Why is the shaft power at its maximum when the axial flow pump has zero flow rate, when theoretically flow rate and shaft power are proportional to each other?

Figure 1 Characteristic curves of axial flow pumps The way in which the characteristic curves of axial flow pumps are represented is the same as that of centrifugal pumps; the horizontal axis represents flow rate Q, while the vertical axis represents parameters such as head H, efficiency η, and shaft power P. However, the trends of the H-Q curve and the P-Q curve are quite different from those of centrifugal pumps, as shown in Figure 1. It can be seen that on the head curve, as the flow rate decreases starting from the point of highest efficiency (the optimal operating condition), the head gradually increases. When the flow rate drops to Q2, the head reaches the inflection point B; as the flow rate continues to decrease, the head also decreases until it reaches the second inflection point C. Starting from point C, the head increases rapidly as the flow rate further decreases, and when the flow rate is zero, the head can reach approximately twice the value at the optimal operating condition. For the condition where the flow rate is zero, it is commonly referred to as the shut-off condition; at this point, the pump achieves its highest head and maximum power. The corresponding head is called the shut-off point head, and the power is called the shut-off point power. Figure 2 Schematic diagram of secondary recirculation within the impeller of an axial flow pump. The characteristic curve of an axial flow pump has this shape because, as the flow rate decreases, the angle between the relative velocity of the fluid flow and the circumferential direction decreases, while the blade installation angle remains unchanged; as a result, the attack angle of the cascade airfoil increases. The smaller the flow rate, the greater the shock angle; when this shock angle increases to a certain level, flow separation occurs on the surface of the airfoil. Therefore, as the flow rate decreases from the optimal value Qe, the angle of attack of the airfoil increases, which raises the lift coefficient and thus increases the head pressure. When the flow rate decreases to less than Q2, flow separation occurs on the airfoil due to the excessive angle of attack, the lift coefficient drops, and the head loss increases. When the flow rate decreases to less than Q1, the head does not decrease but increases instead, due to secondary backflow of the liquid within the impeller. When the flow rate decreases to less than Q1, the head generated by each computational flow surface of the impeller is different, resulting in secondary backflow of the liquid (Figure 2); part of the liquid that flows out of the impeller returns to it again to receive energy once more. Secondary recirculation relies on impact to transfer energy, resulting in high hydraulic losses. As the head increases, it also consumes a large amount of power from the pump, causing the efficiency to drop sharply. Therefore, on the efficiency curve of an axial flow pump, the efficiency drops rapidly as the flow rate decreases, resulting in a narrow high-efficiency operating range. Given the characteristics of the axial flow pump’s performance curve, starting the pump with the outlet valve closed often makes it difficult to start the pump, and there is a risk of overload for the motor. Therefore, when starting an axial flow pump, the outlet pipe valve must be fully open in order to reduce the starting power required.