Critical pressure in liquids due to dynamic choking

The existence of a critical pressure ratio due to gas-dynamic choking is well known for an ideal gas. It is reasonable to assume that liquids whose compressibility is defined by the bulk modulus also have a critical pressure ratio. The problem discussed here is a fundamental one because it deals with the basic principles of the compressible flow of liquids. It has been shown that even though an ideal gas with a constant heat capacity ratio value has a critical pressure ratio, liquid with a constant bulk modulus value experiences a critical pressure difference. As the outlet pressure gradually decreases, the liquid reaches the local speed of sound, and further reduction of this pressure does not lead to an increase in mass flow. This phenomenon occurs in liquids without considering the change from a liquid to a gaseous phase. Behavior is confirmed analytically for different bulk modulus models, and for a constant bulk modulus value, the phenomenon is verified by numerical simulation using computational fluid dynamics. The conclusions published in this work point to striking analogies between the behavior of liquids and ideal gas. The equations governing the motion of liquids derived in this work, thus complete the fundamental description of the critical flow of fluids.

Figure 1

Function $\psi$ of an ideal gas for heat capacity ratio $\kappa =1.2$ and $\kappa =1.4$.

Figure 2

The course of function ${\psi }_{K_{\mathrm{const}}}$ of a liquid with a constant bulk modulus $K_{\mathrm{const}}$.

Figure 3

The course of function ${\psi }_{K_{\mathrm{lin}}}$ of a liquid having linear dependence on the bulk modulus $K_{\mathrm{lin}}$ of pressure depends on the ratio of the bulk moduli $K_{\mathrm{lin}}/K_0$ for selected coefficients $a$.

Figure 4

Computational domain representing the flow of liquid from the reservoir to the nozzle.

Figure 5

Distribution of Mach number in the computational domain for selected values of ratio $\mathrm{\Delta }p/K_{\mathrm{const}}$ obtained by CFD analysis performed for a constant bulk modulus.

Figure 6

The flow through the nozzle during an increase in the pressure difference between the inlet and outlet based on CFD analysis for the constant value of bulk modulus $K_{\mathrm{const}}$.

Figure 7

Dependence of water bulk modulus on pressure at constant temperatures 25 °C, 40 °C, and 60 °C.

Figure 8

The course of function ${\psi }_{K_{\mathrm{lin}}}$ on pressure $p$ for water at 25 °C for three values of reservoir pressure $p_0=1500,2000,$ and $3000\mathrm{MPa}$.

Figure 9

Water: the course of critical pressure $p_{\psi \max }$ for different values of inlet pressure $p_0$ at 25 °C when considering $K_{\mathrm{lin}}$ and $K_{\mathrm{const}}$.

Figure 10

The influence of parameter $a$ in equation $K_{\mathrm{lin}}=ap+b$ on the value of the minimum pressure in the liquid reservoir when liquid-dynamic choking may occur.