Heat Transfer 1 - Abdelkhalak El Hami

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      Let us imagine a series of flatcar trains on a set of parallel tracks. The train furthest to the side is fixed to a platform. All of these trains are full of sailors. Let us suppose that our train follows the outside, parallel rail tracks. No brakes will prevent these trains from moving. Only the last train, at the platform, is stuck.

      For a reason we do not need to analyze (cinema allows all kinds of fantasy), “clusters” of fighting sailors jump from one wagon to the next. These “clusters” contain a component of speed that is parallel to the train, which will communicate information about the quantity of movement to the adjacent train. These trains will then start to move, more quickly the closer they are to the outside train. And the same occurs up to the train at the platform. This train will not move, but a force will be applied to its brakes.

      We have just discovered the mechanism of dynamic viscosity. At the same time, the parallel trains in relative movement give us a picture of the notion of boundary layers.

      At the same time, these agitated clusters carry their disordered kinetic energy, “thermal” agitation. We have just seen the mechanism of the thermal boundary layer.

      Finally, let us include a few red mariners in the crowd of white. They will be carried with the clusters, and we have just invented the limit layer of diffusion of a species.

      We are in a fantasy, and let us benefit from it as far as we can. To finish, let us suppose that this is carnival day; each sailor has a belt equipped with bells.

      All the individuals have a different speed, and the impacts are random, all the bells start to jingle, each with a different frequency. The distribution of frequencies will depend on the statistical distribution of speeds (Boltzmann statistics), and the intensity of noise produced will depend on the total agitation energy of the sailors.

      NOTE.– the model is certainly simplistic. The emission comes from quantum transitions in the gas atoms.

      Here, we have already deviated from the pure substance of the book, but we could go even further.

      Let us suppose that our agitated sailors are in a room with one mobile wall (a nightmare scenario frequently seen on the silver screen).

      The incessant impacts of the fighters on this wall create a force that pushes it. This force, reduced to a surface unit, explains the notion of pressure.

      By pushing against this wall, our crowd applies work that is greater than the resistance.

      Here we see an equivalence spring up between work and heat that, at a fundamental level, are simply two mechanical energies: one ordered and the other disordered. The first principle of thermodynamics is illustrated by this.

      We can see that the incidence of an average blow on the wall is rarely normal.

      Therefore, an average fighter will have a trajectory that will be reflected off the wall. And only the normal component of its speed will be able to push (or transfer work to) the wall.

      Thus, we see that it will be impossible for the crowd (taken to mean a gas) to give all its energy to a mobile wall.

      The fundamental mechanism that leads to the second principle of thermodynamics has just been demonstrated.

      And if this text has a moral, it would be: Writing down thermodynamics, just like thermal science, is based on continuous equations. The fundamentals of physics that determine these phenomena arise from the field of the discontinuous: discontinuity of matter, divided into particles; discontinuity of light, divided into photons.

      The Problem of Thermal Conduction: General Comments

      1.1. The fundamental problem of thermal conduction

      The fundamental problem of thermal conduction involves the determination of temperature domains and flows across particular surfaces, for a given physical situation, in one or several given environments.

      The resolution of a thermal conduction problem involves:

      1 a) for all problems, a heat equation is considered, resulting from a local heat balance;

      2 b) for specific conditions of the problem, the conditions are constituted at the limits that are applied to the heat equation.

      In the most general case, these temperature domains T are not homogeneous. They can be three-dimensional and variable over time:

      In other systems, we note that T = T (r, θ, z, t) or T = T (r, θ, Φ, t).

      Temperature is expressed in degrees Celsius (°C), formerly degrees centigrade, or in Kelvin (K).

      We know that the temperature, expressed in Kelvin, is measured from absolute zero. The conversion rule is known as: T K = T C + 273.15.

      Fortunately, significant simplifications are possible for many problems.

      The temperature field can only depend on a spatial variable. We say that conduction is monodimensional or bidimensional.

      The temperature can remain fixed at each point as time progresses.

      This last remark divides the approach that we will adopt into two parts:

       – stationary conductive heat transfer;

       – non-stationary conductive heat transfer.

      Two important categories will be examined in this chapter:

       – problems of stationary conduction with a single dimension: T = T(x) or T = T(r)

       – problems of non-stationary conduction with a single dimension: T = T (x,t) or T =T (r,t)

      For many problems, the analytical approach is possible. This will be the case, in particular, for the stationary or non-stationary problems with a single dimension. In (most) other cases, a numerical approach is required.

      1.2.1. Temperature, isothermal surface and gradient

      The temperature T is a parameter defined in all thermodynamics classes.

      As of now, we can define isothermal surfaces in space. An isothermal surface is a surface on which the temperature is constant:

      [1.1]

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