FIGURE 6.5 Normalization of geometric measurements into dimensionless ratios.

the fluid, the harder it is to push it, thus requiring more power. In addition, the power requirement must also increase as the speed of rotation is increased. Note that N is expressed in radians per second.

As shown in Figure 6.3, a vortex is being formed, raising the level of water higher on the wall and lower at the center. This rising of the level at one end and lowering at the other is has to do with the weight of the water. Because the weight of any substance is a function of gravity, the gravity g must enter into the functionality of the power given to the fluid. The shape factors have already been nondimension-alized, so we will ignore them for the time being and consider only the diameter Da of the impeller as their representative in the functional expression for the power. Letting the power be P,

Now, to continue with our dimensional analysis, let us break down the variables of the previous equation into their respective dimensions using the force-length-time (FLT) system as follows:

Variable Dimensions, FLT


N 1/T only if N is in radians per unit time; a radian by definition is dimensionless

Da L

p M/iL = FT2/L4, M is the dimension of mass in the MLT system

By inspection, the number of reference dimensions is 3; thus, the number of pi variables is 6 - 3 = 3 (the number of variables minus the number of reference dimensions). Reference dimension is the smallest number of groupings obtained from grouping the basic dimensions of the variables in a given physical problem. Call the pi variables nb n2, and n3, respectively. Letting nj contain P, write [PIN] = (FL/T)/ (1/T) = FL to eliminate T. [ ] is read as "the dimensions of." To eliminate L, write [P/NDa] = FL/L = F. To (1/T)2L4| = 1. Therefore,

[P/NDa] = FL/L = F. To eliminate F, write [P/NDa(1/pN2D4a)] = F(1/(FT2/L4)

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