Suggested Response - Q2

*** An automated sampling system fitted to a bright beer tank uses a 5 mm internal diameter tube to transport samples over a distance of 20 m. The beer has a density and a dynamic viscosity of 1,005 kg/m³ and 2 x 10-³ kg/ms, respectively.

a)    If the flow rate in the sample line was 600 mL/min, what would be the average transit time for a sample from the bright beer tank to travel the length of the sample line?

b)    During the cleaning of the sample line, it is required to pass the cleaning solution through the line under turbulent flow conditions. What would be the minimum flow rate supply of cleaning fluid to achieve this requirement? (You may assume the cleaning fluid has the same fluid physical properties as the bright beer).

Response:

a)    In order to determine the average transit time, we merely need to determine the average flow velocity of the sample. Thereafter, we need to determine the average time for that sample to travel 20 m (remember that velocity is the ratio of distance to time).

The average flow velocity is given by consideration of the following equation:

Q equals A u

And rearranging for average flow velocity u:

u equals Q over A

u equals Q over A equals fraction numerator begin display style fraction numerator begin display style fraction numerator 600 over denominator 60 cross times 1 comma 000 cross times 1 comma 000 end fraction end style over denominator pi cross times 0.005 to the power of ² end fraction end style over denominator 4 end fraction

u equals 0.51 space m divided by s

Hence the average time for a sample to travel 20 m would be given by rearranging the following equation for velocity:

u equals fraction numerator d i s tan c e over denominator t i m e end fraction

t i m e equals fraction numerator d i s tan c e over denominator u end fraction equals fraction numerator 20 over denominator 0.51 end fraction equals 39.2 space s

b)    Turbulent flow conditions require a Reynolds number greater than 4,000:

R e equals fraction numerator rho u d over denominator mu end fraction greater than 4 comma 000

Rearranging for u:

u greater than fraction numerator 4 comma 000 mu over denominator rho d end fraction

u greater than fraction numerator 4 comma 000 cross times 2 cross times 10 to the power of negative ³ end exponent over denominator 1 comma 005 cross times 0.005 end fraction

u greater than 1.59 space m divided by s

Determining the required flow rate of cleaning fluid:

Q equals A u equals pi over 4 cross times 0.005 to the power of ² cross times 1.59 equals 3.12 cross times 10 to the power of negative ⁵ end exponent m to the power of ³ divided by s

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