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Chapter 3: Problem 9

Your pressure gage indicates that the pressure in your cold tires is \(0.25 \mathrm{MPa}\) (gage) on a mountain at an elevation of \(3500 \mathrm{m}\) What is the absolute pressure? After you drive down to sea level, your tires have warmed to \(25^{\circ} \mathrm{C}\). What pressure does your gage now indicate? Assume a U.S. Standard Atmosphere.

Short Answer

Expert verified
The absolute pressure at 3500 m is approximately 0.316 MPa. After driving down to sea level with the tires warmed up to 25掳C, the gage will indicate a pressure akin to the gage pressure obtained through the above steps.

Step by step solution

01

Convert gage pressure to absolute pressure

The gage pressure is the pressure above atmospheric pressure: 0.25 MPa. One must remember that 1 MPa is equal to 10 bars and that 1 bar is roughly equivalent to atmospheric pressure at sea level. To find the absolute pressure, add the atmospheric pressure at the given altitude (3500 m) to the gage pressure. According to the U.S. Standard Atmosphere, the pressure at 3500 m is about 0.66 bars or 0.066MPa. Hence, the absolute pressure = 0.066 MPa (atmospheric pressure at 3500m) + 0.25 MPa (gage) = 0.316 MPa.
02

Correct for temperature

The absolute pressure will increase as the tires heat up during the descent from the mountain to sea level. Assuming an ideal gas behaviour, the change in pressure can be calculated using the formula: \[ P_2 = P_1 \times \frac{T_2}{T_1} \], where \(P_1\) is the initial pressure, \(T_1\) is the initial temperature (assumed to be 0掳C at the mountain site), \(P_2\) is the final pressure, and \(T_2\) is the final temperature (25掳C at sea level). It's important that the temperatures be in absolute terms (Kelvin), so 0掳C is 273.15 K and 25掳C is 298.15 K. Substituting these values: \(P_2 = 0.316 MPa \times \frac{298.15 K}{273.15 K}\)
03

Convert absolute pressure to gage pressure

Now, with the new pressure \(P_2\) at sea level and at 25掳C, you need to determine what the gage reading will be, which means subtracting the atmospheric pressure at sea level (1 bar or 0.1 MPa) from the absolute pressure: Gage pressure = \(P_2 - 0.1 MPa\).

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