Heat Transfer: Model Show Slower Flux Retains Larger Percent Hot Volume. Why?

[ThePinkGeologist]

Imagine magma flowing into two chambers in the Earth from some deep source. It's hot (1500 K).

For one source (SOURCE 1), the flow of magma into the chamber is 10 m/s

For the other (SOURCE 2), it flows in at 5 m/s

After 5000 years, we measure how much of the magma is still above 1150 K

Although the total volume of magma over 1150 K is larger in SOURCE 1, the PERCENT VOLUME that is hotter is larger in SOURCE 2

I've attached a simple diagram so you can visualize it

(VOLUME 1.2/VOLUME 1.1) < (VOLUME 2.2/VOLUME2.1)

Why would that be?

 

[IRstuff]

Is this for school? Student postings are not allowed.

TTFN
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[ThePinkGeologist]

It's not homework It's the initially confusing results of a numerical analysis I'm doing on the thermal evolution of magma sill stacks ... but I think I've satisfied myself that this is a product of reduced surface area for the smaller sill stack (my spheroidal picture is kind of distracting since the emplacement geometry is actually a cylinder that grows taller as magma flows in).

 

[IRstuff]

The other issue is that your flow rate as a velocity lends nothing to the problem, since it's the mass flow that matters.

Your graph units and scaling make no sense; what are m/a? Why does your vertical scale go to 1000% Also what are the relative diameters of the stacks?

TTFN
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Need help writing a question or understanding a reply? forum1529


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There is a homework forum hosted by engineering.com:

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[btrueblood]

I'm not sure that flow rate matters at all, but the relative sizes. Smaller chambers will have a larger surface area to volume ratio, and thus a higher heat loss rate (so either your diagram is exactly backwards relative to the volume of the chamber, or something else is happening in your model that you aren't telling us about).

 

[ThePinkGeologist]

It's really just heat loss by conduction in a cylinder with a height determined by the rate of hot magma added to the system (really, you can say the magma is added in in meters/time or cubic meters/time ... it's done in 2D but axially symmetric).

And your right - once I really thought about surface area the results look even more wrong because of the area/volume problem ... I am going to have to re-calculate the set myself and see what the problem is because I can't think of a good reason for this.

And the y-axis went to 1000 because I was in a hurry and made a mistake. You can read it as "100"