Combining the Darcy’s Law Variables

Let’s use the values we derived earlier in the lesson, and use Darcy’s Law to predict the flow rates of our wort separation devices:

For the mash tun

Factor

k

m²

A

m²

DP

bar

DP

mPa

µ

mPa.s

L

m

Mash tun

9.72 x 10-11

20

0.1

1 x 107

1.85

1

 

 Inputting our figures into Darcy's equation and calculating gives us:

straight Q equals fraction numerator KA increment straight P over denominator straight mu straight L end fraction equals fraction numerator 9.72 cross times 10 to the power of negative 11 end exponent space cross times space 20 space cross times space 1 cross times 10 to the power of 7 over denominator 1.85 space cross times space 1 end fraction equals fraction numerator 0.01944 over denominator 1.85 end fraction equals 0.0105 space straight m cubed divided by straight s



For the lauter tun

Factor

k

m²

A

m²

DP

bar

DP

mPa

µ

mPa.s

L

m

Lauter tun

3.084 x 10-11

50

0.1

1 x 107

1.25

0.5

 

Inputting our figures into the equation and calculating gives us:

straight Q equals fraction numerator KA increment straight P over denominator straight mu straight L end fraction equals fraction numerator 3.084 cross times 10 to the power of negative 11 end exponent space cross times space 50 space cross times space 1 cross times 10 to the power of 7 space over denominator 1.25 space cross times space 0.5 end fraction equals fraction numerator 0.01542 over denominator 0.625 end fraction equals 0.0247 space straight m cubed divided by straight s



And finally for the mash filter:

Factor

k

m²

A

m²

DP 

bar

DP

mPa

µ

mPa.s

L

m

Mash  filter

3.525 x 10-13

330

0.3

3 x 107

1.65

0.05

 Inputting our figures into the equation and calculating gives us:

straight Q equals fraction numerator KA increment straight P over denominator straight mu straight L end fraction space equals space fraction numerator 3.525 cross times 10 to the power of negative 13 end exponent space cross times space 330 space cross times space 3 cross times 10 to the power of 7 over denominator 1.65 space cross times space 0.05 end fraction equals fraction numerator 0.00348988 over denominator 0.0825 end fraction equals 0.0423 space straight m cubed divided by straight s



To convert flow rate from m3/s to hL/hr, we multiply by 3,600 (seconds into hours) and multiply by 10 (m3 into hL). The table below shows the figures for each device along with their comparative flow rates.

Table: Comparison of flow rates for the different wort separation devices

Factor

k

m²

A

m²

DP

bar

DP

mPa

µ

mPa.s

L

m

Q

m³/s

Q

hL/h

Comparative Q

 

Mash tun

9.72 x 10-11

20

0.1

1 x 107

1.85

1

 

0.0105

378

1

Lauter tun

3.084 x 10-11

50

0.1

1 x 107

1.25

0.5

 

0.0247

889.2

2.3

Mash  filter

3.525 x 10-13

330

0.3

3 x 107

1.65

0.05

0.0423

1,522.9

4

The mash tun is predicted to be the slowest. It does benefit from the highest bed permeability, but counteracting this, it has the highest bed loading (the deepest bed and lowest cross-sectional area).

The lauter tun is predicted to run over twice as fast as the mash tun. Its bed permeability is lower and would give a lower flow. However, this is more than compensated for by its bed loading being 2.5 times lower (shallower bed and larger cross-sectional area). The flow achieved may be higher than our prediction, as the lauter tun has a set of knives which can be used to cut the bed and increase the bed permeability.

The mash filter is predicted to run four times faster than the mash tun. Its bed loading is fifteen times lower but the benefit of this is reduced by a very low bed permeability, resulting from the fine comminution provided by the hammer mill.

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