Calculating drop out in manifold systems

A manifold in a particle counting system allows one particle counter to monitor many (typically more than 30) sample points. This reduces the costs of such monitoring and is often adequate for use in non-critical areas.

A manifold system consists of a particle counter that uses the aerosol manifold to multiplex its input to one of several locations. The air from the location is drawn through a length of tubing at a typical flow rate of 3 cfm. The particle counter then samples a proportion of this flow (usually 1 cfm) to measure the particle counts at the location. In most pharmaceutical cleanrooms, the concentrations of particles more than 0.5?m and less than 5.0?m in size are of interest. However, particles of more than 0.2?m are subject to gravitational sedimentation.¹ This means fewer particles reach the particle counter than were present at the sample inlet. Investigations into the loss of particles in manifold sample tubes² show that particles of less than 1.0μm (nominal) show no transport loss, whereas those of greater than 1.0μm are attenuated. In particular, the counts at 5.0μm are significantly reduced over short distances.

Gravitational effects There are studies of particle deposition in tubing (e.g. human lungs) that model not only gravitational sedimentation but also the effects of collisions between particles and tube walls.³ These are useful to review to appreciate the full range of factors that influence particle transport, though here we are interested only in the effects of gravitational sedimentation. This article provides a basic, simplified, mathematical and physical framework for understanding influences on particle drop out. The gravitational force on a particle is:

F_gravity = mg (1)

where F is the force, m is the mass of the particle and g is the gravitational acceleration. The mass of the particle, assumed to be a sphere, is:

m = (4 π ρ+r³)/3 (2)

where ρ is the density of the particle and r is the radius of the particle. Gravitational fields can also be generated by rapid changes in direction (centripetal acceleration) by bends in tubing. A particle moving through a fluid (such as air) meets a resistive force. For spheres, the Stokes-Navir equation can be used to determine the frictional coefficient, hence the resistive force:

F _fluid = 6πηr v (3)

where η is the viscosity of the fluid, r is the radius of the particle and v is the velocity of the particle. The terminal velocity of the particle (the speed at which it falls when the force of gravity balances the frictional forces) will be when:

F_gravity = F _fluid (4)

The terminal velocity is then:

v = (2g ρ r²)/9η (5)

The time it takes a particle to fall a distance d is:

t = d/v (6)

It will be noted that for 5.0μm particles the loss with tube length appears to be an exponential decay curve rather than a straight line. This is because the sample tubing has a circular cross-section and not a rectangular cross-section. That is, the maximum distance a particle has to fall before it hits the tube wall is not constant across the tube's horizontal diameter, thus more particles fall shorter distances than longer distances.

For a circular cross section (see figure 1) the maximum distance a particle can fall for a given position is:

D = 2√R²-x² (7)

where D is the maximum distance a particle can travel for a given x, the distance from the centre along the horizontal axis, and R is the radius of the tubing. This explains the observed form of the initial particle drop out with distance curve.

Density dependent The above equations explain the factors affecting particle transport down a sample tube. Equation 5 shows that the transport of a particle depends on that particle's density and the square of its radius. This predicts that 0.5μm particles are transported 100 times more effectively than 5.0μm particles. This fits with the observation that whereas 5.0μm particle loss is noticeable, 0.5μm loss is apparently zero. We can estimate the sedimentation velocity of 5μm polystyrene latex (PSL) particles in air at about 25°C as follows: where:

r = 2.5x10^-6m, ρ = 2x10³kgm^-3, η =1x10^-5Nm^-1s^-1, g =10ms^-2 (8)

then: v = 2.8x10^-3ms^-1 (9)

At 3 cfm the air velocity through a 9mm (Internal Diameter) tubing is 22ms^-1. After half a second (11m) approximately 50% of the 5.0μm particles fall to the tube wall. This agrees well with experiment², where it is seen that about 50% of (PSL) 5.0μm particles are lost with a sample tube length of about 10 metres. If the particles are glass then close to 100% of the particles are predicted to be lost at this distance. If the particles are lead oxide then all are expected to be lost within two metres.

Velocity variation The model presented here appears to be useful and it is tempting to apply it to correct the measurements made using manifolds for losses due to sedimentation. However, by reference to equation 5 it will be noted that the rate of sedimentation depends on the density of the particles and the viscosity of the air. Densities of common materials (relative to water) vary from that of oils (less than one) to metals (more than 10). Although it is possible for particle counters to determine the density of each particle⁴ this is not done, and the best that can be attempted is to use a typical value of between two and five. The viscosity of air is sensitive to temperature and pressure. Temperature variations from any assumed norm along the sample tube will cause differences in particle drop out rate. Variations in air velocity clearly alter the probability of a particle being transported. This model presented here assumes the tubes are straight and there is no turbulent flow. Bends in tubing increase particle loss through centrifugal sedimentation and turbulence reduces the loss by keeping particles in the air stream. Any elasticity in the collisions between a particle and the tube wall will also increase the number of particles being transported. This appears to depend on temperature.³ A further complication to estimating particle sedimentation is that real particles are seldom smooth spheres; they have uneven surfaces and are not spherical. It can be shown⁵ that the frictional coefficient of a rough sphere is significantly larger than a similar sized smooth sphere. This in turn means the optical radius (from a particle counter) is unlikely to correspond to the hydrodynamic radius. As it must be the optical radius that is used to correct for particle drop out, but it is the hydrodynamic radius that determines the rate of sedimentation, there is a clear difficulty. The optical radius cannot be estimated to better than 17%. As the drop out depends on the square of the hydrodynamic radius, it is evident that any correction factor based on the optical radius has poor accuracy. It is aslo clear that although the model presented here is helpful in understanding particle drop out, it shows that to predict and so correct for particle loss requires additional information that is often impossible to obtain. In essence, it is necessary to know the original particle size distribution at the sample inlet, which defeats the point of using a manifold. The end result of any attempted correction will be to replace one measurement that has understandable uncertainties with a result that has been corrected using a range of unprovable assumptions and is often a worse, or no better, representation of the "true" value than the uncorrected value.

Quantifiable limits To conclude, the use of aerosol manifolds to economically measure particle counts is helpful. The limitations when used to measure 5.0μm particles are well understood, easily modelled and so are manageable without the need to resort to unnecessary, and in the end arbitrary, corrections. If knowing the concentrations of large particles is important, then manifolds should not be used, but in situ (point of use) counters instead. This is well known and often stated, but with the absence of reasoned, quantified, argument. It is the author's intention that this article provide this knowledge so the choice of particle monitoring method is a rational one, based on sound scientific principles.