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

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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.

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Abbreviations

A :

Area (m2)

B :

Virial coefficients

C :

Sonic velocity (m s−1)

C L :

Specific heat capacity of liquid (J/kg K)

d :

Number of main dimensions

D :

Diffusion coefficient

D e :

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)

h fg :

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)

m L :

Droplet mass (kg)

M L :

Liquid mass flow rate (kg s−1)

M T :

Total mass flow rate (kg s−1)

n :

Total number of dimensional (physical) variables

N :

Number of molecules per unit mass

P :

Pressure (kPa)

P r :

Pressure ratio

\(\dot{P}\) :

Dry expansion rate

q c :

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)

α r :

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

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

*:

Critical condition

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Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, Hakim Sabzevari University, Sabzevar, Iran

    Fahime Salmani

  2. Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

    Mohammad Reza Mahpeykar

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  1. Fahime Salmani
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Correspondence to Mohammad Reza Mahpeykar.

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Appendix

Appendix

Nozzle Young A

  • Converging length: L = 81.724 mm

  • Converging radius: R = 127.0 mm

  • Nozzle depth: depth = 12.395 mm = constant

  • Diverging section: W = 24.926 + 0.041358 x2 0 < x < 2.541 mm

  • W = 24.650 + 0.2101x 2.541 < x < 100.0 mm

Nozzle Young B

  • Converging length: L = 83.0 mm

  • Converging radius: R = 150.11 mm

  • Nozzle depth: depth = 12.395 mm = constant

  • Diverging section: W = 24.9869 + 0.007923 x2 + 0.0001637 x3 0 < x < 10 mm

  • W = 23.749 + 0.20757x 10 < x < 117.0 mm

Nozzle Moore

figure a

Nozzle Krol

  • Converging length: L = 43.18 mm

  • Nozzle depth: depth = 25 mm

  • Converging section: width = 30.58 mm − 0.02269x

  • Diverging section: W = 29.6 + 0.002578 x2 0 < x < 15.24 mm

  • W = 30.18 + 0.07857x 15.24 < x < 254.0 mm

Nozzle Binnie and Green

  • Converging length: L = 50.39 mm

  • Converging radius: R = 124.87 mm

  • Nozzle depth: depth = 22.20 mm = constant

  • Diverging section: W = 19.075 + 0.02758 x2 0 < x < 1.27 mm

  • W = 19.032 + 0.07005x 1.27 < x < 157.0 mm

Nozzle Binnie and Woods

  • Converging length: L = 49.784 mm

  • Converging radius: R = 156.53 mm

  • Nozzle depth: depth = 9.625 mm = constant

  • Diverging section: W = 35.382 + 0.0002546 x2 + 0.003148 x3 0 < x < 3.81 mm

  • W = 35.030 + 0.13903x 3.81 < x < 158.0 mm

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Salmani, F., Mahpeykar, M.R. Dimensional analysis for estimating wetness terms of condensing steam using dry flow data. J Therm Anal Calorim 137, 2121–2134 (2019). https://doi.org/10.1007/s10973-019-08108-9

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  • DOI: https://doi.org/10.1007/s10973-019-08108-9

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