Heating a Trailer, Part 1

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

Background:
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
1/U2=1/hgi+deltag/kg+1/hgi.

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.

goodbye black background :(

Some people feel that it is hard to read the white text on the black background, so I am changing. I think the current template is maybe a bit cartoony, but if I tell myself the circles are really representations of spheres, I like it pretty well. Of course, I still think white text on black is easiest on my eyes, but I've kind of given up on it in general, so I may as well go with a black text blog. When I have tried to change my computer settings to white on black I become more annoyed because only some programs, such as word processors, will understand, while others will never work right as white on black and just seem even worse when you have to flip to them from a white on black screen.

go science!

I just went to an amazing thesis defense, and learned a lot. I was annoyed that I didn't have time to read the thesis ahead of time, but now I'm glad because I don't think I would have understood it as well the other way around. Now I have to read the thesis before I forget everything I learned (or lose motivation)! Also, there were good snacks, coffee (and I brought my own mug) and the presenter's new baby there. The baby did not stay for the whole thing, but was making adorable baby noises :)

"real world problems"

Here is a (slightly paraphrased) snippet from my application essay to grad school, which I am reading now due to a sequence of events that started with a dying computer 3 years ago. (It's a long story that I could summarize, but I'm not in the mood.)

"I chose to pursue a doctorate in this field to apply my basic science knowledge to solve real world problems".

That is HILARIOUS!

2 lines!

I finally took the test this morning, and I am pregnant! I hope everything is going alright, because I don't feel sick at all. It is nice to be able to eat at will, though. I wonder why this one is so different? Maybe it's a girl.

Now . . . when to tell the adviser?

Checkup!

Several interesting things at LP's checkup today:

1) I was lollygagging this morning, not realizing the time, and TE was playing with LP, so we didn't get going to the appointment on time . . . we started about 10 minutes late. We had planned a 40 min commute (the doctor recently moved far away), which would have been barely possible with the tandem, but today we needed separate bikes. I was happy to take out my recumbent, as it has been sitting unused for awhile. It's really nice how fast it accelerates in response to my pedaling versus a huge tandem recumbent with trailer. TE took the trailer today, and since it was also windy, for the first time I got to be faster than him--I was enjoying it. Even though we were sprinting to get there in time, we got further behind schedule due to wind and rain. When we got there, I was soaking wet with sweat (I wore one too many layers!) and we were 15 minutes late. The doctor was not ready yet anyway, though.

2) You know how you can be somewhat oblivious to something that is culturally or socially supposed to bother you, but then when people keep pointing it out it can start to bother you? LP has been saying "dada" for months now, but still does not say "mama". Recently, he started saying "mo" or "ma" to mean "more", to rub it in that he can pronounce the "ma", but I still didn't really care. Today, though, the doctor was trying be nice to me and told LP he needs to start saying "mama", and pointed to me. Then, ScienceWoman wrote that her daughter, Minnow, said "I wuv do Mommy" and apparently it was really heartwarming. (Of course, Minnow is a bit older than LP.) Now I am a little (just a little bit) sad now that LP does not say "mama". He does sort of think I'm "dada", but he knows TE is the real "dada" and seems to suspect I'm some other type of "dada". I guess it will come eventually.

3) LP is developing fine.

4) LP has gained only an ounce since last time. Actually, I think he weighed in more during the finger incident doctor's appointments than he did today. No wonder he's been so hungry recently--he must be growing very fast and burning lots of calories running around like a crazy person. The doctor was not concerned, though, and neither am I since he still looks to be well within the normal range in his daycare cohort.

5) When we dropped LP off at daycare, it was later than usual, so breakfast was over. It was STORYTIME! They were all (technically, most) sitting in the story area and watching the teacher read a book. It makes it seem like LP is a real kid now! I wish I had storytime. I think he will learn a lot really fast now that he's moved up to the new room at daycare. He already uses a spoon better. They say he is doing very well there. Now that there are all these activities and different things to see him play with, watching the webcam is a lot more fun. Yesterday he was playing on a baby sized set of stairs, and bumping butts with another kid, who then decided to go down the stairs on his back and head first. (I was glad LP didn't try that, because I wouldn't want him doing it on real stairs!) Today I got to see TE pick him up, as I was still here in the office.

I don't even want to take the test now

We may have ruined our perfect record, because I think I am not pregnant. My cold is getting better and I still don't feel pregnant. I looked at my notes from LP's pregnancy and I felt pregnant by this time. Maybe I am just missing it because I am sick and still nursing.
Not that it is a big problem as compared to other peoples' problems, but I am bummed and I don't want to take the test now because I don't think I will get the desired result and it seems like a waste of resources. The tests come free from the health center in packs of 2, and the one I have now is the second one from when LP was getting started.
I just thought of something hopeful, though! Maybe I'm pregnant but it is different than last time because this one is going to be calmer than the LP, either because it is a girl or because of the nursing evening out the hormones (or some other reason). That would be nice, to be able to actually eat what I want (without throwing up) as a hungry pregnant person. Maybe I should take the test.

getting sick, and voting

The LP has been cranky for a couple of days at night, and has a runny nose, so I suspected for several days that he was getting sick. I should have known that there was impending doom for me (and probably TE), but I guess it's better that I didn't realize what was coming until I actually started feeling crappy yesterday. Maybe I can be sick twice as much later this week if there is an ELP on the way . . . with LP the sickest I got the whole pregnancy was the day I found out for sure he was there. Hopefully I'm not messing up the ELP with the cold medicine. . . I took some last night to try to breathe better in my sleep after being up late with LP because he couldn't get to sleep. At least we got to watch McCain's speech and part of Obama's between crying bouts.
Speaking of the election, I stood in line to vote for the first time (the line was a first, I mean) yesterday. Despite leaving the house more than an hour earlier than usual so that we could vote on the way to daycare, we were later than usual dropping LP off. I spent somewhat over a half hour in line, but TE spent about 45 minutes because he was in the longest line (due to different last names). It was fun, in a way, because it made me feel like I was part of history (I think at least one friend was disappointed not to have to wait in line). It would've been even more fun if the line was only, say, 10 minutes, and the LP was happier, but at least he wasn't screaming.