kN/m gage. The discharge flow is 0.15 m /s and the outlet diameter of the discharge pipe is 375 mm. The motor driving the pump is 50 hp. Calculate TDH.
Vb = -t-—-j-""-— = -1-T- = 1.36 m/s cross sectional area of pipe n( 0.375 )2/4
Assume temperature of water = 25°C; therefore, density of water = 997 kg/m 196( 1000), 1.362 . t .
4.4 pump scaling laws
When designing a pumping station or specifying sizes of pumps, the engineer refers to a pump characteristic curve that defines the performance of a pump. Several different sizes of pumps are used, so theoretically, there should also be a number of these curves to correspond to each pump. In practice, however, this is not done. The characteristic performance of any other pump can be obtained from the curves of any one pump by the use of pump scaling laws, provided the pumps are similar. The word similar will become clear later.
The following dependent variables are produced as a result of independent variables either applied to a pump or are characteristics of the pump, itself: the pressure developed AP (corresponding to TDH), the power given to the fluid P, and the efficiency n. The independent variables applied to the pump are the discharge Q, the viscosity of the fluid f, and the mass density of the fluid p. These are variables applied, since they came from outside of and are introduced (applied) to the pump. The independent variables that are characteristics of the pump are the diameter of impeller or length of stroke D, the rotational speed or stroking speed o, some roughness e of the chamber, and some characteristic length € of the chamber space. There may still be other independent variables, but experience has shown that the forgoing items are the major ones. Although they are considered major, however, some of them may still be considered redundant and can be eliminated as will be shown in the succeeding analysis.
For aP the functional relationship may be written as
At large Reynolds numbers the effect of viscosity f is constant. For example, consider the Moody diagram. At large Reynolds numbers, the plot of the friction factor f and the Reynolds number, with f as the ordinate, is horizontal. Both f and f are measures of resistance to flow; thus, they are directly related. Because the effect of f at large Reynolds numbers is constant, the effect of f at large Reynolds numbers must also be constant. The rotation of the impeller or the movement of the stroke occurs at an extremely rapid rate; consequently, the flow conditions inside the pump casing are turbulent or are at large Reynolds numbers. Hence, since f is constant at high Reynolds numbers, it does not have any functional relationship with aP and may be removed from Equation (4.21). € as a measure of the pump chamber space is already included in D. It may also be dropped. Lastly, since the casing is too short, the effect of roughness e is too small compared to the other causes of the aP. It may therefore be also dropped. Equation (4.21) now takes the form
Applying dimensional analysis, let [x] be read "the dimensions of x." Thus, [AP] = F/L2, [p] = Ft2/L4 , [o] = 1/t, [D] = L, and [Q] = L3/t. By inspection of these dimensions, the number of reference dimensions is three. Because five variables are used, by the pi theorem, the number of n terms is two (number of variables minus number of reference dimensions, 5 - 3 = 2). Let the n terms be nj and n2, respectively, and proceed with the dimensional analysis.
To eliminate the dimension F, divide AP by p. Thus, aP
To eliminate t, divide by œ as follows:
To completely eliminate dimensions, divide by D as follows:
Therefore ni =
To get n2, operate on Q to obtain
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