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Momentum Heat Mass TransferMHMT1
Introduction and rules (form of lectures, presentations of papers by students, requirements for exam). Preacquaintance (photography). Working with databases and primary sources. Tensors and tensorial calculusRudolf Žitný, Ústav procesní a
zpracovatelské techniky ČVUT FS 2010
Rules, database, Tensorial calculus
sourceDt
D
3P/2C, 4 credits, exam, LS2015, room C-434• Lectures Prof.Ing.Rudolf Žitný, CSc. (Wednesday 9:00-11:30)• Tutorials Ing.Karel Petera, PhD. First tutorial at 25.2.2015
• Rating 30(quise in test)+30(example in test)+40(oral exam) points
A B C D E F
90+ 80+ 70+ 60+ 50+ ..49
Momentum Heat Mass TransferMHMT1
excellent very good good satisfactory sufficient failed
3P/2C, 4 credits, exam, LS2015, room C-434
Momentum Heat Mass TransferMHMT1
MHMT1 Momentum Heat Mass Transfer
• Textbook: Šesták J., Rieger F.: Přenos hybnosti tepla a hmoty, ČVUT Praha, 2004
• Monography: available at NTL Bird, Stewart, Lightfoot: Transport Phenomena. Wiley, 2nd edition, 2006Welty J.R.: Fundamental of momentum, heat and mass transfer,Wiley,2008
• Databases of primary sources Direct access to databases (WoS, Elsevier, Springer,…)knihovny.cvut.cz
Jméno DUPS
Literature - sourcesMHMT1
Information about peoples (publications)
database of papers used in presentations.
EES Electronic sourcesMHMT1
Key words
Full text available in pdf
Science DirectMHMT1
Prerequisities: Tensors
Bailey
MHMT1
Transfer phenomena operate with the following properties of solids and fluids (determining state at a point in space x,y,z):
Scalars T (temperature), p (pressure), (density), h (enthalpy), cA (concentration), k (kinetic energy)
Vectors (velocity), (forces), (gradient of scalar), and others like vorticity, displacement…
Tensors (stress), (rate of deformation), (gradient of vector) , deformation tensor…
Prerequisities: Tensors
u
f
T
u
Scalars are determined by 1 number.
Vectors are determined by 3 numbers
Tensors are determined by 9 numbers
333231
232221
131211
321 ),,(),,(
zzzyzx
yzyyyx
xzxyxx
zyx uuuuuuu
Scalars, vectors and tensors are independent of coordinate systems (they are objective properties). However, components of vectors and tensors depend upon the coordinate system. Rotation of axis has no effect upon a vector (its magnitude and the arrow direction), but coordinates of the vector are changed (coordinates ui are projections to coordinate axis). See next slide…
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Rotation of cartesian coordinate systemMHMT1
Three components of a vector represent complete description (length of an arrow and its directions), but these components depend upon the choice of coordinate system. Rotation of axis of a cartesian coordinate system is represented by transformation of the vector coordinates by the matrix product
'1 1 2 3
'2 1 2 3
'3 1 2 3
cos(1',1) cos(1', 2) cos(1',3)
cos(2 ',1) cos(2 ', 2) cos(2 ',3)
cos(3',1) cos(3', 2) cos(3',3)
a a a a
a a a a
a a a a
a1
a2
a’1
a’2
1
2
1’
2’
2 cos(1 ', 2 )a
1 cos(1 ',1)a
'1 1'2 2'3 3
cos(1',1) cos(1', 2) cos(1',3)
cos(2 ',1) cos(2 ', 2) cos(2 ',3)
cos(3',1) cos(3', 2) cos(3',3)
a a
a a
a a
[ '] [[R]][ ]a a
Rotation matrix (Rij is cosine of angle between axis i’ and j’)
a
Rotation of cartesian coordinate systemMHMT1
Example: Rotation only along the axis 3 by the angle (positive for counter-clockwise direction)
1
'1 1'2 2
cos(1',1) cos cos(1', 2) sin
cos(2 ',1) sin cos(2 ', 2) cos
a a
a a
1[[R]] [[R]]=[[I]] [[R]] [[ ]]T TR
2
1’
2’
2,'12
1 ',1
( ) 2 ',12
2 ', 2
Properties of goniometric functions sin))2
(cos( sin)2
cos( cos)cos(
2 2
2 2
cos sin cos sin
sin cos sin cos
cos sin cos sin sin cos
sin cos cos sin sin cos
1 0
0 1
therefore the rotation matrix is orthogonal and can be inverted just only by simple transposition (overturning along the main diagonal). Proof:
a
Stresses describe complete stress state at a point x,y,z
ij
x
y
z
x
y
z
x
y
z
zz
zy
zxyz
yy
yxxz
xy
xx
Index of plane index of force component (cross section) (force acting upon the cross section i)
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Stress tensor is a typical example of the second order tensor with a pair of indices having the following meaning
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Later on we shall use another tensors of the second order describing kinematics of deformation (deformation tensors, rate of deformation,…)
Nine components of a tensor represent complete description of state (e.g. distribution of stresses at a point), but these components depend upon the choice of coordinate system, the same situation like with vectors. The transformation of components corresponding to the rotation of the cartesian coordinate system is given by the matrix product
Tensor rotation of cartesian coordinate system
[[ ']] [[R]][[ ]][[R]]T
cos(1',1) cos(1', 2) cos(1',3)
[[ ]] cos(2 ',1) cos(2 ', 2) cos(2 ',3)
cos(3',1) cos(3', 2) cos(3',3)
R
where the rotation matrix [[R]] is the same as previously
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Orthogonal matrix of the rotation of coordinate system [[R]] is fully determined by 3 parameters, by subsequent rotations around the x,y,z axis (therefore by 3 angles of rotations). The rotations can be selected in such a way that 3 components of the stress tensor in the new coordinate system disappear (are zero). Because the stress tensor is symmetric (usually) it is possible to anihilate all off-diagonal components
Tensor rotation of cartesian coordinate system
'1
'2
'3
0 0
0 0 [[R]][[ ]][[R]]
0 0
T
Diagonal terms are normal (principal) stresses and the axis of the rotated coordinate systems are principal directions (there are no shear stresses in the cross-sections oriented in the principal directions).
ji
ji
ij
ij
for 1
for 0
100
010
001
Special tensorsKronecker delta (unit tensor, components independent of rotation)
Levi Civita tensor is antisymmetric unit tensor of the third order (with 3 indices)
In terms of the epsilon tensor the vector product will be defined
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Scalar productMHMT1
a
b
Scalar product (operator ) of two vectors is a scalar
3' '
1 1 2 2 3 31
i i i i i ii
a b a b a b a b a b a b a b
aibi is abbreviated Einstein notation. Repeated indices are summing (dummy) indices.
' '
' ' ' ' ' ' ' '1 1 2 2 3 3
i im m j jk k
Ti i im ik m k mi ik m k mk m k m m
a R a b R b
a b a b a b a b R R a b R R a b a b a b
Proof that
cos|||| baba
Remark: there were used dummy indices m and k in these relations. Letters of the dummy indices can be selected arbitrary, but in this case they must be different, so that to avoid appearance of four equal indices in a tensorial term in the following product a.b (there can be always max. two indices with the same name indicating a summation)
' 'i i i ia b a b
Scalar productMHMT1
Example: Scalar product of velocity [m/s] and the normal vector of an oriented surface [m2] is the volumetric flowrate Q [m3/s] through the surface
u
sd
usdQ
sd
u
Example: scalar product of velocity [m/s] and force [N] acting at a point is power P [W] (scalar)
u
F
i i x x y y z zP u F u F u F u F u F
jijiiji fnnfn
3
1i
Scalar product
x
y
z
nf
Scalar product can be applied also between tensors or between vector and tensor
i-is summation (dummy) index, while j-is free index
This case explains how it is possible to calculate internal stresses acting at an arbitrary cross section (determined by outer normal vector n) knowing the stress tensor.
MHMT1
Scalar product - examples
Dot product of delta tensors
Scalar product of tensors is a tensor
Double dot product of tensors is a scalar
Trace of a tensor (tensor contraction)
MHMT1
ijmjim 1 0 0 1 0 0 1 0 0
0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
im mj ij
: km mk
( ) example ( ) 3mmtr tr
233223132321231 babababac
kjijkj k
kjijki babac
abbac
3
1
3
1
)(
Vector product
Vector product (operator x) of two vectors is a vector
MHMT1
Important relationship between the Levi Civita and the Kronecker delta special tensors
for example
b
a
c
sin|||||| bac
jminjnimkmnijk
check for i=1,j=2,(k=3),m=1,n=2 123 312 = 11 22 - 12 21 = 1
check for i=1,j=2,(k=3),m=2,n=1 123 321 = 12 21 - 11 22 = -1
Vector productMHMT1
F
r
Moment of force (torque) FrM
umF 2
Coriolis force
uF
application: Coriolis flowmeter
Examples of applications
i j ijab a b
Diadic product
Diadic product (no operator) of two vectors is a second order tensor
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Differential operator (Nabla)MHMT1
Bailey
ixxyx
i ),,(
Differential operator (Nabla)
Symbolic operator represents a vector of first derivatives with respect x,y,z.
applied to scalar is a vector (gradient of scalar)
ii x
TT
z
T
y
T
x
TT
),,(
applied to vector is a tensor (for example gradient of velocity is a tensor)
i
jj
zyx
zyx
zyx
x
uu
z
u
z
u
z
uy
u
y
u
y
ux
u
x
u
x
u
u
i
GRADIENT – measure of spatial changes
MHMT1
),( yxu
( , )u x dx y
dxx
uudu xxx
xu xu
dx
source 0 u
3
1 i
i
i i
izyx
x
u
x
u
z
u
y
u
x
uu
sink 0 u
Differential operator
The vector fi represents resulting force of stresses acting at the surface of an infinitely small volume (the force is related to this volume therefore unit is N/m3)
Scalar product represents intensity of source/sink of a vector quantity at a point
i-dummy index, result is a scalar
jizzyzxzzyyyxyzxyxxx
zyxzyxzyxf jif ,,
DIVERGENCY – magnitude of sources/sinks
Scalar product can be applied also to a tensor giving a vector (e.g. source/sink of momentum in the direction x,y,z)
onconservati 0 u
MHMT1
ii xx
T
z
T
y
T
x
TTT
2
2
2
2
2
2
22
Laplace operator 2
Scalar product =2 is the operator of second derivatives (when applied to scalar it gives a scalar, applied to a vector gives a vector,…). Laplace operator is divergence of a gradient (gradient of temperature, gradient of velocity…)
),,(23
12
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
ii
j
i i
jzzzyyyxxx
xx
u
x
u
z
u
y
u
x
u
z
u
y
u
x
u
z
u
y
u
x
uu
Physical interpretation: 2 describes diffusion processes (random molecular motion). The term 2 appears in the transport equations for temperatures, momentum, concentrations and its role is to smooth out all spatial nonuniformities of the transported properties.
i-dummy index
MHMT1
Divergency of gradient – measure of nonuniformity
Laplace operator 2MHMT1
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
)exp()( 2xxT
T2
Negative value of 2T tries to suppress the peak of the
temperature profile
Positive value of 2T tries to enhance the
decreasing part
Integrals – Gauss theorem
V S
Pdv n Pds
Variable P can beScalar (typically pressure p)
Vector (vector of velocity, momentum, heat flux). Surface integral represents flux of vector in the direction of outer normal.
Tensor (tensor of stresses). In this case the Gauss theorem represents the balance between inner stresses and outer forces acting upon the surface,
Divergence of P projection of P to outer normal
MHMT1
V S
pdv npds
V S
udv n uds
V S
dv n ds
Volume integrals with nabla operator can be converted to surface integrals (just only by replacing nabla with unit normal vector )
V ds
n
n
Physical interpretation: accumulation in volume V is overall flux through boundary
TranscriptionsMHMT1
Bailey
Symbolic notation, for example , is compact, unique and suitable for definition of problems in terms of tensorial equations. However, if you need to solve these equation (for example ) you have to rewrite symbolic form into index notation, giving equations (usually differential equations) for components of vectors or tensors, which are expressed by numbers (may be complex numbers).
T2
2T f
Symbolic indicial notation
General procedure how to rewrite symbolic formula to index notation
Replace each arrow by an empty place for index
Replace each vector operator by (-1)
Replace each dot by a pair of dummy indices in the first free position left and right
Write free indices into remaining positions
Practice examples!!
MHMT1
Coordinate systemsPrevious conversion procedure can be applied only in a cartesian coordinate systems
MHMT1
Formulation in the x,y,z cartesian system is not always convenient, especially if the geometry of region is cylindrical or spherical. For example the boundary condition of a constant temperature is difficult to prescribe on the curved surface of sphere in the rectangular cartesian system. In the case that the problem formulation suppose a rotational or spherical symmetry, the number of spatial coordinates can be reduced and the problem is simplified to 1D or 2D problem, but in a new coordinate system.
In the following we shall demonstrate how to convert tensor terms from symbolic notation (which is independent to a specific coordinate system) into the cylindrical coordinate system.
1 1 1 1
T T r T T z T T sc
x r x x z x r r
2 2 2 2
T T r T T z T T cs
x r x x z x r r
Coordinate systems (cylindrical)
Cylindrical (and spherical) systems are defined by transformations
rx1
x2
(x1 x2 x3)
(r,,z)
zx
rsx
rcx
3
2
1 1
2
3
0
0
0 0 1
dx c rs dr
dx s rc d
dx dz
where sin coss c
1
2
3
0
0
0 0 1
c sdr dx
s cd dx
r rdz dx
Using this it is possible to express partial derivatives with respect x1,x2,x3 in terms of derivatives with respect the coordinates of cylindrical system
3 3 3 3
T T r T T z T
x r x x z x z
MHMT1
dzz
xd
xdr
r
xdx iii
i
r
c
xr
s
xs
x
rc
x
r
2121
, , ,
MHMT1
In the same way also the second derivatives can be expressed
2 2 2 2 22
2 2 2 21
2( )
T T cs T T T s T sc
x r r r r r r r
2 2
2 23
T T
x z
2 2 2 2 22
2 2 2 22
2( )
T T cs T T T c T cs
x r r r r r r r
giving expression for the Laplace operator in the cylindrical coordinate system
2 2 2 2 2 2
2 2 2 2 2 2 21 2 3
1 1T T T T T T T
x x x r r r r z
Coordinate systems (cylindrical)
(use goniometric identity s2+c2=1)
MHMT1
Previous example demonstrated how to solve the problem of transformations to cylindrical coordinate system with scalars. However, how to calculate gradients or divergence of a vector and a tensor field? Vectors and tensors are described by 3 or 3x3 values of components now expressed in terms of new unit vectors
( ) ( )( ) zrm m r zu u i u e u e u e
Einstein summation applied in the cartesian coordinate system
unit vectors in the cylindrical coordinate system
( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ...z z
mn m n
r r r r zrr r rz zz
i i
e e e e e e e e
Unit vectors in orthogonal coordinate system are orthogonal, which means that( ) ( )( ) ( ) ( ) ( )1 0 ... 1zr r r ze e e e e e
and therefore the components in the new coordinate system are for example( ) ( ) ( )( ) ( ) ( ) r z r z
m m rz m mn nu u e u e e e e e
( ) ( )( ) zri r i i z iu u e u e u e
izz
i iee
)()( this is i-th cartesian component of the unit vector
Coordinate systems (cylindrical)
( )1
( )2
( )3
0
0
0 0 1
r
z
i c s e
i s c e
i e
Transformation of unit vectors
( )1
( )2
( )3
0
0
0 0 1
r
z
e c s i
e s c i
e i
Example: gradient of temperature can be written in the following way (alternatively in the cartesian and the cylindrical coordinate system)
( ) ( ) ( )1 2 3
1 2 3 1 2 1 2 3
( ) ( ) ( )
( ) ( )
1
r z
r z
T T T T T T T TT i i i c s e s c e e
x x x x x x x x
T T Te e er r z
rx1
x2
(x1 x2 x3)
(r,,z)( )e
1i
2i
MHMT1
( )re
Coordinate systems (cylindrical)
Nabla operator in the cylindrical coordinate system
( ) ( ) ( )1r ze e er r z
1
T T T sc
x r r
substitute by using previously derived
MHMT1 Coordinate systems (cylindrical)
Example: Divergence of a vector ( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( )
( )
r zr m m z mm
m m
r zr zm m mr zm r m m z
m m m m m m
u e u e u euu
x x
ue e eu ue u e u e u
x x x x x x
( ) ( )( ) zrm m r zu u i u e u e u e
components of unit vectors follow from the previously derived ( )
1( )
2( )
3
0
0
0 0 1
r
z
e c s i
e s c i
e i
1 0 0
0
0
)(3
)(3
)(3
)(2
)(2
)(2
)(1
)(1
)(1
zr
zr
zr
eee
ecese
esece
Note the fact, that the partial derivatives of these components with respect to xm are not zero and can be calculated using the previously derived relationships
( )1
1 1 1
re c c ss
x x x r
1 2
, s c
x r x r
giving
( )1
1 1 1
e s s sc
x x x r
…and so on
MHMT1 Coordinate systems (cylindrical)
Substituting these expression we obtain
r
csu
r
sucs
r
u
r
su
r
csuc
r
u
x
ur
rr
22
2
1
1
2 222
2
r rr
u uu u u cs c c css u cs u
x r r r r r r
3
3
zu u
x z
Summing together, the final form of divergence in the cylindrical coordinate system is obtained
z
uu
rr
u
r
uu zrr
1
MHMT1 Coordinate systems (cylindrical)
Example: Gradient of a vector
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )
...r z r z
r n n z n r n n z nr r rrr m n r m n
m m
u
u e u e u e u e u e u ee e e e
x x
( ) ( )( ) zrm m r zu u i u e u e u e
Substituting previous expressions for unit vector and their derivatives results to final expression for the velocity gradient tensor in a cylindrical coordinate system
m and n are dummy indices (summing is required)
z
u
z
u
z
u
u
ru
u
ru
u
r
r
u
r
u
r
u
u
zr
zr
r
zr
1
)(1
)(1
MHMT1 Coordinate systems (general)
Procedure how to derive tensorial equations in a general coordinate system.
2. Define transformation x1(r1,r2,r3),… r1(x1,x2,x3),… and therefore also the first and the second derivatives of scalar values, for example
3. Define unit vectors of coordinate systems r i and express their cartesian coordinates Calculate their derivatives with respect to the cartesian coordinates
4. In case that the result is a vector, for example the gradient of scalar, calculate its components from
1. Rewrite equation from the symbolic notation to the index notation for cartesian coordinate system, for example j
iji
uu
x
),,( 321 rrrfx
rij
j
i
11 21 311 1 2 3
,...j j j ju u u uf f f
x r r r
( ) ( )
1 2 3( , , )ri rim me g r r r
( )31 2
1 1 2 3
r r r r rm m m m m
i i i i
e g g g g rr r
x x r x r x r x
In the case that result is a tensor calculate its components
( ) ( )m mu u e u e
( ) ( ) ( ) ( )r z r zrz i ij je e e e
Remark: This suggested procedure (transform everything, including new unit vectors, to the cartesian coordinate system) is straightforward and seemingly easy. This is not so, it is „crude“, lenghty (the derivation of velocity gradient is on several lists of paper) and without finesses. Better and more sophisticated procedures are described in standard books, e.g. Aris R: Vectors, tensors…N.J.1962, or Bird,Stewart,Lightfoot:Transport phenomena.
EXAMMHMT1
Tensors
What is important (at least for exam)MHMT1
You should know what is it scalar, vector, tensor and transformations at rotation of coordinate system
[ '] [[R]][ ]a a [[ ']] [[R]][[ ]][[R]]T
3
1i i i i
i
a b a b a b
Scalar and vector products
kjijkj k
kjijki babac
abbac
3
1
3
1
)(
(and what is it Kronecker delta and Levi Civita tensors?)
(and how is defined the rotation matrix R?)
What is important (at least for exam)MHMT1
Nabla operator. Gradient
ixxyx
i ),,(
Divergence
3
1 i
i
i i
izyx
x
u
x
u
z
u
y
u
x
uu
Laplace operator
ii xx
T
z
T
y
T
x
TTT
2
2
2
2
2
2
22
What is important (at least for exam)MHMT1
V S
Pdv n Pds
Gauss integral theorem
(demonstrate for the case that P is scalar, vector, tensor)