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.