momentum heat mass transfer mhmt4 newtonian fluids and navier stokes equations. steady and transient...

Momentum Heat Mass TransferMHMT4

Newtonian fluids and Navier Stokes equations. Steady and transient flow between parallel plates, flow in pipe, annular gap, hydraulic diameter. Non-Newtonian fluids. RMW equation. Thixotropic fluids.

Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

Navier-Stokes equations Non-Newtonian fluids

sourceDt

D

Unknowns / Equations

There are 10 unknowns (assuming isothermal flow):

u,v,w, (3 velocities), p, xx, xy,…(6 components of symmetric stress tensor)

And the same number of equations

Continuity equation

3 Cauchy’s equations

6 Constitutive equations

( ) 0ut

Dup g

Dt

2 ( )( )3

uII

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MHMT4

Substituting the constitutive equation for viscous stresses (Generalised Newtonian Fluid)

into the divergency term of the Cauchy’s equation

Dup g

Dt

2 ( )( )3

uII

Navier Stokes equations

gives, see the next slide…


22 ( ( )( )) ( ( )( ( ) )) ( ( )( ) )

3 3Tu

II II u u II u

2 2

2( ) ( ) ( ( ) )

3

2( ) ( ) ( ( ))

3

2( ) ( )

3

( )3

ij j i kij

i i i i j i k

j i i

i i i j j i

j i i i i

i i i j j i j i j i

j i

i i i j

u u u

x x x x x x x

u u u

x x x x x x

u u u u u

x x x x x x x x x x

u u

x x x x

2 2

3i i

j i j i

u u

x x x x

( ) 2( ( ) ) ( ) ( ) ( ) ( ) ( )

3 3T II

II u II u u u II

These terms are ZERO for incompressible fluids

These terms are small and will be replaced by a parameter sm

Divergence of viscous stresses

This is the same, but written in the index notation (you cannot make mistakes when calculating derivatives)

MHMT4


( ( ) ) m

Dup II u s g

Dt

General form of Navier Stokes equations valid for compressible/incompressible Non-Newtonian (with the exception of viscoelastic or thixotropic) fluids

Special case – Newtonian fluids with constant viscosity (compressible)

2 21( ) ( )

3i i i k

k ik i k k i k

u u u upu g

t x x x x x x

MHMT4

2 1( ( ))

3

Dup u u g

Dt

This term is zero for incompressible liquids

Navier Stokes equationsSpecial cases

2D flow (liquids) in Cartesian coordinate system2 2

2 2

2 2

2 2

( ) ( )

( ) ( )

x x x x xx y x

y y y y yx y y

u u u u upu u g

t x y x x y

u u u u upu u g

t x y y x y

MHMT4

2D flow in cylindrical coordinate system

2

2

2

2

1( ) ( ( ) )

1( ) ( ( ) )

r r r r rr z r

z z z z zr z z

u u u ru upu u g

t r z r r r r z

u u u u upu u r g

t r z z r r r z

Navier Stokes equationsSpecial cases

2D compressible flow formulated in terms of stream function and vorticity reduces number of equations (continuity equation is automatically satisfied) and eliminates pressure.

MHMT4

2 2

2 2

2 2

2 2

( ( )) ( ) ( )y yx xx y

u gu gu u

t x y x y x y x y

x y

This term is zero for

incompressible liquids

y xu u

x y

, x yu u

y x

These equations follow from the Navier Stokes equations using vorticity and stream function according to the previously introduced definitions

Navier Stokes solutionsMHMT4

Modigliani

The convective acceleration term makes Navier Stokes nonlinear and therefore analytical solutions can be found only when this term disappears (flow in straight pipes) or is very small comparing with the viscous term (Re<1, creeping flow).

uu

Drag flowLaminar flow between parallel plates

2

2( )x x x

x

u u upu

t x x y

MHMT4

x

ux(y)

y U

H

Steady drag flow (no pressure gradient)2

20 xu

y

21 cycux x

Uu y

H

Shear stress (constant in the whole gap)yx

U

H

Transient drag flow (U-unit step of velocity of plate)2

2x xu u

t y

x

ux(t,y)

yU

H

(t)Momentum of -layer

1( )

2U t

Stress in -layer( )

U

t

Stress is a flux of momentum 1( )

( ) 2

Udt Ud t

t

21( )

4dt d t

Penetration depthMHMT4

x

ux(t,y)

yU

H

(t)

2

0 0

1( )

4

t

dt d t

( ) 4 4t t t

Integration

yields expression for thickness of the accelarated fluid layer (penetration depth)

This solution is only an approximation, because the linear velocity profile with a turning point at is not an exact solution of the Navier Stokes equation. Exact solution exists in form of an infinite series for a finite thickness of gap H and for the case that H is defined by error function, giving more accurate prediction of the penetration depth

( )t t

http://en.wikipedia.org/wiki/Error_function



Extensional flowMHMT4

In the simple shear flow between parallel plates velocities are defined in terms of the rate of shear

In the simple extensional flow (uniform stretching of incompressible fluid in the x-direction) velocities are defined in terms of rate of elongation

0 zyx uuyu [1/s]

1 1

2 2x y zu x u y u z

[1/s]

x

z

y

Constitutive equation for Newtonian liquid gives

3 timeselongationalshear viscosityviscosity(Trouton's ratio)

2

3

xx yy zz

xx yye

Flow in a circular pipeMHMT4

D=2Rz

uz(r)dr

dz

r

Steady fully developed laminar flow – only one non-zero velocity component uz(r)

Balance of forces for control volume (ring dr x dz)

0

))()((2)(2

z

p

rrz

rdzdrrdzr

z

p

zrdrdz

rzrzzz

rzrzrz

zz

Constitutive equation (Newtonian liquid)

0 ( )z z r zzz rz

u u u u

z r z r

Navier Stokes in z-direction

z

p

r

ur

rr

z

p

r

u

r

u

r

z

zz

))((1

)1

(2

2

21

2

ln4

)( CrCdz

dprruz

2

2( ) (1 ( ) )4z

R dp ru r

dz R

General solution

Boundary conditions r=0,R


12 4 42 * *2 *

0 0 0

2 ( ) 2 (1 ( ) ) (1 )4 2 8

R R

z

R dp r R dp R dpV ru r dr r dr r r dr

dz R dz dz

Volumetric flowrate

Hagen Poisseuille law 4

8

R dpV

dz

Darcy Weisbach equation for calculation of pressure drop in channels

DuuL

pD

D

Lup ff

642

2

12

2

2

32

D dpu

dz

Re

64f Re

uD

Reynolds number is an indicator of laminar/turbulent flow regime. Above the value Re=2300 (in US) or 2100 (in EU ) the flow is mostly turbulent.

The complete profile including the turbulent flow regime is presented in the Moody diagram (see next page) or described by using correlation Churchill S.W.: Friction factor equation spans all fluid-flow regimes. Chemical Engineering, 1977, 84 pp.91-92.

http://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_equation




21

2 f

Lp u

D

Frinction factor f depends upon Re

and relative roughness

Pressure drop in a circular pipe Darcy Weisbach equation

http://en.wikipedia.org/wiki/Darcy%E2%80%93Weisbach_equation






Flow in an annular gapMHMT4

Navier Stokes equation for flow between two concentric pipes is the same and its general solution is also the same as with the circular pipe

21

2

ln4

)( CrCdz

dprruz

2 222

2 2

1( ) (1 ( ) ln )

14 lnz

R dp r ru r

dz R R

z

D2=2R2

uz(r)

D1=2R1

Only the boundary conditions are different, giving

4 2 242 (1 )

(1 )18 ln

R dpV

dz

60

70

80

90

100

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

R1/R2

f.R

e

64-circular pipe

96-parallel plates

(use per partes for integration r lnr)

2

22

64 (1 ) 64( )

1Re Re1

1ln

f f

verify that the limit for 1 is 3/2 using expansion

...32

)1ln(32

Equivalent diameterMHMT4

General cross section of a channel can be characterized by equivalent hydraulic diameter Dh, that is used in definition of Reynolds number.

4 4h

A VD

P S

Cross section surface

Volume of channel

Perimeter of cross section

Surface of wall

At turbulent flows the same correlations for pressure drop (friction factor) can be used. Correlations for circular pipe are usually used, however the cross sections with sharp corners (triangles, cusped ducts) lead to error up to 35% .

Equivalent diameter is used also in laminar flows, but different correlations for different cross sections must be used (from this point of view the laminar regime is more complicated).

Modified definitions of equivalent diameter exist for specific classes of cross sections (e.g. average distance from the point of maximum velocity in triangles, or square root of the cross section area, see next slides).

a

b

Increased f when compared with circular pipe

Equivalent diameter and fReMHMT4

Equivalent diameter for rectangular cross section

2 3

4 2

2( ) 1 /

Re 96(1 1.355 1.9467( ) 1.701( ) ...)

h

f

ab bD

a b b a

a a a

b b b

Equivalent diameter for excentric inner tube is independent of excentricity!

2

2 2

64(1 )Re

11 (1 ) / ln

f

2 22 1

2 12 1

4 ( )2( )

2 ( )h

R RD R R

R R

e

However fRe decreases with the increasing

excentricity!

Re 28f

fRe varies from the values about 30 (corners) to about 130 (bundle of pipes)

Non-Newtonian fluid flowMHMT4


How to calculate the volumetric flowrate in a circular pipe as a function of pressure drop in the case of non-Newtonian fluids?

In 1D case (fully developed unidirectional axial flows) the constitutive equations for incompressible generalised Newtonian fluids (GNF) can be expressed as

)( fdr

duz

shear rate

shear stress

p

y

n

f

Kf

f

)(

)()(

)(

/1

for Newtonian fluid

for Power law fluid

for Bingham fluid

RR

zR

zRz

R

z drrfdrrdr

rdudrr

dr

rduru

rdrrruV

0

2

0

2

0

20

2

0

)()(

))(

2

1)](

2([2)(2

Volumetric flowrate for quite arbitrary radial velocity profile is


2 2 3 2

0 0 0

( ) ( )( ) ( ) ( )wR R

w w w

R R RV f r dr f d f d

Radial shear stress profile follows from the equilibrium of forces

wzRpR

zrpr

2

22

2

w

Rr Rr

w

(r)

This linear shear stress profile holds for any fluid and can be used for replacement of the integration variable r

This is Rabinowitsch Mooney Weissenberg (RMW) equation

23 3

0

1( )

w

w

Vf d

R

giving volumetric flowrate regardless of

specific model as a function of wall shear stress

z

pRw

2


Application RMW equation for power law fluid

2 1/ 1/ 1/3 1/ 3

0

1( ) ( )

3 1 3 1 2

w

n n nwn

w

V n n R pd

R K n K n LK

Friction factor

2

2 64

Refg

D p

Lu

232

Reg

Lu

D p

3 1 2

( )nu n LK

pR n R

2

Re 8( )2(3 1)

n nn

g

n D u

n K

Modified Reynolds number (reduces to standard Re for n=1)

1/( ) ( ) nfK


Application RMW equation for Bingham fluid ( ) y

p

f

4)(3

1)(

3

41

Re

64

w

y

w

y

f

4 4 3 32 4

3 3 3

1 1 4 1( ) (1 ( ) )

4 3 4 3 3

w

y

y w y w y y ywy

w p p w p w w

Vd

R

Introducing the friction factor and expression for the wall shear stress

2 2

82 wf

D p

Lu u

we obtain

Rep

uD

Remark: Given pressure drop (therefore w) it is quite easy to calculate flowrate. Reversely: given flowrate the pressure drop must be calculated by solution of algebraic equation of the 4 th order (but there exists graphical representation of the previous equation).


Exercise: Derive the RMW equation for the Herschel Bulkley model

1/( ) ( )y nfK

1 3 1 2 1 12

3

2( ) [ (1 ) (1 ) ( ) (1 ) ]

3 1 2 1 1

n n ny y y y yw n n n n

w w w w w

V n n n

R K n n n

K-consistency, n-power law index, y-yield stress

….verify that the n=1 reduces to the previously derived Bingham model.


Practical importance of RMW equation is in the fact that it enables generalization of experiments with arbitrary liquid, without necessity to identify a specific rheological model.

L

2R

differential pressure transducer

flow meterpump

p

V

2w

R p

L

3

V

R

T

Diagram of consistency variables

Experimentally determined curves w, are independent of the pipe dimensions, therefore can be used for design of pipelines (with the same liquid and at the same temperature).

Remark: different w, curves recorded at different diameters of pipes indicate anomalies, for example wall slip (Mooney analysis), different flow regime (turbulent flow) or experimental errors (insufficient stabilization length of pipe, Bagley correction).

Thixotropic fluid in a pipeMHMT4

flow meter

pump

L

2R


pV

fluid X

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

4

5

6

7

8

9

10

11

12x 10

6

flowrateV

pressure dropp

Research described in the following pages was motivated by experimentally determined strange behaviour of a secret fluid X:

How is it possible that one and the same fluid at the same temperature and at the same flowrate exhibits different pressure drops ?


Thixotropic fluids are characterised by viscosity, which depends upon the deformation history (structure and consistency is affected by shear rate at previous times). Example of constitutive equation of a thixotropic liquid was presented as the HZS model (see previous lecture). Problem of pressure drop of a thixotropic fluid in laminar flow in a circular pipe is usually solved numerically, see the list of relevant papers

Sestak J., Zitny R., Houska M.: Změny vlastností tixotropních látek v průběhu zpracování. In 21.konf.SSChI, Vyhne °1994 (HZS model, evaluation of structural parameter change in a continuous system of series of ideally mixed vessels)

Sestak J., Zitny R., Houska M.: Dynamika tixoropnich kapalin. Rozpravy CSAV Praha 1990

Sestak J., Zitny R., Houska M.: Simple rheological models of food liquids for process design and quality assessment. Journal of Food Engineering, 1983, pp.35-49 (thixotropic integral model Zitny, and differential model HZS)

Sestak J., Houska M., Zitny R.: Mixing of thixotropic fluids. Journal of Rheology, 1982, pp. 459-475

Zitny R.: Nestacionární tok tixotropní kapaliny v trubce. Acta polytechnica, 1977, pp.95-102 (integral model of thixotropy solved by assuming that the structural parameter depends only upon time and axial coordinate)

Zitny R.: Vliv dissipace a tixotropie na tok nenewtonskych kapalin v trubce. Disertation CVUT 1977

Sestak J., Zitny R.: Tok tixotropni kapaliny v trubce. Acta Polytechnica, 1976, pp.45 (integral model of thixotropy, pressure drop calculated from correlation where is Deborah number, ratio of relaxation and process time)

0

1/

structural number (K / )Re Reynolds for De Deborah numberpower law fluids

Re1 ( 1) (1 )

64m

f m De

SN K

SN De e

-time constantof thixotropy

uDe

L

Ahmadpour A., Sadeghy K.: An exact solution for laminar, unidirectional flow of Houska thixotropic fluids in a circular pipe. J. of Non-Newtonian Fluid Mechanics, 194 (2013), pp.23-31

Corvisier P., Nouar C., Devienne R., Lebouché M.: Development of a thixotropic fluid flow in a pipe. Experiments in Fluids, 31 (2001), pp.579-587

Schmitt L., Ghnassia G., Bimbenet J.J., Cuvelier G.: Flow properties of stirred Youghurt: Calculation of the pressure drop for a thixotropic fluid. J.Food Eng. 37 (1998), pp.367-388

Escudier M.P., Presti F.: Pipe flow of a thixotropic liquid. J. Non-Newtonian Fluid Mech., 62 (1996), pp.291-306

Billingham J., Fergusson J.W.J.: Laminar unidirectional flow of a thixotropic fluid in a circular pipe. J. of Non-Newtonian Fluid Mech., 47 (1993), pp.21-55

Kemblowski Z., Petera J.: Memory effects during the flow of thixotropic fluids in pipes. Rheol.Acta 20, (1981), pp. 311-323


The HZS model of thixotropic fluids is represented by the Herschel Bulkley constitutive equation (a combination of power law liquid with a consistency coefficient K and power law index n and Bingham liquid with a yield stress y )

1

2(( ) 2 : )2 :

ny yK K

Structural parameter =1 describes fully recovered inner structure (and high consistency of liquid), while =0 corresponds to completely destroyed structure (and minimum consistency K and yield stress). Time changes of are described by

regenerationdiffusion structure decay (usually neglected)

(1 ) 2 :m

DD a b

Dt

There exist many different modifications and interpretations, for example the latest work, Ahmadpour (2013), assumes only partial and not the material time derivative on the left side, the diffusion term on the right side is considered only by Billingham and Fergusson (1993) in a generalized Moore’s model of thixotropy.


( ) ny y K K

The tensorial form of the HZS model can be simplified for the special case of unidirectional simple shear flow to scalar equations for the shear stress and the shear rate

2

decay regenerationdiffusion

(1 ) mDD a b

Dt

u

r

Complete solution of a creeping laminar flow in a pipe should calculate axial as well as radial profiles of velocity and structure parameter based upon linearity of radial shear stress profile

Ru(t,r,x) (t,r,x) =w r/R

A great simplification would be assumption that the depends only upon the axial coordinate and time, (t,x). This assumption can be accepted only if the flow is so slow that the diffusion in the radial direction has enough time to equalize the radial profile of and that the problem of -transport with a nonuniform radial velocity profile can be substituted by a model with “plug flow” (constant velocity) and modified diffusion in the axial direction (model of axial dispersion, which will be discussed later – lecture on mass transport)

2

2

decay regenerationconstant dispersion velocity

(1 )a

mu D a bt x x


For independent of radial coordinate it is possible to apply RMW (Rabinowitsch, Mooney, Weissenberg) equation with

*1/ 2

3 3 *

* * * * *1 3 1 2 1 12

*

1( )

2( ) ( (1 ) (1 ) ( ) (1 ) )

3 1 2 1 1

w

y

y n

w

n n ny y y y yw n n n n

w w w w w

Vd

R K

n n n

K n n n

yyyKKK ** ,

We need to calculate pressure drop (dp/dx=2w/R) for given flowrate therefore it is necessary to invert the previous equation

))()(()1)(12(

2

12

11(

13

))(

(

3*

2**

3**

w

y

w

y

w

y

n

wyw

nnn

n

nn

n

R

VK

Remark: Iterative evaluation is necessary, but is fast and convergent for arbitrary n,K,y


If the structural parameter depends only upon x and time and if we neglect the axial dispersion term the evolution of is described by hyperbolic partial differential equation

(1 ) mu a bt x

which can be integrated analytically along characteristic dx=u.dt as soon as the velocity and shear rate are constant

rateshear sticcharacteri

32 ,

R

V

R

Vu

]))(([1

)( ))((0

0ttbamm

m

ebaaaba

t

(t) is value of structural parameter of a fluid particle having value 0 at time t0 (assuming that the particle is under action of constant shear rate).

0,46

0,47

0,48

0,49

0,50

0,51

0,52

0,53

0,54

0,00E+00 5,00E-05 1,00E-04 1,50E-04 2,00E-04 2,50E-04 3,00E-04

flowrate

lam

bda


In our experiment the fresh fluid at the inlet to pipe has (pressumably) fully recovered structure, therefore 0=1. At a distance x from inlet depends of a fluid particle upon the time of action (t=x/u) and upon the intensity of action (m). The time of action decreases with the increasing flowrate, while the intensity of decomposition increases with flowrate:

flow meter

pump

2R


pV

fluid X

x=?=1

2

( )1( , ) ( )

m x Ra BVm V

mx V a BV e

a BV

It is interesting that there exists a flowrate when is minimum, see graph. This extreme (maximum of thixotropy effect) is determined by equation

0),,,,,(

V

mRxbaV

…that must be solved numerically.


Resulting expressions (for example expressions for friction factor, for structural parameter, etc) are usually formulated in terms of dimensionless parameters, Debora number, thixotropy number, Bingham number and other, see for example Billingham, Fergusson (1993) .

1 1( (1 ) )De Tx

e m m mDe De

))(( 32 m

RV

baxR

VDe

3( )m

b VTx

a R

Deborah number De is the ratio of thixotropic time scale and the process time. The thixotropic number Tx is just the ratio of decay and bildup terms.

The „optimum flowrate“ condition can be expressed generally as0V

…and for the special case m=1, the Debora number at maximum tixotropy goes to infinity. Nontrivial optimum exists only for m<1.

Example: a=b=0.01, m=0.9, R=0.01, x=1 optimum at De=1, Tx=5.3 (therefore flowrate 2.10 -5 m3/s)


flow meter

pump

L

2R


pV

fluid X

=1

The following MATLAB program calculates pressures p corresponding to the stepwise increase and decrease of flowrate

timeVmin=V1

Vmax=Vnstep/2

t1=0 tnstep

duration of time steps tistep is determined according to flowrate (velocity u) and constant x

x

x

t

characteristics

x

2RV

xt

istepistep


% simulation (flowrate up and down)l=3; %length of piper=0.01; %radiusnx=50;nstep=150;k=50; dk=500; ty=3000; dty=3000; n=0.8 % Herschel Bulkleya=0.000001; b=0.01; m=0.9; % HZS thixotropy modelvmin=1e-5; vmax=1e-4; % min,max.flowratex=linspace(0,l,nx);dx=l/(nx-1);% time sequence of flowrate and corresponding timesnstep2=nstep/2;dv=(vmax-vmin)/nstep2; v(1)=vmin;time(1)=0;for i=2:nstep2 v(i)=v(i-1)+dv; time(i)=time(i-1)+dx*3.141*r^2/v(i-1); endfor i=nstep2+1:nstep v(i)=v(i-1)-dv; time(i)=time(i-1)+dx*3.141*r^2/v(i-1); endlamx(1:nx)=1;[dp(1),taux]=dpa(k,dk,ty,dty,n,v(1),r,lamx,nx,dx);[dt,lam]=lamnew(a,b,m,v(1),r,lamx,nx,dx);figure(3)hold offfor istep=2:nstep lamx(1:nx)=lam(1:nx); [dp(istep),taux]=dpa(k,dk,ty,dty,n,v(istep),r,lamx,nx,dx); [dt,lam]=lamnew(a,b,m,v(istep),r,lamx,nx,dx); if mod(istep,40)==0 plot(x(1:nx),lam(1:nx)) hold on endendfigure(1)plot(time(1:nstep),v(1:nstep))figure(2)hold offplot(v(1:nstep),dp(1:nstep))hold onplot(v(1),dp(1),'rd')

function [dt,lam]=lamnew(a,b,m,vdot,r,lamx,nx,dx)% vector of structural parameters lam(1..nx) at time tnew% inlet value is always lam(1)=1 (fresh fluid)% a,b,m model parameters (see model HZS)u=vdot/(3.141*r^2);gamma=vdot/(3.141*r^3);lam(1)=1;dt=dx/u;gam=gamma^m;for i=1:nx-1lam(i+1)=(a-(a-(a+b*gam)*lamx(i))*exp(-(a+b*gam)*dt))/(a+b*gam);end

function tauw=hb(ty,k,n,vdot,r)% ty-yield stress, k-consistency, n-flow index, vdot-flowrate, r-radiuskappa=n/(3*n+1);eps=1e10;iter=0;while (iter<50 & eps>1e-4) iter=iter+1; tauw=ty+k*(vdot/(kappa*3.141*r^3))^n; t=ty/tauw; kappan=n/(3*n+1)*(1-t/(2*n+1)-2*n/((n+1)*(2*n+1))*(t^2+n*t^3)); eps=abs(kappa-kappan); kappa=kappan;end

function [dp,taux]=dpa(k,dk,ty,dty,n,vdot,r,lamx,nx,dx)% pipe length (nx-1).dx , radius r, flowrate vdot, strict parameter% lam(1),...lam(nx)% result: pressure drop dp, vector of wall stresses taux(1)....taux(nx)for i=1:nx taux(i)=hb(ty+lamx(i)*dty,k+lamx(i)*dk,n,vdot,r);enddp=0;for i=1:nx-1 dp=dp+taux(i)*2*dx/r;end

]))(([1

)( ))((0

0ttbamm

m

ebaaaba

t

))()(()1)(12(

2

12

11(

13

))(

(

3*

2**

3**

w

y

w

y

w

y

n

wyw

nnn

n

nn

n

R

VK


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

4

5

6

7

8

9

10

11

12x 10

6

V flowrate

pres

sure

dro

p

n=0.8, m=0.9

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

-4

time [s]

V f

low

rate

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

3

4

5

6

7

8

9

10

11

12x 10

6

V flowrate

pres

sure

dro

p

n=0.8, m=0.9

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

-4

time [s]

V f

low

rate

0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

-4

time [s]

V f

low

rate

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

4

5

6

7

8

9

10

11

12x 10

6

V flowrate

pres

sure

dro

p

n=0.8, m=0.9

Very slow (180 s) Medium (70 s) Fast changes (25 s)

k=50 dk=500 ty=3000 dty=3000 n=0.8 a=0.000001 b=0.01 m=0.9

red diamond is starting point (zero time,

minimum flowrate)


k=50 dk=500 ty=3000 dty=3000 n=0.8 medium rate of flowrate changes

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

7

V flowrate

pres

sure

dro

p

n=0.8, m=0.9

b=0.001

a=0.000001

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

4

5

6

7

8

9

10

11

12x 10

6

V flowrate

pres

sure

dro

p

n=0.8, m=0.9

b=0.01

a=0.000001

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5x 10

6

V flowrate

pres

sure

dro

p

n=0.8, m=0.9

b=0.05

a=0.000001

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

4

5

6

7

8

9

10

11

12x 10

6

V flowrate

pres

sure

dro

p

n=0.8, m=0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

5

6

7

8

9

10

11

12

13

14x 10

6

V flowrate

pres

sure

dro

p

n=0.8, m=0.9

a=0.01

b=0.01

a=0.1

b=0.01

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

7

V flowrate

pres

sure

dro

p

n=0.8, m=0.9

a=1

b=0.01


k=50 dk=500 ty=3000 dty=3000 n=0.8 a=0.001 b=0.01 m=0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

4

5

6

7

8

9

10

11

12x 10

6

V flowrate

pres

sure

dro

p

n=0.8, m=0.9

m=0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

0.5

1

1.5

2x 10

7

V flowrate

pres

sure

dro

p

m=0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

7

V flowrate

pres

sure

dro

p

m=0.5

EXAMMHMT4


What is important (at least for exam)MHMT4

Navier Stokes equations (2D incompressible)

2 2

2 2

2 2

2 2

( ) ( )

( ) ( )

x x x x xx y x

y y y y yx y y

u u u u upu u g

t x y x x y

u u u u upu u g

t x y y x y

2 2

2 2

2 2

2 2

( ( )) ( ) ( )y yx xx y

u gu gu u

t x y x y x y x y

x y

Formulation with vorticity and stream function


Drag flow between parallel plates (steady and transient)

Penetration depth( )t t

22( ) (1 ( ) )

4z

R dp ru r

dz R

Flow in a circular pipe

21

2 f

Lp u

D

Darcy Weisbach, Reynolds number and Mooney diagram

Re

64f Re

uD


Non Newtonian flows

RMW equation

23 3

0

1( )

w

w

Vf d

R

z

pRw

2

momentum heat mass transfer mhmt4 newtonian fluids and navier stokes equations. steady and transient...

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