Hello friends of steemit, in this opportunity I want to talk about a subject that I consider very important for its technological applications, I refer to the transport phenomenons. Today we will talk about heat transfer.

![conducemetal.jpg](https://steemitimages.com/DQmV8hfczYeRa3FAwC7r4Se7zgheU6ebYrG4xnepXZPMqNY/conducemetal.jpg) [Source](http://termo2transfer.blogspot.com/2015/12/ley-de-fourier-de-conduccion-de-calor.html) _____________

Heat transfer can be defined as the transmission of energy from one region to another, as a result of a temperature difference. Conduction, convection and thermal radiation represent the three mechanisms by which heat transfer occurs.

In conduction the transfer of heat is made through the material, without net mass flow, while in thermal radiation the bodies are separated in space and the change of thermal energy obeys the laws of electromagnetism.

The transfer of heat by conduction from a solid surface to a fluid in motion is known as heat transfer by convection.

In this first part, we will focus on the thermal conduction process, later, we will talk about convection and thermal radiation.

##### THERMAL CONDUCTION

Thermal conduction refers to the transfer of energy from the most energetic particles to the least energetic particles of a substance, due to the interactions between them. Higher temperatures are associated with higher molecular energies and when neighboring molecules collide, as they constantly do, a transfer of energy from the more energetic to the less energetic molecules must occur.

In the presence of a temperature gradient, the energy transfer per conduction must occur in the direction of the decreasing temperature.

In the conduction heat flow, the energy is transmitted by direct molecular communication (elastic shock in case of liquids) without appreciable displacement of the molecules.

The equations that describe this process are called Fourier Laws. ![imn.png](https://steemitimages.com/DQmVPaGoJFaaZSLzjnEecHGAVQBmC7bxxtJeF9Xs5M8jSkc/imn.png) [Heat flow in the decreasing direction of temperature](https://es.wikipedia.org/wiki/Conducci%C3%B3n_de_calor) ##### First Fourier Law

The first Fourier Law states that there is a proportionality between the flow of heat transfer by conduction (https://steemitimages.com/DQmfLRjkUcLEjsaRhZnvGZUkgadH8ijHvTGd7mN37U5AZcY/1.png) and the temperature gradient.

![4.png](https://steemitimages.com/DQmaPomMkEeNz1CmaKvcHwxihmhqex1J8v5UtNLmgGEhBeA/4.png) eq. (1)

Where (https://steemitimages.com/DQmfLRjkUcLEjsaRhZnvGZUkgadH8ijHvTGd7mN37U5AZcY/1.png) is the heat flow per unit area in the spatial coordinates, which is established in the body due to temperature differences between two points of the same. The minus sign indicates that the flow is inverse to the direction of the gradient.

K is a constant characteristic called thermal conductivity. ##### Second Fourier Law: Energy Equation.

The energy equation can be found by considering the balance on a control volume and then differentially obtaining said equation, which is the following:

![33.png](https://steemitimages.com/DQmThk3ELXaVyzfARKvUDmHUtwJx24ZrBgsE7oMjDezVFFC/33.png)eq. (2) Where: e is the energy. ![8.png](https://steemitimages.com/DQmevww1KQosJUgMuev1DPxhxFqbJfYtvMHFny3LTvioHLD/8.png) is the heat flow by conduction. ![9.png](https://steemitimages.com/DQmevYvcRYN634vCyArbi9g19Bdc18sy5VJnnTgdQgvrpqC/9.png) is the flow of heat by radiation. The last two terms of the equation represent the power developed by the forces of body and surface.

The above equation states that the temporal change rate of the total energy of the control volume is equal to the net power developed by the force acting on the control volume, plus the net supply of energy that occurs across the surface of the control.

If we group the terms and apply the Gaussian divergence theorem, we obtain: ![10.png](https://steemitimages.com/DQma8xrDguyyFoGzGCEWNMDorgNTjx8RXG5sXxYVDEyY4t5/10.png) eq. (3) Since the volume is arbitrary (it is not zero), the integrand must be equal to zero. Combining with the continuity equation we obtain: ![11.png](https://steemitimages.com/DQmUxR3gFHUX1pVDybW67BtYmgpsGL6GYiHxzZYpY6GXyfC/11.png) eq. (4)

Where the mechanical energy comes from taking the scalar product between the velocity vector and the Cauchy equation:

![12.png](https://steemitimages.com/DQmUKxfTPKUGgQSzgsXSyEyHzYPtWj8vwjjPXeobj6YzLbj/12.png) eq. (5)

If we subtract the equations (4) and (5), where the stress tensor decomposes in pressure and viscous effort, we obtain:

![13.png](https://steemitimages.com/DQmazbV83Ps6V95zkegy8mq9SPdPQUNsRotZDrWZ9GJGx7W/13.png) eq. (6)

We can express this last equation in another way. If we introduce the enthalpy (https://steemitimages.com/DQmRBP348XBJE6eq4o1ug75CmznsbQNXzLR7JxnS2Nzu5RS/14.png), after algebraic manipulations, we obtain the expression of the energy equation as a function of temperature, which is the following:

![15.png](https://steemitimages.com/DQmUdWbQEZhcqEiYxdyGb2jXV8Mbhv8qPtagzBoC6NStxtz/15.png) eq. (7) ![17.png](https://steemitimages.com/DQmPNTJhbfbnPggMMPr7NDonNhrsmoZzXqof8a8RGnXrR1N/17.png) is the specific heat at constant pressure. ![18.png](https://steemitimages.com/DQmNzU9L6L3wxQx69C6UvcwvyAsoEpLDZhUjJ4b7SSE1z44/18.png) are the contributions of the terms of expansion and compression. ![19.png](https://steemitimages.com/DQmSw34yfs9tzmBcdRX8KxrcwzV7pEfZeFGPWV6udx1WD3X/19.png) represents the viscous dissipation. ![Transferencia-de-calor-Conduccion.png](https://steemitimages.com/DQmcoXkxWQPyuoQ4DkFsS3sckgFEv4pDSBuQXZzGmCim4N6/Transferencia-de-calor-Conduccion.png) [Source](https://www.vix.com/es/btg/curiosidades/5077/transferencia-de-calor-conduccion)

Now, if we assume that the contributions of the terms of expansion and compression and viscous dissipation are small, and further that the terms of heat transfer by radiation and by convection are negligible, the above equation is considerably simplified.

Under these conditions and using the first Fourier law, we obtain the equation for conduction heat flow: ![20.png](https://steemitimages.com/DQmNfQZ1iviqrdhrLuYQ3YRMjCtfGhTZw7w6hbTn1sDB3uq/20.png) eq. (8) Which is called Second Fourier Law _____ _**References:**_ Nozad, I., Carbonell, R. G., and Whitaker, S. (1985). Heat conduction in multiphase systems: Theory and experiments for two-phase systems. Bird, R.B. y Stewart, W.e., Lightfoot, E. N, 1996. Fenómenos de transporte, Reverté S.A., México D.C. Welty, J. R.; wicks, C. E.; Wilson, R. E., (1984). Fundamentos de transferencia de momento, calor y masa. Edit. Limusa, México D.C.