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Modeling heat flow -- from a chain of blocks to heat in the earth

Modeling heat flow -- from a chain of blocks to heat in the earth. Used to understand: The flow of heat in the Earth. Spread of pollution in ground water. Flow in the Everglades. The one block problem from last time: Heat flow proportional to temperature difference

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Modeling heat flow -- from a chain of blocks to heat in the earth

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  1. Modeling heat flow -- from a chain of blocks to heat in the earth Used to understand: The flow of heat in the Earth. Spread of pollution in ground water. Flow in the Everglades.

  2. The one block problem from last time: • Heat flow proportional to temperature difference •  is constant relating temperature difference to heatflux. • Toutside is the "Boundary Condition". Every problem like this needs them! • You also need an "Initial Condition"

  3. Now lets expand this problem to multiple blocks. • Heat flow between blocks scales as temperature difference • What is Tbc? Why the name?

  4. We can write this • The form of the interior equation we will see again and again.

  5. BC Interior BC • Lets look at the general structure of these equations • You will usually get something like this

  6. How do we understand this? • Lets look at interior, with no radioactivity. • If T changes linearly with space, is there any change in T with time? (on board) • Why? • What drives change?

  7. Lets do steady state for two cases: • Tbc1=T1 and TbcN=TN • What does this mean? • Tbc1=(one temperature) and TbcN=(another temperature) • What does this mean?

  8. Why have we done all of this • because we canthink aboutwhat happenswhen theblock sizebecomes small!

  9. How do we think about the continous problem? • think about "heat flux;" proportional to • so • so what is dT/dt? • difference between at either end • this is • if k is constant, so:

  10. The same as the "chain of blocks" with no Q • If T changes linearly with space, is there any change in T with time? Now look at the continuous problem without radioactivity: • Is there any difference?

  11. Wait a minute! why do we write • And not • This is a Partial Differential Equation (PDE) • Derivatives of more than 1 variable (t,x) • Why a ∂? why not a d ? What does it mean?

  12. Lets do steady state for two cases: • Tbc1=T1 and TbcN=TN • What does this mean? • Tbc1=(one temperature) and TbcN=(another temperature) • What does this mean?

  13. How do we solve numerically

  14. Why is it an ODE again? • Now we can solve it with Runge-Kutta • Note it is the (nearly) the same as the block model • Why? because it is fundamentally the same problem in the limit of small blocks!

  15. Next time: How to apply this heat equation to real world problems • And how to solve it in Matlab.

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