The mathematical model of pumping liquid foodstuffs by a centrifugal pump with an open impeller was proposed in this article. The liquid levels difference in the initial and final tank changes during pumping. Therefore, the pumping process is non-stationary with a slowly changing fluid flow rate. The mathematical model includes two differential equations: the balance of the pumped liquid and the dynamics of the fluid flow in the pipeline. The operating characteristics of the centrifugal pump were approximated by second-order polynomials according to the test data of the manufacturers of this equipment. The need to take into account the influence of fluid viscosity on the pump performance is shown in the article. The pump pressure is noticeably reduced, and the power consumed increases for concentrated apple juice compared to pumping water. The system of differential equations with corresponding boundary conditions was solved numerically. The flow rate very quickly reaches a value corresponding to the difference in levels at the beginning of pumping. Therefore, the problem can be solved in a quasi-stationary approximation. The change in fluid flow over time is close to linear. Therefore, it is possible to find the pumping time of a given volume by an approximate method. The approximate calculation algorithm has been proposed. It was used to calculate the pumping of a concentrated apple juice. The effect of changes in the diameter and length of the pipeline on the performance and energy efficiency of the pumping unit has been investigated.