- #1
Soren4
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- 2
Homework Statement
Consider a cylindrical tank closed by a movable piston with mass ##m=3 kg##. The radius of the cyclinder is ##r=7.5 cm##. In the tank there is a mass ##m'=2 kg## o water at temperature just below ##100°C##. At the base of the cyclindrical tank there is an electrical heater which provides heat to the water with power ##P=100W##. Treat vapor as an ideal gas.
a) When water starts boiling, what is the velocity of raising of the piston?
b) After all the water becomes vapor, what is the velocity of raising of the piston?
Answers : ##\bigg[ v_a=4.19 mm/s \, , \, v_b=12.6 mm/s \bigg]##
Homework Equations
The Attempt at a Solution
I have less problems with part b). In part b) it should just be the isobaric expansion of an ideal gas, so I could write
$$P=\frac{dV}{dt} \frac{c_p}{R} p=\frac{7}{2} \frac{dh}{dt} (3 \cdot 9.81)=\frac{7}{2} v_b (3 \cdot 9.81)\implies v_b=0.97 m/s$$
Which is not the correct result (I suppose vapor biatomic, but also trying with monoatomic does not give the correct answer)
I also tried for point a) but I'm quite unsure of the procedure. In that case I think that
$$P=\lambda \frac{dm}{dt}=\lambda \rho_{vapor} \frac{dV}{dt}=\frac{\lambda p \mathscr{M}}{R T} \frac{dh}{dt} \pi r^2=\frac{\lambda (3 \cdot 9.81) \mathscr{M}}{R T}v_a \implies v_a=0.26 m/s$$
Where ##\frac{dm}{dt}## is the mass of water per unit time that becomes vapor, ##\mathscr{M}## is the molecular mass of water and ##\lambda## is the enthalpy of vaporization .
Also in this case the result is not correct.
Could you highlight the conceptual mistakes in what I have tried for this exercise?
Thanks a lot in advice