# notions of heat exchange

Reading time:**Conduction **is heat transported between two bodies in contact that are at different temperatures or between two parts of the same body that are at different temperatures.

The heat flow Ø transmitted by conduction over a length x across an area S perpendicular to this flow is given by the Fourier law :

θ_{1} – θ_{2} representing the drop in temperature over the distance x, λ being the heat conduction coefficient for the material expressed in practice in W·m^{–1} by °Kelvin.

For most solids, λ is an almost linear function of the temperature: a λ = λ_{} (1 + aθ); l is usually positive for insulators and negative for metals except aluminium and brass. However, heat conduction coefficient variation based on temperature is relatively low. Between 0 and 100 °C, the following values can be used as an initial approximation (table 88) :

**Heat conduction coefficient variation based on temperature**

At ambient temperature, the following values can be used :

- still water = 0.58 W·m
^{–1;} - calm air = 0.027 W·m
^{–1}·K^{–1}.

The Fourier equation can also be written as follows :

With regard to conduction through several materials in series, for a total temperature drop Δθ, we can write :

**Convection **is heat transferred in a fluid from a solid under the effect of motion that is caused by density differentials (natural convection) or that is mechanically induced (forced convection).

In practice, heat transfer from a solid body at temperature θ and a fluid at temperature θ_{1} involves both the convection and the conduction processes, making this a particularly complex phenomenon. We can then establish a** global transmission coefficient** k such that :

Using the same system of units, the value of k depends on certain physical properties of the fluid, on its circulation velocity, on the geometry of the solid. This means that the values for k can vary widely. For example, the following values are feasible (table 89) :

**The values for k vary widely**

For additional information, our readers might find the following of use :

- the Aide-mémoire du thermicien (see heat transfer) publ. Elsevier;
- the Techniques de l’Ingénieur (see energy engineering) vol. BE
_{1}.

**Radiation** is the transmission of heat as radiated energy. This phenomenon takes place without material support and, therefore, can take place in a vacuum.

The Stefan-Boltzmann law provides the heat flow emitted by radiation :

T being the absolute temperature of the radiant body, e an emission factor equal to 0 for a perfect reflector and to 1 for a black body and k being a dimensional constant.

## heat exchangers

The quantity of heat passing through a wall can be written as follows :

- S is the exchange surface area in m
^{2;} - d
_{m}is the mean temperature difference between the two sides of the wall, characterised by the logarithmic mean of fluid incoming and discharge temperatures; - k is the global transmission coefficient in W·m
^{–2}·K^{–1}or in mth·m^{–2}·h^{–1}. °C according to the nature and conditions applicable to fluid flow and to wall characteristics - Q is expressed in watts or in mth·h
^{–1}depending on the system of units selected.

In the case of complex media such as sludge, the transfer coefficient will be mainly experimental. For example, through tube heat exchangers :

- in sludge digestion, the exchange coefficient can reach values of up to 1,300 W·m
^{–2}·K^{–1}[1,100 mth·m^{–2}·h^{–1}°C] for fluid velocities of between 1 and 2 m · s^{–1;} - in sludge heat treatment, with sludge-sludge exchangers, the exchange coefficient can reach 350 W·m
^{–2}·K^{–1}[300 mth·m^{–2}·h^{–1}°C] for fluid velocities of up to between 0.5 and 1 m·s^{–1}.

#### determination of the logarithmic mean of temperatures

Let us assume a backflow heat exchanger within which two fluids are circulating.

The logarithmic mean is provided by the following relation :

**Hausband table**

*Note : mathematically, it can be demonstraded that if*

*, the difference between the logarithmic mean and the arithmetical mean is less than 5 %. This comment justifies the fact that the arithmetical mean is often used in most exchangers used in sludge treatment applications**.*