Pore Network Simulations of Heat and Mass Transfer inside an Unsaturated Capillary Porous Wick in the Dry-out Regime

Abstract

In this work, a two-dimensional pore network model is developed to study the heat and mass transfer inside a capillary porous wick with opposite replenishment in the dry-out regime. The mass flow rate in each throat of the pore network is computed according to the Hagen–Poiseuille law, and the heat flux is calculated based on Fourier’s law with an effective local thermal conductivity. By coupling the heat and the mass transfer, a numerical method is devised to determine the evolution of the liquid–vapor interface. The model is verified by comparing the effective heat transfer coefficient versus heat load with experimental observations. For increasing heat load, an inflation/deflation of the vapor pocket is observed. The influences of microstructural properties on the vapor pocket pattern and on the effective heat transfer coefficient are discussed: A porous wick with a non-uniform or bimodal pore size distribution results in a larger heat transfer coefficient compared to a porous wick with a uniform pore size distribution. The heat and mass transfer efficiency of a porous wick comprised of two connected regions of small and large pores is also examined. The simulation results indicate that the introduction of a coarse layer with a suitable thickness strongly enhances the heat transfer coefficient.

Appendices

Appendix 1: Influence of the Convergence Parameter $\xi$ (cf. Sect. 3.5) on the Vapor Pocket Pattern and on the Computation Time

The simulations are performed for four different values of the convergence parameter $\xi$. All other parameters used in the simulations are given in Table 1. Figure 17 shows the dependency of the normalized computation time $\tau$ and of the vapor area fraction $f_\mathrm{v}$ on the values of $\xi$. The normalized time is defined as the ratio of the time required for the simulation with a given value of $\xi$ to the simulation time with $\xi = 0.00$, which varies from 4 to 7 h depending on throat size distribution. Due to less iterations for a larger value of $\xi$, the computation time can be reduced by about 10 % by increasing $\xi$ from 0.00 to 0.01. Since the results obtained for these two values of $\xi$ do not deviate too much, $\xi = 0.01$ is chosen for all simulations presented in this study. As can also be seen, the vapor pocket patterns obtained for the values of $\xi = 0.00$ and 0.01 slightly differ.

Fig. 17

Influence of convergence parameter $\xi$ on the vapor area fraction and on the normalized computation time $\tau$. The inset figure shows the vapor pocket patterns obtained from simulations with different values of $\xi$

Appendix 2: Supplementary Investigation on Convective Heat Transfer and Viscous Dissipation

To develop the present model, several simplifications are made. The convective heat transfer and the viscous dissipation are assumed to be negligible. In this appendix, a supplementary investigation is made to examine the validity of these simplifications. The simulations are performed with water as a working fluid and with the input parameters stated in Table 1. From the pressure and phase distributions, the mass flow rate $\dot{M}_{ij}$ in each throat is calculated by Eq. 1. The fluid interstitial velocity in each throat is computed as

$$\begin{aligned} v_{0, ij} =\frac{\dot{M}_{ij} }{\rho _\mathrm{f} \pi r_{ij}^2 } \end{aligned}$$

(24)

With the transfer area between two neighboring pores $A_{\mathrm{cv}, ij} ={ WL }$, the superficial velocity v is calculated by Eq. 25.

$$\begin{aligned} v_{ij} =\frac{v_{0,ij} \pi r_{ij}^2 }{ WL } \end{aligned}$$

(25)

To evaluate the contribution of heat transported by convection to the total heat flux, the Péclet number Pe, which is a dimensionless value representing the ratio between convective heat transfer and conductive heat transfer is used.

$$\begin{aligned} Pe=\frac{\hbox {convective}\; \hbox {heat}\; \hbox {flux}}{\hbox {conductive}\; \hbox {heat}\; \hbox {flux}}\approx \frac{v\rho _\mathrm{f} c_{p, f} }{\lambda _\mathrm{eff} /L} \end{aligned}$$

(26)

Fig. 18

Variation of the averaged Péclet number for different values of the heat load $\dot{q}$

Fig. 19

Variation of the temperature gradient due to viscous heating for different values of the heat load $\dot{q}$

The superficial velocity used in this calculation is the average value for the vapor and liquid regions. The evolution of the Péclet number in vapor and liquid regions for different values of the heat load $\dot{q}$ is presented in Fig. 18. As can be seen, the Péclet number in both vapor and liquid regions is very small (lower than 0.002). It implies that the convective heat transfer is very small (approximately 0.2 %) compared to the conductive heat transfer, and thus the heat transported by convection is negligible in the heat flux calculations with the network configurations used in these simulations. This result also indicates that with the wick composed of non-insulated solid material, i.e., $\lambda _\mathrm{s} > 1$ W/mK, the heat transported by convection can be neglected (lower than 2 %). However, in the case of a solid skeleton made by low thermal conductivity material such as a polymer or plastic (for which the thermal conductivity varies from 0.5 to 0.2 W/m K), the contribution of the convective heat transfer to the total heat flux is significant and this term should be considered in a more comprehensive pore network model.

In viscous flow, the mechanical energy is converted into thermal energy by viscous dissipation; thus, the temperature of the fluid increases. Morini (2005) and Han and Lee (2007) proposed an equation to compute the fluid temperature gradient along a smooth cylindrical throat due to the viscous heating as follows:

$$\begin{aligned} \frac{\mathrm{d}T}{\mathrm{d}L}=\frac{8v_o \upsilon _\mathrm{f} }{c_{p, f} r^{2}} \end{aligned}$$

(27)

The computational results are presented in Fig. 19. The temperature gradient in the vapor flow is significantly higher than in the liquid flow because of the higher kinematic viscosity and lower density. The temperature gradient in vapor throats varies from 0.63 to 11.61 K/m in a heat load range from $\dot{q} = 20\,\hbox {kW/m}^{2}$ to $\dot{q} = 140\,\hbox {kW/m}^{2}$. The maximum increase of the vapor temperature (for a heat load of $\dot{q} = 140\,\hbox {kW/m}^{2})$ due to the viscous heating in a throat with length of $L = 1$ mm and $\hbox {L}_{\mathrm{y}} = 44$ mm is 0.011 and 0.51 K, respectively. These temperature differences are extremely small compared to the fin superheat value (164.32 K) at these operating conditions. Therefore, the influence of viscous dissipation on the fluid flow is truly negligible in the pore network configurations used in this study.