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Please consider a physics question that I am asking here because I have no ability to calculate the answer myself.

Given a large cubical mass of water ice is deposited on a desert location on Earth, how long would it take to melt?

IDEA SPARK In a SF story, I was considering brining in an icy body, roughly 1 km in diameter, to a desert location to provide water for drinking/irrigation. I'm assuming no radioactivity & dangerous contamination for the icy body: this seems to be a reasonable assumption, from what I have read. The ice contamination (if any) involve dust/rocks, which can just be filtered by the locals if necessary using cloths, etc.

All the details below are just me putting numbers and variations to the core idea.

To extend the core idea, I included ice bodies from glaciers, in recognition of an old, old Saudi plan to ship an iceberg to their land.

EXTERNAL TEMPERATURE To keep it simple, assume a steady environmental temperature of 35 C. No rain.

INTERNAL TEMPERATURES We'll divide the mass by initial location, with a from space (asteroid), and g from the ice cap (glaciers).

Space ice has an initial internal temperature of -100 C. Glacial ice has an initial internal temperature of -20 C.

So, re-categorizing the masses:

  • Mass 1s - 100 m diameter, internal temp -100 C;
  • Mass 1g - 100 m diameter, internal temp -20 C;
  • Mass 2a - 1000 m diameter, internal temp -100 C;
  • Mass 2g - 1000 m diameter, internal temp -20 C;
  • Mass 3a - 3000 m diameter, internal temp -100 C;
  • Mass 3g - 3000 m diameter, internal temp -20 C;

How long would it take, for these masses to melt?

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    $\begingroup$ 1. There is a Physics stack; 2. we don't do math homework; 3. you're asking too many questions at once; and 4. this is not a question that helps you make a fictional world or understand how the one you're making works. Check our our tour and help center and find out what Worldbuilding is all about $\endgroup$
    – elemtilas
    Jul 23 '20 at 21:04
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Your question is not directly answerable because You need energy transfer to melt ice an temperature alone will not give you relevant figures.

I will give you a bit of background to help you understand the physics of what you ask.

  • Fusion latent heat for ice melting is ~330kJ/kg.
  • Specific heat of ice (heat to increase temperature 1°C) is only ~2kJ / °C Kg This means it takes less heat to go from -100°C -> 0°C than going 0°C (ice) -> 1°C (water).

Heat transfer is essentially through surface and that increases with the square of dimensions while mass increases with cube of dimensions, this means a mass twice as big (as linear dimensions) will take about four times the time to melt.

Energy transfer is dependent on conditions on the surface, much more than pure external temperature You can easily multiply tenfold "life" of your mass simply putting a reflective Mylar sheet over it, OTOH you can shorten considerably melting time putting black gravel on the ice (I assume it is in full sunshine).

Irradiation is ~5kw/day (tropical areas), which translates to ~18000kJoule

Basic math is as follows:

1'000m cube:
    1'000'000'000 Kg     -> 330*10^9 kJoule to melt
    1'000'000 m2 (area)  -> 18*10^9 kJoule / day from sun
    Your Ice cube will be gone in ~20 days (~30 if it started at -100*C) 

... this IF and ONLY IF the whole heat from the sun will be absorbed, i.e.: if the ice behaves as a "black body", which obviously isn't the case.

In former times ice was actually snow fallen on mountains, amassed in lumps (~10m cubes) and simply covered with a leaf top; it lasted the whole summer (even in rather hot climate) and was cut in bars and sold.

Other things that may affect melting are:

  • hot winds (bringing in heat from a larger area)
  • geography (ice in a small lake formed by its own melting will last much longer than ice on a sand desert absorbing all water).
  • if ice is a compact lump or it breaks in smaller masses (increasing surface).
  • etc. etc.

A really big mass (1cube km) should live over a year even if no special measures are taken to preserve it. To have it melt faster you need to be sure solar heat is efficiently absorbed.

Note: I did most computations in my head, so I my have goofed somewhere, please cross-check.

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