My experience with air purifiers in polluted Delhi

My experience with air purifiers in polluted Delhi

Since the year ends today, I would like to share my experience and simple analysis of two air purifiers that I own, in Delhi: one from Philips and the other from Smart Air.

Background & CADR:

http://www.oransi.com/beginners-guide-air-purifier-s/168.htm

As pointed out in the above website, air purifiers come in two styles: tower-type (the Philips) and box-type (Smart Air):

“Tower Air Purifier - the tower design, as the name suggests is shaped like a tower. These tend to be taller with a smaller footprint, so they take up less floor space. Given the design, the filters are smaller and often have lower air flows.

Box Style - the box shaped air purifier has more of a square shape. The advantage of the box style is the larger surface area of the filter. This allows it to more easily move more air with a lower pressure drop. As a result, the box style can clean larger room sizes and provide cleaner air.”

My experience is similar, but I should add that the higher the speed setting, the more efficient is the air purifier. And you should opt for an air purifier with the highest CADR (see below). Unfortunately, the higher the speed, the noisier the purifier – which is a problem if you want to sleep and are sensitive to even white noise.

The main issue for me is: how effective is the air purifier? To answer this question, I took readings of PM2.5 and PM10 with my AirVeda particle monitor (in mg/m³), and tried to compare them with reported values. Unfortunately, the air purifiers are not as effective as claimed/reported.

https://smartairfilters.com/en/blog/purifiers-useless-windows-opened/

For example, Smart Air reports in Beijing, that their air purifier reduced the particle concentration from 1,000 mg/m³ to 100 mg/m³ in 20 mins with all doors and windows closed, but by only 60% (to 400 mg/m³) if one window was left open. My experience in Delhi is that on Diwali day, you cannot get the particle concentration below 300 mg/m³ no matter how long you wait. A caveat, however: I used a Philips air purifier and I did not use the highest speed setting. Next time I won’t make the same mistake. I am left with a doubt: is it that rooms in Beijing are more airtight than those in Delhi, since it is a lot colder in Beijing? Is that what makes the difference?

https://en.wikipedia.org/wiki/Clean_Air_Delivery_Rate

I should note that there is a difference between the cfm (cubic feet per minute) of the air purifier and its CADR (clean air delivery rate):

CADR = (filter efficiency) X (cfm)

For a HEPA (high efficiency particulate air) filter with 99.7% efficiency, they are almost the same, but for a 40% efficiency filter, the CADR is that much smaller (by 2.5X). Both the air purifiers that I have use HEPA filters.

https://gadgets.ndtv.com/others/features/how-to-buy-the-best-air-purifier-in-india-1622333

"The CADR shows you how much air is coming out of the purifier," says Bapat, "and it should change the air in the room at least five times per hour. Otherwise, you're not going to be breathing clean air." (Girish Bapat, Director, West and South Asia for the Swedish company Blueair).

https://www.epa.gov/indoor-air-quality-iaq/residential-air-cleaners-second-edition-summary-available-information

“The CADR is a measure of a portable air cleaner’s delivery of contaminant-free air, expressed in cubic feet per minute. For example, an air cleaner that has a CADR of 250 for dust particles can reduce dust particle levels to the same concentration as would be achieved by adding 250 cfm of clean air. The portable air cleaner’s removal rate competes with other removal processes occurring in the space, including deposition of particles on surfaces, sorption of gases, indoor air chemical reactions, and outdoor air exchange. While a portable air cleaner may not achieve its rated CADR under all circumstances, the CADR value does allow comparisons among portable air cleaners.”

Simple model:

My experience with plotting the PM10 & PM2.5 concentrations with respect to time is that they decrease exponentially. So I am following a simple model, as below:

Let r be the PM concentration in mg/m³ in a closed room, and r_out be the PM outside.

The rate of change of PM is:

                                dr/dt = b - ar

Where b is the in-leak rate into the room (sealing is imperfect), in units of mg/m³/min.

The solution is:

                                r = (b/a) – (b/a - r₀) exp(-at) = bt - (bt - r₀) exp( - t/t)

Where r₀ is the initial PM value. The parameter a = 1/t, in the fits to experimental data.

The parameter b is given by:

                                 b = r_out (1/V)(dV/dt)

Where dV/dt is the volume in-leak rate and V is the volume of the room.

The parameter t is given by:

                                t = khV/V_ap

Where V_ap is the volume rate at which the air purifier fan pushes air through it, h is the filter efficiency of the air purifier (close to unity) and k is a constant of proportionality that accounts for the fact that t is much larger than V/V_ap.

Note: I have not specifically considered particle generation within the room, nor have I taken into account the settling of particles, mainly because the settling time is inversely proportional to the square of the particle radius. For PM2.5 particles this goes into hours.

Calculations & readings:

Room Volume = (169)(9) = 1521 ft³ = 43.1 m³. The Philips air purifier has V_ap = 279 m³/hr = 4.65 m³/min. So V/V_ap = 43.1/4.65 = 9.3 mins. However, experimentally t is greater than this value (up to 3X).

Note: a) The AirVeda particle monitor reading for PM10 is only for particles in the size range 2.5 -10 microns and does not include those that are less than 2.5 microns 

b) the readings are stable values, but ideally one should take a large number of readings and then smooth the curve. I have not done this.

The PM2.5 on 5Nov.17 is fitted to:  13.63 + 147.8 exp(-t/27.83)

And the PM10 to: 23.09 + 220.7 exp(-t/30.61)

Unfortunately the concentrations outside changed during the afternoon:

t	PM2.5	PM10
13:45	153	239
14:40	132	202
16:00	115	175

The limiting value should be determined by the last set of PM values.

For PM2.5, the intercept is 13.63 = b (27.83), so b = 0.49 mg/m³/min.

Similarly, for PM10,  b = 23.09/30.61 = 0.75.

Using the equation for b, for PM2.5, dV/dt = (43.1)(0.49)/115 = 0.18  m³/min.

For PM10, dV/dt = (43.1)(0.75)/175 = 0.18 m³/min.

The fact that we get the same estimated dV/dt for both particle sizes is nice.

This number 0.18 m³/min equates to 10.8 m³/hr, while the room volume is 43 m³. That is, an air change of the whole room due to the in-leak would need about 4 hrs. The AHAM (Association of Home Appliances Manufacturers) criterion is that the in-leak should not be more than 1 air change per hour (1 ACH), used to calculate the maximum room size for which an air purifier is rated.

I won’t go into detail, but I will just tabulate the time constants for PM2.5 and PM10 for several days, as well as the other coefficients y₀ (bt) and A₀ (r₀ - bt). There is day-to-day variation, and that is even disregarding the fact that I changed the speed setting (to the highest setting, speed 3) or used a different room or used two air purifiers at once (on 8^th Nov, which almost doubled the speed):

Date	Conditions	PM2.5 (mins)	PM10 (mins)	b2.5, b10 (mg/m3/min.)
31st Oct17	Philips	27 + 117 exp(-t/27.6)	48 + 177 exp(-t/29.3)	0.98; 1.64
3rd Nov17	Philips	24 + 111 exp(-t/23.7)	40 + 179 exp(-t/24.0)	1.01 ; 0.99
5th Nov17	Smart Air	14 + 148 exp(-t/27.8)	23 +221 exp(-t/30.6)	1.01; 0.75
8th Nov17	Smart Air & Philips	15+ 96 exp(-t/14.5)	33+200 exp(-t/17.0)	1.03; 1.94
9th Nov17	Smart Air (living room)	36 + 268 exp(-t/36.5)	77 + 424 exp(-t/36.8)	0.99; 2.09
21st Nov17	Philips	31 + 81 exp(-t/26.3)	57 + 176 exp(-t/30.7)	1.18; 1.86
29th Nov17	Philips speed 3	15 + 136 exp(-t/14.8)	38 + 226 exp(-t/14.2)	1.01; 2.67
30th Nov17	Philips speed 3	22 + 139 exp(-t/12.6)	37 + 233 exp(-t/12.6)	1.74; 2.93
21st Dec17	Philips speed 3	56 + 263 exp(-t/20.7)	104 + 354 exp(-t/21.7)	2.70; 4.79

The living room volume (9^th Nov) is almost the same as the bedroom, but it has a sliding door. Clearly there is significant variation in the parameter b (which is the product of the in-leak rate dV/dt and the outside concentration r_out, divided by the room volume V) as can be seen from the last column values for PM2.5 and PM10 (b_2.5, b₁₀). That is, you may get high ingress of pollutant particles if either the outside air has a high concentration of PM or if the in-leak rate into your room is high. Like if you leave a window or door open. But even if doors and windows are closed, there is space under the door or between windows and window-frames for air to get in. All the readings above are with all doors and windows closed – and with a filler shoved under the doors to my bedroom.

Which is why I am skeptical of the claim of reducing particle concentration from 1,000 mg/m³ to 100 mg/m³ in 20 mins, and even lower with greater time. For example, today’s readings: outside PM2.5 is 447 mg/m³ and PM10 is 677 mg/m³ which is high even for Delhi, but lower than Diwali levels. At the highest speed setting, all doors and windows closed, under-door filler in place, inside my bedroom the stabilized value of PM2.5 is 65 and PM10 is 107. This is just barely acceptable – even by lax Indian standards for PM2.5 (60 mg/m³). WHO recommends 20 mg/m³.  And some recent studies suggest even lower values. So don’t even ask what will happen on Diwali… I will probably need two air purifiers running full blast – and then to wear a N99 mask! Or, I should get hold of room-sealing experts...

Factors influencing In-leak rate:

There are three driving factors that influence in-leak (apart from geometry), in descending order of strength:

a a)      Temperature

b b)    Pressure

c c)    Wind speed

Considering that 300 K corresponds to 1013 mb, 1K (or 1 °C) corresponds to 3.38 mb.

And a wind speed of 10 m/s (36 km/hr) corresponds to (1.2)(100)/2 = 60 Pa or 0.6 mb.

In Delhi, wind speed mostly is around, or below, 10 m/s.

On a typical day, atmospheric pressure may move up or down by about ±5 mb in Delhi.

(i)                   In the course of the day, if the pressure is falling, the inside pressure will always lag behind the outside pressure, so air will flow outwards. Similarly, if the outside pressure increases over a period of time, air will flow into a building. How much flow happens will depend upon the size of openings (doors, windows, cracks, etc).

(ii)                The temperature of rooms inside lags the outside temperature by a few hours. Starting in the morning, as outside temperature increases, air will flow into the building. After sunset, similarly, outside temperature will drop and air will flow outwards.

(iii)               But what happens if both outside temperature and pressure are varying? There are four possibilities:

Outside temperature	Outside pressure	Flow
Increasing	Increasing	Inwards
Increasing	Decreasing	Depends on relative changes
Decreasing	Increasing	Depends on relative changes
Decreasing	Decreasing	Outwards

Estimating the time constant:

I am trying to estimate the time required to purify the room air, so here goes with a fairly crude argument:

Assume V = 43 m³ and air purifier (AP) flow-rate of 4.3 m³/min. That means 10 volumes or ‘packets’ of air.

Suppose we start when the air purifier is switched on: all 10 packets of air are ‘dirty’, so the time required for one 4.3 m³ packet to reach the air purifier and cross it is 1 min. At the other end, suppose all packets have been cleaned up except one. Then the time required for it to reach the air purifier varies between 1 and 10 mins, with an average of about 5.5 mins – but the maximum time is 10 mins. If there were 2 dirty packets of air, the time required would vary between 1 & 5 mins with an average of 3 mins, and max of 5 mins.

Extending the same logic, the maximum time for cleaning all packets = [(10/1) + (10/2) + (10/3) + … + (10/10)] @ 10 [0.5772 + ln(10)] = 10 [ 0.5772 + 2.301] = 28.78 mins, where 0.5772 is Euler’s constant, using the approximation for the harmonic series. Without the approximation it is: 29.24 mins (by just adding up the numbers).

The average time would be: [5.5 + 3 + 2.1 + 1.7 + 1.5 + 1.3 + 1.2 + 1.1 + 1.05 +1] = 19.5 mins.

But 10 mins is what it would take with perfect efficiency, 19.5 mins on an average. The best time constant I got is 12.6 mins on 30^th Nov17. But that is just the 1/e time constant (63% clean-up), whereas the above argument refers to a 100% clean-up… that should be roughly 3 time constants…but it does not take into account the in-leak so the comparison is iffy…

Also the air, after being purified, is expelled from the air purifier with some speed. If that speed is high, the air packet may not come back to the air purifier quickly, reducing the random factor and the probability of double- or triple-purification…So the airflow is neither perfectly 1-D  nor perfectly random.

This argument assumes randomness – but the air flow is probably chaotic, not random, so the time may work out to be between 10 and 19.5 mins (‘chaos’ means that there are some elements of order in the air flows). However, looking at chaos is above my pay grade! Maybe somebody can enlighten me…

The analysis of the in-leak rate is, as indicated above, dependent on data of wind-speed, temperature (inside and outside) and atmospheric pressure (inside and outside). I tried assuming that, after the air purifier is switched off, to fit the increase in particle concentration to a simple exponential:

 Y = y₀ + A₀ ( 1 – exp(-t/t))

But it did not work. The situation with variable v, T_in and T_out, p_in and p_out, is more complicated. I have to look for a better model.

Vegetarianism

Vegetarianism

“A perfect Hindu must be a vegetarian”.

The above statement could easily have been declaimed by any of the new 'gurus'/'leaders' who inhabit India today. (As it happens, I heard a woman say it while I was dreaming. I guess my subconscious wants to blog about vegetarianism!).

At one level, the statement follows directly from the idea of non-violence (ahimsa) that is associated with Hinduism.

However, apart from the current Hindutva strategy of forcing vegetarianism down unwilling throats, one may also ask whether sacrificing animals was a common practice in Vedic times? That history, of course, is bound to be contentious. And multiple, as well as contradictory, strands of thought may well have co-existed in ancient times. The correlation between vegetarianism and Hinduism is probably not one-to-one.

But what does one mean by a perfect vegetarian? Obviously one would have to give up eggs (I’m not there yet!). But a vegan may well insist that one must also eschew (and not chew) all dairy products. This injunction may not sit well with orthodox Hindus: how does one conduct all those rituals without ghee? (As for me, I am unwilling to sacrifice milk chocolate!). And: would one be guilty of smelling chicken soup?

Another argument for vegetarianism is that the animals we eat are fed food crops grown from land in the Amazon, the Congo basin and the Himalayas, and this habitat loss causes a 60% reduction in ‘global biodiversity’ [1].  This argument sounds quite convincing to me – although it may not be to the taste of a committed non-vegetarian. However, a non-vegetarian may well argue that in cold climates you must consume animal protein to survive. WWF probably counters that we don’t eat, we over-eat.

A few years ago, biotech companies have come up with lab-grown meat for hamburgers. When first announced, it was unaffordable – but improved tech has brought the cost down from $325,000 to $12! [2]. However, it is at least a decade away from being commercially viable. Presumably, even a quasi-modern Hindu who does not want to eat beef could eat that hamburger.

This reminds me, naturally, of Arthur C. Clarke’s macabre SF story, first published in 1964, ”The Food of the Gods” [3], in which it turns out that the lab-grown ‘ethical’ meat was actually cloned from human cells – making the customers, who loved it without knowing what it was, cannibals – in some sense of the word. But suppose that the initial cells had not been cloned from human cells, but grown ab initio from chemicals. Would that make it ok?

Leaving such grave difficulties aside, another ethical argument used by vegetarians is the avoidance of suffering, and we associate that with any living organism that has a central nervous system (CNS) – excluding plants and jellyfish. This is not entirely clear. Does a jellyfish not suffer? What about a plant? Does suffering have to be ‘centralized’ and not ‘distributed’? Can one speak of suffering at a cellular level?

Contrary to popular belief, it seems that jellyfish actually does have something like a CNS [4]. What brains jellyfish have are more akin to ‘neural nets’ or a ‘ring’ CNS – but that should suffice to stop jellyfish-eaters (if any)!

The Biblical commandment is: ‘thou shall not kill’. This refers to humans – but it could just as well be extended to plants, jellyfish and even cells. If we go so far as to include cells in a blanket prohibition, then even biotech companies would not get a free pass - and strict vegetarians would starve to death. However, the biotech way would still be ethically better than the alternative.

A Hindu might also ask: do you get worse karma from eating a goat (that eats grass) or a shark (that is at the top of its food chain)? Do you have to pay for the ‘sins’ of the shark, if any?

1   1)      http://www.independent.co.uk/news/science/meat-eating-destroying-planet-report-warning-a7985071.html

2   2)      https://www.fastcompany.com/3044572/the-325000-lab-grown-hamburger-now-costs-less-than-12

     3)      Arthur C. Clarke “Food of the Gods” https://en.wikipedia.org/wiki/The_Food_of_the_Gods_(short_story)

    4)      R.A.Satterlie Journal of Experimental Biology 214 (2011) 1215-1223

Green & Blue Flowers Should Be Less Common in Nature

Green & Blue Flowers Should Be Less Common in Nature

One would expect a relatively low frequency of green and blue flowers because they would unnecessarily confuse insect and bird pollinators. The problem of distinguishing flowers may be more acute for insects than for birds, because the former have lower visual acuity (VA) [1]. The selection pressure should be greater against green flowers because they could get missed in the midst of green leaves, and sometimes green trunks or stalks. Blue flowers could get missed against the background of the blue sky, at some angles and times of the day. However, the selection pressure against blue flowers should be weaker because the sky is in the background and because its color is much more variable due to clouds, rain, fog etc. (Nevertheless, in spring, when a lot of pollination takes place, skies are largely blue.) So the frequency of blue flowers should be higher than that of green flowers. This argument assumes that the visual system of the pollinators could respond to both these colors (and to other alternative colors), and that sight is the dominant mechanism for finding the flowers.

No comprehensive database of flower colors:

According to Joanna Klein in the NYT [2], “Less than 10 percent of 400,000 floral species bear blue flowers. It’s unclear why.” This article was written in the context of Japanese researchers recently creating the first blue chrysanthemums by genetic engineering [3].

“Less than 10 percent of the 280,000 species of flowering plants produce blue flowers,” Prof.David Lee [4] said. He added that true blue (as opposed to structural blue [5]) is rare in nature, and explained the chemistry: ‘The key ingredients for making blue flowers are the red anthocyanin pigments. “Plants tweak, or modify, the red anthocyanin pigments to make blue flowers,” Lee said. “They do this through a variety of modifications involving pH shifts and mixing of pigments, molecules and ions.”’ A different argument for the rareness of blue flowers is that most plants rely on chlorophyll which strongly absorbs blue, which is useful since it is a high energy (and high photon flux) part of the solar spectrum [6]. Plants that have preferred chlorophyll find it difficult to come up with the chemistry for blue.

There is agreement on the proportion of blue flowers: less than 10% of the total - whatever that is. Probably the total number of ‘floral species’ is 400,000, because it includes 120,000 flowering trees as well.  I Googled the internet but could not find a corresponding number for green flowers. Nor could I find any number in David Lee’s book [7]. But then I just glanced though the chapter on flowers… Absent this critical bit of information, the argument about green flowers cannot really go much further.

Another expert [8] has this to say:

“No one has surveyed all of the world's flowers. All of the world's plant species haven't even been discovered and named yet. Further, flower color statistics have not compiled anywhere for the majority of the earth's plant species. We know of no central repository of flower color information… Green may actually be the most common flower color… If we had to rank the four colors you asked about in order from most common to least, we would guess -- and we emphasize guess, here -- that they would line up like this: 1. white, 2. yellow, 3. blue, 4. Red. ” 

Note that this expert states that green may be the most common color – but, oddly, when rank ordering the flowers does not put green in the first four.

Pollinators and their preferences:

One might add that bees respond to ultraviolet light, so what is a white flower to us, has clear ‘nectar guides (like landing strips)’ on it for the bee [1]. That is not really relevant, but what is apposite is a study mentioned by Joanna Klein [2]: Australian researchers, Adrian Dyer et al [9], found that bees native to Australia (T.Carbonaria) prefer blue flowers.

 Dyer et al [9] state that: “insect-pollinated flowering plants often generate spectral signals that suit the color capabilities of important bees, or other potential pollinators in an environment”. Many studies show that: “bumblebees show a preference for blue stimuli across a wide geographic range”. 

 “T. carbonaria bees showed a significant preference for stimuli from the blue and blue–green regions of the colour hexagon, consistent with findings that honeybees and bumblebees

tend to prefer flowers with such spectral characteristics. While it remains to be definitively shown whether bee innate colour preferences may drive flower evolution, there is evidence that such preferences are linked to flowers of these hues having higher nectar rewards.” (emphasis added by me). That is, although it would be expected that ‘innate color preference’ of bees drives flower color evolution, it has not yet been definitively proven.

Those ‘higher nectar rewards’ for blue flowers: it looks like sheer bribery to me! I would speculate that flowering plants added a nectar bonus to these blue flowers to compensate for the slight disadvantage of a blue sky. However, I would have to mention that Dyer et al nowhere mention a ‘blue sky’ problem, and they have replicated the natural environment in their lab for testing.

Dyer also gave an interview with ABC [10] in which he pointed out that there are two different types of birds in terms of their visual systems: some birds see violet, green, blue & red and these are called ‘violet-sensitive’, whereas there are others that also see ultraviolet (and these are called ‘UV-sensitive). Bees can see UV, blue and green – but not red. Pollinating flies can see four different types of color i.e. they are tetrachromats (but they prefer yellow).

Dark adaptation and visual acuity of pollinators:

Moths, which operate at night, do not see color at all, and the plants they visit are white. Oops! The hawkmoth, it seems, does see in color at night… [11a]. It seems that the compensation for the lack of photons at night occurs in one of three ways [11b]:

a     a)      Decreased focal length (distance between the aperture and the light-sensitive cells), f

b     b)      Increased diameter of the ommatidium or eye, D

c     c)      Special light-trapping structures that cause a ‘double pass’ of the photons, increasing the quantum efficiency h(probability of detecting the photons).

So, the number of photons detected is increased since it is proportional to h(D/f)².

In  Kelber’s words: “A closer inspection of the geckos' eyes revealed that, with no rods to fall back on, the cones in their eyes had evolved to become more rod-like, longer and more sensitive. Like the hawkmoths, they also had large lenses and a shorter focal distance to cut down how far the light had to travel through the eye.” Kelber  also points out that many nocturnal eyes have [11c]: “a mirror-like structure at the base of the eye, which reflects the light across the photoreceptors for a second time.” With this adaptation, Kelber et al showed that nocturnal hawkmoths have color vision even under dim starlight (10^-4 cd/m²). At such light levels, humans are blind - not just color-blind!

Some insects are crepuscular i.e. they are active at dawn and dusk. They have a special adaptation, called a neural superposition compound eye (as opposed to a simple apposition compound eye), in which the photons from 7 adjacent ommatidia are summed up in the neural layer, so that the number of photons detected is increased by 7X – without sacrificing spatial resolution. This type of neural superposition gives them an advantage during twilight over predators and competitors who have apposition eyes and allows them to detect small objects [12].

And we have not even taken up bats, butterflies, midges, mosquitoes and wasps yet! Ok, forget the bats…? Well, you cannot: they are dichromats. So if coevolution of flowers and their pollinators is occurring, then it must be specific to the species involved. In addition, Dyer adds that bees cannot see very well (their bad VA was mentioned earlier [1]), and they can only see flowers when they get quite close: “maybe 50-60 cms”, and probably use the scent of flowers to find them from afar. This argument might imply that flower color should not really be important for small pollinators! Dyer also states that primeval plants were probably dull, pale yellow or green, until about 100 million years ago, when they evolved the more vibrant hues that we see today.

Considering that bees do not see very well, the smallest object that they can resolve with an ommatidium is about 6.7 mm (at a distance of 50 cms, corresponding to an angular resolution Dq of 13.3 mrad for a 30 micron diameter ommatidium d and 400 nm light wavelength l) since a) diffraction constrains the angular resolution, and b) the diameter of the ommatidium d is related to the radius of curvature r of the bee’s head (about 3 mm) [1]:

                                                       Dq = l/d = d/r

Thus, the optimum diameter of the ommatidium is given by: d_opt = (lr)^1/2, which is fixed by the wavelength of light and the radius of the bee’s head [1].

Feynman [1] also states that bees have 30X worse spatial resolution than humans: this would suggest that they can resolve down to about 1.5- 3.0 mm (assuming the human eye can resolve between 50 - 100 microns). In that case, the scent of the flower would have to guide them to its vicinity, and then sight would do the rest in the final (terminal), homing-in on the target.

The blog [12a] gives the ommatidium diameter of A.Bilineata as about 4 microns. As pointed out by Feynman [1] and by Ref.14, diffraction limits the diameter of the ommatidium to above this limit. A ommatidium of 2 micron diameter would be of no use because diffraction would blur any image beyond usefulness. For smaller insects, such as flies, the ommatidium diameter may be somewhat smaller (but it has to be > 4 microns anyway to avoid diffraction), so a significant fraction of the surface of the head is covered by ommatidia!

This blog [12a], which gives a very detailed description of the insect eye, both day- and night-adapted, mentions that the visual acuity of the compound eye of the insect is about 100X worse than that of humans( compared to 30X quoted in [1]).

Rigosi et al [13] studied the honeybee A.Mellifera. Previous studies had shown that: “bees could not discriminate a dark object smaller than 3 deg”. Rigosi et al measure better angular resolution when the bees are light-adapted, a value that is 30% lower than the above value. They also found that: “in both frontal and lateral regions, responses saturate for large objects that fill the receptive field but

decrease linearly as the object area falls below 1 deg² . As features fall below the size of the receptive field they are increasingly blurred to a lower effective contrast until the response is indistinguishable from noise”. This amounts to an area of about 9 x 9 mm² at a distance of 50 cms.

A standard definition of a point target is when it occupies one pixel. The distance at which it occupies on pixel depends upon its width (let us assume width w = 5 cms, arbitrarily) and the minimum angular resolution (MAR) (or Dq), taken here as 1 deg. Then the distance d above which it is seen as a point target by the pollinator is:

                         d = w/(MAR) = 5/0.017 = 287 cms. 

The minimum requirement for the target to be resolved is that it is covered by 6 x 6 = 36 pixels, which will occur at a distance of about d/6 = 48 cms, which roughly agrees with the number mentioned by Dyer [9,10]. This criterion of needing 3 line-pairs across the target (or 6 pixels) was first proposed by J.Johnson of the Army Night Vision Lab [14]. However, the size of the target (the flower) will vary depending upon the angle of approach. The terminal homing-in phase mentioned above refers to distances less than d, where the target is rsolved.

An interesting question is whether at distances at which the flower is a point target, could the pollinator use its color (apart from its scent) to detect it? Something like the fact that we can see the color of a distant star.

Color response, and the ambiguity in determining the color of flowers:

Since the spatial resolution is in the range of about 2 - 7 mm, the color of the flower may actually become more important to the bee than its shape (for identification, not landing). This problem would become even more acute for butterflies which are smaller than bees, or for smaller size bees, that also have smaller ommatidia. Interestingly, one butterfly, G.Sarpedon, has been found that responds to 15 colors (i.e. it has more than fifteen types of photoreceptors) – whereas we can only see three (trichromats) [15]. Note that this is in the same league as the other record-holder, the mantis shrimp (H.Trispinosa), which has 12-16 distinct photoreceptors [16].

According to Virginia Morell [17]: “Butterflies need only four receptor classes for color vision, including spectra in the UV region. So why did this species (G.Sarpedon) evolve 11 more? The scientists suspect that some of the receptors must be tuned to perceive specific things of great ecological importance to these iridescent butterflies—such as sex. For instance, with eyes alert to the slightest variation in the blue-green spectrum, male bluebottles can spot and chase their rivals, even when they’re flying against a blue sky.” (emphasis added). Well, the much-awaited blue sky finally showed up somewhere (even if it leaves much to be desired)!

Feynman [1] mentions that bees cannot see red (like bulls!), so they do not visit ‘true red’ flowers (that do not have any other tinge of color to which bees might be sensitive) – but these flowers are visited by hummingbirds which do see red. That sounds suspiciously like coevolution to me – or is it just rank avian opportunism? – or is it both?

Arnold et al [18] hypothesized that there may be seasonal variation in flower color to better attract pollinator insects in a ‘market’. Bees generally prefer blue, green and UV, while butterflies prefer pink/purple and hoverflies prefer yellow and white. “The pollination syndrome hypothesis might lead us to expect that if particular pollinator guilds constitute a larger proportion of the total pollinators at certain times of year, then those plant species blooming at that time should be more likely to possess the flower colors associated with those pollinators.” However, pollinators do not exhibit strong brand loyalty and are quite capable of learning new behavior, “able to associ­ate almost any color with reward. “  In the presence of such perfidy, coevolution sounds impossible! 

Arnold et al  [18] conclude:

“…(although) data collected appears to suggest that in some habi­tats, certain colors of flowers bloom at particular times of year…we found no statistically signifi­cant evidence that the colors of flowers change throughout the year.” Apparently, a lot of hypotheses just do not pan out! They add: “During much of the year, pollinators in woodland must forage under lower light levels, and also under light that is spectrally different from normal daylight (with a spectral peak around 550 nm owing to filtering through green leaves)”.

After all this argument about the supposed disadvantages suffered by blue and green flowers, it is useful to add a caveat. Even a disadvantageous situation may be exploited by some plants and animals because it serves as an ecological niche in which there is lower competition. In fact, one might well say that Nature abhors a vacant ecological niche! This would imply that, even if the frequency of blue flowers is low, it will never be negligibly small. Another problem is with the definition of ‘blue’. The researchers who tweaked the color of the chrysanthemum were concerned that it should be ‘true blue’, without any tinges of other colors. So even the estimates of 10% for the fraction of blue flowers could be dependent on its precise definition. The expert [8] adds: “Many flowers are multi-colored. Some species feature flowers that change in color as they age. Other plants bear flowers of different colors on the same plant.”  No wonder there is no exhaustive database! And the color seen depends not only on the eye of the beholder (the visual system), but the type of illumination (solar, lunar, direct, reflected, polarized…).

 Another authority [19] also stated:

“We simply have no idea what the most common flower color is in the world but it's probably green. Big or small, we like bright colors and we like weird colors. All the rest just get glazed over. In reality, many plant species, especially trees, produce small, nondescript green flowers.”

He adds:

“It is actually an easier question to ask ‘what is the rarest flower color?’ To that, most botanists will probably say black.”

I started with an idea that green and blue flowers should be less common, but the literature does not support any such easy generalization. In fact some experts claim that green may be the most common flower color. The literature says that blue flowers are less than 10%. Does that compare with 1/7 or 1/3 – if we assume all colors have equal weight? But color is clearly important to pollinators, since their visual systems have even adapted to provide color sensitivity at night.

To summarize:

a)      No complete database exists of flower colors – and green is probably under-counted, since humans may just not notice them

b  a)      Flower color itself is not clearly defined, since it depends on many environmental variables such as lighting in the day, twilight or night, angle, texture etc.

c  b)      Different pollinators have very different visual systems, so coevolution is specific to the flower-pollinator pair

d  c)     Small pollinators have low visual acuity (e.g. distance to flower should be less than about 50 cms for clear vision for bees), and they home in on flowers from a distance using scent, but use color for the final phase, to actually make contact. Even nocturnal moths do this! This highlights the importance of color for the pollinator – even at night!

e  d)      There is no conclusive proof for coevolution of bees (that are the most well-studied pollinators) and flowers - although it is generally accepted that coevolution happens.

f   e)       Blue flowers are less than 10% of the total, maybe because chlorophyll absorbs blue, and because anthocyanins are mostly red (but change to blue by changing pH), The ‘blue sky’ argument that I have suggested is not mentioned by anyone.

References:

1     1)      Feynman lectures Vol.1 Ch.36 (Addison & Wesley, 1963)

2     2)      https://www.nytimes.com/2017/07/28/science/scientists-give-a-chrysanthemum-the-blues.html

3     3)      N.Noda et al., Sci. Adv. 2017;3: e1602785 26 July 2017

4      4)      https://www.mnn.com/your-home/organic-farming-gardening/stories/the-science-of-blue-flowers

5)    www.npr.org/sections/.../how-animals-hacked-the-rainbow-and-got-stumped-on-blue

6)    https://biology.stackexchange.com/questions/56476/why-are-so-few-foods-blue

7     7)      David Lee “Nature’s palette: the science of plant color” (Univ. of Chicago Press, 2007)

8     8)      http://www.wildflower.org/expert/show.php?id=1456

9      9)      A.G.Dyer et al J.Comp.Physiol.A (2016) DOI 10.1007/s00359-016-1101-4

1    10)   http://www.abc.net.au/news/science/2016-11-16/birds-and-bees-prefer-have-flower-colours-preferences/7959382

1    11)   a)  http://www.bbc.com/earth/story/20141128-these-animals-see-colour-at-night

b) A.Kelber Nature 419 (2002) 922

c) A.Kelber & L.S.V.Roth J.Expt.Biol.209 (2006) 781-88

12)    a) http://cronodon.com/BioTech/Insect_Vision.html  

b) Michael Land & Dan-Eric Nilsson “Animal eyes” (Oxford University Press, 2012)

13)   E.Rigosi et al Sci.Rep. (2016) DOI: 10.1038/srep45972

14)   https://www.hgh-infrared.com/FAQ/Perimeter-Security/Definition-of-DRI-Detection-Recognition-Identification-ranges

1        15)   P.-J. Chen et al Frontiers in Ecology & Evolution 4 (2016) doi: 10.3389/fevo.2016.00018

1        16)   http://www.sci-news.com/biology/science-color-vision-mantis-shrimp-01719.html

1        17)   http://www.sciencemag.org/news/2016/03/butterfly-has-extreme-color-vision

1       18)   S.E.J.Arnold et al Israel J.Plant Sciences 57 (2009) 211

1       19)   http://www.indefenseofplants.com/blog/2016/4/26/what-is-the-most-common-flower-color