2.3 Heat transfer model in the membrane module.
As a cooling process, the heat transfer process determines the membrane surface temperature, which also determines the nucleation driving force. As shown in Figure 1d, the ideal hollow fiber membrane can be regarded as a cylindrical wall. The inner and outer radius of the cylindrical wall are respectively r 1 andr 2; the membrane length is L ; the temperatures of the inner and outer surfaces are maintained at constant temperatures t 1 and t 2, respectively.
As a simplified model, for the membrane length exceeds more than ten times of the outer diameter of the membrane outer diameter, both ends of the heat transfer area of the cylindrical wall is negligible, and the axial direction and the circumferential direction thermal conductivity is also negligible. The temperature varies only on the radial direction, named one-dimensional steady-state heat conduction of the long cylindrical wall. Unlike flat wall heat transfer, the heat transfer area of the cylinder wall is not constant and varies with radius. When solving the radial heat conduction problem of the cylindrical wall, it is convenient to apply the cylindrical coordinate system. A one-dimensional steady-state heat transfer equation describing the absence of an internal heat source is44,45,
\(\frac{d}{\text{dr}}\left(r\frac{\text{dt}}{\text{dr}}\right)=0\)(18)
The boundary conditions are shown as followed45,
\(r=r_{1},\ t=t_{1}\)\(r=r_{2},\ t=t_{2}\) (19)
The solution satisfying the above boundary conditions is45,
\(t=t_{1}-\frac{t_{1}-t_{2}}{\ln\left(\frac{r_{2}}{r_{1}}\right)}\ln\frac{r}{r_{1}}\)(20)
The above equation shows that the temperature distribution is the logarithm of \(r\) when radial heat conduction through the cylinder wall function. With the decreasing r, the temperature gradient can also be modified with a high accuracy on the one-dimensional, which is a nonnegligible features of heat transfer process via a hollow fiber membrane module.
According to Fourier’s law, the heat transfer rate through the wall of the radius \(r\) is,
\(\Phi=-\text{λA}\frac{\text{dt}}{\text{dr}}\) (21)
where \(A\) represents the surface area of the cylinder wall,\(A=2\pi rL\). \(\frac{\text{dt}}{\text{dr}}\) represents temperature gradient.
A single-layer cylindrical wall steady-state heat transfer rate equation can be obtained,
\(\Phi=\frac{2\pi L\left(r_{2}-r_{1}\right)\lambda\left(t_{1}-t_{2}\right)}{\left(r_{2}-r_{1}\right)\ln\frac{2\pi r_{2}L}{2\pi r_{1}L}}=\frac{\left(A_{2}-A_{1}\right)\lambda\left(t_{1}-t_{2}\right)}{\left(r_{2}-r_{1}\right)\ln\frac{A_{2}}{A_{1}}}=\lambda A_{m}\frac{t_{1}-t_{2}}{b}=\frac{\text{Δt}}{R}\)(22)
where \(b\) represents thickness of the cylinder wall, m. \(R\)represents thermal resistance of the cylindrical wall,\(R=\frac{b}{\lambda A_{m}}\), °C /w. \(A_{m}\) represents Logarithmic average area, m2. The hollow fiber membrane is compared to a hollow cylinder having a thickness. The thermal conductivity and the thickness of the cylinder are jointly considered to illustrate the heat transfer in the hollow fiber membrane module. With the model above and known thermal conductivity of the membrane materials, we can predict the interfacial supercooling degree and then carry out further nucleation and growth kinetic simulation.