On a winter day when the atmospheric temperature drops to -10^∘ C, ice forms on the surface of a

On a winter day when the atmospheric temperature drops to...

On a winter day when the atmospheric temperature drops to $-10^{\circ} \mathrm{C}$, ice forms on the surface of a lake. (a) Calculate the rate of increase of thickness of the ice when $10 \mathrm{~cm}$ of ice is already formed. (b) Calculate the total time taken in forming $10 \mathrm{~cm}$ of ice. Assume that the temperature of the entire water reaches $0^{\circ} \mathrm{C}$ before the ice starts forming. Density of water $=1000 \mathrm{~kg} \mathrm{~m}^{-3}$, latent heat of fusion of ice $=3.36 \times 10^{5} \mathrm{~J} \mathrm{~kg}^{-1}$ and thermal conductivity of ice $=1.7 \mathrm{~W} \mathrm{~m}^{-1}{ }^{\circ} \mathrm{C}^{-1}$. Neglect the expansion of water on freezing.

On a winter day when the atmospheric temperature drops to $-10^{\circ} \mathrm{C}$, ice forms on the surface of a lake.
(a) Calculate the rate of increase of thickness of the ice when $10 \mathrm{~cm}$ of ice is already formed. (b) Calculate the total time taken in forming $10 \mathrm{~cm}$ of ice. Assume that the temperature of the entire water reaches $0^{\circ} \mathrm{C}$ before the ice starts forming. Density of water $=1000 \mathrm{~kg} \mathrm{~m}^{-3}$, latent heat of fusion of ice $=3.36 \times 10^{5} \mathrm{~J} \mathrm{~kg}^{-1}$ and thermal conductivity of ice $=1.7 \mathrm{~W} \mathrm{~m}^{-1}{ }^{\circ} \mathrm{C}^{-1}$. Neglect the expansion of water on freezing.

Key Concepts

Stefan Problem

The Stefan problem is a class of phase change problems that deals with the moving boundary during processes like freezing and melting. It applies an energy balance that couples conductive heat transfer with latent heat effects to determine the dynamics of the phase change interface, such as the growth rate of ice thickness on a body of water.

Energy Balance at the Phase Change Interface

The energy balance at the phase change interface involves equating the rate of thermal energy removal by conduction through an existing solid layer to the rate of latent heat release during the solidification of water. This balance is crucial for establishing the rate at which the freezing front advances in the formation of ice.

Heat Conduction

Heat conduction is the process of energy transfer within a material or between materials that are in direct contact, driven by a spatial temperature gradient. In the analysis of ice formation, heat conduction governs how thermal energy is removed through the ice layer, influencing the rate at which the underlying water loses heat and freezes.

Latent Heat of Fusion

Latent heat of fusion refers to the energy that must be extracted from or provided to a substance per unit mass to change its phase between solid and liquid at a constant temperature. In the process of water freezing, the removal of this energy is a key factor that determines the speed of the phase change, thereby affecting the overall thickness growth of the ice.

Transcript

00:01 In this question, it is told that on a winter day when temperature drops to minus 10 degrees celsius, ice is formed on the surface of a lake.

00:08 So first of all, i am writing the given data for this question.

00:12 We have given the thermal conductivity of ice that is 1 .7 watt per meter degrees celsius.

00:18 We have given the density of water that is 10 to the power 3 kilograms per meter cube.

00:23 And latent heat of fusion of ice that is alice is given to us 3 .36 into 10 to the power 5.

00:30 Jule per kg and length of ice form that is small l is 10 semi or i can say 10 into 10 to the power minus 2 meter now we in the first part of the question we have to find the rate of increase of thickness of the ice so we know that the rate of flow of heat is given by that is dlq by dlt is equals to this is k a into t1 minus t2 upon l.

01:13 Now dlq is equal to this is mass into latent heat or fusion of ice upon dlt is equal to this is ka t1 minus t2 upon small l.

01:28 Now from here we can say that l upon delta t is equal to this is ka t1 minus t2 upon mass can be written as row of water into area into small l.

01:43 This is mass into this is latent heat of fusion of ice now put all the data in this equation so rate of increase of thickness that is l upon delta t is given by this is 1 .7 into 0 minus this is minus 10 upon this is 10 to power 3 into this area and this area will cancel out this is 10 into 10 to the power minus 2 into 3 .36 into 10 to the power 5 after solver this we get that l upon delta t is equal to this is 5 into 10 to the power minus 7 meter per second so this is the rate of increase of forming of ice and this is our answer of first part now in the second part we have to find the total time taken to form a 10 semi -thick ice so let to form a thin layer of thickness dt to time is required.

03:04 So we can say that mass of thin layer is that is dm is equal to row of water into area into dx...

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