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The MBR Book: Principles and Applications of Membrane

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282 <strong>The</strong> <strong>MBR</strong> <strong>Book</strong><br />

C.1 Separation distance D<br />

Consider three equally spaced hollow fibres (HFs) in an element <strong>of</strong> internal volume<br />

V (Fig. C.1). <strong>The</strong> fibres <strong>of</strong> diameter d are separated by a distance <strong>of</strong> D, where D represents<br />

the diameter <strong>of</strong> a virtual cylinder surrounding each HF. <strong>The</strong> total volume <strong>of</strong><br />

these virtual cylinders is equal to V minus the interstitial volume created by three<br />

touching cylinders. Each HF is then associated with one interstitial volume.<br />

Given the axial symmetry <strong>and</strong> from basic trigonometry, the % volume occupied<br />

by the interstitial volume is given by the ratio:<br />

Thus:<br />

area <strong>of</strong> equilateral triangle <strong>of</strong> sides D � area <strong>of</strong> semicircle <strong>of</strong> side D<br />

R �<br />

area <strong>of</strong> equilateral<br />

triangle <strong>of</strong> sides D<br />

( D/ 2) Dsin 60�( p/ 8)<br />

D2<br />

0.886/2 � 0. 3923<br />

R �<br />

�<br />

� 0.0460<br />

( D/ 2) Dsin60<br />

0.886/2<br />

<strong>The</strong> volume occupied <strong>and</strong> the surface area provided by the fibres are given by:<br />

Vf�d2NL 4<br />

p<br />

Af�pdNL D<br />

D sin 60<br />

Figure C.1 Cross-section <strong>of</strong> three HFs in module<br />

d<br />

(C.1)<br />

(C.2)<br />

(C.3)

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