# liquid chromatography column - Scott R.P.W.

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Chapter 12

LC COLUMN DESIGNTHE DESIGN PROCESS Packed Columns

The design process involves the use of a number of specific equations (most of which having been previously derived and/or discussed) to identify the column parameters, the operating conditions and the resulting analytical specifications necessary to achieve a particular separation. The characteristics of the separation will be defined by the reduced chromatogram of the particular sample of interest

The first equation to be employed will be that of Purnell (I), which is used to calculate the efficiency required to separate the sample into its constituents. The data used is the separation ratio of the critical pair and the capacity ratio of the first eluted peak of the critical pair. The Purnell equation is reiterated as follows,

where (a) is the separation ratio of the critical pair, and (k') is the capacity ratio of the first eluted peak of the critical pair.

The next equation of importance is the relationship between the column length and the height of the theoretical plate of the column (H),

Now, as has been previously shown, from the Van Deemter equation (2), the value of (H) is given by,

•2 7 êä (a-l)

(I)

I = nH

(2)

H = A + - +Cu

(3)

u

where,

(4)

1Ë6

â = 2yDr

and

Ñ =

Î rl2

1+6ê‘+11ê ð 8 ê' df

24 (l+k1 )2 Dfri n2 (t+k-)2 DS

(5)

(6)

Now, in LC, (Dm) and (Ds) are commensurate and, if the stationary phase is silica gel or a bonded phase, then df << dp> and thus, the second function in the equation can be ignored,

Consequently,

Ñ =

l+6k'+i lk'

,2 d

2

24 (l+k1)

¦*2 Dn

(7)

Differentiating equation (3) and equating to zero, to obtain an expression for the optimum velocity it has been shown that,

uopt

(8)

Substituting for (B) and (C), from equations (5) and (7) in equation (8) and simplifying,

-.0.5

uopt

4 Dm 3Y(i+kf

dp (l+6k +l Ik'2)

(9)

By substituting for (U0pt) from equation (8) in equation (3) and simplifying an expression for (Hrnin) can be obtained,

Hmin = A+2^BC ................................................. (10)

Again substituting for (A), (B), and (C) from equations (4), (5) and (7),

¦Hrnin — 2Xdp + 2

2 Y^rn

( ~ ,.2\ +6k' + 1 lk'^up

k 24 (l + k’)

Ë2 D„

...(II)

187

Thus,

Hfnin - 2Xdp + 2

(1 + 6K + I lk'2)d2 \ I p

(12)

Now, Giddings (3), has calculated that for a well packed column (ß.) and (y) would take values of 0.5 and 0.6 respectively.

and, when k' = infinity Hmin = 2.48dp

It can be seen that the numerical value of (k'), the capacity ratio of the first eluted peak of the critical pair, can make a significant difference to the value of Hmin. To simplify expressions in chromatographic theory it is often assumed that, to the first approximation, (Hmin) can be taken as 2dp It is clearly seen that, in column design, such assumptions could lead to significant errors particularly at extreme values of (k'X

Substituting for (Hmin) and (n) from equations (12) and (1) respectively, in equation (2), an expression for the column length (l) can be obtained,

Thus, when k' = 0,

Hmin - 1 45dp

I =

(13)

166

It should be pointed out that equation (13) does not give an expression for the optimum column length as the optimum particle diameter has yet to be identified

The Optimum Particle Diameter

The column length is also defined by the D'Arcy equation that describes the flow of a fluid through a packed bed in terms of the particle diameter, the pressure applied across the bed, the viscosity of the fluid and the linear velocity of the fluid. The D'Arcy equation is given as follows,

_ <pPdp2

(14)

where (P) is the inlet pressure to the column,

(T|) is the viscosity of the mobile phase,

(<p) is the D'Arcy's constant which for a well packed LC column takes a value of about 35 when the pressure is measured in p.s.i..

Equating equation (2) to equation (I A),

*p<|p2 _ „í

Substituting for (uopt) and (Hmin) from equations (9) and (10) respectively,

2

and rearranging.

"Vc

= n|A

2B

(15)

Substituting for (A), (B) and (C) from equations (4), (5) and (7) respectively, in equation (15)

2 n0.5

3 Y (1 + k‘)

<pPdp

-— = n

ß

2ia

4Dn

p dr

(l + 6k' + 11k'2

4 Y^m

....(16)

169

It is seen that there will be a unique value for (dp), the optimum particle diameter, that will allow the minimum HETP to be realized when operating at an inlet pressure (P). Rearranging and solving for dp(0pt)

dpfopt) ~

4ntiDn

<pP

2Ë

3Y(l + kf

0.5

(l+ 6k'+ 11k'2) V' 4

+ Y

0.5

(17)

Substituting for (n) in equation (17) from equation (I)

dp(opt) - -

8(1 +k') Ï^ãï

k'( a -1) <pP

2ß

3Y(i + kf

(l + 6k' + l Ik'2) v' ' i

0.5

+ Y

0.5

.(18)

Equation (18) allows the optimum particle diameter to be calculated that will allow the separation to be achieved in the minimum time by utilizing the maximum available inlet pressure and operating at the optimum mobile phase velocity. It is one of the most important equations in column design

The characteristics of many of the equations discussed in this chapter will be tested against realistic chromatographic conditions and the typical conditions chosen are given in table 1.

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