Sunday, November 30, 2008

Heating a Trailer, Part 1

Electric Heating of a Bike Trailer for an Infant/Toddler, Part 1: Problem Setup

It is getting cold, and TE has built a suspended bike trailer with solid walls (as opposed to the fabric covered one we have now) which should be better for winter use. (I helped only a little, because he's been getting up at 4 AM every morning to work on it, but I need my sleep since I feed LP at night). It is well insulated, but LP is still very small, so there has been considerable discusson on how/whether to heat the trailer for the very coldest days. Of course, there is a safety issue either way: a heating system must be implemented cautiously to prevent burns, but if there is no heating system we are worried about frostbite. So far, we have been bundling up LP with multiple layers, including a hood and hat under his helmet, socks on his hands, and blankets on top. He knows how to take off the socks, though, and TE is concerned that if his fingers get cold he won't realize it because they will also get numb. I think he'll know if his fingers are too cold or too hot, but it's hard to tell the difference between the "I'm a bit cold" cry and the "I'm bored" cry. TE wants to buy a big battery anyway for eventual use with a motor as an assist when I am very pregnant this summer. (I don't think we need a motor since we didn't have that much trouble last pregnancy, but this time we have to pull a trailer and have less time on our hands to take the trip slowly.)

Problem Statement:
Before I talk about heaters, I want to find out the scale of the problem to zeroth order. Let's assume the trailer is an enclosed box (assume little leaking of air from outside for now) of dimension 25x25x30 inches. The majority of the box is 0.125 inch plywood with 0.5 inch foam insulation. There are also 6 windows of average size 6x18 inches made of 0.118 inch plexiglass. A small person is inside but may weigh anywhere from 8-50 lbs (we want the new baby to be able to use it too). If the outside temperature is 0 deg. F and wind speed relative to the trailer may be 10-30 mph, how much energy is needed to keep a constant steady-state inside temperature of 40 deg. F (he will still be wearing layers)?

Human Heat Output:
One could do a calculation based on the amount of heat lost to the environment, but I saw a Google Books result from Human Vitality and Efficiency Under Prolonged Restricted Diet, Francis Gano Benedict et al., which claims that under normal diet conditions the average heat output of their subjects was 25.2 calories/kg/(24 hours). From the text it is clear that they mean food Calories, or (25200 cal*4.18 cal/J)/kg/(24hr*60min/hr*60s/min)=1.21 W/kg. Babies are probably outputting more heat per kg than adults, so let's say the likely range of occupant wattage is 5-50 W.

Equation for Heat Loss from Trailer Walls:
We have a total surface area of 3600 square inches of wall and 648 of window. Newton's law of cooling states Q=h*A*deltaT, where Q is the rate of heat transfer, h the heat transfer coefficient, A the area, and deltaT the temperature difference. We can write this equation for the inside and outside walls, and write the conduction equation Q=k/deltax*A*deltaT for the plywood, foam, and plexiglass, where deltax is the thickness and k the conductivity. Of course, then we have many different temperatures to solve for (for instance the deltaT across the plywood is the temperature difference between the outside 0 deg. F and inside of plywood, which unknown but a bit colder than the inside of the trailer). We can simplify by analogy to electrical circuits: each heat transfer coefficient or conductivity divided by thickness is a resistance to heat transfer, and they can be added up like electrical resistors. Then you can have an overall heat transfer coefficient U, for the equation Q=U*A*deltaT. In series are the heat transfer from plywood to outside hwo, plywood conductivity kw/deltaxw, foam conductivity kf/deltaxf, and heat transfer from foam to inside hfi, which will combine to make an overall heat transfer coefficient U1. In parallel to all that, and with a different area, so we'll make a separate overall heat transfer coefficient U2, is the heat transfer from plexiglass to outside hgo, heat transfer across plexiglass kg/deltaxg, and heat transfer to plexiglass from inside hgi. So we have
Q=(U1A1+U2A2)deltaT, where
1/U1=1/hwo+deltaxw/kw+deltaxf/kf+1/hfi and

Actually, there would be different heat transfer coefficients for the different walls, as some will be facing the wind more than others. I will neglect this, and assuming a worst case scenario of 30 mph wind on all walls. The heat transfer coefficients for either the wood or plexiglass in contact with 30 mph air should be similar since both are relatively smooth flat plates, so hgo=hwo and hgi=hwi.

(Sorry about the lack of pretty equations--perhaps I will fix them later, someday after I get a full night's sleep . . .)

Next steps:
Now all we need are to look up the constants, calculate, and compare to the heat output of various devices that are being considered: nothing, warm thermal masses (such as those that can be microwaved to keep food hot at potlucks), electric blankets, and other small electric heating devices, preferably those designed for human use in a related application.
The suspense is killing me but this post is becoming long and I must eat dinner!

Feel free to leave ideas on trailer heating or knowledge of any of the constants in the comments.

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