Thermodynamics: h vs u and Quality?

[swraman]

Hi,

I am trying to understand the solution to a quiz I had, and am having trouble.

The question was:

Two well-insulated rigid tanks are connected by a valve. Tank A contains 5 kg of superheated steam at 800oC and 800kPa. Tank B contains 1 kg of saturated water mixture at 150oC and 30% quality. The valve is opened and the two tanks eventually come to thermodynamic equilibrium. Perform a thermodynamics analysis based on conservation of mass and energy to determine if there is any liquid in the final state.

The solution says:

KE=PE=W=Q=0
Conservation of Energy: Ei-Ef=ΔEsys=0 Therefore Ei=Ef. This becomes Ui=Uf or MA*UA+MB*UB=Mf*Uf

Tank A: VA=MA*vA=5kg x 0.618 m3/kg =3.09 m3
UA = 5kg x 3662.5 kJ/kg =18312.5 kJ

Tank B: VB=MB*vB=1kg x(0.3* 0.3924 m3/kg+0.7* 0.00109 m3/kg)=0.1184 m3
UB = 0.3x 2559.1 kJ/kg +0.7 *631.66 kJ/kg =1209.89 kJ

Total volume at equilibrium = 3.208 m3
Total mass= 6 kg

Specific volume= 3.208m3/6kg=0.5346 m3/kg

Total internal energy= 19522.39 kJ specific energy=3253.7 kJ/kg
(v, u) states falls into the superheated regime; there will be no liquid in the final state.

First off, why (when they solve for internal energy and specific volume) do thy use the formula (Quality)*Uvapor + (1-quality)*Uliquid ? This makes sense to me in my head, but it says no our text that the generic equation for u,h,or v is:
Y = Yliq + (Quality)*Ysat.vapor

so why don't they use it in this scenario?

Second, why do they use internal energy and not enthalpy?

Thanks

--a lost Mechanical engineering student

 

[Q_Goest]

First off, why (when they solve for internal energy and specific volume) do thy use the formula (Quality)*Uvapor + (1-quality)*Uliquid ? This makes sense to me in my head, but it says no our text that the generic equation for u,h,or v is:
Y = Yliq + (Quality)*Ysat.vapor

so why don't they use it in this scenario?

Second, why do they use internal energy and not enthalpy?

Thanks

--a lost Mechanical engineering student

The equation you're trying to write: Y = Yliq + (Quality)*Ysat.vapor

... is off just a bit. The correct equation is: Y = Y_f + X Y_fg
Where Y_f is the fluid property at 100% saturated liquid
and
Y_fg is the difference between the fluid property at 100% saturated gas and 100% saturated liquid.

Regarding why not enthalpy, if you rewrite the first law it helps to clear this up. Normally, we write:
dU = dQ + dW

But for any control volume with flow in and out, we can also rewrite it:
dU = dQ + dW + dH
where
dH = (U_in + pV_in) - (U_out + pV_out)
Note that gas flowing into or out of a control volume does work pV so we say H = U + pV
(See ref: http://web.mit.edu/16.unified/www/FALL/thermodynamics/chapter_4.htm )

So if we apply this to the system with two separated tanks that has a control surface around both tanks, we find no heat or work crossing the control surface (dQ=0 and dW=0), nor do we find any flow across the control surface, so dH = 0. All we're left with is dU = 0 (or U1 + U2 = U3).