Heat Flow in Three Identical Rods of Metal

[zorro]

Homework Statement

Three rods of identical cross-sectional area are made from the same metal, form the sides of an isosceles triangle ABC right angled at B. The points A and B are maintained at temperature T and √2 T respectively in steady state. Assume that only heat conduction takes place. Then

a) Rate of heat flow in BA is equal to rate of heat flow in CA

b) Temperature gradient across CA is equal to the temperature gradient across BA

The Attempt at a Solution

I got the temperature at C as 3T/1+√2.

Temperature at B > Temperature at A.
Rate of flow in BA will be equal to the rate of heat flow in BCA (not CA)
Again temperature gradient will be same across BCA

Both a and b are correct. I don't understand how. Help.

 

[AC130Nav]

Homework Statement

Three rods of identical cross-sectional area are made from the same metal, form the sides of an isosceles triangle ABC right angled at B.

How can isosceles triangle ABC be right angled at B?

 

[rl.bhat]

Your calculation of temperature at C is wrong.
The two statements are true for BA and BC, rather BA and CA.
Check the problem.

 

[zorro]

How can isosceles triangle ABC be right angled at B?

Take a look at the figure.

Your calculation of temperature at C is wrong.

I found the temperature at C by equating the rate of heat flow through BC and CA.
I got the same answer after checking.

 

[rwisz]

I would first acknowledge that the statements a and b are indeed correct, based on the given information. I would then explain the reasons behind these statements using principles of heat transfer and thermodynamics.

Firstly, the rate of heat flow in a material is directly proportional to the temperature difference across that material. In this case, both BA and CA are made of the same metal and have the same cross-sectional area, therefore the only difference in temperature is between points A and B. Since the temperature at point B is √2 times higher than at point A, the rate of heat flow in BA will also be √2 times higher than in CA, making the rates of heat flow in both directions equal.

Secondly, the temperature gradient across a material is defined as the change in temperature per unit length. In this case, both BA and CA have the same length and are made of the same metal, therefore the temperature gradient will be the same for both rods. This means that the temperature difference between points A and B will be evenly distributed across both rods, resulting in the same temperature gradient across both BA and CA.

In conclusion, the principles of heat transfer and thermodynamics explain why the rate of heat flow and the temperature gradient across BA and CA are equal in this scenario. This also highlights the importance of understanding the properties of materials and how they affect heat transfer processes.