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.

**What is heat transfer by conduction?****Answer**: Conduction is the transfer of heat through a material without any movement of the material itself. It occurs when heat is passed from one particle of the substance to another adjacent particle through molecular vibration or electron movement.

**How does the nature of a material affect its conductive properties?****Answer**: Different materials have varying abilities to conduct heat, known as their thermal conductivity. Materials with high thermal conductivity, like metals, transfer heat quickly, while those with low thermal conductivity, like wood or rubber, transfer heat slowly.

**Why do metals generally conduct heat better than non-metals?****Answer**: Metals have a lattice of closely packed atoms with free-moving electrons. These free electrons can move rapidly through the metal and transfer energy efficiently from the hotter to cooler regions, making metals good conductors of heat.

**How does temperature difference influence the rate of heat conduction?****Answer**: The greater the temperature difference between two ends of a material, the faster the rate of heat transfer by conduction. Heat always flows from the hotter end to the cooler end until equilibrium is reached.

**What is the relationship between the cross-sectional area of a material and the rate of heat conduction?****Answer**: The rate of heat conduction is directly proportional to the cross-sectional area of the material. This means if you double the cross-sectional area, the rate of heat conduction will also double, given other factors remain constant.

**How does the thickness or length of a material affect heat conduction?****Answer**: The rate of heat conduction is inversely proportional to the thickness or length of the material. That is, the thicker or longer the material, the slower the rate of heat conduction and vice versa.

**What is “thermal resistance”?****Answer**: Thermal resistance is a measure of a material’s resistance to the flow of heat. It is the reciprocal of thermal conductivity. Materials with high thermal resistance are good insulators as they oppose the transfer of heat through them.

**How do composite materials, like walls with insulation, affect heat conduction?****Answer**: Composite materials or layered structures combine the properties of their individual components. A wall with insulation, for example, uses the insulating properties of the insulation layer to reduce the rate of heat conduction, thereby retaining heat inside a room or keeping external heat out.

**Why do objects feel cold or hot to touch even though they are at room temperature?****Answer**: The sensation of cold or hot when touching an object is related to its thermal conductivity. Objects with high thermal conductivity, like metals, rapidly draw heat from our skin, making them feel cold, whereas objects with low thermal conductivity do not transfer heat as effectively, and thus might feel warmer in comparison.

**How does the presence of impurities or defects in a material impact its conductive properties?****Answer**: Impurities or defects can disrupt the regular arrangement of particles in a material, which can hinder the efficient transfer of heat through it. Depending on the nature and distribution of the impurities or defects, the material’s thermal conductivity may decrease, making it a poorer conductor of heat.