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QUESTION

For a business project, I'm looking into the viability of a greenhouse situated in Canada's far northern town of Iqaluit. The problem is how to mathematically simulate the internal air temperature of the greenhouse.

WHAT I ALREADY HAVE

  • detailed 3D renderings of the greenhouse
  • all materials with their corresponding thermodynamic values such as K-values
  • climate data that shows the approximate intensity of solar radiation in Iqaluit, angles of sunlight throughout the day and hourly temperature data.
  • data for the required environment for certain crops to thrive, such as humidity, required sunlight, temperature fluctuations, pH, etc.

CONTEXT

What make this greenhouse investigation unique is that the greenhouse is a much smaller affordable design that is meant to grow a small supplementary source of fresh produce for low-income families which often suffer from food insecurity in northern towns such as Iqaluit due to prohibitively high costs for fresh produce.

We would also like to strive for simplistic, mechanical systems that are easier to maintain and less prone to failure. A major concern of the group as of now is by how much could the internal air temperature plummet by at night?

GOAL

The goal in mind is to use the information we have so far to determine:

  1. Is it possible to grow vegetables such as leafy greens within the current design WITHOUT an external power source other than sunlight? (A greenhouse that does not require electric heating is a major goal) How does the internal air temperature vary with the current design?

  2. In the case that the current design is not sufficient for healthy crop growth, what modifications are necessary? (Many potential solutions have been considered):

    • Have a solar powered heater that charges through the day and intermittently provides heat through the night.
    • Greatly increase thermal insulation surrounding the greenhouse through the use of a thick air-inflated wrap that surrounds the entire structure in addition to thicker polycarbonate panels for construction.
    • Use a setup of concave reflective panels surrounding the greenhouse to increase the amount of heat collected during the day. I have already done some calculations but my numbers ended up far off. Equations I have worked with so far are $Q = mc\Delta T$ and $\dfrac{Q}{t} = \dfrac{A\Delta T}{\text{Thermal Heat Transfer Coefficient}}$.

CONCLUSION

I think the problem demands looking at every single material and also examining the characteristics of the air inside as an ideal gas, as well as the poly-carbonate and it's emissivity/albedo (black body) characteristics.

Any steps or hints in the right direction with equations or methods of calculation would be greatly appreciated!

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    $\begingroup$ BTW, this is a greenhouse design that has been sucessfully used in China as far north as 43° (still south to Iqaluit) lowtechmagazine.com/2015/12/reinventing-the-greenhouse.html $\endgroup$ – mart Aug 30 '16 at 12:50
  • $\begingroup$ @mart: 43 deg north isn't a big deal. I live at 42.5 deg N, and grow tomatoes outside in the summer. Put another way, a greenhouse that does absolutely nothing works fine at this latitude. $\endgroup$ – Olin Lathrop Aug 30 '16 at 17:41
  • $\begingroup$ @OlinLathrop your keyword is summer. Upon re-skimming my article, I see they also tried these in Manitoba (50°N) but needed extra heating in the winter months to grow tomatoes. Iqualuit is 60°N or so. $\endgroup$ – mart Aug 31 '16 at 7:30
  • $\begingroup$ @mart: Your question doesn't say anything about wanting this greenhouse to work year-around. That would be a tough problem even for significantly more equatorial latitudes. I thought, and what seems reasonable, is that you want to get growing conditions in a greenhouse in summer like would be outdoors at a few 10s of degrees lower latitude. Even that may be a stretch for solar-only. Doing it in winter seems totally unreasonable at first knee-jerk reaction. $\endgroup$ – Olin Lathrop Sep 3 '16 at 13:11
  • $\begingroup$ It's not my question, I posted the link because the OP might find it helpful. $\endgroup$ – mart Sep 3 '16 at 21:01
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There are way too many details to get into here. However, do the basic first order analysis. Determine whether there is enough solar power over a day to keep the greenhouse at some temperature.

You say you know all the dimensions and insulation values. Pick a plausible temperature differential between inside and outside, and compute the power required to maintain that temperature differential. This is all straight forward.

Now compare that to the area required to intercept that amount of solar power. At your latitude, use maybe 800 W per square meter from unobstructed sun perpendicular to a flat black panel. Then consider expected number of sunlight hours per day, and cosine of angle to the sun over the day.

If the average sun power received doesn't meet or exceed the average power required, the system simply won't work and there is no point going further. Redesign or go home.

If the average received power is sufficient, then you look into whether sufficient heat storage is reasonable for what is required. You first have to decide how long of a no-sun interval you want to design for. Obviously there won't be sun every night, but you also have to consider cloudy days. What length of bad weather occurs infrequently enough for it to be acceptable for the plants to die or supplemental heat to be required?

I suspect that overall, any useful solution won't be cheap or small due to the large thermal storage required. You can store more heat in the same volume if it can be stored as a phase change in some material. This would need to be a liquid-solid transition to be practical. Unfortunately, the water liquid-solid transition temperature is not convenient. I've heard of paraffin used for this, but a few cubic meters of paraffin is probably too expensive.

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