Show that Temperature remains unchanged in phase changes

[dRic2]

Well, I have to show mathematically that T remains unchanged in phase change. I know it comes from sperimental evidences, but my professor asked to show it with "numbers". So I thought to go like this:

$$(dH_l)/dt ≈ (dm)/dt) h_v + Q$$

Energy balance for system (let's say it is liquid and changes into vapor). h_v is the entalpy flow-rate leaving the system (liquid) so it's the entalpy of the vapor. Q is the heat.

(dm)/(dt) is the mass (of vapor) leaving the system. In fact (mass balance) $$(dm)/(dt) = - m_o$$ (the mass can only leave the system)

So:

$$(dH_l)/(dt) = (dh_l m)/(dt) = m(dh_l)/(dt) + h_l(dm)/(dt) ≈ (dm)/(dt) h_v + Q$$
$$m(dh_l)/(dt) + h_l(dm)/(dt) ≈ (dm)/(dt) h_v + Q$$

$$m(dh_l)/(dt) ≈ (dm)/(dt) h_v - h_l(dm)/(dt) + Q = -Δh_ev * (dm)/(dt) + Q$$

Now I have to show that, since dh = cp*dT, the equation gives 0 so that dT/dt = 0 -> T = cost.

I suppose the only way to show this is to say that these $$Δh_ev * (dm)/(dt) + Q$$ are equal because of the definition latent heat.

Is this right, or should I go for an other way?

ps: How to us fractions? It would be mush easier to visualize

Thanks!

 

[Chestermiller]

Are you allowed to use the phase rule?

 

[Tio Barnabe]

ps: How to us fractions?

If you are asking how to call fractions in LaTeX, then just type \frac{A}{B}, where A goes into the numerator and B goes into the denominator. Here is what shows up: $\frac{A}{B}$.

 

[dRic2]

Are you allowed to use the phase rule?

I think no. I mean, the course is "heat transfer" so I suppose I have to use energy balance equation. And by the way I think Gibb's rule is derived by the sperimental evidence that a one-component system's "state" is determined by defining 2 variables (like T and P) so I don't think I can use it here. Sorry for my English, I'm trying to translate and I hope I used the correct words.

If you are asking how to call fractions in LaTeX, then just type \frac{A}{B}, where A goes into the numerator and B goes into the denominator. Here is what shows up: $\frac{A}{B}$.

is there no other way? it takes to much to write frac{}{} every time... :(

 

[Tio Barnabe]

is there no other way? it takes to much to write frac{}{} every time... :(

Unfortunately, no.

 

[Chestermiller]

I think no. I mean, the course is "heat transfer" so I suppose I have to use energy balance equation. And by the way I think Gibb's rule is derived by the sperimental evidence that a one-component system's "state" is determined by defining 2 variables (like T and P) so I don't think I can use it here. Sorry for my English, I'm trying to translate and I hope I used the correct words.

Not if there are 2 phases.

 

[dRic2]

Not if there are 2 phases.

Yes, I know. I said it's derived. I know the rule for a generic system with N components and F phases, but the "first" case (with F = 1 and N = 1) can not be stated other than by evidence. Anyway, suppose I can not use it. Any other suggestions?

 

[Chestermiller]

Yes, I know. I said it's derived. I know the rule for a generic system with N components and F phases, but the "first" case (with F = 1 and N = 1) can not be stated other than by evidence. Anyway, suppose I can not use it. Any other suggestions?

Model the heat capacity by including a Dirac delta function in temperature at the phase transition. Of course, all this does is cause the enthalpy to undergo a discontinuous change equal to the heat of vaporization at the phase transition, while the temperature remains constant. So it doesn't really prove anything.

 

[dRic2]

Model the heat capacity by including a Dirac delta function in temperature at the phase transition. Of course, all this does is cause the enthalpy to undergo a discontinuous change equal to the heat of vaporization at the phase transition, while the temperature remains constant. So it doesn't really prove anything.

Mhm sounds too much... Can you tell my, by the way, what's wrong with my original thinking?