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 :Heat transfer conduction – problems and solutions 1

kQ = k

kP = 2k

Wanted: Temperature between the two metals

Solution :

The equation of the heat conduction :

Heat transfer conduction – problems and solutions 2

Q/t = The rate of the heat conduction, k = thermal conductivity, A = the cross-sectional area of the object, T1 = high temperature, T2 = low temperature, l = the length of metal.

Heat transfer conduction – problems and solutions 3

Temperature = 60 oC.

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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 (kB) = k

The thermal conductivity of metal A (kA) = 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 :

Heat transfer conduction – problems and solutions 4

Q/t = the rate of the heat conduction, k = thermal conductivity, A = the cross-sectional area, T2 = high temperature, T1 = low temperature, l = length of metal

Heat transfer conduction – problems and solutions 5

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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.

Heat transfer conduction – problems and solutions 7

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

Heat transfer conduction – problems and solutions 8

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 = 210oC

The temperature of one end of metal B = 30oC

Wanted: Temperature between metal A and B

Solution :

The equation of the rate of the heat conduction :

Heat transfer conduction – problems and solutions 11

Q/t = the rate of the heat conduction, k = the thermal conductivity, A = the cross-sectional area, T1-T2 = the change in temperature, l = length of metal

The temperature between P and Q :

Heat transfer conduction – problems and solutions 9

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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 T1 = 100 oC and one end of metal II is heated so T2 = 25 oC , then what is the temperature between both metals.

Known :

The heat conductivity of metal I = kHeat transfer conduction – problems and solutions 10

The heat conductivity of metal II = 2k

The temperature at one end of metal I = 100oC

The temperature at one end of metal II = 25oC

Wanted: Temperature between metal I and II

Solution :

The equation of the heat conduction :

Heat transfer conduction – problems and solutions 12

Q/t = the rate of the heat conduction, k = the thermal conductivity, A = the cross-sectional area, T1-T2 = the change in temperature, l = the length of metal

Temperature between both metals :

Temperature between metal I and II is 50oC.

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 = 4kHeat transfer conduction – problems and solutions 11

The thermal conductivity of metal I = k

The temperature of one end of the metal I = 50oC

The temperature of one end of the metal II = 0oC

Wanted: the temperature at the junction of the two metals

Solution :

The equation of the heat transfer conduction :

Heat transfer conduction – problems and solutions 12

Q/t = the rate of the heat conduction, k = thermal conductivity, A = the cross-sectional area, T2 = high temperature, T1 = low temperature, T1-T2 = The change in temperature, l = length of metal

Heat transfer conduction – problems and solutions 13

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

Heat transfer conduction – problems and solutions 14

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 = 3kHeat transfer conduction – problems and solutions 15

The thermal conductivity of metal B = k

Length of metal A = 2l

Length of metal B = l

The temperature of one end metal A = 100oC

The temperature of one end metal B = 40oC

Wanted: The temperature at the junction

Solution :

The equation of the heat transfer conduction :

Heat transfer conduction – problems and solutions 16

Q/t = the rate of the heat conduction, k = thermal conductivity, A = the cross-sectional area, T2 = high temperature, T1 = low temperature, T1-T2 = The change in temperature, l = length of metal

Heat transfer conduction – problems and solutions 17

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

Heat transfer conduction – problems and solutions 18