Page 253 - Fluid mechanics, heat transfer, and mass transfer

CONDUCTION HEAT TRANSFER 233

the equation of the form, conductive heat transfer rate; k is the thermal con-
ductivity; a is the thermal diffusivity, k/rc p , where r
k ¼ a þ bT; ð8:3Þ is the density and c p is the heat capacity; and t is the
time.
where a and b are constants.
. Give an example of heat transfer by conduction with
& dT/dx is temperature gradient.
internal heat generation.
& The negative sign on the right-hand side signifies that
& Electrical resistance heaters.
conduction is in the direction of decreasing temper-
. Define steady-state heat conduction.
ature. It indicates that thermal energy flows from hot
& Steady-state conduction is said to exist when the
regions to cold regions.
temperature at all locations in a substance is constant
& k, thermal conductivity, is the rate of thermal energy
with time, as in the case of heat flow through a
transfer per unit area and per unit temperature gra-
uniform wall.

dient. Units are W/(m- C).
& In other words, temperature is a function of position
& A is the area of the surface that is perpendicular to the
only and rate of heat transfer at any point is constant.
flow direction for heat energy.
. What happens in unsteady-state heat conduction?
. Define similar laws for (i) momentum transfer, (ii) mass
& In unsteady-state heat conduction, temperature var-
transfer, and (iii) electrical energy transfer.
ies with both time and location.
(i) Momentum Transfer:
& Momentum transfer is described by Newton’s . Give examples of materials having (i) high thermal
law that relates shear stress to velocity gradient, conductivity and (ii) low thermal conductivity.
employing a proportionality constant called (i) Diamond (900–2320), silver (429–415), gold
viscosity. (318), and copper (401).
(ii) Air (0.025), polyurethane foam (0.026), and glass
Shear stress ¼ m du=dy: ð1:2Þ

fiber (0.043) (values in brackets are in W/(m- C)).
(ii) Mass Transfer: . “Thermal conductivities of metals can significantly be
& Fick’s first law relates flux of a component to its affected by the presence of impurities in them.” True/
composition gradient, employing a constant of False?
proportionality called diffusivity. & True. Impurities in metals can give rise to variations
& Rate of mass transfer is in thermal conductivity by as much as 50–75%.
N A ¼ D AB dc A =dx; ð8:4Þ . “Thermal conductivity of an alloy is usually much lower
than that of either metal of which it is composed.” True/
whereD AB istheproportionalityconstant,desig- False? Give examples.
nated as diffusivity, which is a measure of ability
& True. For example, k for copper is 401, for nickel is
of transfer of mass.
91; for constantan (55% Cu þ 45% Ni) is 23, and for
& Mass transfer is in the direction decreasing
aluminium is 237; and for bronze (90% Cu þ 10%
concentration, which explains the negative sign. Al) is 52.
. “Ice has a thermal conductivity much higher than
(iii) Electrical Energy Transfer: water.” True/False?
& Ohm’s law is expressed as
& True.
I ¼ V=R; ð8:5Þ . What is the effect of temperature on thermal conduc-
tivity of solids?
where I is the current in amperes, V is the & The conductivity of solids changes mildly with tem-
voltage, and R is the resistance. perature except at very low temperatures where it can
. Write three-dimensional conduction equation. acquireverylargevalues. Forinstance, pure copper at

& Three-dimensional conduction equation is 10K has a conductivity of about 20,000 W/(m- C),
whereas its conductivity at normal temperatures is

2 2 2 2 2 2 401 W/(m- C).
q T=qx þ q T=qy þ q T=qz þ
. How does thermal conductivity vary with temperature
q=k ¼ð1=aÞðqT=qtÞ; ð8:6Þ
for (i) liquids and (ii) gases?
where T is the absolute temperature; x, y, and z are the & For most liquids, k is lower than that for solids,

directions of flow in the three dimensions; q is the typical values being about 0.17 W/(m- C).