HEAT TRANSFER - brief notes

Introduction

As long as there is a temperature driving force between two points heat will be transferred down the temperature gradient (from hot to cold). Everyday examples we encounter are central heating, air conditioning, double glazing, refrigeration, etc. In an engineering context heat transfer is important in many unit operations. Reactants need to be heated to reaction temperature, temperatures in reactors need to be controlled, distillation requires both boiling and condensation, etc. In this course, we aim to study heat transfer in a chemical engineering context, so that the process of heat transfer may be undertood and that initial design of a heat exchanger may be done.

The rate of heat transfer depends on the temperature driving force, the area available for heat transfer, the nature of the material and the mode of heat transfer (conduction, convection or radiation).

Conduction

Vibrational energy is transferred between atoms or molecules that do not themselves move. It is the basic transfer mechanism for heat transfer in solids but can also occur through layers of liquids and gases that are not highly mobile. The thickness of the material is important to the rate of heat transfer giving the basic equation:

\begin{displaymath}Q\;{\propto}\;\frac{A}{x}{\Delta\theta}\end{displaymath}

Thermal conductivity is the constant of proportionality.

\begin{displaymath}Q=\frac{kA}{x}{\Delta\theta}\end{displaymath}

More generally,

\begin{displaymath}Q=-kA\frac{d\theta}{dx}\end{displaymath}

This is Fourier's law of heat conduction.

Typical values of thermal conductivity for a range of material types are given below:

Copper 390 $\frac{W}{mK}$ at 300 K

Glass

 1.05 $\frac{W}{mK}$ at 300 K

Asbestos

 0.088 $\frac{W}{mK}$ at 300 K

Air

 0.026 $\frac{W}{mK}$ at 300 K

Water

 0.61 $\frac{W}{mK}$ at 300 K

For composite planar materials the heat flow through each slab is the same:

\begin{eqnarray*}Q & = &\frac{k_1}{x_1}A(\theta_{hot}-\theta_i) = \frac{k_2}{x_2... ...ght) A}{\frac{x_1}{k_1}+\frac{x_2}{k_2}+\frac{x_3}{k_3}+ \cdots} \end{eqnarray*}


Convection

Energy is transferred by large scale (macroscopic) motions. Therefore it is limited to liquids and gases where the atoms or molecules are free to move. As the fluid is a conducting medium, conduction still occurs. Forced convection occurs when an exterior agent drives the fluid motion, natural convection occurs when buoyancy differences caused by local temperature differences drive the flow.

Convective heat transfer normally takes place from a solid surface to a fluid. Adjacent to a solid wall large scale motions die away leaving a virtually still layer of fluid. Conductive transfer occurs across this film which is of unknown thickness.

\begin{displaymath}Q=hA{\Theta}\end{displaymath}

where h is the heat transfer coefficient for the film. Values depend on the material and on the way it is being handled.

We could write both equations for heat transfer by conduction and convection making use of the resistance to heat transfer, R:

\begin{displaymath}Q=\frac{A}{R}{\Delta\theta}\end{displaymath}

For conduction $R=\frac{x}{k}$


For a fluid film $R=\frac{1}{h}$
Where heat transfer processes occur in series (as in double glazing), we may add resistances

\begin{displaymath}{R_T}={\displaystyle \sum_{i=1}^{i=n}}{R_i}\end{displaymath}

Radiation

All bodies emit energy in the form of electro-magnetic waves. When this falls on a second body some is reflected, some passes through and some is absorbed. The proportions of each depend on the physical properties of the material. The transfer does not depend on the presence of any physical medium between the two bodies.

\begin{displaymath}Q\;{\propto}\;{T^4}\end{displaymath}

Radiant heat transfer becomes increasingly important with increasing surface temperature.

The governing equation is:

\begin{displaymath}Q=A{\epsilon}{\sigma}({T_s}^4-{T_{sur}}^4)\end{displaymath}


${\sigma}$ is the Stefan-Boltzmann constant with a value of 5.67 ${\times}10^{-8}\frac{W}{{m^2}{K^4}}$.

The emissivity, ${\epsilon}$, is a property of a surface.

    ${\displaystyle \epsilon}$
Black body = 1
Polished metal = 0.21
Oxidised metal = 0.64 to 0.78
Strongly oxidised metal = 0.95
Matt black paint = 0.91