Theoretical Analysis of Cryogenic Fluid Evaporation in Sintered Microporous Structures

Abstract

Evaporation in microstructures has been proven to be a heat transfer enhancement compared with evaporation on flat surfaces due to the thin films formed in the porous or grooved surfaces. To investigate the evaporation characteristics in sintered porous wicks for cryogenic loop heat pipes, a more realistic geometric model to characterize the porous skeleton is developed. Then, a mathematical model considering the Marangoni effect and slip velocity effect to predict the liquid–vapor interface is built. The evaporation interface can be divided into the thin-film region and intrinsic meniscus region. A new solution method calculating from the intersection of these two regions is established and validated, and the initial boundary conditions are not dependent on the contact angle whose value is small and ambiguous for cryogenic fluids. Then the evaporation characteristics of the thin-film region are discussed. About 40% of the temperature drop occurs in the thin-film region, and the evaporation thermal resistance should be considered in this region. Finally, the parametric analysis is conducted to study the effects of superheat, pressure difference at the interface, accommodation coefficient and pore radius on the interface profile and evaporation heat transfer process in sintered microporous structures. The liquid–vapor interface recedes with the superheat increasing and the mean heat transfer coefficient of the interface decreases. With the increase of the pressure difference at the interface, the liquid–vapor interface recedes and the mean heat transfer coefficient decreases. The mean heat transfer coefficient increases with the accommodation coefficient despite the coverage of the liquid–vapor interface reducing and small pores would be helpful to enhance heat transfer in porous media.

Appendix A

The liquid velocity profile can be written as,

$$u\left(r,\theta \right)=\frac{r}{2\mu }\frac{{\text{d}}{p}_{l}}{{\text{d}}\theta }-\frac{R\;\mathrm{ln}\left(R-r\;\mathrm{cos}\;\theta \right)}{2\;\mathrm{cos}\;\theta \mu }\frac{{\text{d}}{p}_{l}}{{\text{d}}\theta }\text{+}\frac{{C}_{1}}{R}\mathrm{ln}\left(\frac{r}{R-r\;\mathrm{cos}\;\theta }\right)+{C}_{2}$$

(33)

Then, the first derivative of the liquid velocity can be expressed as,

$$\frac{\partial u}{\partial r}=\frac{1}{2\mu }\frac{{\text{d}}{p}_{l}}{{\text{d}}\theta }+\frac{R}{2\mu }\frac{1}{R-r\;\mathrm{cos}\;\theta }\frac{{\text{d}}{p}_{l}}{{\text{d}}\theta }\text{+}\frac{{C}_{1}}{r}\frac{1}{R-r\;\mathrm{cos}\;\theta }$$

(34)

The boundary conditions are as follows.

$$\left\{\begin{array}{c}{\left.u\right|}_{r={r}_{0}}={\left.\beta \frac{\partial u}{\partial r}\right|}_{r={r}_{0}}\\ {\left.\frac{\partial u}{\partial r}\right|}_{r={r}_{0}+\delta }=\frac{1}{\mu }\frac{{\text{d}}\sigma }{r{\text{d}}\theta }\end{array}\right.$$

(35)

Substitute Eqs. (33) and (34) into Eq. (35),

$$\left\{\begin{array}{l}\frac{{r}_{0}}{2\mu }\frac{{\text{d}}{p}_{l}}{{\text{d}}\theta }-\frac{R\;\mathrm{ln}\;\left(R-{r}_{0}\mathrm{cos}\;\theta \right)}{2\;\mathrm{cos}\;\theta \mu }\frac{{\text{d}}{p}_{l}}{{\text{d}}\theta }\text{+}\frac{{C}_{1}}{R}ln\left(\frac{{r}_{0}}{R-{r}_{0}\mathrm{cos}\theta }\right)+{C}_{2}\\ =\beta \left(\frac{1}{2\mu }\frac{{\text{d}}{p}_{l}}{{\text{d}}\theta }+\frac{R}{2\mu }\frac{1}{R-{r}_{0}\mathrm{cos}\;\theta }\frac{{\text{d}}{p}_{l}}{{\text{d}}\theta }\text{+}\frac{{C}_{1}}{{r}_{0}}\frac{1}{R-{r}_{0}\mathrm{cos}\;\theta }\right)\\ \frac{1}{2\mu }\frac{{\text{d}}{p}_{l}}{{\text{d}}\theta }+\frac{R}{2\mu }\frac{1}{R-\left({r}_{0}+\delta \right)\mathrm{cos}\;\theta }\frac{{\text{d}}{p}_{l}}{{\text{d}}\theta }\text{+}\frac{{C}_{1}}{{r}_{0}+\delta }\frac{1}{R-\left({r}_{0}+\delta \right)\mathrm{cos}\;\theta }=\frac{1}{\mu }\frac{{\text{d}}\sigma }{\left({r}_{0}+\delta \right){\text{d}}\theta }\end{array}\right.$$

(36)

According to the second equation in Eq. (36), C₁ can be written as,

$${C}_{1}=\left(\begin{array}{c}-\frac{1}{2\mu }\left({r}_{0}+\delta \right)\left(2R-\left({r}_{0}+\delta \right)\mathrm{cos}\;\theta \right)\frac{{\text{d}}{p}_{l}}{{\text{d}}\theta }\\ \text{+}\frac{1}{\mu }\left(R-\left({r}_{0}+\delta \right)\mathrm{cos}\;\theta \right)\frac{{\text{d}}\sigma }{{\text{d}}\theta }\end{array}\right)$$

(37)

Then, take the expression of C₁ into the first equation in Eq. (36), and C₂ can be solved:

$$\begin{array}{l}{C}_{2}=\left(\begin{array}{c}\beta \frac{1}{2\mu }\frac{2R-{r}_{0}\mathrm{cos}\;\theta }{R-{r}_{0}\mathrm{cos}\;\theta }-\frac{1}{2}\frac{1}{\mu }{r}_{0}+\frac{R}{2\;\mathrm{cos}\;\theta }\frac{1}{\mu }ln\left(R-{r}_{0}\mathrm{cos}\;\theta \right)\\ -\left(\begin{array}{c}\beta \frac{1}{{r}_{0}\left(R-{r}_{0}\mathrm{cos}\;\theta \right)}\\ -\frac{1}{R}ln\left(\frac{{r}_{0}}{R-{r}_{0}\mathrm{cos}\;\theta }\right)\end{array}\right)\frac{1}{2\mu }\left({r}_{0}+\delta \right)\left(2R-\left({r}_{0}+\delta \right)\mathrm{cos}\;\theta \right)\end{array}\right)\frac{{\text{d}}{p}_{l}}{{\text{d}}\theta }\\ \text{+}\left(\beta \frac{1}{{r}_{0}\left(R-{r}_{0}\mathrm{cos}\;\theta \right)}-\frac{1}{R}\mathrm{ln}\left(\frac{{r}_{0}}{R-{r}_{0}\mathrm{cos}\;\theta }\right)\right)\frac{1}{\mu }\left(R-\left({r}_{0}+\delta \right)\mathrm{cos}\;\theta \right)\frac{{\text{d}}\sigma }{{\text{d}}\theta }\end{array}$$

(38)