Assessment of the inclination surface on the microlayer behavior during nucleate boiling, a numerical study

Abstract

Nucleate boiling is an important part of pool boiling process. Heat transfer from the microlayer plays a considerable role in heat transfer to the fluid. The axisymmetric assumption of the microlayer for a horizontal surface needs to be evaluated for an inclined one. In this study, the effect of surface orientation on the microlayer thickness and the heat transfer rate are investigated numerically. The governing equations are simplified employing scaling analysis. The results for the microlayer thickness, the heat flux and the total heat transfer rate for the heated surface are obtained and presented. The asymmetry of the microlayer increases as the surface inclination angle varies from horizontal to vertical. Even though, the driving force due to gravity in the microlayer is negligible, however its effect on the macro region changes the microlayer parameters. The results show that the maximum microlayer heat transfer rate for the vertical surface increases by 28.8% compared to that for a horizontal surface. The proposed model, which is capable of evaluating the microlayer thickness and its surface heat transfer rate, can be employed as a surface boundary condition in the macro region simulations of the nucleate boiling.

Appendices

Appendix 1

$$ {A}_1={h}_{ev}{\rho}_v{h}_{fg}+\frac{k_l}{\delta}\kern0.5em {P}_1={\rho}_v{h}_{fg}{T}_v\kern0.5em {K}_1=\frac{\sigma {\delta}^{{\prime\prime} }}{{\left({\delta^{\prime}}^2+1\right)}^{5/2}} $$

$$ {A}_2=3\sigma {\delta^{{\prime\prime}}}^2\frac{\delta^{\prime }}{{\left({\delta^{\prime}}^2+1\right)}^{5/2}}\kern0.5em {p}_g=g\sin \alpha \left({\rho}_l-{\rho}_v\right)\left(\delta -{\delta}_e\right)\kern0.5em {h}_{ev}={a}_1{h}_{fg}\sqrt{\frac{M}{2\pi R{T}_{sat}^3}} $$

$$ {A}_3=\frac{1}{{\left({\delta^{\prime}}^2+1\right)}^{3/2}}\kern0.5em {p}_d=\frac{A}{\delta^3}\kern0.5em {P}_1={\rho}_v{h}_{fg}{T}_v $$

$$ {A}_4=\frac{1}{{\left({\delta^{\prime}}^2+1\right)}^{7/2}}\kern0.5em {C}_1=\frac{\mu_l{p}_v}{\rho_vR}\kern0.5em {P}_2=g{p}_1-{K}_1-{p}_d $$

$$ {A}_5=\frac{h_{ev}\left(-{C}_1{P}_2+{P}_1\right)+\frac{k_l{T}_w}{\delta }}{A_1}\kern0.5em {C}_2=\frac{1}{\rho_v{h}_{fg}^2}\kern0.5em {C}_3={\rho}_v{h}_{fg} $$

$$ {A}_6={\rho}_v{h}_{fg}\left({A}_5-{T}_v\right)+{C}_1{P}_1\kern0.5em {A}_7={h}_{ev}\left(-{C}_1{P}_2+{P}_1\right)+\frac{k_l{T}_w}{\delta}\kern0.5em {A}_8=\frac{9\sigma {\delta}^{{\prime\prime\prime} }{\delta}^{{\prime\prime} }{\delta}^{\prime }}{{\left({\delta^{\prime}}^2+1\right)}^{5/2}} $$

$$ {A}_9=-\frac{15\sigma {\delta^{{\prime\prime}}}^3{\delta^{\prime}}^2}{{\left({\delta^{\prime}}^2+1\right)}^{7/2}}\kern0.5em {A}_{10}=\frac{3\sigma {\delta^{{\prime\prime}}}^3}{{\left({\delta^{\prime}}^2+1\right)}^{5/2}}\kern0.5em {A}_{11}=g\left({\rho}_l-{\rho}_v\right){\delta}^{{\prime\prime} } $$

$$ {A}_{12}=-\frac{12A{\delta^{{\prime\prime}}}^2}{\delta^5}\kern0.5em {A}_{13}=\frac{3A{\delta}^{{\prime\prime} }}{\delta^4}\kern0.5em {A}_{14}=g\left({\rho}_l-{\rho}_v\right){\delta}^{\prime } $$

$$ {A}_{15}=\frac{3A{\delta}^{\prime }}{\delta^4}\kern0.5em {A}_{16}=-\frac{\sigma {\delta}^{{\prime\prime\prime} }}{{\left({\delta^{\prime}}^2+1\right)}^{5/2}}+{A}_2+{A}_{14}+{A}_{15} $$

$$ {A}_{17}={A}_8+{A}_9+{A}_{10}+{A}_{11}+{A}_{12}+{A}_{13}\kern0.5em {A}_{18}=-{h}_{ev}{C}_1{A}_{16}-\frac{k_l{T}_w{\delta}^{\prime }}{\delta^2} $$

$$ {A}_{19}=\frac{h_{ev}^2{A}_6\Big({\rho}_v{h}_{fg}\left(\frac{A_{18}}{A_1}+\frac{\delta^{\prime }{k}_l{A}_7}{\delta^2{A}_1^2}+{C}_1{A}_{16}\right)}{\rho_v{h}_{fg}^2} $$

$$ {A}_{20}=\frac{1}{2}\left({A}_2-{A}_{19}+{A}_{15}-{A}_{16}\right)-\frac{3}{2}\left(\frac{\sigma {\delta^{{\prime\prime}}}^3{\delta}^{\prime }}{{\left({\delta^{\prime}}^2+1\right)}^{5/2}}+\frac{A{\delta}^{\prime }}{\delta^4}\right)+\frac{1}{2}{\rho}_lg\sin \alpha \cos \varphi $$

$$ {A}_{21}=\frac{2{\delta}^{\prime }{k}_l}{\delta^2{A}_1^2}\left({A}_{18}+\frac{\delta^{\prime }{k}_l{A}_7}{\delta^2{A}_1}-\frac{\delta^{\prime }{A}_7}{\delta }+{\delta}^{{\prime\prime} }{A}_7\right) $$

$$ {A}_{22}=\frac{6A{\delta^{\prime}}^2}{\delta^5}-\frac{3}{2}\frac{A{\delta}^{{\prime\prime} }}{\delta^4}-\frac{1}{2}{A}_{11}\kern0.5em {\mathrm{A}}_{23}=-\frac{9}{2}\frac{\sigma {\delta}^{{\prime\prime} }{\delta}^{{\prime\prime} }{\delta}^{\prime }}{{\left({\delta^{\prime}}^2+1\right)}^{5/2}}\kern0.5em {\mathrm{A}}_{24}=\frac{2{k}_l{\mathrm{T}}_{\mathrm{w}}{\delta^{\prime}}^2}{\delta^3} $$

$$ {\mathrm{A}}_{25}=-\frac{k_l{\mathrm{T}}_{\mathrm{w}}{\delta}^{{\prime\prime} }}{\delta^2}\kern0.5em {\mathrm{A}}_{26}=3\left({\mathrm{A}}_{16}+{\mathrm{A}}_{19}\right)+{2\mathrm{A}}_{20}-3{\rho}_lg\sin \alpha \cos \varphi $$

$$ {A}_{27}=\frac{1}{3}\frac{C_1{A}_1^2{A}_{26}{\delta}^2+3{\rho}_v{h}_{fg}{A}_1{A}_{18}{\delta}^2+3{\rho}_v{h}_{fg}{k}_l{A}_7{\delta}^{\prime }}{A_1^2{\delta}^2} $$

$$ {A}_{28}=\frac{h_{ev}^2\Big({\rho}_v{h}_{fg}{\left(\frac{A_{18}}{A_1}+\frac{\delta^{\prime }{k}_l{A}_7}{\delta^2{A}_1^2}+{C}_1{A}_{16}\right)}^2}{\rho_v{h}_{fg}^2} $$

$$ {A}_{29}={A}_{17}{\left({\delta^{\prime}}^2+1\right)}^{3/2}\kern0.5em {A}_{30}=-\frac{1}{2}\frac{A_{26}{\delta}^2{\delta}^{\prime }}{\mu_l}\kern0.5em {A}_{31}=-\frac{1}{6}\frac{A_{26}{\delta}^3}{\mu_l} $$

$$ {A}_{32}=-\frac{r\left({A}_5-{T}_w\right){k}_l}{\rho_l{h}_{fg}\delta}\kern0.5em {A}_{33}={A}_1{A}_{21}+{A}_{24}+{A}_{25}-{A}_{29}{A}_3{C}_1{h}_{ev} $$

$$ {A}_{34}=-3{\delta^{{\prime\prime}}}^3\left({\delta^{\prime}}^2+1\right)\kern0.5em {A}_{35}=15{\delta^{{\prime\prime}}}^3{\delta^{\prime}}^2\kern0.5em {A}_{36}={C}_1{A}_1{A}_{29}{A}_3+{C}_3{A}_{35} $$

Appendix 2

Macro region equations:

The macro region equations including continuity, momentum and energy equations are defined as follows:

$$ \frac{\partial \rho }{\partial t}+\nabla \cdot \left(\rho u\right)={\dot{m}}_{masstransfer}, $$

(1)

$$ \frac{\partial \left(\rho u\right)}{\partial t}+\nabla \cdot \left(\rho\;u\;u\right)=-\nabla p+\nabla \cdot \left[\mu \left(\nabla u+\nabla {u}^T\right)\right]+\rho g+{F}_{\sigma }, $$

(2)

$$ \frac{\partial \left(\rho E\right)}{\partial t}+\nabla \cdot \left(u\left(\rho E+p\right)\right)=\nabla \cdot \left[{k}_{eff}\left(\nabla T\right)\right]+{S}_h, $$

(3)

The additional equation for tracking the interphase between the phases is the continuity equation for the volume fraction of one of the phases.

$$ \frac{\partial \left(\rho \alpha \right)}{\partial t}+\nabla \cdot \left(\rho u\;\alpha \right)={\dot{m}}_{mass\kern0.17em transfer}, $$

(4)

The mass transfer between phases has two parts. The first part is due to the overall mass transfer of vaporization around the bubble perimeter. The second part is due to the effect of microlayer. The first part is accounted by the employed boiling model automatically, while the second part should be added to the model. Following the method introduced by Son et al. [18], the additional mass transfer due to the microlayer ($\rho {\dot{V}}_{micro}$) is defined as follows:

$$ {\dot{V}}_{micro}=\underset{R_0}{\overset{R_l}{\int }}\frac{k_1\left({T}_w-{T}_{\mathrm{int}}\right)}{\rho_v{h}_{fg}\delta \varDelta {V}_{micro}} rdr. $$

(5)

Equation (5) is considered as a source term in the continuity equation of vapor phase.