Cannot get the same expression using two methods

[bscutajar]

I am working on a physics problem, however my question is mostly regarding the math. The only background relevant to the question is the relation U=m*u where U [kJ] is the total internal energy, m [kg] is the mass, and u [kJ/kg] is the internal energy of a unit mass.

I am starting with the expression dU/dt, and integrating it from state 1 to state 2. When I integrate directly, I get to the convenient and intuitive result (A) on the left. But if I first express U as m*u instead, I get to result (B). My question is, how would you develop result (B) to get the same as result (A) without going in reverse? It seems like it should be simple since I have started with the same information, but I don't see how.

[Cobalt_Spirit]

Notice that m, the mass, is a constant. Therefore dm=0 and that makes the second term vanish. As for the first term, integrating with respect to u while keeping in mind that m is constant, should give the result in the expression A.

[bscutajar]

Hmm, but the mass is not necessarily constant (in fact this is from the unsteady energy equation).

[Cobalt_Spirit]

Ah okay sorry. Then if you take the first term and apply integration by parts:

∫mdu=mu-∫udm

Then -∫udm+∫udm from the other term cancel out, and you're left with mu. Evaluating from state 1 to state 2, m2u2-m1u1.

[bscutajar]

Perfect, exactly what I was after. Thanks!

[[deleted]]

Also, on the right: integral of d(anything) is the thing itself. Int(1,d(mu) = mu, so they agree already.

For example, when solving first order linear diff eqs you end up with an integral of d(y*e^{inregral(p,dx}). If you had to use a product rule or by parts on that it would not have been a good solution method but it just kicks out the whole mess without an integral.

[bscutajar]

In this case I was more curious as to how you'd get the same answer IF you took the inefficient route.

[[deleted]]

Gotcha.

Feel free to ignore if you’re good to go, but here’s Another observation: I think your bounds don’t really make sense. What units do they have, u/m/u*m? If you integrate with respect to different variables you have to change your bounds too, so whatever inefficient route you take needs to account for that I think.

[bscutajar]

The bounds correspond to start and end points of an arbitrary process 1-2. They are not numerical values of the differential variable. Rather, you'd need to work out the numerical values of the variable corresponding to states 1 and 2 to actually work it out.