Module 2. Convection heat transfer

Lesson 8

SHAPE AND SIZE OF HEAT TRANSFER SURFACES, HEAT TRANSFER CO-EFFICIENT AND ITS INTERPRETATIONS

8.1  Shape and Size of Heat Transfer Surfaces

A large number of diverse heat transfer surfaces may be composed of bodies of the simplest shape i.e. a plate or a tube, further a plate may possess one or two heat transfer surfaces. Plates may be arranged vertically, horizontally or in an inclined position. In the case of horizontally arranged plate with only one heat transfer surface the later may be turned up or down, further heat transfer surfaces may consist of several plates. A similar variety of heat transfer surfaces may be composed of tubes. Each such surface offers specific condition for flow and heat transfers. Therefore the shape and size of heat transfer surfaces substantially affect heat transfer. It is also important whether the fluid moves inside an enclosed space or washes all the sides of heating surface.

8.1.1  Heat transfer co-efficient

According to Fourier’s Law the relationship between the rate of heat transfer through contact and the conditions in which heat transfer occurs can be expressed by the following equation:

This relationship, however, cannot be used in practical calculations. Because to solve this equation one must know the temperature gradient at the wall and its variation over the entire heat transfer surface (F), which is impossible. Therefore for the sake of facilitating calculation, Newtonian formula is considered Q = αF (t_F – t_W) [kcal/hr]

According to this relationship, the amount of heat transferred from the fluid to the wall or vice versa, is proportional to the heat transfer surface F and temperature difference (t_F – t_W). The conditions of heat transfer between the fluid and solid are characterized by the factor of proportionality α, which is called heat transfer co-efficient. The co-efficient ‘α’ determines intensity of heat transfer and is expressed in kcal/m²-hr-°C units.

 8.1.2  Interpretations

Numerically heat transfer co-efficient is equal to amount of heat transferred in unit time through unit area at a temperature difference of 1°C between the solid and fluid.

Newtonian formula does not mean any fundamental simplification because the complexity of heat transfer and difficulties in calculation in this case are all embodied in a single value heat transfer co-efficient.

The accepted method of calculating is usually employed to study the phenomenon as well; attention is concentrated only on the determination of the heat-transfer coefficient and on the establishment of its dependence on different factors. Earlier treatises analysed only the influence of the most important factors, primarily the temperature difference and velocity of fluid flow. Subsequent studies have shown, however, that the heat transfer coefficient is a complex function of many variables upon which the process as a whole depends. In general, the heat transfer coefficient is a function of shape ∅, dimension l₁, l₂, l₃ and temperature t_w of the heating surface, fluid flow velocity ν, fluid temperature t_f and physical properties of the fluids, such as thermal conductivity λ; heat capacity c_p, density ϱ, viscosity μ and other factors. Thus

In general the heat transfer co-efficient

The process of heat transfer is being studied both theoretically and experimentally. In the first case, the problems are solved analytically, in the second, by direct experiment.

8.2  Differential Equations of Heat Transfer

To study any phenomenon means to establish the relationship between the variables characterizing the phenomenon. In the case of complex phenomena whose defining properties vary both in time and space. It is very difficult to determine the relationship between the variables. The general laws of physics enable us only to establish the relationship between variables i.e. between co-ordinates, time and physical properties and the relation developed covers a brief time interval and that two only on elementary volume of the entire space: In doing so we ignore some values or replace a complex relationship by a simpler one.

The relationship thus obtained is the general differential equation of the process considered. Integrating this equation we obtain the analytical relationship between the values over the entire range of integration and for the entire time interval under consideration. Such differential equations can be worked out for any process and notably here, the process of heat transfer. Since heat transfer involves both thermal and hydrodynamic phenomena, there should be a whole series of differential equations developed for the aggregate process:

i. Equation of heat transfer

ii. Equation of Conduction

iii. Equation of flow

iv. Continuity Equation