Pharmaceutical Manufacturing Handbook: Production and

TABLE 7 Parameters Calculated from Force – Time Profi les
Source
Parameters
Heckel [118, 119] (Figure 18 )
− ln = ln(
− ) = +
ε
1
Kp A
1 D
where K = deformation parameter
A = powder bed densifi cation
Yield pressure [120]
1
Yield pressure =
Heckel slope
Yield strength [118]
Kawakita [121, 122]
Walker/Balshin [124, 125]
S ø nnergard [126]
Cooper – Eaton [123]
Cooper – Eaton (linearized) [123]
ANALYSIS OF TABLETING PROCESS 1077
1
Yield strength =
3 × Heckel slope
p
C ab a p
1 1
= +
where a = porosity of powder bed
b = compression parameter
V
100Vrel = 100 × = − Wlog p+ C
V∞
where W = compressibility coeffi cient
− ppm /
Vrel = V1− Wlog p+ Ve e
where W = compressibility coeffi cient
n V0−V −k1/ p −k2/
p
V* = ∑ aV i i*
= = ae 1 + ae 2
i=
1 V0−V∞ where k 1 = deformation pressure for fraction part 1
k 2 = deformation pressure for fraction part 2
a 1 = fraction part 1 of deformation
a 2 = fraction part 2 of deformation
V0−V Q
lnV* = ln =− + ln R
V0 − V∞
p
where Q = extent of compressibility
R = sum of fraction parts
The equation of Kawakita describes volume reduction with pressure in the form
of a hyperbolic equation. Walker and Bal ’ shin [125] postulated a logarithmic relation
between applied pressure and volume reduction, which was further modifi ed
by S ø nnergard [126] . Cooper and Eaton [123] use an exponential function, which
can also be linearized. Pressure thresholds for deformation mechanisms are determined.
It should be noted that all of these equations and tableting models determine
descriptive parameters.
The equation of Heckel is the most extensively used model and the underlying
porosity – pressure plot is called a Heckel plot (Figure 18 ). The equation for
the linear compression process follows fi st - order kinetics (Table 7 ). Heckel