1. Momentum Heat Mass Transfer
MHMT9
Energy balances. FK equations
Mechanical, internal energy and enthalpy balance and heat transfer. Fourier´s law of heat conduction. Fourier-Kirchhoff ´s equation. Steady-state heat conduction. Thermal resistance (shape factor).
Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT 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 Cauchy’s equation for velocities (kinetic energy is square of velocities, so that it is sufficient to multiply the Cauchy’s 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
Proof follows from the previous equation rewritten to the index notation
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)
U=ux(H) y

6. Example: Dissipation MHMT9
Generated heating power in a gap between rotating shaft and casing
Rotating shaft at 3820 rpm
D=5cm
L=5cm
U=10 m/s y
H=0.1 mm
Gap width H=0.1mm, U=10 m/s, oil M9ADS-II at 00C
=3.4 Pa.s, =105 1/s, = Pa, = W/m3
At contact surface S= m2 the dissipated heat is 26.7 kW !!!!

7. Internal energy balance
MHMT9
The equation of kinetic energy transport is a direct consequence of Cauchy’s 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
Heat flux by conduction
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/Dt=0.

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 h=uE+pv (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/Dt=Dp/Dt=0 during evaporation but Dh/Dt>0).

14. Fourier Kirchhoff equation
MHMT9
Diffusive (molecular) heat flux is proportional to the gradient of temperature according to the Fourier’s 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. 15

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 q=0 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 16

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 
BC of the first kind (prescribed temperatures)
BC of the second kind (prescribed flux q0 in one point)
BC of the third kind (prescribed heat transfer coefficient)
Tw2
these boundary conditions with fixed values are called Dirichlet boundary conditions x
the boundary conditions of the second kind is called Neumann’s boundary condition h
the boundary condition of the third kind is called Newton’s or Robin’s boundary condition 17

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
cylinder R2
Sphere (bubble) 18

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)
R0,T0 L
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 Q>0 represents a line heat source, while negative value heat sink. 19

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. 20

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 rQ h y x
T(x,y)
Sink -Q rS
Source Q
Lines characterised by constant values k=rS/rQ are isotherms.
This equation describes a circle of radius R and with center at position m T1 T2 x y m R T1 T2 x y
Resulting temperature field describes for example the following cases: (see also the next slide) 21

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 …
h1 h2 S T1 T2 h S1 S2
T1 T2 L R1 R2
T1 T2 Q
Serial Parallel Tube wall Pipe burried under surface 22

23. EXAM
MHMT9
Energy transport

24. What is important (at least for exam)
MHMT9
Kinetic energy
Conservation of all forms of energy
Internal energy (by subtraction kinetic energy, see also 1st law duE=dq-dw)
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