Coaxing Water From Desert Air

From the windtraps and stillsuits of Dune’s Arrakis, to the moisture vaporators of Tatooine, science fiction has invented fantastic ways to collect the water necessary for life on desert worlds. On Earth we generally have an easier go of it, but water supply in arid climates is still an important issue. Addressing this obstacle, a team of researchers from MIT and the University of California at Berkeley have developed a method to tease moisture out of thin air.

A year after the team first published their idea, they have successfully field-tested their method on an Arizona State University rooftop in Tempe, proving the concept and the potential for scaling up the technology. The device takes advantage of metal-organic framework(MOF) materials with high surface area that are able to trap moisture in air with as little as 10% humidity — even at sub-zero dewpoints. Dispensing with the need for power-hungry refrigeration techniques to condense moisture, this technique instead relies on the heat of the sun — although low-grade heat sources are also a possibility.

Before you ask, the team has also analyzed the water trapped by the MOF and found it to be free of impurities. That said, there is always room for improvement and experimentation with different materials — the researchers tout even higher yields than the quarter-litre of water per day per kilogram of MOF at present. Still, these results are exciting enough to keep us watching to see the next phase.

[Thanks for the tip, Qes!]

75 thoughts on “Coaxing Water From Desert Air

  1. Very interesting results and, I am sure they will improve as it develops. Now, if they can figure a way to get beer out of the air, then I believe they would really have something. All kidding aside, this could save a lot of lives. (The water, not the beer)

      1. It was funny, but beer actually did save lives.

        Since part of beer making is boiling water, stuff like typhoid would be killed as part of the beer making process. This is back when clean water supplies didn’t exist and what water was available was normally contaiminated.

          1. >”this was usually served as “small beer””

            When you make beer out of grains, the sugars are washed out of the grist in the mash tun, basically you have this thick wet mass like boiled müesli, and hot water is drizzled on top while the sugary water drips off down the bottom. This is called sparging.

            The first runs collected are high in sugars and gets fermented into strong beer, and the diluted leftovers are separated and made into small beer for the servants. The leftovers have a wooden tea-like taste because the boiling water added to the grist extracts tannins. Both can be mixed together to make “middle” beer, but this was not commonly done as it didn’t taste very good, nor did it get you drunk, so it was basically a waste of grain.

            But, it can’t be properly called beer for two reasons. First, the medieval -ales- didn’t use hops, secondly, the wort was not boiled after the sparging. Boiling increases the sugar concentration to make the beer stronger, so the middle beer can be upgraded to strong beer and so the process efficiency increases, and it eliminates bacteria, which with the addition of hops increased the shelf-life of the beer considerably.

            The reason why the ales were drunk fresh was because in the mashing process, the temperature doesn’t necessarily go high enough to fully pasteurize the wort. The temperature is brought to certain “stops” at around 30, 40, 65 C by mixing in a known quantity of boiling water, kept at that temperature to allow the enzymes to work, and then the resulting liquid is drained off as explained above. That killed the worst pathogens, but once the yeast stops working all the other bacteria that still survive the process come out of hibernation and the ale goes off.

        1. It also provided self-generated abiotic inert atmosphere with overpressure in the vessels where it was kept, so not only it was pathogen-free, but it tended to stay that way for prolonged time.

          1. When making ale the old fashioned way, it’s not exactly a sterile process. It’s mainly based on the fact that the yeast is overgrowing all the nasty stuff and eating up all the free sugars before the pathogens have a chance to spoil it. The alcohol helps, but it’s not until the concentration grows to about 22% that all other bacterial activity stops, and you can’t reach that high by fermentation.

            Medieval ales were drunk fresh, because once the yeast stops then other bacteria set in and start to consume the alcohol, and the dead yeast, and the drink goes off. It didn’t keep for very long.

    1. Hmm, good question Ostracus and especially if very many of these types
      installed across large areas and where that water not evaporated so
      the atmosphere depleted in those regions…

      One would think not much effect though as CO2 is lifting H2O via Psychrometry
      and H2O’s retention time short anyway ~ 7 days so given ~70% Earth’s surface
      covered with water one would expect more evaporation as equilibrium patterns shift.

      So there might be a tiny drop in H2O’s partial pressure if millions of large scale devices
      Water Extraction Tech (WET) devices were implemented were implemented on a grand
      scale but, the partial pressure of H2O would soon stablise though if these devices
      stayed in use one would expect a new equilibrium as the average would dip and
      even then I expect only slightly.

      Depending on where they are used if on large scale however, might result in local
      impacts re changes in rainfall Eg as water is extracted across large areas, resulting in
      vapour pressure shifts in the local atmospheric column perhaps offering H2O rich air at
      higher altitudes where rain forms moving to lower altitudes thus depleting rain formation.

      This suggests areas were rain storms are common and chaotic have these
      devices implemented upstream (mostly) from those regular downpour areas so
      as to moderate effects Eg, Floods, land-slides etc
      Even then the weather/climate system has immense chaotic potential so to
      avoid inflexions perhaps distribute these or something similar sparsely…

      FWIW: A quick search found this, see the truck’s cabin, not relevant of course ;-)

        1. No Mark, because the key funny odd ball bit is the trucks
          cabin being almost totally immersed within a few secs of the
          start of the video. ie The video starts a few seconds before
          although your browser might miss that, the start time ref is 51:32

          Sorry it was your choice to start at the beginning and watch
          it up to 1 hr 23 mins and spend 7 mins writing your retort ;-)

          All good fun, some amazing other examples of odd ball behaviours :D

          1. Hmm Kiwi Joker, when you claim:-
            “Engine sound and horn is fake, though. That truck isn’t moving.”

            I watched this a few times and doubt your claim has validity but, yah never know…
            if at least for these reasons:-
            1. Its a slow moving wide river which offers a bow wave on the cabin
            more consistent with the truck moving
            2. A slow moving river wouldnt necessarily offer the pattern of water
            movement around the truck body ie wheels behind articulation
            3. The engine sound change as the boat passes is consistent so
            there is either a lot of trouble to fake it or engine running.
            4. There is a knocking sound of the engine consistent with some
            load though there isnt much exhaust. ie as result of water ingress
            5. There is no evidence on the truck it flipped as it would very likely
            as water has great force even if moving slowly able to turn it around
            and tumble it as few rivers if any have laminar flow. ie Its in line (still)
            with river flow which would be a dynamic re flows at that depth.

            Hey, it is possible its not moving, we can’t really know for sure unless we
            have more info, do you have a reference other than the comments
            in the you tube video ?

    2. I imagine that it could be as harmful as massive fields of dark solar panels that change the local climate by causing fixed position thermals and therefore can disrupt natural wind patterns and create heat islands. Sometimes those changes may be good, but unless you can prove that you can model the process accurately and that the model indicates a benefit you are risking doing harm to the current balances in surrounding ecosystems which have adapted over huge timescales to the local natural conditions and their cyclic patterns of change.

        1. Since the vast bulk of available wind generators are below about 100m or so
          high then very unlikely as the bulk of atmospheric mass flow re wind is in the
          1000’s of meters above the local ground level, consequently effect negligible…

          1. But the hurricane is powered by the warm moist air from the surface of the water. So changing the conditions at that boundary could actually have a noticeable effect.

      1. Ordinary roofs create such thermals as well. And yes, that’s well known to glider pilots, which can take advantage of that: thermals have some preference to release there instead of more random locations.

        1. Hmm thanks Traumflug,
          Though wouldn’t solar panels not produce as much re thermals as they
          do remove 15 to 20% (maybe) of that incident energy for use elsewhere
          and some of that energy as “stored” in you like as potential energy in
          any final product which as Solarex has offered arose from solar panels
          initially producing more solar panels…

          If I perceive the progress of solar panels re efficiency and the comparative
          economics overall then cascade commercial issues re deferred implementation
          energy costing intrinsic to devices as subsequent many solar panels – much like
          a taylors/mclaurens numerical series could well exponentially displace all other
          energy sources more rapidly in conjunction with wind and including nuclear sources
          unless energy density/security exceeded net present costs by a high margin.

          So the move to integrate solar panel type power sources into all materials re large
          area building surfaces seems to be approaching an inflection of a most interesting
          type if you were already positioned in this area. If not already, then if you do have
          capital concerned re its max utility – then what the heck should you do ;-)


          1. So…
            We place large windmills horizontally over solar farms to harvest energy from the rising thermals!
            And since a lot of solar farms are located in desert locations, we placed these water collectors under the solar panels for even more benefit!

          2. Ah Ren,
            Np, in respect of your comment/question:-
            you lost me at
            ” taylors/mclaurens numerical series”.

            Those names refer to people who long ago offered the paradigm of infinite
            series of terms in an equation for a result that converges or diverges. The
            math(s) has an almost universal value across many branches of math(s).
            Useful to explore digits of Pi and in all sorts of practical pursuits too such
            as in engineering, commerce, biochemistry, genetics ie in terms of equilibria
            calculations for various reaction constants. Its also oddly applicable to
            share market trading if you have spare capital and not shy exposing risk
            and being attentive to perturbations ah lah the trumps of the world or
            accommodating a probability value term handling random elements

            I included that phrase in case any applied math devotees observing might be
            provoked to apply it to the paradigm where solar panels are produced via
            the power generated primarily from solar panels earlier on in the process
            as part of potential for a never ending cascade of utility…

            So to re-word that para & expand mostly for Luke & despite his chagrin :-)

            1. If I perceive the progress of solar panels regarding efficiency and the
            comparative economics overall could offer a cascade in power production.
            2. Then the flow on effect in terms of commercial issues where some of
            the power used is deferred or rather included in the product and might
            be crafted into an infinite series such as Taylors and analysed/simulated.
            3. Then development of such an equation might allow increased power
            production from solar (and wind) to exponentially displace other sources.
            4. Also more rapidly against nuclear where the capital cost is so high
            since the capital regarding (re) solar is so very small and more efficient
            than ever.
            5. By proper (and full wider) application of “Net Present” costing methods
            the reality of those economics can be brought to bear to an earlier decision
            process whether to consider fossil or nuclear energy production methods.

            Most of the forums I am on are via subscription behind paywalls thus
            almost all are peers who also have lengthy uni exposure so I don’t have
            to be effusive by way of habit as they know I aim to communicate efficiently.
            Frankly I’m not comfortable dumbing things down automatically just in case
            some can’t easily connect the dots. ie If they can’t or don’t care I don’t
            hear from them that’s fine as they might have looked up the names
            and had an “ah ha” moment – that worked well for me decades ago,
            so far be it for me to remove that potential for neural joy from any readers ;-)

            I’m not in favour of limiting that also as its good often to provoke imagination
            as to how these things are or could be connected even if no “ah ha” expectation.

            In any case I sometimes enjoy expanding on issues when polite questioned,
            so thanks Ren, np – as it also reminds me to reflect on linguistic issues,
            that is, if there is anything I can include towards inference code in some
            distant AI type approach to manage all the forums/feedback tracking potentials
            for the simple sake of efficiency and unlikely an AI wrote this ;-)

            Besides why is Luke reviewing videos on mental health issues (& how many?)
            and wading through at least seven plus minutes just to try to locate something
            to support complaint when his time would be far better spent looking up the names
            Taylor/Maclauren in context with engineering learning something new ?

            Simple google search on “taylors/mclaurens” despite mis-spelling gets to
            this link at the top of the first page – LoL !

            No human or AI were harmed, exploited or charged $ in the above crafting,
            I promise they were only consulted at their discretion and pleasure.

          3. Don’t worry, nobody else got it either.

            Mike’s just on his homemade “nootropic” drug regiment again, which most likely consists of weed and amphetamines.

          4. No Luke,
            You can’t intelligently claim “nobody else got it either” as there’s no evidence
            of any survey, besides what makes you imagine its that hard, please see to
            improving your nutrition significantly as it does improve attention span
            as well as increase ability to understand more interesting paradigms.

            Do you consider unconnected prejudice on a public forum somehow smart ?

            No Luke to your claim:-
            “on his homemade “nootropic” drug regiment again, which most likely consists of weed and amphetamines.”

            I design a formulation for friends and family under professional circumstances
            in a laboratory and whether attached to a residence is irrelevant. Logistics
            of convenience, security and efficiency, surely you can understand that ?

            Since I graduated in Food Science with a distinction in food chemistry in 2010 after
            two previous degrees I can advise there is a sound basis for development of GRAS
            classed mineral supplementation. It does take a few months to start to ‘kick in’ and not
            for the faint hearted regarding (re) discipline who arbitrarily bark critique from the sidelines.

            Perhaps you haven’t noticed Luke your anonymity & uninformed critique reflects on you :/

            How about dealing with the topic, isnt that more productive ?
            Do you really need little old me to spell out the details. For some reason the reply link to
            your earlier outburst is missing – why is that ?

            Have you checked out my hyper text link on my name, can you understand that perhaps ?

    3. Please define “larger”
      Because it’s all a matter of scale, to have any meaningfull impact you need to have a huge scaling of device.
      Even our huge carbon burning frenzy since more than one century and many many billions of energy burning devices have a not so evident effect on global scale, so a device like that…

      1. Hmm mac0123345
        When you arbitrarily claim (or are susceptible to odd ball emotive propaganda – source?):-
        “a not so evident effect on global scale,” in respect of CO2 delta then you are incorrect !

        The Evidence is directly related to the so far not yet refuted Physics of Radiative Transfer
        of which an aspect re the Beer Lambert law/principle has been tested to the extremes.
        I’d suggest you also look up this link and the references and tell us where they are wrong
        and which experiments prove that and the base for *any* Experimental Method which even
        comes close,

        ie CO2 Since the start of the Industrial Revolution is adding some 1.7 Watts per cubic meter
        energy flux as averaged on a global scale and distributed by atmosphere and ocean currents
        then why would this *not* be a global effect…

        Obviously the distribution is a varying mix of laminar (like) and turbulent distribution re
        currents in air and oceans and subject to inflections ie Chaotic. Add to that the immense
        facile confusion between weather and climate by the not so well educated who are malleable
        to emotive propaganda raising straw man like paradigms.

        IOW: Only measuring air temperatures misses a factor of at least a 1000 re overall heat flow !

        Also worth looking up the aspect of H2O offers more than one level of amplification from CO2

        The tertiary amplification effect is then summed (CO2+H2O) effect upon increased CH4
        and N2O emissions as well as even further CO2 emissions from perishable organic materials
        and the third world attempting to reverse crop yields re more fertiliser.

        Also look up actuarial tables re CO2’s effect on increasing cyanogens as in Casava where
        the major insurance co’s manage risk through hedge funds. Apply this to economic indicators
        re those hedge funds affecting crop yields re clover as food for cattle as well as the secondary
        effects re low cyanogen levels affecting meats before it paralyses the cattle etc..
        My understanding is only some food crops are not significantly affected by changes in CO2
        levels and those have doubtful protein value. Though cattle will eat lettuce and tomatoes as
        my cats do, since I exercised some operant conditioning getting their attention ;-)

        As we can infer from your claim then its clear you’re not acquainted with key base high school
        Physics then please review the issue of comparative overall heat capacity of
        oceans vs atmosphere of some 3000+ to 1 (Which some suggest can be 4000:1 when including
        other water and ice globally)

        Is there any new Experimental Method or rational hypothesis which might come close to proving
        the base Physics which has never been refuted is any way, shape of form even a little wrong ?

        I’m seriously interested in what contrary evidentiary based positions there are, whether a planned
        Physics experiment, maths re integration of gas/vapour absorbance/emission especially re any
        applicable differential equations or derived integrations re atmosphere layers and columns etc,

        So please tell us then maybe we will both discover something new ;-)

        1. Few types, apologies busy here:-

          1. Should be square meter per earths surface equivalent at ground level,
          sorry Not cubic meters…

          There is a cumulative aspect to this 1.7 W per square meter which I guess
          is also on a log scale as it also increases overall IR emissions too but, still
          less than the radiative forcing as equivalent to sun’s output having gone
          up by that amount.

          The problem we now have is the last 20yrs or so Sol’s output has been a bit
          lower than normal – its bound to cycle up – then we are in for a double whammy :/

        2. uhhh, you’re going balistic for nothing… It took decade for general consensus on global warning, and you’ve got the first industrial country gouvernment still in full denial mode.
          So back to the OP, no these little devices will have mostly null effect on a global scale, even per billions.

          1. Beg Pardon mac012345 ? you irrefutably claimed:-
            “Even our huge carbon burning frenzy since more than one century and many many billions of energy burning devices have a not so evident effect on global scale, so a device like that…”

            So now appears you shift ground to after the event amend your
            intent to suggest you were “only” referring to the added issue
            “so a device like that” as tail end, pray tell how that works,
            well unless you hope most won’t notice the faux par ?

            My response is directly in response to your definitive claim:-
            “Even our huge carbon burning frenzy since more than one century and many many billions of energy burning devices have a not so evident effect on global scale..”

            Which I have directly addressed head on to trounce inappropriate &
            uneducated base for idle claim – no politics here mate – basic Physics,
            your attempt to distract doesn’t help your case one bit, Integrity is to
            apologise for (accidentally) misleading in case you only meant
            the device noted not the “..huge carbon burning frenzy..” – capisce ?

            Can we interpret that, other than a facile back pedaling attempt you
            fully and completely accept the Physics which has never been refuted ?

            I will take non-response on that specific issue over a 7 day span as tacit
            acceptance reverting the implication in your initial claim, np :-)

            PS: Yeah an edit option would be good though you did have the option
            to reply to your own comment to offer correction – upon subsequent review…

      2. So the increase in the number of hurricanes/cyclones is not in any way related to the carbon burning frenzy ?

        There are two global heat sinks, one on the north and one on the south pole (gigantic blocks of ice) which through their latent heat of fusion can absorb a lot of thermal energy.

        These two large dampeners stabilize global weather, so fewer massive hurricanes (Northern Hemisphere) and massive tropical cyclones (Southern Hemisphere).

        In many ways it is like the moderator rods in a nuclear reactor slowly melting and dripping away. Everything looks fine until it is not, and there is no easy way to fix things once they have gone too far.

          1. Eh RunnerPack ?
            You are quoting an organisation that states because nobody
            got fried by the Fukushima Explosions its not a disaster if
            at least for the massive radiation release with unknown cancer
            mutation consequences, displacing residents, setting the industry
            back and destroying trust in technical skill and leaving a huge cleanup
            for decades ?

            Really ?

            Credibility, reputation, propaganda issues need to be addressed lest
            they reflect on proponent’s pattern of copying links without consideration :/

      1. That would increase atmospheric temperature but the atmosphere is just one way the earth absorbs heat.
        It takes heat to convert to a gas and vapor gives that heat off as it condenses, the aptly named ‘heat of vaporization’.

        1. For Leithoa which appears as reply to RoboMonkey,
          1. Why would removing water vapour increase atmospheric temperature ?
          Because removing H2O reduces absorbance of infra red, thus removes
          the mechanism of holding up radiative transfer so More radiation bescapes to space
          2. In respect of your second sentence yes it does take heat to phase change liquid
          to gas but according to the issue of Psychrometry in relation to CO2 adding radiative
          forcing then, although some H2o will precipitate the equilibrium is still raised so the
          average H2O in atmosphere rises. This means More heat is retained. This extra
          radiative forcing of H2O adds to that of CO2.
          ie The H2O vaporised doesnt reduce CO2 radiative forcing and the H2O doesnt
          escape to space, some falls back down and is re-evaporated…

          And this is the key issue, I left the best for end, so far never refuted :D

          1. >>Why would removing water vapour increase atmospheric temperature ?
            Because that’s what weather does, it moves heat around the atmosphere using water vapor. Water vapor is far from the only IR absorbing gas in our atmosphere.

            >>The H2O vaporized doesn’t reduce CO2 radiative forcing and the H2O doesn’t
            escape to space, some falls back down and is re-evaporated…

            Why would it? Water vapor and CO2 have no bearing on each other when it comes to absorbing IR and for water vapor to have any chance of escaping into space it would need to have insane amounts of energy. Generally it’s density keeps it at the lower levels of the atmosphere.

            >>And this is the key issue, I left the best for end, so far never refuted :D

            I’m not sure why you present this as some ‘gotcha’, argument. Literally no one is saying the sun or insolation don’t play a role in earth’s climate.
            You gotta work on your communication skills & cut back on the stream of consciousness writing if you don’t want to sound like a crack pot who throws jargon around randomly.

      2. But you see [RoboMonkey],
        once the water is collected, it is drunk or used to water plants, (well even after it is drunk, it can be used to water plants), so through respiration, perspiration, inspiration, and aspiration, equilibrium is maintained.

      3. Yes indeed RoboMonkey
        It would reduce radiative forcing due to H2O but, unless one
        also reduces CO2 it will just get back up again via evaporation.
        The self positive feedback effect from H2O doesnt seem to last for
        any more than a few days. The retention time of H2O i about 7 days.

        ie If you locally increase humidity the as that air mass moves around
        it encounters lower temps horizontally as well as vertically and then
        precipitates out as rain due that window the reduced radiative forcing
        allows more infra red to escape space.

        The problem is and in relation (re) Psychrometry H2O goes back
        up again when that dray mass encounters higher temps in its cycle
        whilst passing over water (as 70% Earth’s surface H2O) and so it goes…

        Maybe if we had many water from air devices strategically situated
        maybe coupled with CO2 sequestering in those same devices, even if
        to offer soda (lol) then we might get somewhere. Though I far prefer
        that CO2 go to cement type products for long term removal, I read
        there are other possibilities too and I hope for convergence globally
        to address this.

        There is no doubt re CO2 and climate change as the base is irrefutable
        in respect of radiative transfer contributing to radiative forcing. The problem
        is many are easily swayed by weather even over a decade or so and unclear
        on the issue of chaotic fluid flows (liquid as per oceans and gas as per atmosphere)
        all coupled together and complicated by Psychrometry re CO2 lifting H2O.

        Thanks for comment and opportunity to augment :-)

  2. It would also make a very energy efficient dehumidifier. And since dehumidification is often a large part of what makes air conditioning work, it could potentially make for very efficient air conditioners that only use the heat pump for cooling.

    1. Yeah, let’s get one of these in my basement. Dehumidifiers are used around here to prevent mold and the power draw is noticeable. Higher efficiency is always appreciated.

  3. When they claim extraction at 10% humidity, I would wonder if moisture in the air at this level could be naturally extracted by any process. For example, placing these somewhere in a desert and then crying that the air is a bit drier downwind does not seem to be a big issue. Or is it?
    Can they be talored to extract only a certain percentage… say takng humidity from 25% down to 20%?

    1. Thanks for comment Fred,
      There are so many methods such as hygroscopic salts etc
      The problem is going the other way in respect of or regarding (re) separation
      back to liquid water which has good utility for the safety & low maintenance issues.
      My eldest son is a chemical engineer now at MIT on a PhD and I have my own lab.
      We worked on this and arrived at a system worthy of the next step re pilot plant.
      The design for such a system needs no external power source or even solar panels
      but, the logistics of going from a lab version to a product is complex. Fortunately
      we are not in need of capital, just need time and competent staff both of
      which are the hardest to find as well as risk assessment issue re disclosures.
      Though I do like the idea of just giving the idea to the 3rd world but, even that
      has “unusual repercussions” I can’t yet go into. Again, need time and people.

      I wouldnt worry about the downwind issue as air movements are chaotic and
      partial pressures even out really well vertically as well as horizontally. If the scale
      were really high and upstream of a large crop which needed high humidity air whilst
      in/near a desert then maybe, so far I havent seen that scenario arise.

      Indeed its easy to arrange parameters in respect of air flow rates via various
      mechanisms and this could be dynamic then need a power source to move/close
      the baffles or alter the chemicals to suit over the selected temperature ranges
      for the period of time of absorption ie the classical issues re time/temp surface area…

        1. Although I do like the 1984 film, I would have loved to see the film David Lynch had actually intended to make before it became a “Alan Smithee” film (because he was denied the privilege of final cut). Herbert, who was alive at the time, said he was pleased with Lynch’s film. But he also said “Paul was a man playing god, not a god who could make it rain”

  4. nothing new or interesting, the Incas did this thousands of years ago, by channeling dew to underground systems, and many indigenous tribes use variations of the “fog fence to collect water from the air. their methods are entirely passive, and require zero energy or intervention, and once built continue to function unattended. the only question, is why did MIT bother to develop a device which has been also “invented” by many other universities, and never found a market?.

    1. “the only question, is why did MIT bother to develop a device which has been also “invented” by many other universities, and never found a market?.”

      For education?

    2. Because it’s much more compact and effective than fog walls or underground caverns, and works down to 10% relative humidity while the others need high relative humidity (like the morning dew) to collect any water.

  5. James,
    I know I’m not at the right place but I cannot post in your capacitance MKII’s page, it seems I never have the right CAPTCHA answer …. The post my be closed ???
    Could there be a mismatch betwin the formula to solve C between wiki and the sketch ? In wiki they multiply R by ln(…). In the sketch you divide R by (ln…) ???

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