The time in which a layer of ice of thickness h will grow on...

The time in which a layer of ice of thickness $h$ will grow on the surface of the pond of surface area $A$, when the surrounding temperature falls to $-T^{\circ} \mathrm{C}$ is : (Assume $K=$ thermal conductivity of ice, $\rho=$ density fo water, $L=$ latent heat of fusion) (a) $t=\frac{p L}{2 K T} h^{2}$ (b) $t=\frac{\rho L}{K T} h^{2}$ (c) $t:=\frac{\rho L h^{2}}{3 K T}$ (d) $t=\frac{\rho L h^{2}}{4 K T}$

The time in which a layer of ice of thickness $h$ will grow on the surface of the pond of surface area $A$, when the surrounding temperature falls to $-T^{\circ} \mathrm{C}$ is :
(Assume $K=$ thermal conductivity of ice, $\rho=$ density fo water, $L=$ latent heat of fusion)
(a) $t=\frac{p L}{2 K T} h^{2}$
(b) $t=\frac{\rho L}{K T} h^{2}$
(c) $t:=\frac{\rho L h^{2}}{3 K T}$
(d) $t=\frac{\rho L h^{2}}{4 K T}$

Key Concepts

Energy Balance Approach

The energy balance approach involves equating the rate at which heat is removed from the system via conduction to the energy needed to freeze the water, as dictated by the latent heat of fusion. This balance is used to derive relationships between the growth of the ice layer over time and physical parameters such as thermal conductivity, latent heat, and the temperature difference driving the heat flow.

Heat Conduction

Heat conduction is the process by which thermal energy is transmitted through a material from regions of higher temperature to regions of lower temperature. In the context of forming an ice layer, the conduction through the ice determines the rate at which heat is extracted from the water beneath. This process is governed by Fourier's Law, which relates the heat flux to the temperature gradient and the thermal conductivity of the material.

Latent Heat of Fusion

The latent heat of fusion is the amount of energy per unit mass that must be removed from a substance to change its phase from liquid to solid at its melting/freezing point without a change in temperature. In ice formation, this energy must be extracted from the water in order for it to freeze, linking the energy required for the phase change to the rate of heat removal by conduction.

Thermal Conductivity

Thermal conductivity is a property of a material that indicates its ability to conduct heat. In the context of ice formation, the thermal conductivity of ice determines how effectively the latent heat is conducted away from the freezing interface. Higher thermal conductivity facilitates quicker removal of heat, thereby affecting the rate at which ice can grow.

Transcript

Please give Ace some feedback