# Conductive and viscous sub-layers on forced convection and mechanism of critical heat flux during flow boiling of subcooled water in a circular tube at high liquid Reynolds number

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## Abstract

The turbulent heat transfer, the subcooled boiling heat transfer and the steady state CHF for a Pt-circular test tube of a 3 mm inner diameter and a 100 mm heated length are measured with a wide range of inlet subcooling and flow velocity at high liquid Reynolds number, i.e. *Re*_{d} = 3.01×10^{4} to 1.43×10^{5}. The inner surface temperature of the Pt-circular test tube calculated by the steady one-dimensional heat conduction equation is compared with the values derived from authors’ turbulent heat transfer correlation and with the numerical solutions of the RANS equations (Reynolds Averaged Navier-Stokes Simulation) of *k-ε* turbulence model for the flow velocities ranging from 4 to 21 m/s. The thicknesses of conductive sub-layer from non-boiling regime to CHF are measured by numerically analyzing the heat transfers with conductive sub-layer on forced convection and with thinner one dissipated by the evaporation on nucleate boiling. The thicknesses of viscous sub-layer on forced convection are estimated from the thicknesses of the conductive sub-layer and Prandtl numbers of the surface temperature on the heated surface. Furthermore, the thicknesses of conductive sub-layer at the CHF point are extrapolated from the measured values at various flow velocities. The experimental values of the CHF are also compared with authors’ widely and precisely predictable correlations of critical heat flux during flow boiling of subcooled water and the corresponding theoretical values of the liquid sub-layer dry-out models suggested by other researchers, respectively. The authors’ correlations and other researchers’ theoretical values can represent the subcooled boiling CHFs obtained in this study within the ranges of −13.27 to 6.76% difference and − 32.51 to 13.16% one, respectively. A suggestion based on the experimental data as to what the dominant mechanism is for critical heat flux during flow boiling of subcooled water on a vertical circular tube is confirmed again at high liquid Reynolds number. The transitions to film boiling at the subcooled water flow boiling on the Pt test tube of *d* = 3 mm and *L* = 100 mm would occur due to the liquid sub-layer dry-out model at the steady-state CHF as well as those on the Pt test tube of *d* = 3 mm and *L* = 66.5 mm, but not due to the heterogeneous spontaneous nucleation and the hydro-dynamic instability.

## Nomenclature

*Bo**q/Gh*_{fg}, boiling number*Bo*_{cr}*q*_{cr,sub,st}*/Gh*_{fg}, boiling number at CHF point*C*_{1},*C*_{2},*C*_{3},*C*_{4},*C*_{5},*C*_{6}*c*_{pl}Specific heat at constant pressure, J/kgK

*d*Test tube inner diameter, m

*f*_{F}Fanning friction factor

*G**ρ*_{l}*u*, mass velocity, kg/m^{2}s*h*_{fg}Latent heat of vaporization, J/kg

*L*Heated length, m

*L*_{eff}Effective length, m

*Nu*_{d}*hd*/*λ*_{l}, nusselt number*P*_{cr}22064 kPa, critical pressure, kPa

*P*_{in}Pressure at inlet of heated section, kPa

*P*_{ipt}Pressure measured by inlet pressure transducer, kPa

*P*_{out}Pressure at outlet of heated section, kPa

*P*_{opt}Pressure measured by outlet pressure ransducer, kPa

*Pr**c*_{p}*μ/λ*, Prandtl number*(Pr)*_{Ts}Prandtl number of the surface temperature on the heated surface under forced convection

*Q*Heat input per unit volume, W/m

^{3}*Q*_{0}Initial exponential heat input, W/m

^{3}*q*Heat flux, W/m

^{2}*q*_{cr,sub,st}Steady-state CHF for subcooled condition, W/m

^{2}*Ra*Average roughness, μm

*Re*_{d}*Gd/μ*_{l}, Reynolds number*Rmax*Maximum roughness depth, μm

*Rz*Mean roughness depth, μm

*r*_{i}Test tube inner radius, m

*r*_{o}Test tube outer radius, m

*(Δr)*_{out}Outer control volume width for

*r*-component, m*TEM*Calculated temperature of the outer control volume, K

*T*Water temperature, C

- \( \overline{T} \)
Average temperature of test tube, K

*T*_{f,av}Average liquid temperature, K

*T*_{in}Inlet liquid temperature, K

*T*_{L}(

*T*_{in}+*T*_{out})/2, liquid bulk mean temperature, K*T*_{out}Outlet liquid temperature, K

*T*_{s}Heater inner surface temperature, K

*T*_{sat}Saturation temperature, K

*T*_{so}Heater outer surface temperatures, K

*T*_{s,av}Average inner surface temperature, K

*ΔT*_{L}*(T*_{s,av}*-T*_{L}*)*, temperature difference between average inner surface temperature and liquid bulk mean temperature, K*ΔT*_{sat}*T*_{s}*-T*_{sat}, inner surface superheat, K*u*Flow velocity, m/s

*y*^{+}*y(τ*_{w}*ρ*_{l}*)*^{0.5}*/ν*_{l}, dimensionless normal-distance coordinate*y*^{+}_{CSL}*(f*_{F}*/2)*^{0.5}*ρ*_{l}*uδ*_{CSL}*/*μ_{l}, non-dimensional thickness of conductive sub-layer*δ*Conductive sub-layer on nucleate boiling heat transfer

*δ*_{CSL}*(Δr)*_{out}*/2*, thickness of conductive sub-layer and conductive sub-layer on forced convection*δ*_{VSL}Thickness of viscous sub-layer on forced convection

*ε*Rate of dissipation of turbulent energy, m

^{2}/s^{3}- μ
_{l} Viscosity, Ns/m

^{2}- μ
_{w} Viscosity at tube wall temperature, Ns/m

^{2}*ν*_{l}*μ*_{l}*/ρ*_{l}, kinematic viscosity of fluid, Ns m/kg*ρ*_{l}Density of fluid, kg/m

^{3}*τ*_{w}shear stress at the wall, N/m

^{2}*χ*Vaper quality

## Notes

### Acknowledgements

This research was performed as a LHD joint research project of NIFS (National Institute for Fusion Science), Japan, NIFS15KEMF066, 2015, 2016 and 2017.

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