10.2 Transport Processes
445
at a steady speed. Since Newton’s second law relates force and the rate of change of
velocity (and thus the rate of change of momentum) the same information is contained
in either choice.
Driving Forces and Linear Laws
To a first approximation each rate variable depends on a single state variable that we
call the driving force for that variable. The temperature gradient is the driving force
for heat conduction, the concentration gradient of substance i is the driving force for
diffusion of that substance, and the rate of shear is the driving force for viscous flow. The
empirical formulas that describe these phenomena specify that the rate of the process is
directly proportional to the driving force, so that these formulas are called linear laws.
In a thorough treatment of irreversible thermodynamics, more carefully defined driving
forces are used and the possibility is included that the driving force for one transport
process can make a contribution to the rate of another transport process. An example
is thermal diffusion, in which a diffusion flow is driven by a temperature gradient. We
will not discuss this and other cross-effects, although there is a considerable literature
involving them.
Fourier’s Law of Heat Conduction
The flow of heat in a solid, liquid, or gas system is described by a linear law known as
Fourier’s law of heat conduction:
q −κ∇T (Fourier’s law)
(10.2-1)
where κ is a coefficient called the thermal conductivity. If the temperature varies only
in the z direction, this equation is
q z −κ
∂T
∂z
(10.2-2)
Fourier’s law is named for Jean Baptiste
Joseph Fourier, 1768–1830, a famous
French mathematician and physicist
who also invented Fourier series and
Fourier transforms.
Fourier’s law is called a linear law because the rate is directly proportional to
(linearly dependent on) the gradient of the temperature, ∇T , which is the driving
force. The thermal conductivity κ can depend on the composition, the temperature,
the pressure, and on the identities of the substances present, but does not depend on
the temperature gradient. Fourier’s law is also called a phenomenological law, which
means that it is not derived theoretically but simply describes a phenomenon. Fourier’s
law holds quite accurately for gases, liquids, and solids. Table A.16 in Appendix A
gives the values of the thermal conductivity for several substances.
E X A M P L E 10.2
A cubical cell 0.100 m on a side is filled with benzene. The top surface is maintained at
25.0 ◦ C and the bottom surface is maintained at 15.0 ◦ C. After some time, the system will
reach a steady state in which the state of the system does not change with time although heat
is flowing. Calculate the amount of heat flowing through the benzene per hour after a steady
state is achieved, neglecting convection.
445
at a steady speed. Since Newton’s second law relates force and the rate of change of
velocity (and thus the rate of change of momentum) the same information is contained
in either choice.
Driving Forces and Linear Laws
To a first approximation each rate variable depends on a single state variable that we
call the driving force for that variable. The temperature gradient is the driving force
for heat conduction, the concentration gradient of substance i is the driving force for
diffusion of that substance, and the rate of shear is the driving force for viscous flow. The
empirical formulas that describe these phenomena specify that the rate of the process is
directly proportional to the driving force, so that these formulas are called linear laws.
In a thorough treatment of irreversible thermodynamics, more carefully defined driving
forces are used and the possibility is included that the driving force for one transport
process can make a contribution to the rate of another transport process. An example
is thermal diffusion, in which a diffusion flow is driven by a temperature gradient. We
will not discuss this and other cross-effects, although there is a considerable literature
involving them.
Fourier’s Law of Heat Conduction
The flow of heat in a solid, liquid, or gas system is described by a linear law known as
Fourier’s law of heat conduction:
q −κ∇T (Fourier’s law)
(10.2-1)
where κ is a coefficient called the thermal conductivity. If the temperature varies only
in the z direction, this equation is
q z −κ
∂T
∂z
(10.2-2)
Fourier’s law is named for Jean Baptiste
Joseph Fourier, 1768–1830, a famous
French mathematician and physicist
who also invented Fourier series and
Fourier transforms.
Fourier’s law is called a linear law because the rate is directly proportional to
(linearly dependent on) the gradient of the temperature, ∇T , which is the driving
force. The thermal conductivity κ can depend on the composition, the temperature,
the pressure, and on the identities of the substances present, but does not depend on
the temperature gradient. Fourier’s law is also called a phenomenological law, which
means that it is not derived theoretically but simply describes a phenomenon. Fourier’s
law holds quite accurately for gases, liquids, and solids. Table A.16 in Appendix A
gives the values of the thermal conductivity for several substances.
E X A M P L E 10.2
A cubical cell 0.100 m on a side is filled with benzene. The top surface is maintained at
25.0 ◦ C and the bottom surface is maintained at 15.0 ◦ C. After some time, the system will
reach a steady state in which the state of the system does not change with time although heat
is flowing. Calculate the amount of heat flowing through the benzene per hour after a steady
state is achieved, neglecting convection.