Problem 2 - Biofluid Mechanics: An Introduction to Fluid Mechanics, Macrocirculation, and Microcirculation

Question

NASA is planning a mission to a newly found planet and will monitor the density of the new planet's atmosphere. Assume that NASA knows that atmosphere behaves as an ideal gas and that the planet's gravitational force is a function of altitude \(\left(g(z)=\frac{18.7 m}{s^{2}\left(1-\frac{z}{10,000 m}\right)}\right)\), where \(z\) is in m). The temperature of the atmosphere is constant at \(250 \mathrm{~K}\), and the gas constant is \(340 \mathrm{Nm} / \mathrm{kg} \mathrm{K}\). Assume that the pressure at the planet's surface is \(2 \mathrm{~atm}\). Calculate the pressure and density at an altitude of 1,5 , and \(9 \mathrm{~km}\).

Step-by-Step Solution

Answer

Calculate pressure and density for 1 km, 5 km, 9 km using the barometric formula with given gravity function.

Step 1: Understanding the Given Information

We have a planet's atmosphere that behaves as an ideal gas, with gravitational acceleration given by \(g(z)=\frac{18.7}{1-\frac{z}{10,000}} \ m/s^2\). The temperature is constant at 250 K, and the gas constant \(R = 340 \ Nm/ kg \ K\). The surface pressure \(P_0 = 2 \ atm\). Our task is to find the pressure and density at altitudes of 1 km, 5 km, and 9 km.

Step 2: Convert Atmospheric Pressure to SI Units

First, convert the given surface pressure from atmospheres to Pascals (Pa) since the ideal gas law uses these units. \(1 \ atm = 101325 \ Pa\), thus, for surface pressure, \(P_0 = 2 \times 101325 = 202650 \ Pa\).

Step 3: Calculate Pressure at Each Altitude

To find the pressure at a given altitude, use the barometric formula: \( P(z) = P_0 \exp\left(-\frac{M g(z)z}{RT}\right) \), where \(M\) is the molar mass of the atmosphere's gas. Here, replace \( g(z) \) with the given function and calculate at 1 km, 5 km, and 9 km.

Step 4: Calculate Density Using Ideal Gas Law

Using the ideal gas law, \( \rho(z) = \frac{P(z)M}{RT} \), substitute the calculated \(P(z)\) values from the previous step to find the density \(\rho(z)\) at each altitude. Make sure to use the correct gravitational force \( g(z) \) at each altitude.

Step 5: Solve for Molar Mass if Needed

If the molar mass \(M\) is not given, assume the composition is similar to Earth's atmosphere. Typically, air has a molar mass of 28.97 g/mol. Convert this to kg/mol for the calculations.

Step 6: Compute Results for Each Altitude

Go through the calculation process for 1 km, 5 km, and 9 km. Plug in each altitude into the equations for pressure and density using the given expressions for gravity and known gas law constants.

Key Concepts

Ideal Gas Law

In biofluid mechanics, the ideal gas law is a critical concept when studying atmospheres, like the one on the newly discovered planet. It relates the pressure, volume, and temperature of a gas, making it ideal for modeling atmospheric conditions. The equation is: \[ PV = nRT \] where:

• \( P \) is the pressure
• \( V \) is the volume
• \( n \) is the number of moles of the gas
• \( R \) is the specific gas constant
• \( T \) is the temperature in Kelvin

For our planet, this equation allows us to link atmospheric pressure and density with temperature and molar mass. We primarily use it to express how a change in altitude can affect these variables.

Gravitational Force

Gravitational force on the new planet varies with altitude, which is reasonably unusual compared to Earth where gravity is nearly constant at the surface level. The given function for gravitational force is:\[ g(z) = \frac{18.7}{1-\frac{z}{10,000}} \, m/s^2 \]This formula shows that the gravitational pull weakens as you ascend the planet's atmosphere. Gravitational force is crucial because, in the atmosphere, it affects how pressure decreases with an increase in altitude, impacting the calculations for both atmospheric pressure and density.

Atmospheric Pressure

Atmospheric pressure decreases with altitude due to gravitational forces pulling the air molecules closer to the planet's surface. To find the pressure at various altitudes, we use the barometric formula:\[ P(z) = P_0 \exp\left(-\frac{M g(z)z}{RT}\right) \]where:

• \( P_0 \) is the surface pressure
• \( M \) is the molar mass of the atmospheric gas
• \( g(z) \) is the gravitational acceleration
• \( R \) and \( T \) are the gas constant and temperature

This formula accounts for the gravitational force's variability in high altitudes, which significantly influences the atmospheric pressure on this planet.

Density Calculation

After obtaining the pressure data at different altitudes on the new planet, calculating the atmosphere's density becomes feasible with the ideal gas law rearranged:\[ \rho(z) = \frac{P(z)M}{RT} \]Understanding density is vital since it informs us about the mass per unit volume of the atmosphere at various altitudes, influencing potential habitability or flight conditions. To ensure accurate results, plug in the appropriate pressure \( P(z) \), calculated earlier, along with the known values for temperature \( T \) and the molar mass \( M \).

Altitude Effects on Atmosphere

As altitude increases, several changes occur due to the reduced atmospheric pressure and density. With this planet's atmosphere modeled as an ideal gas, we see distinct effects:

• Reduced pressure and density contribute to lower aerodynamic resistance, impacting any landers or probes.
• Decreased gravitational pull affects gas behavior, leading to variations in chemical compositions.
• Constant temperature assumption simplifies calculations, minimizing complications regarding thermodynamics outside this exercise's scope.

Understanding these variations helps in designing equipment and predicting atmospheric behavior in different planetary explorations.