Modeling Time to Reach Specific Pressure in Adiabatic Tire Inflation

[tashkent]

Homework Statement

Hi, I need a big help on how to carry on with this problem:

You witnessed the failure of a nylon rim which exploded injuring the person filling the tire with air. Develop a documented mathematical model that predicts the time required to reach a specific pressure. Your results are to be tabulated and plotted. Specifically predict the time to change from 100 kPa to 800 kPa. Assume the process is adiabatic and that the tire volume may be represented by a Torus (doughnut). In order work this problem you have to solve for pressure by iterating forwards in time.

Given:

Tire size 25 cm x 8 cm (outside diameter x width)
Rim diameter 11 cm
Air line pressure 1380 kPa
Initial tire pressure 100 kPa
Airline pipe area 32 mm2
Valve orifice area 6 mm2
Initial tire temp 16°C

Homework Equations

Volume of torus = pi^2/4 D d^2

Area = pi^2 D d

The Attempt at a Solution

I already got confused with what D is. Id D (25 + 11) = 36 inches or (25 - 11) = 14 inches? Then d is 16 inches because of 8 + 8 inches. HELP!

q = 0 since its adiabatic. But I can't find a formula that will harmonize pressure with time...

 

[Astronuc]

One needs to show more work.

Relate the dimensions D and d of a torus with those given for the tire. They may or may not be on the same basis. Most torus formulae would use the mean radius from central axis of rotation to the center of the toroidal chamber, and the minor diameter is the diameter of the circular cross-section of the toroidal chamber.

Does the tire expand? If so, some elastic energy will be stored in the tirewall.

One can increase pressure by adding mass (moles) of gas and/or increasing temperature. Adiabatic means no heat transfer across the boundary. What happens to the work done on the gas to compress it?

 

[tashkent]

Hello, astronuc.

I already got the volume of the torus: V = pi^2/4 D d^2 where D is 19 cm and d is 8 cm. V is then 375 cm3.

I am assuming the tire will expand. But the big problem here is finding a math model to incorporate all the variables. These requires knowledge of thermodynamics and fluid mechanics. I'm sorry, I'm not a Mech'l. Engg. major. These students will excel in this problem.

I have a typo: air line pressure should be 1380 kPa. But I need to do time iteration to solve for pressure. I am having a hard time on figuring out what formula to use. This is not ideal gas law, its more complicated than that. Need help.

 

[Astronuc]

Thinking about this problem, I realized that the tire tube cross section may not be the stardard circular cross-section. The tire is 8 cm wide by 25 cm OD and 11 cm ID. So it may be more square. The volume can be found by the Theorem of Pappus, in which the volume is mean circumference (which is 2 pi * mean radius) times the area being rotated around an axis.

In this case, it is 2 pi R * A = pi D * A, where D is the average of the OD and ID, and A = 8 cm * (OD - ID)/2.

In order to calculate time, one has to calculate the rate of pressure change dP(t)/dt of the tire as the pressure increases to 800 kPa. From the ideal gas law, PV = nRT, the pressure will change (increase) as n increases, so one has to find n(t), and that depends on the mass flow rate, which itself depends in the instantaneous differential pressure between the line pressure P_l and the tire pressure P(t). One may assume the line pressure is constant.

See if one can develop the solution.

There should be a pressure drop coefficient for the area reduction associated with the valve and line.

 

[tashkent]

Hello astronuc.

I don't know what mass flow rate equation to use to utilize the areas of the valve orifice and the airline pipe...

 

[Astronuc]

Hello tashkent,

What text is one using for this coursework? Or what lecture notes does one have that would provide some idea of flows.