Momentum Heat Mass Transfer

1
Momentum Heat Mass Transfer
MHMT9
Energy balances. FK equations
Mechanical, internal energy and enthalpy balance
and heat transfer. Fouriers law of heat
conduction. Fourier-Kirchhoff s equation.
Steady-state heat conduction. Thermal resistance
(shape factor).
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
2
Power of forces
MHMT9
When analysing energy balances I recommend you to
imagine a fixed control volume (box) and forces
acting on the outer surface
Power of surface force W
3
Kinetic energy balance
MHMT9
Kinetic energy balance follows directly from the
Cauchys equation for velocities (kinetic energy
is square of velocities, so that it is sufficient
to multiply the Cauchys equation by velocity
vector scalar product results to scalar energy)
and this equation can be rearranged (how? see the
next slide) to the final form of the kinetic
energy transport
4
Kinetic energy balance
MHMT9
When considering different forms of energy
transport equations it is important to correctly
interpret decomposition of energies to reversible
and irreversible parts
5
Dissipation of kinetic energy
MHMT9
This identity follows from the stress tensor
symmetry
Rate of deformation tensor
Example Simple shear flow (flow in a gap between
two plates, lubrication)
6
Example Dissipation
MHMT9
Generated heating power in a gap between rotating
shaft and casing
Rotating shaft at 3820 rpm
D5cm
L5cm
Gap width H0.1mm, U10 m/s, oil M9ADS-II at
00C ?3.4 Pa.s, ?105 1/s, ?3.4.105 Pa, ??
3.4.1010 W/m3 At contact surface S0.0079 m2 the
dissipated heat is 26.7 kW !!!!
7
Internal energy balance
MHMT9
The equation of kinetic energy transport is a
direct consequence of Cauchys equations and does
not bring a new physical information (this is not
a new law). Such a new law is the equation of
internal energy balance which represents the
first law of thermodynamics stating, that the
internal energy increase is determined by the
heat delivery by conduction and by the mechanical
work. This statement can be expressed in form of
a general transport equation for ?uE
Internal energy uE is defined as the sum of all
energies (thermal energy, energy of phase
changes, chemical energy) with the exception of
kinetic energy u2/2 and this is the reason why
not all mechanical work terms are included in the
transport equation and why the reaction heat is
not included into the source term Q(g).
8
Internal energy balance
MHMT9
Interpretation using the First law of
thermodynamics du dq -
p dv
Dissipation of mechanical energy to heat by
viscous friction
Heat transferred by conduction into FE
Expansion cools down working fluid This term
is zero for incompressible fluid
9
Internal energy balance
MHMT9
Remark Confusion exists due to definition of the
internal energy itself. In some books the energy
associated with phase changes and chemical
reactions is included into the production term
Q(g) (see textbook Sestak et al on transport
phenomena) and therefore the energy related to
intermolecular and molecular forces could not be
included into the internal energy. This view
reduces the internal energy only to the thermal
energy (kinetic energy of random molecular
motion). Example Consider exothermic chemical
reaction proceeding inside a closed and thermally
insulated vessel. Chemical energy decreases
during the reaction (energy of bonds of products
is lower than the energy of reactants), no
mechanical work is done (constant volume) and
heat flux is also zero. Temperature increases. As
soon as the internal energy is defined as the sum
of chemical and thermal energy, the decrease of
chemical energy is compensated by the increased
temperature and DuE/Dt0.
10
Total energy balance
MHMT9
The internal energy transport equation is in fact
the transport of total energy (this is expression
of the law of energy conservation) from which the
transport equation for the kinetic energy is
subtracted
The term ? is potential of conservative external
volumetric forces (for example potential energy
in gravitational field gh).
11
Enthalpy balance
MHMT9
Thermal engineers prefer balancing of steady
continuous systems in terms of enthalpies instead
of internal energies. The transport equation
follows from the equation for internal energy
introducing specific enthalpy huEpv (v is
specific volume)
giving the final form of the enthalpy balance
This enthalpy balance (like all the previous
transport equations for different forms of
energy) is quite general and holds for
compressible/incompressible fluids or solids with
variable transport properties (density, heat
capacity, )
12
Fourier Kirchhoff equation
MHMT9
Feininger
13
Fourier Kirchhoff equation
MHMT9
Primary aim of the energy transport equations is
calculation of temperature field given
velocities, pressures and boundary conditions.
Temperatures can be derived from the calculated
enthalpy (or internal energy) using thermodynamic
relationship
giving the transport equation for temperature
The reason why reaction enthalpy and enthalpy of
phase changes had to be included into the source
term Q(R) is the consequence of limited
applicability of thermodynamic relationships
between enthalpy and temperature (for example
DT/DtDp/Dt0 during evaporation but Dh/Dtgt0).
14
Fourier Kirchhoff equation
MHMT9
Diffusive (molecular) heat flux is proportional
to the gradient of temperature according to the
Fouriers law
where ? is thermal conductivity of
material. Fourier Kirchhoff equation for
temperature field reads like this
As soon as thermal conductivity is constant the
FK equation is
15
Thermal conductivity ?
MHMT9
Thermal and electrical conductivities are
similar they are large for metals (electron
conductivity) and small for organic materials.
Temperature diffusivity a is closely related with
the thermal conductivity Memorize some typical
values
Material ? W/(m.K) a m2/s
Aluminium Al 200 80E-6
Carbon steel 50 14E-6
Stainless steel 15 4E-6
Glas 0.8 0.35E-6
Water 0.6 0.14E-6
Polyethylen 0.4 0.16E-6
Air 0.025 20E-6
Thermal conductivity of nonmetals and gases
increases with temperature (by about 10 at
heating by 100K), at liquids and metals ? usually
decreases.
16
Conduction - stationary
MHMT9
Let us consider special case Solid homogeneous
body (constant thermal conductivity and without
internal heat sources). Fourier Kirchhoff
equation for steady state reduces to the Laplace
equation for T(x,y,z) Boundary conditions
at each point of surface must be prescribed
either the temperature T or the heat flux (for
example q0 at an insulated surface). Solution of
T(x,y,z) can be found for simple geometries in an
analytical form (see next slide) or numerically
(using finite difference method, finite
elements,) for more complicated geometry.
The same equation written in cylindrical and
spherical coordinate system (assuming axial
symmetry)
cylinder
sphere
17
1D conduction (plate)
MHMT9
Steady transversal temperature profile in a plate
is described by FK equation which reduces to
with a general solution (linear temperature
profile)
Differential equations of the second order
require two boundary conditions (one BC in each
point on boundary)
Tf1
Tw1
?

1. BC of the first kind (prescribed temperatures)
2. BC of the second kind (prescribed flux q0 in one
   point)
3. BC of the third kind (prescribed heat transfer
   coefficient)

Tw2
these boundary conditions with fixed values are
called Dirichlet boundary conditions
the boundary conditions of the second kind is
called Neumanns boundary condition
x
h
the boundary condition of the third kind is
called Newtons or Robins boundary condition
18
1D conduction (sphere and cylinder)
MHMT9
Steady radial temperature profile in a cylinder
and sphere (for fixed temperatures T1 T2 at inner
and outer surface)
R1
R2
cylinder
Sphere (bubble)
19
1D conduction (cylinder-heating power)
MHMT9
Knowing temperature profiles it is possible to
calculate heat flux q W/m2 and the heat flow Q
W. The heat flow can be also specified as a
boundary condition (thus the radial temperature
profile is determined by one temperature T0 at
radius R0 and by the heat flow Q)
For cylinder with thermal conductivity ? and for
specified heat flow Q related to length L, the
logarithmic temperature profile can be expressed
in the following form
Positive value Qgt0 represents a line heat source,
while negative value heat sink.
20
2D conduction (superposition)
MHMT9

• Temperature distribution is a solution of a
  linear partial differential equation (Laplace
  equation) and therefore is additive. It means
  that any combination of simple solutions also
  satisfies the Laplace equation and represents
  some stationary temperature field. For example
  any previously discussed solution of potential
  flows (flow around cylinder, sphere, see chapter
  2) represents also some temperature field
  (streamlines are heat flux lines, and lines of
  constant velocity potential are isotherms).
  Therefore the same mathematical techniques
  (conformal mapping, tables of complex functions
  w(z) describing dipoles, sources, sinks,
  circulations,) are used also for solution of
• temperature fields
• electric potential field
• velocity potential field
• concentration fields
• The principle aim is thermal resistance,
  electrical resistance Thus it is possible to
  evaluate for example the effect of particles
  (spheres) or obstacles (cylinders) to the
  resistivity of inhomogeneous materials.

21
2D conduction (superposition)
MHMT9
As an example we shall analyse superposition of
two parallel linear sources/sinks of heat Q
Temperature at an arbitrary point (x,y) is the
sum of temperatures emitted by source Q and sink
-Q
Lines characterised by constant values krS/rQ
are isotherms.
This equation describes a circle of radius R and
with center at position m
Resulting temperature field describes for example
the following cases (see also the next slide)
22
Thermal resistance
MHMT9
Knowing temperature field and thermal
conductivity ? it is possible to calculate heat
fluxes and total thermal power Q transferred
between two surfaces with different (but
constant) temperatures T1 a T2
RT K/W thermal resistance
In this way it is possible to express thermal
resistance of windows, walls, heat transfer
surfaces
Serial Parallel
Tube wall Pipe burried under surface
23
EXAM
MHMT9
Energy transport
24
What is important (at least for exam)
MHMT9
Kinetic energy
Internal energy
Fourier Kirchhoff
25
What is important (at least for exam)
MHMT9
Steady state heat conduction (cartesian,
cylindrical, spherical coordinates)
1D temperature profiles (cartesian, cylindrical,
spherical coordinates)
26
What is important (at least for exam)
MHMT9
Thermal resistance
Serially connected plates
Cylinder