## Monday, January 9, 2017

### Transmission of Heat Problems with Solutions Three

We are solving series of problems on transmission of heat. Here we are talking about conduction, convection and radiation. Each method is little different and radiation is the fastest way of transmission of heat and it does not need any medium for the propagation. To know the magnitude of transfer of heat energy via radiation, we have Stephen’s law. As per it the rate of flow of heat energy is directly proportional to forth power of absolute temperature of the body. According Wien’s displacement law, the wave length corresponding to temperature of maximum heat energy are inversely proportional to the absolute temperature and the their product is constant and it is called Wien’s constant.

Problem

Three identical rods are connected in Y shape as shown in the diagram below. The temperature at each end is given to us and we need to find the temperature of the system at the junction.

Solution

We know that heat flows from a body of higher temperature to a body of lower temperature. Here heat energy starts flowing from 90 degree side to zero temperature. The sum of the heat that flows from two bodies of higher temperature shall be the heat through the third rod. Taking that into consideration and using the definition of thermal conductivity, we can solve the problem as shown in the diagram below.

Problem

One cylinder of less radius is kept in the hallow cylinder of higher radius as shown in the diagram below. The two materials are different and they have different coefficients of thermal conductivity. Each end of the system is at a different temperature and there is no loss of heat energy in the system and we need to find the effective thermal conductivity of the system.

Solution

These two cylinders acts as if like they are connected in parallel. The total flow of the heat is the sum of the heat flows through both the rods. Taking that into consideration and applying the definition, we can solve the problem as shown in the diagram below.

Problem

Three rods of same materiel are and dimensions are connected in the shape of a triangle as shown in the diagram and at each corner temperature is different. We need to find the temperature at the third corner and respective ratio.

Solution

Corner B is being at high temperature heat flows from B to both A and C. The heat that flows from B will first go to C and then the same heat will go through the rod and reach the point A. So the heat flow with the two rods is same. Taking that into consideration, we can solve the problem as shown in the diagram below.

Problem

Six identical conducting rods are connected in the shape given below. Temperature at the beginning and end of the system is given to us and we need to measure temperature at a given junction.

Solution

We can identify that at the junction upper two rods and lower two rods are connected in series and their combination of top and bottom are connected in parallel. Further taking the definition into consideration, we can solve the problem as shown in the diagram below.

Problem

A sphere and cube are made up of same material and they have equal volume. They are heated to the same temperature and allowed to cool in the same surroundings as shown in the diagram below. We need to find the ratio of rate of loss of heat of both of them.

Solution

As the volume of sphere and cube are same, we can find the relation between the radius of the sphere and side of the cube by applying their volume formula as shown in the diagram below. Taking the Stefen’s law into consideration, we can solve the problem as shown in the diagram below.

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