9.5 Heat Transfer Mechanisms
333
The behavior of the material in conducting heat is incorporated into the number
κ H , called the heat conductivity, introduced as the proportionality constant in our
statements above about the conduction rate. Written quantitatively, the statements
reduce to:
δQ
δt
= κ H A
dT
dx
.
(9.7)
The heat conductivity of pure diamond is 2000 W/m-K. Cast iron is 55 W/m-K,
human skin 14 about 0.1 W/m-K, and air at STP 0.025 W/m-K.
The three-dimensional form of Eq. (9.7) in an isotropic medium is 15
J Q = −κ H ∇T ,
(9.8)
in which J Q is the flux of heat per unit area per unit time, δQ/δA δt, pointing in the
direction of the greatest downward change in temperature. Eq. (9.8) is referred to as
Fourier’s law of heat flow. 16
If there is a net heat flow into a system, the extra energy can cause an increase
in the kinetic and vibrational energy of the molecules. This extra energy may also
be sufficient to overcome any weak binding of the molecules. If so, that unbinding
will dominate the energy absorption. The most likely state of the molecules has
a balanced distribution of available energy. 17 Unbinding a macroscopic system of
molecules without breaking the molecules themselves is a phase change. When
water changes from ice to liquid, or liquid to gas, the water molecules held together
largely by hydrogen bonds are being unbound. Water takes 80 cal/g to change from
solid to liquid phase, and 540 cal/g to change from liquid to gas at 100 ◦ C. As the
temperature will not change during a thermodynamic phase change of a material, ice
water will remain at 0 ◦ C as long as the ice has not completely melted. When rain
turns to sleet or snow, it releases heat, slowing the drop in temperature that caused
the freezing.
If the residual energy deposited in a material by heat conduction does not do
work such as producing a phase or chemical change, the temperature of the material
increases. As we will see in the Chapter on statistical mechanics (Chap. 10), the
ability of a material to absorb heat depends on the variety of ways that energy can
be re-distributed among the available quantum states.
14 Skin tissue is a dynamic organ, with mechanisms to control heat loss by controlling the
circulation of blood, making the effective thickness of the skin between outside temperatures and
body temperatures variable.
15 In material such as a crystal or a mixture with oriented molecules, the flow of heat may not be
in the direction of the thermal gradient. In these cases, Eq. (9.8) is replaced by J Q = −κ H · ∇T ,
where κ H is a 3 × 3 positive-definite symmetric tensor.
16 Joseph Fourier, Théorie Analytique de Chalier [Paris] (1822).
17 This point will be made more exact when we expand on the subject of the statistical basis of
bioenergetics in Chap. 10.
333
The behavior of the material in conducting heat is incorporated into the number
κ H , called the heat conductivity, introduced as the proportionality constant in our
statements above about the conduction rate. Written quantitatively, the statements
reduce to:
δQ
δt
= κ H A
dT
dx
.
(9.7)
The heat conductivity of pure diamond is 2000 W/m-K. Cast iron is 55 W/m-K,
human skin 14 about 0.1 W/m-K, and air at STP 0.025 W/m-K.
The three-dimensional form of Eq. (9.7) in an isotropic medium is 15
J Q = −κ H ∇T ,
(9.8)
in which J Q is the flux of heat per unit area per unit time, δQ/δA δt, pointing in the
direction of the greatest downward change in temperature. Eq. (9.8) is referred to as
Fourier’s law of heat flow. 16
If there is a net heat flow into a system, the extra energy can cause an increase
in the kinetic and vibrational energy of the molecules. This extra energy may also
be sufficient to overcome any weak binding of the molecules. If so, that unbinding
will dominate the energy absorption. The most likely state of the molecules has
a balanced distribution of available energy. 17 Unbinding a macroscopic system of
molecules without breaking the molecules themselves is a phase change. When
water changes from ice to liquid, or liquid to gas, the water molecules held together
largely by hydrogen bonds are being unbound. Water takes 80 cal/g to change from
solid to liquid phase, and 540 cal/g to change from liquid to gas at 100 ◦ C. As the
temperature will not change during a thermodynamic phase change of a material, ice
water will remain at 0 ◦ C as long as the ice has not completely melted. When rain
turns to sleet or snow, it releases heat, slowing the drop in temperature that caused
the freezing.
If the residual energy deposited in a material by heat conduction does not do
work such as producing a phase or chemical change, the temperature of the material
increases. As we will see in the Chapter on statistical mechanics (Chap. 10), the
ability of a material to absorb heat depends on the variety of ways that energy can
be re-distributed among the available quantum states.
14 Skin tissue is a dynamic organ, with mechanisms to control heat loss by controlling the
circulation of blood, making the effective thickness of the skin between outside temperatures and
body temperatures variable.
15 In material such as a crystal or a mixture with oriented molecules, the flow of heat may not be
in the direction of the thermal gradient. In these cases, Eq. (9.8) is replaced by J Q = −κ H · ∇T ,
where κ H is a 3 × 3 positive-definite symmetric tensor.
16 Joseph Fourier, Théorie Analytique de Chalier [Paris] (1822).
17 This point will be made more exact when we expand on the subject of the statistical basis of
bioenergetics in Chap. 10.