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Non-axisymmetric three-dimensional stagnation-point flow and heat transfer under a jet impingement boiling

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Abstract

The non-axisymmetric three-dimensional flow and heat transfer in the stagnation-point region of a planar jet impingement boiling on a flat surface has been investigated by using similarity solution approach, considering additional diffusivity terms in momentum and energy equations as a result of bubble-induced mixing in flow. The free jet stream along z direction impinges on the surface and produces a flow with different velocity components. This situation may happen if the flow pattern on the plate is bounded from both sides in one of the directions, because of any physical limitation or due to conditions of the surface such as moving plates or stretching sheets with different values of stretching velocities in the x and y directions. The governing equations have been transformed into ordinary differential equations by introducing appropriate similarity variables, and an exact solution has been obtained for three-dimensional boiling problem for the first time. The similarity variables have been presented based on non-axisymmetric three-dimensional and additional diffusivity effects. The bubble-induced diffusion due to bubble formation, growth, departure and collapse causes an enhancement in heat transfer rate from the surface to the bulk flow. The total heat flux transferred from the surface to the flow has been estimated as summation of the single-phase heat transfer due to forced convection and the nucleate boiling heat flux due to bubble-induced diffusion. The effects of the velocity components ratio and the ratio between the maximum total diffusivity to the molecular diffusivity on the flow field and heat transfer characteristics have been obtained and discussed and illustrated graphically. A comparison of the predicted heat flux has been made with previously published experimental data. As expected, the average deviation values show relatively more accurate results for the three-dimensional simulation than the two-dimensional one because of being closer to the experimental conditions.

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Abbreviations

c p :

Specific heat (J kg−1 °C−1)

C :

Velocity gradient

f :

Dimensionless function used in Eq. (10)

g :

Dimensionless function used in Eq. (10)

h :

Heat transfer coefficient (W m−2 °C−1)

Ja:

Jacob number

k :

Thermal conductivity (W m−1 °C−1)

M :

Decay function of the thermal additional diffusivity, defined in Eq. (15)

N :

Decay function of the momentum additional diffusivity, defined in Eq. (14)

p :

Pressure (N m−2)

Pr:

Prandtl number

\( \Pr_{\text{t}} \) :

Bubble-induced Prandtl number

\( q^{\prime \prime } \) :

Heat flux (W m−2)

\( \text{Re}_{\text{j}} \) :

Jet Reynolds number

T :

Temperature (°C or K)

\( \Delta T \) :

Temperature difference (°C or K)

u :

Velocity component in x direction (m s−1)

v :

Velocity component y direction (m s−1)

\( V_{\text{j}} \) :

Jet velocity

w :

Velocity component in z direction (m s−1)

\( w_{\text{j}} \) :

Jet width (m)

We:

Weber number

\( \alpha \) :

Molecular thermal diffusivity (m2 s−1)

\( \nu \) :

Molecular kinematic diffusivity (m2 s−1)

\( \mu \) :

Molecular dynamic diffusivity (kg ms−1)

\( \rho \) :

Density

\( \lambda \) :

Velocity components ratio

\( \varepsilon \) :

Additional diffusivity (m2 s−1)

\( \varepsilon^{ + } \) :

Dimensionless total diffusivity

\( \eta \) :

Dimensionless distance from surface

\( \theta \) :

Dimensionless temperature

axi:

Axisymmetric

b:

Bubble

cr:

Critical value

FDB:

Fully developed boiling

j:

Jet related value

h:

Thermal energy

l:

Liquid

m:

Momentum

max:

Maximum value

model:

Model-predicted value

nb:

Nucleate boiling

s:

Surface (wall)

sp:

Single phase

sub:

Sub-cooled

sup:

Superheat

v:

Vapor

\( \upvarepsilon \) :

Related to the bubble-induced diffusivity

w:

Related to the jet width

2-D:

Two-dimensional value

3-D:

Three-dimensional value

\( \infty \) :

Free stream related value

′:

First derivative

″:

Second derivative

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Acknowledgements

Hereby, the financial support of Ferdowsi University of Mashhad under contract No. 2/47377 is acknowledged during accomplishment of this research.

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Authors and Affiliations

  1. School of Engineering, University of Guelph, Guelph, ON, Canada

    M. R. Mohaghegh & Shohel Mahmud

  2. Faculty of Engineering, Ferdowsi University of Mashhad, P.O. Box No. 91775-1111, Mashhad, Iran

    Asghar B. Rahimi

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  1. M. R. Mohaghegh
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Correspondence to Asghar B. Rahimi.

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Mohaghegh, M.R., Rahimi, A.B. & Mahmud, S. Non-axisymmetric three-dimensional stagnation-point flow and heat transfer under a jet impingement boiling. J Therm Anal Calorim 145, 211–224 (2021). https://doi.org/10.1007/s10973-020-09694-9

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