I would like to ask about the Bernoulli equation analysis regarding the increase in pressure behind the pipe of a centrifugal pump

The principle of Bernoulli’s equation primarily describes the conservation of fluid energy (including pressure energy, kinetic energy, and potential energy) as the fluid moves. Its formula is: P + ρgh + 0.5ρv^2 = constant, where P is the pressure energy, ρgh is the potential energy of the fluid, 0.5ρv^2 is the kinetic energy, ρ is the density of the fluid, g is the acceleration due to gravity, h is the horizontal height, and v is the flow velocity. The function of a centrifugal pump is to convert electrical energy into kinetic energy and pressure energy of the fluid through a motor. As the speed of a centrifugal pump increases, electrical energy is converted more effectively into kinetic energy of the fluid; the flow velocity rises and thus the kinetic energy increases. According to Bernoulli’s equation, the pressure of the fluid also increases at this point, meaning that the pressure downstream in the pipe rises as well. Specifically, we denote the states before and after the acceleration of rotation speed as State 1 (lower rotation speed) and State 2 (higher rotation speed), with the corresponding pressures, velocities, and heights being P1, v1, h1 respectively, and P2, v2, h2 respectively. Since the heights of the pump’s inlet and outlet remain essentially constant, h1 = h2. By applying Bernoulli’s equation, we obtain: P1 + 0.5ρv1^2 = P2 + 0.5ρv2^2. As the rotational speed increases, v2 > v1; therefore, it follows that P2 > P1. In other words, increasing the rotational speed of a centrifugal pump leads to an increase in the pressure at the outlet side of the pump. .

Bro, I’d like to ask: what exactly is this equation that shows the flow from one point to another? Is it from before the pump to after the pump? Why isn’t the head value associated with the centrifugal pump taken into account in this equation? Also, according to the equation you’ve formulated, when v2 is greater than v1, shouldn’t p1 be greater than p2 so that both sides of the equation remain equal? Please provide an explanation

With the impeller remaining constant, H/n^2 is a constant value; in other words, the head is proportional to the square of the rotational speed. This is related to the characteristics of the pump, and there is no direct relationship with Bernoulli’s equation.