Page 53 - Mechanical Engineers Reference Book

1/42 Mechanical engineering principles
For a tubular surface The only simple radiation problem that can be solved with
the simple approach above is that of a grey body in large
Q = U'l(Tf, - Tf2) where 1 is the length surroundings. The word 'large' implies that radiation not
The situations in which this technique is commonly used incident on the grey body which will be incident on the
include walls of buildings, double glazing and hot water pipes, surroundings and will therefore be reflected will not be
and the method is also used in heat exchanger design. It re-incident on the grey body. Thus the surroundings are
should suffice for simple calculations provided suitable values effectively black. (This might be true with a linear size factor
of the surface heat transfer coefficients for convection can be greater than 10.) In this simple case it can be shown that the
obtained. Values of U can be found for buildings, but care heat transfer rate is
should be taken to ensure that the quoted figures include
convection effects, as some only account for conduction and
will need the effect of convection to be added.17 If the where E is the emissivity of the body, A is the area of the body,
temperature difference is not constant then a mean value Tis the temperature of the body in K and T, is the tempera-
should be used. A suitable equation for a mean can be found ture of the large (black) surroundings in K.
in the heat exchanger section which follows.

1.7.2.4 Radiation
Heat transfer due to radiation effects is of increasing impor-
tance as temperature increases because the rate of energy
emitted by an ideal black body is given by the Ste-
fan-Boltzmann law, in which the absolute temperature (Cel-
sius + 273) is raised to the fourth power:
8'- a74
-
b
where a is the Stefan-Boltzmann constant 5.67 X
W m-* K-4 and the subscript b refers to the ideal black body.
Real bodies emit less radiation and the monochromatic Figure 1.65 A small grey body in large (black) surroundings
&A = [$I
emissivity is defined by
T I. 7.2.5 Simple transient problems
The value of E varies with A and T because real bodies are If a body is being cooled or heated by convection or radiation
selective emitters, but for simple calculations it is often and the thermal conductivity is large so that the rapid heat
assumed that emissivity is constant. The calculations asso- transfer rates within the body enable it to be assumed that the
ciated with this assumption are based on grey body theory, for body temperature distribution is uniform, then the situation is
which the rate of energy emission is given by known as a lumped capacity system. For such a system the
complex methods of transient heat transfer are not required
&; = EO74
and a simple energy balance equation may be drawn up and
Some emissivity values are shown in Table 1.10, but it must integrated. The most common case is quenching, a convective
be clearly understood that with such wide-ranging values it boundary problem for which in time dt a small heat transfer
would be unwise to estimate unknown emissivities, and mea- SQ occurs when the body temperature changes from T by an
surements would need to be made unless suitable data could amount dT (Figure 1.66). Thus
be found.
Radiation incident on a body may be absorbed, reflected or SQ = pc,VdT = -hA(T - Tf)
transmitted. Thus we write a + p + T = 1 where a, p and r where Ti is the fluid temperature, A the body surface area, V
are the absorbtivity, reflectivity and transmissivity, respect- the body volume, p the body density, cp the body specific heat
ively. Ideal black bodies absorb all incident radiation but real and h is the surface heat transfer coefficient. Integration gives
bodies do not. Gases are often assumed to transmit all
radiation, but this is not always true, particularly with hydro-
carbon combustion products. Solids have a transmissivity of
zero. With these simple ideas it is necessary to know the values
of only a and p. It can be shown that a grey body has where 00 is the initial temperature difference between fluid
absorbtivity equal to emissivity, (Y = E. Thus, provided the and body and 0 is the temperature difference at any future
transmissivity is zero, a knowledge of the grey body emissivity time t. The quantity (pc,V/hA) may be regarded as the time
enables reflectivity to be determined, since constant of the system.
p=l--E

Table 1 .I 0
Fluid ,--Body
~
Material Emissivity Temperature, T, Volume, V
Surface heat Density, p
Rusted iron plate 0.69 at 19°C transfer Area, A
Molten iron 0.29 at 1300-1400°C coefficient, h Specific heat, C,
Polished brass 0.06 at 100°C 0 = (T- T,) Temperature, T
Asbestos board 0.96 at 21°C
Figure 1.66 A lumped-capacity system with convection