If I am heating water in an open container with constant rate of heat supply (q) (water initially at 30°C ) until entire water converts to steam, the temperature (T) vs time (t) graph in sensible heating region must be non-linear with increasing slope because q (dt)= mc (dT) so (dT/dt) must increase with time as mass of water left in container decreases with increase of temperature because rate of evaporation increases with increase in temperature.
But in most of the textbooks it is drawn linear in sensible heating region. I am not able to understand where am I wrong?
And if I draw mass of water left vs time graph then in sensible region, the rate of decrease of water increases with temperature because rate of evaporation increases so the graph in this region must be nonlinear but in latent heating region, the mass must be decreasing with constant slope as temperature is not changing. Any suggestions where am I wrong?
Best Answer
Let the heat energy balance be: $$Q=Q_h+Q_e\tag{1}$$ where $h$ stands for 'heating up' and $e$ for 'evaporation'.
The mass balance is: $$m=m_0-m_e\tag{2}$$
where $_0$ stand for initial mass.
Differentiate $(2)$: $$\mathrm{d}m=-\mathrm{d}m_e$$ Differentiate $(1)$: $$0=\mathrm{d}Q_h+\mathrm{d}Q_e$$ Use: $$\mathrm{d}Q_h=-\mathrm{d}m_ec_p\frac{\mathrm{d}T}{\mathrm{d}t}$$
Insert:
$$-\mathrm{d}m_ec_p\frac{\mathrm{d}T}{\mathrm{d}t}+\alpha L_e\mathrm{dm}_e=0$$ where $\alpha$ is a constant and $L_e$ the latent heat of vapourisation. $$-c_p\frac{\mathrm{d}T}{\mathrm{d}t}+\alpha L_e=0$$
Integrate: $$(T_2-T_1)=\frac{\alpha L_e}{c_p}t$$
Which strongly suggest linearity (up to the boiling point)