Why would hot fluid retain more % heat at lower flow rates?

[@PinkGeology]

Homework Statement

Hot magma (1500 K) is flowing into the Earth at rates of 0.05 meters/year, 0.5 meters per years and 1 meter per year.

Although more total volume of magma will retain a temperature at or above 1150 K at higher rates of flow for any given time (say, at 1000 years), a higher PERCENT of the magma that has flowed in will remain at or above 1150 K for the lower rates of flow.

Why is that the case?

Homework Equations

I have a graph to show. The one of the left shows TOTAL volume retained at or above 1150 K for each rate (red being the fastest rate and green being the slowest)

On the right it shows the PERCENT of the inject magma at or above 1100 K as a function of time for each rate (same colors)

The Attempt at a Solution

This is an intuitive question but my intuition feels like the results are wrong! :)

 

[DaveC426913]

I can't be much help because
a] I'm having trouble picturing the scenario. Where is the magma flowing from? Where is it flowing to?
b] The images are too small to read. And there's insufficient context to interpret them (see a]).

 

[@PinkGeology]

I'm sorry - I don't know why the images are showing so small ... magma is flowing from some hot reservoir deep in the Earth (say, at about 30 km depth) into a shallower reservoir (maybe 8 km depth).

Here it is cooling to the temperature of the crust, which is set by the geothermal gradient of 20 K/km

It flows into the Earth in cylindrical shape with a radius = 3km and a height that grows over time at the rate of flow ... (0.005 m/yr, 0.02 m/yr, etc).

I just find it strange that if it flows in very very slow, a larger PERCENT of the magma that flows in will remain at or above some reference temperature (we are using 1150 K). .

 

[@PinkGeology]

Here is a crude picture I drew ... imagine both of the cylinders were filled up steadily in the same amount of time (say, 1000 years) ... the smaller one just grew at a slower rate.

For some reason, the results show that a greater PERCENT of the smaller volume is hot ( > 1150 K) compared to the larger cylinder. (i drew the crude internal boxes with rounded corners to represent the theoretical zone that is > 1150 K)

 

[insightful]

If the flow rate were zero and the heat from below equaled the heat loss above, wouldn't 100% of the magma remain above 1150K? If the flow rate were very fast, then more reaches the "cooling zone" and a higher percentage is cooled. (This is a guess, but I like to use exagerated high and low limits to see if it provides a clue.)

 

[haruspex]

If the flow rate were zero and the heat from below equaled the heat loss above, wouldn't 100% of the magma remain above 1150K? If the flow rate were very fast, then more reaches the "cooling zone" and a higher percentage is cooled. (This is a guess, but I like to use exagerated high and low limits to see if it provides a clue.)

It doesn't sound like you have the right model. The heat arriving in the upper chamber only comes from the magma flowing into it. The heat flowing out will depend on the temperature difference between what is there and the surface temperature. If heat is only flowing in slowly then to get a correspondingly low rate of outflow you need the magma in the chamber to have cooled a lot.

Since we are discussing what percentage is above some threshold, it seems we should not assume much mixing within the chamber. To understand the behaviour we need a model of how heat flows within and out of the layers in the upper chamber. Is a larger chamber wider, taller, or both? Does the incoming magma push the existing magma up?

I can see how the percentages could be about the same, but I agree it is surprising that the lower flow rate would give the higher percentage.
The images are still too poor. The legend is illegible. The forum upload process does some image resolution reduction I've never understood.

 

[mfb]

What happens to the material that was in there before? Where exactly does your magma enter and what happens to the magma around it when it does?

It looks like a simulation error but it is unclear what the simulation is actually doing.

 

[256bits]

If the flow rate were zero and the heat from below equaled the heat loss above, wouldn't 100% of the magma remain above 1150K? If the flow rate were very fast, then more reaches the "cooling zone" and a higher percentage is cooled. (This is a guess, but I like to use exagerated high and low limits to see if it provides a clue.)

I take another view from haruspex, which may give a clue to the problem with the model.

For one thing we do not know that actual mathematical model used, so we have no way to verify if correct assumptions have been made or correct formulas used, but it seems to be linear first order ( what else ! ) from the 1st graph ie Volume above 1150K versus Time, where there is dynamic time frame followed by a more linear steady state heat flow approaching an asymptote ( at least evident for the slower flows). If the time frame is measured in 10^3 years, around the tick 1, or 1000 years, most of the curves have reached some sort of steady state, with the slower growth having heat flux in = heat flux out, showing that the 1150 K maximum volume has been reached; but with larger flows still showing a growth in 1150 K volume, or depth, of the sill.

The 20K/km is most likely the key and could be inappropriately used.
For 30 km down from the Earth surface, assuming a 300 K surface temp, the temperature of the surrounding rock would be 900 K, 8 km down 460 K. ( Of course, no magma around to change the temperature gradient ). Back to this later.

But the 2nd graph, for the sill, the ratio 1150 K volume / total volume does not reflect the dynamic nor linear portions of the first graph, except it seems for the slowest growth (green ), which takes the turn in the exponential from vertical to more horizontal around the 1000 year mark. The faster growths seem to take the turn much earlier , around 50 years, which seems odd.

Since we do not know the diffusivity of the surrounding rock, nor of the magma, it could be that for greater injections of magma, the volume growth of the 1150 K front is outstripping the thermal growth in some manner by the mode, and always in contact with the Earth's thermal gradient temperature, producing a greater heat conduction to outlying regions. The volume/surface area aspects seem to be OK for the first graph( Volume at 1150K ) but not the second( ratio of 1150k/total injection).

 

[haruspex]

I take another view from haruspex, which may give a clue to the problem with the model.

For one thing we do not know that actual mathematical model used, so we have no way to verify if correct assumptions have been made or correct formulas used, but it seems to be linear first order ( what else ! ) from the 1st graph ie Volume above 1150K versus Time, where there is dynamic time frame followed by a more linear steady state heat flow approaching an asymptote ( at least evident for the slower flows). If the time frame is measured in 10^3 years, around the tick 1, or 1000 years, most of the curves have reached some sort of steady state, with the slower growth having heat flux in = heat flux out, showing that the 1150 K maximum volume has been reached; but with larger flows still showing a growth in 1150 K volume, or depth, of the sill.

The 20K/km is most likely the key and could be inappropriately used.
For 30 km down from the Earth surface, assuming a 300 K surface temp, the temperature of the surrounding rock would be 900 K, 8 km down 460 K. ( Of course, no magma around to change the temperature gradient ). Back to this later.

But the 2nd graph, for the sill, the ratio 1150 K volume / total volume does not reflect the dynamic nor linear portions of the first graph, except it seems for the slowest growth (green ), which takes the turn in the exponential from vertical to more horizontal around the 1000 year mark. The faster growths seem to take the turn much earlier , around 50 years, which seems odd.

Since we do not know the diffusivity of the surrounding rock, nor of the magma, it could be that for greater injections of magma, the volume growth of the 1150 K front is outstripping the thermal growth in some manner by the mode, and always in contact with the Earth's thermal gradient temperature, producing a greater heat conduction to outlying regions. The volume/surface area aspects seem to be OK for the first graph( Volume at 1150K ) but not the second( ratio of 1150k/total injection).

Where do these graphs come from? Is there any online reference you can link to?
It seems to me that the way the graphs curve is also paradoxical. The magma should cool relatively rapidly in the early years of the flow, more slowly later. Are you sure you have it the right way up, that these represent the volume of magma above a threshold temperature, not the volume below the threshold?

 

[256bits]

@haruspex
These graphs come from, as per the OP ( a previous thread )
"Ok, I've built a numerical model to show the cooling of hot magma sills entered into the crust over time.
...snip ...
I need to develop this for a journal paper I am working on, ..."


I copied and pasted into another - Microsoft Office Picture Manager.

Enlarged, I can barely make out the features ( except one , or two)

Graph 1:
- - - - - - - -
Heading -> "Change in volume over time for different xxxx rates" where xxx looks like "time"
( Here I am making the assumption thing the assumption that the ordinate represents the volume with a greater temperature than 1150K. Total injected volume would be a linear lline with constant slope of rate of growth x area / time )
Ordinate-> "Volume (km^3) graduations of 100 to 1000
Abscissa -> "Time(years) x 10^4) graduations 1 to 10 ( that's the one it may be x 10^3 )

Legend :
red - 4 x 10^-2 ma^-1
blue - 3 x 10^-2 ma^-1
brown - 2 x 10^-2 ma^-1
black - 1 x 10^-2 ma^-1
pink - 5 x 10^-3 ma^-1
green - 6 x 10^-2 ma^-1

graph 2:
- - - - - - - -
Heading -> "Percent of injected volume versus time"
Ordinate -> "Percent of injected volume % " graduations of 100 to 1000
Abcissa -> "Time(years) x 10^4) graduations 1 to 10 ( that's the one it may be x 10^3 )

Legend :
as previous

- - - - - -
Notes:
Green, the faster flow, is what the OP is worried about versus the progressively faster flows.

Previous threads on this subject
https://www.physicsforums.com/threads/numerical-modeling-of-hot-magma.814056/
https://www.physicsforums.com/threads/counterintuitive-results-in-conduction-cooling-model.809628/

The png files are old and non-existant.
The pdf creates an error due to the version I have.

It seems to me that the way the graphs curve is also paradoxical.

These two graphs may not represent that from the previous threads, as the data may have changed during the time period of posting. OP would have to verify
The percentage graph ordinate seems to be in error. --> 1000 % seems unlikely.
Green on percentage seems to extend to infinity in the y-direction. That's odd.

This part from the initial thread on the subject, explains the total magma injected ( 16km ) versus height of each sill.

For instance, for a total of 16 km of magma ...
1a-n. 40, 400m high sills emplaced at rate of 5e-3 ma-1, 5e-4 ma-1, 1e-2 ma-1, , 2e-2 ma-1, 3e-2 ma-1, and 4e-2 ma-1
1b-n. 160, 100m high sills emplaced at the same set of rates
1c-n. 320 sills, 50 m high at each of the rates and ...
1d-n. 640 sills (25 m high) at each rate.

My " outstripping the thermal growth" which is a hand waving way of putting it, comes from the fact that subsequent injections ie 40, 400m high sills for example, may be laying one on top of the other, in which case the Earth's thermal gradient would be affected for each subsequent injection in the model.

I really doubt if the actual solution can be obtained without more info about the model being used.

 

[haruspex]

@haruspex
These graphs come from, as per the OP ( a previous thread )
"Ok, I've built a numerical model to show the cooling of hot magma sills entered into the crust over time.
...snip ...
I need to develop this for a journal paper I am working on, ..."I copied and pasted into another - Microsoft Office Picture Manager.

Enlarged, I can barely make out the features ( except one , or two)

Graph 1:
- - - - - - - -
Heading -> "Change in volume over time for different xxxx rates" where xxx looks like "time"
( Here I am making the assumption thing the assumption that the ordinate represents the volume with a greater temperature than 1150K. Total injected volume would be a linear lline with constant slope of rate of growth x area / time )
Ordinate-> "Volume (km^3) graduations of 100 to 1000
Abscissa -> "Time(years) x 10^4) graduations 1 to 10 ( that's the one it may be x 10^3 )

Legend :
red - 4 x 10^-2 ma^-1
blue - 3 x 10^-2 ma^-1
brown - 2 x 10^-2 ma^-1
black - 1 x 10^-2 ma^-1
pink - 5 x 10^-3 ma^-1
green - 6 x 10^-2 ma^-1

graph 2:
- - - - - - - -
Heading -> "Percent of injected volume versus time"
Ordinate -> "Percent of injected volume % " graduations of 100 to 1000
Abcissa -> "Time(years) x 10^4) graduations 1 to 10 ( that's the one it may be x 10^3 )

Legend :
as previous

- - - - - -
Notes:
Green, the faster flow, is what the OP is worried about versus the progressively faster flows.

Previous threads on this subject
https://www.physicsforums.com/threads/numerical-modeling-of-hot-magma.814056/
https://www.physicsforums.com/threads/counterintuitive-results-in-conduction-cooling-model.809628/

The png files are old and non-existant.
The pdf creates an error due to the version I have.These two graphs may not represent that from the previous threads, as the data may have changed during the time period of posting. OP would have to verify
The percentage graph ordinate seems to be in error. --> 1000 % seems unlikely.
Green on percentage seems to extend to infinity in the y-direction. That's odd.

This part from the initial thread on the subject, explains the total magma injected ( 16km ) versus height of each sill.My " outstripping the thermal growth" which is a hand waving way of putting it, comes from the fact that subsequent injections ie 40, 400m high sills for example, may be laying one on top of the other, in which case the Earth's thermal gradient would be affected for each subsequent injection in the model.

I really doubt if the actual solution can be obtained without more info about the model being used.

From what I can make out of those prior threads, the issue is the effect of the pattern of injection, not the rate:

a larger hot zone is created with 160 100m sills than for 320 50m sills or for 640 25m sills emplaced at the same overall rate

...the volume of the "hot" zone when the emplacement of a constant volume of hot sills is all done will vary as a matter of two things: the overall rate at which the magma is emplaced (duh) and the thickness of the sills

I don't see the graphs you posted in this thread in either of the earlier threads. They're not in the pdf. Did you capture them from the dropbox files before they disappeared?

Think about what I said regarding the way the graphs curve. Surely you would expect that the average cooling would be fastest during the early stages, whatever the rate?
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