CH.5. BALANCE PRINCIPLES Continuum Mechanics Course (MMC) - ETSECCPB - - PowerPoint PPT Presentation
CH.5. BALANCE PRINCIPLES Continuum Mechanics Course (MMC) - ETSECCPB - UPC Overview Balance Principles Convective Flux or Flux by Mass Transport Local and Material Derivative of a Volume Integral Conservation of Mass Spatial
Continuum Mechanics Course (MMC) - ETSECCPB - UPC
Balance Principles Convective Flux or Flux by Mass Transport Local and Material Derivative of a Volume Integral Conservation of Mass
Spatial Form Material Form
Reynolds Transport Theorem
Reynolds Lemma
General Balance Equation Linear Momentum Balance
Global Form Local Form
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Angular Momentum Balance
Global Spatial Local Form
Mechanical Energy Balance
External Mechanical Power Mechanical Energy Balance External Thermal Power
Energy Balance
Thermodynamic Concepts First Law of Thermodynamics Internal Energy Balance in Local and Global Forms Reversible and Irreversible Processes Second Law of Thermodynamics Clausius-Planck Inequality
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Governing Equations
Governing Equations Constitutive Equations The Uncoupled Thermo-mechanical Problem
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The conservation/balance principles:
Conservation of mass Linear momentum balance principle Angular momentum balance principle Energy balance principle or first thermodynamic balance principle
The restriction principle:
Second thermodynamic law
The mathematical expressions of these principles will be given in,
Global (or integral) form Local (or strong) form
REMARK These principles are always valid, regardless of the type of material and the range of displacements or deformations.
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The term convection is associated to mass transport, i.e., particle
Properties associated to mass will be transported with the mass when
there is mass transport (particles motion)
Convective flux of an arbitrary property through a control
S
S amountof crossing unitoftime A convective transport
A S
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Consider: An arbitrary property of a continuum medium (of any tensor order) The description of the amount of the property per unit of mass,
(specific content of the property ) .
The volume of particles crossing a
differential surface during the interval is
Then, The amount of the property per unit of mass crossing the differential
surface per unit of time is:
,t x
A dV dS dh dt dS dm dV dSdt v n v n
S
dm d dS dt v n
dV dS
, t t dt
A
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inflow
v n
v n Consider: An arbitrary property of a
continuum medium (of any tensor order)
The specific content of (the amount
per unit of mass) .
Then, The convective flux of through a spatial surface, , with unit
normal is:
If the surface is a closed surface, , the net convective flux is:
A
,t x
A
S
n
S s
t dS
v n
V V
t dS
v n
S V
= outflow - inflow Where: is velocity is density
A
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REMARK 1 The convective flux through a material surface is always null. REMARK 2 Non-convective flux (advection, diffusion, conduction). Some properties can be transported without being associated to a certain mass of particles. Examples of non-convective transport are: heat transfer by conduction, electric current flow, etc. Non-convective transport of a certain property is characterized by the non- convective flux vector (or tensor) :
,t q x ;
s s
dS dS
q n v n convectivefl non-convectiveflu u x x
convective flux vector
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non-convective flux vector
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Compute the magnitude and the convective flux which correspond to the following properties:
a) volume b) mass c) linear momentum d) kinetic energy
S
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a) If the arbitrary property is the volume of the particles: The magnitude “property content per unit of mass” is volume per unit of mass, i.e., the inverse of density: The convective flux of the volume of the particles through the surface is:
V A
1 V M
1
S s s
dS dS
v n v n
S V
VOLUME FLUX
S s
t dS
v n
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b) If the arbitrary property is the mass of the particles: The magnitude “property per unit of mass” is mass per unit of mass, i.e., the unit value: The convective flux of the mass of the particles through the surface is:
M A
1 M M
1
S s s
dS dS
v n v n
S M
MASS FLUX
S s
t dS
v n
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c) If the arbitrary property is the linear momentum of the particles: The magnitude “property per unit of mass” is mass times velocity per unit of mass, i.e., velocity: The convective flux of the linear momentum of the particles through the surface is:
v M A
M M v v
S s
dS
v v n
S M v
MOMENTUM FLUX
S s
t dS
v n
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d) If the arbitrary property is the kinetic energy of the particles: The magnitude “property per unit of mass” is kinetic energy per unit of mass, i.e.: The convective flux of the kinetic energy of the particles through the surface is:
2
1 2 v M A
2 2
1 1 2 2 M M v v
2
1 2
S s
dS
v v n S
2
1 2 M v
KINETIC ENERGY FLUX
S s
t dS
v n
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Consider: An arbitrary property of a continuum medium (of any tensor order) The description of the amount of the property per unit of volume
(density of the property ),
The total amount of the property
The time derivative of this volume integral is:
A
,t x
REMARK and are related through .
V
,
V
Q t t dV x
lim
t
Q t t Q t Q t t
A
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Q t
Q t t
, , , , lim lim [ , , ] , , , lim lim
V V t t V V t t V V
t t t t dV t dV Q t t Q t t dV t t t t t t dV t t t t dV dV t t t
x x x x x x x x x
Q t
Q t t
Control Volume, V
Consider: The volume integral The local derivative of is: It can be computed as:
REMARK The volume is fixed in space (control volume).
,
V
Q t t dV x
, , , lim t
not V V t V
t t dV t dV t dV t
x x x local derivative
Q t
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Q t
Q t t
Consider: The volume integral The material derivative of is: It can be proven that:
REMARK The volume is mobile in space and can move, rotate and deform (material volume).
,
V
Q t t dV x
( ) ( )
, , , lim x x x
t
not V V V t t V t t
d t dV dt t t dV t dV t
material derivative
Q t
, x v v v
V V V V V t
V
d d t dV dV dV dV dt t dt
dV t
derivative of derivative of the integral the integral
derivative of the integral
convective local
material
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It is postulated that during a motion there are neither mass
The total mass of
Where:
t
t t V
t t
t t t t V
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Conservation of mass requires that the material time derivative of
The global or integral spatial form of mass conservation principle: By a localization process we obtain the local or differential
lim ,
t t
V V V t
t t t d t dV V V t t dt
M M M ( , ) ,
t t
V V V V V
d d t dV dV V V t dt dt
x v
( , ) ( , ) ( , ) ( )( , ) ( )( , ) , x x x v x v x x for V dV t d t t t t V t dt t (localization process)
CONTINUITY EQUATION
, ( )
V V V t
d d t dV dV dt dt
x v
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Consider the relations: The global or integral material form of mass conservation
The local material form of mass conservation principle reads :
1 ( ) F v F d dt
( , ) 1 ( , ) ( ) ( ( , ) ) , ,
| |( , )
F F X X v F X F F X
F X
V V V V V
t d t d d t dV dV t dV dt dt dt t t t dV V V t t
t
0 , t t
V t X F
1
,
t t t t
t t
F X X F X F X
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( , ) t t X
dV F
F F v F d dt dV dV
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Consider: An arbitrary property of a continuum medium (of any tensor order) The spatial description of the amount of the property per unit of
mass,
The amount of the property in the continuum body at time
Using the material time derivative leads to, Thus,
,t x
t
V V
Q t dV
V V V t
d d dV dV dt dt
REYNOLDS LEMMA
d dt v
( ) ( )
V V V V t
d d d d Q t dV dV dV dt dt dt dt
v v
=0 (continuity equation)
d d dt dt
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V
dV
V dV
d dt
1
ˆ e
2
ˆ e
3
ˆ e
The amount of the property in the continuum body at time for
Using the material time derivative leads to, And, introducing the Reynolds Lemma
V
Q t dV
t
V V V V
d dV dV dV dt t
v
v n
V V V
d dV dV dS dt t
REMARK The Divergence Theorem:
v n v v n
V V V
dV dS dS
, x v
V V V V t
d t dV dV dV dt t
V
dV
n
v
V
d dV dt
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V
dV
V dV
d dt
1
ˆ e
2
ˆ e
3
ˆ e
The eq. can be rewritten as:
V V V
d dV dV dS t dt
v n REYNOLDS TRANSPORT THEOREM Rate of change of the total amount of . within the control volume V at time t. A Rate of change of the amount of in a material volume which instantaneously coincides with the control volume V.
A
Net outward flux of through the surface that surrounds the control volume V. A
v n
V V V
d dV dV dS dt t
V
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V
dV
V dV
d dt
1
ˆ e
2
ˆ e
3
ˆ e
V V V
d dV dV dS t dt
v n REYNOLDS TRANSPORT THEOREM (integral form)
V V V
d dV dV dS t dt
v n
( ) ( ) d V t t dt v x
REYNOLDS TRANSPORT THEOREM (local form) ( ) [ ( )]
V V V V
d dV dV V V t t dt
v
( )
V
dV
v ( )
V
dV t
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Consider: An arbitrary property of a
continuum medium (of any tensor order)
The amount of the property per
unit of mass,
The rate of change per unit of time
a) Generation of the property per unit mas and time time due to a source: b) The convective (net incoming) flux across the surface of the volume. c) The non-convective (net incoming) flux across the surface of the volume:
So, the global form of the general balance equation is:
,t x
v n j n
V V V V
dV k dV dS dS t
A A
a c b
( , ) x k t
A
( , ) j x t
non-convective flux vector
A
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The global form is rewritten using the Divergence Theorem and
The local spatial form of the general balance equation is:
j
d dt
k
A A
v n v j j
V V V V V V V V
dV dS t dV k dV t d dV k dV V V t dt
A A A A
v n j n
V V V V
dV k dV dS dS t
A A
REMARK For only convective transport then and the variation of the contents of in a given particle is only due to the internal generation .
d dt
k
A
( ) j
A
k
A
d dt
(Reynolds Theorem)
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If the property is associated to mass , then:
The amount of the property per unit of mass is . The mass generation source term is .
The mass conservation principle states mass cannot be generated.
The non-convective flux vector is .
Mass cannot be transported in a non-convective form.
Then, the local spatial form of the general balance equation is:
A M
1
1 1
( ) ( )
d dt
t
v
k
M
j
M
( ) t v
j
d dt
k
A A
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( ) d V t t dt v v x
Two equivalent forms of the continuity equation.
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Applying Newton’s 2nd Law to the discrete system formed by n
For a system in equilibrium, :
1 1 1 1 1
v R f a v v
n n n i i i i i i i i n n i i i i i i
d t m m dt d t dm d m dt dt dt
Resulting force
P
mass conservation principle:
i
dm dt
0, t R
d t dt P
t cnt P
CONSERVATION OF THE LINEAR MOMENTUM
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t P
linear momentum
The linear momentum of a material volume of a continuum
M
t
V
V
M
1
v
n i i i
t m
P
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The time-variation of the linear momentum of a material volume is
Where: If the body is in equilibrium, the linear momentum is conserved:
t
V
V V
t dV dS
R b t
body forces surface forces
t R
d t t cnt dt P P
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The global form of the linear momentum balance principle: Introducing and using the Divergence Theorem, So, the global form is rewritten:
t t
V V V V V V V
t
P
V V V
t t
V V V V V V V V V
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Applying Reynolds Lemma to the global form of the principle: Localizing, the local spatial form of the linear momentum
t t
V V V V V V V
LOCAL FORM OF THE LINEAR MOMENTUM BALANCE (CAUCHY’S EQUATION OF MOTION)
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Applying Newton’s 2nd Law to the discrete system formed by n
For a system in equilibrium, :
1 1 1 1 1
v M r f r r r v v r v
n n i O i i i i i i n n n i i i i i i i i i i i i
d t m dt d d d d m m m dt dt dt dt
L 0,
O
t M
d t t dt L
t cnt L
CONSERVATION OF THE ANGULAR MOMENTUM
i
=0
MO d t t dt L
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t L
angular momentum
The angular momentum of a material volume of a continuum
Where is the position vector
with respect to a fixed point.
M
t
V
V
M
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The time-variation of the angular momentum of a material volume
Where:
t
O V V
O V V
t dV dS
M r b r t
torque due to body forces torque due to surface forces
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The global form of the angular momentum balance principle: Introducing and using the Divergence Theorem, It can be proven that,
t
V V V V
T T V V V V T V
dS dS dS dS dV
r t r n r n r n r
ˆ ; r r m m e
T i i i ijk jk
m m e
REMARK is the Levi-Civita permutation symbol.
ijk
e
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Applying Reynolds Lemma to the right-hand term of the global
Then, the global form is rewritten:
ijk jk i V V
t t
V V V V V V V
Reynold's Lemma
=0
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Rearranging the equation: Localizing
( , ) ,
V V V V
d dV t dV V V t dt
v r b m m x
=0 (Cauchy’s Eq.)
( , ) ; , , 1,2,3 ; , m x x
i ijk jk t
t m i j k V t e
123 23 132 32 23 32 231 31 213 13 31 13 312 12 321 21 12 21
1 1 1 1 1 1
1 2 3 i i i
e e e e e e
( , ) ( , ) ,
T t
t t V t x x x
SYMMETRY OF THE CAUCHY’S STRESS TENSOR
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Power, , is the work performed in the system per unit of
In some cases, the power is an exact time-differential of a
It will be assumed that the continuous medium absorbs power
Mechanical Power: the work performed by the mechanical actions
(body and surface forces) acting on the medium.
Thermal Power: the heat entering the medium.
E
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The external mechanical power is the work done by the body
In spatial form it is defined as:
e V V
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Using and the Divergence Theorem, the traction
Taking into account the identity : So,
t n
n v n v
V V V V
dS dS dV dV
spatial velocity gradient tensor
:l :d :w
=0
:
V V V
dS dV dV
t v v d
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Divergence Theorem
Substituting and collecting terms, the external mechanical power
:
V V
e V V V V V
V
dV dV
v d
2
v
1 1 ( v ) 2 2 d d dt dt
v
v v d dt v
2 2
e V V V V
Reynold's Lemma
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2
t
e V V V V V
external mechanical power entering the medium stress power kinetic energy
e
REMARK The stress power is the mechanical power entering the system which is not spent in changing the kinetic energy. It can be interpreted as the work by unit of time done by the stress in the deformation process of the medium. A rigid solid will produce zero stress power ( ) .
d
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Theorem of the expended mechanical power
The external thermal power is incoming heat in the continuum
The incoming heat can be due to: Non-convective heat transfer across the
volume’s surface.
Internal heat sources
( , )
V
t dS
q x n
heat conduction flux vector
incoming heat unit of time ( , )
V
r t dV
x
specific internal heat production
heat generated by an internal source unit of time
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The external thermal power is incoming heat in the continuum
In spatial form it is defined as:
where: is the heat flux per unit of spatial surface area. is an internal heat source rate per unit of mass.
)
e V V V
V V
dS dV
nq q
, r t x
,t q x
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The total power entering the continuous medium is:
2
e e V V V V V t
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A thermodynamic system is a macroscopic region of the continuous
medium, always formed by the same collection of continuous matter (material volume). It can be:
A thermodynamic system is characterized and defined by a set of
thermodynamic variables which define the thermodynamic space
The set of thermodynamic variables necessary to uniquely define a
system is called the thermodynamic state of a system.
HEAT MATTER
ISOLATED SYSTEM OPEN SYSTEM
1,2,....n
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Thermodynamic space
A
thermodynamic process is the energetic development
a thermodynamic system which undergoes successive thermodynamic states, changing from an initial state to a final state Trajectory in the thermodynamic space.
If the final state coincides with the initial state, it is a closed cycle process.
A state function is a scalar, vector or tensor entity defined univocally as a
function of the thermodynamic variables for a given system.
It is a property whose value does not depend on the path taken to reach that
specific value.
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Is a function uniquely valued in terms of the “thermodynamic state”
Consider a function , that is not a state function, implicitly defined in
the thermodynamic space by the differential form:
The thermodynamic processes and yield: For to be a state function, the differential form must
an exact differential: , i.e., must be integrable:
The necessary and sufficient condition for this is the equality of cross-derivatives:
1 2
,
1 2
, , ,
n
1 1 2 1 2 1 2 2
, , f d f d
d
1 1 ' 1 2 2 2
2 1 2 2 ' 2 1 2 2
( , ) ( , )
B A B B B A
f f
1 1
,..., ,..., , 1,...
j n i n j i
f f i j n d
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1,..., n
1
2
1. There exists a state function named total energy of the system, such that its material time derivative is equal to the total power entering the system: 2. There exists a function named the internal energy of the system, such that:
It is an extensive property, so it can be defined in terms of a specific internal energy (or
internal energy per unit of mass) :
The variation of the total energy of the system is:
t E
2
( )
( ) 1 : v 2 :d q n
e e V V V V V t
e
Q t e
P t d d t P t Q t dV dV r dV dS dt dt
E
t U
, u t x
:
V
t u dV U
d d d t t t dt dt dt E K U REMARK and are exact differentials, therefore, so is . Then, the internal energy is a state function.
d K dE
d d d U E K
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Introducing the expression for the total power into the first
Comparing this to the expression in the second postulate: The internal energy of the system must be:
2
1 v 2 :d q n
V V V V V t
d d t dV dV r dV dS dt dt
E
K
:d q n
t
V V V V V
d d t u dV dV r dV dS dt dt
U
GLOBAL FORM OF THE INTERNAL ENERGY BALANCE , external thermal power
e
Q t
stress power
P t
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d d d t t t dt dt dt E K U
Applying Reynolds Lemma to the global form of the balance
Then, the local spatial form of the linear momentum balance
, du r V t dt :d q x
LOCAL FORM OF THE ENERGY BALANCE (Energy equation)
( )
q
:d q n :d q
t t t t
V V V V V V V V V V V V V V V V V V V V
V
dV
d d du t u dV dV dV r dV dS dt dt dt du dV dV r dV dV V V t dt
t
U
U
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( , ) V dV t x
The total energy is balanced in all thermodynamics processes
In an isolated system (no work can enter or exit the system) However, it is not established if the energy exchange can happen
There is no restriction indicating if an imagined arbitrary process is
physically possible or not.
e e
d d d P t Q t dt dt dt E K U
e e
d P t Q t dt E d d dt dt U K d d dt dt U K d d dt dt U K
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If a brake is applied on a spinning wheel, the
speed is reduced due to the conversion of kinetic energy into heat (internal energy). This process never occurs the other way round.
Spontaneously, heat always flows to regions of
lower temperature, never to regions of higher temperature.
The concept of energy in the first law does not account for
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A reversible process can be “reversed” by means of infinitesimal
It is possible to return from the final state to the initial state along the same path.
A process that is not reversible is termed irreversible. The second law of thermodynamics allows discriminating:
REVERSIBLE PROCESS IRREVERSIBLE PROCESS
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REVERSIBLE IRREVERSIBLE IMPOSSIBLE POSSIBLE
thermodynamic processes
1. There exists a state function denoted absolute temperature, which is always positive. 2. There exists a state function named entropy, such that:
It is an extensive property, so it can be defined in terms of a specific entropy
The following inequality holds true:
,t x
S
s ( ) s( , )
V
S t t dV x ( ) s
V V V
d d r S t dV dV dS dt dt
q n
Global form of the 2nd Law of Thermodynamics = reversible process > irreversible process
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( ) s
V V V
d d r S t dV dV dS dt dt
q n
Global form of the 2nd Law of Thermodynamics = reversible process > irreversible process
e V V
Q t r dV dS
rate of the total amount of the entity heat, per unit
system
e V V
r t dV dS
q n
rate of the total amount of the entity heat per unit
heat/unit of temperature power) entering into the system
e t
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The rate of the total entropy of the system is equal o greater than the rate of heat per unit of temperature
Consider the decomposition of entropy into two (extensive) counterparts: Entropy generated inside the continuous medium: Entropy generated by interaction with the outside medium:
s ,
i i V
S t dV x
s ,
e e V
S t dV x
i e i e
S t S t S t dS dS dS dt dt dt
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If one establishes, Then the following must hold true: And thus,
i e V V V V
dS dS dS dS r dV dS V V t dt dt dt dt
q n
e e V V
dS r dV dS dt
q n
i e V V e
dS dt
dS dS dS r dV dS dt dt dt
q n
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The internally generated entropy of the system , , never decreases along time
( )
i
S t
The previous eq. can be rewritten as: Applying the Reynolds Lemma and the Divergence Theorem: Then, the local spatial form of the second law of thermodynamics is:
t t
i V V V V V V V V V V t t
d d r s dV s dV dV dS V V t dt dt
q n
i V V V V V V V V
ds ds r dV dV dV dV V V t dt dt
q
,
i
ds ds r V t dt dt q x = reversible process > irreversible process
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Local (spatial) form of the 2nd Law of Thermodynamics (Clausius-Duhem inequality)
Considering that, The Clausius-Duhem inequality can be written as
2
1 1 q q q
2
1 1
i
ds ds r dt dt q q
i
s s
i local
s
i cond
s REMARK (Stronger postulate) Internally generated entropy can be generated locally, , or by thermal conduction, , and both must be non-negative.
i cond
s
i local
s
Because density and absolute temperature are always positive, it is deduced that , which is the mathematical expression for the fact that heat flows by conduction from the hot parts of the medium to the cold ones. q
1 r s q
CLAUSIUS-PLANCK INEQUALITY
2
1 q
HEAT FLOW INEQUALITY
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Substituting the internal energy balance equation given by
:
not
du u r dt d q
:
i local
s s r q
: u s d : r u q d
: s u d
Clausius-Planck Inequality in terms of the specific internal energy
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The Helmholtz free energy per unit of mass or specific free
Taking its material time derivative,
Clausius-Planck Inequality in terms of the specific free energy
: u s : u s s u s s
: u s d
: s d
REMARK For infinitesimal deformation, , and the Clausius-Planck inequality becomes:
d
( ) s :
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Conservation of Mass. Continuity Equation. 1 eqn.
v
Linear Momentum Balance. First Cauchy’s Motion Equation. 3 eqns.
b v
Angular Momentum Balance. Symmetry of Cauchy Stress Tensor. 3 eqns.
T
Energy Balance. First Law of Thermodynamics. 1 eqn.
: u r d q
Second Law of Thermodynamics. Clausius-Planck Inequality. Heat flow inequality 2 restrictions
u s :d
2
1 q
8 PDE + 2 restrictions
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The fundamental governing equations involve the following variables: At least 11 equations more (assuming they do not involve new unknowns),
are needed to solve the problem, plus a suitable set of boundary and initial conditions.
Cauchy’s stress tensor field
v
u s q
density 1 variable
velocity vector field 3 variables 9 variables specific internal energy 1 variable absolute temperature heat flux per unit of surface vector field 3 variables 1 variable specific entropy 1 variable 19 scalar unknowns
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Thermo-Mechanical Constitutive Equations. 6 eqns.
Thermal Constitutive Equation. Fourier’s Law of Conduction. 3 eqns. State Equations. (1+p) eqns. (19+p) PDE + (19+p) unknowns
, , v
, , s s v
1 eqn.
, K q q v
, , 1,2,...,
i
F i p
, , , u f v
Kinetic Heat Entropy Constitutive Equation. set of new thermodynamic variables: .
1 2
, ,...,
p
REMARK 1 The strain tensor is not considered an unknown as they can be obtained through the motion equations, i.e., .
v
REMARK 2 These equations are specific to each material.
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Conservation of Mass. Continuity Mass Equation. 1 eqn.
v
Linear Momentum Balance. First Cauchy’s Motion Equation. 3 eqns. Energy Balance. First Law of Thermodynamics. 1 eqn. Second Law of Thermodynamics. Clausius-Planck Inequality. 2 restrictions. Mechanical constitutive equations. 6 eqns.
( ( ), ) v
16 scalar unknowns 10 equations
MMC - ETSECCPB - UPC 78
The mechanical and thermal problem can be uncoupled if
The temperature distribution is known a priori or does not intervene in
the thermo-mechanical constitutive equations.
The constitutive equations involved do not introduce new thermodynamic
variables, .
Then, the mechanical problem can be solved independently.
,t x
79
Conservation of Mass. Continuity Mass Equation. 1 eqn.
v
Linear Momentum Balance. First Cauchy’s Motion Equation. 3 eqns. Energy Balance. First Law of Thermodynamics. 1 eqn. Second Law of Thermodynamics. Clausius-Planck Inequality. 2 restrictions. Mechanical problem Thermal problem Mechanical constitutive equations. 6 eqns. 10 scalar unknowns
( ( ), v )
80
Then, the variables involved in the mechanical problem are:
Cauchy’s stress tensor field
v
density 1 variable
velocity vector field 3 variables 6 variables
u s q
specific internal energy 1 variable absolute temperature heat flux per unit of surface vector field 3 variables 1 variable specific entropy 1 variable Mechanical variables Thermal variables
81
82
The convective flux of through a spatial surface with unit normal
is:
Time derivatives of a volume integral:
t A S
n
S s
t dS
v n
Where: is an arbitrary property is the description of the amount
,t x
t A
inflow
v n
v n
, x
not V
d t dV dt
material derivative
,
V V V V t
d t dV dV dV dt t
x v
, t x
not V
t dV local derivative
83
Conservation of mass: the mass of a continuum body is a conserved
quantity.
Reynolds Lemma: Reynolds Transport Theorem:
Global spatial form Local spatial form (Continuity Equation)
V V
d dV dV dt
v v
V V V t
d d dV dV dt dt
V V V V V
dV dV dS dV dS t
v v n
Divergence Theorem
84
Linear Momentum Balance: Angular Momentum Balance:
Global spatial form Local spatial form (Cauchy’s Equation of Motion) Global spatial form Local spatial form (Symmetry of the Cauchy stress tensor)
t
V V V V
d dV dS dV dt
b t v + , d V t dt v b x
,
T
V t x
t
V V V V
d dV dS dV dt
r b r t r v
85
Mechanical Energy Balance: External Thermal Power: Total Power
2
1 v 2
t
e V V V V V
d P t dV dS dV dV dt
b v t v :d
external mechanical power entering the medium stress power kinetic energy
K P
e V V
Q t r dV dS
is the heat flux per unit of spatial surface area. is an internal heat source rate per unit of mass.
, r t x
,t q x
Where:
e e
P Q
86
First Law of Thermodynamics. Internal Energy Balance. Second Law of Thermodynamics.
Global spatial form Local spatial form (Energy Equation)
:d q n
t
V V V V V
d d t u dV dV r dV dS dt dt
e
Q t
P t
, du r V t dt :d q x Global spatial form Local spatial form (Clausius-Duhem inequality)
s
V V V
d d r S dV dV dS dt dt
q n
,
i
ds ds r V t dt dt q x
= reversible process > irreversible process
1 q r s
CLAUSIUS-PLANK INEQUALITY
87
Governing equations of the thermo-mechanical problem: 19 scalar unknowns: , , , , , , .
Conservation of Mass. Continuity Mass Equation. 1 eqn.
v Linear Momentum Balance. First Cauchy’s Motion Equation. 3 eqns. b v Angular Momentum Balance. Symmetry
3 eqns.
T
Energy Balance. First Law of Thermodynamics. 1 eqn. :d q u r Second Law of Thermodynamics. Clausius-Planck Inequality. 2 restrictions
:d u s
2
1 q
8 PDE + 2 restrictions
v
u q s
88
Constitutive equations of the thermo-mechanical problem:
The mechanical and thermal problem can be uncoupled if the temperature
distribution is known a priori or does not intervene in the constitutive eqns. and if the constitutive eqns. involved do not introduce new thermodynamic variables.
Thermo-Mechanical Constitutive Equations. 6 eqns.
Thermal Constitutive Equation. Fourier’s Law of Conduction. 3 eqns. State Equations. (1+p) eqns. (19+p) PDE + (19+p) unknowns
, , v
, , s s v
1 eqn.
, K q q v
, , 1,2,...,
i
F i p
, , , u f v
Kinetic Heat Entropy Constitutive Equation. set of new thermodynamic variables: .
1 2
, ,...,
p
89