Why Halving a Greenhouse's Heat Takes 2.83 Times the Vent

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Before reaching for evaporative cooling, there is an opening to size, and the arithmetic for it is unkind in a specific way. The flow through a vent driven by warm air rising goes as the square root of the temperature difference, which means the heat it carries out goes as the difference to the power one and a half. Turn that around and the vent area you need goes as the temperature rise to the power of minus one and a half.

So halving the temperature rise takes 2.828 times the vent area, quartering it takes 8 times, and going from twelve degrees to one takes 41.57 times. That is why passive venting gets a greenhouse from unusable to tolerable and never to comfortable, and it is why fans exist.

There is a second problem underneath it, and it is a unit problem. Ventilation gets specified in air changes, which is a volume rate. The heat that has to leave is a glazed area times an insolation. Volume does not appear in the gain at all, and on a small frame the usual one air change a minute is out by a factor of 13.7.

The frame, the sun, and what a wax hinge is worth

A Backyard Discovery Aggie cedar cold frame greenhouse publishes a complete geometry, which makes it the right place to start: 48 inches long by 28.5 wide and 32.25 high, 94.79 pounds, 100 percent cedar with powder coated steel legs, wide bottom slats with a mesh liner for drainage, a double wall polycarbonate lid, and a heat sensitive wax hinge that opens the lid above 70 Fahrenheit and closes it again as things cool.

Those dimensions give a lid and footprint of 0.8825 square metres and four walls of 3.1834, and an internal volume of 0.7229 cubic metres, which is 25.53 cubic feet. Our page on why a winter greenhouse needs mass, not glazing works the night time half of this frame's behaviour and establishes the two loss coefficients the category runs on, 2.9 watts per square metre per kelvin for three quarter inch cedar and 3.6 for double wall polycarbonate. This page uses them on purpose so the two agree. The frame loses 12.409 watts for every kelvin it sits above the garden.

That page computes the winter day, at 300 watts per square metre, and gets 17.07 kelvin, which lands within two degrees of where the wax hinge is set. Now run the summer day. Clear midday on a horizontal surface is nearer 900 watts per square metre, and polycarbonate passes about 80 percent, so 635.4 watts arrives inside.

With the lid shut, 635.4 divided by 12.409 is 51.21 kelvin above ambient. On a 25 degree Celsius day that is 76 degrees inside, which is not a growing environment, it is a cooking one. The winter figure and the summer figure are the same frame with the same arithmetic and they differ by a factor of three, entirely because of the sun.

Now crack the lid. Say four inches along the 48 inch edge, which is an opening of 0.12385 square metres, with an equal low level inlet through those bottom slats, so the effective area for a two opening stack is the area divided by the square root of two, 0.08757 square metres. Take the height between the openings as 0.6 metres and a discharge coefficient of 0.6.

Solve 12.409 times the rise plus 12.702 times the rise to the power 1.5 equals 635.4, and the answer is 11.46 kelvin. Four inches of lid takes 51.21 degrees down to 11.46. The wax hinge is worth 39.75 kelvin, and the airflow doing it is 0.0357 cubic metres a second, which is 75.6 cubic feet a minute, produced by nothing except warm air wanting to be higher up than cool air.

Our own record calls the hinge "the feature everything else supports" and says 70 Fahrenheit is "warm enough that the frame genuinely needs venting, cool enough that it opens before damage rather than after". Both correct, and 39.75 kelvin is the size of what it is preventing. Full details in our review of the Aggie cedar cold frame.

Where the 1.5 power bites

Open the lid wider and watch what you get back. Twelve inches instead of four is three times the area, 0.37155 square metres, and an effective area of 0.26272. Run the same balance and the rise falls to 6.00 kelvin.

So tripling the opening bought a factor of 1.910 in temperature. Pure vent theory predicts three to the power two thirds, which is 2.080, and the shortfall is the conduction term: the 12.409 watts per kelvin through the cedar and the polycarbonate does not care how far the lid is open, so as the vent takes over, the fixed part of the loss becomes the thing holding you back.

The general shape is worth memorising because it applies to every vented enclosure, from a cold frame to a loft. Heat removed by buoyancy is proportional to the area times the temperature difference to the power one and a half, so at a fixed heat input the area needed is proportional to the temperature rise to the power minus one and a half. Halving the rise is 2 to the 1.5, which is 2.828. Quartering it is 8. Taking a frame from 12 kelvin to 1 kelvin is 12 to the 1.5, which is 41.57 times the opening, and there is no cold frame in which 41.57 times the lid is a lid.

A fan removes heat in proportion to its flow times the temperature difference, with no exponent on the difference at all, so the flow needed scales as the inverse of the rise rather than the inverse 1.5 power. That single difference in exponent is the entire case for powered ventilation, and it only starts to matter once you want a small rise. At twelve degrees the lid wins easily, for free. At three it does not.

The frame that has the same arithmetic and no automation

A Beipoo 2-tier wooden greenhouse cold frame publishes 35.43 inches long by 14.57 deep and 27.56 high, solid fir, high transparency PVC panels, an openable roof vent, removable slatted shelves, drainage holes, a magnetic door latch, a slanted roof and model BPG5004-007.

Its footprint is 0.3330 square metres and its walls 1.7781, with a volume of 0.2331 cubic metres. PVC single sheet runs near 5.5 watts per square metre per kelvin and fir near 2.9, so taking the walls at a representative mixed 4.5 gives 8.00 watts per kelvin, plus 1.83 for the roof, so 9.83 watts per kelvin. PVC passes about 85 percent, so the gain on the same summer day is 254.7 watts and the shut frame runs 25.9 kelvin above ambient. Half the cedar frame's figure, because it has a smaller glazed roof relative to its losses, and still far too much.

Its own drawbacks name the problem exactly: "the roof vent is opened by hand, so a hot day while you are out is still a risk." That is the honest sentence and 25.9 kelvin is the size of the risk. On a 25 degree afternoon the inside reaches 51. The cedar frame's wax hinge is a mechanical solution to precisely this and costs no power, which is the comparison the two listings do not make with each other.

There is a limit on the hinge too, and it is a fixed threshold. It opens above 70 Fahrenheit, which means on a 90 Fahrenheit day it is already fully open at dawn and has no authority left for the rest of the afternoon. From that point the vent area is the only control and it is not adjustable. Which is the moment a fan becomes the answer rather than a luxury.

What the fans publish, and what the frames need

A VEVOR greenhouse air circulation fan rated 1800 CFM publishes a 12 inch aluminium alloy blade, 13.78 by 8.27 by 24.02 inches, 9.41 pounds, a Q235 steel frame, AC110 to 230 volts, ten speeds, an external probe on an 8.2 foot lead, automatic start and stop on temperature from 33 to 176 Fahrenheit or humidity from 1 to 90 percent, a cycle timer from 1 to 600 minutes across up to 100 cycles, and noise at or below 60 decibels.

An iLiving 18 inch wall mounted fan rated 6360 CFM publishes an 18 inch blade, 6,360 cubic feet a minute, a sealed 168 watt motor, 22.5 by 8.25 by 28 inches, 20 pounds, UL listing, a weatherproof corrosion resistant finish, an OSHA approved grill, an adjustable bracket, three pull chain speeds and compatibility with a separately sold misting kit.

Now set those against the frame. To hold the cedar frame 6 kelvin above ambient needs 164.2 cubic feet a minute. To hold 3 kelvin needs 350.3. To hold 1 kelvin needs 1,094.5.

So the 1,800 unit, if it exhausted, would hold 0.613 kelvin, which is 70.5 air changes a minute through a 25.53 cubic foot box. The 6,360 unit would hold 0.175 kelvin at 249 air changes a minute. Both are machines for a building, which is what their records say: iLiving names "garages, workshops, patios, greenhouses, loading docks and livestock barns", and the record correctly observes that the airflow figure "is consistent with that list rather than with a domestic room".

A VIVOSUN AeroWave E6 Gen2 oscillating clip fan publishes a 6 inch blade, 320 cubic feet a minute at a stated 33 decibels A, a dual ball bearing EC motor, five level auto oscillation, a natural wind mode, an IP54 rating, 5.58 by 9.15 by 9.88 inches at 1.6 pounds, ten speed levels of which its own drawbacks admit only five work without a separate hub, and a two year warranty.

320 cubic feet a minute is, for this frame, almost exactly the right number. It would hold 3.27 kelvin, which is within a tenth of a degree of the 350 cubic feet a minute a 3 kelvin target asks for. That pairing is not something either listing could know, because neither publishes the quantity that connects them.

A pair of PGQUY 5 inch dual grow tent clip fans publishes two fixed heads rather than one oscillating one, a brushless DC motor below 40 decibels A, an IP54 rating, an 8 foot USB cord with a two port plug, 5.25 by 5.25 by 2.25 inches each, three speeds, and model GTCF-PPACK. It publishes no airflow at all, which is the one figure a fan exists to have, and without it nothing on this page can be said about it.

The distinction that none of this works without

Everything above assumed the fan moves air across the boundary. Three of these four do not. A circulation fan inside a closed frame removes exactly zero heat, because the air it moves comes back. Its airflow does not appear in the balance, and its contribution to the temperature rise is not small, it is nil.

That is not a criticism of circulation, which does a real and separate job: it breaks up the still layer on a leaf, it evens out the temperature so that the top shelf and the bottom shelf agree, and it carries moisture off foliage. All of that is worth having and none of it is cooling. Our page on why a fan's CFM rating is not the breeze you feel works through what an airflow figure does and does not promise a person standing in front of one, and the same gap applies here with a plant in the chair.

One sentence in our records blurs the two. The VEVOR record says ten speeds "let you run gentle circulation in spring and autumn and step up to faster air exchange through summer". Circulation and air exchange are different quantities with different units of effect, and a circulation fan cannot do the second one no matter what speed it runs at. Exchange requires an opening to outside and a pressure difference across it. The fan's own automatic temperature control makes this pointed: thresholds from 33 to 176 Fahrenheit will switch it on when the frame heats up, and switching on a circulation fan when a sealed frame is at 76 degrees homogenises 76 degrees.

And then the term that runs the wrong way. A fan inside a closed space is a heater of exactly its own electrical input. The iLiving unit publishes 168 watts, and it is the only one of the four that publishes a wattage at all. Against the cedar frame's 12.409 watts per kelvin, 168 watts is 13.54 kelvin of temperature rise from the motor by itself, which is 18 percent more than the 11.46 kelvin the sun manages through a cracked lid. In a walk in greenhouse of 8 by 10 feet the same 168 watts is worth about 1.5 kelvin, which is negligible. The figure is not the fan's fault; it is a statement about which box you put it in.

The misting kit is the interesting exception, and it belongs to a different mechanism. Pairing the wall fan with the separately sold kit turns air movement into evaporative cooling, which can take air below ambient rather than merely towards it. Our page on why an evaporative cooler runs out of degrees in humid air sets the ceiling on that, which is the wet bulb temperature, and the record is right that this fan is "rated for that use rather than merely surviving it", which is not a small distinction on a sealed motor.

Why air changes is the wrong unit

Here is the unit error, stated plainly. Greenhouse ventilation is conventionally specified at about one air change a minute at peak. On this frame, one air change a minute is 25.53 cubic feet a minute, which contributes 14.53 watts per kelvin of flow against 12.409 watts per kelvin of fabric. Total 26.94, so the rise is 635.4 divided by 26.94, which is 23.6 kelvin.

Twenty three and a half degrees above ambient, following the rule. The frame needs 13.7 air changes a minute to reach 3 kelvin, so the rule is out by a factor of 13.7.

The reason is the same one our heater page identifies for the opposite problem. An air change rate is a volume per unit time, and the quantity it has to carry is a glazed area times an insolation. Those two scale differently. Make a frame half as tall and you halve the volume, so one air change a minute halves, while the glazed lid and the sun have not changed at all. A rule expressed in air changes gives different answers for the same roof depending on how deep the box beneath it is, which is not a property the sun has.

The correct form is not an air change rate. It is a flow derived from the heat: take the glazed area, multiply by the insolation and the transmittance, divide by the temperature rise you will accept, and divide again by 1,206 watts per cubic metre per kelvin to get cubic metres a second. That calculation needs a glazed area, and neither of these frames publishes one, though both publish enough dimensions to recover it.

What this category does not publish

Neither cold frame publishes a glazed area or a vent area. Those are the two numbers the whole balance runs on. Three overall dimensions get you the first by multiplication, which is why the cedar frame can be worked at all, and nothing gets you the second: how far a wax hinge lifts a lid is not stated anywhere, and it is the single most consequential figure on the product.

Three of the four fans publish no wattage, on a machine whose job is removing heat and which makes heat doing it. Only one publishes an airflow with a noise figure beside it, and one publishes no airflow at all. None publishes a static pressure, which is what decides whether an extract fan can actually pull air through a glazing gap rather than just spin in free air, and that absence is the reason this page has to say "if it exhausted" rather than "it exhausts".

Our roundup of greenhouse fans covers the category, and the figure to carry into it is the one derived above rather than a rating in air changes: the flow that holds the rise you want, which for a frame of this size is a couple of hundred cubic feet a minute and not a couple of thousand.

The honest limit, and the three numbers to carry

What is exact here is the structure. A steady state rise is a gain divided by a total loss coefficient. Buoyancy flow goes as the square root of the temperature difference, which follows from equating a pressure head to a velocity head and needs no assumption, so the heat removed goes as the 1.5 power and the inverse 1.5 power law for area follows by inversion. A fan's removal is linear. The areas and the volume come from published dimensions by multiplication. The 2.828, the 8 and the 41.57 are powers of two, four and twelve and nothing else.

What is representative is the physical furniture. The 900 watts per square metre of summer midday insolation sets every absolute watt and kelvin on the page: at 700 it falls by 22 percent and at 1,000 it rises by 11, while every ratio and every power law is untouched. The 80 and 85 percent transmittances, the 5.5 and 4.5 watts per square metre per kelvin for PVC and mixed fir, the 0.6 discharge coefficient, the 0.6 metre stack height, the four inch and twelve inch lid gaps and the equal inlet area are all in the same category. The discharge coefficient is the one worth flagging: a sharp edged slot runs nearer 0.6 and a well shaped one nearer 0.8, which moves the 11.46 kelvin to about 9.3, and does not touch the exponent.

What this page deliberately does not compute is three things. It does not compute a temperature any plant needs, because that is a biology question and no listing here carries a figure for it; all that is claimed is where the inside sits relative to outside. It does not compute what the misting kit achieves, because that needs the local wet bulb depression and a water flow rate, and the kit publishes neither and is a separate purchase. And it does not compute the snow load that the pop up greenhouse question was really asking about, because that needs a member section, a span and a fixing detail, and a cold frame listing publishes a total weight and a timber species and nothing structural. What can be said is that the thermal argument and the structural one are unrelated, and a frame that vents well in July says nothing about a frame that holds up in January.

Three numbers to carry. Passive vent area goes as the temperature rise to the power minus one and a half, so halving the rise takes 2.828 times the opening and there is no lid large enough to reach ambient. Required airflow is a gain divided by a temperature rise, not a volume times an air change rate, and on a small frame the conventional rule is out by a factor of 13.7. And a circulation fan removes no heat at all, while adding its own wattage, which on a box this small is worth 13.54 kelvin for the one fan here that tells you what it draws.