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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