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1、2.2.4.2 Centrifugal pump theoryTheoretical headFor an ideal centrifugal pump the equation interrelating the developed head and capacity is called basic equation. The basic equation can be derived from fundamental principles of fluid dynamics. In an ideal pump, liquid is assumed to flow through the u
2、nit without friction, and at any given cross section all to flow at the same velocity. As shown in the vector diagram for a single vane, Fig. This assumption in turn is equivalent to assumption that there are an infinite number of vanes, of zero thickness, at an infinitesimal distance apart.The powe
3、r input to the impeller, and therefore the power required by the pump, can be calculated from the angular-momentum equation for steady flow. The velocity V has radial and tangential components Vr and Vt, respectively. The angular momentum of a mass m of fluid is therefore rmVt. 2.2-13 Since work don
4、e per second N=T ( angular velocity), the power equation for an ideal pump is (where =90,Vt1=0) Work done per second by the impeller on the fluid equates the rate of energy transfer 2.2-14 2.2-15 Therefore, combining Eq(2.2-14) and Eq.(2.2-15) gives Since the relationship between the angular velocit
5、y and the tangential velocityr2=u2 2.2-16 2.2-17 From Fig.3.3-10 (vector diagram at tip of vane)From Fig.3.3-10 (vector diagram at tip of vane)From Fig. 2.20 andandandand H is the ideal theoretical head developedIn practice, the total head across the pump is less than this due to energy dissipation
6、in eddies and in friction. The relating the ideal theoretical developed head and the capacity of the pumpThe volumetric flow rate Q Where A is the cross section area of the impeller periphery, therefore the relation between head and volumetric flow. Substituting from Eq.(2.2-20 ) into Eq.(2.2-19 ) 2
7、.2-20 Since u2, A, and 2 are constant for a given rotating speed and a given size of the pump.Eq(2.2-21 ) shows that the relation between head and volumetric flow rate is linear, then 2.2-21 2.2-22 The equation is for a ideal pump. The plot of theoretical head H versus capacity Q is shown in figure.
8、 Lower curve indicates the relation between the actual head and capacity of centrifugal pump The developed head of an actual pump is considerably less than that calculated from the ideal pump relation.2 90o are forward facing and Head increases with Q, Pump performance also depends on 2.ideal HQ 2 9
9、0=90 90oDifference between Theoretical and Actual Head Developed The effects of speed on capacity, head and power - The Affinity Laws The affinity laws are mathematical expressions that define changes in pump capacity, head, and power when a change is made to pump speed, impeller diameter, or both.
10、When the rotational speed n of a pump is increased, tip speed u2 rises proportionally; in an ideal pump velocity V2, Vt2, and Vr2 also increase directly with n. As rough approximations, the following relationships, called affinity laws, can be used for a given pump. The capacity Q is directly propor
11、tional to the rpm n. According to the equation 2.2-20 : Head H changes in direct proportion to the square of Q, or the square of speed n:2.2-22 2.2-23 The power consumed W is proportional to the product of H and Q, or2.2-24 A given pump can be modified when needed for a different capacity by changing the impeller size. Then the affinity laws for a constant-rpm n are as follows: l The capacity Q is proportional to the diameter D, l the head H is proportional to D2, l and the brake horsepower W is proportional to D3.
