Suggested Response - Q4

**During a design meeting for refitted CO2 bung lines (the CO2 vent lines from a fermenter), you are offered two sizes (a 50 mm or a 65 mm internal diameter). The maximum gas flowrate is expected to be 180 kg/h, and the maximum line length will be 65 m. The frictional pressure drop over the line must be kept below 5 kPa to limit excessive fermenter headspace pressure. Which line diameter would you choose and why?

Assume a surface roughness of 0.05 mm; ρ = 1.9 kg/m3; and µ = 15 x 10-6 kg/ms.

Response:

The volumetric flowrate of CO2 would be determined from the mass flowrate and the density:

rho equals G over Q

The volumetric flowrate  would be given by:

Q equals G over rho equals fraction numerator begin display style fraction numerator 180 over denominator 3 comma 600 end fraction end style over denominator 1.9 end fraction equals 2.63 cross times 10 to the power of negative 2 end exponent space m cubed divided by s

Let us consider the 50 mm line in detail. In order to calculate the Reynolds number we need to know the line velocity:

Q equals A u

u equals Q over A equals fraction numerator begin display style fraction numerator 2.63 cross times 10 to the power of negative 2 end exponent over denominator pi cross times 0.05 squared end fraction end style over denominator 4 end fraction equals 13.4 m divided by s

Consequently, the Re is given by:

R e equals fraction numerator rho u d over denominator mu end fraction equals fraction numerator 1.9 cross times 13.4 cross times 0.05 over denominator 15 cross times 10 to the power of negative 6 end exponent end fraction equals 84 comma 867 space equals space 8.5 cross times 10 to the power of 4

Therefore, Re > 4,000, which confirms the flow is turbulent and we need to use the Moody chart to determine the friction factor:

R e l a t i v e space R o u g h n e s s space ϵ equals fraction numerator A b s o l u t e space R o u g h n e s s space e over denominator P i p e space I n s i d e space D i a m e t e r space d end fraction equals fraction numerator 0.05 over denominator 50 end fraction equals 0.001

From the Moody diagram, we can determine that the friction factor is 0.006. So, using the Darcy-Weisbach equation:

increment P subscript f equals 4 phi L over d rho u squared

increment P subscript f equals 4 cross times 0.006 cross times fraction numerator 65 over denominator 0.05 end fraction cross times 1.9 cross times 13.4 squared

increment P subscript f equals 10 comma 644 space P a space equals space 10.6 space k P a

This frictional pressure drop exceeds our required maximum of 5 kPa, so we would reject the 50 mm bung line.

Repeating the same consideration for the 65 mm options gives the following parameters (check these for yourself on the Moody chart above):

u space equals space 7.93 space m divided by s semicolon space R e space equals space 6.53 space x space 10 to the power of 4 semicolon space epsilon space equals space 0.0008 semicolon space empty set space equals space 0.0055 semicolon space increment P f space equals space 2.8 space k P a

Therefore, with the 65 mm line, the frictional pressure drop is 2.8 kPa, which is below the maximum permitted, so we would choose this line size.

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