Page 51 - Mechanical Engineers Reference Book

1/40 Mechanical engineering principles
1.7 Heat transfer Single layer Plane surface

1.7.1 Introduction
Whenever a temperature difference occurs there is an energy
flow from the higher temperature to the lower. A study of heat
transfer is concerned with the determination of the instana-
neous rates of energy flow in all situations. We determine heat
transfer rates in watts. These rates will be constant in situa-
tions where the temperature difference remains constant but
variable (transient) when the temperature difference varies I k'z I
either due to the heat transfers or to other energy changes
such as internal chemical reaction. x1 x2
There are three modes of heat transfer:
Figure 1.59 One-dimensional conduction through a single-layer
1. Conduction, which is of greatest interest in solid bodies plane wall
but also occurs in fluids where it is often overshadowed by
convection;
2. Convection, which occurs in fluids when energy is trans- Multi layer plane surface
ferred due to the motion of the fluid:
3. Radiation, which occurs between two systems at different
temperatures which need not be in contact provided any
intervening medium is transparent to the radiation.
In practice, all three modes occur simultaneously and it is
necessary to draw up a balance at a boundary. For example,
energy may be conducted to the surface of an electric storage
heater and is then convected and radiated to the surroundings.
Thus calculations can become complex, and in this particular
case where energy is added at certain times this is a conti-
nuously varying situation.
Three approaches to heat transfer will be discussed below: x3 x4
1. A simple method suitable for many estimations; Figure 1.60 One-dimensional conduction through a multi-layer plane
2. A more detailed appraisal of the field; wall
3. Comments on the use of computers.

1.7.2 Basic principles of heat transfer'"16 similar way to electrical resistances so that for a multilayer
plane surface there are a number of resistances in series
1.7.2.1 Conduction (Figure 1.60). Thus we can write
Fourier's law for conduction states Ax
Q' = (Tl - T4)/-&-
dT
@'= -k-
dx Cylindrical surfaces For tubes it is more convenient to eva-
luate heat transfer rates per unit length, and integration gives
The thermal conductivity k(Wm-'K-l) is a property of the
material which varies with temperature but for small tempera- Q' = (TI - T2)/- In r21rl = AT/- In r21rl
ture ranges is usually considered constant. Typical values are
shown in Table 1.8. With constant k Fourier's equation can be 2nk12 2ak12
integrated for four common situations. (see Figure 1.61), and in this case the thermal resistance is
in
m
K
Table 1.8 (2) w-1.
Q' = (Tl - 7.d x(
Substance Thermal conductivity, Wm-' K-', at 20" C For a multilayer tube, thermal resistances are added to give
Aluminium 204 In (ro"terlrln"er)
Iron 52 2ak
Facing brick 1.3 (see Figure 1.62).
Water 0.597
Air 0.026 (100 kPa) I. 7.2.2 Convection
Glass wool 0.04
The fundamental equation for convective heat transfer at a
solid-fluid interface is
Plane surfaces Integration gives
where 0 is the temperature difference between surface and
fluid. The surface heat transfer coefficient h (W m-' K-') is
not a property of the fluid or the surface but depends on the
(see Figure 1.59). The quantity (Axlk) is known as the thermal flow pattern, the fluid properties md the surface shape. The
resistance in m2 K W-l. Thermal resistances can be added in a coefficient has to be determined for each situation and can