RV Maintenance Thermal Efficiency 3 |

While I was out at Black Butte Lake, I decided to see what the insulation characteristics were on the trailer. Before I began, I let the sun come through the windows and warm the RV up to about 85 degrees. Ya, a little warm, I know. But the greater the temperature difference the more accurate. At sundown, the windows were insulated with Reflectix and the trailer was allowed to cool. I waited a while, to let everything settle, and started the run at 17:00. During this measurement, no furnace heat. And the Hot Water Heater was off. And the refrigerator was turned off.

However, I forgot about the stupid pilot light, which later turned out to be a big issue. But because of that mistake, I was able to write equations that explained internal heat sources. So, it actually turned out to be a really super "good" thing. Now, I would not trade that mistake for anything.

The inside temperature started out much higher than the ambient outside temperature at 83.8 degrees. And I started taking readings at 17:03 with a GPS clock, accurate withen 1 second.

Yellow data points are Inside Temperatures.

Blue data points are Outside Temperatures.

time inside outside

17:03 83.8F 65.1F

18:10 81.2F 56.2F

19:14 75.2F

20:18 75.7F 52.7F

22:32 69.7F 47.5F

00:59 65.3F 45.9F

05:16 58.2F 39.4F

It is interesting to note a slight rise in slope of temperature at 01:00 that is reflected in both curves.

Also, the two Woofers had to go pee in the middle of the night. In fact, I had to open the door twice for them, and this let in plenty of cold air.

Also, the pilot light was lit in the stove.

The starting temperature a 17:03 is 65.1F degrees: TAH. The "ground" temperature TGL is estimated at 35F degrees.

The Curve seems to fit of this logarithmic form.

The Asymptote Line

It is in "Cyan" color, and constantly about 12.5 degrees above the Outside Temperature. If the outside temperature levels-off and remains stable without change, heat sources in the RV will always sustain a stable temperature inside the RV that is ABOVE the outside temperature. It is a set amount. In this case it is 12.5 degrees. It is the delta Temp determined by the Qp, Area, Rval. The inside temperature will "approach" this asymptote line, but will never reach it. Well, actually, it could reach it if the Outside temperature slope suddenly turned positive. But if the outside temperature slope continues to decline, as it behaves here, or if it is even zero (level), the inside temperature will never reach it.

Asymptote curve. |

It should be possible to write the equation with the pilot light, and other stuff, included.

The added energy is continuing to add energy every hour. It is a function of t.

To write this, I took liberties, and I had to make assumptions. But I have a creative license, and the computer stayed with me, and wrote equations that give a very accurate graph.

I began by taking the basic definition of the Rval. And trying to relate it to the "Decay Function". That decay function has an exponent, k. And there MUST be a correlation between how a thing cools down in the Decay Function and its Rval. There is NOTHING else to control its behavior. I multiply both sides of the equation by 1/t. These are the same units as k, so I will assume that it is possible that k could equal this. Later, it will be clear that it is indeed true.

Once that is established in what I call an Equivalent, then I turn my attention to creating an Active Heat term, to explain the pilot light.

And then two terms materialize, including my Active Heat term. Th Heat Source term is colored "red". I invented this term, a function of t, to continually supply new heat, and raise the curve.

But I have a problem with my fresh water tank, a 60 gallon tank, which has a dual purpose. It is also used as a heat reservoir.

My RV, unlike other RVs, was designed with the Fresh Water Tank as not only supplying warm water, but also as a heat reservoir. The reservoir holds a BTU per degree per pound. The warm water, just under the floor, would supply more thermal inertia, and more heat reserves.

At the time, it was 35 gallons full.

It is basic to see:

8.34 lb/gallon, 1BTU/degree/lb, = 291.9 BTU/degree

But then again, the water does not have easy access to the inside, as it is just beneath the floor. The tank is insulated from below to the outside, which is great. But also, unfortunately, insulated somewhat from the inside area. Probably, the tank does not need to be analysed in isolation. Certainly, it will be added in automatically, so forget that I mentioned it.

It is NOT part of a "closed" system. An example would be an active heat source such as the stove pilot light, which is supplying aprox 650 BTU/hr. As for the BTUs of the pilot...

That is not known exactly. The spec is 700 BTU/hr. But my pilot light gasses are vented to the outside. This was a modification that I made to the stove. Thus, the heat into the RV can not be 100% efficient. So the value should be somewhere less than 700.

In addition there is body heat. I estimated 300 BTUs/hr for me, and about 100 for the doggies. Just guesses.

The units for this term are Q per temperature, the same as the other term.

I should talk about S...

At first, I invented S as an inverse "slope", a true deferential. But it did not work. Then I realized that it was a "delta". The denominator as the difference between the start inside Temperature and the outside Temperature. And the numerator as the difference between the Start time of zero, and the active time. Actually, reverse that order: S is positive.

I put those values in the equations and the curve lined up with the data points. I was surprised! I retained the misleading name of "S", although it has nothing to do with "slope".

Internal Stores, mass energy reserves:

mCp=374 BTU/deg

The old value was 144 BTU/deg, which I think was too low and not enough "stuff". I like the new value better...

mCp is the Specific Heat, Cp, times mass (m). Things in the RV "hold" heat. These include cans of food in the cupboards, the commode and tub, the steel of the stove, and the wood of the walls. This is an average of all the masses all over inside of the RV. mCp is Q per degree. So the real essence of mCp is Q (in BTU) per degree.

The units of mCp are Q/(delta Temperature), or BTU per degree.

Rval:

The Old Rval value was Rval=-6.447. New value is -6.89,

There has been no great improvement due to the insulation itself.

The new Rval is very much a guess. The Rval can only be determined accurately by "the definition" method. And that was not done this time. To measure Rval directly, the furnace needs to produce a lot of BTUs, and the furnace needs to come on and off to maintain a constant temperature.

Here is the "main" exponent: kb(t)

I have always been suspicious, and wanted a way to express, a relationship of R-Value to the natural decay function.

So, I created two terms withen my "equivalent".

The old Time Constant was -0.5 to -0.6 per hour. That was when the RV was new, and represents a fast decay and a lot of losses. The NEW Decay Time Constant k -0.1269 is a wonderful difference. It is a long way from -0.6s and -0.5s. The units of K are 1/t, or one-over-time.

The better k value is from my RV work of running around stuffing insulation, and from my work on the stove.

(Just as a note, I need to compare the Vans Insulation:

Rval=-4.642

mCp=685.7

The mCp is over twice that of the RV! So much more steel and thermal inertia in the Van, especially from the engine and tranny and steel floor. Dispite it's poor insulation, it is slow to change temperature.)

Q total is 1100 BTU/hr.

Certainly, 1100 was an educated guess, but it tracks well on the graph. The human body produces about half of the pilot light energy. About 300 BTU/hr or 100W.

I did work on the stove. The pilot light gasses are vented to the outside: thus, the pilot light can not be 100% efficient. It has to be less than the 700 BTU/hr. I never imagined that such heat sources would be significant and actually show up in the calculations. But they do! Although all these things started out as mere guesses, - there they ARE!

As plain as day!

The value is responsible for the asymptote curve, at 12.5 degrees above the outside temperature, as well as raising the final curve.

A is 605 sq ft, surface area. A is the surface area of the RV: the four walls, ceiling and floor.

The Outside temperature Line

This will go into the final, equation. I conjured this up without any formal proof. As the time goes to infinity, the asymptotic nature of the final equation becomes clear.

Without internal heat, the "normal" asymptote line would simply be the Outside Temperature. But my RV has continuous internal heat generation. The Q heat will create a temperature difference. In this case about 12.5 degrees. An imaginary line will run 12.5 degrees above the outside temperature.

Tfin is the Inside Temperature.

Tr(t) is the Outside Temperature Ta(t), plus the heat rise amount; A changing function of t.

TBH is the specific (Hottest) Starting Inside Temperature. A constant.

kb(t) is the main exponent. And contains my two Equivalent terms.

83.8 to 82.8 Inside almost steady state

35 outside steady state

24 hours/day

4143BTUs/hr Total - 1220 BTUs/hr body heat = 2913 BTU/hr = 853.7 Watts

2913 BTU/hr:

650000BTU/Canister, = 223 hrs = 9.3 days, 9.3days/$25, = $80/month

853 Watts:

0.853 kW * $0.15= 12.8 cents/hr =$3.07/day = $92/month

72 Inside

50 outside

659 BTU/hr during only 12 hours, at night. 659 BTU/hr = 193 Watts

659 BTU/hr * 12hr/day = 7908 BTUs/day , 650000/7908 = 82.2 days, =$9/month

193Watts, 0.193*.15= , $10/month

Utility Electricity is more expensive than my onboard Propane.

But at most full hookups, electricity is free, and it's use is a no-brainer.

Until the insulation characteristics change in my RV, I can now calculate the inside temperature; using changing outside temperatures, and different inside heat sources. ...And I can also see the insulation improvements.

Extended graph |

Extended graph

Now clearly, with the graphing, my trailer is fixed and it is working great as it should, at least on paper. Can't stand to have a broken trailer due to a faulty cooling curve. With the equations done, I have the perfect excuse to test out the equations and the RV at the ocean or a lake. It is just the right amount of excuse. Time to go. Gota get it hitched up first though, then decide where...