Heat loss from a Newtonian animal

J. theor. Biol. (1971) 33, 35-61

Heat Loss from a Newtonian Animal THOMASH. STRUNK Biophysical Laboratory, Harvard Medical School, 25 Shattuck Street, Boston, Mass. 02115, U.S.A. (Received 20 November 1970)

The conditions necessary for true Newtonian cooling to occur are unrealistic for an animal, but Newton’s law of cooling is used overwhelmingly in the physiological literature as the sole description of heat loss. For purposes of developing fundamental relationships between the animal and its environment, these constraints of Newtonian cooling are imposed upon an animal and the process is discussed within the framework of Fourier’s law and the general heat equation. The arguments are related to Newton’s law and to some physiological observations. The dependence of the cooling rate and cumulative heat loss on the animal’s metabolic and circulatory rates, area-volume ratio, surface conductance, mass, heat capacity, insulation thickness and conductivity and on physical factors of the environment is given explicitly. 1. Introduction The traditional treatment of heat loss and thermoregulation in animals (Crosbie, Hardy & Fessenden, 1963; Davson, 1964; Hardy, 1949; Irving, 1964; King & Farner, 1964) employs simple empirical expressions that approximately fit the experimental data. Despite these significant advantages, this approach offers little insight into the basic phenomena of thermal regulation. Newton’s law of cooling, for example, is an empirical expression describing heat loss under the simultaneous conditions of conduction, forced convection and radiation. It does not describe any single process nor does it illustrate the parameters controlling the process. The use of Newton’s law of cooling in the physiological literature seldom bears much resemblance to its current function in engineering heat transfer. There are even instances (Hart, 1963 ; Irving, 1964) where it has been used to describe conduction within an animal’s body. The preponderant use of Newton’s law, exclusively, to describe all modes of heat exchange in animals is an unfortunate practice for it obscures important processes. In conjunction with the heat equation, however, its true value may be realized. 35

36

T. fi.

STRUNK

2. Newtonian Conditions In 1701 Newton published, anonymously, his observations on the timetemperature relationship for cooling of a solid, red-hot, uninsulated block of iron. Newton’s law of cooling was stated in his publication as follows: “With the thermometer I found the measure of all heats up to that at which lead melts and on the hot iron I found the measure of the other heats. For the heat which the iron communicates in a given time to cold bodies which are near it, that is, the heat which the iron loses in a given time, is proportional to the whole heat of the iron. And so if the times of cooling are taken as equal, the heats will be in a geometrical progression and consequently can easily be found with a table of logarithms.” (Scott, 1967.) His observations may be expressed as

- deK 0 dt ’ with 6 equal to the excess of temperature surroundings, or

of the iron over that of the

where c is a constant of proportionality. The important point here is that for the red-hot block the “bulk” temperature of the block and its surface temperature were equal, to a very good approximation. Therefore Newton’s law relating the rate of cooling to the difference in surface temperature and environmental temperature would still conform to observation if one chose to use the “bulk” temperature of the iron rather than the surface temperature. Newton made no connection to the area or nature of the surface of the cooling block of iron. Apparently Newton’s concepts of heat and temperature were no clearer than anyone else’s at that time, for although he measured temperature he referred to it as heat. To apply Newton’s proportionality to heat requires extra(post)-Newtonian observations and several specific assumptions. The necessary observations stem from the fact that prior to the discoveries of Joseph Black in the years 1759 to 1762, the distinction between the concepts of heat and degree of hotness (temperature) had not been clearly made. George Martine (1701-1741) discovered that with a constant supply of heat the quantity of heat absorbed by a substance is directly proportional to the time of exposure. The time required to heat substances to the same temperature did not agree with the prevalent volume or weight hypotheses.

HEAT

LOSS

FROM

A

NEWTONIAN

ANIMAL

37

Black understood this to mean, for instance, that less heat was required to warm quicksilver than to warm water to the same temperature. As Black said : “The quicksilver therefore may be said to have less capacity for the matter of heat.” Furthermore, he proposed that the time of heating (and therefore the amount of heat added) is directly proportional to both weight and rise of temperature. H cc wAT (3 or H = SWAT, where His the amount of heat, iv is the weight, ATis the temperature rise and S, the constant of proportionality, is different for each different substance and was called the “capacity for heat” (Allen & Maxwell, 1948; Roller, 1961). Subsequent use of the method of cooling to determine specific heats by Regnault in 1861 illustrated its inadequacy for solids. This was due primarily to lack of a uniform temperature throughout the sample so that the thermometer registered a temperature different from the temperature of the radiating surface. In addition, the area and nature of the surface must be identical for the method of cooling (by radiation heat loss only) to yield reliable specific heats and this is much more difficult to achieve with powdered solids than with liquids. Newton’s law was formulated under conditions of conductive, convective and radiative heat loss and will hold fairly well for pure radiation losses only if 6 is small. The later work of Rumford, Davy and Joule made possible the thermodynamic formulation that, together with Newton’s observations and certain assumptions, yields the familiar, contemporary Newtonian law of cooling. The enthalpy is defined for a closed system as H = E+pV.

Then the first law of thermodynamics, dE = dQ-pdl’, may be rewritten as dQ = dH - Vdp, whence, at constant pressure

(3)

(4) Differentiating

AQ = j C,dT. (4) with aspect to time

dQ =d;JC,dT d dt

(5)

38

T. 11. STRUNK

and assuming a constant heat capacity gives

dQvC dt

fT ’ dt’

Using Newton’s relationship for dT/dt, the total heat lost from the system to the surroundings across the surface area A of the system takes the usual form (e.g. Grober & Erk, 1961; Schneider, 1955) of the cooling law dQidt = - c,ctl = h, AO.

This is also encountered in a less presumptuous dQ = h:O dA dt,

(7) expression

where dQ is the quantity of heat passing from an area dA to the surroundings in the time dt and hf is the local coefficient. All further reference to Newton’s law of cooling will mean equation (7) and not Newton’s original proportionality. Newton’s law of cooling then is a linear relationship between the flux of heat energy across the boundary of an object and the difference in “bulk” temperatures between the object and its surroundings. In applying Newton’s law as the sole description of heat loss one must assume (i) that such bulk temperatures do exist, that they are uniform and represent closely the real temperature throughout the object and throughout the surroundings. Uniformity of these temperatures further requires (ii) that the conductivity of the object be extremely large relative to the surroundings. Furthermore, (iii) there can be no heat sources or sinks within the object, thus (iv) only transient-state systems may be considered and (v) the heat capacity must be constant. If conditions (i) and (ii) are not met, the temperature at any point within the body and the surface temperature are no longer equal and Newton’s law will then apply to the heat transfer from the surface of the body to the surroundings (e.g. the boundary layer heat transfer). Condition (iii) will always obtain because neither Newton’s experiments nor the development of equation (7) had any relationship to sources or sinks. If 8 # 0, condition (iv) follows necessarily from (iii). The difficulties of a truly analytical treatment of heat exchange in animals can be appreciated by examination of a few of the standard engineering models (Carslaw & Jaeger, 1959; Kays, 1966) with the concurrent realization that much of the physiology one might hope to incorporate is not completely understood. Conversely, the purely empirical approach of lumping everything in the conductance h obscures mechanisms and important physiological and physical variables. A compromise can be rewarding in its ability to predict the behavior of the system as well as indicate the measurements needed and their relative importance. Examples of this technique are the efforts of Wissler (1961, 1963, 1966) and Crosbie et al. (1963).

HEAT

LOSS

FROM

A

NEWTONIAN

ANIMAL

39

Some of the following observations have been known to physiologists for many years but are seldom expressed in analytical relation to the modes of heat transfer. Other very important empirical relationships have emerged only through physiological investigation (such as the heat production-body weight correlation) and no analytic description is available. Rather than engage in the complexities of biological reality, I felt that some observations based upon a few simple models would emphasize the generality of certain relationships between the animal and its environment and illustrate the proper use of Newton’s law. To this end, after imposing conditions (i) to (v), the ensuing arguments construct a “Newtonian animal” by adding circulation, metabolism and insulation and discussing the result as though it were alive. To those of a more biological persuasion who may wisely have reservations over such an artless approach to living systems, I would like to remark that there is historical precedent in the work of Morrison & Tietz (1958), who actually constructed Newtonian animals. Their technique consisted of skinning dead mice and shrews, casting the bodies in Wood’s metal and stitching the original skins back on the metal casts. Morrison’s metal mouse and the Newtonian animal model real “dissipative structures” in biology. In this respect they are not of the genre currently envisaged by some physical chemists, for the biological structures they model have already evolved-as jackrabbit ears, bat wings and the feet of the Laysan albatross. 3. Conduction

If, in an arbitrary volume V of the system under consideration with a surface area A, a molecule in V collides at A with a molecule outside of V and transfers kinetic energy by collision, with neither molecule crossing A, then heat flow by conduction has occurred. The animal is considered as a solid volume V with a surface area A and with a thermal conductivity k. It is assumed initially that: (a) there is no bulk motion of the surroundings in which the animal is immersed, or of the animal; (b) there are no gradients other than temperature; (c) composition of both phases are uniform; (d) the surroundings (e.g. air) are at a lower temperature than the animal; and (e) there is an absence of external forces. The physical situation in which Newton observed heat loss was roughly one of suddenly immersing a very hot, solid object into relatively cool surroundings (air at room temperature). For these conditions the temperature of the object and “most” of the surroundings can be assumed to be uniform. To construct a purely conductional analogy to the Newtonian situation, though physiologically unrealistic, is instructive and is begun here by defining a bulk, spatially “uniform” body temperature T. This temperature is assumed to be

40

T.

H.

STRUNK

a fairly accurate representation of the actual temperature existing throughout most of the body when the following conditions prevail. To maintain this relatively uniform temperature and yet allow heat energy to cross the surface, the conductivity of the body, k, is assumed finite but very large relative to the surroundings. This means that within I/ there exist small gradients of temperature, which are responsible for the heat flux and the nearly uniform temperature. T may vary monotonically with time. The bulk temperature of the surroundings, T, (ambient temperature), is taken as the fixed, uniform temperature of the surroundings at large distances from the body. For this temperature to closely approximate the actual temperature of the bulk of the surroundings the boundary layer heat flux must be proportional to T-T, and T, must be the actual bulk temperature of the surroundings except for the vicinity close to the body. This will be true if cP is very large so that a steady loss of heat from the body will change the temperature in the immediate surroundings, Tdr slowly. Thus V’T, z 0, with Td = T on the surface A. For pure conduction T, drops from T to T, over a distance d. The linear nature of the problem means that the flux across d is proportional to (T- T,) and, except for the vicinity close to the body, T, is the temperature of the surroundings. Let T (x, y, z, t) be the temperature at the point (x, y, z) of the body. If heat is being conducted away from the body, the flow of heat can be represented by a vector q such that the flux integral for the oriented surface A represents the number of calories crossing A in the direction of the given unit normal, n, per unit time. (8)

The appropriate transport equations are: q=

-L,,VlnT= pcpa;

= -v-q.

-NT,

(9) (10)

The sign convention used here is consistent with the second law of thermodynamics wherein the heat flux q is positive when VT is negative. Equation (9) is Fourier’s law of heat conduction and (lo), the divergence of q, is the net rate at which heat is being transported away from (x, y, z). The partial derivative in equation (10) has its origin in the barycentric time derivative dT/dr = aT/at+u*VT. Since constraint (a) fixes the mass of V in space, dT/dt = aT/at and the total derivative will be used hereafter with this understanding.

HEAT

LOSS

FROM

A

NEWTONAN

ANIMAL

41

The total uniform rate of heat generation per unit volume due to metabolism is expressed by (see Appendix)

(11) Substitution of (9) into (10) and inclusion of the metabolic sources gives the general heat conduction equation

-PC, \; = V. (-kVT)- t: = -kVZT- dQ' .dt~.

(12)

In the absence of sources or sinks equation (I 2) becomes the Fourier equation PC, ‘$

= kV2T.

Tn the steady state dT/dr = 0 and equation (12) gives the Poisson equation

For steady conduction in a source and sink-free system, equation (12) now becomes the Laplace equation, kV2T = 0. (1% The problem at hand will initially neglect sources so the expression for pure conductional cooling may be obtained by integrating (13) over the volume of the animal

I pc$,ddV=

j kV2TdT/,

(16)

V

which becomes, with Gauss’ theorem, s,PC,,dd;dl/=jkVT*dA.

A Since T is approximately uniform, the effective temperature gradient occurs at the surface, across d. Hence the entire time-temperature history of the body is controlled by the surface resistance. Therefore, the relevant conductivity is the conductivity of the surroundings, k,. Because the effective temperature gradient occurs across the layer of thickness d, which may be difficult to measure, k, and dare combined into a parameter h, the unit surface conductance (surface film coefficient), h = k,/d. The conditions discussed here are equivalent to heat conduction in a stagnant air layer of thickness d and are characterized by a value of the dimensionless Nusselet number, NNu, of unity. Since

42

T.

H.

STRUNK

where (dT/dn),,,, is the actual temperature distribution at the surface, this means VT.n = -0/d and upon integration the right-hand side of (13) becomes JkVT.dA=

-@d/l

= -It,Aejd=

--hAtI.

(18)

A

where li, is an average over the surface A. Often a “surface conductance” K, is defined as hA. The integration of (17), assuming the left-hand side is uniform in V, and using (18) gives dT Vpc = -hAO. ’ dr After rearrangement equation (19) may be integrated, i dInU= - VTL [dl, Gv &=0 P to give the cooling equation In 0 = -(Ah/Vpc,)t+In O”, (21) where 0” is the initial temperature difference. Equation (21) illustrates that for pure heat conduction, a plot of In 8 vs. time is linear with a slope of -(Ah/pc,V) and a In 0 intercept at In 8”. The cumulative heat loss is obtained by rewriting (21) as O/O” = exp [-(Ah/ Vpc,)t]. (22) The instantaneous rate of temperature change is dT/dt = -0” (Ah/I/p,) exp [-(Alt/Vpc,)t]. (23) Since

‘PC,!!&V.qdV, dt V use of Gauss’ theorem again gives dT =Jq*dA=

dQ dr.

(25)

exp [ -(Ah/Vpc,)t].

(26)

vpc,&

Substitution

of (23) into (25) yields

dQ - =

Integration time t.

(24)

-&Ah

dt of (26) from t = 0 to

t gives the amount of heat conducted in

AQ = AhO” 4 exp [ -(Ah/Vpc,)t]

dt

0

= PVpc,(exp

[ -(Ah/Vpc,)f]

- l}.

(27)

HEAT

LOSS

FROM

A NEWTONIAN

43

ANIMAL

The mistaken notion that Newton’s law applies to conduction within the body probably results from the fact that, in the steady state, conduction may also be a linear function of the temperature difference and thus has the appearance of Newton’s law. This point is discussed in section 7(c). 4. Convection

Heat convection is a combination of two processes: molecular heat conduction into the surroundings and heat transfer owing to the motion of the surroundings. The conduction is usually considered as taking place into a thin, “stagnant” boundary layer next to the conducting surface with subsequent transport by movement of the gas or liquid surroundings. This is the situation to which Newton’s law pertains. The boundary layer heat flux is described by a complex set of partial differential equations. Because of the mathematical complexity involved, the simple Newtonian law of cooling is usually assumed, and herein lies its great value. Newton’s law does not apply to conduction within the body. The description of heat transfer, by Newton’s law alone, demands the assumption that the thermal conductivity of the body is so high that virtually all of the resistance to heat flow is confined to the boundary layer, for it was from exactly such a system that it originated. When the resistance to heat flow of the body is significant, heat loss or gain produces appreciable thermal gradients within the body. The temperature within can no longer be considered uniform and conduction within the body can be rate limiting. Then the heat equation must be used. The model used here is identical with that above and imposes the same constraints with the exception of the first part of constraint (a) pertaining to the surroundings. This is relaxed for convection, and since the boundary layer thickness will now decrease due to motion of the surroundings, T approaches T, over a shorter distance than in the absence of convection and hence is an even better choice for T,. The heat flux across the boundary of Vis still a linear function of (T-T,) and the major temperature gradient occurs across the boundary layer. For points just in the surroundings and immediately adjacent to the surface of the body, heat energy can be transferred from the body to the surroundings only by conduction. This heat transfer is z=jq.ndA=

-IkJT+dA=

-Ak,(g)

, surf

where k, is the thermal conductivity of the surroundings and (i?T/h~)~~~~. is an average value of the temperature gradient in the surroundings, aT/an, evaluated at the surface in a direction normal to the surface. The first integral

44

T.

II.

STRUNK

may be regarded as taken over the surface, approached from within the body, and the second integral as over A, approached from without. The heat conducted into the boundary layer equals the heat loss from it by convection. The description of the latter by Newton’s law of cooling permits a definition of the convection conductance h, dT -h,A(T-T,)= -Ak,( af, > . surf

For the Newtonian animal the bulk temperature of the body is, to a good approximation, the same as the surface temperature. Therefore dQ = 11,AU. (29) (0dt surface-surroundings Equation (19) would now apply to the convective model with h, of equation (29) replacing h. The analytical approach to h, is concerned with determining the temperature distribution in the boundary layer by considering the dynamics of the surroundings. For many situations there are a number of largely empirical relations for h, that give excellent results. By way of example, forced convection over a cylinder for a Reynolds number lo3 < NRE. < 5 x 104, may be characterized by a value of h, of ltc = (0*2%/4(p, Udl~c~)~‘~(c~~,/k,)~‘~, where d = a significant dimension (diameter here), U = velocity of surroundings approaching cylinder normal to the axis, p, = viscosity of the surroundings and cP and p are the heat capacity and density of the surroundings (Chapman, 1960). The methods used to develop equation (21) now give, for the cvlinder.

This is more appropriately a Newtonian cooling curve and illustrates the manner by which parameters of the cooling body and the environment influence the process. 5. Insulation (A)

THE

CORE-SHELL

MODEL

Addition of insulation to the Newtonian animal produces the basic coreshell model that is frequently used to describe thermoregulation. In the case of a Newtonian core covered with several layers of poor insulation, we have for the flux across layer i qi = - kiVT, (31)

HEAT

LOSS

FROM

A

NEWTONIAN

ANIMAL

45

where i= 1,2,3, . . . i,j,k, . . . n and ki and Ti are the conductivity and temperature of layer i. If the layers are not uniform, then kl represents appropriate averages through the layer. With ct phases of different k, per layer, Ei = C k,VT, C VT,. (32) 01 I d It will be assumed that heat is lost from the outer surface of the insulation by convection and the heat capacities of the layers, C,i, are negligible. In the steady state I q.dA = 5 IE,VT, .dA, = . . . = S liiVTi,dAi A Al = .a. = ,I:iVT;dA. = 1. I;,VT,*dA,, (33) where A is the surface area of the uninsulated Newtonian core, Ai is the surface area between the first and second insulation layer, etc. and A, is the area of the outer surface of insulation. The temperature gradients refer to the inner and outer surfaces of the respective layers and VT, goes from T, on the outer surface of the insulation to Ta at large distances from A,. With Gauss’s theorem JpcPd;dl/=jqdA=

. . . = j IisVTd * dA,. A A‘ This becomes, with equation (28) and the corresponding discussion, V

or h.b + C Li/Ei Ai ‘2s I ~ + A, C Li/Ei Ai , = --AU- TJ ,t, I >

(35)

Should A z AI z . . . z Ai = . . . x A,, then (36) and the cooling equation takes the form

Equation (37) would now describe conduction-convection cooling of an insulated Newtonian animal, subject to the above conditions. The rate of cooling increases with increasing thermal conductivity, increasing values of

46

T.

H.

STRUNK

h,, increasing area/volume ratio, decreasing thickness of insulation, decreasing total heat capacity of the animal, pc,V, and decreasing mass. The rate at which this animal cools is now an explicit function of well defined physical factors and not obscured in an empirical constant. The time constant for the system, or the time required for (T-

T,) = O*367(T0 - T,),

is

Vpc,

(

,I + 1 LJEi A* I )I

The time constant is also called the capacity lig and illustrates how the heating and cooling time of the system depends on the total heat capacity and surface conductivity. Within the system, the thermal conductivity determines the difference in point to point response rate, or transfer lag. In Newtonian cooling, of course, the transfer lag is virtually zero. The cumulative heat loss, following the argument of equations (22) to (27) is given by AQ = (T,- T”)Vpc, (9)

INSULATION

[ I -exp (-At/VP,, THICKNESS

AND

(If, + T b/h))]* BODY

(38)

GEOMETRY

The relationship between thickness and body geometry may be very important in understanding an animal’s response to severe thermal stress. Consider two radial geometries, the cylinder and the sphere. This corresponds to insulating a cylindrical Newtonian core and a spherical Newtonian core. For the cylinder the radial heat flow through the tubular shell of insulation is inversely proportional to the log of the outer radius, r, and the surface heat loss is directly proportional to r. As r increases, the radial heat conduction decreases but the cooling surface area increases. Thus, at some value of r the heat loss is a maximum. For a fixed core diameter the insulation thickness is determined by the outer radius and increasing the outer radius can therefore lead to the situation where the heat loss is increased. Schneider (1955) has shown that this critical value of r, where heat loss or gain is a maximum, is given for the cylinder by rc = k/h, and for the sphere by rc = 2klh,, where k is the thermal conductivity of the insulation and h, is the unit surface conductance. Increasing k or decreasing h, extend the critical radius. In the preceeding discussion of the uninsulated Newtonian animal, the influence of geometry on heat loss appeared in the area-volume ratio, which may simply be different for different animals or else may be regulated through behavior. The concept of a critical radius is an additional geometric parameter that may also differ among animals and/or be subject to behavioral regulation. Thus an essentially cylindrical animal might readily increase its critical radius by assuming a relatively spherical form, and thereby be able to withstand environmental extremes (increased values of h,) that it otherwise would not

HEAT

LOSS

FROM

A

NEWTONIAN

ANIMAL

47

have survived. On the other hand if the outer radius is initially less than rc, then for a fixed core radius it is possible to increase surface heat loss by increasing the insulation thickness. (C)

INSULATION

THICKNESS

AND

HEAT

CAPACITY

When the heat capacity of the insulating layers is not negligible, the problem is more complicated. By way of example, a single layer of such insulation will now be placed upon the surface of the Newtonian animal and the cooling rate and cumulative heat loss examined for the one dimensional case. The insulation thickness L extends from x = a to x = b and to finite but very large distances, relative to x, in the y and z direction. At x = a the insulation makes perfect thermal contact with the surface of the almost perfect conductor, the Newtonian animal. At x = b heat is lost across the surface of the insulation by convection to the surroundings. For t < 0, the temperature of the body (T), insulation (Ti) and surroundings (T,) are uniformly the same, T”. At t = 0 the ambient temperature is suddenly dropped to T,. The bulk temperature of the Newtonian animal will change as Ti, evaluated at x = a, changes because the almost perfect conductor can lose heat only as the insulation does. Let 0(t) = (T-T,) and Bi(x, r) = ( Ti - T,) so that e(r) = Qi(a, t). The initial and boundary conditions are : t = 0, O(r) = 0” = (T” - TJ ; t > 0, o(t) = Oi(a, t); x = a,

d0i

dOi = 0 (U = -Idf xzn +a ax IX=R

lii/pC,);

x = b,

Subject to these conditions one must solve t > 0,

a
uiv2ei =

dOi (7,

Cai =

Ei/Picpi)*

This is done by the application of the separation of variables method (Carslaw & Jaeger, 1959; Jakob, 1949; Rohsenow & Choi, 1961; Schneider, 1955). Due to the boundary conditions the particular solution contains two non-vanishing terms 0 = Ae-n2ait sin (nx)+ Be-“2ait cos (FIX). (40)

T. H. STRUNK

48

Substitution of this equation into the boundary conditions gives two simultaneous equations involving the coefficients A and B. For non-vanishing values of A and B the determinant of their coefficients must vanish and this produces the secular equation O!llCOS (NJ)- n’cli sin (W) WI COS (nb) - he/Pi sin (Fib)

--an sin (na)--~~~n’ cos (na) = 0, -an sin (nb)+ /~,/l;, cos (rib)

(41)

where the roots are the eigenvalues of the resulting transcendental equation

To each eigenvalue, n, corresponds an eigenfunction and these are orthonormal. The general solution to equation (39) may now be written as O,(X, t) = 2 e-nJZu”(Aj sin (njX)+ Bj COS (njX)>, j=

(43)

1

where the coefficients Aj and B, are Aj = j 8” sin (nix) dx i sin’ (njx) dx,

(t

I (I

Bj = ~ 8” COS (nix) dx i COS’ (nix) dx.

Ia To obtain the cumulative heat loss, consideration of equation (10) provides 0

;, pi cpi “d”: d v = ii V . qid F = j qi. dAi = ;yj = “ds,

(44)

Ai

where A, is the closed surface bounding Vi. The cumulative heat loss is AQ = f d;dt Substitution

= - j j Pic,,idl/i “d”; dt = - j pic,&--8,“) Yi f

Vi

d&.

(45)

1

(46)

from equation (43) gives

AQ = -piCliysi

[

jzL (ePJzaif -l)(Aj

sin (njX)+Bj

COS (njX)

d&.

Since term by term integration is possible, the summation integration may be interchanged. Thus AQ = -piC,i

jg, (e-nJ2u1’- 1) ( AjfJJsin(njx)dxdydz+ XYZ

+ B, j s j cos (njx) dx dy dz *Yz

>I

(47)

HEAT

LOSS

AQ = picPiAi i

FROM

A NEWTONIAN

(1 -e-“jzai’)2@

49

ANIMAL

x

j=l (cos

x

(nja)-cos

(?ljb))Z

cos (nja) sin (nja)-COS

(---~

(fljb)

~-- -+

sin (njb)+(b-U)

+ _ --L!k!?~!!~s’n
-y*l+a,

(49)

where j3 = pcPV, a = VdQ’/dr and 0r is the difference between the bulk temperature of the core, T, and the surface temperature T,. Heat is lost by convection from the surface so that we also have

with t12 = (T,- T,). Addition

of these two equations then gives

d0 ~--..---- e

a .z + j?(L/kA + 1/II, A) = c~kTTT~lt,i)*

(51)

The general solution to this differential equation is

When t = 0 the constant c is seen to be c = 9” - aL/kA,

so that after rearrangement,

equation (52) becomes (53)

T.B.

4

50

T.

H.

STRUNK

As I becomes very large 8 approaches U”, the temperature differential maintained by the constant metabolic rate (0” = uL/kA). The cooling curve may now be written in the form

and it has been assumed that 8” > aL/kA. The constancy of the metabolic rate means that there will always be a constant, finite value of 8” for any T, < T. By way of previous arguments [equations (22) to (27)] the cummulative heat loss is -At B(Llk+ llh,) - 1 . (55)

>1

It is perhaps a little more germane to biology to consider a temperature dependent heat source. A linear relationship such as V dQ’/dt = a + bT, (56) where, for comparative purposes, a is given the same value as in equation (49), gives d0 -- (kA-b?E-e (y+ .--..tL--(57) iii= WLlk + 1lU WW + 1lh,)’ The general solution of this expression is the cooling equation

and we see that for t > 0 and kA > bL, cooling occurs. The rate of cooling is now a function of metabolic source strength and is less than in either the sourceless model or the previous example of constant sources. Furthermore, the temperature differential maintained by metabolism, 8” = aL/(kA- bL), is larger here than for the case of constant sources. The discussion so far is incomplete to the extent that it does not account for the case where, with a finite rate of heat production, 8” = 0. In this instance a heat sink, such as evaporative water loss, is required in the heat equation. This is easily added (e.g. as a constant or a linear function of T) and permits 8” = 0 through a balanced heat production and heat loss. In torpid hummingbirds Lasiewski & Lasiewski (1967) have measured a small but significant water loss which could offset a major portion of the torpid heat production. One of the purposes of this report was to keep the models as simple as possible in the interests of clarity, and it goes beyond this intent to enter into the various thermoregulatory responses observed in torpid animals. However, it, may be remarked that in torpid pocket mice Wang & Hudson (1970) have observed null values for 8” down to an ambient temperature of 14°C. For

HEAT

LOSS

FROM

A

NEWTONIAN

ANIMAL

51

temperatures Mow this, ~9” increased-indicating active thermoregulation. Hainsworth & Wolf (1970) have reported essentially the same behavior in torpid hummingbirds with regulation of body temperature below 18°C. Thus a single cooling curve yields rather minimal information and a series of curves for various values of T, is needed to establish the behavior of 6”. 7. Physiological Correlations In the following discussion I will try to relate some ideas extracted from the physiological literature to the previous comments. The purpose is twofold: first, some of the ideas are perfectly valid but appear to me to be a little vague in both their origin and limitations; second, some are not valid. (A)

TORPOR

In a study of the physiology of hummingbirds, Lasiewski & Lasiewski (1967) observe that the rate of entry into torpor is inversely related to the body weight of the bird. Hudson & Bartholomew (1964, p. 545) state that one can assume the rate of decline in body temperature during entry into torpor will be in part a function of body size and that, generally, large animals cool slower than small ones. They further indicate that rate of arousal from torpor is inversely related to body size. These conclusions are derived from experimental observations and are not predictable from Newton’s law. They are, however, stated explicitly in equation (21) for constant h,, and equations (30) and (48) where, neglecting other modes of heat exchange, one sees that for a given A/V ratio a heavier animal cools or warms more slowly than a lighter one. Also, for a similar body geometry, a larger animal has a lower A/V ratio and hence will cool or warm more slowly than a smaller one. (B)

METALBOLISM

AND

CIRCULATION

In dealing with heat loss in agamid lizards Bartholomew & Tucker (1963) began with Newton’s law of cooling in the form of equation (7). They next wrote an expression similar to equation (19) dT/dt = (C/K)0 (4 and integrated it to obtain In 0 = (C/K)t + In E, @I where C is the thermal conductance &A), K is the specific heat of the animal (Vpc,) and E is an integration constant (P). By plotting in 8 vs. t they determined a value for C. After deciding that this value was only an “apparent” thermal conductance (C = C’) due to the presence of metabolism, they attempted to derive a correction which would yield a meta-

52

f.

H.

STRUNK

bolically independent value. In assuming that the metabolic sources, M, were constant they wrote an equation comparable to equation (49), dT/dt = (M+ CY)/K, (cl so that by equating equations (a) and (c) they obtained C’ = c+M/e.

(4

The claim was next made that the quantity M/O was the required correction factor and, by subtracting it from the experimental or apparent value C’, they could determine C. While there is nothing mathematically wrong with equation (d), the interpretation of its meaning and the use of it by these workers is very much in error. This is easily seen by recalling the method used to obtain a value of C’ wherein equation (a) was integrated directly and In 0 was plotted as a function oft in equation (b). The comparison between equations (a) and (c) should not have been made because the experimental thermal conductance is obtained from a plot of equation (b) and not from equations (a) or (c). The relationships that should have been compared are equation (b) and its counterpart (obtained by separating the variables in equation (d) and integrating). It is then immediately obvious that a conslant metabolic rate, such as M, can in no way affect the rate of cooling or heating-the slopes of In 0 vs. t are identical in both equation (b) and the equation resulting from integration of equation (c). Their thermal conductances were not properly corrected for metabolism and their conclusions based upon these results were not warranted. The rate of cooling or warming will be influenced by the metabolic rate only when the latter is a function of T, as in equation (56). Bartholomew & Tucker in fact determined that the log of the metabolic rate was a linear function of T, but their use of this in their corrections compounds the error and does not restore validity to their method. Furthermore, these investigators concluded that the probable reason the lizards warmed faster than they cooled and their observation that dead lizards cooled and warmed more slowly than living ones was, in part, due to the presence of circulatory regulation in the latter. Basically circulation brings a mass of blood at core temperature to the peripheral capillary beds where heat can be transferred between the blood and the surrounding tissues. For this heat transfer to occur these tissues must be at a temperature different from the core-and therefore Newton’s law of cooling, alone, does not suffice to describe the situation. The Newtonian model can accommodate blood circulation in a manner that illustrates the observations of Bartholomew & Tucker. For simplicity the Newtonian core is covered with a layer of poor insulation of thickness L and thermal conductivity k. Just under the outer surface of this insulation

HEAT

LOSS

FROM

A

NEWTONIAN

ANIMAL

53

are a number of capillary beds with a total heat transfer area A’. Heat is transferred between the capillary blood and the environment through a thin overlying layer of this insulation of thickness d, where d Q L. The capillary beds are supplied with arterial blood at core temperature (T) and heat transfer between larger vessels and the surrounding insulation is neglected. Thus the blood enters the capillaries at T, exchanges heat across A’ with the surface tissue at virtually T, and returns to the core through the venous supply at T,. Rewarming occurs in the core by heat conduction across the walls of the larger vessels. The circulatory heat exchange is given by dQ”/dt = A’ti~;( T - T,), (59) where ti is the mass transfer rate (blood flow in cm3/sec times the blood density in g/cm3) and cj is the specific heat capacity of the blood. We now define 8, = (T- T,), t$ = (T,- T,) and j? equals the total heat capacity of the blood and the Newtonian core, remembering that the heat capacity of the insulation is very small. For a constant metabolic rate, a, we then have fidB/dt = :;”

0, -c&, +a,

(SI= A’rlic;),

(60)

/?dO/dt = -/I, A&. Assuming that ti is constant with time, distance and temperature, addition of these two equations and rearrangement gives the differential equation for 0 as a function of t. “~(kAll+aj+h!A=

-e+(kA~~+u).

(61)

c

The general solution of this equation is In (O-fP)

= ln(P-P)-t//3

m-* h,A For constant values of CL, increasing the blood flow rate increases the rate of cooling for T > T,, or of warming for T c T,. The rather simplified treatment considered here also allows combination of a temperature dependent metabolic rate [equation (56)] and circulation. After solving the two equations (with u constant) pi;=

-kfe,-u0,+(u+bD), (63)

/II;

= -h,AO,,

the final cooling curve has the form [l-b/(kA/L+u)]t

tw

54

T.

H.

STRUNK

where 0” is now given by 0” = ai(kA/L+cr-b). Thus, considering only (kA/L+a) > 6, smaller cooling rates are to be expected for larger values of b, and greater values of 0”. But as c1increases so does the cooling rate, while 8” decreases, in agreement with the experimental observations. (C)

ORIGIN

OF HARDY’S

LAW

Hardy (1949) presents a “law of steady state heat conduction” H, = KdA (Tll- w,

where H,, is the quantity of heat conducted, K is the thermal conductivity, A is the area, t is time and d is the thickness. The term “steady state” in physiological heat transfer is an implicit statement of Poisson’s equation wherein the heat generated by internal sources equals the heat lost by conduction. Hardy’s expression then, comes directly from equation (16) and is really an integration of Fourier’s law, equation (9), q = -kVT.

The term H,, which Hardy used, is -AQ,

dQ -di=jq*dA=

where

-JkVT*dA.

A

(65)

A

Assuming k is not a function of VT (although it may be a function of T), that VT is not a function of time and VT*n = -(Th- T,)/d is a good approximation, then two integrations yield AQ = k;’ (T,, - T,)t. When an animal cools or warms, VT is no longer independent of time as equations (21) and (48) indicate. Hence the quantity of heat conducted is no longer given by equation (66) and an expression such as equations (27), (48) or (55) is now necessary. (D)

IRVING’S

MODEL

Irving (1964, p. 364) states: “The Newtonian formula describing loss of heat by conduction (italics mine) from a warm object to cooler air is T,-T,=K.Z. H in which Tb is the temperature of the body (a specific character), T, is the temperature of the surrounding air, K is a constant for the units of measurement, Z is the resistance to flow of heat or insulation and H is the metabolic production of heat.”

HEAT

LOSS

FROM

A NEWTONIAN

ANIMAL

55

If what Irving means by “Newtonian” is that there exist two bulk temperatures T, and Tb to which the flux of heat by conduction is linearly related, then the formula does have the appearance of Newton’s law of cooling. However, since he says it describes conduction, it would be more correctly related to the integrated Fourier law equation (66) similar to Hardy’s expression. But if Irving means that his expression is identically Newton’s law then its application is in error for (i) Newton’s law does not describe conduction (Fourier’s law does that) and (ii) the red-hot block of Newton’s experiment was neither insulated nor metabolizing. (E)

HART’S

MODEL

Also somewhat confusing are the statements of Hart (1963): “The flow from a heated body cooling in air is usually represented by Newton’s law of cooling which states simply that the heat loss H = C x S(T-TT,),

(4

where C is a cooling constant, S is the surface area, T is the temperature of the body, and T, is the temperature of the air. If there is no heat production of the body, Newton’s law in its form as a differential equation predicts fall of temperature in an exponential manner to reach asymptotically the environmental temperature. If there is heat production, the above equation is applicable in the steady state in which heat production equals heat loss. Newton’s law has been applied to homeotherms in the steady state, particularly by Scholander and associates, who demonstrated its general validity. With a constant core temperature, heat loss to the surface (mean surface temperature) can be represented by the equations H/S = (Tb - T,)I,

(b)

where Tb and T, represent body and skin temperatures, and the cooling constant C is given as the reciprocal of I,, the thermal insulation of the tissues; and HIS = (T, - T&z, w where T., T. and I2 represent temperatures of skin, air and insulation external medium, respectively.” Hart’s equation (a) will be Newton’s law only if C = h, and T is the temperature of the surface of the body. For the surface to have the same temperature as the bulk of the body at all times implies a “Newtonian” animal. The reference to heat production would seem inappropriate for it has no relation to Newton’s law and its proper place is in the general heat equation. Then, assumption of the steady state yields the Poisson equation which can be integrated over the volume of the animal

56

T.

H.

STRUNK

and, with appropriate assumptions, this gives (I

UL

an explicit statement that the heat generated by M:reactions equals the heat lost to the surroundings. In equation (b) Hart has used Newton’s law to describe conduction, which it does not. If we recognize equation (a) as Newton’s law, with C = h,, then in equation (b) the same C is certainly not the thermal conductance (k/L = L/Ii) of the tissues involved, as it must be if equation (b) describes conduction. Equation (b) should be identified as an integration of Fourier’s law. In equation (c) we have Newton’s law again, almost in its proper place. Hart’s term “insulation external medium” is an unnecessarily confusing alternative to h,. (F)

PHYSIOLOGICAL

CONDUCTANCE

It will be helpful for a moment to assume that the metabolic heat production is adequately given by the rate of oxygen consumption, dO,/dt, times the calorific equivalent of a given volume of O2 consumed, H’. Furthermore, suppose that the heat loss is by conduction only, through a single, uniform layer of insulation of thickness d and conductivity k so that h = k/d is the conductance of the animal. Then in the steady state

VH’ ddqJ= hA(T- TJ In the usual treatment of thermoregulation the calorific equivalent of the O2 consumption rate per unit mass is plotted against ambient temperature, T, (Davson, 1964). For homeotherms in general, between the upper and lower critical temperatures the basal metabolic rate is independent of T,. For temperatures below the lower critical temperature, equation (67) describes the linear increase in metabolic heat production. Since pV is the total mass of the animal, then per unit mass equation (67) becomes H’d02 hA w

P dt F(T-T,)* Division by (T-T.) gives the slope of the linear relationship, the quantity referred to in the physiological literature as the conductance of the animal, H’ dOz _~ = 1~4 -. - cal. -- .---..-.----.pV g set deg C’ PV-T,) dt This is not a conductance in the sense of heat transfer. The reason for this choice of units lies in the difficulty of measuring the area of the animal. There would be less problem if it were always made clear that this “conductance”

HEAT

LOSS

FROM

A NEWTONIAN

ANIMAL

57

contained, in addition to the conductance h, the A/V ratio. However, the impression usually conveyed is that differences in slope are the result of differences in h, i.e. the slope of the line through the data, hA/pV, is interpreted as h alone. While this is most likely valid in comparing results from members of the same species (except where behavior modifies A/V), it could be in error in comparisons among different species. Two animals with identical conductances but differing A/V ratios will exhibit two different slopes. From Scholander’s classic data (Davson, 1964, p. 204) although there is not much chance of misinterpreting the differences between Arctic foxes and weasels, the nature of the difference between weasels and lemmings is not immediately clear. In accord with the previous comments then, given a fixed upper critical temperature and basal metabolic rate, the lower critical temperature and the breadth of the thermoneutral zone are determined by the A/V ratio as well as h. More recently Barnett & Mount (1967) (along with Kleiber) created their own special problem: “Heat flow from an animal to its environment obeys Fourier’s law, which states that heat flow between two regions is proportional to the temperature difference and to the thermal conductance. (Kleiber, 1961). Kleiber disagrees with the commonly held view that heat loss takes place according to Newton’s law of cooling, which states that the rate of cooling is proportional to the temperature difference between the body’s surface and the surroundings. He points out that, since the body’s temperature is well regulated, a simple law of cooling does not apply. In practice the difference between these two laws is not great: but when, for example, thermal insulation is changing, Fourier’s law is superior, since it deals with a regulated flow rather than one which is diminishing as the body cools.”

There is no reason why the use of one law should exclude use of the other. Fourier’s law applies to both transient (dT/dt # 0) and steady-state (dT’,dt = 0) heat conduction. Newton’s law alone applies only to transient heat loss by conduction-convection-radiation. Kleiber (1961) has unnecessarily limited his argument, for animals may lose heat by modes other than conduction. Also Fourier’s law deals with the temperature gradient, not the temperature difference and special assumptions are needed to effect the simplification. In practice the difference between the two laws is great and Fourier’s law readily deals with a heat flow that changes as the body temperature changes. In transient conduction one uses Fourier’s equation which equates the time rate of change of internal energy to Fourier’s law. It is not a question of which law is “superior” because they describe two different physical processes and one must use the correct law for each process. Since the two processes often occur together the two laws are often used together

58

T.

H.

STRUNK

and thus provide a very adequate description of many heat transfer problems. Many other important contributions to heat exchange have been excluded from this discussion because I wished to show how fundamental considerations might lead to predictable behavior. It would thus appear that in the biological literature the laws governing heat transfer are often used incorrectly. Furthermore, there seems to be some tendency to use Newton’s law, exclusively (and inconsistently), to describe all physiological heat transfer. This neglects other modes of heat transfer and obscures important parameters. The value of Newton’s law of cooling lies in its use as a boundary condition because of the mathematical simplification it allows. And, except for the most simple cases, with the proper heat equation the solution for T(x, y, z, t) is generally not easy. I think the problem is to define the modes of heat exchange and to find the correct and experimentally complete set of variables that will allow a consistent prediction of thermoregulatory processes. I contend that Newton’s law alone is not suited for this, and doubly so when incorrectly applied, I thank Russel M. Holdredge, Professor of Mechanical Engineering, Utah State University and Donald C. Mikulecky, Lecturer in Biophysics, Biophysical Laboratory, Harvard Medical School, for reading the manuscript and for valuable discussions. REFERENCES ALLEN,H. S. & MAXWELL,R. S. (1948). A Texfbook ofHeat, part 1. London: Macmillan & co. BARNETT, S. A. & MOUNT,L. E. (1967).In Thermobiology.(A. H. Rose, ed.) p, 411. New York: Academic Press. BARTHOLOMEW, G. A. & TUCKER,V. A. (1963). Physiol. 2001. 36, 199. CARSLAW, H. S. & JAEOER, J. C. (1959). Conduction of Heat in Solids. London: Oxford University Press. CHAPMAN, A. J. (1960). Heat Trun&r. New York: Macmillan & Co. CR~S~IE, R. J., HARDY,J. D. & FESSENDEN, E. (1963). In Temperature, Its Measurement and Control in Science and Industry, Vol. 3. (J. D. Hardy, ed.) p. 627. New York: Reinhold. DAV~~N,H. (1964). A Textbook of General Physiology. Boston: Little, Brown & Co. GROBER, H. & ERK, S. (l%l). Fundamentals of Heat Transfer. (Revisedby U. Grigull.) New York : McGraw-Hill. HAINSWORTH, F. R. & WOLF,L. L. (1970). Science, N. Y. 162, 368. HARDY,J. D. (1949). In Physiology of Heat Regulation and the Science of Clothing. (L. H. Newburg, ed.) p. 88. Philadelphia: SaundersCo. HART,J. S. (1963). In Temperature, Its Measurement and Control in Science and Industry, Vol. 3. (J. D. Hardy, ed.) p. 374. New York: Reinhold. HUDSON, J. W. & BARTHOLOMEW, G. A. (1964). In Handbaok of Physiology, Vol. 4, p. 541. Washington, DC. : American PhysiologicalSociety. IRVINO,L. I. (1964). In Handbook of Physiology, Vol. 4, p. 361. Washington, D.C.: Ainerican PhysiologicalSociety. J~;Ko~,M. (1949). Heat Transfer, Vol. 1. New York: John Wiley.

HEAT

LOSS

FROM

A NEWTONIAN

59

ANIMAL

KAYS, W. M. (1966). Convective Heat and Muss Transfer. New York: McGraw-Hill. KING, J. R. & FARNER, D. S. (1964). In Handbook of Physiology, Vol. 4, p. 602. Washington, D.C. : American Physiologicat Society. KLEIBER, M. (1961). Quoted in Bamett & Mount (1967). LASIEWSKI, R. C. & LASIEWSKI, R. S. (1967). Auk 84, 34. MORRISON, P. R. & TIETZ, W. J. (1958). J. Mammal. 33, 78. Scorr, J. F. (Ed.) (1967). The Correspondence ofIsaac Newton, Vol. 4, 1694-1709, Letter 636. Cambridge: Cambridge University Press. PRIGOGINE, I. & DEFAY, R. (1954). Chemical Theromodynamics. London: Longmans Green & Co. ROHSENOW, W. M. & CHOI, H. Y. (1961). Heat, Mass and Momentum Transfer. Englewood Cliffs, N.J. : Prentice-Hall. ROLLER, D. (1961). Case 3 in Harvard Case Histories in Experimental Science. (J. B. Conant, ed.) Cambridge, Mass.: Harvard University Press. SCHNEIDER,P. J. (1955). Conduction Heat Transfer. Reading, Mass.: Addison-Wesley. WANG, L. C. & HUDSON, J. W. (1970). Comp. Biochem. Physiol. 32, 275. WISSLER, E. H. (1961). J. appl. Physiof. 76, 734. WISSLER,E. H. (1963). In Temperature, Its Measurement and Control in Science and Industry, Vol. 3. (J. D. Hardy, ed.) p. 603. New York: Reinhold. WISSLER, E. H. (1966). Chem. Engng Prog. Symp. Ser. 62, 66.

APPENDIX

A

Chemical Thermodynamics

When chemical reactions occur in the system, the enthalpy is a function of temperature, pressure and composition H = W’,

P, 0,

(Al)

where t is the extent of the reaction (Prigogine & Defay, 1954). The total differential of H is

dH= (g),, dT+(“a’,‘),;dp+(;),,, dt.

(A21

The first law then becomes

dT+ [(!&

dQ=(;;),,,

-Y] dp+(f$)r,p

dt,

(A31

and at constant p, the apparent heat capacity is seen to be

C,(app.) = :QT= (g),,, + (g),,, ($)p

C-44)

where

dH

(--> al-

= C,,, = heat capacity at “frozen” composition.

PVC

(A5)

60

T.

H.

STRUNK

When a number of reactions occur in the system the superscript c( will designate a general reaction, and u = 1, 2, 3, . . . r. Thus,

= heat of the a reaction. But the heat capacity at frozen composition the empirical relationship

(A6)

is a function of T, given by

C,, = C;,, + C;,, T + C;,< T2 + . . . ,

which means

aQ

C---J ‘it

at

dT

---+C;
(A@

P.< =

Now the time rate of change of total heat at constant p is

Since the reaction rate is generally some exponential Newtonian approximation rapidly becomes worse.

function

APPENDIX B Notation dA

CP

V’ V”

d k k. L h:” H e nt VP P iQ,dt

ndA = area element vector n = unit outward vector normal to dA dA = surface area element specific heat at constant pressure of tissue in Y divergence Laplacian operator boundary layer thickness L,,/T = thermal conductivity of tissue in V thermal conductivity of surroundings coefficient of heat conduction unit surface conductance enthalpy degree of advancement, d
of T, the

HEAT P

t

T

Td T, R U

V

LOSS

FROM

A NEWTONIAN

ANIMAL

density of tissues in V time temperature of body of volume V temperature of surroundini in boundary layer temperature! of surroundings at huge distances from body (T---J, excess of body temperature over that of surroundings velocity of center of mass of V volume of body

61