Laboratory work №1 Construction of dynamic mathematical models of simple hydraulic systems The purpose of the work

Laboratory work № 1
Construction of dynamic mathematical models of simple hydraulic systems
The purpose of the work: to determine, using simulation, the optimal temperature for pumping a viscous liquid to achieve the optimal power of the pump motor.
To study the influence of the pipeline diameter on the pump power, if oil is pumped at different temperatures and different amounts of petroleum oil obtained at the process unit of an oil refinery, and draw technological conclusions about the choice of pipeline diameter.


Theoretical description
All basic processes (hydrodynamic, thermal, mass transfer, etc.) can proceed only under the action of a certain driving force, which for hydromechanical processes is determined by the pressure difference, for heat exchange processes - by the temperature difference, for mass transfer processes - by the difference in substance concentrations, etc.
Carrying out chemical technology processes is usually associated with the movement of liquids, gases or vapors in pipelines and apparatuses, the formation or separation of heterogeneous systems (mixing, dispersion, settling, filtering, etc.).
Hydraulics is divided into hydrostatics (the laws of equilibrium of fluids at rest) and hydrodynamics (the laws of fluid motion). At the same time, it is customary to combine liquids, vapors and gases under a single name - liquids, since at flow velocities much lower than the speed of sound, the laws of motion of liquids without significant amendments are valid for gases and vapors. Therefore, liquids are understood to mean all substances that have fluidity.
The mathematical model of the process of pumping a viscous liquid is built on the basis of the laws of hydraulics and is presented in Table. 1.

Except for the cross section of the pipeline and the linear pumping speed, all other determined variables are functions of temperature. It is clear that with increasing temperature, the viscosity of the oil decreases, which leads to a decrease in pressure losses due to friction and a decrease in the power of the pump motor. The higher the oil heating temperature, the lower the required pump power. However, these considerations do not answer the question of the optimal power of the pump motor.

If we translate our reasoning into the economic plane, then we will be able to formulate an optimization criterion. With an increase in temperature, the pump power decreases, but the cost of heating the oil increases, while the total costs should first decrease with an increase in temperature and then increase. The optimum transfer temperature will be the temperature at which the minimum of the total cost function is reached. The same temperature will correspond to the optimal power of the pump motor. In the Excel program, you can solve this problem graphically (approximately) and numerically (with the accuracy specified by the numerical method).
Technique for solving a task in Excel
The graphic solution of the problem is shown in fig. 2.1.

Below, comments will be given on entering formulas in the corresponding cells of the table. However, it is highly desirable that students create such a table on their own, which is why in Fig. 2.1 once again presents the equations of the mathematical model.

In the block of cells A3:C15, the initial data is entered. The solution is presented in cell block E3:P14. In the block of cells E8:E14, the research range is entered: the transfer temperature is from 20 to 70 °C. In cell F8, formula 3 is entered to calculate the density, which is then copied to cell F13 by dragging the autofill marker (this operation will be repeated for all subsequent cells in row 8).

Spreadsheet Design Guidelines
To enter Greek characters, use the command Insert -> Symbol.
To plot the dependence of pump power on temperature, it is necessary to select non-adjacent blocks E9:E14; K9:K14.
To plot the dependence of costs on temperature, non-adjacent blocks E9:E14 are selected; N9: P14. Using the Insert -> Scatter Plot command, a graph is built.
Diagrams are formatted to the form accepted in the technical literature. Formatting is conveniently done using menu commands, called by double-clicking the left mouse button on the corresponding diagram element.
It can be seen from the calculation table and graphs that the optimal temperature for pumping is about 40 °C.

Assignment for independent work
Study the effect of the pipeline diameter on the pump power, if the oil is pumped at the appropriate temperature, and draw technological conclusions about the choice of pipeline diameter.
Make the necessary edits to the copy of sheet 1 using the values ​​of your choice. In this case, be guided by the sample solution presented in Fig. 2.1. Make changes to the original data.