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9 Bioenergetics
If the heat flow into a material only causes a redistribution of the molecules
among the possible energy states, and does not change the energy states themselves,
we define the ability of the material to take in heat while changing temperature one
degree to be its heat capacity. The specific 18 heat is the heat capacity c p per unit
mass of material, held at constant pressure:
Q = c p mmT .
(9.9)
Water has a relatively high specific heat, as far as common substances go. Because
we contain a lot of water (about 70% of our body mass), our tissue correspondingly
has a relatively high specific heat. For example, we lose more heat by conduction
from our skin for a given drop in temperature than dry wood under the same
conditions. In our environment, air with high humidity becomes far more difficult
to change in temperature from night to day than dry air. It is the moisture in the air
which requires more heat to change the air temperature. In this respect, Arizona,
with very dry air most of the year, has much larger variations in temperature from
night to day than Washington, D.C., which often has high humidity, particularly in
the summer.
If the material being heated is a layer within a thicker object, we can analyze the
heat transport in the material by first imaging a division of the material into thin
layers. Start with a thin layer of thickness δx, with adjacent layers left and right.
Take the layer on the left hotter than the middle layer, and the middle hotter than
the one on the right, making a flow of heat from left to right. Any energy left in the
middle layer will then be accumulating at a rate
δQ L
dt
−
δQ R
dt
= c p δm
dT
dt
.
(9.10)
From Eq. (9.7), the rates in and out on the left can be written as shown below, and
we use δm = ρAdx to write
− κ H A
∂T
∂x
L
−
∂T
∂x
R
= c p ρAδx
∂T
∂t
,
(9.11)
where ρ is the density of the material. The minus sign on the left takes care of the
fact that each of the two gradients of the temperature are negative if heat is flowing
to the right. Now the parenthetical factor on left-hand side, divided by δx, is the
second derivative. There results the rather famous ‘heat equation’:
18 Formally, the modifier ‘specific’ in front of a measurable quantity means the measure for the
given material divided by the measure for a standard material, usually water. The calorie was
defined using water, so the value of the specific heat of a substance is the heat capacity divided by
approximately one calorie per degree. Rather than dividing, most researchers and references report
the specific heat in calories per gram degree.
9 Bioenergetics
If the heat flow into a material only causes a redistribution of the molecules
among the possible energy states, and does not change the energy states themselves,
we define the ability of the material to take in heat while changing temperature one
degree to be its heat capacity. The specific 18 heat is the heat capacity c p per unit
mass of material, held at constant pressure:
Q = c p mmT .
(9.9)
Water has a relatively high specific heat, as far as common substances go. Because
we contain a lot of water (about 70% of our body mass), our tissue correspondingly
has a relatively high specific heat. For example, we lose more heat by conduction
from our skin for a given drop in temperature than dry wood under the same
conditions. In our environment, air with high humidity becomes far more difficult
to change in temperature from night to day than dry air. It is the moisture in the air
which requires more heat to change the air temperature. In this respect, Arizona,
with very dry air most of the year, has much larger variations in temperature from
night to day than Washington, D.C., which often has high humidity, particularly in
the summer.
If the material being heated is a layer within a thicker object, we can analyze the
heat transport in the material by first imaging a division of the material into thin
layers. Start with a thin layer of thickness δx, with adjacent layers left and right.
Take the layer on the left hotter than the middle layer, and the middle hotter than
the one on the right, making a flow of heat from left to right. Any energy left in the
middle layer will then be accumulating at a rate
δQ L
dt
−
δQ R
dt
= c p δm
dT
dt
.
(9.10)
From Eq. (9.7), the rates in and out on the left can be written as shown below, and
we use δm = ρAdx to write
− κ H A
∂T
∂x
L
−
∂T
∂x
R
= c p ρAδx
∂T
∂t
,
(9.11)
where ρ is the density of the material. The minus sign on the left takes care of the
fact that each of the two gradients of the temperature are negative if heat is flowing
to the right. Now the parenthetical factor on left-hand side, divided by δx, is the
second derivative. There results the rather famous ‘heat equation’:
18 Formally, the modifier ‘specific’ in front of a measurable quantity means the measure for the
given material divided by the measure for a standard material, usually water. The calorie was
defined using water, so the value of the specific heat of a substance is the heat capacity divided by
approximately one calorie per degree. Rather than dividing, most researchers and references report
the specific heat in calories per gram degree.