Air at an absolute pressure of 1500 kPa and 15°C flows in a 1.5 cm diameter tube. What is the maximum mass flow rate to keep the flow laminar?

To determine the maximum mass flow rate to keep the flow within a tube laminar, we can use the Reynolds number \( Re \), which is a dimensionless number used to predict flow patterns in different fluid flow situations. For flow in a pipe or tube, laminar flow tends to occur at a Reynolds number of less than 2300.

The Reynolds number for a pipe flow can be calculated using the formula:

\[ Re = \frac{{\rho V D}}{\mu} \]

Where: - \( \rho \) is the density of the air (kg/m³), - \( V \) is the mean velocity of the air (m/s), - \( D \) is the diameter of the tube (m), - \( \mu \) is the dynamic viscosity of the air (Pa·s or N·s/m²).

To keep the flow laminar, we need to ensure that: \( Re < 2300 \).

Given: - Absolute pressure \( P = 1500 \) kPa, - Temperature \( T = 15 \)°C, - Diameter \( D = 1.5 \) cm = 0.015 m.

First, we need to find the density of the air (\( \rho \)), using the ideal gas law:

\[ PV = nRT \]

Where \( V \) is the volume, \( n \) is the number of moles, \( R \) is the gas constant = 287 J/(kg·K) for air, and \( T \) is the temperature in Kelvin.

To find \( T \) in Kelvin, add 273 to the Celsius temperature: \( T = 15 \)°C + 273 = 288 K

Rearranging the ideal gas law to solve for density, \( \rho \), we get:

\[ \rho = \frac{P}{RT} \]

Using the given pressure (converting kPa to Pa by multiplying by 1000), we calculate the density:

\[ \rho = \frac{1500 \times 10^3 \text{ Pa}}{287 \text{ J/(kg·K)} \times 288 \text{ K}} \]

Now, we need to find the dynamic viscosity \( \mu \) of air at 15°C, which can typically be found in a table in standard textbooks or online sources. At 15°C, the viscosity of air is around \( 1.78 \times 10^{-5} \) Pa·s.

Using the criterion that \( Re < 2300 \) for laminar flow, we can rearrange the Reynolds number equation to solve for the maximum velocity \( V \):

\[ V = \frac{Re \cdot \mu}{\rho \cdot D} \]

Now plugging in the maximum Reynolds number for laminar flow, we find \( V \):

\[ V = \frac{2300 \cdot 1.78 \times 10^{-5} \text{ Pa·s}}{\rho \cdot 0.015 \text{ m}} \]

With \( V \) calculated, we can then determine the mass flow rate \( \dot{m} \), using the formula:

\[ \dot{m} = \rho A V \]

Where \( A \) is the cross-sectional area of the tube:

\[ A = \frac{\pi D^2}{4} \]

By substituting the area and calculated \( V \) into the mass flow rate equation, we will have the maximum mass flow rate that keeps the flow laminar.

Since the specifics calculations require numeric input (especially the air density calculation which depends on the precise value of the gas constant and the dynamic viscosity), please proceed with the actual numbers or look them up in tables to find the maximum mass flow rate.

Extra: The Reynolds number is crucial for understanding fluid dynamics and characterizing the nature of flow within a tube or around objects. Laminar flow is characterized by smooth, orderly motion of fluid particles, often in parallel lines, with less mixing and lower friction than turbulent flow. Turbulent flow, on the other hand, is chaotic with lots of mixing and higher friction losses.

For pipelines and tube flow, engineers strive to maintain laminar flow to minimize pressure drops and energy losses. In practical applications such as ventilation systems, water piping, and blood flow in medical devices, knowing when and how to keep flow laminar is important for design and operation efficiency.

It is also important to recognize that, while keeping flow laminar reduces friction losses, in some industrial processes such as heat exchangers, turbulent flow is preferred because it can enhance heat transfer between the fluid and the pipe wall. The balance between maintaining laminar flow for reduced friction and promoting turbulent flow for increased heat transfer is a key element of mechanical and chemical engineering design.