9
votes
6answers
1k views

Why is it natural to look for solutions involving dimensionless quantities?

While studying the Heat Equation, I got stuck in a statement in my book. It says: We have seen that the combination of variables $\displaystyle \frac{x}{\sqrt{Dt}}$ is not only invariant with ...
0
votes
1answer
148 views

Heat equation : boundary conditions?

Say we have a bar centered at $x=0$, that is heated. We have the 1D Heat equation which we can solve to find a parabolic temperature profile : $$\kappa \frac{d^2 T}{d x^2}=-\frac{Q}{L S}$$ with ...
1
vote
0answers
48 views

Derivation of cylindrical line heat source problem?

I have a line heat source embedded inside a cylinder and I am trying to find the temperature distribution T(r,t). By using the similarity variable, the solution to the differential equation is ...
0
votes
1answer
73 views

Showing specififc internal energy $e$ is a perfect differential

I am reading a book on Gas dynamics and there is a small section on thermodynamics before the conservation laws of mass momentum and energy are introduced. The book says $$ p = R \rho T$$ where ...
0
votes
1answer
258 views

Problem involving 1st law of thermo and ideal gas law

Problem: $1.0 \text{ kg}$ of air at pressure $10^6 \text{ Pa}$ and temperature $398 \text{ K}$ expands to a five times greater volume. The expansion occurs such that in every instance the added ...
0
votes
1answer
422 views

Temperature Vs. Volume of Water

Here and here it states that water is at its highest density around $4^\circ$ Celsius. I know very little physics and a Google search has left me without an answer. I am teaching an ODE class in the ...
8
votes
1answer
418 views

Modification of Newton's Law of Cooling

Yesterday I randomly started thinking about Newton's Law of Cooling. The problem I realized is that it assumes the ambient temperature stays constant over time, which is obviously not true. So what I ...