User:Softwarestorage/Tapped density

Tapped density is the term used to describe the bulk density of a powder (or granular solid) after consolidation/compression prescribed in terms of "tapping" the container of powder a measured number of times, usually from a predetermined height. The method of "tapping" is best described as "lifting and dropping". Tapping in this context is not to be confused with tamping, sideways hitting or vibration.

Tapped density is an important experimentally determined value in those industries producing and packaging powdered and granular materials such as beverages (dry form) & processed foods   , and to those industries where powder packing is a critical process parameter, particularly pharmaceuticals  , powder metallurgy  and industrial catalysis  (for example in the petrochemical industry).

Tapped density is variously reported in units such as g/cm3, g/cc, lb/cu.ft.

Powder bulk density
The bulk density of a powder simply expresses the amount, usually weight or mass, of a powder in a specified volume. However, since powders are composed of particles and voids, the volume occupied by a given number of particles depends on how closely they are packed. The packing of particles depends on their shape, cohesiveness, short-range motion and external forces. In practical terms, the bulk density of a powder tends to increase the more it is subjected to tapping, vibration and other mechanical action which causes particles to move relative to one another in a way that allows smaller particles to occupy the voids between larger particles.

Tapped density or tapped bulk density is one formal expression of bulk density obtained under specified conditions. The "end point" used for measurement of tapped volume and calculation of tapped density can be defined by total number of taps (either explicitly or the product of tapping rate and time or duration of tapping), or some defined amount of small change in tapped volume from time to time, since the theoretical final volume, at infinite time/taps is approached asymptotically in an approximately logarithmic manner.

Measuring tapped density
In general, any graduated container can serve as a means to determine tapped density. In practice, graduated glass measuring cylinders are most often used. In the standard methods below, the total capacity of the cylinder to be used, and the readability of its scale are stated. The cylinder can be tapped manually or by mechanical device.



Manual tapping
The raising and lowering of the cylinder by hand is done either (i) without reference to the height traversed and arbitrary acceleration in both upward and downward directions; the hand remaining in contact with the cylinder at all times (hand tapping), or (ii) by constraining the upwards distance traveled and allowing free-fall of the cylinder under gravity (drop box).In hand tapping the cylinder containing the powder is tapped by repeatedly striking its base down onto a hard surface. A drop box has a hole in its lid, large enough for the body of the graduated cylinder to pass though it, but too small for the base of the cylinder. The distance between the bottom of the box and the underside of the lid defines the height through which the cylinder can be lifted and dropped.

Mechanical devices
Tap density analyzers (tap density testers) use an electric motor to turn a cam under a specially constructed cylinder holder. The holder secures the cylinder to a vertical shaft which runs in a low friction bearing. The tapping rate is normally expressed in taps per minute; the rate being typically a few hundred. The actual rate is determined by the rotational speed of the cam under the shaft/platform. Digital or electromechanical counters are usually incorporated in the device to automatically stop the cam rotation after a predetermined (yet adjustable) number of taps. The height through which the container falls is known as the drop height or stroke. It is set by the distance between the highest point on the cam and the striking surface. In the standard methods referred to below, the drop height is one of two values, 3mm (or 1/8") or 14mm, within tolerances specified therein. These values merely reflect drop heights in commercial equipment and have no special physical meaning otherwise.

Standard methods
The measurement of tapped (tap) density has been formalized in a number of standard methods particularly for metal powders, catalysts  , pigments , dried food powders   and carbon.

Tapped density and powder flowability
The change in tapped powder volume has been related to flow properties of powders.

Carr's compressibility index and Hausner ratio
Tapped density is used in the calculation of Carr's Compressibility Index (Carr index), the percentage change in powder bulk volume (from its initial "loose" condition) upon tapping and of Hausner ratio, the fractional change in bulk volume from "loose" to tapped". Since any value of tapped bulk density strictly depends upon experimental details as drop height, number of taps, container geometry, sample mass etc, so must the "Carr" and "Hausner" values, notwithstanding any uncertainty in the starting "loose" or "freely settled" density value.

Descriptive equations
Measuring the change in powder bed porosity as a function of the number of taps (N), Kawakita described the compression of a powder as follows:
 * Kawakita


 * $$ \frac{N}{C} = \frac{N}{a} + \frac{1}{ab} $$

where C is equal to the Carr index (the percentage change in initial volume) and a and b are evaluated graphically by plotting N/C vs. N.

The Cooper-Eaton model of compaction was modified by Mohammadi and Harnby
 * Cooper-Eaton


 * $$\,\! D_N = D_t-(D_t-D_a)exp^{-N/T} $$

where Dt and Da are the theoretical tapped and aerated bulk densities respectively, DN is the observed bulk density after N taps. T is a so-called "compaction constant" and is inversely related to the rate of densification.

Heckel described the plastic compression of powder as a change in relative density, D, as a function of applied pressure. A plot of the natural logarithm of the ratio (1/1-DN) vs. N, the number of taps (substituting for pressure), where DN is the relative powder density for a given number of taps, yields slope of value k which is related to the yield strength of the powder bed.
 * Heckel


 * $$ ln \frac{1}{1-D_N} = kN + A $$