1. Basic concepts of heat transfer: Heat Conduction
TEM – Lecture 2
Basic concepts of heat transfer: Heat Conduction

2. Summary of Last Lecture
The Course Organization
The importance of Energy and Mass Transfer at global and Engineering scale.
The close linkage between Mass and Energy Transfer
Heat Conduction and Radiation as the only energy transport processes not associated to mas transfer. This is why the course will start with heat conduction.

3. Heat Transfer Processes
Why doesn’t a stationary fluid support a horizontal temperature gradient?

4. Temperature vs Heat
Temperature is a measure of the kinetic energy of a molecule. Is Energy per molecule.
Heat is the Energy of an ensemble of molecules. It can be expressed per unit of volume or unit of mass.
Heat and Temperature are related by the Heat capacity:
Heat capacity depends on the number of molecules per unit of volume and on the molecular mass.
The energy necessary to change the temperature of a body of mass M is:
To store a big heat quantity I need a large Mass and a large “c” .
What is the material with higher “c” on earth?

5. Heat Conduction Process
Fourier Law of
heat conduction

6. Heat Conduction: Thermal conductivity
(Lei de Fourier)
Units:

7. Conductivity of different materials
Why are thermal and electrical conductivity well correlated?

8. Why do metals usually touch “cold” while other materials can touch “hot”?

9. The Physical meaning of k
High conductivity means that one can move a large amount of heat, to a long distance, i.e. if the conductivity of a wall material is high, heat can cross it rapidly.
Conductivity is the product of density (mass per volume) by the heat capacity (heat per mass) that gives heat per unit of volume, by a velocity and by a distance.

10. Diffusivity
These are the units of any property diffusivity (heat, mass, momentum, …).
In Fluid Mechanics we have seen that these units are also the units of the kinematic viscosity and the units of mass diffusivity. Why is this so?

11. Why is diffusivity identical for most properties?
This is so because diffusivity depends on:
The velocity of molecules (m/s), the larger this is, the faster they transfer their properties (kinetic energy in case of heat, momentum in case of the momentum transport or mass in case of mass transport) and
On the distance (m) traveled by each molecule up to collide with another molecule (the larger is the distance, the higher is diffusivity).
The product of these two properties is the diffusivity. In fluids Diffusivity is associated to the displacement of molecules (gases) or groups of molecules (liquids) and thus is about the same for any property. In case of gases diffusivity increases with temperature, while in case of liquids increasing temperature reduces the size of the groups of molecules and consequently it reduces diffusivity.

12. Diffusivity and Diffusion
Figures below represent 2 material systems, one fully white and the other fully Black separated by a diaphragm. The top figures represent the molecules (microscopic view) and the figures below the macroscopic view. When the diaphragm is removed the molecules from both systems start to mix and we start to see a grey zone between the two systems (b) at the end everything will be grey (c). During situation (b) we there is a diffusive flux of black molecules crossing the diaphragm section. This flux cannot be advective because velocity is null.
(a)
(b)
(c)

13. Ver texto sobre propriedades dos fluidos e do campo de velocidades
Diffusivity
When the diaphragm is removed molecules move randomly. The net flux is the diffusive flux.
The flux of molecules in each sense is proportional to the concentration and to the individual random velocity:
But,
Diffusivity is the product of the displacement length and the molecule velocity. This velocity is in fact the difference between the molecule velocity and the average velocity of the molecules accounted for in the advective term.
Ver texto sobre propriedades dos fluidos e do campo de velocidades

14. Diffusivity Diffusivity is definide as:
Where is the molecule velocity part not resolved (or included) in our velocity definition. In a laminar flow is the brownian velocity while in a turbulent flow is the turbulent velocity, a macroscopic velocity that we can see in the tubulent eddies.
is the lenght of the displacement of a molecule before being disturbed by another molecule (or of a portion of fluid in a turbulent flow). When the molecule hits another molecule it gets a new velocity.
Diffusivity dimensions are:

15. Rate of heat accumulation
Per unit of volume:
The rate of accumulation of heat is the symmetric of the flux divergence:
using

16. The stationary temperature profile is linear

17. Heat Budget: Energy Conservation Principle
In case of thermal energy:
The rate of accumulation of heat inside a control volume, is the difference of the amount of heat entering and leaving per unit of time (heat flux) plus the heat production/consumption.

18. Heat budget (Fluxes 1/2)

19. Heat budget (Fluxes 2/2)
In case of heat generation:

20. Diffusion equation
The resolution of this equation needs initial conditions and boundary conditions.

21. Initial conditions
Can be any. A hot bar left outside will cool down, while a cold bar will warm.
Whatever are the initial conditions, after some time the solution will depend only on the boundary conditions.
Nowadays one would like the earth to loose into space as much heat as that received from the sun. Can we say that the earth has forgotten the initial conditions?

23. Momentum Equation for viscous flows with null pressure gradient
If applied forces (e.g. pressure gradient or gravity)
It is identical to the temperature evolution equation for conduction.
And the momentum diffusivity units are the same…
The same would hold for diffusion of a matter inside another
(e.g. nitrate in water):

24. Summary of the Lecture Forms of Heat Transfer
Conduction, thermal conductivity (heat diffusivity)
Heat budget and temperature evolution equation.
General concept of diffusivity.
Reading: Chapter 2 of the book