# Setting up an equation for calculating how long it takes a body to change temperature in its sorroundings

The United States has just recently been hit by a massive vortex of Arctic air. These unusually bitter temperatures have sparked my interests to ask the following rheotical question:

How much time would it take for a naked human standing outside at the south pole to reach a temp. of 77 F (hypothermic death) given a wind speed of 40 mph, a resultant wind chill temp. of ~ -200 F (derived from the world record low at Vostok,) and various other variables such as body temp., surface area, etc., that I will estimate.

So, how would I set up an equation to calculate the time it takes the body to go from T1 to T2 given (I assume) the bodies' radiated power P?

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The Stefan-Boltzmann law doesn't have much to do with it, since the vast majority of the heat loss would be due to forced convection in this case. –  Nathaniel Jan 7 '14 at 8:15
Not through radiation? Because of the winds? –  Armend Veseli Jan 7 '14 at 8:24
The wind doesn't change how much you radiate, it's just that the rate at which you lose heat due to the wind is much higher than the rate at which you radiate it in this situation. –  Nathaniel Jan 7 '14 at 8:38

Brief note before answering, @Nathanial was absolutely correct. The temperature differences are so small here that radiation is negligible. You typically need temperature differences of 1000s of degrees in order for it to be the dominating effect (or be in a vacuum with no convection). Also, "wind chill" is not actually a temperature. It's a measure of how cold it would have to be in the absence of wind to lose heat at the same rate. Therefore, in all of the ensuing calculations, use the real temperature.

The approach to take setting up the problem strongly depends on how you want to solve it. The geometry of the human body (or any non-simple geometry) makes an exact analytical solution impossible. Therefore, you have pretty much 2 options:

## Simple Method

Simplify the problem A LOT and treat the problem as a simple body with uniform temperature inside. This method is called "Lumped Parameter Analysis", since you're effectively lumping the entire body (over which the PDEs that govern heat transfer apply pointwise) into a single object. This method is valid when the rate of heat conduction inside the body is very large compared to the rate of heat loss to the surroundings. The non-dimensional number that allows you to make this determination is called the "Biot Number" (which has a lovely Wikipedia page of its own that you should read). Since the body has blood pumping around in it to evenly distribute the heat (ignoring the fact that the fingers and skin will actually be a bit colder than the core temperature), it is a reasonable first-order approximation to assume that the body temperature is constant-ish.

By taking this approach, you are left with the following differential equation:

$$mc_p\frac{dT}{dt} = Vq_{vol} - A_{s}h(T-T_\infty)$$

In this equation, $m$ is the mass of the body, $c_p$ is the specific heat of the body (you could approximate this with that of water), $V$ is the volume of the body, $q_{vol}$ is the volumetric heating term (due to your body's ability to produce heat), $h$ is the heat transfer coefficient for convection, and $A_s$ is the surface area of the body.

Approximating $h$ is a bit more involved, so I would direct you to the convection chapters of Incropera's book Fundamentals of Heat and Mass Transfer. http://www.amazon.com/Fundamentals-Heat-Mass-Transfer-Guide/dp/0470055545

Once you have the information you need in that ODE (including your initial conditions), you can solve it any way you'd like and then find how long it takes for the body to reach 77 F.

## Harder Method

In the other approach, you do not make nearly so many simplifications, but it is much harder to solve and requires the use of a computer and either expensive software or solid programming skills.

First, you need to know the governing equations of heat transfer. Since we're doing it the hard way, we won't simplify to 1D or 2D. We will also be solving for the temperature field, not just a single number that describes the temperature of the entire body. To complicate matters, the heat transfer coefficient also becomes (in general) a function of position due to the non-uniform fluid flow around the body:

$$T=\hat T(\mathbf{x}) \\ h=\hat h(\mathbf{x})$$

The main equation is conservation of energy (1st law of thermodynamics) expressed in differential form:

$$\rho c_p\frac{\partial T}{\partial t} = -k \nabla^2T + q_{vol}$$

This equation describes heat conduction within the body, and does not yet account for the convection at the boundary. Next, we have to specify the initial temperature field $T_0(\mathbf{x})$ and the boundary conditions. The only boundary condition is convection (you could account for conduction through the feet into the ice if you're really ambitious too). The mathematical statement for heat flux due to convection is:

$$q''= -k \frac{\partial T}{\partial \mathbf{n}} = h(T - T_\infty)$$

Here, $\frac{\partial T}{\partial \mathbf{n}}$ is the temperature gradient in a direction normal to the boundary. How you handle this depends strongly on your solution method, which goes beyond the scope of this post. The heat transfer coefficient $h$ can be determined by a) solving the fluid dynamics problem to find the flow field around the body and using this to solve for $h(\mathbf{x})$, or b) simplifying your life (since fluids problems are much harder computationally than heat transfer ones) and just approximating $h$ as being uniform over the entire body and approximating it based on the far-field wind speed.

The last bit of complication is creating the geometry of your problem. You will solve this numerically, so you will need a mesh of the body.

At this point, you need to pick your favorite numerical method (personally, I like the finite element method) and solve it. Finite differences will likely be difficult since you're dealing with a non-rectangular geometry, so finite elements or finite volumes are your best bet.

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There you have it. The problem is set up and is ready to be solved. In the simple method, you have an ODE that you can solve analytically or numerically for the body temperature. In the harder method, you have a parabolic PDE that will yield the temperature field as a function of time. Given this, you can find the time when the average temperature reaches your target temperature.

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+1 to you (if I could, but I can't yet.) I appreciate the lengthy in-depth answer. A question like this one showed up in a text book during an AP physics class in high school, although it didn't tackle the problem in the same manner. I could of sworn it used the Stefan-Boltzmann law or some other black body radiation relationship. –  Armend Veseli Jan 15 '14 at 3:57