momentum heat mass transfer
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Momentum Heat Mass Transfer. MHMT2. Balance equations. Mass and momentum balances. - PowerPoint PPT PresentationTRANSCRIPT
Momentum Heat Mass TransferMHMT2
Fundamental balance equations. General transport equation, material derivative. Equation of continuity. Momentum balance - Cauchy´s equation of dynamical equilibrium in continua. Euler equations and potential flows. Conformal mapping.Rudolf Žitný, Ústav procesní a
zpracovatelské techniky ČVUT FS 2010
Balance equations. Mass and momentum balances.
sourceDtD
Mass-Momentum-Energy
Conservation laws-conservation of mass-conservation of momentum M.du/dt=F (second Newton’s law)-conservation of energy dq=du+pdv (first law of thermodynamics)
Transfer phenomena summarize these conservation laws and applies them to a continuous system described by macroscopic variables distributed in space (x,y,z) and time (t) hpu ,,,
MHMT2
Mechanics and thermodynamics are based upon the
Description of kinematics and dynamics of discrete mass points is recasted to consistent tensor form of integral or partial differential equations for velocity, temperature, pressure and concentration fields.
Transported property MHMT2
Convective fluxes ( transported by velocity of fluid)
Diffusive fluxes ( transported by molecular diffusion)
Driving forces = gradients of transported properties
Transfer phenomena looks for analogies between transport of mass, momentum and energy. Transported properties are scalars (density, energy) or vectors (momentum). Fluxes are amount of passing through a unit surface at unit time (fluxes are tensors of one order higher than the corresponding property , therefore vectors or tensors).
Transported property
u u
q
Aj
MHMT2
dyud x
yx)(
( )py
d c Tq a
dy
( )AAy AB
dj Ddy
]/[ 2 Pasm
smkg
2 2[ ]J Wm s m
2[ ]kgAm s
This table presents nomenclature of transported properties for specific cases of mass, momentum, energy and component transport. Similarity of constitutive equations (Newton,Fourier,Fick) is basis for unified formulation of transport equations.
Mass conservation (fixed fluid element)MHMT2
Mass conservation principle can be expressed by balancing of a control volume (rate of mass accumulation inside the control volume is the sum of convective fluxes through the control volume surface). Analysis is simplified by the fact that the molecular fluxes are zero when considering homogeneous fluid.
Control volumes can be fixed in space or moving. The simplest case, directly leading to the differential transport equations, is based upon identification of fluxes through sides of an infinitely small FLUID ELEMENT fixed in space.
Mass conservation (fixed fluid element)
......)]21()
21[(
)()()()(
zyxxuzyx
xuux
xuu
yxBTzxSNzyWEzyxt
0u v wt x y z
Accumulation of mass Mass flowrate through sides W and E
xx
y
z
zSouthWest
Top
East
North
Bottom
x
y
MHMT2
Using the control volume in form of a brick is straightforward but clumsy. However, tensor calculus is not necessary. ),,( wvuu
0i
i
ut x
sometimes written as( )
( ) 0
div u
ut
Continuity equation written in the index notation (Einstein summation is used)
Continuity equation written in the symbolic form (Gibbs notation)
MHMT2
Using index or symbolic notation makes equations more compact
Example: Continuity equation for an incompressible liquid is very simple
0 0i
i
uux
Mass conservation (fixed fluid element)
1 2 3( , , ) ( , , ) ( , , )x y zu u v w u u u u u u
Time rate changes of MHMT2
Rate of change of property (t,x,y,z) recorded by the observer moving at velocity
( ) ( ) ( ) ( )s s s s
d x y zdt t x t y t z t
u v w ut x y z t
Total derivativeTime changes of recorded by observer moving at velocity
Material derivative is a special case of the total derivative, corresponding to the observer moving with the particle (with the same velocity as the fluid particle)
)(su
)(su
u
tDtD
Observer (an instrument measuring the property ) can be fixed in space and then the recorded rate od change is
t
0 km/h
20 km/h
10 km/hfixed observer measuring velocity of wind
running observer
observer in a balloon
( ) div ut
( (
( )
( )
( )
))
div u u grad
div u u grad
u g
t
di
t
tD
t Dt
u
rad
v
This follows from the mass balance
These terms are cancelled
Balancing in a fixed fluid element and material derivativeMHMT2
[Accumulation in FE ] + [Outflow of from FE by convection] =intensity of inner sources or diffusional fluxes across the fluid element boundary
Integral balance of (fixed CV)MHMT2
V ds
n
uIntegral balance in a fixed control volume has the advantage that it is possible to exchange a time derivative and integration operator (V is independent of time)
VV
dvt
dvdtd
rate of accumulation Convective transport Diffusive flux of Internal (projection velocity superposed to the to outer normal) fluid velocity u
( )
V S S V
dv n u ds n ds dvt
volumetric sources of e.g. gravity, microwave
( )( ( )) =
V
V S VD
dvDt
u dv n ds dvt
apply Gauss theorem (conversion of surface to volume integral)
Differential balance of MHMT2
Integral balance should be satisfied for arbitrary volume V
( ( ) ( ) ) =0
( ) ( ) 0
V
u dvt
ut
Therefore integrand must be identically zero
Remark: special case is the mass conservation for =1 and zero source term
( ) 0ut
and using this the differential balance can be expressed in the alternative form
( ) 0DDt
Momentum conservationMHMT2
Modigiani
Momentum balance = balance of forces is nothing else than the Newton’s law m.du/dt=F applied to continuous distribution of matter, forces and momentum.
Newton’s law expressed in terms of differential equations is called
CAUCHY’S equation
valid for fluids and solids (exactly the same Cauchy’s equations hold in solid and fluid mechanics).
Momentum integral balance
MOMENTUM integral balances follow from the general integral balances
( )u
MHMT2
( )( ( ))V S V
u uu dv n ds fdvt
u
( )( ( ))V S V
u dv n ds dvt
for
total stress
f
external forces,
like gravitysource
flux
p
viscous stress
Momentum conservationMHMT2
Differential equations of momentum conservation can be derived directly from the previous integral balance
( )V V
Du dv p f dvDt
which must be satisfied for any control volume V, therefore also for any infinitely small volume surrounding the point (x,y,z) and
viscous forcespressure volumetricon surface ofacceleration forces forcesfluid particleof fluid particle
Du p fDt
This is the fundamental result, Cauchy’s equation (partial parabolic differential equations of the second order). You can skip the following shaded pages, showing that the same result can be obtained by the balance of forces.
[N/m3]
Cauchy’s Equations
Du p fDt
MHMT2
( )D uDt t
Making use the previously derived relationship
the Cauchy’s equation can be expressed in form
Cauchy’s equation holds for solid and fluids (compressible and incompressible)
( )u uu p ft
These formulations are quite equivalent (mathematically) but not from the point of view of numerical solution – CFD.
formulation with primitive variables,u,v,w,p. Suitable for numerical
solution of incompressible flows (Ma<0.3)
conservative formulation using momentum as the unknown variable is suitable for
compressible flows, shocks…. Passage through a shock wave is accompanied by jump of p,,u
but (u) is continuous.
Ma-Mach number (velocity related to speed of sound)
Euler’s Equations inviscid flowsMHMT2
Inviscid flow theory of ideal fluids is very highly mathematically developed and predicts successfully flows around bodies, airfoils, wave motion, Karman vortex street, jets. It fails in the prediction of drag forces.
Euler’s Equations and velocity potential
1u u u p ft
MHMT2
Eulers’s equations are special case of Cauchy’s equations for inviscible fluids (therefore for zero viscous stresses)
Vorticity vector describes rotation of velocity field and is defined as
ni imn
m
uu
x
2 1 2 13 312 321
1 2 1 2
nz mn
m
u u u u ux x x x x
for example the z-coordinate of vorticity is
Using vorticity the Euler equation can be written in the alternative form
1 1( )2
u u u u p ft
Proof is based upon identity: see lecture 1.jminjnimkmnkij
nk
nm
m
kj
m
nkmjnknjmj
m
nimnijkj
m
nimnkijjikijk u
xuu
xuu
xuu
xuu
xuuu
)(|
1( ) ( )2
u u u u u u u u u
this formulation shows, that for zero vorticity the Euler’s equation reduces to Bernoulli’s equation: acceleration+kinetic energy=pressure drop+external forces
Euler’s Equations and velocity potentialMHMT2
Inviscid flows are frequently solved by assuming that velocity fields and volumetric forces f can be expressed as gradients of scalar functions (velocity potential)
fu
… integrating along a streamline gives Bernoulli’s equation
1( ) 02
pu ut
Vorticity vector of any potential velocity field is zero (potential flow is curl-free) because 2
0 0i imnm nx x
Velocities defined as gradients of potential automatically satisfy Kelvins theorem stating that if the fluid is irrotational at any instant, it remains irrotational thereafter (holds only for inviscible fluids!).
Because vorticity is zero the Euler equation is simplified
to understand why, remember that for the Levi Civita tensor holds imn= - inm
Euler’s Equations and stream functionMHMT2
In 2D flows it is convenient to introduce another scalar function, stream function
x
y
uy x
ux y
Velocity derived from the scalar stream function automatically satisfies the continuity equation (divergence free or solenoidal flow) because
2 2
0yxuu
x y x y y x
Curves =const are streamlines, trajectories of flowing particles. For example solid boundaries are also streamlines. Difference is the fluid flowrate between two streamlines.
Advantages of the stream function appear in the cases that the flow is rotational due to viscous effects (for example solid walls are generators of vorticity). In this case the dynamics of flow can be described by a pair of equations for vorticity and stream function
In this way the unknown pressure is eliminated and instead of 3 equations for 3 unknowns ux uy p it is sufficient to solve 2 equations for and .
2 2
2 2 zx y
Euler’s Equations vorticity and stream functionMHMT2
Let us summarize:
For incompressible (divergence-free) flows the velocity potential distribution is described by the Laplace equation (ensures continuity equation)
2 22
2 20 0ux y
For irrotational (curl-free) flow the stream function should also satisfy the Laplace equation 2 2
2 2 0y xz
u ux y x y
Problem of inviscid incompressible flows can be reduced to the solution of two Laplace equations for stream and potential functions, satisfying boundary conditions of impermeable walls ( ) and zero vorticity at inlet/outlet ().0n
Euler’s Equations flow around sphereMHMT2
Example: Velocity field of inviscid incompressible flow around a sphere of radius R is a good approximation of flows around gas bubbles, when velocity slips at the sphere surface. Velocity potential can be obtained as a solution of the Laplace equation written in the spherical coordinate system (r,,)
0)(sinsin
1)( 2
rr
r
r
UVelocity potencial satisfying boundary condition at r and zero radial velocity at surface is
)2
(cos 2
3
rRrU
and velocities (gradient of )
))(211(sin ))(1(cos 33
rRUu
rRUur
Velocity profile at surface (r=R) determines pressure profile (Bernoulli’s equation)
2 20
3 9sin ( ) sin2 8
u U p p U
The solution is found by factorisation to functions
depending on r and on only
Euler’s Equations flow around cylinderMHMT2
r
U2
2( ) 0r rr r
Example: Potencial flow around cylinder can be solved by using velocity potencial function or by stream function. Both these functions have to satisfy Laplace equation written in the cylindrical coordinate system (the only difference is in boundary conditions).
Stream function satisfying boundary condition at r (uniform velocity U) and constant at surface is
2
sin ( )RU rr
giving radial and tangential velocities
2 2cos (1 ( ) ) sin (1 ( ) )rR Ru U u Ur r
Compare with the previous result for sphere: the velocity decays with the second power of radius for cylinder, while with the third power at sphere (which could have been expected).
see the result obtained by using complex functions
Many interesting solutions of Euler’s equations can be obtained from the fact that the real and imaginary parts of ANY analytical function
satisfy the Laplace equation (see next page).
z=x+iy is a complex variable (i-imaginary unit) and w(z)=(x,y)+i(x,y) is also a complex variable (complex function), for example
Euler’s Equations and complex functionsMHMT2
( ) ( , ) ( , )w z x y i x y
,...ln)(,)(,)(,)( 2 zazwzazwazzwazzw
Simple analytical functions describe for example sinks, sources, dipoles. In this way it is possible to solve problems with more complicated geometries, for example free surface flows, flow around airfoils, see applications of conformal mapping.
x
yConformal mapping =const streamlines
=const Equipotential lines
w(z)z(w)
This is important statement: Quite arbitrary analytical function describes some flow-field. Real part of the complex variable w is velocity potential and the imaginary part Im(w) is stream function!
Euler’s Equations and complex functionsMHMT2
Derivative dw/dz of a complex function w(z=x+iy)=+i with respect to z can be a complex analytical function as soon as both Re(w), Im(w) satisfy the Laplace equation
22
2222
0
00
))(()(lim
)(lim
)()(lim
dydx
dxdyxy
dyy
dxx
idxdyxy
dyy
dxx
idydx
dyy
dxx
idyy
dxx
dzzwdzzw
dzdw
dz
dzdz
dy=0 x ydw i u iudz x x
dx=0 x ydw i u iudz y y
Result should be independent of the dx, dy selection, therefore
and this requirement is fulfilled only if both functions , satisfy Cauchy-Riemann conditions
2 2 2 2
2 2 2 20, 0x y x y
, x y y x
and therefore
Euler’s Equations and complex functionsMHMT2
x ydw u iudz
The real and the imaginary part of derivative dw/dz determine components of velocity field
w(z)=+i ux=/x uy =/y streamlines
az2az
a z
/a z
lna z
lnia z
az
2ax 2ay
0
2 2
2 22 2( )x x ya
x y
2 2
2 22 2( )x x ya
x y
2 2
2 2 2( )y xax y 2 2 22
( )xya
x y
2 2
axx y 2 2
ayx y
2 2
ayx y 2 2
axx y
x
x
x
x
y
y
y
y
y
source
circulation
dipole
x
x
Euler’s Equations and complex functionsMHMT2
Example: Let’s consider the transformation w(z)=az2 in more details
0
0,01
0,02
0,03
0,04
0,05
0,06
0,07
0,08
0,09
0,1
0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,1
x
y
=-3
=3
=1
=0.5
=0.1
=0
iaxyyxaiyxaazzw
2)()()( 2222
Equipotential lines2 2 2( )a x y y x
a
Stream lines
2 , 2x yu ax u ayx y
22
axy yax
The same graph can be obtained from inverse transformation z(w)
( ) wz w x iya
Euler’s Equations and complex functionsMHMT2
The following examples demonstrate the most important techniques used for construction of conformal mappings
Potential flow around circular cylinder with circulation (using directory of basic transformations, see previous slide – application of superposition principle: sum of analytical functions is also an analytical function)
Potential flow around an elliptical cylinder (making use conformal mapping of ellipse to circle, based upon Laurent series expansion – this is application of the substitution principle: analytical function of an analytical function is also an analytical function)
Cross flow around a plate (or how to transform an arbitrary polygonal region into upper half plane of complex potential – Schwarz Christoffel theorem)
Flows with free surface (contraction flow from an infinitely large reservoire through a slit)
Euler’s Equations cylinder with circulationMHMT2
Example: Potential flow around cylinder with circulation can be assumed as superposition of linear parallel flow w1(z)=Uz, dipole w2(z)=UR2/z and potential swirl w3(z)=/(2i) ln z (see the previous table).
22 2
uniform flow inthe x-direction circulationdipole
( ) ( ) ln( )2
x iyw z U x iy UR x iyx y i
Substituting coordinates x,y by radius r and angle results into (x+iy=r ei) 2
( ) (cos sin ) (cos sin ) ln( )2 2
R iw z Ur i U i rr
Comparing real and imaginary part potential and stream functions are identified2
( , ) cos ( )2
Rr U rr
2
( , ) sin ( ) ln2
Rr U r rr
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
without circulation, I have a problem in Matlab
velocity potential is the real part of the analytical function w(z)
stream function is the imaginary part of the analytical function w
Euler’s Equations elliptic cylinderMHMT2
Example: Potential flow around elliptic cylinder. Previous example solved the problem of potential flow around a cylinder with radius R, described by the conformal mapping 2
( ) ( )Rw z U zz
There exist many techniques how to identify the conformal mapping (z) transforming a general closed region in the z=x+iy plane into a unit circle, for example numerically or in terms of Laurent series
The analytical function transforming outside of an elliptical cylinder to the plane of complex potential w= +i can be obtained in two steps: First step is a conformal mapping (z) transforming ellipse with principal axis a,b to a cylinder with radius a+b. The second step is substitution of the mapping (z) to the velocity potential
2( )( ) ( ( ) )( )
a bw z U zz
1 2
1 0 21
this series converge outside a circle
...nn
n afinnetransform
a a az a a
…this is the way how to solve the problems of flow around profiles, for example airfoils. It is just only necessary to find out a conformal mapping transforming the profile to a circle.
Euler’s Equations elliptic cylinderMHMT2
11 0
az a a
Re
Im
x
y
aa+b
b
Z-plane-plane
For the conformal mapping of ellipse only three terms of Laurent’s series are sufficient
with 2 21 0 1
1 1, 0, ( )2 2
a a a a b
Inversion mapping (z) is the solution of quadratic equation2 2 2( )z z z a b
Complex potential (potential and stream function) is therefore2
2 2 2
2 2 2conformal mapping of ellipticalregion to circular region
( )( ) ( )a bw z U z z a bz z a b
Euler’s Equations conformal mappingMHMT2
Generally speaking it does not matter if we select analytical function w(z) mapping the spatial region (z=x+iy) to complex potential region w=+i, or vice versa.
This is because inverse mapping is also conformal mapping.
Euler’s Equations cross flow around a plateMHMT2
2 2( ) 1 ( ) 1w z x iy h z h i
See M.Sulista: Analyza v komplexnim oboru, MVST, XXIII, 1985, pp.100-101.
fi=linspace(-10,10,1000);for psi=0.1:0.1:1z=complex(fi,psi);w=(z.^2-1).^0.5;plot(w);hold on;end
0 1 2 3 4 5-1.5
-1
-0.5
0
0.5
1
1.5
=0.1
=1
Solution for h=1 by MATLAB
Please notice the fact, that in this case the role of z and w is exchanged, complex variable w is spatial coordinates x,y, while z=+i is complex potential of velocity field.
Euler’s Equations 3D stream functionMHMT2
Disadvantage of the approach using stream function, complex variables and conformal mapping is its limitation to 2D flows. While in the 3D flow the irrotational velocity field can be described by only one scalar function , description of 3D solenoidal field (satisfying continuity equation) by stream function is not so simple. It is necessary to use a generalized stream function vector and to decompose velocities into curl free and solenoidal components (dual potential approach)
u
Curl free (potential flow)
Divergence free (solenoidal flow)
Vorticity vector is expressed in terms of the stream function vector
2( )
jminjnimkmnkij using identity
The dual potential approach increases number of unknowns (3 stream functions and 2 vorticity transport equations are to be solved) and is not so frequently used.
Euler’s EquationsMHMT2
Simple questions
1. Let the scalar function (t,x1,x2) satisfies Laplace equation. Does it mean that the gradient of this function represents velocity field satisfying both Euler equations in the directions x1,x2? The answer is positive.
2. Is it possible that a velocity field satisfying the Euler’s equations and the continuity equation is rotational (therefore cannot be expressed as a gradient of potential)? Answer is positive again.
EXAMMHMT2
Transport equations
What is important (at least for exam)MHMT2
You should know what is it material derivative
u
tDtD
Balancing of fluid particle Balancing of fixed fluid element
( )D uDt t
Reynolds transport theorem
( ( ))fix fixfluid V V
particle
d Ddv dv u dvdt Dt t
What is important (at least for exam)MHMT2
( ) 0ut
Continuity equation and Cauchy’s equations
Du p fDt
1 1( )2
u u u u p ft
Euler’s equation
Bernoulli’s equation1( ) 02
pu ut
What is important (at least for exam)MHMT2
u
What is it vorticity, stream function and velocity potential
u
u
x
y
uy x
ux y
Special case for 2D flows
( ) ( , ) ( , )w z x y i x y
Complex potential, analytical functions and conformal mapping