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between conductivity (rate of spatial passage of heat
energy) and thermal energy storage (product of specific
heat capacity per unit mass and density, that is, specific
heat per unit volume). Thermal diffusivity gives an idea of
how long a material takes to respond to imposed tempera-
ture changes, for example, air has a rapid response and
mantle rock a slow one. This leads to a useful concept con-
cerning the characteristic time it takes for a system that has
been heated up to return to thermal equilibrium. Any
system has a characteristic length, l , across which the heat
energy must be transferred. This might be the thickness of
a lava flow or dyke, the whole Earth's crust, an ocean cur-
rent, or air mass. The conductive time constant,
, is then
given by l 2 /
.
4.18.4 Molecular diffusion of heat and
concentration in fluids
In fluids it is the net transport of individual molecules down
the gradient of temperature or concentration that is respon-
sible for the transfer; the process is known as molecular
diffusion . As before, the process acts from areas of high to
low temperature or concentration so as to reduce gradients
and equalize the overall value (Fig. 4.148). For temperature
the rate of transfer depends upon the thermal conductivity,
as for solids, but the process now occurs by collisions
between molecules in net motion, the exact rate depending
upon the molecular speed of a particular liquid or gas at par-
ticular temperatures. For the case of concentration the over-
all rate depends on both the concentration gradient and
upon molecular collision frequency and is expressed as a dif-
fusion coefficient . The rate of molecular diffusion in gases is
rapid, reflecting the high mean molecular speeds in these
substances, of the order several hundred meters per second.
The rapidity of the process is best illustrated by the passage
of smell in the atmosphere. By way of contrast the rate of
molecular diffusion in liquids is extremely slow.
Heat axis
HIGH
LOW
For 1D variation of heat at any instant
the flux, Q , goes from high to low
temperature.
T
T + δ T
Q = heat flux
k = thermal conductivity
Q
Q = -k δ T/ δ x
This is the heat conduction equation
Applies when conditions do not
change with time.
x
x + δ x
x -axis
Fig. 4.147
ID heat conduction.
(a)
(b)
concentration axis
n
n + δ n
δ
n
For 1D variation of molecular
concentration at any instant
the flux J , goes from high to low
concentration
HIGH
LOW
HIGH
LOW
n
n + δ n
δ
n/
δ
t = 0
J in
J out
J in = J out
n = no mols./unit vol. = conc.
J = - D δ n/ δ x
J = no particles crossing
unit area per sec. in direction >x
(c)
J
D = a diffusion coefficient
measuring the rate of diffusion
HIGH
LOW
/
δ n/ δ t = 0
J in = J out
J x
J x + dx
J = -D δ n/ δ x
/
D δ 2 n x 2 = δ n t
This is Fick´s law of diffusion.
Applies when conditions do not
change with time.
x
x + δ x
x
x + δ x
Particles can accumulate or be lost;
there may be a gradient of J across x
x-axis
Fig. 4.148 Molecular diffusion occurs in liquids and gases as translation of molecules from high concentration/temperature areas to low
concentration/temperature areas so as to eliminate gradients. The rate of diffusion is rapid for gases and slow for liquids (a) Fick's law of 1D
diffusion, (b) Derivation: Steady state diffusion (time independent), and (c) time variant diffusion (time/space dependent).
 
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