Extra Supplementary Material
Heat transfer model of emperor penguin
Surface area and characteristic dimensions
A simple geometric model of an emperor penguin Aptenodytes forsteri was represented by
main body (prolate spheroid), head (sphere), beak (cone), flippers and feet (flat plates) (Fig.
1). A specimen (Hunterian Museum, Catalogue: Z1125) mounted in a realistic upright stance
was used to provide representative body dimensions and body surface areas for heat
transfer calculations.
The surface area of the main body trunk Atrunk (m2) was determined by:
Atrunk  2
ab 1
sin e  2b 2
e
e
where
a
2
 b2
 and a and b are the semi-major and
minor axes lengths, respectively [1]. a was taken to be half the body length from the black
neck collar to the base of abdominal feathers (0.34 m), b was half the maximum diameter of
the body trunk (0.16 m) determined from a measurement of girth taken under the flippers.
The beak area was a cone
Abeak  rs
where the radius r was half the maximum width at the
base of the beak (0.01 m) and hypotenuse s was approximated by beak length (0.11m). The
area of the head was approximated by a sphere minus the base of a cone (beak):
Ahead  d 2  r 2
where the diameter d was taken to be the mean (± standard error) of
head height and width (0.11±0.004 m) measured from the bottom of black head plumage
and r as above. Flippers were approximated as flat rectangles of length l and width w,
A flipper  l  w
and were determined by tracing outlines onto 1mm squared graph paper.
The flipper length was the maximum diagonal length from top to bottom of flipper
(0.28±0.011 m) and mean flipper width (0.065±0.0002 m).
The surface area of each foot was obtained by tracing its outline onto 1 mm2 graph paper.
The mean surface area of a single foot in contact with the ground was 0.0036±0.0004m2.
The total surface area of a foot was therefore approximated as twice the measured area. As
emperor penguins often rest on their tarsometatarsus joint, this area was also traced from
the museum specimen and averaged 0.0006±0.00008 m2 or 17% of the lower surface area
of the foot. For radiation and convective calculations the characteristic dimension of the
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foot was taken to be maximum foot width, f (0.056±0.0025 m). The thickness of the foot, t
was determined from the mean thickness of metatarsi (mean = 0.014 m SE = 0.001). The
total surface area of an average emperor penguin was calculated to be 0.56 m 2 which was
within previously measured values [1, 2].
Surface
% Total
d (m)
Nusselt Relationship
Area
surface
(m2)
area
Trunk (excluding flippers)
0.471
83.8
0.32
Prolate Nu =0.24Re0.6
Head and beak
0.040
7.2
0.11
Sphere Nu =0.34Re0.6
Flippers (outside surface only)
0.036
6.5
0.065
Flat plate Nu =0.032Re0.8
Feet
0.014
2.5
0.056
Flat plate Nu = 0.032Re0.8
Total
0.562
Table 1. Calculated surface area, percentage of total surface area, characteristic dimensions
for heat transfer calculations of emperor penguin and relationship between Nusselt and
Reynolds numbers [7].
Heat transfer
A distributed parameter heat transfer model was used to estimate total heat exchange, qtot
(W) for an emperor penguin by summing heat transfer from each body region (head, trunk,
flippers and feet) assuming that the penguin was in thermal equilibrium with its
surroundings [3]:
qtot  q head  qtrunk  q flippers  q feet
(1)
Radiation
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As measurements were undertaken during mid-winter (1h50-2h50 light/24h) solar heat gain
was assumed to make a trivial contribution to heat transfer. Radiative heat loss q rad was
determined by solving the radiation balance at the surface. Heat loss by radiation was the
difference between radiation emitted from the penguin’s surface, q bird and radiation gained
from the environment, q env such that:
q rad  qbird  qenv
(2)
Radiation emitted from each body part of surface area, A with surface temperature Ts (K) was
determined according to:
qbird  ATs
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(3)
Where  is emissivity of bird plumage (=0.98, [4]) and  is the Stefan-Boltzmann constant
(5.67 x10-8 Wm-2K-1). We assumed that each part of the body exchanged radiation equally with
sky and surroundings such that amount of radiation absorbed was equal to the mean flux from
sky and ground surface:
 Ld  Lu 
q env  Aal 

2


(4)
where A (m²) is the radiative area and, al is the long wave absorptivity (=emissivity) of the
penguin. Ld (Wm-2) and Lu (Wm-2) are the downward and upward radiative heat fluxes from
sky and snow surface, respectively. The downward radiative flux was estimated using the
empirical relationship measured in Antarctica [5]:
Ld  0.665Ta  18.175h  8.003c  14.088
(5)
where Ta is air temperature (K), h the specific humidity (gkg-1) and c the cloud cover (oktas).
The upward radiative flux was determined by:

Lu   g Tg
4

(6)
where  g is the emissivity of ground surface (snow = 0.97, [6]) and Tg (K) the ground ice
temperature.
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Convection
Heat transfer by convection, qconv from each region of the body was calculated by:
qconv  hATs  Ta 
(7)
Forced convection is the dominant mode of heat transfer in wind (≥ 0.5 ms-1 in this study) such
that the heat transfer coefficient was determined by:
h  Nu
k
d
(8)
where k is the thermal conductivity of air (0.0225 Wm-1oC-1 at -20oC), d (m) is the characteristic
dimension of each body part in the direction of air flow and Nu is the dimensionless Nusselt
number. The Nusselt number is a measure of the ratio of buoyant to viscous forces. It depends
on shape and can be related to the dimensionless Reynolds number, Re from Re  ud  ,
where u is the wind speed (ms-1), d the characteristic dimension (m) and  the kinematic
viscosity of air (11.6 x 10-6 m2s-1 at -20oC). The relationship between Nu and Re has been
determined empirically for a range of geometric shapes and flow regimes (Table 1).
Conduction
Emperor penguins commonly cover the upper surface of their feet by abdominal feathers
and therefore a bird will lose heat by conduction, qcon from the lower surface of its feet to
the snow surface such that:
qcon  Ak
(T feet  Tg )
x
(9)
where k is the thermal conductivity of the foot tissue (Wm-1oC-1) of thickness x (m). The
thermal conductivity of foot tissue was taken to equal skin conductivity (0.502 Wm-1oC-1) [8].
Emperor penguin feet remain above freezing and the minimum heat loss by conduction was
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estimated from minimum foot temperature of 3.3°C [9]. The temperature of ground surface
underlying the foot was assumed to equal the surface temperature of surrounding snow
surface. If feet were not visible it was assumed birds were resting on their tarsometatarsus
where area in contact with ground was 17% of foot area. When standing or walking, the
foot will also lose heat by radiation and convection from the upper surface (as above).
Latent heat loss
Latent heat loss for emperor penguins was estimated from previous measurements of the
evaporative water loss that remained constant between -47 and 20 oC and averaged 5.85 g
h-1 [1]. The vaporisation of 1g water requires 2.43 kJ, therefore latent heat loss for an
emperor penguin is equivalent to 4.0 W.
Data
Input data for the model was taken from surface temperature measurements of emperor
penguins and meteorological data recorded at the breeding colony of Pointe Géologie in
Terre Adélie (66o40’S 140o 01’E), Antarctica in June 2008 (ESM2). Where surface
temperature data were missing for a particular body part for an individual, the missing value
was computed using regression with air temperature from GLM models (see paper).
References
1.
Pinshow, B., Fedak, M.A., Battles, D.R., Schmidt Nielsen, K. 1976 Energy-expenditure
for thermoregulation and locomotion in emperor penguins. Am. J. Physiol. 231,903-12.
2.
Le Maho, Y., Delclitte, P., Chatonnet, J. 1976. Thermoregulation in fasting emperor
penguins. Am. J. Physiol. 231,913-22.
3.
McCafferty, DJ, Gilbert, C, Paterson, W, Pomeroy, PP, Thompson, D, Currie, J, Ancel,
A. 2011 Estimating metabolic heat loss in birds and mammals by combining infrared
thermography with biophysical modelling. Comp. Biochem. Physiol.A-Molec. Integ. Physiol.
158, 337-345.
4.
Hammel, H.T. 1956 Infrared emissivities of some arctic fauna. J. Mammal. 37, 375-8.
5.
Cho, H.K., Kim, J., Jung, Y., Lee, Y.G., Lee, B.Y. 2008 Recent changes in downward
longwave radiation at King Sejong Station, Antarctica. J. Climate. 21, 5764-76.
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6.
Kondo, J., Yamazawa, H. 1986. Measurement of snow surface emissivity. Boundary-
Layer Met. 34,415-6.
7.
Monteith, J.L., Unsworth, M.H. 1990 Principles of environmental physics. London:
Edward Arnold.
8.
Gates, D.M. 1980 Biophysical Ecology. Berlin: Springer – Verlag.
9.
Prévost, J, Sapin-Jaloustre, J. 1964 A propos des premieres mesures de topographie
thermique chez les Spheniscides de la Terre Adelie. Oiseau. 34, 52-90.
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Fig.1. Geometric model of emperor penguin. Definition of terms given in text.
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