Question

On a winter day when the atmospheric temperature drops to `-10^(@)C` , ice forms on the surface of a lajke. (a) Calculate the rate of increases of thickness of the ice when 10cm of ice is already formed. (b) Calculate the total time taken in forming 10cm of ice. Assume that the temperature of the entire water reaches `0^(@)C` before the ice starts forming. Density of water `=1000kgm^(-3)` , latent heat of fusion of ice `=3.36xx10^(5)Jkg^(-1)` and thermal conductivity of ice `=1.7Wm^(-1)`^(@)C^(-1)` . Neglect the expension of water on freezing.

Answer 1

Correct Answer - A
`A=1.7 W//m-^(@)C`
`(rho_w)=1000kg//m^(3)`
`L_(ice)=3.36xx10^(5) J/kg`
`L=10xx10^(-2) m`.
(a)`(Q)/(T)=(kA(theta_1-theta_2))/(l)`
implies .(Q)/(T)=(kA(theta_1-theta_2))/(Q)` ltbrge(kA(theta_1-theta_2))/(mL)`
`=(kA(theta_1-theta_2))/(Al rho_w L)`
`(1.7xx|(0-10)|)/(10xx1000xx3.36xx10^(5)xx10^(-2))`
`=(17xx10)/(3.36xx10^(7))`
`(17)/(3.36)=5.059xx10^(-7) m/sec`.ltbrge5xx10^(7)m/sec.`
(b) Let us assume taht x length of ice has been formed. To form a small strip of ice of length dx, dt time is required.
` (dQ)/(dt)= (KA(Delta theta ))/x`
`implies (dmL)/(dt)=(KA(Delta theta ))/x`
implies (Adx (rho_w) L)/(dt)=(KA(Delta theta))/x`
`implies (dx (rho_w) L)/(dt)=(K(Delta theta))/x`
implies dt=(xdx (rho_w)L)/(K(Delta theta)`
`implies int_(0)^(1)dt=((rho_w)L)/(K(Delta theta) int_(0)^(1)xdx`
`implies t=((rho_w)L)/(K(Delta theta)[(x^(2))/(2)]_(0)^(1)=((rho_w)L)/(K(Delta theta).1/2`
Putting values
`t=(1000xx3.36xx10^(5)xx(10xx10^(-2))/(1.4xx10xx2)`
`=(3.36)/(2xx17)xx10^(-2)xx10^(5)xx10^(3)xx10`
`=(3.36)/(2xx17)xx10^(7)sec.`
`=(3.36xx10^(7))/(2xx17xx3600) hrs.`
`=27.45 hr=27.5 hrs`.