CALCULATION OF DEPOSITION VELOCITY AND POSITION OF PARTICLE
Forces on an Indoor Aerosol
When an aerosol particle inside a room is considered, the airflow patterns and the forces that act on the particle, based on its diameter, influence its behavior. There are four major forces, which act on an aerosol particle inside a room either individually or in a combination depending on the diameter range in which the particle falls. These forces are
Drag force
Brownian force
Lift force
Gravitational force
Let us take a look at each of these forces; when and how they have an influence on the behavior of the particles.
Drag force
Aerosols consist of two components, a gas or gas mixture, which most commonly is air, and the particles suspended in it. The behavior of particles within the aerosols depends to a large extent on the motion and the intrinsic properties of the suspending gas. The motion of individual gas molecules affects particles whose diameter is less than 0.1 m and hence the kinetic energy of gases is useful in understanding the behavior of these particles. Larger particles can be treated as being submersed in a fluid (the continuum regime). Intermediate particles can be treated by the adjustments of the equations from the continuum regime. This intermediate range is termed as a transition or slip regime. The equations for various parameters that affect the drag force have been taken from Paul [14].
Let us look at the concepts and parameters that affect gas and particle motion:
1. Reynolds number
When measuring an aerosol, it is important to understand what happens to an aerosol in the environment on it’s way into the sensor of a measuring instrument. Due to various external forces, the trajectories of the aerosol particles can deviate from the gas flow. The flow pattern of the gas, whether smooth or turbulent, is governed by the ratio of inertial force of the gas to the friction force of the gas moving over the surface. This ratio is expressed as the Reynolds number Re
The Reynolds number characterizes the flow and hence depends on the gas density g and not on the particle density. At NTP i.e., 200C and 101 KPa (1 atm)
g = 1.192 * 10-3 g/cm3 and = 1.833 * 10-4 dynes/cm2
Re = 6.5 Vd …………………………………………………(2.2)
2. Gas density and Mach number
The density of a gas g is related to its temperature T and pressure P through the equation of state
When the gas moves at a high velocity relative to the acoustic velocity Vg in that gas, the gas becomes compressed. The degree of compression depends on Mach number
where U is gas velocity in air and Usonic the sonic velocity at ambient temperature is about 340 m/s.
3. Knudsen number
Large aerosol particles are bombarded constantly from all directions by a large number of gas molecules. But when a particle is small, less than 1m in size, its location in space may be affected by bombardment of individual gas molecules and then its motion is no longer determined by continuum flow considerations, but by gas kinetics. The average velocity of a molecule V is a function of its molecular weight M and the gas temperature T. In air at normal temperature and pressure, this molecular velocity Vr is 463 m/s. Using these air reference values, the average velocity can be estimated for other gases and temperatures.
The mean free path is the mean distance a molecule travels before colliding with another molecule. In air at 200C and atmospheric pressure, the mean free path r is 0.0665 m. Using these reference values is determined for other pressures and temperatures.
The gas properties for several gases at NTP i.e. 293.15 0K and 101.3Kpa is given in Table 1.
Table 1: Properties of various gases at NTP
Gas / Dynamic viscosity (P) / Density g (10-3g/cm3) / Mean free path (m)Air / 182.03 / 1.205 / 0.0665
Ar / 222.92 / 1.662 / 0.0694
He / 195.71 / 0.167 / 0.192
H2 / 87.99 / 0.835 / 0.123
CH4 / 109.77 / 0.668 / 0.0537
C2H6 / 92.49 / 1.264 / 0.0328
Iso-C4H10 / 74.33 / 2.431 / 0.0190
N2O / 146.46 / 1.837 / 0.0433
CO2 / 146.73 / 1.842 / 0.0432
Source: Paul [14]
The Knudsen number Kn relates the gas molecular free path to the physical dimension of the particle, usually the particle radius
where dp is the average diameter of the particle.
Kn<1 indicates continuum flow and Kn>1 indicates free molecular flow. If Kn is approximately in the range of 0.4-20, then the flow is referred to as transition or slip flow regime.
If the particle is much smaller than the gas molecular mean free path (Kn>1) it can travel past an obstacle at a very small distance from the object since no gas particle may impede it. If the particle is very large (Kn<1), many gas collisions occur near the surface and the particle is decelerated. When the Knudsen number is of the order of unity, the particle may slip by the obstacle. When the particle size is in this slip flow regime, it is convenient to assume that the particle is still moving in a continuum gas flow. To accommodate the difference, a slip correction factor Cc also known as the Cunningham slip correction factor is introduced into the equations.
Using the Millikan cell apparatus , and are constants determined taking =6.73*10-8 m at 296.15K and 760 torr. The values for , and for solid particles and oil droplets are given in Table 2.
Based on the application of Modulated Dynamic Light Scattering (MDLS) method [12] a range of different values of , and were obtained which are given in Table 3 and Table 4. The air pressure is maintained as 760 to 0.2 torr and the Knudsen number ranges from 0.06 to 500.
Table 2: Values for , and using Millikan Cell apparatus
Particle type / / / Solid particles / 1.142 / 0.558 / 0.999
Oil droplets / 1.207 / 0.440 / 0.596
Source : Paul [12]
Table 3: Values of , and by direct computation and nonlinear fit , and by direct computation and nonlinear fit
Parametes / Direct computation / Nonlinear fit / 1.2310 / 1.2291
/ 0.4695 / 0.4711
/ 1.1783 / 1.1705
Source : Paul [14]
Table 4: Values of , and as calculated by different authors
Author / Mean free path (m) / / / Knudsen and Weber (1911) / 0.09417 / 0.772 / 0.40 / 1.63
Millikan (1923) / 0.09417 / 0.864 / 0.29 / 1.25
Langmuir (1942) / 0.133 / 0.62 / 0.22 / 1.25
Davies (1945) / 0.066 / 1.257 / 0.40 / 1.10
DeMarcus and Thomas (1952) / 0.0655 / 1.25 / 0.44 / 1.09
Reif (1958) / 0.0652 / 1.26 / 0.45 / 1.08
Fuchs (1964) / 0.0653 / 1.246 / 0.42 / 0.87
Dahneke (1972) / 0.066 / 1.234 / 0.414 / 0.87
Allen and Raabe (1982) / 0.0673 / 1.155 / 0.471 / 0.596
Allen and Raabe (1985) / 0.0673 / 1.207 / 0.440 / 0.596
Hutchins and Harper (1995) / 0.0673 / 1.231 / 0.4695 / 1.1783
For pressures other than atmospheric, the slip correction factor changes because of the pressure dependence of mean free path in Kn and the following can be used for solid particles.
where P is pressure in KPa and dp is particle diameter in m.
Cc equals 1 in continuum regime and becomes greater than 1 for decreasing particle diameter in the transition regime.
4. Aerodynamic drag on particles
The externally applied forces on an aerosol particle are opposed and balanced by the aerodynamic drag force. As an example, consider a skydiver. The air resistance eventually balances the gravitational force pulling the skydiver towards the earth. A particle’s drag force Fdrag relates the resistive pressure of the gas to the velocity pressure and is determined by the relative motion between the particle and the surrounding gas. When the particle dimensions are much larger than the distance between the gas molecules, the surrounding gas can be considered as a continuous fluid.
Under this condition, the drag force is given by
The aerodynamic drag force is related to the gas density and not the particle density. The coefficient Cd relates the drag force to the velocity pressure. When the inertial force pushing the gas aside, due to the velocity difference between the gas and the particle, is smaller than the viscous resistance force, the drag coefficient Cd is given by
Substituting equation (2.1) in equation (2.11)
Substituting this value of Cd in equation (2.10) we get
This equation is also known as the Stokes law. The particle drag for shapes other than spheres is usually difficult to predict theoretically. Therefore for particles of other shapes, a dynamic shape factor , is introduced that relates the motion of the particle under consideration to that of a spherical particle.
where dm is mass equivalent diameter
The gases are not continuous fluids as indicated above, but consist of discrete molecules. Therefore, when the particle size approaches the mean free path of the gas molecular motion, we can apply a correction that takes the slip between the gas and the particle into account. Thus, the Cunningham slip correction factor Cc is introduced into equation (2.14)
Equation (2.15) assumes that the flow around the particles is laminar. For larger Rep, empirical relationships for Cd have been developed to extend Stokes law.
For Rep above 0.1, Paul [14] gives the following equations.
Thus, using the appropriate form of the drag coefficient (equations 2.11, 2.16 or 2.17) and including the shape factor and slip coefficient we can calculate the drag force calculated over a wide range of particles and conditions.
Table 5: The shape factors for various types of compact particles
Shape / Dynamic shape factor, Sphere / 1.0
Cluster of spheres
2-sphere chain
3-sphere chain
4-sphere chain / 1.12
1.27
1.32
Glass fiber / 1.71
Dusts
Bituminous coal
High-ash soft coal
Quartz
Sand
Talc
UO2
ThO2 / 1.05-1.11
1.95
1.36-1.82
1.57
2.04
1.28
0.99
Source : Paul [14]
Brownian Force
Suppose that a particle, about 0.001m in size, is suspended in a liquid at room temperature. Then, its thermal motion may be observed, under a microscope, to display a random zig-zagged path due to random thermal collisions with liquid molecules. The collisions give rise to unbalanced forces, called Brownian forces, acting on the particle. This random behavior of the particle is called Brownian motion. This Brownian force causes an abrupt change of the particle velocity in both magnitude and direction. Hence, the Brownian force is very important for sub micron particles.
If a number of particles subject to Brownian motion are present in a given medium and there is no preferred direction for the random oscillations, then over a period of time the particles will tend to spread evenly throughout the medium. Suppose if A and B are two adjacent regions, and at time t, A contains twice as many particles as B, at that instant the probability of a particle leaving A to enter B is twice as great as the probability that a particle will leave B to enter A.
Brownian force is modeled as a Guassian white noise random process as described by Ahmadi and Smith [6]. The Brownian force is represented by ni (t).
where
and
Lift Force
The Staffman’s lift force becomes important for particles that are not too small i.e. particles between 2m and 4m in size with high shear rates. It is defined by Ahmadi and Chen [3] as
The deformation rate tensor dij is defined as being equal to. K is the constant coefficient of Staffman’s lift force having a value of 2.594.
When distances very near to the wall are considered, the particle is assumed to be primarily translating parallel to the wall. The lift force is mainly in the direction perpendicular to the wall and is given by Ahmadi and Chen [3] as
The lift forces in other directions are small and hence are neglected. The Staffman’s expression for the lift forces is restricted to the condition that the particle Reynolds number based on the particle-fluid slip velocity is much smaller than the square root of the particle Reynolds number based on the shear field. McLaughlin (1991,1993) evaluated the correction factor Jm and relaxed these restrictions. When gravity is in the direction of flow, the lift force enhances the particle deposition rate. For upward flow, however, the lift forces will move particles away from the wall and reduces the deposition rate.
Gravitational Force
When indoor aerosol particles enter the room, the larger particles
(particle size > 4m) deposit on internal surfaces quickly under the influence of gravity. For such particle deposition, gravitational force has a major role to play. Gravitational force also has the influence on the deposition of the indoor particles when the size of the particles is between 2m and 4m. When gravity is in the same direction as the flow, particles move much faster than the flow due to gravitational sedimentation effect. The gravitational force per unit mass is given by Ahmadi and Chen [3] as
Velocity Distribution
Doors and windows can be used to supply fresh air to rooms of a building but to guarantee energy efficient air exchange rate mechanical ventilation is preferred. For mechanical ventilation either displacement ventilation or mixing flow systems can be used with supply flow rate reduced to meet hygienical requirements. For comfort ventilation in rooms, vertical displacement flow systems have become very popular. Wall mounted diffusers supply air directly into the occupied zone at low velocity. The hot air released by the people entrains to the occupied zone and creates an upward flow. To know the velocity distribution in a room, it becomes important to examine the flow in front of an air terminal device and to investigate whether this flow can be treated independent of parameters such as room geometry, heat source location, location of exhaust opening etc. In order to simplify the design procedure it is assumed that the flow depends only on certain parameters such as type of diffuser, obstacles on floor, flow rate and Archimedes number.
In our study it is assumed that the flow is radial flow from a single diffuser wherein the air movement is not influenced by the sidewalls. Radial flow is the flow close to the floor that has a virtual origin close to the diffuser. Radial flow at the diffuser is common and is generated by blades in the diffuser in such a way that the velocity obtains a radial distribution at the surface.
The velocity distribution near the floor is described by the decay of the maximum velocity ux as a function of distance x. The velocity decay as given by Nielsen [7] is
where
h is the height of the diffuser and x0 is the distance to a virtual origin for self-preserving flow, which in the study is assumed to be 1 feet.
The face velocity, uf, is given as the supply flow rate divided by the face area, af, of the diffuser
The equation was further modified by Nielsen [8] to
where
e is a factor that represents the initial increase in flow rate due to entrainment in the accelerating flow close to the opening.
bm is a factor which adjusts the flow in the direction = 0 to the flow profile generated by the diffuser.
KDr is independent of the distance x+ x0, but is a function of the Archimedes number.
The velocity in most cases will be proportional to 1/(x+ x0) and the value of KDr is adjusted to the situation to predict the velocity. A lot of experiments have been conducted to establish a relationship between KDr and Archimedes number. KDr is different for different products as it varies from 5 to 13 ms-1 at high Archimedes number. From the experiments it was found that KDr increases with increasing Archimedes number. The reason for this being that the gravity will accelerate the vertical flow close to the opening and this will generate an air movement in a relatively thin layer along the floor. The design of a diffuser is influenced by the parameters in equation (2.21). A KDr value of 7 is obtained as given by Nielsen [8],
when
If the value of bm is changed to 1.5 the KDr value changes to 11 as explained in Nielsen [8]. However in order to obtain more information about the general air movement, more measurements have to be taken and more experiments have to be performed in this field.
Figure 1 shows the difference in the distribution of the indoor air contaminant in the zones having displacement ventilation and mixed ventilation.
Supply air inlet
Supply air inlet
Displacement Ventilation Mixed Ventilation
Figure 1: Displacement Ventilation and Mixed Ventilation
Assumptions made in the development of the model
When an aerosol particle enters a room, it has an initial velocity uo because of which it travels a distance xi. Based on the distance it travels within a specific period of time, the velocity distribution in the room under displacement ventilation is studied from which the deposition velocity of the aerosol particle is calculated. A model has been formulated to study the aerosol dynamics based on the diameter of the particles. In order to develop this model certain assumptions have been made. They are
(a)No heat and mass transfer between the air and the particles.
(b)Rebound effect is neglected.
(c)When a particle reaches a wall it will stick to the surface.
(d)Negligible forces acting on particles of varying diameters are neglected.
(e)Aerosol size distribution is pretty constant.
(f)Aerosol particles are spherical in shape.
(g)Airflow is homogeneously and isotropically turbulent.
(h)There is no generation of particles indoors.
The size of the aerosol particle is taken as the prime criterion to develop the model. Based on the size of the aerosols, the prominent forces acting are considered and from that the various parameters such as distance traveled, deposition velocity of the particle, velocity distribution etc. influencing aerosol dynamics are calculated. Sections 2.8, 2.9, 2.10, 2.11 and 2.12 show the various steps involved in the development of the model.
Equations for the deposition velocity and distance traveled when the diameter of the particle is less than 0.01m (d<0.01m)
The basic governing equation as given by Chen and Ahmadi [3] gives the rate of change of velocity as a function of the Brownian force per unit mass.
where
and
Its known that
Substituting the value of uip from equation (2.24) in equation (2.25) we get
Rearranging the terms we get
Further investigations have to be done to get the value of Gi.