Fluid flow in process units - mycourses.aalto.fi

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Fluid flow in process units

1 Introduction ......................................................................................................................................... 3

2 Mechanical energy balances ................................................................................................................ 3

2.1 Friction ......................................................................................................................................... 4

2.2 Friction in empty pipes ................................................................................................................. 4

2.3 Local friction losses....................................................................................................................... 6

2.4 System curve ................................................................................................................................ 6

2.5 Pump curve .................................................................................................................................. 6

2.6 Pump cavitation ........................................................................................................................... 7

2.7 Fans, Blowers and Compressors.................................................................................................... 7

2.8 Flow in porous media ................................................................................................................... 8

3 Momentum balances ..........................................................................................................................10

3.1 Mechanical energy vs. momentum balances ................................................................................10

3.2 Newton's viscosity law .................................................................................................................11

3.3 Laminar flow in a pipe .................................................................................................................12

3.4 General momentum balances ......................................................................................................15

4 Non-Newtonian fluids .........................................................................................................................17

5 Turbulence ..........................................................................................................................................19

5.1 Turbulent energy spectrum .........................................................................................................20

5.2 Turbulence modeling ...................................................................................................................21

5.3 Turbulence near walls..................................................................................................................23

6 Multiphase flow ..................................................................................................................................25

6.1 Flow regime maps and pressure drop ..........................................................................................25

6.2 Interphase forces.........................................................................................................................26

6.3 Fluidization ..................................................................................................................................29

6.4 Two-phase flow in porous media .................................................................................................30

6.5 Capillary pressure and surface wetting ........................................................................................31

7 Mixing .................................................................................................................................................33

7.1 Mixing in stirred tanks .................................................................................................................33

7.2 Multiphase mixing in stirred tanks ...............................................................................................34

7.3 Static mixers ................................................................................................................................36

8 Computational fluid dynamics (CFD)....................................................................................................38

8.1 Evaluation of the grid ..................................................................................................................39

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8.2 Modeling of equipment with moving parts ..................................................................................40

8.3 Computational modeling of multiphase systems ..........................................................................41

8.4 Modeling of moving fluid interfaces ............................................................................................41

9 Flow measurement .............................................................................................................................42

10 Symbols ..............................................................................................................................................43

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1 IntroductionThis course gives the basics for fluid flow on master level studies. It first reviews basics of pump/piping systemdesign using mechanical energy balances. This is typically a BSc level topic, but due to its importance inchemical engineering design, it is reviewed on this course. Most of the fluid flow problems in practice aresolved with this approach appended with a general engineering judgement. In addition to that, this coursegives overview on various fluid flow related issues, such as non-Newtonian flow, flow regime maps in two-phase flows, mixing in stirred tanks and static mixers, and flow in porous media. In addition, turbulence andcomputational fluid dynamics are briefly introduced.

These lecture notes follow the same themes as the course, but are not necessarily presented in precisely thesame order. This lecture note is not either designed to be fully stand-alone; some things are explained verybriefly here and should be explained in more detail during the lectures. You are also by no means restrictedto find information in this lecture note; this is intended to give an overview to help following the lectures andto understand other information sources (books, articles, web pages, presentations, discussions withcolleagues etc.) better.

2 Mechanical energy balancesGeneral energy balance can be considered to be “always valid”, provided all the relevant source terms areproperly taken into account. The other balances always needed are the material balances (total mass orchemical component masses or moles). In many fluid flow related cases also momentum balances areneeded, although this is true for mainly in such situations, where precise flow patterns are needed. In manycases this is not necessary. Equivalence of mass and energy as treated by high energy physics is not eitherneeded in chemical engineering applications; mass and energy can be considered completely independentwith their own balances.

General energy balance is typically divided in two parts: mechanical energy and thermal energy. The reasonto do so is that in different applications one or the other form is only needed. There is naturallytransformation of mechanical energy to thermal energy (due to friction losses) or in some casestransformation of thermal energy to mechanical energy (in heat engines), but these transformations aretypically taken into account by specific source terms.

Mechanical energy balance can be written in the following way:

( ) ( ) ( ) f2aa

2bbababpp hvv21zzgppW r+a-ar+-r+-=rh (1.)

Where the first term on the left is (shaft) work done by the pump, the first term on the right is pressuredifference between the end and the source point, the second term is hydrostatic pressure due to heightdifference between end and source points, the third is pressure change due to velocity (also called dynamicpressure), and the last term on the right is due to friction losses.

This equation without the shaft work by the pump and the friction loss term is called Bernoulli equation,which is then a mechanical energy balance with some assumptions (no work or friction). It is applicable tosome specific cases, although the general form, which is sometimes called Extended Bernoulli equation, ismuch more general and should be used in most cases.

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Out of the various terms, the dynamic pressure is very small compared to the others in almost all cases. Forconstant flow cross-sectional area and incompressible flow, it is precisely zero. Relative difference of theother terms depend on the case; for horizontal pipes the hydrostatic pressure difference is zero, and for flowbetween two vessels open to the same gas space (atmosphere etc.), the pressure difference term is zero.

Note that the term "pressure drop" is often used to estimate mechanical energy loss when it dissipates toheat. This is not necessarily the same as pressure difference, as the latter may include hydrostatic head.Pressure drop in a straight vertical pipe is the same irrespective of flow direction (in one-phase flow), butpressure difference between beginning and end of the pipe depend on the flow direction.

2.1 FrictionThe friction term is typically divided into friction caused by empty pipe (viscous dissipation at pipe walls) andlocal friction losses:

2v

DLh

2

if ÷øö

çèæ z+

Dxr=r å (2.)

The first term in parenthesis describes friction caused by empty pipe (it is proportional to the pipe lengthDL).The proportionality coefficient x is called the Darcy friction factor. The second term in the parenthesis is asum of all the local friction losses caused by pipe bends, valves, instrumentation etc. Even heat exchangers,reactors and other process equipment can be taken into account in a similar way. The formulation is writtenso that dependency on velocity is visible. For a fully turbulent flow, the friction factors in the parenthesis arerelatively constant (independent on velocity), so that pressure drop will be proportional to the velocitysquared.

2.2 Friction in empty pipesFriction (Darcy friction factor) in empty pipes can be estimated from the Moody diagram:

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As can be seen, for a fully turbulent regime, the friction factor becomes constant after sufficiently high flowrate. This depends also on the surface roughness of the pipes, which is a property of the material andmanufacturing method.

As these diagrams are not very convenient for normal design calculations which require iterative solutionsto various variables, equations are typically used. The fully turbulent regime friction can be well estimatedusing the Colebrook equation:

÷÷ø

öççè

æ

x+-=

x Re51.2

D7.3klog0.21

(3.)

As can be seen, the friction factor x is implicit and can be solved only iteratively. There are various explicitapproximations that are of sufficient accuracy for most engineering purposes, for example:

÷÷ø

öççè

æ÷øö

çèæ +--=

x Re13

7.3D/klog

Re02.5

D7.3klog0.21

(4.)

For laminar pipe flow, there is an analytical solution to the friction term. This will be discussed later with themomentum balances.

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2.3 Local friction lossesAll additional disturbances to well-developed pipe flow cause additional pressure losses. These are casedependent, and are calculated as local frictional losses or resistance coefficients. In principle it is ratherstraightforward to look at appropriate contributions and add them for the summation term in equation 2. Itdoes not matter in which order the local losses are in the direction of flow, and how long empty pipes are inbetween. Resistance coefficients may depend on the pipe diameter. Sometimes pipe entry from a vessel isincluded as local friction loss; sometimes it is included in the dynamic pressure discussed earlier. Typicallythis term is small, but should be considered consistently. The largest local pressure drop values are typicallycaused by valves, where significant (typically at least tens of kPa) pressure drop is required even for a fullyopen valve due to controllability reasons. If the pressure drop is higher somewhere else, changing the valveposition would not make a significant modification to the flow rate, which is not desirable situation. If wholeequipment (heat exchangers etc.) are included as local pressure drops, these could be of comparable tocontrol valve pressure drop values, but preferably not higher.

2.4 System curveWhen pressure change is plotted against flow rate, the so called system curve is obtained. This curvedescribes pressure changes in the whole piping system as a function of flow rate. For turbulent flow (as istypical in chemical engineering industrial applications), the curve follows a second order polynomial form Dp= a + bv2, where a contains pressure difference and hydrostatic head (see mechanical energy balance,equation 1), and b contains possible dynamic pressure effect and most notably the friction terms. Typicallythe system curve is plotted in the same figure together with the pump curve shown in the next chapter.

2.5 Pump curvePressure increase in the pump as a function of flow rate follows a characteristic shape for the selected pumptype. The most typical pump type in chemical industries is centrifugal pump. Hypothetical centrifugal pumpand two piping curves are drawn in the same figure here:

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Y-axis is often plotted in terms of height (head), so that its dimension is meters (or feet) instead of pressure.Pressure can be changed to head by dividing by fluid density and gravitational acceleration. The pump curve(black) crosses system curve (red or blue) at the operating point. At this point the left and right hand sides ofthe mechanical energy balance are equal.

The two system curves in the figure could represent situations with different openings of a control valve.Valve is originally fully open (flow v1 and pressure difference p1), and then partially closed (v2 and p2).Naturally a fully closed valve would lead to infinite local resistance coefficient with zero flow rate. In thatsituation the pressure next to the pump is at maximum, corresponding to the point where pump curvecrosses y-axis. The piping system must be designed to mechanically tolerate at least this pressure and somesafety margin.

2.6 Pump cavitationIn some cases there is a danger that the pumped liquid will form bubbles inside the pump. This can happenif the liquid is relatively hot compared to its bubble point, i.e. close to boiling. As there is some pressure dropin the pump feed piping, the pressure inside the pump may be lower than the bubble point pressure, leadingto bubble formation. These bubbles collapse again as the pressure in the pump is increased. This processcauses noise and harms the pump. It is important to prevent this by designing the pump feed piping systemin an appropriate way. The most common way is to position the pump below the feed vessel, so that thepressure in the pump is higher than in the feed vessel due to hydrostatic head.

It is important to prevent also gas bubbles from the feed vessel to enter the pump, at least continuously. Thiscan be done by proper design of the vessel itself. This topic may seem trivial, but is essential in all chemicalengineering applications; control of the phases in the sense that phases are either in good contact (mixedwell enough for the purpose but not more) or separated and allowed to flow at desired destinations (e.g.settling vessels).

2.7 Fans, Blowers and CompressorsThese are equipment that move and compress gases. Fans discharge large volumes of gas into open spacesor large ducts. They generate low pressure increases, in the order of 0.04 atm. Blowers are high-speed rotarydevices that develop maximum pressure up to 2 atm. Compressors can compress gases into very highpressures using several stages.

In calculations, fans can be treated as pumps as the density of gas is not changing. The main principaldifference between pumps and compressors is that compressors are pumping gases, which are compressible.As with pumps, there are many different compressor types. Compressors can be crudely divided into positivedisplacement and dynamic compressors. Positive displacement compressors can further be divided intorotary and reciprocating compressors. Dynamic compressors are either axial or centrifugal type.

The most important design variable for compressors is perhaps the required compressing power. Typicallycompressors are assumed either isentropic (zero entropy generation, no cooling) or polytropic (some cooling,can be used for large compressors with cooling that is not perfect). For small and well cooled compressors,isothermal assumption can also be used, but not needed so often in practice. Since it is difficult to increasepressure very much in a single compressor stage (except for reciprocating compressors), they are often builtas multi-stage apparatus.

Power requirements (J/mol) for single-stage compressors can be estimated from:

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( )

úúû

ù

êêë

é-÷÷

ø

öççè

æ-

=-

1pp

1nnRTW

n/1n

1

21compr (5.)

This gives the ideal compression work, which needs to be corrected by dividing with compressor efficiencyhcompr (e.g. 0.7). In order to estimate power (W), this must be multiplied by the molar flow of gas (mol/s).

For isentropic compressors, n is the ratio of heat capacities, n=k=Cp/Cv (for air this is approximately 1.4). Thiscan be also used for reciprocating compressors. For polytropic compression of ideal gases, n can be estimatedwith the following correlation:

comprk1k1

1n

h--

= (6.)

In practical compressor calculations, especially at high pressures, it is important to specify alsothermodynamics (equation of state) properly, as gases may be far from ideal at high pressures. In these cases,correlation for n is more complicated than above.

2.8 Flow in porous mediaIn many cases, solid materials are either treated in chemical engineering applications or used as catalyst orother processing media. For example biomass treatment, filtration, packed bed reactors etc. are verycommon. This chapter considers only situations where the solid phase is stagnant, i.e. solid particles are notmoving. Fluidization and other applications where solids are moving are treated separately. Only one fluidphase is also considered here, multiphase flow is discussed in other chapters.

Typical to these situations is that the precise structure of the porous media is not known, but averageproperties are known instead. The known properties typically include porosity (empty space available for thefluid to flow) and particle size and shape. The latter can be tedious in case of wide distributions both in sizeand shape, as is often the case in biomaterial treatment.

A layer of porous material in the fluid flow path can be considered as a local friction loss. However, in mostcases the unit consisting the porous bed is designed separately during the preliminary design phase, so thatits pressure drop is evaluated separately from the other pump and piping system. Later on, the porous bedcan be accounted in the system curve discussed earlier.

The most classical pressure drop equation for flow in porous media is the Ergun equation:

( ) ( ) 23

p32

p

2

jd1Ej

d1E

Lp

ee-r

+ee-m

=D

rm (7.)

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Here the parameters Em and Er are empirical constants. In the original paper by Ergun, Em =150 and Er = 1.75were recommended, but later on based on more experimental data, more conservative values of Em =180and Er = 1.8 are proposed. j is the superficial velocity, i.e. linear average velocity of the fluid if it would flowthrough the empty vessel without solids. If particles are not completely similar (spherical with constantdiameter), an effective diameter needs to be used.

Another way of expressing pressure drop – flow rate relationship in porous media is to use permeabilityconcept. It is typically used for relatively slow flow rates (laminar flow), and in materials that are notcomposed of separate particles, but materials with natural pores in which fluid may flow. Permeability canbe defined by the Darcy's law:

LpkAQ

DD

m= (8.)

where k is permeability and A is cross-sectional area. If the relationship j = Q/A is used, it is easy to see thatpermeability for laminar flow in porous material can also be predicted with the Ergun's equation.

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3 Momentum balancesIn some cases, the pressure drop alone is not sufficient, but we want to know more about the flow patterns.In these cases mechanical energy balance alone is not sufficient. Momentum balances are in many respectsmore demanding to formulate properly and especially to solve. One fundamental challenge comes frommomentum transfer due to viscous forces. Mass transfer (diffusion) and heat transfer (thermal conduction)can often be analyzed with a one-dimensional model: heat is conducted from hot to cold and diffusiontransports molecules from high concentration to low (statistically). Instead, momentum is transferredperpendicular to the flow direction. For example linear momentum in x -direction is transferred by viscousforces in y- and z –directions. Flow therefore "grabs" fluid elements next to it to move in the same direction.Near the walls the fluid is stagnant (so called no slip condition), and this process transports linear momentumfrom the main flow towards the stagnant walls. In this process, part of the fluid mechanical energy is lost dueto dissipation to heat. The correlations in the previous chapter estimate this in some well-defined systems orbased on empirical correlations. If these are not available or the precise flow pattern needs to be known fordesign purposes, linear momentum balances need to be used.

3.1 Mechanical energy vs. momentum balancesTypically mechanical energy balance, or energy balance in general, seems to be easier to comprehend thanlinear momentum balances. This may be due to their wider use in classical problem solving in schools. Heresome simple relations are first discussed to clarify the two concepts.

The dimension of energy (extensive property for which a conservation equation or balance can be written) isJoule, so that time dependent energy balance has terms that are in the dimension of J/s, or W. Sometimesenergy balances are called therefore power balances, although the conserved property is energy. For fluidflow applications, power is simply a product of pressure drop and volumetric flow rate:

pQP D= (9.)

Power also equals to force times velocity.

Linear momentum is mass times velocity, dimension (kg m/s). Time dependent momentum balance containterms that are of the same dimension than rate of change of linear momentum (kg m/s2). This is the same asforce (N). Actually many mechanics problems are classically solved with "force balances", which can beunderstood as time independent linear momentum balances, where the system is not accelerating. Thenvarious forces compensate each other. The conserved extensive property, however, is linear momentum, notforce. Force and pressure applied to a surface are related as

AFp = (10.)

This is one of the terms appearing in the linear momentum balances for fluid flow problems as shown later.

In addition to the linear momentum, also angular momentum is conserved. However, angular momentumbalances are not typically separately formulated in chemical engineering fluid flow problems and not

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discussed further here. For differential momentum balances, symmetry of the stress tensor impliesconservation of angular momentum on small scales.

3.2 Newton's viscosity lawLet's start analyzing linear momentum balances by reviewing Newton's viscosity law. Perhaps the mostclassical way of explaining this is to consider liquid layer between two parallel plates that are very large. Largeplates are assumed just to neglect any boundary effects. The plates are initially at rest, and the fluid stagnant.At some time moment, one of the plates is being pulled with a force F:

The other plate remains stagnant. The fluid elements just next to the moving plate move with the samevelocity as the plate, and fluid elements just next to the stagnant plate remain stagnant. After a while, avelocity profile is formed in the fluid between the plates:

The force divided by plate area is called shear stress:

t=AF

(11.)

It is experimentally found that the shear stress is proportional to the shear rate (velocity gradient):

dydvx

yx m-=t (12.)

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This proportionality is called Newton's viscosity law. Here subscript yx for the shear stress refer to x-directional linear momentum being transferred into y-direction. In principle it is transferred also to z-direction (the third dimension not shown in the figure), but as the system is symmetric in that direction inthis example (constant velocity), the net transfer in that direction is zero and can be excluded from thisanalysis. Minus sign refers to the fact that momentum is transferred to the direction of lower velocity.

Newton’s law for viscosity could be compared to Fick’s law for diffusion and Fourier’s law for heat conduction:

Diffusion (Fick):dy

dcDJ AABA -= Heat conduction (Fourier):

dydTq l-=

It is clear that these laws are mathematically quite similar including flux, a proportionality coefficient, andassumed linear dependency on gradient related to the modeled variable. Therefore, the solutions are also inmany respects similar. The challenge in fluid flow is that there are always more than one spatial coordinateinvolved, as linear momentum in x direction (in this case) is transported in y –direction. For diffusion andheat conduction, often one-dimensional treatment is sufficient.

The proportionality coefficient in the Newton's viscosity law is viscosity. For many fluids it does not dependon the shear rate or time, i.e. is practically constant. These are called Newtonian fluids. If the proportionalitycoefficient depends on the shear rate or time, the fluid is called non-Newtonian. In all cases, viscosity is afunction of temperature.

Rheology is a study discussing properties of flowing substances, and characterization of non-Newtonian fluidsis an essential part of it. These will be discussed later.

In general three-dimensional cases, collection of each shear stress component is called stress tensor.

3.3 Laminar flow in a pipeA classical application example of Newton's viscosity law is a well-developed laminar flow in a circular pipe.Well-developed here means, that any entry effects at the pipe feed have been disappeared from the flowprofile. We need to formulate a linear momentum balance for a control volume shown in the next figure:

The control volume is between the two cylindrical regions at arbitrary distance r from the pipe center axis.The control volume is of thickness Dr and of length DL. Flow is assumed to be from left to right.

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Linear momentum flow in to the control volume through the control volume boundary on the left (ring at theleft end of the cylinder) and linear momentum flow out from the right end are

( )( ) 0zzz vvrr2in =rDp=

( )( ) Lzzz vvrr2out =rDp=

Since the cross-sectional area of the pipe is constant and the fluid here assumed incompressible (constantdensity), these two terms are equal and cancel each other out from the balance.

Another source term in the linear momentum balance is due to pressure at the same two ends. Force equalspressure times area, so the pressure terms are:

( ) 0prr2in Dp=

( ) Lprr2out Dp=

Now pressure at the inlet (p0) is not the same as pressure at the outlet (pL). This pressure difference is thedriving force for the fluid to move, and the related energy is dissipated due to viscous forces. In order todescribe these, we need the shear stress definition:

( ) rrzLr2in tDp=

( ) drrrzLr2out +tDp=

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where the cylindrical part area is multiplied by the shear stress at the corresponding location (r and r+Dr).The linear momentum balance thus reads

( ) ( ) Lrrrz0rrz rpr2Lr2rpr2Lr20 Dp-tDp-Dp+tDp= D+ (13.)

In order to close the equation, we need to use the Newton's viscosity law for the shear stress. After insertingit and allowing Dr to approach differentially small thickness, we end up with the following equation:

rLpp

drdvr

drd L0 ÷

øö

çèæ

D-

=÷øö

çèæ m- (14.)

This second order ordinary differential equation can be solved with two boundary conditions for the variabledepending on r (velocity). Note that pressure drop per length is assumed constant and thus the equationhere is not a partial differential equation.

The two conditions are no slip at the wall (discussed earlier), and symmetry at the center. The latter boundarycondition results from the fact that for the shear stress to be defined at the center, the shear rate needs tobe smooth (i.e. zero). The solution to this is

( ) ( ) ÷÷ø

öççè

æ÷øö

çèæ-÷÷

ø

öççè

æDm-

=-÷÷ø

öççè

æDm-

-=2

2L022L0

Rr1R

L4ppRr

L4pprv (15.)

This gives velocity at any distance from the center of the pipe. It can be seen, that the velocity is a secondorder polynomial form. Maximum velocity is at the center and minimum (zero) near the walls.

We are often interested in the average velocity, as it is related to the volumetric flow rate. Unless otherwisespecified, it is always the average velocity that is referred to when fluid flow rate in pipes is expressed in m/s.Average velocity can be obtained by integrating the local velocity profile:

( ) 2R

0

Rvrdr2rvQ p=p= ò (16.)

The result is

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2vR

L8pp

RQv max2L0

2 =m-

=p

= (17.)

Here it can be seen that the maximum velocity at the centerline is twice the average velocity. The previousequation is often written in the form explicit to pressure drop:

vD

L32vR

L8p 22

Dm=

Dm=D (18.)

This equation is called Hagen-Poiseuille law. Often it is also given in terms of volumetric flow instead of linearflow velocity. Note that this is to some extent similar than the laminar contribution of the Ergun equation,but restricted to circular straight pipes.

When this is compared to the definition of Darcy friction factor (part of equation 2), we see that

Re64

vD64

=rm

=x (19.)

This is an analytical solution for the Darcy friction factor for purely laminar flow in circular pipes. It can beseen also in the Moody diagram for laminar flow velocities (Reynolds number below approximately 2000).This is one of the few cases where momentum balances can be solved analytically. In most cases a numericalsolution is needed. There are well established methods and software for this. This field of fluid flow study iscalled Computational Fluid Dynamics (CFD), and is briefly introduced later.

3.4 General momentum balancesUnfortunately the geometry is not always as simple as a circular pipe. If we want to calculate flow patternsin more complicated situations, we need general momentum balances. There will be three of them, one foreach spatial coordinate. General differential momentum balances can be formulated based on the followingcontrol volume sketch:

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When all the directions are taken into account and all the relevant source terms, the following equationsresult in:

( ) ( ) ( )f+

¶t¶

=¶r¶

+¶r¶ S

xxUU

tU

jj

jii(20.)

Note that this is the same form as a general transport equation, e.g. differential material or energy balance.Here the transported property is linear momentum divided by volume, rU. Note that in this context symbolU is used for velocity (for historical reasons).

Here so called Einstein notation is used, so there are actually three separate equations for the linearmomentum; subscript i here refers to the separate equation whereas subscript j refers to repeated indices(summation).

When Newton's law is used for shear stress tensor t, and gravity and pressure gradients are observed to bea source term, we end up with

( ) ( )i

i

j

j

i

ji

i

j

ij

i gxU

xU

xxp

xUU

tU

r+÷÷ø

öççè

æ

¶

¶+

¶¶

¶¶

m+¶¶

-=¶¶

r+¶

¶r (21.)

In the previous equation, also differential material balance is used, and viscosity is assumed constant.Detailed derivation of these equations can be found in the fluid flow literature.

The previous equations, along with the differential material balance, are called Navier-Stokes (N-S)equations. The first term on the left describes rate of change of linear momentum, the second termconvection of momentum, the first term on the right is pressure gradient source term, the second viscousdissipation and the last term source term due to gravity. In this course, we are not extensively working withthe N-S equations by hand, but later on discuss how it is typically solved with proper algorithms.

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4 Non-Newtonian fluidsFor Newtonian fluids, the shear stress is linearly proportional to the shear rate, and this relation does notdepend on time. For non-Newtonian fluids, the dependency is more complicated. Non-Newtonian fluids canbe characterized based on their shear stress – shear rate dependency, shown in the following figure

Typically, pure fluids with low molecular weights have close to Newtonian behavior, e.g. water or lighthydrocarbons. Pseudoplastic fluids are also common, e.g. syrup, pulp in water, and many fermentationbroths are pseudoplastic. Cornstarch and some colloidial systems are dilatant. Drilling mud, toothpaste,mustard are some examples of Bingham –type behavior where the fluid has a yield stress before it starts toflow at all.

Viscosity in these systems can be characterized by shear rate dependent apparent viscosity. It is the slope ofsuch curve plotted through measured shear stress vs. shear rate point in the figure above and origin.

Further complication is that in some cases apparent viscosity is time dependent. This can occur e.g. insituations where the fluid consists of long chain macromolecules, which take time to orient in the directionof the flow, or in cases where other reversible structural changes happen in the fluid once subjected to shear.

There are several models for non-Newtonian fluid viscosities. Perhaps the simplest is the power law:

n

dydvm ÷÷

ø

öççè

æ=t (22.)

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which leads to the following apparent viscosity model:

1n

dydvm

-

÷÷ø

öççè

æ=m (23.)

Another, more versatile is Carreau model, which is suitable for a wide range of shear rates, but requires alsoexperimental apparent viscosity measurements over a very wide range of shear rates in order to identify theunknown four parameters.

( ) 2/1n2

0 dydv1

-

¥

¥

÷÷

ø

ö

çç

è

æ÷÷ø

öççè

æl+=

m-mm-m

(24.)

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5 TurbulenceWhen the flow rate is increased, the destabilizing inertial forces become larger than the stabilizing viscousforces, and the flow becomes unstable. The threshold for this can be estimated with the Reynolds number:

mr

=vDRe . There is no precise value for this transformation, and it depends also on the definition of the

characteristic length used in the Reynolds number. There is also a transitional regime between the laminarand fully turbulent flow, where the flow is not stable, but there are not yet such statistical variations in thelocal flow velocities that are characteristic to fully turbulent flow. Depending on the case, above Reynoldsnumber around 4000 – 10 000, the flow can be assumed to be fully turbulent. Interestingly, the Reynoldsnumber appears also naturally when the Navier-Stokes equations are made dimensionless. This underlinesthe fundamental nature of the Reynolds number in the field of fluid dynamics.

Although the N-S equations basically can be used to model precisely all possible fluid dynamics problems,turbulent fluctuations are typically too rapid and small for carrying direct numerical simulation in practice(more about this later). Therefore the turbulent flow is typically modeled with the so called Reynoldsdecomposition, where the instantaneous velocity is split into time-averaged and fluctuating part:

'iii uUU += (25.)

This division is illustrated in the next figure.

Note that the time-averaged part iU does not imply a steady-state assumption, but the changes in theaverage flows are just assumed to be much slower than turbulent fluctuations.

When the Reynolds decomposition is applied to N-S equations, a time-averaged form is obtained:

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( ) ( )i

'j

'i

i

j

j

i

ji

i

j

ij

i guuxU

xU

xxp

xUU

tU r+÷

÷ø

öççè

ær-÷

÷ø

öççè

æ

¶¶+

¶¶m

¶¶+

¶¶

-=¶¶r+

¶¶r (26.)

Here it can be seen, that the equation is otherwise similar than the original N-S, but there is an extra term

'j

'i

'ij uur=t (27.)

This term is called Reynolds stress tensor (Reynolds stresses), and it describes turbulent momentumtransport. Note that it appears in the same part of the equation as viscous dissipation. It takes into accounta process where eddies (small scale liquid vortices) carry linear momentum to slower moving regimes of thefluid in a similar fashion as viscous forces do (see discussion related to the Newton's viscosity law). As a result,it gives additional friction-like effect to the flow.

As small scale fluctuations are not known (unless the full non-decomposed N-S equation is solved, which wetry to avoid in this whole process), the Reynolds stress term needs to be modeled somehow. There arevarious attempts to do this especially as related to the computational fluid dynamics. These models are oneexample of so called “closure models”, which are in practice often needed in physical modeling. These arediscussed later.

5.1 Turbulent energy spectrumVarious eddies in the flow can be characterized based on their size and the typical kinetic energy they contain.A classical scheme to illustrate this is the following:

21

This is often called Kolmogorov energy spectrum or energy cascade. Largest eddies that are created from themain flow are on the left and smallest dissipating kinetic energy to heat are on the right. The largest eddiesmove faster and contain more energy, and the small eddies contain less energy (per eddy). However, thereare much more small eddies, so on average the total energy in various eddy sizes is rather constant until thesmallest eddies which are no longer broken into smaller ones but the energy is dissipated to heat. Smallesteddies are characterized by the so called Kolmogorov length and time scales:

Length scale ߟ = ቀ ఓయ

ఌఘయቁଵ/ସ

(28.)

Time scale ߬ఎ = ቀ ఓఌఘቁଵ/ଶ

(29.)

Where e is the local energy dissipation to heat (W/kg). It is a very useful variable also when total energy inputto a system is evaluated. For example in stirred tanks, average e value is the total mixing power per fluidmass. It is a quick and convenient value to evaluate how strong mixing is used. Typically its values are around0.01-10 W/kg in liquid mixing. This energy is then dissipated all around the mixing tank, but not equally. Nearthe impeller the dissipation is much more intense, and based on Kolmogorov time and length scales, alsoeddies are smaller and more rapid.

5.2 Turbulence modelingTurbulence modeling typically refers to various efforts to model Reynolds stress tensor empirically. This isneeded as there is no analytical solution to the turbulent flow problem. The most typical approaches arebased on the so called Boussinesq approximation. Based on that assumption, turbulent contribution is addedto the viscosity:

Tmolecular m+m®m (30.)

Note that the turbulent viscosity mT is a property of the flow, not of the fluid. When turbulent viscosityapproach is used, the time averaged N-S equations appear essentially the same as the original ones, butviscosity replaced with the combination of molecular and turbulent contributions. In fully developedturbulent flows (high Reynolds numbers), the turbulent viscosity is much higher than the molecular one.

Turbulence models are often characterized by the number of additional transport equations needed to modelthe Reynolds stress tensor, or turbulent viscosity if that is assumed. Some of them are listed in the followingtable:

22

In practice, 0 and 1 equation models are not used. Two-equation models are currently perhaps the mostoften used, as they are a convenient compromise between computational burden and accuracy. Perhaps themost typical is the so called k-e model (at least for chemical engineering applications), where one transportequation (new differential balance equation for a modeled property) is formulated for kinetic energy of theturbulent fluctuations

å=

=3

1i

2'iu

21k (31.)

And one for the turbulent kinetic energy dissipation rate e discussed earlier. Formally the latter is not aconserved property, but nevertheless a transport equation can be formulated it with apparently goodpredictive capabilities. There are specific source terms (production and disappearance) for k and e as well asconvection and other terms. These equations can model so called history effects, where turbulentfluctuations move with the main flow while the eddies are dissipating fluctuation energy. Eddy viscosity canbe directly then calculated with the two additional variables. Note that the total set of equations to modelthree-dimensional turbulent flow would in this case be six: one material balance, three linear momentumbalances (one for each direction), one equation for k and one for e. Turbulence is thus assumed isotropic asthere is no preferred direction in the modeled eddies.

In more rigorous turbulence models, such as Reynolds stress model and large-eddy simulations (LES), thedirectional character of larger eddies are taken into account. In Reynolds stress model, this is done by usingseveral additional transport equations, and in LES by modeling numerically all except the smallest eddies.

The most rigorous model is Direct numerical simulation (DNS), where no turbulence model is needed. Thenumerical solution needs to be so accurate that all the smallest eddies are resolved. This requires a

23

computational grid where the grid size is smaller than the Kolmogorov length scale and time dependentsolution where the time step is shorter than the Kolmogorov time scale. Even then the solution is chaotic innature; we cannot say for sure that at a certain point in space at a given time the flow is precisely as predicted,the result is just one possible realization of the flow.

5.3 Turbulence near wallsNear solid walls, and also near fluid interfaces, turbulence tends to decay. Large eddies simply cannot existvery close to the wall for physical reasons. Near the walls velocity gradients can be very steep, and molecularviscosity becomes significant even for fully turbulent flow. Near phase boundaries (solid-fluid or fluid-fluid)are of high importance in chemical engineering, as profiles close to the walls or interfaces determine massand heat transfer rates. The following figure illustrates the processes near boundaries, and revises theclassical film theory which can be applied both to heat and mass transfer.

In the film model, there is a hypothetical film thickness next to a wall or phase boundary. This thickness isdetermined so that the gradient next to the wall is the same as in the “true” profile, so that mass or heattransfer rate due to diffusion or thermal conduction is the same in both models. The whole film is assumedlaminar, and its thickness is typically not needed separately, but combined with diffusion coefficient orthermal conductivity into mass or heat transfer coefficient.

In fluid flow boundary layer, also a laminar film is assumed. However, this is true laminar layer next to thewall, and its thickness is not the same as in the film model (it is thinner). Next to it, a so called “universalvelocity profile” is assumed, where decay of turbulence and other flow properties can be modeled withcorrelation approach.

It is well known that some turbulence models do not predict well fluid flow in the boundary layer. Mostnotably k-e model is not good at this region. One of the reasons is that near the wall the turbulence is no

24

longer isotropic, i.e. flow fluctuations are not similar in directions perpendicular to the wall and adjacent tothe wall. In order to help in this, so called wall functions are used along with such turbulence models, so thatthe flow near the walls is predicted with empirical wall functions and the turbulence model is applied outsidethis region.

25

6 Multiphase flowIn many chemical engineering applications, multiple phases flow simultaneously. This could be just gas andliquid phases flowing together in pipes, flow in multiphase reactors, such as gassed stirred tanks, bubblecolumns, or trickle beds, or in most separation processes, where the process performance is based on goodcontact between two immiscible phases and distribution of the separated components between thesephases, such as in distillation or extraction columns. The first, and perhaps even the most important, skill fora chemical engineer in these cases is to be able to visualize the situation. This is required for sensible designof the units so that the phases flow as designed; either providing good contact for mass transfer, or providegood separation of the phases e.g. in settlers. For good contact, typically energy input is needed to disperseone phase into another. In many cases, this energy is provided by pressure drop (power equals pressure droptimes volumetric flow), and in some cases it is provided mechanically, such as in stirred tanks. For properseparation of the phases, the simplest approach is to allow slow enough flow so that dispersed phase bubblesor droplets have sufficient time to settle. A classical design approach for this kind of operation would becalculating fluid velocity for self-venting flow, where liquid velocity downwards should not be faster than theassumed bubble rise velocity, if the bubbles need to be separated into a gas space on top of the liquid. Inthese cases, the cross-sectional area available for the liquid (or dispersion) flow should be large enough sothat the velocity is below the rise velocity of bubbles. In these cases, design requires that a threshold bubblesize is assumed. Smaller than this threshold sized bubbles will not be properly separated since they riseslower than the liquid velocity, but larger ones should separate. Another example is gas-liquid flow invertically placed reactors: in downward flow arrangement gas accumulation in the reactor is relatively largeand cannot be avoided, whereas in upwards flow arrangement gas escapes more rapidly from the reactorspace. Which one is the desired operation, depends on the case. There are several such examples whererather simple common sense helps designing units, at least giving qualitative reasoning for the design. Whenthis common sense is combined with quantitative models capable of predicting the flow, proper design canbe achieved.

6.1 Flow regime maps and pressure dropTwo (or more) phases may flow in completely different flow patterns, or textures, depending on theconditions. The most important parameter is probably flow rate of each of the phases, flow direction, andthe cross-sectional area available for the flow. Also physical properties, such as liquid phase viscosity andsurface tension, may affect flow pattern. Based on the flow rates, various patterns can be found fromavailable diagrams. Note that the limits are not necessarily precise. In the figure below, schematic flowpatterns are shown on the left, and one available flow regime map on the right for horizontal flow.

26

One problem in these maps is that in many cases they are based on experimental data with rather narrowvalidity range. Most of the data is measured by using air as gas and water as liquid, and in pipes withdiameters typical for laboratory equipment. These are sometimes corrected for varying properties withseparate correction factors. It is, however, clear that for example slug occurs much more easily in very smallpipes where capillary forces are important than in very large pipes, where very large continuous slugs areless probable. On the other hand, slug formation in small scale does not necessarily lead to any problems (itcan sometimes be even desired), but slugs in large industrial pipes may lead to failure in piping constructionsdue to pressure shocks and heavy vibrations.

Pressure drop for two-phase flows is often calculated with such a procedure, where first flow regime isidentified based on superficial velocities of the two phases, then pressure drop is calculated for the twophases separately (as if they would be flowing without the other phase), and then empirical correlations isused to combine these information for a two-phase pressure drop and holdup. One classical approach forthis is by Lockhart and Martinelli (not reported here in detail). Many others exist as well. Many of them arealso implemented in process flowsheet simulators, such as Aspen Plus. Naturally care must be taken whenusing these correlations, and combined with common sense.

In some cases, the two phases may be separated and fed to separate gas and liquid pipes. This requires aseparate settling tank with liquid level control, but for example when distillation column feed is partiallyvaporized liquid, this arrangement could in some cases give more stable feed distributor system inside thecolumn as splashes due to slugs are avoided.

6.2 Interphase forcesForces between dispersed phase entities (bubbles, droplets or particles) are needed in order to be able topredict their movement with respect to the continuous phase. In many cases, the final outcome is formulatedin terms of terminal velocity, i.e. velocity of the particle if it would rise or drop freely in a large volume ofcontinuous fluid. Even in these cases, the result can be simply a steady state solution of momentum balancewritten for an individual particle. The momentum balance relevant for this situation can be written as

( ) ( ) Dfp FgVdtmUd

-r-r= (32.)

The term on the left is rate of change of momentum, which in practice means acceleration. The first term onthe right hand side is buoyancy, i.e. force resulting from the density differences. This is basically the drivingforce for separation: if the densities of the two phases are similar, they flow together. This may be probleme.g. in extraction where the densities of the two phases may be close to each other. In extraction, it isimportant to select the solvent so that there is a density difference between the phases.

The last term on the right is interfacial force. This is actually a combination of several force terms, but inmany cases drag is the most relevant term. Drag can be calculated with definition of the drag coefficient:

27

2pDD U

21ACF r= (33.)

Here CD is the drag coefficient, similar in nature than friction coefficients discussed earlier. Ap is the cross-sectional area of the particle as seen by the continuous phase. For example for spherical bubbles or dropletsof diameter d, cross-sectional area would be Ap = p/4d2. U is the velocity difference between the particle andthe continuous phase.

There are several empirical correlations and graphs available for the drag coefficient, for example:

For this, the following three regime correlation gives reasonable approximation for drag coefficients forspherical particles with relevant Reynolds numbers in chemical engineering applications:

Re < 1 CD = 24/Re

1 < Re < 103 CD = 18*Re-0.6

103 < Re CD = 0.44

For a particle Reynolds number, the characteristic diameter is always the particle diameter.

When the first of these is inserted into the momentum equation, spherical particle and steady state isassumed, the so called Stokes law for terminal velocity is obtained. Stokes' law is valid for very small bubblesor droplets. When the last of the regimes is assumed (constant drag), the so called Newtonian settling law isobtained. That is valid for relatively large particles, where flow around the particle is turbulent, but boundarylayer is not yet separated from the particle surface (drop at high Re values in the graph). There are severalother correlations also available.

One thing worth noting here is that the previous analysis is valid only for spherical particles, although evensolid particles are often of different shape. For relatively large bubbles and droplets, their shape starts todeviate from a spherical one. There are also flow regime maps for fluid particle shapes. One such is presentedin the following figure, where bubble shape as a function of its physical properties is given. Based on the

28

graph, bubble Reynolds number and thus rising velocity can be estimated when its Eotvos and Mortonnumbers are known.

where

srD

=2gDEo (34.)

32f

4gMosrmrD

= (35.)

With given physical properties (specified chemical system), a simpler graph relating bubble diameter to itsterminal velocity can be obtained. One such is in the following figure for air-water system. One importantpoint is that bubble rise velocity depends on whether the liquid is contaminated (contains surfactants) ornot. Fluid inside pure droplets and bubbles circulate with moving interface. This reduces drag (increasesterminal velocity) as compared to solid particles. Even small amounts of surfactants concentrate on gas-liquidsurfaces, and stop surface movement. Besides bubble rise velocity, this has an impact on gas-liquid masstransfer and bubble coalescence tendency. In practice water contains almost always some surfactants so inmost practical cases it can be considered as contaminated from this perspective (although it would beperfectly fine as a drinking water).

29

One noteworthy point in gas bubble rise is that due to onset of oscillations, terminal velocity is often ratherconstant over a wide range of bubble sizes relevant for chemical engineering applications (from 1 to 20 mm).For rough calculations, bubbles can often be assumed to rise approximately 20-30 cm/s. The same is not truefor liquid droplets, where terminal velocities for relevant droplet sizes in chemical engineering applications,such as in extraction, are not constant.

6.3 FluidizationFluidization is an operation, where upwards flowing gas or liquid overcomes gravity for individual particles.Limiting velocity for this is called minimum fluidization velocity. If the flow is slower, the particles settle onthe bottom of the equipment, typically on top of a grid preventing particles to drop to the gas or liquid feedingsystem. Above the minimum fluidization velocity, particles start to float freely in the vessel. Typically thesystem is relatively well mixed, which is preferable e.g. in cases where catalytic exothermic reaction occur inthe particles. In fluidized mode, it is also possible to continuously remove solids from the vessel. If fluidvelocity is increased further, particles do not stay in the vessel anymore, and pneumatic conveying results in.

Fluidization is extensively used in oil refineries, where the core of modern refineries is FCC (fluid catalyticcracking) unit. Less valuable heavy oils are converted into light components more suitable to be used asgasoline compounds. Catalyst in these operations deactivate due to coke formation very rapidly, so catalystparticles are allowed to flow continuously through the reactor part into a regenerator where coke is burnedaway. After this, regenerated catalyst is returned to the reactor.

Fluidization can be used also for other processes, such as in drying. Typical flow patterns that can be foundin fluidization are shown in the following figure.

30

6.4 Two-phase flow in porous mediaIn some cases, the flow in porous media contains two liquid phases in addition to the solid matrix throughwhich the fluids flow. For example in trickle bed reactor, gas and liquid flow co-currently through fixed bedof solid catalyst particles. In these cases, both pressure drop and liquid hold-up in the reactor are relevantdesign parameters. In order to predict these, momentum balances for both gas and liquid needs to beformulated. After simplification, pressure drop correlations for both phases are obtained, with appropriateinteraction terms between gas and liquid, liquid and solid, and perhaps between gas and solid in cases wherepart of the solid is not wetted. The calculated pressure drop for both phases needs to be the same in a givenbed. This can be achieved since the volume fraction of the flowing fluids give one additional degree offreedom. One set of equations suitable for predicting two-phase flow in packed beds is the following.

Pressure drops for both of the phases:

gfLp

GGL r+a

-=D(36.)

( ) g1

ffLp

LLSGL r+

a--=D

(37.)

gas-liquid and liquid-solid interactions (fully wetted solid phase assumed here):

( )rrGGLrGGLGL jjBjAf r+ma= (38.)

( )( )LLLLSLLLSLS jjBjA1f r+ma-= (39.)

31

Relative velocity of the phases in the pores:

( ) LG

r j1

jja-

a-a

= (40.)

Interaction terms:

( )2p

33

2

GLGL d1EAeaae-= m (41.)

( )p

33GLGL d1EBeaae-= r (42.)

( )( ) 2

p33

2

LSLS d11EA

ea-e-

= m (43.)

( )( ) p

33LSLS d11EB

ea-e-

= r (44.)

This system of equation can be solved iteratively to give pressure drop and void fraction (gas saturation).

6.5 Capillary pressure and surface wettingOne further thing to consider in multi-phase flows is capillary pressure. It affects wetting properties of porousmaterial, formation of small bubbles (bubble nucleates), and in many other relevant phenomena in chemicalengineering. The following figure illustrates capillary pressure in simple capillaries immersed into liquid, andhow capillary pressure affects fluid saturation in porous media.

32

With narrow capillaries or small particles with small openings in between them, capillary pressure could bestrong. For larger pipes or flow in between large particles, it is less significant. Capillary pressure can beestimated, if the curvature of the surface is known from the Young-Laplace equation:

R2pcs= (45.)

Here R is the radius of curvature, and s is the surface tension.

Capillary pressure is often advantageous, as it helps to properly wet solid surfaces, provided the surfacematerial is such that it is preferably wetted by the liquid. This is important e.g. in distillation or absorptionwith packed columns, in coalescers helping to increase drop size before settling, or in trickle bed reactors,where proper wetting of catalyst is important for good reactivity and to avoid hot spots in the reactor.

33

7 MixingOne particularly important process in chemical engineering where fluid flow aspects are important is mixing.Mixing may be needed e.g. in chemical reactors to get reactants in one or several phases into good contactwith each other, facilitate mass transfer between the phases by increasing interfacial area (breaking largebubbles or droplets) and preventing settling, or homogenizing products. Although mixing may appear as asimple operation, it is important to optimize mixing processes in order to minimize energy input.

7.1 Mixing in stirred tanksIn many cases, it is convenient to carry out mixing in vessels with impeller(s). These impellers mainly act tomove (blend) fluid and disperse phases for a good contact. For various purposes, there are a large numberof different impeller and mixing tank geometries. For example laminar mixing requires typically completelydifferent impeller geometry than turbulent mixing. Mixing of various non-Newtonian fluids may also requirespecial designs.

Mixing can be characterized with various parameters. The first to be calculated is the Reynolds number. Instirred tanks, it is calculated based on impeller diameter and its tip speed. Typically the constant factor p isnot included in the impeller Reynolds number definition, so it becomes

mr

=2NDRe (46.)

If impeller Re is above 104, the flow can be considered turbulent, if Re is much below 1000, flow is laminar.In between there is a transitional regime similarly as in the pipe flow. Also for impellers, these limits cannotbe defined very precisely.

Besides the Reynolds number, perhaps the most important factor characterizing mixing is power input. It canbe estimated when impeller power number is known:

53P DNNP r= (47.)

Again there are graphs available to estimate impeller power numbers. One such is shown below.

34

Note that for laminar flow, power number is inversely proportional to the impeller Reynolds number, and forfully turbulent flow, it is practically constant (but different for each impeller type). For example for acommonly used Rushton turbine (impeller 1), the power number is approximately 5 in the turbulent regime.Similar behavior was earlier found in drag coefficient in pipes, or drag of particles in free rise or fall incontinuous fluid.

Other important parameters for mixing in stirred tanks are flow from the impeller (based on the pumpingnumber) and blending time, i.e. time required to achieve certain degree of homogeneity after a tracer feed.These can be estimated with the following equations:

3QNDNQ = (48.)

3/1p

2

5.05.1

NDHT2.5N =q (49.)

The first of these is a definition of a pumping number NQ, and the second is an empirical correlation tomeasured blending times in various stirred tanks. Pumping numbers can be found for different impeller typesin similar graphs and correlations as the power number.

7.2 Multiphase mixing in stirred tanksAs discussed earlier, in many chemical engineering applications processes contain multiple phases. This isalso true for mixing in stirred tanks. Some examples are aerated fermenters or other gas-liquid reactors,stirred extraction columns with or without chemical reaction, crystallization, most polymer reactors, and soon. In some cases there may be even more than two phases, such as in flotation. In these cases it is important

35

that the phases are in good contact, does not separate unintentionally, and that the whole vessel contain allthe phases.

For mixing power, the simplest approach is to calculate just effective dispersion viscosity and density, anduse that in one-phase correlations. Especially for gas-liquid mixing, gassing reduces power input by theimpeller, which can be taken into account with specific gassed system power correlations.

There are several empirical correlations available for estimating whether the conditions are suitable for two-phase mixing. For example for solid-liquid mixing, solid phase should be well suspended. If a significantfraction of solids lay at the bottom of the vessel, mass transfer is reduced, and in worst cases the particlesmay form a single solid bulk, e.g. in crystallizers where the particles may form agglomerates.

The figure below describes the situation where (a) particles are not properly suspended, (b) above so calledjust suspended speed, and (c) fully suspended dispersion.

The most important design criterion is to be above just suspended speed, i.e. avoid regime (a). For this, thereare several correlations. One such is by Zwietering:

SXdDFrRe 13.0

2.0

p

45.01.0imp =

úúû

ù

êêë

é(50.)

where X is the mass ratio of suspended solids * 100 (i.e. %), S is a tank geometry specific parameter, typicallybetween 3-8 for most typical cases. Typically when these correlations are formulated, much more data isavailable from relatively standard geometries. For more exotic designs, these correlations are less reliable.

Another typical multi-phase design problem is gas feed and impeller speed to avoid flooding in aeratedreactors. Impeller flooding is illustrated qualitatively in the next figure:

36

When too much gas is fed to the system, it starts to accumulate behind impeller blades. In that situationbubbles coalesce into a single gas cavity behind the impeller, and leave that regime as large bubbles withsmall mass transfer area. Bubble phase is also poorly distributed in the vessel, as gas typically rises just closeto the impeller axis.

In order to avoid this, there are empirical correlations again available. One such is presented here to predictonset of flooding:

5.3

3G TDFr30

NDQFl ÷

øö

çèæ>= (51.)

It can be seen that flooding occurs if gassing rate is very high compared to the impeller speed.

Another gassing related problem for very large vessels is that gas superficial velocity (volume flow divided byvessel cross-sectional area) increases in scale-up if gas feed to vessel volume is kept constant. This leadsultimately to change in flow regime, so that very large vessels may be closer to bubble columns than agitatedvessels. In very large scales, this may lead to different optimal structure than in small scales.

Impeller selection can also partially help to avoid flooding. Instead of flat blades, such as in Rushton turbine,curved impellers can be used so that cavity behind impeller does not form so easily.

7.3 Static mixersIn some cases, mixing is easier to accomplish in a pipe than in separate vessel. Normal pump is used for powerinput, and a static mixer could be inserted in the pipe to enhance mixing. For turbulent flow, the main reasonfor static mixers is to reduce time needed for mixing (flows would mix by themselves anyway), but for laminarflows static mixers may be needed since flow would mix only very slowly with molecular diffusion mechanism.One (out of many) possible static mixer type is shown in the following figure.

37

38

8 Computational fluid dynamics (CFD)Since the Navier-Stokes equations can be solved analytically only in very simple special cases, a numericalsolution is required. One complication in fluid flow modeling compared to many other modeling efforts inchemical engineering is that the geometry where the interesting flow occurs is often relatively complexwhere strong symmetries cannot be assumed. This is very different compared to e.g. reactor or mass transferoperation modeling, where often zero- or one-dimensional models (e.g. plug flow reactor) can be used withsuccess. Sometimes radial or other symmetries can be used for fluid flow, or only part of the unit needs tobe modeled (half or one quarter, but 3D). If these are possible, it is always advisable to take advantage ofsuch symmetries. Note however that although the equipment itself would seem symmetrical, the flow itselfcould be unstable in such a way that the symmetry is broken. One example is bubble column, where thecolumn itself could in some cases be assumed approximately symmetrical, but the bubble swarm swirlsaround the column so that the final flow is not only three-dimensional but also time dependent.

In order to solve the fluid flow, the modeled volume is divided into a computational grid. In the followingfigure, there are some discrete control volumes to be used in computational fluid dynamics

The whole domain must be discretized with elements such as those in this figure. The ones on the left are for2D modeling, and the others for 3D.

Once the domain is discretized, a numerical scheme is needed to solve the N-S equations along with otherpossible equations, such as turbulence model, energy balance or chemical component balances. There arevarious methods for this, but perhaps the most often used in the field of CFD is control volume method. It isalso relatively easy to understand, as separate balances are formulated for each element. Fluxes at theboundaries for all the necessary variables (mass, momentum etc.) are calculated from the cell values withsuitable interpolation schemes. For fluid flow, often so called upwind schemes are used, where informationat the boundaries are taken from the upwind side, i.e. where the flow is coming from. Detailed algorithmsfor solving the equations are outside the scope of this course.

There are also other methods that can be used for solving the discretized balances, for example finite elementmethod (FEM) used by COMSOL software. Origins of FEM are in other fields of physics, and it is usedextensively e.g. in structural mechanics.

39

8.1 Evaluation of the gridOne of the most crucial aspects in CFD is to evaluate whether the discretization, i.e. the grid, is good for thepurpose. A poor grid may lead to completely erroneous results, and a "too good" grid with excessive numberof grid points leads to very slow solution of the model. Nowadays 3D CFD grids typically contain around 104-106 cells in engineering applications, so that the number of variables is typically at least six times larger thanthat (see discussion related to N-S equations and turbulence modeling). Each variable, e.g. new chemicalcompound which concentration profiles we wish to follow, increases the total number of variables by thenumber of cells.

Even if the number of grid points is optimal, the solution may be slow if the grid topology is poor. This maybe simply a result of a wish to model very small details in large equipment, requiring very small cells near thesmall details, and large cells elsewhere. Then the small cells may be crucial to the computational speedalthough the total number of cells is reasonable.

In the following figure one typical CFD grid is shown. The modeled piece of equipment is part of a laminarmixer, where two flows coming from the left are divided and recombined in order to form a lamella kind ofstructure for the flow, where diffusion lengths would be short implying fast mixing. These kinds of structuresare used in microprocess technology, where the small scales result in laminar flow. In laminar flows, mixingis always much slower compared to similar situation with turbulence.

In the figure, the edges of the computational cells at the walls are shown. The cells continue and fill the spacewhere fluid flows. It can be seen that the cell size is not the same everywhere; near the corners there aresmaller cells than elsewhere. This is necessarily due to numerical reasons; otherwise the whole solutionwould suffer from inaccuracy. In general, the grid shown in the figure above is not particularly dense, andstill suffers probably from numerical diffusion (one form of inaccuracy smoothing sharp gradients).

One good way of evaluating grid is to check the solutions with different number of cells. Then a suitablevalue, such as overall flow rate or total energy dissipated to the system is calculated, and checked whetherthis value changes as a function of grid size. In the following figure a typical situation is shown, where thenumber of cells is on the x-axis and a modeled property on y-axis. Sometimes there are different ways tocalculate the modeled property, and predictions with the different methods do not give the same result until

40

the grid is good enough. One example is mixing power in stirred tanks. It can be predicted from the turbulentenergy dissipation averaged over all the cells, or from angular moment imposed to the impeller blade.Typically the former is converged to the final value with a higher number of cells than the latter. Thendepending on the primary reason for the CFD analysis, the optimal number of cells may also vary.

One further issue that needs to be taken into account is the stability of time dependent CFD solution. If thetime step is too long compared to the flow, the solution is not stable. In practice, the flow should not proceedmore than one cell interval at a given time step (the flow should not "jump" over a cell). This limitation isknown as Courant-Friedrichs-Lewy criterion.

8.2 Modeling of equipment with moving partsIn many cases, there are moving parts (walls from the CFD point of view) in the equipment. For example instirred tanks impeller moves with respect to the equipment. Then the grid needs to be somehow adjusted totake this into account. There are two commonly used methods: Multiple Reference Frames (MRF) and movinggrid. In the first, the vessel is divided into two sub-regions, one describing the volume close to the impellerand one the volume outside it. The impeller region moves with the impeller so that the grid is stagnant withrespect to the impeller. The outside region is stagnant with respect to the other vessel. As the two grids movewith respect to each other, the MRF method needs to distribute information between the two regions.

The moving grid method is, according to its name, such that the grid is adjusted at each time step to the newgeometry. This is very versatile method and can be used for any changing geometry. However, it iscomputationally more challenging than MRF. The ideas of the two methods are illustrated in the followingfigure:

MRF with two regions Moving grid

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8.3 Computational modeling of multiphase systemsMultiphase fluid flow is very typical in chemical engineering applications, which adds complexity to manyother fluid flow modeling tasks for example in mechanical engineering problems. This is the case inmultiphase reactors, separation processes, agglomeration, filtration and so on. In many cases the structureof the two phase system is not well defined, for example in randomly packed beds (catalyst particles orrandom packings in mass transfer operations). In those cases, the region can be assumed porous, and frictionterms for CFD can be taken from the similar terms discussed in the mechanical energy balances.

In other cases, there is a flowing dispersed phase, for example bubbles, droplets or solid particles. In thosecases, there are typically two options. The first is when both cases are modeled with their own momentumequations, and the interaction between them with drag laws. This is called Euler-Euler, or interpenetratingcontinua approach. Here the precise locations of the dispersed phase elements (bubbles, droplets...) are notresolved, but just a volume fraction of each phase is assumed for each computational cell. Another option isso called Lagrangian approach, or Euler-Lagrange as continuous phase is modeled normally. In this case,individual particles (or particle swarms) are modeled so that their locations and velocities are known.

Typically Euler-Euler approach is used for high volume fractions and large systems, and Euler-Lagrange forrelatively dilute systems and when particle impacts to solid structures are important. Even for Euler-Eulerapproach, approximate fluid element paths can be estimated with tracer particles after a solved CFDsimulation.

Essential in a successful two-phase simulation is correct modeling of interphase momentum transfer.Typically drag force is the most important, but there are also other possible relevant forces, such as lift forceor virtual mass. Even for the drag force, there are several possible models available, and it is not always self-evident which one to select. Sensitivity and case studies are often needed if the results cannot be validateddirectly based on experimental data.

8.4 Modeling of moving fluid interfacesIn some cases, the location of the fluid-fluid interface is important, so that the interface itself needs to beresolved. In academic research, bubble or droplet behavior (shape, breakage etc.) can be simulated directlyto get more insight to the system. This is often more of academic interest and to develop more practicalcorrelations for the modeled phenomenon. Sometimes the interface is also of practical interest, for examplein settling units and in case of slugs or other large fluid elements compared to the equipment size. Alsosurface shape may be of interest, e.g. in mixing whether there is such a surface vortex that enters the impellerregime leading to excessive entrainment of gas to the liquid.

There are different methods to model the moving fluid interface:

- Volume of Fluid (VOF). A volume fraction ("color function") is used to describe fractions of bothphases in a cell. The interface lies in cells where this variable is not 0 or 1. Various algorithms existfor trying to avoid interface smearing (it should be sharp).

- Level-set. A smooth scalar function is added to describe the interface location. The scalar is movedaccording to a transport equation.

- Front-tracking. The phase interface is tracked with a set of marker particles.

In all these cases the grid should be dense enough to capture the interface curvature so that surface tensionrelated forces can be taken into account properly.

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9 Flow measurementIn order to control processes, various measurements are needed. The most common ones are perhapstemperature and pressure measurements, but there are several equally important, such as fluid levelmeasurement and flow rate measurements.

There are numerous flow measurement devices based on different operating principles. One possibleclassification is the following (in parentheses some flowmeter types in each category):

1. Flowmeters with wetted moving parts (positive displacement, hydraulic Wheatstone bridge, turbine,variable area)

- sensitive to wear, mechanical failure can occur- typically only for clean fluids

2. Flowmeters with no wetted moving parts (differential pressure, oscillatory, target, thermal)

- plugging or wear can occur- adds pressure drop

3. Obstructionless flowmeters (Coriolis mass, magnetic, ultrasonic)

- basically a subset of 2, but fluid flows through freely

4. Flowmeters with sensors mounted external to the pipe (clamp-on ultrasonic, correlation)

- Sensor material not sensitive to fluid properties

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10 SymbolsA area (m2)c concentration (mol/m3)CD drag coefficient ( )Cp, Cv specific heat capacities at constant pressure and volume (J/molK) or (J/kgK)D diameter (m)DAB diffusion coefficient (m2/s)dp particle size (m)F force (N)f force term (various) (N)FD interfacial force (N)g gravitational acceleration constant 9.81 (m/s2)H stirred tank height (or liquid level) (m)hf friction loss (J/kg)j superficial velocity (m/s)k ratio of specific heat capacities ( )k turbulent kinetic energy (J/kg = m2/s2)k permeability (m2)L length (m)m, n parameters of viscosity models ( )n polytropic exponent ( )N impeller speed (1/s)NP power number ( )NQ flow number ( )p pressure (Pa)P power WQ flow rate (m3/s)R gas constant 8.314 (J/molK)r radius (m)S source term (various)S tank geometry parameter ( )t time (s)T temperature (K)T stirred tank diameter (m)U velocity (m/s)v velocity (m/s)V volume (m3)W pumping work (J/kg)Wcompr compressor work (J/mol)x length coordinate (m)X mass ratio of suspended solids (%)y length coordinate (m)z height position, coordinate (m)

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a flow profile correction factor ( )a gas saturation (gas fraction in void space) ( )s surface (interface) tension (N/m)r density (kg/m3)x pipe friction factor (Darcy) ( )l thermal conductivity (W/mK)e porosity ( )t shear stress (N/m2 = Pa)th Kolmogorov time scale (s)h Kolmogorov length scale (m)h efficiency ( )z local friction coefficient ( )µ viscosity (kg/ms = Pas)