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How do we model heat flow in a chain of blocks?. Why do we care? Stepping stone to larger problems! The flow of heat in the Earth. Spread of pollution in ground water. Flow in the Everglades. First, lets do the problem of one block! What do we need to solve this problem?.
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How do we model heat flow in a chain of blocks? Why do we care? Stepping stone to larger problems! The flow of heat in the Earth. Spread of pollution in ground water. Flow in the Everglades.
First, lets do the problem of one block! • What do we need to solve this problem?
First, what is heat? • Proportional to energy of vibration of molecules (roughly) • Change in energy proportional to change in temperature (if no phase change) E=Cp*Mass*T • Units of Cp are J/kg/oC
First, we need a governing equation • Tb is temperature of the block • Cp is heat capacity in J/kg/oC • M is mass, in kilograms • Do units work in equation?
1: How does radioactive heating enter equation? • Q in J/kg/s; heat generated per kilogram per second by radioactive decay • Give example of insulated granite; how long to melt? • Equation is now:
2: You need to know transfer of heat between block and surroundings • Heat flow proportional to temperature difference • is constant relating temperature difference to heatflux. • This is a "Boundary Condition". Every problem like this needs them!
3: You need to know what the temperature is when you start! • "Initial Condition" • Why?
How do we start to think about this system? • Often useful to look at "long time" solution • What is a long time? • steady • or the whole system is changing uniformly • details of initial condition not important
For example, what if there is no radioactivity? • What happens if Tb > Toutside? • where does the heat go? • what is sign of dTb/dt? • when will it become zero? • what is steady solution? • does it depend on the initial condition? • does it depend on the boundary condition?
What if block is insulated (=0) and radioactive? • How do we start? • where does the heat go? • what is sign of dTb/dt? • will it become zero? • is their steady solution? • what is the solution? • does it depend on the initial condition? • does it depend on the boundary condition?
What if it is both radioactive and leaks heat into surrounding? • How do we start? • where does the heat go? • what is sign of dTb/dt if Tb<Toutside? • will it become zero? • is their steady solution? • what is the solution? • does it depend on the initial condition? • does it depend on the boundary condition?
Now lets expand this problem to multiple blocks. • Heat flow between blocks scales as temperature difference • What is Tbc? Why the name?
We can write this • The form of the interior equation we will see again and again.
BC Interior BC • Lets look at the general structure of these equations • You will usually get something like this
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? • Why? • What drives change?
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?
Why have we done all of this • because we canthink aboutwhat happenswhen theblock sizebecomes small! • The diffusion of heat in the Earth • The diffusion of pollutants in ground water • Many other problems! • Next time -- what equations are we starting to solve; the beginning of PDE’s.
