Question

A silver rod having a length of 83.0 cm and a cross-sectional diameter of 2.40 cm is used to conduct heat from a reservoir at a temperature of 540 oC into an otherwise completely thermally insulated chamber that contains 1.43 kg of ice at 0 oC. How much time is required for the ice to melt completely

Answer 1

3985 s or 66.42 mins

Step-by-step explanation:

Given:-

- The length of the rod, L = 83.0 cm

- The cross sectional diameter of rod , d = 2.4 cm

- The temperature of reservoir, Tr = 540°C

- The amount of ice in chamber, m = 1.43 kg

- The temperature of ice, Ti = 0°C

- Thermal conductivity of silver, k = 406 W / m.K

- The latent heat of fusion of water, Lf = 3.33 * 10^5 J / kg

Find:-

How much time is required for the ice to melt completely

Solution:-

- We will first determine the amount of heat ( Q ) required to melt 1.43 kg of ice.

- The heat required would be used as latent heat for which we require the latent heat of fusion of ice ( Lf ). We will employ the first law of thermodynamics assuming no heat is lost from the chamber ( perfectly insulated ):

`Q = m*L_f\\\\Q = ( 1.43 ) * ( 3.33 * 10 ^5 )\\\\Q = 476190 J`

- The heat is supplied from the hot reservoir at the temperature of 540°C by conduction through the silver rod.

- We will assume that the heat transfer through the silver rod is one dimensional i.e along the length ( L ) of the rod.

- We will employ the ( heat equation ) to determine the rate of heat transfer through the rod as follows:

`(dQ)/(dt) = (k.A.dT)/(dx)`

Where,

A: the cross sectional area of the rod

dT: The temperature difference at the two ends of the rod

dx: The differential element along the length of rod ( 1 - D )

t: Time ( s )

- The integrated form of the heat equation is expressed as:

`Q = (k*A*( T_r - T_i))/(L)*t`

- Plug in the respective parameters in the equation above and solve for time ( t ):

`476190 = (406*\pi*0.024^2 * ( 540 - 0 ) )/(0.83*4)*t \\\\t = (476190)/(119.49619) \\\\t = 3985 s = 66.42 mins`

Answer: It would take 66.42 minutes to completely melt the ice