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An improved method for determining thermal conductance and equilibrium body temperature with cooling curve experiments
J. Thermal Biology, 1976. Val. 1. pp. 169 to 175. Pergamon Press. Printed in Great Britain
AN IMPROVED METHOD FOR DETERMINING THERMAL CONDUCTANCE AND EQUILIBRIUM BODY TEMPERATURE WITH COOLING CURVE EXPERIMENTS GEORGE S. BAKKEN Department of Life Sciences, Indiana State University, Terre Haute, Indiana 47809, U.S.A.
(Received 29 May 1975; revised 11 November 1975) Al~tmet--l. The analysis of cooling curves to find the thermal conductance by plotting log(body temperature-air temperature) vs time is generally inaccurate. 2. Cooling curves are correctly analyzed by plotting log(body temperature--equilibrium body temperature) vs time. 3. Experiment time may be greatly reduced by using computer analysis to find equilibrium body temperature and the thermal conductance simultaneously. 4. Two computer programs for cooling curve analysis are described, together with representative results. 5. Accurate values of thermal conductance may be determined with cooling ranges of less than 5°C. INTRODUCTION
THE MOSTcommonly used experimental procedure for determining the thermal conductance K of an ectothermic animal has been the cooling (or heating) curve experiment. The animal is placed in an environment with an air temperature T~ different from the initial body temperature. The temperature of the animal, Tb (usually cloacal), is then recorded as a function of time t as T~ comes into equilibrium with the new thermal environment. Assuming that the body temperature follows "Newton's law of cooling", dTddt = - K ~ / C ( T b - T~).
(1)
The heat capacity of the animal is denoted by C, and KN is the Newton's law thermal conductance. The data is most conveniently analyzed by plotting log(T~ - T,) vs t. The slope of the resulting approximately straight line is" d log(Tb dt
Ta)
-0.4343 d i n ( T b - T,)-KN -- 0.4343 --C-. (2)
After C has been measured or calculated from the weight of the animal, K N may be determined from the slope of the plot. This procedure suffers from a number of erroneous assumptions. First, the animal cannot be characterized by a single Newton's law conductance K~; rather, the rate of heat transfer depends on environmental parameters such as wind speed O'racy, 1972). Second, the environmental temperature is not T,, but rather is determined by a combination of conductive, convective and radiative factors (Bakken & Gates, 1975; Bakken, 1976). Third, the endpoint of a cooling curve is not the environmental temperature, but the final equilibrium temperature of the animal (Bakken & Gates, 1975), which depends on metabolism and evaporative water loss. Fourth, the slope of the plot
does not give KN/C if the animal has metabolic heat production which varies with Tb (Bakken & Gates, 1975). Thus, the standard analysis based on equations (1) and (2) is generally inaccurate. However, cooling curve experiments are still valid, provided that a correct energy budget analysis (Birkebak, 1966; Porter & Gates, 1969) is used. Bakken & Gates (1975) and Bakken (1976) applied energy budget analysis to a general animal with metabolic heat production and variable Tb. They presented a linear approximation to overall heat transfer, arranged to be as similar to "Newton's law" as possible: dT b 1 Tb T, + (3) dt z The effective net metabolic heat production, M*, is essentially just the dry metabolic heat production M-E, where M is the metabolic heat produced and E is the heat dissipated by evaporation. A more accurate expression separates E according to the site of evaporation, and includes correction factors for the degree of thermal coupling of each site to the body core so that M* <_ M - E (Bakken, 1976). The factor is the time constant (sometimes called the cooling constant), and is equal to the time required for (l-l/e) 63% of the total response to a sudden change in the thermal environment to occur. The time constant, ~, is equal to C/Ko only if M* is constant and independent of T~, The overall thermal conductance Ko, functionally equivalent to the, Newton's law thermal conductance KN, is: Ko = (Ksf + G) (H + R) + GKsf K,f + H + R
(4)
The thermal conductance to the ground is G, H is the convective conductance to the air, and K,s is the thermal conductance of the integument. The thermal radiation conductance is R = 4A,.aE~ 3. where T, is 169
170
GEORGE
S.
BAKKEN
,
the time-average surface temperature of the animal in kelvins. K--- =C + 273.16: a is the Stefan-Boltzmann constant, 5.67 x 10 -8 W m -2 K-'*; e is the emissivity of the surface of the animal for thermal radiation, typically 0.95-0.98; and A,, is the effective area of the animal for the emission or absorption of radiation, in m-'. In S.I. units, thermal conductances are expressed in W K-~ animal-~, heat capacity in J K - ~ animal- ~, and rates of heat production or loss due to physiological processes in the animal in W animal- ~. The temperature of the environment for a specific animal is
,
.
,
,
,
o
;10
N
\\"-.
,
--
C-T.
2XX,x---
.,
\ X
0 I.¢ o Q. E
"-.,
II .~
T,, = K~s(HT" + RT, + Q, + GTg) + GTg(H + R) (5) (K~s + G) (H + R) + GK~ I.
"1"•',: T a
o IIl
OA
and is called the operative enviromnental temperature. The ground temperature is Tr and net absorbed visible and thermal radiation when the actual surface temperature T~ equals the time-average surface temperature T, is denoted by Q,. (This complex definition arises from the linearization of the Stefan-Boltzmann law for thermal radiation. See Bakken, 1976, for a discussion.) The overall thermal conductance and operative environmental temperature are generalizations of the operative heat exchange coefficient and operative temperature proposed by Winslow, et al. (1937), and Gagge & Hardy 0967). The analysis of Bakken & Gates 0975) and Bakken (1976) leads to the following conclusions about cooling curve experiments: 1. Equation (5) shows that the overall thermal conductance K0 is a function of both the animal and its environment, since G depends on the nature of the surface under the animal and, as noted by Tracy (1972), the coefficient H depends on wind velocity. 2. The reference temperature of the environment is T,,, not T~. The operative environmental temperature is a complex function of both the animal and its environment, and can differ significantly from air temperature. 3. Comparison of equations (1) and (3) shows that may be found from the slope d 1 dT log(Tn - T~h) = --0.4343 -r (6)
,
~
,
Time
Fig. 1. Three plots of log(~ - ~) vs time. assuming equilibrium body temperature T~,~ equal to, g r a t e r than, and
less than T~. The difference is I'C in the cases where T~~ T~. Note that if T~~ ¢ T~ for any reason, such as an effective ambient radiation temperature = 7",, metabolic heat production, or evaporative cooling, the plot is not a straight line. This curvature prevents an accurate determination of the time constant and overall heat transfer coefficient. of a plot of log(Th - T};q) vs t, but not of log (T~ - TD vs t. The equilibrium body temperature, T~,~ = T,, + M*/Ko, is reached as t - ~ :r_. The dependence of Ko in wind speed and surface properties simply implies that a range of wind and substrate conditions should be used to investigate the range of Ko values which may obtain in the natural environment, and presents no difficulty to the cooling curve procedure per se. The problems resulting from the difference between T~ and Tg{ = T,. + M*/Ko are more serious, since they affect the accuracy of the determination of 3. The nature of the problem is shown in Fig. 1. If the actual equilibrium body temperature Tg* is higher than the assumed equilibrium temperature T~, the curve will bend up rather than continuing as a straight line. If T~ is lower than Tr the curve will bend down. Any attempt to assign a single value to the slope of a curved line to obtain r will necessarily
Table 1. Analysis of various fractions of a cooling experiment on a copper-bodied taxidermic model of Passer domesticus, using a straight-line fit to find the time constant T i
iii i ii
ii
"~for * T for *§ Tbeq-o.055 T b e q
X for * dx/x for t Tbeq+0.055 ATbeq-o.1
ATb
Tbi
Tbf
Tbeq
32.50 29.55 22.05
33.89 30.94 23.44
1.39 1.39 1.39
0.44 0.44 0.42
229.2 229.6 231.3
226.4 226.7 228.1
223.6 223.8 224.8
0.022 0.023 0.026
16.33 10.94 4.78
17.72 12.33 6.17
1.39 1.39 1.39
0.40 0.37 0.34
233.4 236.1 240.2
229.8 232.1 234.8
226.3 228.0 229.4
0.028 0.032 0.041
I
I
II
Temperatures are in °C, ~ in set. * These columns show the effect of T of an 0.055°C (0.1°F) difference between air temperature and the equilibrium body temperature TJq. t" Fractional error in z that would result from using air temperature rather than Tb"~ in the cooling curve plot if they differ by 0.1=C. .~The monotonic increase in T for segments of the data nearest Tseq is due to the change in radiative thermal conductance discussed in the text.
Improved method for determining thermal conductance and equilibrium body temperature
171
give poor results. The problem is quite significant; over a small temperature range where an average P Table 1 shows that an error of 0.1°C in the choice can be defined by using a sensitive temperature sensor of an equilibrium temperature results in an error of and a well-controlled environmental _chamber, 2.2-4.1%, depending on the range of cooling used. together with numerical analysis using procedure (3) There are four procedures for eliminating this or (4) above. error: One difficulty in designing a computer algorithm (1). A properly constructed chamber can have the to find T[ ~ and t simultaneously is in deciding how wall temperature equal to T,, which sets T , , - T~ if to weight the data. There are two possible sources the walls are black (Bakken, 1976). Then independent of error: First, the recorder used to measure body knowledge of M* can be used in a procedure de- temperature will introduce uniform variance into the scribed by Bartholomew and Tucker (1963) to correct Th data. These errors will be most significant at the for metabolism and evaporation. This procedure is end of the curve when (Th -- T[ q) is small. Second, inaccurate, since it still attempts to assign a single a live animal does not completely satisfy the assumpvalue to the slope of a curved line. tion that the body is a single, isothermal lump with (2). The experiment can be run until Th = T~~, a single temperature. Different parts of the body cool which typically requires a time equal to 5-7x, depend- at slightly different rates, so that heat will flow from ing on the desired accuracy. While an improvement, one part of the body to another. The actual course this approach also has problems: (A) It uses excessive of body temperature will thus differ somewhat from time, since the required information is obtained in the result of Equation (3). These errors will be unia length of time less than 2T. (B) The animal may form in ln(Tb- T~), and the greatest effect will be change posture, altering H, G, A,~ and thus K0. (C) at the beginning of the experiment. The first source The animal may struggle, changing M* and thus T~q. of error would imply a least-squares routine with (D) The apparatus may not maintain a constant ther- linear weighting; the second source of error implies mal environment for the full 5-7r. (E) The animal weighting by ln(T~ - T[q). may abruptly begin a thermoregulatory response and alter T~a. (F) The metabolic rate of the animal may vary with T~, so that M* and thus T[ q change conMATERIALS AND M E T H O D S tinuously during the experiment. (G) The respiratory To explore the various possible approaches. I decided rate, and thus evaporation, may similarly vary with Th. (H) The vapor pressure of the water on the skin to write two computer programs for numerical coolingand respiratory and buccal mucosa varies with T~, curve analysis. The first program is based on the curvesimilarly altering T~~ continuously during the exper- straightening approach, and weights data as ln(Tb -- T~). The second program is based on a nonlinear least-squares iment. algorithm which weights data uniformly. 3. Since equation (6) shows that a plot of The program for finding Tg~ and z by curve straightenlog(Tb -- T[ q) vs t gives a straight line, the straightness ing is called NEWTON I. The derivation of the algorithm of the line resulting from an estimate of T~,q may be used follows from Fig. 2. I will use In rather than log used as a criterion of the accuracy of the estimate. to avoid the 0.4343 conversion factor. The initial guess Then estimates of Tff can be systematically improved i r , , ~ i to eliminate the curvature of the line and find the correct values of Tff and r. This reduces the experimental time and minimizes the frequency of problems (A)--(E), but is still subject to problems 2(F)--(H). (4). Finally, one of a number of numerical techniques for nonlinear least-squares analysis may be used to find T ~ and x directly, This approach is equivalent to the preceding one in the sources of experimental error. Mathematically, it is easier to compute confidence limits with this approach, but it may be more sensitive to the strong correlation between Tff and x. ., a.CT. -Tj ~ ~ \ . t Although even a correct value of z is not equal to C / K o when M* varies as Tb changes, the overall thermal conductance may still be determined by using the results of Bakken & Gates (1975). If the variation 1 i I i 1 r Time, t in M - E is linear with T~, so that M(Tb) - E(T~) -~ (M - E)v + P(T~ - Tn), (7) Fig. 2. The algorithm for finding the equilibrium body temperature T~q and the time constant z simultaneously then ~ = C / ( K o - P ) and proceeds by estimating Tff as an initial value Tff and plotting In(T~ -- T~l. The line curves up. indicating that T~ Ko = (C/x) + e. (8) is too low. Then a quadratic function, Co + c t t + c2t 2, is Here, T~ is an arbitrary body temperature, (M - E)~ fitted to the plot, and the first two terms, Co + c d . define is the value of [M(T~) - E(T~)] when T~ = T~, and a straight line which gives a fair approximation to the plot of ln(Tb -- T~q). A new estimate for T~. T~q, follows P -- d/dT~[M(T~) - E(Td], averaged for the tempera- from setting ln(Tb -- T ~ ) = co + c,t. as described in the ture range of the cooling curve. A linear variation text. The process is repeated until an estimate of T[q is in M and E is not usually the case over a wide tem- found close enough to the correct value to give a straight perature range. However, the animal can be cooled line in the plot.
172
GEORGE S. BAKKEN
to a linear weighting of data errors for small values of ln(Tb -- ~ 5 . at the expense of considerable complexity. The second program, called N E W T O N II, uses an optimized interpolation between the Taylor series and steepest descent methods for nonlinear least-squares curve fitting developed by Marquardt (1963). The program itself is based on a SHARE library distribution program (Marquardt, 1966), with considerable modification to update the programming and provide specialized input and output. Both programs use the same input file sequence so that no modifications of the input data are needed to change analysis Up to 200 data points may be used, consisting of pairs of time and T~ data, with the temperature scale identified by a F, C. or K designator. Any subset of the data (segment of the cooling curve) m R be selected by entering the initial and final time coordinates of the desired experimental interval This allows the data to be examined for possible changes in T~'~ or z during the experiment resulting from physiological or postural changes of the anim a l The programs print the results of each iteration When the convergence condition is salisfied, the values of Tdt = 0), Tgq, and ~ are printed, together with a semilogarithmic printer plot of the data and fit. and error information. The nonlinear least-squares program provides several statistical error estimates for each parameter, including standard error, one-parameter, support plane, and nonlinear confidence interval estimates. The curvestraightening routine does not lend itself well to standard statistics. Thus, estimates of the quality of the fit are obtained by examining the quadratic coefficient c2, which indicates the curvature of the line fitted to the data, in the results of a systematic variation of T~~ about the convergence value. Accurate results are indicated by a c, coefficient for T~q at least an order of magnitude smaller than
for T~,q. T'~q. is assumed to be below T~J. so that the plot of ln(Tb - Tgq) vs t curves upward. Since the effect of a small difference between T'~q and T~,q is least in the region nearest t = 0, a straight line through a few points near the origin is a fair approximation to the correct slope. A quadratic polynomial may be fitted to the plot, as shown by the dash-dot line in Fig. 2, so that ln(T~ - T~ ~) ~ Co + c l t + c:t:.
(9)
The dashed line. defined by ln(Th - T'l q) = Co + cir. is a good approximation to the line through the first few points on the curve if T~q is not too far from Tr,q. Thus, a better guess for T~,q would be a new trial value T?. chosen so that ln(T. - T~q) = Co + ctt.
(10)
Then using equation (10) to eliminate co + c~t in equation (9) and solving the resulting expression for T~q gives: T~q = T t l t ) -
[ T d t ) - T'~] e -c-'r.
(11)
Equation (!1) is evaluated with t chosen so that ln(Tb--T'~) ~ 0.5. Then T ~ q may be used to plot l n ( T b - T$q), and the procedure repeated to obtain T$L..T~L..T~ ~ until the desired accuracy is achieved. Data resulting in (Tb -- T~'q) less than about ten-times the resolution of the temperature recorder must be excluded from each step of the analysis because the random error scatter in l n ( T b - T~'q) will be excessive for these points. The algorithm will otherwise converge on an excessively low value for T~~ and a consequently excessively large value of z because a random data error of - 6 in T~ results in a larger error in ln(T~ - T~¢~) than an error of + 6. This problem might be avoided by a routine that would shift
Table 2. Analysis of various fractions of a cooling curve experiment on the taxidermic model of Passer domesticus done in an environmental chamber with 156cm/s laminar wind. 5?0 turbulence, air temperature controlled to + 0.05°C i
A.
i
Results from NEWTON I
i
i
c 2 for
Tbl
Tb f
Tneq
T
o /T*
Tneq-.05
Tneq
Tneq+.05
N §
32.50
33.89
1.39
0.44
10.94
12.33
1.39
0.37
226.4
0.000
. l l x l 0 -6
.60x10 - 9
- . l l x l 0 -6
38
232.1
0.025
.15xlO -6
.18xi0 -8
-.16xlO -6
4.77
6.16
1.39
27
0.34
234.8
0.037
.20xlO - 6
.73x10 - 8
- . 2 1 x l O -6
2.66
4.05
19
1.39
0.28
242.2
0.070
.24x10 -6
.31x10 - 7
- . 2 1 x l O -6
1.33
14
2,72
1.39
0.13
264.2
0.167
.25x10 - 6
.45x10 -7
- . 1 8 x l 0 -6
9
21.56
33.89
12.33
0.52
222.3
-0,018
.31x10 -7
.74x10 - 8
- . 2 2 x 1 0 -7
12
8.25
12.33
4.05
0.43
229.4
-0.013
.92x10 -7
, l l x l O -7
~.75x10 -7
14
2.66
4.05
1.39
0.28
242.2
0.070
.24x10 - 7
.31x10 -7
- . 2 1 x 1 0 -7
14
o /~'
N
bTb
B.
Results from NEWTON 11
~T b
Tbi
Tbf
Tneq
s.e.
0.54 0.40 0.39 0.33 0.18
0.017 0.016 0.030 0.057 0.228
223.0 230.5 231.0 237.3 257.6
0.41 0.83 2.49 6.21 32.00
0.43 0.46 Q. 33 I I
0.228 0.089 0.057
223.2 228.3 237.3
2.42 3.02 6.21
32.50 10.94 4.78 2.66 1.33
33.89 12.33 6.16 4.05 2.72
1.39 1.39 1.39 1.39 1.39
21.56 8.27 2.66
33.89 12.33 4.06
12,33 4.06 1.39
T
I
s.e.
0.000 0.030 0,036 0.063 0.152
38 27 19 14 9
0.001 0.024 0.063 III I
12 14 14 II
Temperatures are in °C, ~ in sec. Deviations a are with respect to ¢ for the entire data set. Note that as little as 2.5-YC of cooling is needed to determine the time constant ~. * The deviation a is computed with respect to the value of ~ in the first row of subtable A t The value of ~ is relative to the value of r in the first row of subtable B N is the number of data points in the curve segment analyzed
Improved method for determining thermal conductance and equilibrium body temperature the c, coetScients found for T~q + 0.05. Larger differences indicate better accuracy. RESULTS AND CONCLUSIONS
Tables 2, 3, and 4 present the results of the analysis of three cooling curve experiments. Each table is subdivided, subtable A giving results from NEWTON I, and subtable B giving results from NEWTON II. Table 2 gives an ideal subject (a taxidermic model of Passer domesticus constructed with a hollow copper body for studies of standard operative environmental temperature [Bakken, 1976a-I) under ideal experimental conditions of laminar air flow and closely ( + 0.05°C) controlled air temperature. The first group of data in each sub-table is for successively shorter portions of the first, middle, and final thirds of the data. The results of the two programs given in the two sub-tables are generally separated by less than one standard error. While the two programs are thus of similar accuracy, the nonlinear least-squares routine has the advantage of providing error statistics. Both programs show longer time constants for segments of the cooling curve nearest equilibrium. This is partly a result of the cooling of the surface of the bird model. The radiative thermal conductance R changes by 20% for a 30°C change in surface temperature, resulting in a 3-4°/o change in Ko and 3 (see Equation 4). The remainder of the change is due to the weighting of errors by NEWTON I, and by the failure of the model or animal to satisfy the assumption that it has an isothermal core containing all of the stored heat. Note that values of T for segments of the cooling curve containing as few as 14 data points (corre-
sponding to about 1.33) differ from the value found for the whole curve by less than 7%. This difference drops to the expected 3.5~o for segments with 20 points (1.93). Thus. good data may be obtained from short experiments, and different parts of the cooling curve may be analyzed separately to detect physiological and postural changes in Tgq and ~ as well as the thermal radiation effect noted above. Table 3 illustrates the importance of careful control of environmental conditions during the experiment. The model used in Table 1 is being cooled in a polished metal paint-can respirometer inside a standard temperature-control cabinet, an arrangement commonly used until recently for physiological studies. The model conditions the paint-can environment by warming it above cabinet temperature. As the model cools, so does the paint can. Thus, the experiment shifts from a cooling curve on the model in the can to a cooling curve on the can in the temperature control cabinet, resulting in a monotonic decrease in T~ and increase in ~ as cooling progresses. This tendency is augmented by the effect of thermal radiation reflected by the metal walls of the chamber noted by Porter (1969), and the increase in z with cooling noted earlier. Table 4 shows the results of a cooling curve on a live lizard (Sceloporus occidentalis) in a chamber with a 128cm s- ~ laminar air flow and temperature control to __ 0.0YC. Note that good results may be obtained with 16 points (l.6z) covering a body temperature change of less than 3°C (about 60 x resolution of the recorder). This capability for determining 3 and thus Ko over narrow ranges of body temperature should be valuable in exploring physiological
Table 3. Analysis of various fractions of a cooling curve experiment on the taxidermic model of Passer domesticus done inside a paint can respirometer placed inside a constant-temperature cabinet at - 5 + 0.2°C. Gas flow rate was 350 sccm A.
Results from NENTON I
AT b
Tbi
Tbf
Tneq
T
O/T*
Tneq-.05
c~ for Tneq
Tneq+.05
N
35.00
36.50
1.50
-2.09
366.2
0.000
.41xi0-5
-.25xi0-6 -.36xi0 -5
61
11.00
12.50
1.50
-2.54
389.2
0.062
.46x10-5
-.lOxlO -6 -.53xi0-5
36
4.61
6.11
1.50
-3.61
458.2
0.251
.32xi0-5
-.30xlO -6 -.41xlO -5
21
2.39
3.89
1.50
-7.51
745.2
1.034
.14x10 - 6
- . 9 5 x 1 0 -6 - . 9 6 x 1 0 - 5
13
20.44
36.50
16.05
0.23
326.9
-0.107
.33xi0-5
.86xi0-6 -.10xlO -5
20
9.94
16.05
6.11
-1.64
356.6
0.026
.44x10 - 5
- . 5 5 x 1 0 -6 - . 3 0 x 1 0 -5
22
4.61
6.11
1.50
-3.61
458.2
0.251
.32xi0-5
-.30xlO-6 -.41xlO-5
21
B. Results from NEWTON II ATb
Tbi
Tbf
35.00 11.00 4.61 2.39
36.50 12.50 6.11 3.89
1.50 1.50 1.50 1.50
20.44 9.94 4.61
36.50 16.05 6.11
16.05 6.11 1.50
Tneq
173
s .e.
T
s .e.
o/T t
-1.81 -2.34 -3.38 -4.56
0.059 0.118 0.057 1.38
358.1 378.0 443.6 382.8
1.50 5.43 36.75 103.60
0.000 0.055 0.240 0.069
61 36 21 13
1.13 -2.56 -3.38-
0.394 0.416 0.057
321.5 385.7 443.6
5.23 12.37 36,75
-0.104 0.075 0.240
20 22 . 21
The generally poorer results are due to the poor temperature control of this conventional system. * The deviation a is calculated with respect to the value of ~ in the first row of subtable A. t The deviation a is calculated with respect to the value of ~ in the first row of subtable B.
174
GeorGe S. BAKKEN
Table 4. Analysis of various portions of the data from a cooling curve experiment on a live lizard, Sceloporus occidentalis, conducted in an environmental chamber with a 128 cm/s laminar wind. 5% turbulence, and air temperature controlled to ± 0.05°C I
I
II
A. Results from NENTON I
ATb
Tb£
6.94 2.67
c2 f o r
Tbf
Tneq
•
31.89 27.61
24.94 24.94
24.23 24.22
577.3 582.3
o~ T
Tneq_05
Tn eq
Tneq+.05
N
0.000 0.009
.37x10 - 7 .55xi0 -7
.30x10 - 9 .49x10 - 8
-.41xlO -7 -.54x10 -7
24 16
1.78
26.72
24.94
24.32
534.0* -0.075
.66xi0 -7
-.Ilxl0 -7
-.10xlO -6
13
0.89
25.83
24.94
23.74
874.7
.18xl0 -7
-.44x10 -8
-.30x10 -6
9
5.17
31.89
26.72
23.62
658.3
0.140
.[lxl0 -7
-.llxlO -8
-.13x10 -7
12
1.78
26.72
24.94
24.32
534.0
-0.075
.66xi0 -7
-.llxl0 -7
-.IOxl0 "6
13
B.
Results
ATb
Tb i
6.94 2.67 1.78 0.88 5.16 1.78 C.
0.515
from NEWTON I I Tbf
Tneq
s.e.
r
s.e.
O /x §
31.89 27.61 26.72 25.82
24.94 24.94 24.94 24.94
24.15 24.20 24.34 23.84
0.047 0.089 0.103 0.554
594.7 590.3 520.5* 820.0
9.3 30.5 41.3 303.2
0.000 -0.008 -0.125 0.378
24 16 13 9
31.89 26.72
26.72 24.94
23.34 24.84
0.286 0.103
694.5 520.5*
35.9 41.3
0.167 -0.125
12 13
R e s u l t s u s i n g d a t a n e a r Tneq, from NEWTON I I
ATb
Tbi
Tbf
Tneq
s.e.
7.38 3.I1 2.22 1.32 5.16 2.22
t
31.89 27.61 26.72 25.82
24.50 24.50 24.50 24.50
24.16 24.15 24.17 24.06
0.024 0.038 0.048 0.102
593.8 606.2 587.4* 684.0
6.15 17.7 27.1 76.6
0.002 0.019 -0.013 0.148
33 25 22 18
31.89 26.72
26.72 24.50
23.34 24.17
0.286 0.048
694.5, 587.4
35.9 27.1
0.167 -0.013
12 22
i
s.e.
i
i
o/t §
i
Note that only 2.5-5°C are required to measure r accurately. This is a great advantage when studying thermoregulatory responses of lizards at different body temperatures. * Decrease in t unexplained; apparently due to an unusually large deviation in one data point. i" Deviation a computed relative to z from first row of subtable A. § Deviation a computed relative to x from second row of subtable B. changes of heating and cooling rates in reptiles (Bartholomew & Tucker, 1963; Bartholomew & Lasiewsld, 1965; Morgareidge & White, 1969; Weathers, 1970). Table 4(C) shows that accurate results may be obtained for Tb changes as small as 2°C (40 x resolution) by approaching Tff closely, at the expense of increasing experimental time to 2.31. NEWTON II must be used for the analysis, since (T# - T~) assumes small values. NEWTON I may be slightly more accurate than NEWTON II in analyzing segments of the cooling curve well away from T~ (see Table 4) but the difference is marginal. Generally, NEWTON II will be preferred since it provides detailed error information. The total load and execution times of the two programs are comparable. Source decks and listings for either program are available from the author.
Acknowledgements--This work was supported by a faculty computer grant at Indiana State University, and by NSF Grant GB40980 and AEC Grant AT(11-I)-2164 at the University of Michigan. I wish to express my appre-
ciation to David M. Gates for his support and encouragement of the work done in his lab at the University of Michigan. I also wish to thank Julian P. Adams for valuable discussions, William A. Buttemer for assistance with the experimental work, and my wife Laura for editing and typing the manuscript. REFERENCES
BAKr,~N G. S. (1976) A heat transfer analysis of animals: unifying concepts and the application of metabolism chamber data to field ecology. J. Theor. B/o/. (in press). BAKKENG. S. & GA3~SD. M. (1975) H e a t - t ~ analysis of animals: some implications for field emlogy, physiology and evolution. In Perspectives in Biophysical E.COlogy (Edited by GATESD. M. & ~ g. 13.) pp. 255-290. Springer, New York. BAR'mOt.O~W G. A. & TUCKERV. A. (1963) Control of changes in body temperature, metabolism a n d circulation by the Agamid lizard, Amphibolwna barbaras. Physiol. Zool. 36, 199-218. BAR~IOLOS4mWG. A. & ~ R. C. (1965) Heating and cooling rates, heart rate and simulated diving in the Galapagos Marine Iguana. Comp. Biochem. Physiol. 16, 573-582.
Improved method for determining thermal conductance and equilibrium body temperature BIRKEBAKR. C. (1966) Heat transfer in biological systems. Int. Rer. Gen. Exp. Zoo. 2, 269-344. GAGGE A. P. & HARDY J. D, (1967) Thermal radiation exchange of the human by partitional calorimetry, J. appl. Physiol. 23, 248-258. MARQUARDTD. W. (1963~ An algorithm for least-squares estimation of nonlinear parameters. J. Soc. lndust. Appl. Math. II, 431-441. MARQUARDTO. W. (1966~ Least-squares estimation of nonlinear parameters. SHARE Program Library SDA 3094-01. MORGAREIDGEK. R. t~: ~t,VHITEF. N. (1969) Cutaneous vascular changes during heating and cooling in the Galapagos Marine Iguana. Nature, Lond. 223, 587-591. PORTER W. P. (1969) Thermal radiation in metabolic chambers. Science, Wash. 166, 115-117. PORTER W. P. & GATES D. M. (1969) Thermodynamic
175
equilibria of animals with environment. Ecol. Monogr. 39, 227-244. TaACY C. R. (1972) Newton's law: its application for expressing heat losses from homeotherms. BioSci. 22, 656-659. WEArnERS W. W. (1970) Physiological thermoregulation in the lizard Dipsosaurus dorsalis. Copeia 1970, 549-557. WINSLOW C, E. A., HERRINGTON L. P. & GAGGE A. P. (1937) Physiological reactions of the human body to varying environmental temperatures. Am. d. Physiol. 120(!), 1-22.
Key Word Index--Heat transfer; Newton's law of cooling; cooling curves; conductance; ectotherms; overall heat transfer coefficient; operative environmental temperature; time constant; metabolism and cooling curves.
AN IMPROVED METHOD FOR DETERMINING THERMAL CONDUCTANCE AND EQUILIBRIUM BODY TEMPERATURE WITH COOLING CURVE EXPERIMENTS GEORGE S. BAKKEN Department of Life Sciences, Indiana State University, Terre Haute, Indiana 47809, U.S.A.
(Received 29 May 1975; revised 11 November 1975) Al~tmet--l. The analysis of cooling curves to find the thermal conductance by plotting log(body temperature-air temperature) vs time is generally inaccurate. 2. Cooling curves are correctly analyzed by plotting log(body temperature--equilibrium body temperature) vs time. 3. Experiment time may be greatly reduced by using computer analysis to find equilibrium body temperature and the thermal conductance simultaneously. 4. Two computer programs for cooling curve analysis are described, together with representative results. 5. Accurate values of thermal conductance may be determined with cooling ranges of less than 5°C. INTRODUCTION
THE MOSTcommonly used experimental procedure for determining the thermal conductance K of an ectothermic animal has been the cooling (or heating) curve experiment. The animal is placed in an environment with an air temperature T~ different from the initial body temperature. The temperature of the animal, Tb (usually cloacal), is then recorded as a function of time t as T~ comes into equilibrium with the new thermal environment. Assuming that the body temperature follows "Newton's law of cooling", dTddt = - K ~ / C ( T b - T~).
(1)
The heat capacity of the animal is denoted by C, and KN is the Newton's law thermal conductance. The data is most conveniently analyzed by plotting log(T~ - T,) vs t. The slope of the resulting approximately straight line is" d log(Tb dt
Ta)
-0.4343 d i n ( T b - T,)-KN -- 0.4343 --C-. (2)
After C has been measured or calculated from the weight of the animal, K N may be determined from the slope of the plot. This procedure suffers from a number of erroneous assumptions. First, the animal cannot be characterized by a single Newton's law conductance K~; rather, the rate of heat transfer depends on environmental parameters such as wind speed O'racy, 1972). Second, the environmental temperature is not T,, but rather is determined by a combination of conductive, convective and radiative factors (Bakken & Gates, 1975; Bakken, 1976). Third, the endpoint of a cooling curve is not the environmental temperature, but the final equilibrium temperature of the animal (Bakken & Gates, 1975), which depends on metabolism and evaporative water loss. Fourth, the slope of the plot
does not give KN/C if the animal has metabolic heat production which varies with Tb (Bakken & Gates, 1975). Thus, the standard analysis based on equations (1) and (2) is generally inaccurate. However, cooling curve experiments are still valid, provided that a correct energy budget analysis (Birkebak, 1966; Porter & Gates, 1969) is used. Bakken & Gates (1975) and Bakken (1976) applied energy budget analysis to a general animal with metabolic heat production and variable Tb. They presented a linear approximation to overall heat transfer, arranged to be as similar to "Newton's law" as possible: dT b 1 Tb T, + (3) dt z The effective net metabolic heat production, M*, is essentially just the dry metabolic heat production M-E, where M is the metabolic heat produced and E is the heat dissipated by evaporation. A more accurate expression separates E according to the site of evaporation, and includes correction factors for the degree of thermal coupling of each site to the body core so that M* <_ M - E (Bakken, 1976). The factor is the time constant (sometimes called the cooling constant), and is equal to the time required for (l-l/e) 63% of the total response to a sudden change in the thermal environment to occur. The time constant, ~, is equal to C/Ko only if M* is constant and independent of T~, The overall thermal conductance Ko, functionally equivalent to the, Newton's law thermal conductance KN, is: Ko = (Ksf + G) (H + R) + GKsf K,f + H + R
(4)
The thermal conductance to the ground is G, H is the convective conductance to the air, and K,s is the thermal conductance of the integument. The thermal radiation conductance is R = 4A,.aE~ 3. where T, is 169
170
GEORGE
S.
BAKKEN
,
the time-average surface temperature of the animal in kelvins. K--- =C + 273.16: a is the Stefan-Boltzmann constant, 5.67 x 10 -8 W m -2 K-'*; e is the emissivity of the surface of the animal for thermal radiation, typically 0.95-0.98; and A,, is the effective area of the animal for the emission or absorption of radiation, in m-'. In S.I. units, thermal conductances are expressed in W K-~ animal-~, heat capacity in J K - ~ animal- ~, and rates of heat production or loss due to physiological processes in the animal in W animal- ~. The temperature of the environment for a specific animal is
,
.
,
,
,
o
;10
N
\\"-.
,
--
C-T.
2XX,x---
.,
\ X
0 I.¢ o Q. E
"-.,
II .~
T,, = K~s(HT" + RT, + Q, + GTg) + GTg(H + R) (5) (K~s + G) (H + R) + GK~ I.
"1"•',: T a
o IIl
OA
and is called the operative enviromnental temperature. The ground temperature is Tr and net absorbed visible and thermal radiation when the actual surface temperature T~ equals the time-average surface temperature T, is denoted by Q,. (This complex definition arises from the linearization of the Stefan-Boltzmann law for thermal radiation. See Bakken, 1976, for a discussion.) The overall thermal conductance and operative environmental temperature are generalizations of the operative heat exchange coefficient and operative temperature proposed by Winslow, et al. (1937), and Gagge & Hardy 0967). The analysis of Bakken & Gates 0975) and Bakken (1976) leads to the following conclusions about cooling curve experiments: 1. Equation (5) shows that the overall thermal conductance K0 is a function of both the animal and its environment, since G depends on the nature of the surface under the animal and, as noted by Tracy (1972), the coefficient H depends on wind velocity. 2. The reference temperature of the environment is T,,, not T~. The operative environmental temperature is a complex function of both the animal and its environment, and can differ significantly from air temperature. 3. Comparison of equations (1) and (3) shows that may be found from the slope d 1 dT log(Tn - T~h) = --0.4343 -r (6)
,
~
,
Time
Fig. 1. Three plots of log(~ - ~) vs time. assuming equilibrium body temperature T~,~ equal to, g r a t e r than, and
less than T~. The difference is I'C in the cases where T~~ T~. Note that if T~~ ¢ T~ for any reason, such as an effective ambient radiation temperature = 7",, metabolic heat production, or evaporative cooling, the plot is not a straight line. This curvature prevents an accurate determination of the time constant and overall heat transfer coefficient. of a plot of log(Th - T};q) vs t, but not of log (T~ - TD vs t. The equilibrium body temperature, T~,~ = T,, + M*/Ko, is reached as t - ~ :r_. The dependence of Ko in wind speed and surface properties simply implies that a range of wind and substrate conditions should be used to investigate the range of Ko values which may obtain in the natural environment, and presents no difficulty to the cooling curve procedure per se. The problems resulting from the difference between T~ and Tg{ = T,. + M*/Ko are more serious, since they affect the accuracy of the determination of 3. The nature of the problem is shown in Fig. 1. If the actual equilibrium body temperature Tg* is higher than the assumed equilibrium temperature T~, the curve will bend up rather than continuing as a straight line. If T~ is lower than Tr the curve will bend down. Any attempt to assign a single value to the slope of a curved line to obtain r will necessarily
Table 1. Analysis of various fractions of a cooling experiment on a copper-bodied taxidermic model of Passer domesticus, using a straight-line fit to find the time constant T i
iii i ii
ii
"~for * T for *§ Tbeq-o.055 T b e q
X for * dx/x for t Tbeq+0.055 ATbeq-o.1
ATb
Tbi
Tbf
Tbeq
32.50 29.55 22.05
33.89 30.94 23.44
1.39 1.39 1.39
0.44 0.44 0.42
229.2 229.6 231.3
226.4 226.7 228.1
223.6 223.8 224.8
0.022 0.023 0.026
16.33 10.94 4.78
17.72 12.33 6.17
1.39 1.39 1.39
0.40 0.37 0.34
233.4 236.1 240.2
229.8 232.1 234.8
226.3 228.0 229.4
0.028 0.032 0.041
I
I
II
Temperatures are in °C, ~ in set. * These columns show the effect of T of an 0.055°C (0.1°F) difference between air temperature and the equilibrium body temperature TJq. t" Fractional error in z that would result from using air temperature rather than Tb"~ in the cooling curve plot if they differ by 0.1=C. .~The monotonic increase in T for segments of the data nearest Tseq is due to the change in radiative thermal conductance discussed in the text.
Improved method for determining thermal conductance and equilibrium body temperature
171
give poor results. The problem is quite significant; over a small temperature range where an average P Table 1 shows that an error of 0.1°C in the choice can be defined by using a sensitive temperature sensor of an equilibrium temperature results in an error of and a well-controlled environmental _chamber, 2.2-4.1%, depending on the range of cooling used. together with numerical analysis using procedure (3) There are four procedures for eliminating this or (4) above. error: One difficulty in designing a computer algorithm (1). A properly constructed chamber can have the to find T[ ~ and t simultaneously is in deciding how wall temperature equal to T,, which sets T , , - T~ if to weight the data. There are two possible sources the walls are black (Bakken, 1976). Then independent of error: First, the recorder used to measure body knowledge of M* can be used in a procedure de- temperature will introduce uniform variance into the scribed by Bartholomew and Tucker (1963) to correct Th data. These errors will be most significant at the for metabolism and evaporation. This procedure is end of the curve when (Th -- T[ q) is small. Second, inaccurate, since it still attempts to assign a single a live animal does not completely satisfy the assumpvalue to the slope of a curved line. tion that the body is a single, isothermal lump with (2). The experiment can be run until Th = T~~, a single temperature. Different parts of the body cool which typically requires a time equal to 5-7x, depend- at slightly different rates, so that heat will flow from ing on the desired accuracy. While an improvement, one part of the body to another. The actual course this approach also has problems: (A) It uses excessive of body temperature will thus differ somewhat from time, since the required information is obtained in the result of Equation (3). These errors will be unia length of time less than 2T. (B) The animal may form in ln(Tb- T~), and the greatest effect will be change posture, altering H, G, A,~ and thus K0. (C) at the beginning of the experiment. The first source The animal may struggle, changing M* and thus T~q. of error would imply a least-squares routine with (D) The apparatus may not maintain a constant ther- linear weighting; the second source of error implies mal environment for the full 5-7r. (E) The animal weighting by ln(T~ - T[q). may abruptly begin a thermoregulatory response and alter T~a. (F) The metabolic rate of the animal may vary with T~, so that M* and thus T[ q change conMATERIALS AND M E T H O D S tinuously during the experiment. (G) The respiratory To explore the various possible approaches. I decided rate, and thus evaporation, may similarly vary with Th. (H) The vapor pressure of the water on the skin to write two computer programs for numerical coolingand respiratory and buccal mucosa varies with T~, curve analysis. The first program is based on the curvesimilarly altering T~~ continuously during the exper- straightening approach, and weights data as ln(Tb -- T~). The second program is based on a nonlinear least-squares iment. algorithm which weights data uniformly. 3. Since equation (6) shows that a plot of The program for finding Tg~ and z by curve straightenlog(Tb -- T[ q) vs t gives a straight line, the straightness ing is called NEWTON I. The derivation of the algorithm of the line resulting from an estimate of T~,q may be used follows from Fig. 2. I will use In rather than log used as a criterion of the accuracy of the estimate. to avoid the 0.4343 conversion factor. The initial guess Then estimates of Tff can be systematically improved i r , , ~ i to eliminate the curvature of the line and find the correct values of Tff and r. This reduces the experimental time and minimizes the frequency of problems (A)--(E), but is still subject to problems 2(F)--(H). (4). Finally, one of a number of numerical techniques for nonlinear least-squares analysis may be used to find T ~ and x directly, This approach is equivalent to the preceding one in the sources of experimental error. Mathematically, it is easier to compute confidence limits with this approach, but it may be more sensitive to the strong correlation between Tff and x. ., a.CT. -Tj ~ ~ \ . t Although even a correct value of z is not equal to C / K o when M* varies as Tb changes, the overall thermal conductance may still be determined by using the results of Bakken & Gates (1975). If the variation 1 i I i 1 r Time, t in M - E is linear with T~, so that M(Tb) - E(T~) -~ (M - E)v + P(T~ - Tn), (7) Fig. 2. The algorithm for finding the equilibrium body temperature T~q and the time constant z simultaneously then ~ = C / ( K o - P ) and proceeds by estimating Tff as an initial value Tff and plotting In(T~ -- T~l. The line curves up. indicating that T~ Ko = (C/x) + e. (8) is too low. Then a quadratic function, Co + c t t + c2t 2, is Here, T~ is an arbitrary body temperature, (M - E)~ fitted to the plot, and the first two terms, Co + c d . define is the value of [M(T~) - E(T~)] when T~ = T~, and a straight line which gives a fair approximation to the plot of ln(Tb -- T~q). A new estimate for T~. T~q, follows P -- d/dT~[M(T~) - E(Td], averaged for the tempera- from setting ln(Tb -- T ~ ) = co + c,t. as described in the ture range of the cooling curve. A linear variation text. The process is repeated until an estimate of T[q is in M and E is not usually the case over a wide tem- found close enough to the correct value to give a straight perature range. However, the animal can be cooled line in the plot.
172
GEORGE S. BAKKEN
to a linear weighting of data errors for small values of ln(Tb -- ~ 5 . at the expense of considerable complexity. The second program, called N E W T O N II, uses an optimized interpolation between the Taylor series and steepest descent methods for nonlinear least-squares curve fitting developed by Marquardt (1963). The program itself is based on a SHARE library distribution program (Marquardt, 1966), with considerable modification to update the programming and provide specialized input and output. Both programs use the same input file sequence so that no modifications of the input data are needed to change analysis Up to 200 data points may be used, consisting of pairs of time and T~ data, with the temperature scale identified by a F, C. or K designator. Any subset of the data (segment of the cooling curve) m R be selected by entering the initial and final time coordinates of the desired experimental interval This allows the data to be examined for possible changes in T~'~ or z during the experiment resulting from physiological or postural changes of the anim a l The programs print the results of each iteration When the convergence condition is salisfied, the values of Tdt = 0), Tgq, and ~ are printed, together with a semilogarithmic printer plot of the data and fit. and error information. The nonlinear least-squares program provides several statistical error estimates for each parameter, including standard error, one-parameter, support plane, and nonlinear confidence interval estimates. The curvestraightening routine does not lend itself well to standard statistics. Thus, estimates of the quality of the fit are obtained by examining the quadratic coefficient c2, which indicates the curvature of the line fitted to the data, in the results of a systematic variation of T~~ about the convergence value. Accurate results are indicated by a c, coefficient for T~q at least an order of magnitude smaller than
for T~,q. T'~q. is assumed to be below T~J. so that the plot of ln(Tb - Tgq) vs t curves upward. Since the effect of a small difference between T'~q and T~,q is least in the region nearest t = 0, a straight line through a few points near the origin is a fair approximation to the correct slope. A quadratic polynomial may be fitted to the plot, as shown by the dash-dot line in Fig. 2, so that ln(T~ - T~ ~) ~ Co + c l t + c:t:.
(9)
The dashed line. defined by ln(Th - T'l q) = Co + cir. is a good approximation to the line through the first few points on the curve if T~q is not too far from Tr,q. Thus, a better guess for T~,q would be a new trial value T?. chosen so that ln(T. - T~q) = Co + ctt.
(10)
Then using equation (10) to eliminate co + c~t in equation (9) and solving the resulting expression for T~q gives: T~q = T t l t ) -
[ T d t ) - T'~] e -c-'r.
(11)
Equation (!1) is evaluated with t chosen so that ln(Tb--T'~) ~ 0.5. Then T ~ q may be used to plot l n ( T b - T$q), and the procedure repeated to obtain T$L..T~L..T~ ~ until the desired accuracy is achieved. Data resulting in (Tb -- T~'q) less than about ten-times the resolution of the temperature recorder must be excluded from each step of the analysis because the random error scatter in l n ( T b - T~'q) will be excessive for these points. The algorithm will otherwise converge on an excessively low value for T~~ and a consequently excessively large value of z because a random data error of - 6 in T~ results in a larger error in ln(T~ - T~¢~) than an error of + 6. This problem might be avoided by a routine that would shift
Table 2. Analysis of various fractions of a cooling curve experiment on the taxidermic model of Passer domesticus done in an environmental chamber with 156cm/s laminar wind. 5?0 turbulence, air temperature controlled to + 0.05°C i
A.
i
Results from NEWTON I
i
i
c 2 for
Tbl
Tb f
Tneq
T
o /T*
Tneq-.05
Tneq
Tneq+.05
N §
32.50
33.89
1.39
0.44
10.94
12.33
1.39
0.37
226.4
0.000
. l l x l 0 -6
.60x10 - 9
- . l l x l 0 -6
38
232.1
0.025
.15xlO -6
.18xi0 -8
-.16xlO -6
4.77
6.16
1.39
27
0.34
234.8
0.037
.20xlO - 6
.73x10 - 8
- . 2 1 x l O -6
2.66
4.05
19
1.39
0.28
242.2
0.070
.24x10 -6
.31x10 - 7
- . 2 1 x l O -6
1.33
14
2,72
1.39
0.13
264.2
0.167
.25x10 - 6
.45x10 -7
- . 1 8 x l 0 -6
9
21.56
33.89
12.33
0.52
222.3
-0,018
.31x10 -7
.74x10 - 8
- . 2 2 x 1 0 -7
12
8.25
12.33
4.05
0.43
229.4
-0.013
.92x10 -7
, l l x l O -7
~.75x10 -7
14
2.66
4.05
1.39
0.28
242.2
0.070
.24x10 - 7
.31x10 -7
- . 2 1 x 1 0 -7
14
o /~'
N
bTb
B.
Results from NEWTON 11
~T b
Tbi
Tbf
Tneq
s.e.
0.54 0.40 0.39 0.33 0.18
0.017 0.016 0.030 0.057 0.228
223.0 230.5 231.0 237.3 257.6
0.41 0.83 2.49 6.21 32.00
0.43 0.46 Q. 33 I I
0.228 0.089 0.057
223.2 228.3 237.3
2.42 3.02 6.21
32.50 10.94 4.78 2.66 1.33
33.89 12.33 6.16 4.05 2.72
1.39 1.39 1.39 1.39 1.39
21.56 8.27 2.66
33.89 12.33 4.06
12,33 4.06 1.39
T
I
s.e.
0.000 0.030 0,036 0.063 0.152
38 27 19 14 9
0.001 0.024 0.063 III I
12 14 14 II
Temperatures are in °C, ~ in sec. Deviations a are with respect to ¢ for the entire data set. Note that as little as 2.5-YC of cooling is needed to determine the time constant ~. * The deviation a is computed with respect to the value of ~ in the first row of subtable A t The value of ~ is relative to the value of r in the first row of subtable B N is the number of data points in the curve segment analyzed
Improved method for determining thermal conductance and equilibrium body temperature the c, coetScients found for T~q + 0.05. Larger differences indicate better accuracy. RESULTS AND CONCLUSIONS
Tables 2, 3, and 4 present the results of the analysis of three cooling curve experiments. Each table is subdivided, subtable A giving results from NEWTON I, and subtable B giving results from NEWTON II. Table 2 gives an ideal subject (a taxidermic model of Passer domesticus constructed with a hollow copper body for studies of standard operative environmental temperature [Bakken, 1976a-I) under ideal experimental conditions of laminar air flow and closely ( + 0.05°C) controlled air temperature. The first group of data in each sub-table is for successively shorter portions of the first, middle, and final thirds of the data. The results of the two programs given in the two sub-tables are generally separated by less than one standard error. While the two programs are thus of similar accuracy, the nonlinear least-squares routine has the advantage of providing error statistics. Both programs show longer time constants for segments of the cooling curve nearest equilibrium. This is partly a result of the cooling of the surface of the bird model. The radiative thermal conductance R changes by 20% for a 30°C change in surface temperature, resulting in a 3-4°/o change in Ko and 3 (see Equation 4). The remainder of the change is due to the weighting of errors by NEWTON I, and by the failure of the model or animal to satisfy the assumption that it has an isothermal core containing all of the stored heat. Note that values of T for segments of the cooling curve containing as few as 14 data points (corre-
sponding to about 1.33) differ from the value found for the whole curve by less than 7%. This difference drops to the expected 3.5~o for segments with 20 points (1.93). Thus. good data may be obtained from short experiments, and different parts of the cooling curve may be analyzed separately to detect physiological and postural changes in Tgq and ~ as well as the thermal radiation effect noted above. Table 3 illustrates the importance of careful control of environmental conditions during the experiment. The model used in Table 1 is being cooled in a polished metal paint-can respirometer inside a standard temperature-control cabinet, an arrangement commonly used until recently for physiological studies. The model conditions the paint-can environment by warming it above cabinet temperature. As the model cools, so does the paint can. Thus, the experiment shifts from a cooling curve on the model in the can to a cooling curve on the can in the temperature control cabinet, resulting in a monotonic decrease in T~ and increase in ~ as cooling progresses. This tendency is augmented by the effect of thermal radiation reflected by the metal walls of the chamber noted by Porter (1969), and the increase in z with cooling noted earlier. Table 4 shows the results of a cooling curve on a live lizard (Sceloporus occidentalis) in a chamber with a 128cm s- ~ laminar air flow and temperature control to __ 0.0YC. Note that good results may be obtained with 16 points (l.6z) covering a body temperature change of less than 3°C (about 60 x resolution of the recorder). This capability for determining 3 and thus Ko over narrow ranges of body temperature should be valuable in exploring physiological
Table 3. Analysis of various fractions of a cooling curve experiment on the taxidermic model of Passer domesticus done inside a paint can respirometer placed inside a constant-temperature cabinet at - 5 + 0.2°C. Gas flow rate was 350 sccm A.
Results from NENTON I
AT b
Tbi
Tbf
Tneq
T
O/T*
Tneq-.05
c~ for Tneq
Tneq+.05
N
35.00
36.50
1.50
-2.09
366.2
0.000
.41xi0-5
-.25xi0-6 -.36xi0 -5
61
11.00
12.50
1.50
-2.54
389.2
0.062
.46x10-5
-.lOxlO -6 -.53xi0-5
36
4.61
6.11
1.50
-3.61
458.2
0.251
.32xi0-5
-.30xlO -6 -.41xlO -5
21
2.39
3.89
1.50
-7.51
745.2
1.034
.14x10 - 6
- . 9 5 x 1 0 -6 - . 9 6 x 1 0 - 5
13
20.44
36.50
16.05
0.23
326.9
-0.107
.33xi0-5
.86xi0-6 -.10xlO -5
20
9.94
16.05
6.11
-1.64
356.6
0.026
.44x10 - 5
- . 5 5 x 1 0 -6 - . 3 0 x 1 0 -5
22
4.61
6.11
1.50
-3.61
458.2
0.251
.32xi0-5
-.30xlO-6 -.41xlO-5
21
B. Results from NEWTON II ATb
Tbi
Tbf
35.00 11.00 4.61 2.39
36.50 12.50 6.11 3.89
1.50 1.50 1.50 1.50
20.44 9.94 4.61
36.50 16.05 6.11
16.05 6.11 1.50
Tneq
173
s .e.
T
s .e.
o/T t
-1.81 -2.34 -3.38 -4.56
0.059 0.118 0.057 1.38
358.1 378.0 443.6 382.8
1.50 5.43 36.75 103.60
0.000 0.055 0.240 0.069
61 36 21 13
1.13 -2.56 -3.38-
0.394 0.416 0.057
321.5 385.7 443.6
5.23 12.37 36,75
-0.104 0.075 0.240
20 22 . 21
The generally poorer results are due to the poor temperature control of this conventional system. * The deviation a is calculated with respect to the value of ~ in the first row of subtable A. t The deviation a is calculated with respect to the value of ~ in the first row of subtable B.
174
GeorGe S. BAKKEN
Table 4. Analysis of various portions of the data from a cooling curve experiment on a live lizard, Sceloporus occidentalis, conducted in an environmental chamber with a 128 cm/s laminar wind. 5% turbulence, and air temperature controlled to ± 0.05°C I
I
II
A. Results from NENTON I
ATb
Tb£
6.94 2.67
c2 f o r
Tbf
Tneq
•
31.89 27.61
24.94 24.94
24.23 24.22
577.3 582.3
o~ T
Tneq_05
Tn eq
Tneq+.05
N
0.000 0.009
.37x10 - 7 .55xi0 -7
.30x10 - 9 .49x10 - 8
-.41xlO -7 -.54x10 -7
24 16
1.78
26.72
24.94
24.32
534.0* -0.075
.66xi0 -7
-.Ilxl0 -7
-.10xlO -6
13
0.89
25.83
24.94
23.74
874.7
.18xl0 -7
-.44x10 -8
-.30x10 -6
9
5.17
31.89
26.72
23.62
658.3
0.140
.[lxl0 -7
-.llxlO -8
-.13x10 -7
12
1.78
26.72
24.94
24.32
534.0
-0.075
.66xi0 -7
-.llxl0 -7
-.IOxl0 "6
13
B.
Results
ATb
Tb i
6.94 2.67 1.78 0.88 5.16 1.78 C.
0.515
from NEWTON I I Tbf
Tneq
s.e.
r
s.e.
O /x §
31.89 27.61 26.72 25.82
24.94 24.94 24.94 24.94
24.15 24.20 24.34 23.84
0.047 0.089 0.103 0.554
594.7 590.3 520.5* 820.0
9.3 30.5 41.3 303.2
0.000 -0.008 -0.125 0.378
24 16 13 9
31.89 26.72
26.72 24.94
23.34 24.84
0.286 0.103
694.5 520.5*
35.9 41.3
0.167 -0.125
12 13
R e s u l t s u s i n g d a t a n e a r Tneq, from NEWTON I I
ATb
Tbi
Tbf
Tneq
s.e.
7.38 3.I1 2.22 1.32 5.16 2.22
t
31.89 27.61 26.72 25.82
24.50 24.50 24.50 24.50
24.16 24.15 24.17 24.06
0.024 0.038 0.048 0.102
593.8 606.2 587.4* 684.0
6.15 17.7 27.1 76.6
0.002 0.019 -0.013 0.148
33 25 22 18
31.89 26.72
26.72 24.50
23.34 24.17
0.286 0.048
694.5, 587.4
35.9 27.1
0.167 -0.013
12 22
i
s.e.
i
i
o/t §
i
Note that only 2.5-5°C are required to measure r accurately. This is a great advantage when studying thermoregulatory responses of lizards at different body temperatures. * Decrease in t unexplained; apparently due to an unusually large deviation in one data point. i" Deviation a computed relative to z from first row of subtable A. § Deviation a computed relative to x from second row of subtable B. changes of heating and cooling rates in reptiles (Bartholomew & Tucker, 1963; Bartholomew & Lasiewsld, 1965; Morgareidge & White, 1969; Weathers, 1970). Table 4(C) shows that accurate results may be obtained for Tb changes as small as 2°C (40 x resolution) by approaching Tff closely, at the expense of increasing experimental time to 2.31. NEWTON II must be used for the analysis, since (T# - T~) assumes small values. NEWTON I may be slightly more accurate than NEWTON II in analyzing segments of the cooling curve well away from T~ (see Table 4) but the difference is marginal. Generally, NEWTON II will be preferred since it provides detailed error information. The total load and execution times of the two programs are comparable. Source decks and listings for either program are available from the author.
Acknowledgements--This work was supported by a faculty computer grant at Indiana State University, and by NSF Grant GB40980 and AEC Grant AT(11-I)-2164 at the University of Michigan. I wish to express my appre-
ciation to David M. Gates for his support and encouragement of the work done in his lab at the University of Michigan. I also wish to thank Julian P. Adams for valuable discussions, William A. Buttemer for assistance with the experimental work, and my wife Laura for editing and typing the manuscript. REFERENCES
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175
equilibria of animals with environment. Ecol. Monogr. 39, 227-244. TaACY C. R. (1972) Newton's law: its application for expressing heat losses from homeotherms. BioSci. 22, 656-659. WEArnERS W. W. (1970) Physiological thermoregulation in the lizard Dipsosaurus dorsalis. Copeia 1970, 549-557. WINSLOW C, E. A., HERRINGTON L. P. & GAGGE A. P. (1937) Physiological reactions of the human body to varying environmental temperatures. Am. d. Physiol. 120(!), 1-22.
Key Word Index--Heat transfer; Newton's law of cooling; cooling curves; conductance; ectotherms; overall heat transfer coefficient; operative environmental temperature; time constant; metabolism and cooling curves.