A Scaling Procedure for Predicting Pressure Fluctuations Caused by Fluid Transient in Cryogenic Systems

Abstract

The large-scale cryogenic fluid transfer systems used in various applications are commonly exposed to pressure oscillations caused by fluid transient phenomena. The analysis and experimentation of such system are made easy by developing scale-down models and performing numerous tests. A scaling procedure solely to fluid transient phenomena is developed by non-dimensionalizing the governing equations of fluid transient. The approach for developing scale-down model parameters is formulated using scaling laws and known physical laws. In addition, the method for analyzing the behavior of large-scale system from the scale-down model results is also established. Numerical tests are carried out to vindicate the developed procedure with the data available in the open literature. The methodology is successfully implemented to establish scale-down model for cryogenic systems transferring LOx and LH2.

Appendix A: Deriving Dimensionless Parameters from Governing Equations

First, Eq. (1) is converted into dimensionless form by substituting the above variables as

$$\frac{{\partial \left( {P^{*} \rho v^{2} } \right)}}{{\partial \left( {\frac{{t^{*} L}}{a}} \right)}} + \rho a^{2} \frac{{\partial \left( {\frac{{v^{*} \rho D^{2} a}}{\mu t}} \right)}}{{\partial \left( {x^{*} ta} \right)}} = 0$$

(A.1)

Then, the derivatives are rearranged in the form of original equation as

$$\rho v^{2} *\frac{a}{L}*\frac{{\partial P^{*} }}{{\partial t^{*} }} + \rho a^{2} *\frac{{\rho D^{2} a}}{\mu t}*\frac{1}{ta}*\frac{{\partial v^{*} }}{{\partial x^{*} }} = 0$$

(A.2)

$${\text{Multiplying}}\,{\text{with}}\,\frac{L}{{\rho v^{2} a}} \to \frac{{\partial P^{*} }}{{\partial t^{*} }} + \frac{{\rho^{2} a^{2} D^{2} }}{{t^{2} \mu }}*\frac{L}{{\rho v^{2} a}}\frac{{\partial v^{*} }}{{\partial x^{*} }} = 0$$

(A.3)

$$\frac{{\partial {\text{P}}^{*} }}{{\partial {\text{t}}^{*} }} + \alpha \frac{{\partial {\text{v}}^{*} }}{{\partial {\text{x}}^{*} }} = 0$$

(A.4)

where $\alpha = \frac{{\rho D^{2} aL}}{{\mu {\text{v}}^{2} t^{2} }}$ or $\alpha = Re*\frac{{aD^{2} }}{{{\text{v}}^{3} t^{2} }}$.

Following the same procedure for Eq. (2) results in

$$\frac{{\partial \left( {\frac{{v^{*} \rho D^{2} a}}{\mu t}} \right)}}{{\partial \left( {\frac{{t^{*} L}}{a}} \right)}} + \frac{1}{\rho }\frac{{\partial \left( {P^{*} \rho v^{2} } \right)}}{{\partial \left( {x^{*} ta} \right)}} + \frac{f}{2D}v\left| v \right| = 0$$

(A.5)

$$\frac{{\rho D^{2} a^{3} }}{\mu tL}\frac{{\partial v^{*} }}{{\partial t^{*} }} + \frac{{\rho v^{2} }}{\rho ta}\frac{{\partial P^{*} }}{{\partial x^{*} }} + \frac{f}{2D}v\left| v \right| = 0$$

$$\begin{aligned} & {\text{Multiplying}}\,{\text{with}}\,\frac{{\rho D^{2} a^{3} }}{\mu tL} \to \frac{{\partial {\text{v}}^{*} }}{{\partial {\text{x}}^{*} }} + \frac{{{\upmu }L{\text{v}}^{2} }}{{{\rho D}^{2} a^{3} }}\frac{{\partial {\text{P}}^{*} }}{{\partial {\text{x}}^{*} }} + \frac{{f{\text{v}}^{2} {\upmu }tL}}{{2{\rho D}^{3} a^{2} }} = 0 \\ & \quad \quad \quad \quad \quad \quad \quad \frac{{\partial v^{*} }}{{\partial x^{*} }} + \frac{{\mu Lv^{2} }}{{\rho D^{2} a^{3} }}\frac{{\partial P^{*} }}{{\partial x^{*} }} + \frac{{fv^{2} \mu tL}}{{2\rho D^{3} a^{2} }} = 0 \\ \end{aligned}$$

(A.6)

$$\frac{{\partial v^{*} }}{{\partial x^{*} }} + \beta \frac{{\partial P^{*} }}{{\partial x^{*} }} + \gamma = 0$$

(A.7)

where $\beta = \frac{{\mu Lv^{2} }}{{\rho D^{2} a^{3} }}$ and $\gamma = \frac{{fv^{2} \mu tL}}{{\rho D^{3} a^{2} }}$.