Bringing a liter of water from room temp to a boil will heat the earth how much?

[repoman]

I am trying to get a handle on what is the rough multiplier effect on the heat humans release by burning fossil fuels and the total long term heat trapping just from the released CO2.

I did it a while ago on this subforum and I got that burning natural gas will be a part of trapping heat in the atmosphere that will very roughly trap the same amount of heat in about one year. However, the oceans take up much more heat than the air.

I am trying to get a rough idea as to how many tens, hundreds or thousands of liters of water the CO2 will "in effect" raise from room temp to boiling (energy wise) until the earth reaches the new set point temp all the way down to the depths of the ocean.

Just trying to get a number to make us all feel like shit. Hahaha.

Oh, you like your hot cup of natural gas heated coffee, well that will be in effect like heating the earth up by 437 cups of coffee when all is said and done as a possible example.


So basically if x(t) is all the fossil fuel energy we have released and y(t) is the heat trapped by it (from say 1750) then z(t) = y(t)/(x(t) should be increasing until it will level out.

Anyone here have a guess range for what z(t) is now and what it may be in 50, 100 or 1,000 years from now?

 

[bilby]

Water has a specific heat capacity of 4.2kJ/litre, so heating one litre of water from 298K to 373K requires 75 x 4.2kJ = 315kJ of energy.

A typical electric kettle has an efficiency of around 65%, so to add 315kJ of energy to the water in a kettle, you will use 315kJ / 65 x 100 = 485kJ of electricity.

Transmission and distribution losses between power plant and home average about 6.5% in the USA, so to provide 485kJ at your home, 485kJ / 100 x 106.5 = 517kJ of electricity needs to be generated.

A kWh is equal to 3600kJ, so 517kJ is 0.144kWh.

How much CO₂ that 0.144kWh of generation releases depends on the type of power plant. A country like Norway, where hydroelectricity is the norm, generates around 50gCO₂eq/kWh; France, where most power is nuclear, generates about 100gCO₂eq/kWh; while Germany, with her reliance on coal and gas, generates about 390gCO₂eq/kWh.

The USA, with her heavier reliance on coal than these European nations, generates around 615gCO₂eq/kWh on average, so 0.144kWh will cause the release of approximately 89g of CO₂ into the atmosphere.

The Earth's atmosphere warms by about 1.7K per 10¹⁵kg of additional CO₂ added to the atmosphere, so the warming due to 0.089kg is 0.089kg x 1.7K/10¹⁵kg = 0.15x10^-15K = 1.5x10^-16K

The atmosphere has a mass of about 5x10¹⁸kg, so it is hardly surprising that the total warming from boiling a litre of water only has a tiny effect on the temperature of the entire atmosphere.

Atmospheric warming in Kelvins is approximately 4.28 x ln( C_new / C_old ), where C_new is the final concentration (in ppm) of CO₂ in the atmosphere, and C_old is the starting concentration (also in ppm).

The results of direct and indirect measurements of atmospheric CO₂ level imply that it has increased from preindustrial concentration of ~280 ppm to the benchmark 400 ppm measured in May 2013 by Mauna Loa Observatory; Plugging those numbers into the equation above, we should expect warming of around 4.28 x ln( 400 / 280 ) = 1.5K, and this is pretty much the amount of warming we actually see.

There are now about 3x10¹² tonnes (3x10¹⁵kg) of CO<sub>2[/sup] in the atmosphere; that's around 9x10¹⁴kg more than in the pre-industrial era, or enough to bring to the boil 10¹⁶ litres of water.

Your kettle full of water is not a big deal - it adds about one ten thousand million millionth to the total CO2 generated by the entire history of human industry to date - but all the energy used by humans since the industrial revolution really adds up. It's equivalent to using your electric kettle to bring to the boil the entire Mediterranean Sea, twice over (although I for one am glad that we used it for other things, as it would have been highly disruptive to the Greek fishing industry if we had boiled the Med even once; and the salt water would be very bad for your electric kettle).</sub>

 

[repoman]

I was using the liter of water just for a general example. It could have been a mile by car or air, or the energy required to make a metal can for canned food.

So from skepticalscience.com it said that:

The total heat accumulation in the environment from 1961 to 2003 is estimated as 15.9 x 10^22 Joules.

Strange way of saying ~1.59*10^23 Joules

https://www.skepticalscience.com/Breaking_News_The_Earth_is_Warming_Still_A_LOT.html

I can with some more time get more current numbers and also back before 1961.


Next I went to wiki for the energy consumption of the world, which is vastly from fossil fuels:

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

looked at this chart


and did a rough estimate to get around 10^22 Joules consumed from 1965-2005 which is roughly the same time span. For this time span this is already at least a z(t) =10x multiplier.

Again y(t) = heat trapped from CO2, x(t) = heat released from fossil fuels.

z(t) = y(t)/x(t)

I fully realize that there is a lag and that the CO2 will act for a long time and it has not reachd equilibrium yet.

If we released all that 10^22 Joules of heat with no added greenhouse gases, then earth would really not have heated at all, it would have been shed to get to the temperature/radiation balance that we had when CO2 was 280 ppm.

I guess the number I want now is this: assuming that we burn it all (which is absolutely going to happen) and then given a couple centuries/milennium...

What will y(t) top out at when all is said and done? x(t) should not be too hard to figure out within a margin.

Also a bit curious on the total energy (joules) required to melt all of the ice in antarctica including the heat capacity of the ice (given an average temperature and heat of fusion.

Rough data says:
2.43*10^22 g ice (2.65*10^km^3 * 917 kg/m^3)

assume ice is about -33 Celsius (a bit cheating guess)

Heat of fusion = 334.16 J/g
specific heat capacity for solid water (ice) = 2.06 J/g*K

so need ~400 Joules to melt 1 gram of ice.

rounding up the 2.43 to 2.5, you get about:

roughly 10^25 Joules to melt all of Antarctica. That is a lot, but not that many orders of magnitude about the current yearly heat trapping rate.

Another interesting chart is here:

https://www.epa.gov/climate-indicators/climate-change-indicators-ocean-heat

This is a complicated topic to get solid numbers on, but what do you expect?

A newer link say that the earth (mostly ocean) is trapping 4 hiroshimas/second.
https://www.skepticalscience.com/4-Hiroshima-bombs-worth-of-heat-per-second.html

a good comment says:

According to Wikipedia, the earth receives 119 x 10^15 Joules of energy per second from the sun, discarding the amount that is reflected. http://en.wikipedia.org/wiki/Solar_energy

The point is: how much of that is retained to drive up the earth's temperature, given the energy imbalance in the earth's climate system (Heat In > Heat Out).

The Wikipedia number equates to 1900 Hiroshima bombs (using 6.3 x 10^13 to be the heat released by a single bomb).

According to the above estimation, only 1896 bombs-worth of energy escapes back to outer space per second, leaving 4 bombs-worth per second to accumulate in the earth's climate system, raising the temperature.

An A-bomb's-worth of energy is an easier unit to imagine than 6.3 x 10^13 Joules, so I am good with it.

Discussing this elsewhere, I used the analogy of an electric kettle heating water - the heating element is pumping energy into the water per second, more than is escaping through the kettle sides per second. Result is temperature rise in the water as a fixed amount of energy accumulates per second.

So imagine that this current heating rate only went into melting antarctica (humor me), it would take how long to melt it?

10^25 J / 2.5*10^14 J/s = 4*10^10 seconds * 1year/ 3.2*10^7 seconds = 1,250 years.

 

[Valjean]

This seems way too complicated for any practical application.
If it takes one calorie to raise one ml of water one degree, the number of calories eventually bleeding into the environment should be fairly easy to calculate.

 

[bilby]

This seems way too complicated for any practical application.
If it takes one calorie to raise one ml of water one degree, the number of calories eventually bleeding into the environment should be fairly easy to calculate.

Actually, it seems to me to be rather simple - the waste heat going to the environment is the inefficiency of the kettle, so if the kettle is 65% efficient, the waste heat is 170kJ, and the heat going into the water is 315kJ.

Of course, not only the waste heat goes into the environment; Ultimately, ALL of the energy used ends up heating the environment, so the short term effect is (if we include grid losses) about 517kJ. But then that heat is mostly radiated into space, in the long term. So the long term heating of the planet is almost entirely due to the net radiative forcing from the CO₂ emission, which is per my calculation above.

 

[bilby]

I was using the liter of water just for a general example. It could have been a mile by car or air, or the energy required to make a metal can for canned food.

So from skepticalscience.com it said that:


Strange way of saying ~1.59*10^23 Joules

https://www.skepticalscience.com/Breaking_News_The_Earth_is_Warming_Still_A_LOT.html

I can with some more time get more current numbers and also back before 1961.


Next I went to wiki for the energy consumption of the world, which is vastly from fossil fuels:

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

looked at this chart
View attachment 9991

and did a rough estimate to get around 10^22 Joules consumed from 1965-2005 which is roughly the same time span. For this time span this is already at least a z(t) =10x multiplier.

Again y(t) = heat trapped from CO2, x(t) = heat released from fossil fuels.

z(t) = y(t)/x(t)

I fully realize that there is a lag and that the CO2 will act for a long time and it has not reachd equilibrium yet.

If we released all that 10^22 Joules of heat with no added greenhouse gases, then earth would really not have heated at all, it would have been shed to get to the temperature/radiation balance that we had when CO2 was 280 ppm.

I guess the number I want now is this: assuming that we burn it all (which is absolutely going to happen) and then given a couple centuries/milennium...

What will y(t) top out at when all is said and done? x(t) should not be too hard to figure out within a margin.

Also a bit curious on the total energy (joules) required to melt all of the ice in antarctica including the heat capacity of the ice (given an average temperature and heat of fusion.

Rough data says:
2.43*10^22 g ice (2.65*10^km^3 * 917 kg/m^3)

assume ice is about -33 Celsius (a bit cheating guess)

Heat of fusion = 334.16 J/g
specific heat capacity for solid water (ice) = 2.06 J/g*K

so need ~400 Joules to melt 1 gram of ice.

rounding up the 2.43 to 2.5, you get about:

roughly 10^25 Joules to melt all of Antarctica. That is a lot, but not that many orders of magnitude about the current yearly heat trapping rate.

Another interesting chart is here:

https://www.epa.gov/climate-indicators/climate-change-indicators-ocean-heat

This is a complicated topic to get solid numbers on, but what do you expect?

A newer link say that the earth (mostly ocean) is trapping 4 hiroshimas/second.
https://www.skepticalscience.com/4-Hiroshima-bombs-worth-of-heat-per-second.html

a good comment says:

According to Wikipedia, the earth receives 119 x 10^15 Joules of energy per second from the sun, discarding the amount that is reflected. http://en.wikipedia.org/wiki/Solar_energy

The point is: how much of that is retained to drive up the earth's temperature, given the energy imbalance in the earth's climate system (Heat In > Heat Out).

The Wikipedia number equates to 1900 Hiroshima bombs (using 6.3 x 10^13 to be the heat released by a single bomb).

According to the above estimation, only 1896 bombs-worth of energy escapes back to outer space per second, leaving 4 bombs-worth per second to accumulate in the earth's climate system, raising the temperature.

An A-bomb's-worth of energy is an easier unit to imagine than 6.3 x 10^13 Joules, so I am good with it.

Discussing this elsewhere, I used the analogy of an electric kettle heating water - the heating element is pumping energy into the water per second, more than is escaping through the kettle sides per second. Result is temperature rise in the water as a fixed amount of energy accumulates per second.

So imagine that this current heating rate only went into melting antarctica (humor me), it would take how long to melt it?

10^25 J / 2.5*10^14 J/s = 4*10^10 seconds * 1year/ 3.2*10^7 seconds = 1,250 years.

We did this calculation before:

We can set an upper bound fairly easily though, by calculating the additional energy due to radiative forcing, and plugging in the heat of fusion of ice.

The Antarctic icecap contains 2.65x10¹⁹kg of ice, which Google tells me is enough to raise sea level by 58m if it all melted.

To melt that much ice requires 8.8x10²⁴J of heat; The IPCC estimates that in 2014, the additional radiative forcing due to Greenhouse Gasses (compared to the pre-industrial baseline year of 1750) is 2.936Wm^-2. Plugging in the surface area of the Earth, 5x10¹⁴m², we get an additional insolation due to GHG of 1.5x10¹⁵W, which would take 6x10⁹s to melt that ice, assuming that all of the additional energy went into the Antarctic icecap, and the rest of the world remained at its current temperature.

That gives a fastest possible melt time of ~190 years, and sea level rise at a rate of about 2.25 years per metre.

Of course, the actual time required must be far larger than this, as the heat will not only go into the Antarctic ice cap - a significant fraction will go to melting Arctic ice (which has no effect on sea level as it is already floating), and to warming the oceans (which will increase sea levels slightly due to thermal expansion), the atmosphere, and the Earth's crust. However it is also worth noting that the 2014 value of 481ppmCO₂eq of GHGs has already been exceeded, and likely will continue to rise over the next two centuries, so that will tend to accelerate the rate of ice loss.

I would hazard a guess that we can expect to see, at the absolute worst, about one third of the extra heat going to the Antarctic ice (with a third going to Arctic ice, and another third into the oceans and crust), giving a worst case scenario of ~7 years per metre of sea level rise; So we can guesstimate that 40m of sea level rise will take a minimum of 270 years to occur. That's effectively instantaneous in geological terms, and at 150mm (6 inches, for those watching in black and white) per year, is FAR faster than the current rate of sea level increase, which is about 3mm (0.1 inches) per year.

Of course, this is a VERY crude 'worst case' guesstimate; The real rate could be dramatically slower (if the proportion of the energy that goes to melting Antarctic ice is significantly less than a third of the additional amount due to radiative forcing, which my gut tells me it likely is). But Back-of-the-Napkin-Physics suggests it is not likely to be significantly faster.

Of course, sea level is not the only concern. Ocean acidification will be another major issue, as will increases in severe weather events, and changes in rainfall patterns worldwide. All of these will have major impacts on humans, even if sea level rise were to turn out to be far slower than this guesstimate suggests.

On reflection, it occurs to me that there is a mechanism by which sea level rise could be much faster.

If higher global temperatures allowed the Antarctic and/or Greenland land ice to slide off the land and into the ocean, then it wouldn't be necessary for the ice to melt - as soon as it is floating, it will displace enough water to have the same effect on sea level as if it had melted.

So if warmer climate leads to faster glacial flows into the ocean, sea level rise could be markedly more rapid than my earlier figures imply.

There is some evidence that the floating ice shelves around Antarctica are breaking up, which could lead to more rapid glacial ice flow into the ocean in the areas currently occupied by that sea ice.

Of course, glaciers are notoriously (indeed eponymously) glacial in their flow; a fast glacier is not in the same league as a slow river in terms of bringing water into the sea. But they could certainly bring ice off the land far faster than it could melt.

The bolded is the important part - It's simply not necessary to melt all the ice to cause a catastrophe. When you add an ice-cube to your drink, the liquid level goes up immediately - you don't have to wait for the ice to melt before the level rises. Indeed, the level doesn't change at all after the ice is added, whether it melts or not. Floating ice has already contributed as much to sea level as it ever will.

 

[barbos]

It's not about direct heating of atmosphere by boiling water. Direct heating is inconsequential and tiny anyway.
Solar radiation dwarfs any human caused direct heating by orders and orders and orders of magnitude. It's about CO2 emission which makes solar radiation heating more "efficient" Solar water boilers one of these no-brainer things which should be a law.