Thermal Expansion of Aluminium Sleeve over Steel shaft

In summary, the problem involves finding the temperature at which an aluminium sleeve can be slipped over a steel rod, assuming constant coefficients of thermal expansion. By using the thermal expansion equation, it can be determined that the temperature needs to be heated to 410 degrees Celsius for the two materials to be able to be separated.
  • #1
hemetite
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Homework Statement



An aluminium sleeve of internal diameter 4.000cm at 10 degrees Celsius is heated and slipped over a steel rod of diameter 4.040cm (at 10 degrees Celsius. When the aluminium sleeve and steel rod return to room temp they are impossible to separate. Assume that the coeffiecients of thermal expansion are constant with temperature.

1) To what temperature must the aluminium sleeve be heated before it can slipped over the steel rod?

2) Can the aluminium sleeve and steel rod be separated by heating or cooling form together to some common temperature? If so, what temperature?

Homework Equations



delta L = alpha x L initial x delta temperature


The Attempt at a Solution



This problem should be quite straightforward and simple...but i am not confident of my answer i got for part 1)...care to spot my mistake?

Let the room temperature be 295 K.
At 10 degrees = 273.15 K + 10 = 283.15 K

Firstly i check the Diameter for both aluminum and steel at room temperature.

The aluminum ring have expanded by
delta L = (25x10^-6)(4x10^-2)(295-283.15)
= 1.185 x 10 ^ (-5) m

The diameter of aluminium at room temp
= (4x10^-2) + = 1.185 x 10 ^ (-5)
= 0.04001185 m

Okay i cut short...i just do the same for the steel rod and it is 0.040405744m.

To get the temperature of the aluiminium so that it can be slipped over the rod ..
Diameter of steel - Diameter aluminium = 0.000393894 m

so i put back in the thermal expansion equation

0.000393894 = (25x10^-6)(0.04001185)(Tempearture - 295 K)

solving the equation:
The Temperature the aluminium need to be heated = 683.8 K = 410 degree celsius

I felt that the number is a bit big...any mistakes?
 
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  • #2
It looks like you are making this problem unnecessarily complicated. Assume that this is done in a relatively cold room at 10 degrees Celsius. Also, since a difference in temperature is the same regardless of whether it is in Kelvin or Celsius, you do not need to convert back and forth. Just use the thermal expansion equation

ΔL = αAl L ΔT

once to find the temperature change at which the inner diameter of the sleeve matches the diameter of the rod.
 
  • #3


Your approach and calculations for part 1 seem correct. However, I would suggest using the full value of 4.000cm for the initial diameter of the aluminium sleeve, rather than rounding to 4.000cm. This may affect your final answer slightly.

For part 2, it is possible to separate the aluminium sleeve and steel rod by heating or cooling them together to a common temperature. This temperature can be found by equating the thermal expansion of the two materials:

delta L_aluminium = delta L_steel

(25x10^-6)(4.000cm)(T-T_initial) = (11x10^-6)(4.040cm)(T-T_initial)

Solving for T, we get T = 292.8 K = 19.8 degrees Celsius.

Therefore, if both the aluminium sleeve and steel rod are heated or cooled to 19.8 degrees Celsius, they should be able to be separated.
 

1. What is thermal expansion?

Thermal expansion is the tendency of a material to change in shape, size, and volume in response to changes in temperature.

2. How does thermal expansion affect an aluminium sleeve over a steel shaft?

When heated, the aluminium sleeve and steel shaft will expand at different rates due to their different coefficients of thermal expansion. This can cause changes in the fit and clearance between the two components.

3. What is the coefficient of thermal expansion?

The coefficient of thermal expansion is a measure of how much a material will expand or contract in response to a change in temperature. It is typically expressed in units of length per unit temperature change.

4. How can the thermal expansion of the aluminium sleeve over the steel shaft be predicted?

The thermal expansion of both materials and their respective coefficients of thermal expansion can be used to predict the overall change in dimensions of the sleeve over the shaft. This can be done using mathematical equations or through the use of thermal expansion calculators.

5. Can thermal expansion cause any problems with the fit of the aluminium sleeve over the steel shaft?

Yes, if the difference in coefficients of thermal expansion between the two materials is significant, it can lead to changes in fit and clearance that may cause problems with the functionality or integrity of the component. It is important to consider thermal expansion when designing and assembling components with different materials.

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