Carbon dioxide is often used in biomedical engineering labs for many applications, one of which is cell culturing. UltraSuperCell Services®is a large commercial laboratory, and requires large amounts of carbon dioxide. They have determined that it is much more efficient to pump this carbon dioxide from a neighbouring facility located 1000 m away via a long pipe with an internal diameter of 12 cm and a fanning friction factor of 0.005. The flow is isothermal at 25°C and carbon dioxide has the chemical formula CO2, where carbon has a molecular weight of 12g/mol and oxygen has a molecular weight of 16 g/mol. The neighbouring facility stores the carbon dioxide at a pressure 521 kPa and the lab requires the pressure to be 200 kPa. A. What is the mass flow rate of the carbon dioxide? Show all working.

To solve this problem, we need to use the Bernoulli equation for fluid flow and the ideal gas law.

Step 1: Calculate the density of CO2 using the ideal gas law.

The ideal gas law is PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature.

First, we need to convert the temperature from Celsius to Kelvin. T(K) = T(C) + 273.15 = 25 + 273.15 = 298.15 K

Next, we need to calculate the molar mass of CO2. The molar mass of CO2 = 12 g/mol (for C) + 2*16 g/mol (for O) = 44 g/mol = 44 kg/kmol

Then, we can calculate the density (ρ) using the ideal gas law rearranged to solve for density: ρ = PM/RT

Where: P = pressure = 521 kPa = 521,000 Pa M = molar mass = 44 kg/kmol R = gas constant = 8314 m^2/(s^2Kkmol) T = temperature = 298.15 K

So, ρ = (521,000 * 44) / (8314 * 298.15) = 1.96 kg/m^3

Step 2: Use the Bernoulli equation to calculate the velocity of the CO2.

The Bernoulli equation is P1 + 0.5ρv1^2 = P2 + 0.5ρv2^2

Assuming the velocity at the storage facility (v1) is approximately 0, the equation simplifies to P1 = P2 + 0.5ρv2^2

We can rearrange to solve for v2 (the velocity at the lab): v2 = sqrt((P1 - P2) / (0.5*ρ))

Where: P1 = pressure at the storage facility = 521,000 Pa P2 = pressure at the lab = 200,000 Pa ρ = density = 1.96 kg/m^3

So, v2 = sqrt((521,000 - 200,000) / (0.5*1.96)) = 37.3 m/s

Step 3: Calculate the mass flow rate.

The mass flow rate (ṁ) is given by the equation ṁ = ρAv, where A is the cross-sectional area of the pipe and v is the velocity of the fluid.

The cross-sectional area A = π*(D/2)^2, where D is the diameter of the pipe. D = 12 cm = 0.12 m, so A = π*(0.12/2)^2 = 0.0113 m^2

So, ṁ = 1.96 kg/m^3 * 0.0113 m^2 * 37.3 m/s = 0.83 kg/s

Therefore, the mass flow rate of the carbon dioxide is 0.83 kg/s.