Temperature difference by Dimensional analysis.

[pogs]

Homework Statement

A block of uranium is kept cool in water. How does the difference between the temperature in the centre of the block and the temperature in the water depend on the size of the block?

Relevant Equations

n = n_0*e^(-t/τ) : Radioactive Decay

Attempt at solution:

I wanted to try and solve this with dimensional analysis. I reasoned that I would chose the following dependent variables:
- [V] : Volume ( of the block)
- [Q] : Heat ( the radioactive decay would cause some heating of the water)
- [R]: Radiation
- [Cv]: Heat capacity

I'm thinking the amount of radiation is a factor, it causes the heating and thus the bigger the block the more the radiation, the more heating. Also the heat capacity may play a part.

Taking the dimensions of these I get the following
[V] = L^3
[Q] = M L^2 T ^-2
[R] = L^2 M T^-1
[Cv] = L^2 M K^-1 T

Now we want a temperature so we can set up an equation

Temp(K) = [V]^α * [Q]^β * [R]^γ *[Cv] ^δ = (L^3)^α*(M L^2 T ^-2)^β*(L^2 M T^-1)^γ * (L^2 M K^-1 T)^δ

Taking the dimensions we get

L = 0 = 3α +2β +2γ +2δ
T = 0 = -2β -γ +δ
M = 0 = β +γ +δ
K = 1 = -δ

But when I solve these equations I get
α = 0
β = 2
γ = -1
δ = -1

giving me an equation that says

Q^2/(R*C_v)

Which I'm not to convinced by... Also V disappears as alpha is zero. I guess my dependent variables are wrong somehow. Any ideas/tips?

 

[BvU]

I wanted to try and solve this with dimensional analysis

Not convinced that that is possible.

How about an approach along the line

Uranium decays. Heat generated $\propto$ ...​

Heat exchange with surrounding water $\propto$ ...​

?

$\$

 

[Chestermiller]

Other parameters you should include are thermal conductivity of the uranium and temperature of the water.