Suggested Response - Q7

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a)    A bright beer main of 100 mm internal diameter is 80 m long. The friction factor can be considered a constant at 0.005. (This is not a function of line velocity and is a simplification to assist you in the calculation. We know from our previous studies that friction factor is a function of line velocity). Sketch the characteristic curve for the pipeline between 0 and 1,000 hL/h. Represent the pressure drop as a head in metres of water (ρ = 1,000 kg/m3).

b)  If the pump from question six was used to supply the bright beer main, and a zero-suction head condition existed, what would be the supply flow rate?

Bright beer properties are ρ = 1,010 kg/m3 and µ = 3 cP.

Response:

a)    The characteristic curve is a plot of flow rate (or capacity) against pressure drop (or total head). In order to determine the pressure drop, we will use the Darcy-Weisbach equation and apply it for a range of flow rates 0 to 1,000 hL/h:

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

For a flow rate of 100 hL/h, we need to determine the mean velocity. We will do this from the continuity equation:

Q space equals space A u

So, rearranging for u:

u space equals space Q over A space equals space fraction numerator begin display style fraction numerator 100 space cross times space 100 over denominator 3 comma 600 space cross times space 1 comma 000 end fraction end style over denominator begin display style fraction numerator pi space cross times space 0.1 squared over denominator 4 end fraction end style end fraction space equals space 0.35367 space m divided by s

And substituting this into the Darcy-Weisbach equation gives:

increment P subscript բ space equals space 4 space cross times space 0.005 space cross times space fraction numerator 80 over denominator 0.1 end fraction space cross times space 1 comma 010 space cross times space 0.35367 squared

increment P subscript բ space equals space 2.02 space k P a

Expressing this as a head of water:

increment P subscript բ space equals space increment H rho g

increment H space equals space fraction numerator increment P բ over denominator rho g end fraction space equals space fraction numerator 2 comma 020 over denominator 1 comma 000 space cross times space 9.81 end fraction space equals space 0.2 space m

This set of calculations would be repeated for the complete flow rate range. It is probably easiest to do this is in a spreadsheet to yield the following data:

Q (hL/h)

0

100

200

300

400

500

600

700

800

900

1000

∆H (m)

0

0.2

0.8

1.9

3.3

5.2

7.4

10.1

13.2

16.7

20.6


These data are plotted below:

 

b)   To determine the flow rate that the pump would supply to this line, we need to plot the pump and the pipe characteristic curves. Where they intersect will give us the flow rate. This is shown below:

 

123

The resulting flow rate would be 800 hL/h (with a pump total head of 13 m water).

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