whole of the

space occupied by the fluid; which allows also the small

regions of 'separated flow’ to be simulated.

• However, an accurate calculation of the frictional

forces on the solid surface can be made only by the use

a very fine grid in the boundary layer;

• so, for economy, some element of profile-

presumption is retained, by way of wall functions.

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Oct 17-21, 2007

Xian, ChinaFlows around and inside buildings

• Before CFD, flow prediction was

based on experiments with small

geometrically similar physical

models;

• but this was unreliable , because

the similarity criteria of Reynolds

(viscosity) and Froude (buoyancy)

could not both be satisfied.

• Neither the neglect nor presume approaches had

anything to offer. Therefore, engineers concerned with

heating, ventilating, air-conditioning and fire-

protection of buildings were among the first to turn to

CFD.

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Oct 17-21, 2007

Xian, ChinaFlows around and inside buildings

• CFD has satisfied their requirements; and

• it is for widely used for simulating fires in car-parks and other

buildings;

• BUT, for phenomena such as the fire-ball, it needs to takeaccount of variations in hot-gas-population space.

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Oct 17-21, 2007

Xian, China

Heat exchangers; non-uniformities in space

2.4 Chemical-engineering equipment

No designer can 'neglect' the

temperature variations in heat

exchangers.

Instead, most guess them as

being similar to that calculated for

idealised counter-flow systems.

Since they know that the flow

patterns must differ, they multiplytheir calculated heat-transfer rates

by correction factors like those

on the right.

But these are still guesses, nonethe less.

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Oct 17-21, 2007

Xian, China

Heat exchangers; non-uniformities in

space (end)

These presumption practices derive from the pre-CFD age.

However, it was shown more than thirty years ago (by

Patankar and myself, as it happens), that the calculateapproach is practicable and indeed easy.

It is strange therefore that most heat exchangers

today are still based on presumption rather than

calculation.

Therefore, in section 3.1, below, I shall be

recommending a change of practice.

Sti d h i l t h i

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Oct 17-21, 2007

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Stirred chemical reactors, showing

variations in both space and

population

The process:

Many chemicals products are

created by pumping feedstock

materials (A and B) into a reactor

vessel, where they are stirredtogether by a paddle, in order to

react chemically.

The task is to predict how the rate of production of C from reactants A and B

depends upon the power consumed by

stirring and the rate when mixed in a test-

tube, where: rate/(con cA *con cB ) = k_tube .

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Oct 17-21, 2007

Xian, ChinaStirred chemical reactors

Variations of time-averaged concentration

Before CFD,

the 'neglect' approach had to be used for variations

with position; and it was not bad; for, if the stirring is

vigorous enough, the time-average values of concA and

concB will indeed be almost uniform.

But what about moderate stirring?

The 'presume' approach is not usable in this case; for no

guidance exists as to what profiles should be presumed.

Nowadays, CFD is employed; but it is not enough;

for, if R_ave / (co ncA_ave * concB_ave)= k_reactor ,

it is found experimentally is that

k_reactor is much less than k_tube . Why is this?

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Oct 17-21, 2007

Xian, ChinaStirred chemical reactors

Variations in population space

The answer: non-uniformity in population

space, also called unmixedness, shown here ->

At any point in the reactor, fluid fragments of

many different concentrations can be found.

To calculate their time-average values, one must

know for what proportion of time each is

present.

That means that one needs a probability-

density function, like this --->

Can one calculate it? Yes, as I shall explain later;

and for each location and stirring rate too.

From it can be deduced the C- production rate.

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Oct 17-21, 2007

Xian, China2.5 Simpler non-uniformities in

population: droplet-size

Vaporization of fuel sprays (in Diesels

or gas turbines) consisting of droplets of

various diameters, D, which change size

at a rate governed by :

- dD/dT = const * (1/D) * ln(1+B)where B, the driving force for mass

transfer, depends upon (e.g.) local

temperatures and other gas properties.

This shows that droplets diminish in size

at different rates, the smaller onesdisappearing the more rapidly.

.

The task is to calculate the overall rate of vaporization.

This necessitates knowing the droplet-size distribution at each

location and each time.

V i ti f d l t i

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Oct 17-21, 2007

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Vaporization of a spray; droplet-size

population

The usual three ways are:

1. Neglect variations, i.e. suppose that all the droplets at a

single location in the spray have the same diameter.

2. Presume that the profile isconstant (e. g .) of Rosin-

Rammler form, which cannot

be very accurate.

3. Calculate the ordinates of the

histogram by way of astandard finite-volume

equation, with the source term

dD/dT above.

Use calculate if droplet size is

critical, as in fire extinction.

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Oct 17-21, 2007

Xian, ChinaThe turbulent diffusion flame;

fuel-air-ratio population

Experimentally-observed unmixedness

Hottel, Weddell and Hawthorne drew attention in 1949

to the 'unmixedness' of the gases in a flame

produced by a jet of fuel gas injected into air.

They measured finite time-average concentrations of

both fuel and oxygen at the same location.

That could never be found in a laminar flame.

The first CFD analyses

It was not until 1971 that the first attempt to simulate

this unmixedness numerically was made, on the basis

of a very simple profile presumption.

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Oct 17-21, 2007

Xian, ChinaThe turbulent diffusion flame;

presumed fuel-air-ratio population

The guess was that, at a

point where the time-

average fuel-air ratio was

F, say, the gases

actually present therehad the ratio

F+ g for half the time,

and

F- g for the other half .

Standard CFD calculated F easily.

For g, a new differential equations was invented, having sources

guessed as being proportional to gradients of F- and velocity.

This approach, when appropriate empirical constants were

introduced, allowed turbulent diffusion flames to be simulated.

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Oct 17-21, 2007

Xian, ChinaConfined pre-mixed flame;

reactedness population

In the turbulent diffusion flame,

fuel and air enter separately, and

must be mixed before chemical

reaction can occur, at a rate

limited by the rate of that mixing.

I now consider a flow in which the fuel and air are mixed before they

enter, at uniform and constant velocity, a plane-walled duct in which

is placed a bluff-body 'flame- holder'.

A turbulent wedge-shaped flame spreads across the duct, as the

sketch indicates; and the profile of longitudinal velocity is roughly as

shown.

What then limits its rate? A different kind of mixing: that betweenburned and unburned gases.

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Oct 17-21, 2007


the near-constancy of its angle

When first investigated, this flame showed some

puzzling features, namely that the wedge angle

was almost independent of:

• inlet velocity

• fuel-air ratio;• inlet temperature;

• pressure; and

• inlet turbulence intensity.

But why?

H.S. Tsien, while at CalTech, explained theshape of the profile; but what governed

its angle remained a mystery.

We learned only later

• non-uniformity in space depends on

• non-uniformity in population.

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Oct 17-21, 2007


the first population presumption

The guessed profile

The first idea, embodied in the so-called eddy-break-up model ,

was that the gas population consisted of two components, namely:

The histogram representing the

presumed population thereforeconsisted of two spikes; and

their relative heights dictated

what would be measured as the

time-average temperature.

(1) fragments of wholly un-burned gas which were

too cold to burn; and(2) fragments of hot wholly-burned gas which also

could not burn because either all the fuel or all the

oxygen had been consumed.

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collision between burned and unburned gas

fragments

These latter, being sufficiently hot and also containing

reactants, could burn; and did so very rapidly, thereby increasing

the height of the right-hand spike. Their actual concentration

was considered, implicitly, to be negligibly small.

The rate of collision per unit volume was guessed as proportional

to the rate of dissipation of turbulence energy.

This explained why the flame angle remained almost unchanged

when the inflow velocity was increased.

These two elements of the population were

thought of as colliding with one another and

thereby producing sub-fragments of

intermediate temperature and composition.

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Oct 17-21, 2007


the next presumed reactedness profile

The four-fluid model

The EBU, published in 1970, became very popular; so much so that

25 years passed before the obvious next step was taken;:

to increase the number of presumed components from 2 to 4 !

Collisions between fluids

1 and 3 created fluid 2,

2 and 4 created fluid 3,

1 and 4 created fluid 2

and also fluid 3.

Reaction of fluid 3

created fluid 4

at a chemistry-

controlled rate.Fluids: 1 2 3 4

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Oct 17-21, 2007


applications of the four-fluid model

The chemistry-controlled step (fluid 3 creates fluid 4) explained:

why:

1. the flame angle remained nearly constant, and

2. the flame could be suddenly extinguished by a velocity increase.

The four-fluid model wasused successfully for

simulating flame spread in a

baffled duct and for oil-

platform explosion

simulation.It has been little used; but it

was the first step towards

calculating the reactedness

population,

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Oct 17-21, 2007

Xian, China

In conventional CFD, we

divide space and time into as

many intervals as we need.

Why not do the same for the

reactedness at each point?

The height of each column can

then be deduced from a

From four fluids to many:

the multi-fluid model

Finite-Interval equation’ like this:

height of interval= sum for all faces of coefficient *

height of neighbour interval +

sum of additional sources +

sum for al l other intervals of coeff ic ient *

height o f other interval )

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Oct 17-21, 2007

Xian, ChinaWhat the terms in the finite-interval

equation represent

In: height of interval= sum for all faces of coeff ic ient *

height of neighbou r interval +

the coefficients express rates of convection and diffusion, as in

the the finite-volume equations of conventional CFD.

But in: sum for all other intervals of coeff ic ient *

height of other interval

the coefficients express the physical and chemical processes:

• collision between members of the fluid population, and

• chemical conversion of one member into another.

The finite-interval method is thus merely a natural extension of

the finite-volume method; and its equations can be solved in the

familiar successive-substitution manner.

The calculation of population distributions is easy.

O t 17 21 2007

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Oct 17-21, 2007

Xian, ChinaHow material is distributed after

collision

Here is a diagram from one

of the earliest publications.

It depicts one of the possible

hypotheses, called

'Promiscuous Mendelian'.

The 'colliders' are treated as 'mother' and 'father’; and the word

'promiscous' implies that any two members of the population may

collide.

The word Mendelian, a reference to Gregor

Mendel, the Austrian "father of modern

genetics", implies that the offspring may

appear with equal probability in any

interval between those of the parents.

O t 17 21 2007

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Oct 17-21, 2007

Xian, ChinaA calculated probability-density

function

This hypothesis has been embodied in

the PHOENICS computer code.

Here is one reactedness histogram,

computed with its aid.

As in the the eddy-break-up guess, thereare indeed spikes at zero and unity

reactedness;

but calculation has shown that the

intervals in-between are alsopopulated.

Such probability distributions can to be computed for

each location in the flame. Then the desired reaction

rate for the whole flame can be deduced.

O t 17 21 2007

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Oct 17-21, 2007

Xian, ChinaApplication to gas-turbine

combustion

A three-dimensional gaseous-fuel combustor

I show here one sector of a simple combustor

proposed by Professor Wu Chung-Hua in the

early days of PHOENICS.

O t 17 21 2007

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Oct 17-21, 2007

Xian, ChinaSmoke formation rate is influenced

by turbulent fluctutions

Much later, I used this combustor to

show how one must not neglect

fluctuations of fuel-air ratio when

predicting smoke formation.

The differences, although

small. are significant when

CFD is being used to optimise

the design.

I used a 10-fluid model,

with fuel-air-ratio as the

population-defining

attribute. Each cell had

its own computed

histogram

O t 17 21 2007

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Oct 17-21, 2007

Xian, ChinaConcluding remarks for Part 2

It has been shown that:

1. variations in population space should not be neglected

especially when chemical reaction is involved;

2. they can be presumed;3. but it is better to calculate them.

Why are not crowds of

researchers pouring intothis scarce-explored

territory?

Perhaps because they are

waiting for less-timid

crowds to do so first.

Oct 17 21 20073 Recommendations

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Oct 17-21, 2007

Xian, China3. Recommendations

3.1 To heat-exchanger designers

Current practice

I have already mentioned that heat exchangers are stilldesigned in the basis of presumption.

A shell-and-tube heat

exchanger looking like

this (tubes not shown)can be expected to have

a rather complex flow in

the shell.

So far, I have been discussing general ideas. Now I wish to make

three specific recommendations.

Oct 17 21 2007

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Oct 17-21, 2007

Xian, China

But why presume when one

can calculate, as was shownto be possible by the 35-year-

old publication in which this

image appeared?

or

Yet the software used by designers presumes that the flow in the

shell can be conceptualized thus, and described by very few

parameters.

Oct 17 21 20073 1 T h t h d i

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Oct 17-21, 2007

Xian, China3.1 To heat exchanger designers

The solution

The solution is:

1. do not attempt to calculate the flow pattern

between the tubes in detail, because current

computers are not large or fast enough to

handle the necessary fine grids except for afew tubes at a time.

2. Instead, use the space-averaged approach, with empirically-

based formulae for:

heat-transfer coefficients per unit volume, and

friction factors per unit volume,

as functions of local Reynolds and Prandtl numbers.

3. Then solve the finite-volume equations for (space-averaged)

velocity, pressure, temperature for the shell- and tube-side fluids,

treating both as interpenetrating continua, as is easily possible.

Oct 17 21 20073 1 To heat exchanger designers

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Oct 17-21, 2007

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3.1 To heat exchanger designers

The solution (contd)

I now show some (not new) results for (the central plane of

symmetry of) a particular shell-and-tube heat exchanger.

(a) The shell-side velocity

vectors, when calculated,

appear thus

(b) The consequential shell-side temperatures, are not, as

presumed, a succession of vertical stripes; although the calculated

tube-side temperatures are (very nearly).

Oct 17 21 20073 1 To heat exchanger designers

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Oct 17-21, 2007

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3.1 To heat exchanger designers

The solution (end)

(c) The conventional heat-exchanger-design packages presume

that the shell-side, tube-side and overall heat-transfer coefficients

are uniform throughout; but calculation reveals that they are not,

as the next pictures clearly demonstrate.

Corresponding non-uniformities are exhibited by the calculated

Reynolds- and Prandtl-number values, and the temperature-

dependent fluid properties, from which the heat-transfer coefficients have been computed.

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Oct 17-21 20073 2 To stirred reactor designers and

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Oct 17 21, 2007

Xian, China

But what about the mixture-ratio population grid?

Two distinct cases were considered, namely that:

1. the materials from the entering streams of reactants A and B

were fully mixed at each point in the reactor, which would

correspond to presuming • that its pdf was the single spike shown on the following

diagram, and that

• the amount of product C was as indicated by its horizontal

location;

2. alternatively, at each point there

could be found varying amounts

of 'fluids' (in the multi-fluid sense)

having one of eleven distinct

mixture ratios, so that its pdf

could be that of the histogram

3.2 To stirred-reactor designers and

operators (contd)

Oct 17-21 20073 2 To stirred reactor designers and

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Oct 17 21, 2007

Xian, China3.2 To stirred-reactor designers and

operators (contd)

Case 1 is the conventional-CFD approach which presumes the

state of the mixture-ratio population; and Case 2 represents what is

done by those who recognise that non-uniformities in population

space can be calculated.

The results of the twoapproaches are different. This

is demonstrated by the

following two contour

diagrams showing the product

(i.e. C) concentrations after 10revolutions.

The general patterns are not very dissimilar; but their scales are:

3.2 for the presumption approach and only

2.4 for the calculation approach, at this moment of time.

Oct 17-21 20073 2 To stirred-reactor designers and

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F l u i d F l o w

Oct 17 21, 2007

Xian, China3.2 To stirred-reactor designers and

operators (contd)

The explanation for the difference is to be found in the calculated

mixture-ratio histograms, of which a few will be shown,

corresponding to a single instant of time, a single vertical height and

circumferential angle, and at six different radii, starting near the axis

and moving outward.

These pdf histograms show that:

• detailed information about the micro-mixing can indeed be

obtained by calculation;

• the pdfs vary is shape in a manner that it would be impossible to

guess;

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Oct 17-21, 2007Why one method can suffice for

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,

Xian, ChinaWhy one method can suffice for

both classes of problem

The reasons are:

1. The differential equations for velocities in fluids are very

similar to those for displacements in solids, from which the

stresses can be deduced. Thus

[del**2]* u - [d/dx]* [ p*c1 ] + fx*c2 + convection terms= 0

for velocity, and

[del**2]* U + [d/dx]* [ D*C1 - Te*C3 ] + Fx*C2 = 0

for displacement.

2. The solid-stress equations are indeed the simpler , being linear

where the former are non-linear .

3. Since the solid-stress problem is simpler than the fluid-flow one,

computer codes written for the latter can easily serve for the

former also, as many publications have proved.

Oct 17-21, 2007A thermal-stress example

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The block is heated at

various several points so

that its thermal expansion

is non-uniform.

A thermal-stress example

The three examples which I shall show are several years

old; for I wish to emphasise that my message is not a new

one. But it has suffered from neglect.

First, a cooling fluid flows through

a pressurised curved duct ina solid block.

Oct 17-21, 2007Thermal and mechanical fluid-

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The equations for velocity and

displacement and velocity are so similar

that PHOENICS solves both sets at the

same time. Here the solutions are

presented in terms of vectors.

Thermal and mechanical fluid

structure interactions

In my second example, the fluid-

structure interaction is mechanical rather than thermal. A thin partition

bends as a consequence of the

differences of fluid pressure on

its two sides.

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Oct 17-21, 2007Recommendation number 3

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Xian, ChinaRecommendation number 3

My third recommendation is therefore that:

• Researchers should develop and refine the finite-

volume method for simultaneous fluid-flow and solid-stress calculation;

and

• Engineers concerned with fluid-structure interactions

should demand computer codes which embody

those methods.

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