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Journal of Thermal Analysis and Calorimetry

, Volume 137, Issue 6, pp 2121–2134 | Cite as

Dimensional analysis for estimating wetness terms of condensing steam using dry flow data

  • Fahime Salmani
  • Mohammad Reza MahpeykarEmail author
Article
  • 40 Downloads

Abstract

During rapid expansion in supersonic nozzles and turbine blades, under special conditions, steam may become supercooled vapor, and the heat release rate (\(\dot{Q}\)) due to phase change is substantial. Droplet radius (r) and wetness fraction (WF) are important parameters in designing wet steam equipment. Until now, cost-intensive and complicated methods are applied for designing wet steam equipment. In this paper, an innovative method based on Buckingham Pi dimensional analysis is proposed for predicting r and WF using dry vapor data. A dimensionless droplet radius (DDR) is obtained from the influential parameters at the Wilson point (named DWP). First, DWP, DDR, and WF are obtained from the results of the analytical modeling, and then, two regression equations are proposed for calculating DDR and WF with DWP. Finally, results of the proposed regression relationships are compared for seven analytical cases; the average percent errors associated with the presented equations for the droplet radius or DDR and WF percentage (\(\dot{Q}\)) are found to be less than 30% and 12%, respectively.

Keywords

Nucleating steam flow Wilson point Latent heat Wetness fraction Dimensional analysis Buckingham Pi 

List of symbols

A

Area (m2)

B

Virial coefficients

C

Sonic velocity (m s−1)

CL

Specific heat capacity of liquid (J/kg K)

d

Number of main dimensions

D

Diffusion coefficient

De

Hydraulic diameter

DDR

Dimensionless droplet radius

DWP

Droplet–wetness parameters

E

Total energy

f

Friction coefficient

G

Gibbs free energy change

h

Enthalpy (J kg−1)

hfg

Latent heat (J kg−1)

j

Number of repeated variables

J

Nucleation rate (#/m3 s)

Kn

Knudsen number

L

Divergent section length (m)

Ma

Mach number

\(\dot{m}\)

Mass flow rate (kg s−1)

mL

Droplet mass (kg)

ML

Liquid mass flow rate (kg s−1)

MT

Total mass flow rate (kg s−1)

n

Total number of dimensional (physical) variables

N

Number of molecules per unit mass

P

Pressure (kPa)

Pr

Pressure ratio

\(\dot{P}\)

Dry expansion rate

qc

Condensation coefficient

\(\dot{Q}\)

Heat release rate due to phase change (W)

r

Droplet radius (μm)

R

Gas constant for steam (= 461.4 J/kg K)

Sc

Schmidt number \(\left( {\mu_{\text{G}} /\rho_{\text{G}} D} \right)\)

T

Temperature (K)

U

X-component velocity (m s−1)

V

Velocity (m s−1)

WF

Wetness fraction (ML/MT)

x

Cartesian direction (m)

Greek symbols

αr

Droplet convectional heat transfer coefficient (W/m2 K)

t

Time (s)

δ

Tolman coefficient

ΔT

Degree of supercooling (K)

λ

Thermal conductivity of vapor (W/m K)

μ

Viscosity (m2 s−1)

ρ

Density (kg m−3)

π

Dimensionless group

τ

Viscous stress tensor (N m−2)

θ

Thermal (K)

σ

Liquid surface tension (N m)

Γ

Density of interfacial region

Subscript

0

Stagnation

i

Inlet

K

Kalva’s surface tension correction

L

Liquid

s

Saturation

T

Total

G

Gas (vapor)

WP

Wilson point

Flat surface tension

Superscript

*

Critical condition

Notes

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, Faculty of EngineeringHakim Sabzevari UniversitySabzevarIran
  2. 2.Department of Mechanical Engineering, Faculty of EngineeringFerdowsi University of MashhadMashhadIran

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