### Determine venturi dimensions and expected differential pressure reading for air flow at 50000 Reynolds number, 500 kPa and 50°C, through a circular tube with 50 mm diameter?

__The formulae used in the solution are:__

Reynolds number = (density x velocity x diameter) / viscosity

Delta P = (rho * v^2 / 2) * ((A1 / A2)^2 - 1)

d2 = 0.4 * d1 / sqrt(A1 / A2)

A2 = pi * d2^2 / 4

where:

**Reynolds number**: dimensionless number used to describe the flow regime of a fluid in a pipe or duct

**density**: mass per unit volume of the fluid

**velocity**: speed of the fluid

**diameter:** internal diameter of the pipe or duct

**viscosity**: a measure of a fluid's resistance to flow

**Delta P**: pressure drop across the venturi

**A1**: cross-sectional area of the pipe before the venturi

**A2**: area of the throat of the venturi

**d1**: diameter of the pipe before the venturi

**d2**: diameter of the throat of the venturi

**pi:** mathematical constant representing the ratio of the circumference of a circle to its diameter.

To determine the venturi dimensions and expected differential pressure reading for airflow at 50,000 Reynolds number, 500 kPa, and 50°C, through a circular tube with a 50 mm diameter, we need to use the following formulae:

Reynolds number = (density x velocity x diameter) / viscosity

where density of air at 50°C = 1.058 kg/m^3

viscosity of air at 50°C = 0.0000184 Pa*s

Therefore, velocity = (Reynolds number x viscosity) / (density x diameter) = (50,000 x 0.0000184) / (1.058 x 0.05) = 31.41 m/s

__We can then use the following equation to calculate the expected differential pressure__:

Delta P = (rho * v^2 / 2) * ((A1 / A2)^2 - 1)

where A1 is the cross-sectional area of the pipe before the venturi, and A2 is the area of the throat of the venturi.

Assuming an ideal venturi, we can determine the dimensions of the venturi based on the desired pressure drop. Let's say we want a pressure drop of 10 kPa. Using the equation above, we can solve for the required ratio of the areas:

(A1 / A2)^2 = 1 + (2 * Delta P) / (rho * v^2) = 1 + (2 * 10,000) / (1.058 * 31.41^2) = 1.19

Therefore, A1 / A2 = 1.091

We can assume a standard 45-degree venturi, which has a throat length of about 0.4 times the diameter of the pipe. Therefore, the throat diameter of the venturi can be calculated as:

d2 = 0.4 * d1 / sqrt(A1 / A2) = 0.4 * 50 / sqrt(1.091) = 19.46 mm

Finally, we can calculate the cross-sectional area of the throat:

A2 = pi * d2^2 / 4 = pi * 19.46^2 / 4 = 297.67 mm^2

Using the equation for the expected differential pressure, we can now calculate the expected pressure drop:

Delta P = (rho * v^2 / 2) * ((A1 / A2)^2 - 1) = (1.058 * 31.41^2 / 2) * ((1 / 1.091)^2 - 1) = 10.10 kPa

Therefore, for air flow at 50,000 Reynolds number, 500 kPa, and 50°C, through a circular tube with 50 mm diameter, a 45-degree venturi with a throat diameter of 19.46 mm should produce a pressure drop of about 10.10 kPa.