# Heat transfer conduction – problems and solutions

1. Two metals have the same size but different type. The thermal conductivity of P = 2 times the thermal conductivity of Q. What is the temperature between the two metals, as shown in the figure below.

__Known :__

k_{Q }= k

k_{P }= 2k

__Wanted:__ Temperature between the two metals

__Solution :__

The equation of the heat conduction :

Q/t = The rate of the heat conduction, k = thermal conductivity, A = the cross-sectional area of the object, T_{1 }= high temperature, T_{2} = low temperature, l = the length of metal.

Temperature = 60 ^{o}C.

2. Metal A and metal B have the same length and the same cross-sectional. If the thermal conductivity of metal A = ¼ times the thermal conductivity of metal B. Both metals heated at its end and the change in temperature of both metals are the same. The ratio of the rate of the heat conduction of metal A to metal B.

__Known :__

The cross-sectional of A (A) = The cross-sectional B (A)

The length of A (l) = the length of B (l)

The thermal conductivity of metal B (k_{B}) = k

The thermal conductivity of metal A (k_{A}) = ¼ k

The change in temperature of metal A (ΔT) = the change in temperature of metal B (ΔT)

__Wanted:__ The ratio of the rate of the heat conduction

__Solution :__

The equation of the heat conduction :

*Q/t = **the rate of the heat conduction**, k = **thermal conductivity**, A = **the cross-sectional area**, T*_{2}* = **high temperature**, T*_{1}* = **low temperature**, l = **length of metal*

3. Two metals, A and B, have the same size. The thermal conduction of the metal A = 2k and the thermal conduction of the metal B = k.

As shown in the figure below, the end of both metals has the different temperature.

What is the temperature between both metals?

__Known :__

Two metals, A and B, have the same size.

The thermal conductivity of metal A = 2k

The thermal conductivity of metal B = k

The temperature of one end of metal A = 210^{o}C

The temperature of one end of metal B = 30^{o}C

__Wanted:__ Temperature between metal A and B

__Solution :__

The equation of the rate of the heat conduction :

*Q/t = **the rate of the heat conduction**, k = **the thermal conductivity**, A = **the cross-sectional area**, T*_{1}*-T*_{2}* = **the change in temperature**, l = **length of metal*

The temperature between P and Q :

4. Two metals have the same size. The thermal conductivity of metal II = 2 times the thermal conductivity of metal I. If one end of metal I is heated so T_{1 }= 100 ^{o}C and one end of metal II is heated so T_{2} = 25 ^{o}C , then what is the temperature between both metals.

__Known :__

The heat conductivity of metal I = k

The heat conductivity of metal II = 2k

The temperature at one end of metal I = 100^{o}C

The temperature at one end of metal II = 25^{o}C

__Wanted:__ Temperature between metal I and II

__Solution :__

The equation of the heat conduction :

*Q/t = **the rate of the heat conduction**, k = **the thermal conductivity**, A = **the cross-sectional area**, T*_{1}*-T*_{2}* = **the change in temperature**, l **= the length of metal*

Temperature between both metals :

Temperature between metal I and II is 50^{o}C.

5. Metal rods of the same size, but made of different metals are combined as shown in the figure below. If the thermal conductivity of metal I = 4 times the metal conductivity II, then the temperature at the junction of the two metals is ……

__Known :__

The thermal conductivity of metal I = 4k

The thermal conductivity of metal I = k

The temperature of one end of the metal I = 50^{o}C

The temperature of one end of the metal II = 0^{o}C

__Wanted:__ the temperature at the junction of the two metals

__Solution :__

The equation of the heat transfer conduction :

*Q/t = **the rate of the heat conduction**, k = **thermal conductivity**, A = **the cross-sectional area**, T*_{2}* = **high temperature**, T*_{1}* = **low temperature**, **T*_{1}*-T*_{2}* = **The change in temperature, **l = **length of metal*

Both metals have the same size so that A and I eliminated from the equation :

6. The following figure shows different A and B metal rods connected at one end.

The cross-sectional area of both rods is the same, but the length of A is twice the length of B and the thermal conduction coefficient A is 3 times B. If the free ends A and B are subjected to different temperature, the temperature at the junction is …

__Known :__

The thermal conductivity of metal A = 3k

The thermal conductivity of metal B = k

Length of metal A = 2l

Length of metal B = l

The temperature of one end metal A = 100^{o}C

The temperature of one end metal B = 40^{o}C

__Wanted:__ The temperature at the junction

__Solution :__

The equation of the heat transfer conduction :

*Q/t = **the rate of the heat conduction**, k = **thermal conductivity**, A = **the cross-sectional area**, T*_{2}* = **high temperature**, T*_{1}* = **low temperature**, **T*_{1}*-T*_{2}* = **The change in temperature, **l = **length of metal*

Both rods have the same size so that A eliminated from the equation.