Transient heating with heat loss

Question $10\mathrm{.}1$

An electrical heater is used to heat $1$ litre of water from $14C$ to the boiling temperature at atmospheric

pressure.

$($c$)$ Derive an expression for the water temperature as a function of time

$($d$)$ If heater delivers $1000W$ to the water, how long does it take until the boiling temperature is reached?

Without neglecting heat losses to the environment, and the heat capacities of the heating element and

the container.

Data:

Mass of heater and container $=m_k=400g$

Specific heat capacity of heater and container $=C_{pk}=0\mathrm{.}5kJk{g}^{-1}{K}^{-1}$

Specific heat capacity of water $=C_{pw}=4200Jk{g}^{-1}{K}^{-1}$

Surface area of container $=A=0\mathrm{.}1m^2$

Ambient air temperature $=T_{}=20C$

Overall heat transfer coefficient $(U_{\text{l}\text{o}\text{s}\text{s}})=10W{m}^{-2}{K}^{-1}$