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Formation of gas bubbles of a flowing liquid in a region where the pressure of the liquid falls below its vapor pressure

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• Good approximation to assume that the critical pressure for the onset of cavitation is the usual vapourpressure.

• Mathematical Analysis

classical Bernoulli equation 𝑝 +1

2𝜌𝑣2 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡

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I• High Velocities

II

• Pressure decreases below vapor pressure at that Temperature

III• C A V I TAT I O N O C C U R S

• Cryogenic liquids, including oxygen, nitrogen, and hydrogen, are popular fuels

• Propellants because the by-products are clean and the power/gallon ratio is high

• Issue Rocket fuel and oxidizer pumps is the minimumpressure that the design can tolerate for a given inlettemperature and rotating speed

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C R Y O G E N I C E N G I N E S

Low Inlet Pressure/reduce

Tank Weight

Pump Rotational Speeds High

(Reduce Engine weight)

More chances of Cavitation

(pitting, noise)

• The occurrence of cavitation is determined by the condition that the minimum pressure in the flow pmin is lower or at least equal to the vapour pressure pv

• An index that measures the degree of development of cavitation

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• Cavitation in cryogenic fluids generates substantial thermal effects and strong variations in fluid properties, which in turn alters the cavity characteristics

• Water - very large ratio between liquid and vapor densities, around O(105), Negligible thermal effects

• For cryogenic fluids, the liquid/vapor density ratios are not as high

• Other quantities such as latent heat and thermal conductivity can influence the thermal field more substantially

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• Sensitivity of the vapor pressure with respect to temperature

• Clapeyron Eqn:

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Fluid dPv /dT (KPa / K)

Water 0.19

Nitrogen 20

Hydrogen 28

• Significantly higher slopes of pressure-temperature saturation curve than water, vapor pressure can vary substantially due to the thermal effect

• Local temperature drop due to the evaporative cooling is non-negligible in cryogenic liquids

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SubstanceCp

(KJ/Kg.K)Density

RatioThermal

ConductivityL

(kg/K) ∆v

Water(298K) 4200 43220 681 2442 43.35

N2(83K) 2075 95 134 190 0.12

H2(20k) 9484 57 103 446 0.79

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Variation of physical properties for liquid nitrogen and water along saturation curve

• Homogeneous-fluid modeling

• Favre-averaged Navier-Stokes equations

Continuity Equation

Momentum Conservation

Energy Conservation

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Cavitation Modelling

K-ε Turbulence Model

Assumes that the turbulent viscosity is isotropic

Two Transport Equations – Turbulent kinetic Energy and Dissipation

Mixture property,

Enthalpy

Vapor mass fraction

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Transport based Cavitation Model

Rate of Liq./Vapor Evaporation and Condensation

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Boundary Conditionsσ∞=1.7, Re=9.1 × 106, T∞=83.06K And σ∞=1.61,

Re=1.1 × 107, T∞=88.54K

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Cryogenic cavitation Case 290C, the impact of thermal effect on local

cavitation number (μT / μL =103, σ∞=1.7, Re=9.1 × 106, T∞=83.06K)

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• Computational Framework developed for Cavitation

• Prediction of Cavitation made easier

• Cavitation dependent on the inlet turbulent characteristics

• Larger is the inlet velocity, more is the spread in the cavity

• Differences in Cryogenic and Isothermal cavitation

• Evaporative cooling reduces cavitation intensity and results in a shorter cavity size than that under isothermal conditions

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• Knapp R.T. , Daily J.W. and Hammitt F.G., “Cavitation. McGraw-Hill”, New York, 1970.

• Hord J., “Cavitation in Liquid Cryogens”, II-Hydrofoil. NASA CR-2156 1973

• Tseng, C., and Wei Shyy. "Turbulence Modeling for Isothermal and Cryogenic Cavitation." AIAA Paper No. 2009-1150. In (2009).

• Zhang, X. B., et al. "Computational fluid dynamic study on cavitation in liquid nitrogen." Cryogenics 48.9 (2008): 432-438.

• Franc, Jean-Pierre. Physics and control of cavitation. GRENOBLE UNIV (FRANCE), 2006.

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Contours of Static Pressure

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Contours of Volume Fraction of vapor

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