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Theoretical investigation of effects of local cooling of a nozzle divergent section for controlling condensation shock in a supersonic two-phase flow of steam

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Abstract

In supersonic adiabatic two-phase flows of steam, under the influence of supersonic acceleration, the fluid loses its equilibrium conditions and becomes supersaturated. Following this condition and to restore the fluid to equilibrium, micro droplets of water form in the absence of any surface or foreign particles. This phenomenon is called homogeneous nucleation and the formed minute small droplets grow along the fluid flow path. The formation of these droplets and their growth causes the release of the latent heat of evaporation to the gas phase particularly in the nucleation region, and results in an increase in the flow pressure which is called the condensation shock. In this paper, and in continuation of the series of papers by the authors, in addition to analytically solving the adiabatic gas-liquid supersonic flow of steam in a convergent-divergent channel, a novel solution to controlling the undesired effects of this pressure rise (condensation shock) is presented. In the proposed method, with the help of cooling the divergent section of the nozzle, the analytical model for the 1D non-adiabatic two-phase steam flows is further developed which shows considerable decrease in the intensity of the formed condensation shock. Also the growth rate of the formed droplets due to the cooling of the steam flow has higher importance than the nucleation itself.

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Abbreviations

A :

Area

C P :

Specific heat at constant pressure

D e :

Equivalent diameter

f :

Friction factor

ΔG :

Change in Gibbs free energy

J :

Rate of formation of Critical droplets per unit volume and time

Kn :

Knudsen number (mean free path/droplet diameter)

L :

Latent heat

Ma :

Mach number

m r :

Mass of droplet

P :

Vapour pressure

P 0in :

Inlet Total Pressure

P s (T G ):

Saturation pressure at T G

q :

Condensation coefficient

R :

Gas constant for water vapour

r :

Radius of droplet

T :

Temperature

T s (P):

Saturation temperature at P

ΔT :

Degree of supercooling [T s (P)−T G ]

t :

Time

U :

Velocity

v :

Specific volume

\(\dot{q}\) :

Volumetric heat transfer rate

\(\dot{Q}\) :

Total heat transfer rate

Qc1 & Qc2 :

Volumetric Cooling (Heat rejection)

M :

Total mass flow rate

x :

Distance along Nozzle axis

Xt :

Nozzle throat length

X,Y :

Functions of temperature and density in equation of state

α r :

Coefficient of heat transfer

γ :

Isentropic component

μ G :

Kinematic viscosity of vapour

ζ :

Dryness fraction

ρ :

Density of mixture

λ :

Coefficient of thermal conductivity

σ :

Surface tension

ρ s (T L ,r):

Density corresponding to saturation pressure at temperature T L over a surface of curvature r

Sc :

Schmidt number

G :

Vapour phase

L :

Liquid phase

0:

Stagnation condition

S :

Saturation

∗:

Critical droplet

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Author information

Authors and Affiliations

  1. Ferdowsi University of Mashhad, Mashhad, Iran

    M. R. Mahpeykar & A. R. Teymourtash

  2. Hakim Sabzevari University, Sabzevar, Iran

    E. Amiri Rad

Authors
  1. M. R. Mahpeykar
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  2. A. R. Teymourtash
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  3. E. Amiri Rad
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Corresponding author

Correspondence to M. R. Mahpeykar.

Appendix

Appendix

(I) Gas phase state equation Different methods have been proposed for the state equation of the two-phase flow. In this research considering the differential model of the flow, the virial state equation [21] is used:

(A.1)

where the B 1,B 2 and B 3 factors are known as the virial density factors which are functions of the gas phase temperature [9].

The differential state equation for a real gas is as follows:

(A.2)

where X and Y are defined as:

(A.3)
(A.4)

The Virial coefficients of Vukalovich’s state equation and the thermodynamic properties are as follows:

(A.5)

(II) The equation of specific heat at constant pressure:

(A.6)

(III) The equation of specific heat at constant volume:

(A.7)

(IV) The equation of isentropic index

(A.8)

(V) Vapour phase Mach number

The Mach number used in the governing equations of steam flow can be expressed as:

(A.9)

where C is the frozen speed of sound, i.e., in dry real steam.

Neglecting isentropic index variations, taking the logarithm and differentiating of the above equation gives:

(A.10)

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Mahpeykar, M.R., Teymourtash, A.R. & Amiri Rad, E. Theoretical investigation of effects of local cooling of a nozzle divergent section for controlling condensation shock in a supersonic two-phase flow of steam. Meccanica 48, 815–827 (2013). https://doi.org/10.1007/s11012-012-9634-2

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  • DOI: https://doi.org/10.1007/s11012-012-9634-2

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