Newton's Law: Its Application for
Expressing Heat Losses from Homeotherms
c. Richard Tracy
The applicability of Newton's law of
cooling as a means of expressing heat
losses from homeothermic animals has
geen discussed at length in recent literature ,(Kleiber 1961, McNab 1970,
Strunk 1971). Although this law has
been used extensively (Scholander et al.
1950, Tucker 1965, Dawson and
Schmidt-Nielson 1966, Herried and
Kessel 1967, McNab 1966, Henshaw
1968), the relationship between Newton's original law and the Newton's law
used by biologists today is at best
remote (Strunk 1971, Kleiber 1961).
Strunk (1971) has argued against the
use of Newton's law as a biological
model in favor of models that are based
on the physical principles of heat loss
from animals. McNab (1970) states that
simple biological variability makes impractical the use of the more mechanistically correct equations, and Newton's
law is perhaps a good pragmatic approach for expressing heat loss from
homeotherms. This paper reports a
theoretical examination of some similarities and differences between Newton's
law and the thermodynamic energy
balance as they have been used to
describe heat losses from homeothermic
animals in artificial and natural environments.
the core-shell model (King and Farner
1961) shown in Fig. 1.
If, for illustrative purposes, we consider a very simple hypothetical
homeotherm in a simplified environment in which the principal modes of
heat transfer from the animal are convection and radiation, the animal's energy budget can be written:
the homeotherm; C is a proportionality
constant usually referred to as the
thermal cond uctance ; Tc is the core
body temperature; and Ta is the ambient air temperature.
An alternative equation to Newton's
law which describes the steady-state
heat exchange of a homeotherm has
been given by Porter and Gates (1969).
This energy budget equation is simply
an expression of the first law of thermodynamics which can be stated: The
summed energy into and generated by a
homeotherm (in steady-state) must
equal the energy flowing from the
animal.
The energy budget, however, can be
expressed as a function of the animal
core temperature only if one recognizes
that heat transfer from the core of the
animal to the environment is a complex
function involving heat transfer across
insulating flesh, fur or feathers, as well
as mechanisms for heat transfer from
the animal surface to the environment.
This system of thermal conductors arranged "in series" can be visualized with
]
[
where Eex is the evaporative heat loss
from respiratory surfaces (occurs in all
homeotherms); the bracketed term is
the overall conductance from the
animal's core to the environment; K I is
the "shell" conductance; and K2 is the
surface conductance
[Achc+Ar€a
(T s2+Ta2) (Ts+T a)]. (Ach c) is the
To
SHELL
ENVIRONMENT
I
I
I
1
I
I
I
...
FUR OR
FEATHERS
I
I
I
NEWTON'S LAW AND THE
ENERGY BUDGET
The equation called Newton's law
used by biologists to express the rate of
heat production in homeotherms is
usually written:
(2)
~I
I
I
I
(PROPERTY
OF. ANIMAL)
(ANIMAL- ENVIRONMENT
INTERACTION)
I
I
1
I
I
where M is the net heat production of
I
1
I
14
The author is at the Department of Zoology,
Zoology Research Building, The University of
Wisconsin, Madison, Wisconsin 53706.
656
Kr (Tc - To)
~I
1
I
Fig. 1. Core-shell model. K1 the effective conductance through the insulating "shell." K2
the surface conductance. KT = the overall conductance.
=
=
BioScience Vol. 22 No. 11
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INTRODUCTION
30~--------..-------------,
20
1
10
20
AIR
I
30
TEMPERATURE. ·C
surface conductance due to convection,
where Ac is the surface area from which
convection occurs and h c is the surface
area from which convection occurs and
h c is the average convection coefficient
for the animal. (Areo(Ts2+Ta2)
(Ts-T a)] is the surface conductance due
to radiation, where Ar is the effective
surface area from which radiation occurs, e is the emissivity of the animal's
surface, 0 is the Stefan-Boltzmann constant, and Ts is the animal's surface
temperature. Implicitly assumed in this
presentation of the energy budget equation is that the radiant environment for
the animal is the same temperature as
the ambient air.
Equation (2) is of the same form as
Newton's law where the overall conductance between the animal's core and
its environment (the bracketed term) is
equivalent to Newton's proportionality
constant, C. However, this overall conductance is not usually a constant.
Contained within this term is the convective heat transfer coefficient which is
a variable functionally related to the
wind speed and air properties, as well as
the size, shape, and surface roughness of
the animal. This suggests that there is
neither a unique value for the overall
cond uctance of the animal, nor is the
overall conductance strictly a property
of the animal. The total conductance is
a function of parameters associated with
the environment, and we should expect
it to have different values under different wind speeds.
FIR
(DIMENSIONLESS)
.6 0
2 em ANIMAL
4 em ANI MAL
30
20
(f)
(f)
l4J
-J
Z
o
(f)
Z
l4J
~
o
I
oL=
0
0
0.4
08
.2
4
8 em ANIMAL
....
o
q
roO
12 0
6
0
08
16
2
I
4
16 em ANIMAL
q
roO
30
m
20
120
60
30
15
THERMAL CONDUCTANCE
The typical method for evaluating
Newton's conductance for any homeotherm is to determine the animal's
standard metabolic rates at a variety of
air temperatures below the animal's
November 1972
16
3.2
4.8 0
FUR THICKNESS, em
3.2
6.4
9.6
Fig. 3. Theoretical relationship between the Biot Numbers and fur thickness at several wind
speeds for theoretical animals of different torso diameters. The "shell" is considered to consist
of the fur alone. Fur conductivities were taken from Birkebak (1966). FIR is the ratio of fur
length to body dimension.
657
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Fig. 2. Predicted net metabolic rates as a
function of air temperature at three wind
speeds for E. speciosus.
meters to predict net metabolic rates for
the chipmunk, Eutamias speciosus, as a
function of air temperature at three
different wind speeds (Fig. 2). This
analysis shows that Newton's C varies
with the speed of the wind. Therefore,
one should not expect, for any animal, a
single value of conductance to exist.
Is there any importance to the fact
that Newton's conductances term is
neitlier a constant nor strictly a demonstration of some animal properties (e.g.
conductivities of fat, tissue, and fur, and
heat flow by circulating blood), but
rather a variable functionally related to
an animal's convective environment?
The answer lies in the use of Newton's
C. Many biologists have regarded Newton's C as a measurable property of a
homeotherm that has been thought to
be useful in comparing thermobiological
evolutionary adaptations within and be-
"lower critical temperature" (the
temperature below which a homeotherm appears to respond to a change in
air temperature solely in terms .of a
change in metabolic rate). The regression coefficient from a regression of the
standard metabolic rate on air temperature is then regarded as Newton's C. If,
for the same animal, respiratory water
loss constitutes a negligible heat loss,
this regression coefficient will also be
the overall thermal conductance. However, this regression coefficient may not
be solely a demonstration of some
property of the animal, but it is also
likely to reflect the interaction between
the animal's size, proportions, and the
wind regime in the metabolic chamber.
Heller and Gates (1971) have evaluated the parameters necessary for the
energy budget equations of four species
of chipmunks. I have used these para-
greater, the overall conductance will be
very nearly equal to the internal or
"shell" conductance. This means that
the overall conductance and Newton's C
would be very nearly independent of
wind speed. As an example, consider an
animal whose external conductance, K 2
is 0.01 cal min . a C, and whose internal
conductance, K1, is 0.001 cal min· "C.
This animal's Biot Number is
0.01/0.001 or 10, and its overall conductance is calculated as K I· K2/(K 1 +
K2) or 0.00091. An assumption that
this animal's overall conductance could
be represented by its internal or "shell"
conductance would be in error by only
9%.
To examine the likelihood that certain environments might exist where
homeotherms could have Biot Numbers
equal to or greater than ten, I have
examined the energy balances of four
theoretical mammals varying in torso
diameter from 2 cm (mouse size) to 16
cm (wolf size). The effect of different
fur thicknesses and different wind
speeds on the Biot Numbers of each
theoretical animal is given in Fig. 3. The
fur layer was assumed to be totally
responsible for the "shell" conductance,
and the conductivity of the various fur
lengths was taken from Birkebak
(1966). The analysis shows that only
when these theoretical mammals have
very long fur thicknesses and/or when
they are subjected to very high wind
speeds, do they ever have Biot Numbers
0.012 5 r - - - - - - - - - - - - - - - - - - - - - - - - - - - · - - - - .
• = Slopes of [(M-E' vs TAJ from fig. 2
P
o = Conductance of "shell" only
o
= Slope of (M vs TA) in metabolic chamber
I
0.0100
I
I
I
I
I
I
I
I
u
o
c
E
speCIOSu5
E
N
E
u
<,
o
u
eq ual to or greater than ten. For example, a mouse-sized animal in a 240
cm/sec (5 mph) wind speed environment would need to have a 1 em thick
fur layer to have a Biot Number greater
than ten. This analysis implies that
under most natural circumstances
homeotherms will most likely have Biot
Numbers less than ten or that the
overall conductances or Newton's C will
rarely be constant or strictly reflect
properties of the animals.
To test these conclusions with a live
homeotherm, I analyzed the calculated
overall conductance of E. speciosus as a
function of wind speed. The slopes of
the three lines in Fig, 2 were used as
"data" to obtain the relationship between the chipmunk's overall conductance and wind speed (Fig. 4). The
conductance of the chipmunk's insulating shell, Kj , (Heller and Gates, 1970)
falls on the conductance curve at a wind
speed in excess of 2500 cm/sec (50
mph). This implies that the chipmunk
would have to be in a 50 mph gale
before the overall conductance would
solely reflect properties of the animal.
Another datum that has been used as
a point of reference in Fig. 4 was the
slope obtained by Heller and Gates
(1970) of the relationship between E.
speciosus' metabolic rate and air temperature in their metabolic chamber.
This slope, or overall conductance (assuming respiratory evaporative water
loss to be minimal), falls on the conductance curve at a wind speed of
slightly over 30 cm/sec (-3/4 mph).
Three quarters of a mile per hour then,
is a crude approximation of the wind
speed in the metabolic chamber used by
Heller and Gates. This wind speed is
important only because when one considers the variety of metabolic chambers
found in different laboratories, it seems
likely that for chipmunks of a size
similar to those studied by Heller and
Gates, that wind speeds of from at least
one half to triple that in Heller's
chamber could be expected. Thus, contained in any Newton's C determined
from the slope of a metabolic rate curve
is a component that reflects the wind
speed environment in the chamber.
CONCLUSIONS
10
100
1000
WIND SPEED, em/sec
Fig. 4. Theoretical relationship between the overall conductance and wind speed for E.
speciosus. The line was calculated from the animal's energy balance using parameters evaluated
by Heller and Gates (1970). Conductance of the "shell" and slope of M vs Ta were also taken
from Heller and Gates.
658
The results of the various analyses
presented in this paper indicate that any
study of thermobiological evolutionary
adaptations of homeotherms which considers only Newton's C and does not
separate the effects of animal size and
wind speed on the overall animal conductance, perhaps does injustice to the
theme of the study, as well as leave
room for technical "error." For
example, it has been argued that populations within some species have, for
BioScience Vol. 22 No. 11
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tween species living in different thermal
environments (Scholander et al. 1950,
Tucker 1965, Dawson and SchmidtNielson 1966, Herried and Kessel 1962,
McNab 1966, Henshaw 1968, King and
Farner 1961, Bartholomew 1968).
Another application has been the use of
Newton's law and a constant value of
Newton's C as a predictive model of the
energy budgets of wild mammals in
natural environments (McNab 1963).
Both of these applications of Newton's
law' have depended on a constant value
of Newton's C.
We must, therefore, ask whether or
not environments exist where a homeotherm's overall conductance is constant,
and thus, not a function of the convective environment. Figure 1 reminds
us that the conductance across the
"shell" of a homeotherm is by processes
that strictly reflect properties of the
animal; whereas, conductance from the
animal's surface also involves environmental variables. We need to know
whether environmental conditions exist
under which the animal's surface conductance is of negligible magnitude
compared to the "shell" conductance,
or whether the overall conductance can
ever be approximated by the "shell"
conductance.
For questions such as these, we can
consider the ratio of the external or
surface conductance to the internal or
"shell" conductance. If this ratio,
termed the Biot Number, is ten or
ACKNOWLEDGMENTS
I thank W. P. Porter for his encouragement, constructive criticisms ,
and financial support during the prepa rat ion of this manuscript. Thanks also
go to B. A. Wunder, J . W. Mitchell, H.
C. Heller, T. H. Strunk, D. M. Gates, W.
A. Calder, and M. F. Roberts for their
particularly helpful comments and reviews of the manuscript. I also thank A.
Chambers , J . Curtin, C. Hughes, and
J. Dallman and D. Chandler for
their help on the technical preparation
of the paper. Partial support came from
the Wisconsin Alumni Research
Foundation and NSF grants to W. P.
Porter.
REFERENCES
Bartholomew, G. A. 1968. Animal Function:
Principles and Adaptations. Body temper ature and energy Metabolism. In: Malcolm S.
Gordon (ed.) MacMillan, New York.
Birkebak, R. C. 1966. Heat transfer in biological systems. Int. Rev. Gen. Expt, Zool., 2:
269-344.
Dawson, T. J. and K. Schmidt-Nielson. 1966.
Effect of thermal conductance of water
economy in the antelope jack-rabbit , Lepus
alieni. J. Cell. Physiol. 67 : 463-472.
Heller, H. C. and D. M. Gates. 1971. Altitudinal zonation of chipmunks iEntamias) : Energy budgets. Ecology 52(3) :
424-433 .
Henshaw, R. E. 1968. Thermoregulation
during hibernation: Application of Newton's Law of Cooling. J. Theoretical Bioi.
20: 79-90.
November 1972
Herried, C. F. II, and B. Kessel. 1967.
Thermal conductance in birds and mammals. Compo Biochem. Physiol. 21:
405-414.
Jackson, H. W. 1959. Introduction to Electric
Circuits. Prentice-Hall, New Jersey.
Kendeigh, S. C. 1969. Tolerance of cold and
Bergmann's Rule. Auk. 86: 13-25.
King, J. R. and D. S. Farner. 1961. Energy
metabolism, thermoregulation, and body
temperature. In: A. J. Marshall (ed.) Biology
and Comparative Physiology of Birds.
Academic Press, London.
Kleiber, M. 1961. The Fire of Life. John
Wiley, New York.
McNab, B. K. 1963. A model of the energy
budget of a wild mouse. Ecology 44(3) :
521-532.
_ _ _ 1966. The metabolism of fossorial
rodents: A study of convergence. Ecology
47: 712-733.
____ 1970. Body weight and the energetics of temperature regulation . J. Expt.
Bioi. 53: 329-348.
Porter , W. P. and D. M. Gates. 1969. Thermo. dynamic equilibria between animals and
their enviromnents. Ecol. Monogr. 39:
245-270.
Scholander , P. F., V. Walters, R. Hock, and L.
Irving. 1950. Heat regulation in some arctic
and tropical animals and birds. Bioi. Bull.
99: 236-258.
Strunk, T. H. 1971. Heat loss from a Newtonian animal. J. Theoret. Bioi. 33: 35-61.
Tucker, V. A. 1965. Oxygen consumption,
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California Picket Mouse Perognathus californicus. J. Cell Comp. Physiol. 65: 393-404.
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temperature tolerance, adapted to extreme environments with different body
sizes (see Kendeigh 1969 for discussion
of Bergman's Rule) . Quantitative
analyses of hypotheses such as this will
require the separation of homeotherms'
overall conductances into the component conductances.
Perhaps the most critical pitfall in
the use of Newton's law comes from
any attempt to use Newton's proportionality constant, C, in a prediction of
the heat losses from homeotherms under natural conditions (McNab 1963).
Since the overall conductance of some
homeotherms may vary with wind speed
by as much as 100% (see Fig. 4), we
should expect models that use a constant value of overall conductance to
predict the heat losses from homeotherms in natural environments to be of
extremely limited value. Any such
model that will accurately predict heat
losses from homeotherms will have to
incorporate the functional relationship
between environmental wind speed and
the overall conductance between
homeotherms and their environments
(Porter and Gates 1969) .