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Cutaneous Heat Flux Models do not Reliably Predict Metabolic Rates of Marine Mammals
J. theor. Biol. (2000) 207, 317}323 doi:10.1006/jtbi.2000.2176, available online at http://www.idealibrary.com on
Cutaneous Heat Flux Models do not Reliably Predict Metabolic Rates of Marine Mammals P. BOILY*-, P. H. KVADSHEIM? AND L. P. FOLKOW? *Department of Biological Sciences, ;niversity of New Orleans, New Orleans, ¸A 70148, ;.S.A. and ? Department of Arctic Biology, ;niversity of ¹roms~, N-9037 ¹roms~, Norway (Received on 14 October 1999, Accepted in revised form on 21 August 2000)
Heat #ux models have been used to predict metabolic rates of marine mammals, generally by estimating conductive heat transfer through their blubber layer. Recently, Kvadsheim et al. (1997) found that such models tend to overestimate metabolic rates, and that such errors probably result from the asymmetrical distribution of blubber. This problem may be avoided if reliable estimates of heat #ux through the skin of the animals are obtained by using models that combine calculations of conductive heat #ux through the skin and fur, and convective heat #ux from the surface of the animal to the environment. We evaluated this approach based on simultaneous measurements of metabolic rates and of input parameters necessary for heat #ux calculations, as obtained from four harp seals (Phoca groenlandica) resting in cold water. Heat #ux estimates were made using two free convection models (double-#at-plate and cylindrical geometry) and one forced convection model (single-#at-plate geometry). We found that heat #ux estimates generally underestimated metabolic rates, on average by 26}58%, and that small variations in input parameters caused large variations in these estimates. We conclude that cutaneous heat #ux models are too inaccurate and sensitive to small errors in input parameters to provide reliable estimates of metabolic rates of marine mammals. 2000 Academic Press
Introduction Energetic and behavioral studies of marine mammals often require data on their metabolic rates or their lower critical temperature. For some marine mammals, in particular for large whales, such variables cannot be measured directly. Instead, theoretical heat #ux calculations can be used to estimate metabolic rates and lower critical temperatures of such mammals. Various heat #ux models have been used for this purpose. Their common feature is that they are based on calculations of conductive heat #ux through the
- Author to whom correspondence should be addressed. E-mail: [email protected] 0022}5193/00/230317#07 $35.00/0
blubber layer (e.g. Parry, 1949; Hokkanen, 1990; Lavigne et al., 1990; Folkow & Blix, 1992; Watts et al. 1993). An experimental evaluation by Kvadsheim et al. (1997) showed that such conductive heat #ux models may overestimate metabolic rates by 28}76%, apparently due to their failure to correctly account for the asymmetrical distribution of blubber (Kvadsheim et al., 1997). Moreover, such models can only be used to calculate minimum heat loss rates. This is because the thermal conductivity of blubber, on which all calculations are based, is usually determined in dead blubber samples, and the &&apparent'' conductivity (Kanwisher & Sundnes, 1966) of live blubber is always equal to or higher than this value (Kvadsheim & Folkow, 1997). An 2000 Academic Press
318
P. BOILY E¹ A¸.
alternative to this approach is to calculate heat #ux through the skin of the animals by use of models that combine estimates of conductive heat #ux through the skin with convective heat #ux from the skin into the environment (Gallivan & Ronald, 1979; Ryg et al., 1993; Boily, 1995; Hind & Gurney, 1997). If reliable estimates of heat #ux could be obtained in this way, such estimates are likely to be less sensitive to regional variations in the thickness of the insulative layer, as well as to changes in blubber/skin perfusion. In the present study, we tested the validity of modi"ed versions of previously published cutaneous heat #ux models (Boily, 1995), by comparing measurements of heat production rates with heat #ux estimates based on simultaneous measurements of the input parameters necessary for such calculations, in four harp seals (Phoca groenlandica) resting in cold water.
Materials and Methods EMPIRICAL MEASUREMENTS
Morphometrical, temperature and metabolic rate data were collected as part of previous studies (Kvadsheim et al., 1997; Kvadsheim & Folkow, 1997). In brief, we used four harp seals (Phoca groenlandica) aged 8 months to 5 yr, caught as pups and raised in captivity at the Department of Arctic Biology, University of Troms+. The animals were loosely restrained on a board and were then immersed in a freshwater"lled tank (860 l) with stirred water at constant temperature, so that the entire body trunk was covered by water, while the head could be lifted out of water to breathe. After thermal equilibration in water (1}2 hr of immersion), metabolic rate, subcutaneous temperatures (¹ ) (Fig. 1), 1! and water temperature (¹ ) were recorded. The 5 experiments were performed once with seals 1 and 2, and 3 times with seals 44 and 45. After termination of in vivo experiments, the animals were killed and the surface areas of the trunk and hind-#ippers, the mean outer body radius (R ) M and the standard body length (¸) were measured. Skin thickness (d) (distance from the top of wet fur to the dermal/hypodermal interface), was measured on the dorsal side of the animals using a precision caliper. The &&true'' sensible heat #ux
FIG. 1. Subcutaneous temperature was measured at seven positions on seals 1 and 2 (䊏) and at 2 positions on seals 44 and 45 (䉱). Water #ow speed was measured a posteriori at nine positions (䊉).
from the body surface (Q) was assumed to be equal to the ratio between the metabolic rate minus respiratory heat loss [which was assumed to be 10% of metabolic rate (Folkow & Blix, 1987)], and the combined trunk and hind-#ipper surface area (front-#ipper area was ignored because these appendages were held close to the trunk throughout the experiments). Details on the experimental procedures are described in Kvadsheim et al. (1997) for seals 1 and 2, and in Kvadsheim and Folkow (1997) for seals 44 and 45. Water speed along the surface of the animals (< ) was determined a posteriori as the mean value of the velocity measured at nine positions on the dorsal and lateral surfaces of another harp seal (Fig. 1), by use of a CardioMed medical volume #ow meter (Medi-Stim, Oslo, Norway). The seal was of similar size to those used for the other empirical measurements and was immersed in a circulated water tank lying on a restraint board in exactly the same way as in the previous experiments (Kvadsheim et al., 1997; Kvadsheim & Folkow, 1997). The #ow probe was placed on the surface of the animal and oriented at the angle producing the highest water-speed recordings. Maximum speed at each position was calculated as the time average of 20 s of recordings of stable water #ow. Two di!erent pumps were used during the original experiments with the seals, one with seals 44 and 45 and another with seals 1 and 2, and water speed was measured using both pumps. The skin (wet fur, epidermis and dermis) thermal conductivity (k ) was measured on samples 1 collected from "ve adult harp seals shot in the drift ice o! Jan Mayen in the Greenland Sea, under licences issued by the Norwegian Government. The conductivity measurements were made
319
CUTANEOUS HEAT FLUX IN MARINE MAMMALS
as described by Kvadsheim et al. (1994), with the following modi"cations: the skin samples were placed on top of a polyethylene plate with known conductivity, which in turn was placed on top of a constant temperature heat source, before water was poured over the sample. The skin conductivity was calculated from the steady-state temperature di!erences across the sample and across the polyethylene plate, from the thickness of the skin sample and the polyethylene plate, and from the known conductivity of polyethylene, by use of the Fourier equation. The measurements were repeated 4 times with each sample. THEORETICAL HEAT FLUX CALCULATIONS
Measured sensible heat #ux (Q) was compared to theoretical heat #ux estimates (Q ), calculated from the morphometrical (¸, d, R ) and physical M (¹ , ¹ , <, k ) measurements made during 5 1! 1 experimental trials. Calculations of Q were
performed using theoretical models modi"ed from Boily (1995). A complete list of symbols, units and values is provided in Table 1, and detailed heat #ux equations are provided in Table 2. Since water speed (<) was very low during experiments (0.003}0.013 m s\), we calculated Q using both a forced convection model, assuming single#at-plate geometry, and free convection models, assuming double-#at-plate or cylindrical geometry. The double-#at-plate geometry model was developed by Boily (1995) to take into account the di!erences in heat #ux that occur in horizontal plates depending on their orientation, where it is assumed that half of the animal's surface is facing up and the other half is facing down. Each model consists of two components that must have equal heat #uxes (Table 2). First, there is heat #ux by conduction through the skin (Q ), Q from the skin}blubber interface to the surface of the animal. Second, there is heat #ux by
TABLE 1 ¸ist of symbols and values Symbol
De"nition (units)
Gr Pr Re Nu g ¹ k 5D b5 l k C ¹U ¹5 ¹1 <1! ¸
Grashof number* Prandtl number* Reynolds number* Nusselt number* Acceleration due to gravity (m s\) Film temperature (K) Thermal conductivity of water (W m\ K\) Thermal expansion coe$cient of water (K\) Kinematic viscosity of water (m s\) Dynamic viscosity of water (Kg m\ s\) Speci"c heat of water (J kg\ K\) Water temperature (K) Surface temperature of animal (K) Subcutaneous temperature (K) Water speed (m s\) Overall dimension of animal (m)
k d1 R RM QG
Skin thermal conductivity (W m\ K\) Skin depth (m) Mean body radius of animal (m) Mean subcutaneous radius (m) Measured sensible heat #ux from body surface (W m\) Estimated sensible heat #ux (W m\) by conduction through the skin (Q ) or by Q ) convection of the environment (Q C
Q
* Gr, Pr, Re and Nu are dimensionless. - Modi"ed from Boily (1995).
Value gb (¹ !¹ )¸ l\ 1 5 C kK\ U 5 <¸l\ Table 2 9.8 (¹ #¹ ) 2\ 1 5 !7.51;10\#2.3;10\ ¹ 5D !1.48;10\ ¹ !1.60;10\#9.89;10\ ¹ 5D!6.86;10\!4.36;10\ ¹5D#7.01;10\ ¹ 5D 5D!6.83;10\!4.34;10\ ¹ #6.97;10\ ¹ 5D 5D 8849!31.1 ¹ #5.19;10\ ¹ 5D 5D
Plate models: length Cylinder model: mean diameter 0.42 R !d M
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P. BOILY E¹ A¸.
TABLE 2 Equations used to estimate heat -ux (= m\) through the skin by conduction (Q ) and from the surface of the animal to the environment by convection (Q ) Q C Model
Q * Q
All models
Q C Q "Nu k ¸\ (¹ !¹ ) C 5 1 5 Upper surface: Nu"0.14 (GrPr) Lower surface: Nu"0.25 (GrPr)
Free convection, double-#at-plate geometry
Q "k (¹ !¹ ) d\ Q Q 1! 1
Free convection, cylindrical geometry
k (¹ !¹ ) 1 Q " 1 1! Q ln (R /R ) R M G M
Nu"0.10 (GrPr)
Forced convection, single-#at-plate geometry
Q "k (¹ !¹ ) d\ Q 1 1! 1
Nu"0.036 Re Pr
* Kvadsheim et al. (1997). - From Boily (1995).
convection from the surface of the animal to the environment (Q ) (radiative heat exchange beC tween the surface of the animal and the environment was ignored due to its very minute contribution in water). By combining both heat #ux components (Q and Q ) and knowing that they Q C must be equal, one can solve the equations to obtain Q and Q . However, this process is comQ C plicated by the fact that the physical properties of water used to calculate heat #ux depend on "lm temperature (¹ ), itself dependent on surface UD temperature (¹ ) which is unknown. This leads to 1 a complex system of seven equations and unknowns, that was solved according to methods previously described (Boily, 1995) using the Mathcad software (ver. 8, Mathsoft Inc., Cambridge, MA, U.S.A.). To assess the sensitivity of the models to errors in input parameters, Q was also calculated for one of the animals by varying water temperature (¹ ) and subcutaneous 5 temperature (¹ ) by $0.2 and $0.43C, and 1! by varying water speed (< ) by $0.002 and $0.004 m s\. Results and Discussion Skin thickness (d) was found to be 8.0 and 6.0 mm for seals 1 and 2, respectively, and 4.2 mm for seals 44 and 45. When using the two di!erent water pumps that were used during the original experiments with the seals, mean water speed (< ) was found to be 0.013 m s\ in experiments with
seals 1 and 2, and 0.003 m s\ in experiments with seals 44 and 45. Mean thermal conductivity of skin samples (k ) was 0.42 W m\ K\ (S.D." 1 0.02, n"5). On average the models underestimated Q by 26}58% (Table 3). This is in contrast to traditional blubbler conductive heat #ux models that tend to overestimate metabolic rates (Kvadsheim et al., 1997). However, similar to the results of Kvadsheim et al. (1997), the most accurate model to predict metabolic rates in our experimental validation was based on cylindrical geometry. This model underestimated Q by 3}43% and overestimated Q in one trial by 13% (Table 3). While the free-convection model based on cylindrical geometry was the most accurate model tested by us, it nevertheless is an unreliable predictor of metabolic rates, since it had an average absolute deviation from observed metabolic rates of 26% (Table 3), and was very sensitive to small variations in the input parameters. Thus, heat #ux predictions, using seal 2 as an example, varied by 7 and 15% when ¹ varied by 0.2 and 1! 0.43C, respectively, and by 9 and 17% when ¹ 5 varied by 0.2 and 0.43C, respectively (Fig. 2). The free convection model based on a double#at-plate geometry was the second most accurate model tested in our study. It underestimated Q by 1}49% and had an average absolute deviation from Q of 32%. The sensitivity of that model to variations in input parameters was similar to that of the free convection model based on cylindrical
321
CUTANEOUS HEAT FLUX IN MARINE MAMMALS
TABLE 3 Comparison between measured (Q) and estimated (Q ) heat -ux (= m\) using three theoretical models Seal (trial)
¹ (3C) 5
Q
Q (deviation from Q) Free convection double-#at-plate geometry
1(1) 2(1) 44(1) 44(2) 44(3) 45(1) 45(2) 45(3)
1.2 0.4 1.5 1.4 2.3 1.3 1.8 1.7
55.5 108.9 101.3 95.4 107.2 95.8 104.6 100.8
37.6 92.0 53.2 53.7 54.3 65.7 66.1 99.6
Mean absolute deviation between Q and Q
(!32%) (!16%) (!47%) (!44%) (!49%) (!31%) (!37%) (!1%)
32.1%
Free convection, cylindrical geometry 42.5 105.4 60.4 60.9 61.1 74.7 74.9 113.7
(!23%) (!3%) (!40%) (!36%) (!43%) (!22%) (!28%) (#13%)
26.0%
Forced convection, single-#at-plate geometry 41.5 89.8 24.6 24.6 28.0 29.3 31.0 45.2
(!25%) (!17%) (!76%) (!74%) (!74%) (!69%) (!70%) (!58%)
57.9%
FIG. 2. E!ects of variations in water and subcutaneous temperature (¹ ) on the estimated sensible heat #ux when 1! applying the free convection model based on cylindrical geometry on data from seal 2. (**) ¹ #0.43C; (* *) 1! ¹ #0.23C; (} } } } ) ¹ observed; (- - - - -) ¹ !0.23C; 1! 1! 1! () ) ) ) ) )) ¹ !0.43C. 1!
FIG. 3. E!ects of variations in water and subcutaneous temperature (¹ ) on the estimated sensible heat #ux when 1! applying the free convection model based on double-#atplate geometry on data from seal 2. (**) ¹ #0.43C; 1! (* *) ¹ #0.23C; (} } } } ) ¹ observed; (- - - - -) ¹ ! 1! 1! 1! 0.23C; () ) ) ) ) )) ¹ !0.43C 1!
geometry (Fig. 3). The forced convection model based on single-#at-plate geometry was the least accurate model, underestimating Q on average by 58% (range 17}76%, Table 3). Similar to the free convection models, the forced convection model was sensitive to small variations in input parameters. Heat #ux estimates based on this model varied by 7 and 14% when ¹ varied by 1! 0.2 and 0.43C, respectively, by 7 and 13% when ¹ varied by 0.2 and 0.43C, respectively, and 5
by 6 and 14% when < varied by 0.002 and 0.004 m s\, respectively (Fig. 4). Both the free convection model based on double-#at-plate geometry and the forced convection model based on single-#at-plate geometry were originally proposed by Boily (1995) to estimate heat #ux in molting seals and whales. While the inaccuracy and sensitivity of both models make them inappropriate tools for the "eld estimation of metabolic rates in marine
322
P. BOILY E¹ A¸.
FIG. 4. E!ects of variations in water and subcutaneous temperature (¹ ). (a) (**) ¹ #0.43C; (* *) ¹ # 1! 1! 1! 0.23C; (} } } } ) ¹ observed; (- - - - -) ¹ !0.23C; () ) ) ) ) )) 1! 1! ¹ !0.43C, and in water temperature and water speed (<) 1! (b), on the estimated sensible heat #ux when applying the forced convection model based on single-#at-plate geometry on data from seal 2. (**) <#0.4 cm s\; (* *) <#0.2 cm s\; (} } } } ) < observed; (- - - - -)
the ventral surface of the animals. However, the board was made of aluminum (thickness" 1.5 cm), which has a very low thermal resistance, and therefore is expected to have little in#uence on heat #ux between the animal and the water. In any case, the extra insulation o!ered by the board could only cause a positive deviation between estimated and measured heat #ux, and we observed the opposite. In conclusion, we have found that cutaneous heat #ux models are inaccurate and sensitive to small variations in input parameters. In comparison, traditional calculations of conductive heat #ux through the blubber layer provide more accurate and reliable estimates of minimum heat #ux, provided that the asymmetrical distribution of blubber is taken into account (Kvadsheim et al., 1997). However, in situations where animals actively dissipate heat through increased perfusion of blubber and skin, such calculations are invalid. Studies have shown that in such situations, blood perfusion and subcutaneous temperature may change rapidly both spatially and temporally (Irving & Hart, 1957; Whittow et al., 1972; Kvadsheim & Folkow, 1997), and it is therefore di$cult to obtain accurate measurements of subcutaneous temperature. Consequently, the high sensitivity of cutaneous heat #ux models makes them even more inaccurate in these situations. We thank anonymous reviewers for their constructive comments. This research was funded by the University of New Orleans and by the Norwegian Research Council (grant no 110397/100). REFERENCES
mammals, this is unlikely to a!ect Boily's conclusions regarding habitat choice of marine mammals during the molt. Indeed, since heat #ux estimates overall underestimated Q, the results of the present study re-enforce Boily's argument that the molt of phocid seals is thermally unfavorable in water, and that seasonal movements of beluga whales (Delphinapterus leucas) into warm-water estuaries is thermally favorable during their annual molt. It might be argued that our results are biased because the seals were lying on a board, which prevented free water-#ow along parts of
BOILY, P. (1995). Theoretical heat #ux in water and habitat selection of phocid seals and beluga whales during the annual molt. J. theor. Biol. 172, 235}244. FOLKOW, L. P. & BLIX, A. S. (1987). Nasal heat and water exchange in grey seals. Am. J. Physiol. 253, R883}R889. FOLKOW, L. P. & BLIX, A. S. (1992). Metabolic rates of minke whales (Balaenoptera acutorostrata) in cold water. Acta Physiol. Scand. 146, 141}150. GALLIVAN, G. J. & RONALD, K. (1979). Temperature regulation in freely diving harp seals (Phocagroenlandica). Can. J. Zool. 57, 2256}2263. HIND, A. T. & GURNEY, W. S. C. (1997). The metabolic cost of swimming in marine homeotherms. J. Exp. Biol. 200, 531}542. HOKKANEN, J. E. I. (1990). Temperature regulation in marine mammals. J. theor. Biol. 145, 465}485.
CUTANEOUS HEAT FLUX IN MARINE MAMMALS
IRVING, L. & HART, J. S. (1957). The metabolism and insulation of seals as bare-skinned mammals in cold water. Can. J. Zool. 35, 497}511. KANWISHER, J. &. SUNDNES, G. (1966). Thermal regulation in cetaceans. In: =hales, Dolphins and Porpoises (Norris, K. S., ed.), pp. 397}407. Los Angeles: University of California Press. KVADSHEIM, P., FOLKOW, L. P. & BLIX, A. S. (1994). A new device for the measurement of the thermal conductivity of fur and blubber. J. ¹herm. Biol. 19, 431}435. KVADSHEIM, P. H. & FOLKOW, L. P. (1997). Blubber and #ipper heat transfer in harp seals. Acta Physiol. Scand. 161, 385}395. KVADSHEIM, P., GOTAAS, A. R. L., FOLKOW, L. P. & BLIX, A. S. (1997). An experimental validation of heat loss models for marine mammals. J. theor. Biol. 184, 15}23.
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LAVIGNE, D. M., INNES, S., WORTHY, G. A. J. & EDWARDS, E. F. (1990). Lower critical temperature of blue whales, Balaenoptera musculus. J. theor. Biol. 144, 249}257. PARRY, D. A. (1949). The structure of whale blubber, and a discussion of its thermal properties. Quart J. Microb. Sci. 90, 13}25. RYG, M., LYDERSEN, C., KNUTSEN, L. "., BJ"RGE, A., SMITH, T. G. & "RITSLAND, N. A. (1993). Scaling of insulation in seals and whales. J. Zool. ¸ond. 230, 193}206. WATTS, P., HANSEN, S. & LAVIGNE, D. M. (1993). Models of heat loss by marine mammals: thermoregulation below the zone of irrelevance. J. theor. Biol. 163, 505}525. WHITTOW, G. C., MATSUURA, D. T. & LIN, Y. C. (1972). Temperature regulation in the California sea lion (Zalophus californianus). Physiol. Zool. 45, 68}77.
Cutaneous Heat Flux Models do not Reliably Predict Metabolic Rates of Marine Mammals P. BOILY*-, P. H. KVADSHEIM? AND L. P. FOLKOW? *Department of Biological Sciences, ;niversity of New Orleans, New Orleans, ¸A 70148, ;.S.A. and ? Department of Arctic Biology, ;niversity of ¹roms~, N-9037 ¹roms~, Norway (Received on 14 October 1999, Accepted in revised form on 21 August 2000)
Heat #ux models have been used to predict metabolic rates of marine mammals, generally by estimating conductive heat transfer through their blubber layer. Recently, Kvadsheim et al. (1997) found that such models tend to overestimate metabolic rates, and that such errors probably result from the asymmetrical distribution of blubber. This problem may be avoided if reliable estimates of heat #ux through the skin of the animals are obtained by using models that combine calculations of conductive heat #ux through the skin and fur, and convective heat #ux from the surface of the animal to the environment. We evaluated this approach based on simultaneous measurements of metabolic rates and of input parameters necessary for heat #ux calculations, as obtained from four harp seals (Phoca groenlandica) resting in cold water. Heat #ux estimates were made using two free convection models (double-#at-plate and cylindrical geometry) and one forced convection model (single-#at-plate geometry). We found that heat #ux estimates generally underestimated metabolic rates, on average by 26}58%, and that small variations in input parameters caused large variations in these estimates. We conclude that cutaneous heat #ux models are too inaccurate and sensitive to small errors in input parameters to provide reliable estimates of metabolic rates of marine mammals. 2000 Academic Press
Introduction Energetic and behavioral studies of marine mammals often require data on their metabolic rates or their lower critical temperature. For some marine mammals, in particular for large whales, such variables cannot be measured directly. Instead, theoretical heat #ux calculations can be used to estimate metabolic rates and lower critical temperatures of such mammals. Various heat #ux models have been used for this purpose. Their common feature is that they are based on calculations of conductive heat #ux through the
- Author to whom correspondence should be addressed. E-mail: [email protected] 0022}5193/00/230317#07 $35.00/0
blubber layer (e.g. Parry, 1949; Hokkanen, 1990; Lavigne et al., 1990; Folkow & Blix, 1992; Watts et al. 1993). An experimental evaluation by Kvadsheim et al. (1997) showed that such conductive heat #ux models may overestimate metabolic rates by 28}76%, apparently due to their failure to correctly account for the asymmetrical distribution of blubber (Kvadsheim et al., 1997). Moreover, such models can only be used to calculate minimum heat loss rates. This is because the thermal conductivity of blubber, on which all calculations are based, is usually determined in dead blubber samples, and the &&apparent'' conductivity (Kanwisher & Sundnes, 1966) of live blubber is always equal to or higher than this value (Kvadsheim & Folkow, 1997). An 2000 Academic Press
318
P. BOILY E¹ A¸.
alternative to this approach is to calculate heat #ux through the skin of the animals by use of models that combine estimates of conductive heat #ux through the skin with convective heat #ux from the skin into the environment (Gallivan & Ronald, 1979; Ryg et al., 1993; Boily, 1995; Hind & Gurney, 1997). If reliable estimates of heat #ux could be obtained in this way, such estimates are likely to be less sensitive to regional variations in the thickness of the insulative layer, as well as to changes in blubber/skin perfusion. In the present study, we tested the validity of modi"ed versions of previously published cutaneous heat #ux models (Boily, 1995), by comparing measurements of heat production rates with heat #ux estimates based on simultaneous measurements of the input parameters necessary for such calculations, in four harp seals (Phoca groenlandica) resting in cold water.
Materials and Methods EMPIRICAL MEASUREMENTS
Morphometrical, temperature and metabolic rate data were collected as part of previous studies (Kvadsheim et al., 1997; Kvadsheim & Folkow, 1997). In brief, we used four harp seals (Phoca groenlandica) aged 8 months to 5 yr, caught as pups and raised in captivity at the Department of Arctic Biology, University of Troms+. The animals were loosely restrained on a board and were then immersed in a freshwater"lled tank (860 l) with stirred water at constant temperature, so that the entire body trunk was covered by water, while the head could be lifted out of water to breathe. After thermal equilibration in water (1}2 hr of immersion), metabolic rate, subcutaneous temperatures (¹ ) (Fig. 1), 1! and water temperature (¹ ) were recorded. The 5 experiments were performed once with seals 1 and 2, and 3 times with seals 44 and 45. After termination of in vivo experiments, the animals were killed and the surface areas of the trunk and hind-#ippers, the mean outer body radius (R ) M and the standard body length (¸) were measured. Skin thickness (d) (distance from the top of wet fur to the dermal/hypodermal interface), was measured on the dorsal side of the animals using a precision caliper. The &&true'' sensible heat #ux
FIG. 1. Subcutaneous temperature was measured at seven positions on seals 1 and 2 (䊏) and at 2 positions on seals 44 and 45 (䉱). Water #ow speed was measured a posteriori at nine positions (䊉).
from the body surface (Q) was assumed to be equal to the ratio between the metabolic rate minus respiratory heat loss [which was assumed to be 10% of metabolic rate (Folkow & Blix, 1987)], and the combined trunk and hind-#ipper surface area (front-#ipper area was ignored because these appendages were held close to the trunk throughout the experiments). Details on the experimental procedures are described in Kvadsheim et al. (1997) for seals 1 and 2, and in Kvadsheim and Folkow (1997) for seals 44 and 45. Water speed along the surface of the animals (< ) was determined a posteriori as the mean value of the velocity measured at nine positions on the dorsal and lateral surfaces of another harp seal (Fig. 1), by use of a CardioMed medical volume #ow meter (Medi-Stim, Oslo, Norway). The seal was of similar size to those used for the other empirical measurements and was immersed in a circulated water tank lying on a restraint board in exactly the same way as in the previous experiments (Kvadsheim et al., 1997; Kvadsheim & Folkow, 1997). The #ow probe was placed on the surface of the animal and oriented at the angle producing the highest water-speed recordings. Maximum speed at each position was calculated as the time average of 20 s of recordings of stable water #ow. Two di!erent pumps were used during the original experiments with the seals, one with seals 44 and 45 and another with seals 1 and 2, and water speed was measured using both pumps. The skin (wet fur, epidermis and dermis) thermal conductivity (k ) was measured on samples 1 collected from "ve adult harp seals shot in the drift ice o! Jan Mayen in the Greenland Sea, under licences issued by the Norwegian Government. The conductivity measurements were made
319
CUTANEOUS HEAT FLUX IN MARINE MAMMALS
as described by Kvadsheim et al. (1994), with the following modi"cations: the skin samples were placed on top of a polyethylene plate with known conductivity, which in turn was placed on top of a constant temperature heat source, before water was poured over the sample. The skin conductivity was calculated from the steady-state temperature di!erences across the sample and across the polyethylene plate, from the thickness of the skin sample and the polyethylene plate, and from the known conductivity of polyethylene, by use of the Fourier equation. The measurements were repeated 4 times with each sample. THEORETICAL HEAT FLUX CALCULATIONS
Measured sensible heat #ux (Q) was compared to theoretical heat #ux estimates (Q ), calculated from the morphometrical (¸, d, R ) and physical M (¹ , ¹ , <, k ) measurements made during 5 1! 1 experimental trials. Calculations of Q were
performed using theoretical models modi"ed from Boily (1995). A complete list of symbols, units and values is provided in Table 1, and detailed heat #ux equations are provided in Table 2. Since water speed (<) was very low during experiments (0.003}0.013 m s\), we calculated Q using both a forced convection model, assuming single#at-plate geometry, and free convection models, assuming double-#at-plate or cylindrical geometry. The double-#at-plate geometry model was developed by Boily (1995) to take into account the di!erences in heat #ux that occur in horizontal plates depending on their orientation, where it is assumed that half of the animal's surface is facing up and the other half is facing down. Each model consists of two components that must have equal heat #uxes (Table 2). First, there is heat #ux by conduction through the skin (Q ), Q from the skin}blubber interface to the surface of the animal. Second, there is heat #ux by
TABLE 1 ¸ist of symbols and values Symbol
De"nition (units)
Gr Pr Re Nu g ¹ k 5D b5 l k C ¹U ¹5 ¹1 <1! ¸
Grashof number* Prandtl number* Reynolds number* Nusselt number* Acceleration due to gravity (m s\) Film temperature (K) Thermal conductivity of water (W m\ K\) Thermal expansion coe$cient of water (K\) Kinematic viscosity of water (m s\) Dynamic viscosity of water (Kg m\ s\) Speci"c heat of water (J kg\ K\) Water temperature (K) Surface temperature of animal (K) Subcutaneous temperature (K) Water speed (m s\) Overall dimension of animal (m)
k d1 R RM QG
Skin thermal conductivity (W m\ K\) Skin depth (m) Mean body radius of animal (m) Mean subcutaneous radius (m) Measured sensible heat #ux from body surface (W m\) Estimated sensible heat #ux (W m\) by conduction through the skin (Q ) or by Q ) convection of the environment (Q C
Q
* Gr, Pr, Re and Nu are dimensionless. - Modi"ed from Boily (1995).
Value gb (¹ !¹ )¸ l\ 1 5 C kK\ U 5 <¸l\ Table 2 9.8 (¹ #¹ ) 2\ 1 5 !7.51;10\#2.3;10\ ¹ 5D !1.48;10\ ¹ !1.60;10\#9.89;10\ ¹ 5D!6.86;10\!4.36;10\ ¹5D#7.01;10\ ¹ 5D 5D!6.83;10\!4.34;10\ ¹ #6.97;10\ ¹ 5D 5D 8849!31.1 ¹ #5.19;10\ ¹ 5D 5D
Plate models: length Cylinder model: mean diameter 0.42 R !d M
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P. BOILY E¹ A¸.
TABLE 2 Equations used to estimate heat -ux (= m\) through the skin by conduction (Q ) and from the surface of the animal to the environment by convection (Q ) Q C Model
Q * Q
All models
Q C Q "Nu k ¸\ (¹ !¹ ) C 5 1 5 Upper surface: Nu"0.14 (GrPr) Lower surface: Nu"0.25 (GrPr)
Free convection, double-#at-plate geometry
Q "k (¹ !¹ ) d\ Q Q 1! 1
Free convection, cylindrical geometry
k (¹ !¹ ) 1 Q " 1 1! Q ln (R /R ) R M G M
Nu"0.10 (GrPr)
Forced convection, single-#at-plate geometry
Q "k (¹ !¹ ) d\ Q 1 1! 1
Nu"0.036 Re Pr
* Kvadsheim et al. (1997). - From Boily (1995).
convection from the surface of the animal to the environment (Q ) (radiative heat exchange beC tween the surface of the animal and the environment was ignored due to its very minute contribution in water). By combining both heat #ux components (Q and Q ) and knowing that they Q C must be equal, one can solve the equations to obtain Q and Q . However, this process is comQ C plicated by the fact that the physical properties of water used to calculate heat #ux depend on "lm temperature (¹ ), itself dependent on surface UD temperature (¹ ) which is unknown. This leads to 1 a complex system of seven equations and unknowns, that was solved according to methods previously described (Boily, 1995) using the Mathcad software (ver. 8, Mathsoft Inc., Cambridge, MA, U.S.A.). To assess the sensitivity of the models to errors in input parameters, Q was also calculated for one of the animals by varying water temperature (¹ ) and subcutaneous 5 temperature (¹ ) by $0.2 and $0.43C, and 1! by varying water speed (< ) by $0.002 and $0.004 m s\. Results and Discussion Skin thickness (d) was found to be 8.0 and 6.0 mm for seals 1 and 2, respectively, and 4.2 mm for seals 44 and 45. When using the two di!erent water pumps that were used during the original experiments with the seals, mean water speed (< ) was found to be 0.013 m s\ in experiments with
seals 1 and 2, and 0.003 m s\ in experiments with seals 44 and 45. Mean thermal conductivity of skin samples (k ) was 0.42 W m\ K\ (S.D." 1 0.02, n"5). On average the models underestimated Q by 26}58% (Table 3). This is in contrast to traditional blubbler conductive heat #ux models that tend to overestimate metabolic rates (Kvadsheim et al., 1997). However, similar to the results of Kvadsheim et al. (1997), the most accurate model to predict metabolic rates in our experimental validation was based on cylindrical geometry. This model underestimated Q by 3}43% and overestimated Q in one trial by 13% (Table 3). While the free-convection model based on cylindrical geometry was the most accurate model tested by us, it nevertheless is an unreliable predictor of metabolic rates, since it had an average absolute deviation from observed metabolic rates of 26% (Table 3), and was very sensitive to small variations in the input parameters. Thus, heat #ux predictions, using seal 2 as an example, varied by 7 and 15% when ¹ varied by 0.2 and 1! 0.43C, respectively, and by 9 and 17% when ¹ 5 varied by 0.2 and 0.43C, respectively (Fig. 2). The free convection model based on a double#at-plate geometry was the second most accurate model tested in our study. It underestimated Q by 1}49% and had an average absolute deviation from Q of 32%. The sensitivity of that model to variations in input parameters was similar to that of the free convection model based on cylindrical
321
CUTANEOUS HEAT FLUX IN MARINE MAMMALS
TABLE 3 Comparison between measured (Q) and estimated (Q ) heat -ux (= m\) using three theoretical models Seal (trial)
¹ (3C) 5
Q
Q (deviation from Q) Free convection double-#at-plate geometry
1(1) 2(1) 44(1) 44(2) 44(3) 45(1) 45(2) 45(3)
1.2 0.4 1.5 1.4 2.3 1.3 1.8 1.7
55.5 108.9 101.3 95.4 107.2 95.8 104.6 100.8
37.6 92.0 53.2 53.7 54.3 65.7 66.1 99.6
Mean absolute deviation between Q and Q
(!32%) (!16%) (!47%) (!44%) (!49%) (!31%) (!37%) (!1%)
32.1%
Free convection, cylindrical geometry 42.5 105.4 60.4 60.9 61.1 74.7 74.9 113.7
(!23%) (!3%) (!40%) (!36%) (!43%) (!22%) (!28%) (#13%)
26.0%
Forced convection, single-#at-plate geometry 41.5 89.8 24.6 24.6 28.0 29.3 31.0 45.2
(!25%) (!17%) (!76%) (!74%) (!74%) (!69%) (!70%) (!58%)
57.9%
FIG. 2. E!ects of variations in water and subcutaneous temperature (¹ ) on the estimated sensible heat #ux when 1! applying the free convection model based on cylindrical geometry on data from seal 2. (**) ¹ #0.43C; (* *) 1! ¹ #0.23C; (} } } } ) ¹ observed; (- - - - -) ¹ !0.23C; 1! 1! 1! () ) ) ) ) )) ¹ !0.43C. 1!
FIG. 3. E!ects of variations in water and subcutaneous temperature (¹ ) on the estimated sensible heat #ux when 1! applying the free convection model based on double-#atplate geometry on data from seal 2. (**) ¹ #0.43C; 1! (* *) ¹ #0.23C; (} } } } ) ¹ observed; (- - - - -) ¹ ! 1! 1! 1! 0.23C; () ) ) ) ) )) ¹ !0.43C 1!
geometry (Fig. 3). The forced convection model based on single-#at-plate geometry was the least accurate model, underestimating Q on average by 58% (range 17}76%, Table 3). Similar to the free convection models, the forced convection model was sensitive to small variations in input parameters. Heat #ux estimates based on this model varied by 7 and 14% when ¹ varied by 1! 0.2 and 0.43C, respectively, by 7 and 13% when ¹ varied by 0.2 and 0.43C, respectively, and 5
by 6 and 14% when < varied by 0.002 and 0.004 m s\, respectively (Fig. 4). Both the free convection model based on double-#at-plate geometry and the forced convection model based on single-#at-plate geometry were originally proposed by Boily (1995) to estimate heat #ux in molting seals and whales. While the inaccuracy and sensitivity of both models make them inappropriate tools for the "eld estimation of metabolic rates in marine
322
P. BOILY E¹ A¸.
FIG. 4. E!ects of variations in water and subcutaneous temperature (¹ ). (a) (**) ¹ #0.43C; (* *) ¹ # 1! 1! 1! 0.23C; (} } } } ) ¹ observed; (- - - - -) ¹ !0.23C; () ) ) ) ) )) 1! 1! ¹ !0.43C, and in water temperature and water speed (<) 1! (b), on the estimated sensible heat #ux when applying the forced convection model based on single-#at-plate geometry on data from seal 2. (**) <#0.4 cm s\; (* *) <#0.2 cm s\; (} } } } ) < observed; (- - - - -)
the ventral surface of the animals. However, the board was made of aluminum (thickness" 1.5 cm), which has a very low thermal resistance, and therefore is expected to have little in#uence on heat #ux between the animal and the water. In any case, the extra insulation o!ered by the board could only cause a positive deviation between estimated and measured heat #ux, and we observed the opposite. In conclusion, we have found that cutaneous heat #ux models are inaccurate and sensitive to small variations in input parameters. In comparison, traditional calculations of conductive heat #ux through the blubber layer provide more accurate and reliable estimates of minimum heat #ux, provided that the asymmetrical distribution of blubber is taken into account (Kvadsheim et al., 1997). However, in situations where animals actively dissipate heat through increased perfusion of blubber and skin, such calculations are invalid. Studies have shown that in such situations, blood perfusion and subcutaneous temperature may change rapidly both spatially and temporally (Irving & Hart, 1957; Whittow et al., 1972; Kvadsheim & Folkow, 1997), and it is therefore di$cult to obtain accurate measurements of subcutaneous temperature. Consequently, the high sensitivity of cutaneous heat #ux models makes them even more inaccurate in these situations. We thank anonymous reviewers for their constructive comments. This research was funded by the University of New Orleans and by the Norwegian Research Council (grant no 110397/100). REFERENCES
mammals, this is unlikely to a!ect Boily's conclusions regarding habitat choice of marine mammals during the molt. Indeed, since heat #ux estimates overall underestimated Q, the results of the present study re-enforce Boily's argument that the molt of phocid seals is thermally unfavorable in water, and that seasonal movements of beluga whales (Delphinapterus leucas) into warm-water estuaries is thermally favorable during their annual molt. It might be argued that our results are biased because the seals were lying on a board, which prevented free water-#ow along parts of
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CUTANEOUS HEAT FLUX IN MARINE MAMMALS
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