Basic concepts of heat transfer TEM – Lecture 2 Basic concepts of heat transfer

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.

Heat Transfer Processes Could a stationary fluid fluid support a horizontal temperature gradient?

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. The energy necessary to change the temperature of a body of mass M is: To store a large amount of heat I need a large Mass and a large “c” . What is the material with higher “c” on earth?

Heat Conduction Process Fourier Law of heat conduction

Heat Conduction: Thermal conductivity Units: What is the physical meaning of the termal conductivity?

The Physical meaning of k These are the units of any diffusivity. 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?

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 product of these two properties is the diffusivity. Why are thermal and electrical conductivity well correlated? Why do metals sometimes touch “cold” others touch “hot”?

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

The stationary temperature profile is linear

Heat budget (1/2)

Heat budget (2/2)

Momentum Equation for viscous flows with null pressure gradient 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):

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)

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

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:

Summary of the Lecture Forms of Heat Transfer Conduction, thermal conductivity (heat diffusivity) Heat budget and temperature evolution equation. General concept of diffusivity.