# Heat Loss from a 1/2 Triaxial Ellipsoid (Tortoise)

BACKGROUND (TL;DR = How long does it take to chill a tortoise?)

I'm working on a simple(?) problem in reptile thermoregulation.
Tortoises (family Testudinidae) are common components of terrestrial fossil assemblages dating back as far as the Paleocene. Over the past 61 MA the family has repeatedly evolved species of giant size (100+kg). While extant giant tortoises are found only off the coasts of South America and Africa, at least two giant tortoise species persisted in North America until the terminal Pleistocene. These tortoises, Hesperotestudo crassiscutata and Gopherus hexagonatus, were distributed across the American south and could reach carapace lengths of 1+m. Their presence in fossil assemblages is regarded as evidence of ‘frost-free’ conditions of the local environment. This assumption is based on observations of the climate tolerances and habits of modern species of giant tortoise. In addition, experimentation has shown that the modern Galapagos giant tortoise, Chelonoidis nigra, cannot survive at northern latitudes -- such as the US -- without human aid. The assumption that giant tortoise remains are an indicator of frost-free conditions is so pervasive in the literature that their discovery at a paleontological site is often considered evidence that the site dates to an interglacial oxygen isotope stage. However it is possible that their large body size resulted in high thermal inertia which provided protection from below-freezing temperatures. A good thermal model of the North American giant tortoises is needed to test this hypothesis. Developing such a model, however, requires accurate measurements of the animal’s body mass. Currently available body mass estimates for extinct taxa are based on qualitative comparisons with modern giant tortoises, and are insufficient for modeling purposes. The weight and 68 linear measurements from each of 113 tortoises of 20 different species were obtained and compiled in order to develop linear measurement-versus-weight regressions for tortoises based on carapace shape and taxonomy. Now I need a thermal model to determine rates of heat loss under various scenarios.

MODEL SCENARIO In order to make this problem manageable, I'm making the scenario as simple as possible. If a tortoise is taken from it's optimal operating temperature and placed in an infinitely large cold, dark space, how long does it take before it reaches a non-viable temperature? A real tortoise would have the advantages of being able to bask to heat itself above the ambient temperature, and would have the advantage of heat radiating off the Earth as the atmospheric temperature falls, but I've decided to keep it simple and model this 'worst-case' scenario instead.

AVAILABLE DATA Tortoise Data: Note that tortoises are very nearly perfect 1/2 triaxial ellipsoids.

• Surface area
• Volume
• Mass
• Starting temperature
• Thermal conductivity
• Specific heat

Atmospheric Data:

• Ambient temperature
• Thermal conductivity
• Specific heat
• Density
• Kinematic viscosity
• Thermal expansion coefficient

ATTEMPTED APPROACHES I've already used the program Energy 2D to produce some semi-quantitative answers to this question (which I'll be publishing shortly), but the program is obviously limited in only being able to consider a cross-section of the tortoise.

DESIRED RESPONSE An equation into which I can plug the above variables and model heat loss over time.

You need to perform an energy balance on the turtle.

$m_{t}c_{p, t} \dfrac{dT_t}{dt} = -hA_t (T_{t} - T_{s})$

Where $m_{t}$ is the turtle's mass, $c_{p, t}$ is the turtle's specific heat at constant pressure, $A_{t}$ is the total turtle's surface area, $T_{t}$ is the turtle's temperature, $h$ is the heat transfer coefficient, $T_{s}$ is the surrounding temperature.

https://en.wikipedia.org/wiki/Heat_transfer_coefficient#Convective_heat_transfer_correlations should provide a sufficient heat transfer coefficient.

If you assume all the inputs, aren't functions of temperature.

You will have the following form.

$T(t) = (T(0) - T_{s}) e^{\dfrac{-t *h A_{t}}{m_{t} c_{p,t}}} + T_{s}$

• Presumably, this equation also works for tortoises... – TheTermiteSociety May 9 '17 at 13:04