3.11. DIMENSIONAL ANALYSIS IN MOMENTUM TRANSFER

3.11A. Dimensional Analysis of Differential Equations

In this chapter we have derived several differential equations describing various flow situations. Dimensional homogeneity requires that every term in a given equation have the same units. Then, the ratio of one term in the equation to another term is dimensionless. Knowing the physical meaning of each term in the equation, we are then able to give a physical interpretation to each of the dimensionless parameters or numbers formed. These dimensionless numbers, such as the Reynolds number and others, are useful in correlating and predicting transport phenomena in laminar and turbulent flow.

Often it is not possible to integrate the differential equation describing a flow situation. However, we can use the equation to find out which dimensionless numbers can be used in correlating experimental data for this physical situation.

An important example of this involves the use of the Navier-Stokes equation, which often cannot be integrated for a given physical situation. To start, we use Eq. (3.7-36) for the x component of the Navier-Stokes equation. At steady state this becomes

Equation 3.11-1


Each term in this equation has the units length/time2, or L/t2.

In this equation each term has a physical significance. First we use a single characteristic velocity v and a single characteristic length L for all terms. Then the expression of each term in Eq. (3.11-1) is as follows: The left-hand side can be expressed as v2/L and the right-hand terms, respectively, as g, p/ρL, and μv/ρL2. We then write

Equation 3.11-2


This expresses a dimensional equality and not a numerical equality. Each term has dimensions L/t2.

The left-hand term in Eq. (3.11-2) represents the inertia force and the terms on the right-hand side represent, respectively, the gravity force, pressure force, and viscous force. Dividing each of the terms in Eq. (3.11-2) by the inertia force [v2/L], the following dimensionless groups or their reciprocals are obtained:

Equation 3.11-3


Equation 3.11-4


Equation 3.11-5


Note that this method not only gives the various dimensionless groups for a differential equation but also gives physical meaning to these dimensionless groups. The length, velocity, and so forth to be used in a given case will be that value which is most significant. For example, the length may be the diameter of a sphere, the length of a flat plate, and so on.

Systems that are geometrically similar are said to be dynamically similar if the parameters representing ratios of forces pertinent to the situation are equal. This means that the Reynolds, Euler, or Froude numbers must be equal between the two systems.

This dynamic similarity is an important requirement in obtaining experimental data for a small model and extending these data to scale up to the large prototype. Since experiments with full-scale prototypes would often be difficult and/or expensive, it is customary to study small models. This is done in the scale-up of chemical process equipment and in the design of ships and airplanes.

3.11B. Dimensional Analysis Using the Buckingham Method

The method of obtaining the important dimensionless numbers from the basic differential equations is generally the preferred method. In many cases, however, we are not able to formulate a differential equation which clearly applies. Then a more general procedure is required, known as the Buckingham method. In this method the listing of the important variables in the particular physical problem is done first. Then we determine the number of dimensionless parameters into which the variables may be combined by using the Buckingham pi theorem.

The Buckingham theorem states that the functional relationship among q quantities or variables whose units may be given in terms of u fundamental units or dimensions may be written as (q − u) independent dimensionless groups, often called π's. [This quantity u is actually the maximum number of these variables that will not form a dimensionless group. However, only in a few cases is u not equal to the number of fundamental units (B1).]

Let us consider the following example, to illustrate the use of this method. An incompressible fluid is flowing inside a circular tube of inside diameter D. The significant variables are pressure drop Δp, velocity v, diameter D, tube length L, viscosity μ, and density ρ. The total number of variables is q = 6.

The fundamental units or dimensions are u = 3 and are mass M, length L, and time t. The units of the variables are as follows: Δp in M/Lt2, v in L/t, D in L, L in L, μ in M/Lt, and ρ in M/L3. The number of dimensionless groups or π's is q − u, or 6 − 3 = 3. Thus,

Equation 3.11-6


Next, we must select a core group of u (or 3) variables which will appear in each π group and among them contain all the fundamental dimensions. Also, no two of the variables selected for the core can have the same dimensions. In choosing the core, the variable whose effect one desires to isolate is often excluded (for example, Δp). This leaves us with the variables ν, D, μ and ρ to be used. (L and D have the same dimensions.)

We will select D, ν, and ρ to be the core variables common to all three groups. Then the three dimensionless groups are

Equation 3.11-7


Equation 3.11-8


Equation 3.11-9


To be dimensionless, the variables must be raised to certain exponents a, b, c, and so forth. First we consider the π1 group:

Equation 3.11-7


To evaluate these exponents, we write Eq. (3.11-7) dimensionally by substituting the dimensions for each variable:

Equation 3.11-10


Next we equate the exponents of L on both sides of this equation, of M, and finally of t:

Equation 3.11-11


Solving these equations, a = 0, b = −2, and c = −1. Substituting these values into Eq. (3.11-7),

Equation 3.11-12


Repeating this procedure for π2 and π3,

Equation 3.11-13


Equation 3.11-14


Finally, substituting π1, π2, and π3 into Eq. (3.11-6),

Equation 3.11-15


Combining Eq. (2.10-5) with the left-hand side of Eq. (3.11-15), the result obtained shows that the friction factor is a function of the Reynolds number (as was shown before in the empirical correlation of friction factor and Reynolds number) and of the length/diameter ratio. In pipes with L/D >> 1 or pipes with fully developed flow, the friction factor is found to be independent of L/D.

This type of analysis is useful in empirical correlations of data. However, it does not tell us the importance of each dimensionless group, which must be determined by experimentation, nor does it select the variables to be used.

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