# At what rate does the human body heat itself?

The human body heats itself at some rate, because of this, I'm trying to add a forcing function to the differential equation known as Newtons law of cooling: $$Q = hA(T-T_{env}) + f$$, which I suspect accurate models a very simplified situation.

We have veins and capillaries that almost-evenly distributes heat over ever cubic centimeter of our bodies. Let's assume it's completely even.

Let's say a person presses the entire palm and closed (not spread apart) fingers of their hand on a piece of dry ice, which we'll pretend is a cross-sectional rectangle of surface area, with room temperature air. This coldness will spread throughout the hand.

Then, what forcing function must I add to model the temperature of the hand over time? At what rate does the human body heat its volume to implore Newton's law of cooling?

To the title question: most of the time, between about 50 and 2000 watts if you time-average over a few minutes, depending on level of activity, environmental conditions, and muscle mass.

The content of the question is basically broken at every level. Newton's Law of Cooling is useless here. Even a drastically simplified version of this scenario is a pretty tough heat transport problem. Homogenous heat distribution is a bad model for a human. A human is way too complicated to model from first-principles without considerable experimental data. Dry ice will violently vaporize on contact with your skin, adding another layer of complexity. A hand pushed hard into dry ice for a reasonably long period of time will die, freeze, and probably break off, wrecking whatever remains of the model - and probably causing the human to die of shock, which will also wreck the model.

For an analogous first-principles problem appropriate to beginners who can do differential equations, I'd suggest a warm brick of iron constrained to remain at constant temperature at one end, in thermal contact at the other end with a cold brick of iron, system thermally insulated.

So my thoughts vear more into physics than biology here but this is what I am thinking. It seems you might want to find the rate of heat transfer shown in formula 1 below.

$$R = k A \frac{ \left(T_{1} - T_{2} \right)}{d} \tag{1}\label{1}$$

Where $$k$$ is the thermal conductivity value, $$A$$ is the surface area of the object the heat is being transferred through, $$T_{1}$$ and $$T_{2}$$ are the internal and external temperatures, and $$d$$ is the thickness. So, say you know the surface area of the hand, the thickness of the hand, the starting temperature of the iced hand, and the final temperature of the hand after a given time. So you could possibly use the rate of heat transfer and the time to get the amount of Joules transferred. Or just leave it as the rate of transfer. I'm not sure this will be quite useful in the case of your question but I hope it could possibly lead you in the right direction when looking for an answer.