on a cold winter day, the atmospheric

temp.is t which isbelow 0degree

celcius.acylindrical drum of height h

made of bad conductoris completely

filled with water at0 degree celcius&is

kept outside without any lid.calculate

the tim e taken by the whole of water 2

freezethermal conductuivity of ice is k

&latent heat of fusion is l.neglect the

expansion of water on freezing.

my approach
q/t=ka(t)/h wher t is temp difference
qneeded 2 freeze wateris=ml=d*vol. of

cylinder*l where l islatent heat of

fusion d=density of water

time=qh/ka t
my answer came 2 b dlh(square)/kt where as the answer in book is mine answer/2
what's the fallacy in my approach

To tackle the problem of how long it takes for the water in a cylindrical drum to freeze on a cold winter day, let's break down the concepts and calculations step by step. Your approach is on the right track, but there are some nuances in the heat transfer process that need to be addressed. Let's clarify the situation and identify where the discrepancy arises.

Understanding the Problem

We have a cylindrical drum filled with water at 0 degrees Celsius, and it's exposed to an atmospheric temperature below 0 degrees Celsius. The goal is to calculate the time required for the entire volume of water to freeze. The key factors involved are:

• The thermal conductivity of ice (k)
• The latent heat of fusion (l)
• The density of water (d)
• The dimensions of the cylinder (height h and radius r)

Heat Transfer Mechanism

When the water is exposed to the cold air, heat is lost from the water to the surrounding environment. The rate of heat transfer can be described by Fourier's law of heat conduction, which you correctly noted as:

Q/t = kA(T)

Here, A is the surface area through which heat is being transferred, and T is the temperature difference between the water and the surrounding air. For a cylindrical drum, the surface area A can be calculated as:

A = 2πr(h + r)

where r is the radius and h is the height of the cylinder.

Calculating the Total Heat Required

The total heat required to freeze the water can be expressed as:

Q = m * l

where m is the mass of the water, which can be calculated as:

m = d * V

For a cylinder, the volume V is:

V = πr²h

Thus, the total heat required becomes:

Q = d * πr²h * l

Time Calculation

Now, substituting Q into the heat transfer equation gives us:

Q/t = kA(T)

Rearranging for time t, we have:

t = Q / (kA(T))

Substituting the expressions for Q and A, we get:

t = (d * πr²h * l) / (k * 2πr(h + r)(T))

Notice that the 2π cancels out, simplifying our equation to:

t = (d * r * h * l) / (k * (h + r)(T))

Identifying the Fallacy

Your initial calculation yielded a time expression of dlh²/(kt), which suggests that you might have overlooked the effective area for heat transfer and the temperature difference. The factor of 2 in the book's answer likely arises from a more precise consideration of the heat transfer dynamics, particularly how the temperature gradient affects the freezing process.

Final Thoughts

In summary, the discrepancy in your answer compared to the book's solution likely stems from the assumptions made about heat transfer and the effective area involved. By carefully considering the geometry of the cylinder and the temperature difference, you can arrive at the correct time required for the water to freeze. Always remember to account for the specifics of the physical situation when applying general formulas!