Handout discussing non-dimensionalization1 Non-dimensionalization of the Equations of Motion:

Most all of you have now used the Buckingham Pi Theorem as described by Sections 7.1 7.6 to systematicallydetermine Pi terms for your project, and are currently working on using the ideas of 7.8 to determine the modelscaling. While the Buckingham Pi Theorem is a good method to help you construct a systematically complete set of Piterms, it unfortunately does not always give you the most useful terms directly for your problem, since there are manysets of Pi terms that can legitimately be constructed depending on your choice of repeating variables. How to get pastthis problem? This is where research into the literature on similar problems can provide useful guidance (what didothers find useful?) or prior experience from modeling or experiments can provide insight. If simple analytical modelcan be constructed, non-dimensionalization of this equation can be a useful means of determining what form of the Piterms will be important, as described by section 7.10 of your book.

Non-dimensionalizing the equations that describe your system is one useful means to obtain insight into whatgrouping of dimensional parameters form useful Pi terms for scaling your system. For the current project, the equationof motion for an isolated particle (i.e. mpdVp/dt = ΣF ) is an important component of the model, as this willdetermine the trajectory of a particle within the flow. Integrating this equation for a particle released from a specifiedlocation will determine whether or not it will impact a wall, and hence (if the model is reasonably accurate) wouldallow you to determine the collection efficiency, provided a sufficient number of particle paths were determined acrossthe inlet to the device.

0

-0.5

1 2 3 4 50

0.5y

t-1

1.5

1

2

A = 5, B = 5A = 1, B = 5

Figure 1: Solution to the equation y′′ + Ay′ + By = 0 for(A,B)=(5,5) and (1,5).

Write your equation of motion in a complete asform as possible (keeping all relevant external forceterms), and non-dimensionalize the equation using ap-propriate length, time and/or velocity reference valuesfor your independent variables (x, y, and t). Note thatyou will need to express your force expression on theright hand side in terms of the dimensional constants(such as the density or the size of the particle) and theindependent variables (i.e. the position and time) ortheir derivatives (velocity). Regroup the coefficientsof this ODE on the right-hand side of the equationand examine to see if you can recognize groupings ofyour original Pi terms. These groupings collectivelydictate the behavior of the particle motion. This iswhy if the Pi term coefficients for the model and pro-totype are the same, the behavior will be the same inboth systems. The sensitivity of your model to these coefficients will also dictate how important they are in terms ofmaintaining similarity for that term between the model and the prototype.

A generic example of this is shown in Figure 1, which gives the response to the equation y′′ + Ay′ + By = 0with the initial conditions y(0) = 2 and y′(0) = 0 for two different values of the parameter A. If this were to be yournon-dimensionalized system, A and B would be the Pi terms of the problem, which shows that changing this Pi termwill give a drastically different result, each of which bears little resemblance to the other case.

In terms of completing the dimensional analysis, a very simple but familiar example can be seen by considering theballistic motion of a projectile in a vacuum, specifically desiring to know how far it travels horizontally before landing,L, (which is our dependent unknown parameter) while requiring the projectile to attain a given height h (being a given,this is an independent parameter). For this case, the equation of motion and initial conditions are expressed as:

mpd2~x

dt2= mp~g with ~x|t=0 = (0, 0) and

d~x

dt

∣∣∣∣t=0

= (V0 cos θ, V0 sin θ)

L, h, V0, θ, g andmp are the natural dimensional parameters for the problem. It turns out, however, thatmp completelycancels out of the problem, making it not relevant to the analysis. Taking, h as our reference value for a length, and V0as a reference velocity, we can non-dimensionalize our independent variables ~x and t by:

x? =x

hy? =

y

ht? =

t

h/V0

1

noting that h/V0 is a natural time constant when using this choice of variables. Substituting this in our equation andregrouping on the right-hand side gives the following non-dimensional equations and initial conditions:

d2x?

dt?2= 0 with x?|t?=0 = 0 and

dx?

dt?

∣∣∣∣t=0

= cos θ

d2y?

dt?2= − gh

V 20

with y?|t?=0 = 0 anddy?

dt?

∣∣∣∣t=0

= sin θ

Here we see that the group[gh/V 2

0

]forms a natural Pi group for this problem, and that θ appears only in the boundary

conditions.

2 Particle response time and drag models for particles

Much of what is discussed below, and more, can be found in the useful book: Aerosol Technology: Properties,Behavior, and Measuremetn of Airborne Particles by William C. Hinds, John Wiley and Sons, New York, (1999).

1.0

0.8

0.6

0.4

0.2

01 2 3 4 5

t/τp

Vp(t)U0

Vp=0.63U0

Figure 2: Particle velocity history when released from rest in a con-stant, uniform fluid velocity U0.

For particles particles moving through a fluid, theviscous forces often dominate the drag force. Physi-cally, this give rise to a strong damping force on theparticle motion that constantly tries to force the par-ticle velocity to be equal to the surrounding fluid ve-locity. The relative strength of this viscous drag forcein comparison to the inertia of the particle will controlhow quickly the particle adapts to changes in the flow.If the particle has a very small mass (such as a pieceof dust) the particle will very quickly adapt to changesin the surrounding velocity. If, on the other hand, it isa very massive particle (like say a golf ball) it wouldtake much longer to respond, and will basically ignorechanges in the surrounding fluid. As a consequence,the idea of a viscous particle response time is often a useful parameter in particle motion.

The above concept can be made more quantitative by considering what happens if we take a small particle at rest,and suddenly expose it to a constant, uniform fluid velocity U0. The one-dimensional equation of motion is given bythe first order equation:

mpdVpdt

= Fdrag with Vp|t=0 = 0

For small particles, where the Reynolds number based on particle size is small (i.e. Rep = ρf |Vf − Vp|D/µ < 1),the drag force can be expressed by what is commonly referred to as the Stokes drag force:

Fdrag,Stokes = 3πµD (Vf − Vp) ,

where Vf is the fluid velocity at the particle position. Noting that for a spherical particle mp = ρpπD3/6 and using

the Stokes drag law with the fluid velocity given as a constant U0, one can simplify the above equation to:

dVpdt

=18µ

ρpD2(U0 − Vp) , with Vp|t=0 = 0

Solving the above equation gives the solution

Vp(t) = U0

[1− exp

(− t

τp

)],

where τp ≡ ρpD2/18µ is defined as the viscous particle response time. This is the classical damped solution to a

first-order system response, as shown in Figure 2. Here one can see that τp represents the time it takes for the particleto adjust to 63% of the fluid velocity (assuming a step change in velocity).

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In more complicated flows, this is a convenient time scale to categorize a particles behavior. For example, if wehave an unsteady flow (or at least if the particle see a changing fluid velocity as it moves through a steady flow) thathas a characteristic time scale of τf , then the ratio of τp/τf would give an indication of how well the particle canfollow changes in the flow. If τp/τf << 1, then the particle should follow the flow more or less as a fluid element dueto the fact that it adjusts to the changes in the flow much quicker than they are occurring. If the particle response timeis much larger than the fluid time scale, the opposite it true: the particle can only correct a small faction of its velocitybefore the fluid has already made a large change, rendering it unable to respond.

For particle conditions where Rep is not small, the Stokes drag law is often adjusted with an empirical formula de-veloped by comparing across many experimental tests. Although many such corrections can be found in the literature,one particularly simple and effective one is given by (White, Viscous Fluid Flow, 2nd ed, McGraw-Hill, New York,1991):

Fdrag = 3πµD (Vf −Vp)

[1 +

Rep

4(1 +

√Rep

) +Rep60

]The above is accurate to within several percent for Rep < 100 and within 10% over the particle Reynolds numberrange 100 < Rep < 2× 105. Other empirical approximations are possible and may be a better choice.

3