Question

In an electrically heated boiler, water is boiled at 140°C by a 90 cm long, 8 mm diameter horizontal heating element immersed in water. The heating element is made of mechanically polished stainless steel. The boiler initially contains 0.25 m of water at 20°C. Once boiling starts, it is observed that 25% of the water in the boiler evaporated in 60 min. i. Determine the power rating of the electric heating element immersed in water and the surface temperature of the heating element. [8 marks] ii. Using the above heater, estimate the time required to raise the temperature of 0.25 m2 of cold water from 20°C to 140°C

Answer 1

Step-by-step explanation:

The given data is as follows.

Volume of water = 0.25
`m^(3)`

Density of water = 1000
`kg/m^(3)`

Therefore, mass of water = Density × Volume

=
`1000 kg/m^(3) * 0.25 m^(3)`

= 250 kg

Initial Temperature of water (
`T_(1)`) =
`20^(o)C`

Final temperature of water =
`140^(o)C`

Heat of vaporization of water (
`dH_(v)`) at
`140^(o)C` is 2133 kJ/kg

Specific heat capacity of water = 4.184 kJ/kg/K

As 25% of water got evaporated at its boiling point (
`140^(o)C`) in 60 min.

Therefore, amount of water evaporated = 0.25 × 250 (kg) = 62.5 kg

Heat required to evaporate = Amount of water evapotaed × Heat of vaporization

= 62.5 (kg) × 2133 (kJ/kg)

=
`133.3 * 10^(3)` kJ

All this heat was supplied in 60 min = 60(min) × 60(sec/min) = 3600 sec

Therefore, heat supplied per unit time = Heat required/time =
`(133.3 * 10^(3)kJ)/(3600 s)` = 37 kJ/s or kW

The power rating of electric heating element is 37 kW.

Hence, heat required to raise the temperature from
`20^(o)C` to
`140^(o)C` of 250 kg of water = Mass of water × specific heat capacity × (140 - 20)

= 250 (kg) × 40184 (kJ/kg/K) × (140 - 20) (K)

= 125520 kJ

Time required = Heat required / Power rating

=
`(125520)/(37)`

= 3392 sec

Time required to raise the temperature from
`20^(o)C` to
`140^(o)C` of 0.25
`m^(3)` water is calculated as follows.

`(3392 sec)/(60 sec/min)`

= 56 min

Thus, we can conclude that the time required to raise the temperature is 56 min.