Experimental and numerical study of heat transfer in horizontal concentric annulus containing phase change material

Canadian Journal of Chemical Engineering, August, 2008 by Ritabrata Dutta, Arnab Atta, Kumar Tapas Dutta

Source Terms

In the Equations (4) and (5) the source terms [S.sub.u] and [S.sub.v] have been chosen in a manner such that they can suppress the velocity fields in the solid phase. When the PCM is in solid phase it takes large values, suppressing the velocity field. Also, with the increase of 'f' it should go smaller and finally be 0. One of the most popular and successful methods has been given by Viswanath and Jaluria (1993, 1995). They formulated the source term from the Karman-Kozeny equation for flow in porous media, where the equations resemble a Darcy-type law (where the velocities are proportional to the pressure gradient). Following are the source terms:

[S.sub.u] = -C (1-[f.sup.2]/[f.sup.3] b) and [S.sub.v] = -C (1-[f.sup.2]/[f.sup.3] b) v

The constants C and b can be suitably chosen to simulate the flow characteristics in porous media. From the equations it is clearly evident that when the liquid fraction is 0 then the source term dominate the equation and force the velocities to become zero in the region. The source term becomes negligible when liquid fraction is 1.

Boundary Conditions and Solution Method

In conformity with the experimental conditions the following boundary conditions have been imposed.

(1) No-slip condition at the surfaces of the tubes (inner and outer), forming the annulus.

(2) Isothermal condition at the inner surface of the annulus and adiabatic condition at the outer surface of the annulus.

Solution of the above mentioned Equations ((3)-(6)), after discretising by FVM, have been done by the SIMPLE algorithm (Patankar, 1980), introducing power law scheme and the deferred correction method (Ferziger and Peric, 2002) in the interpolation scheme for the convection term. After proper solution of these equations at each time step, the enthalpy and liquid mass fraction have been updated by the Equations ((1)-(2)).

Code Validation

Our code has been validated with the benchmark problem of natural convection in a square enclosure with a hot and cold wall. Comparisons were made with the previous works on phase-change materials, specifically the vertical annulus case. In all those cases the results were in good agreement.

Time-Efficiency of the Code

The code is also efficient in the sense of time-efficiency. Calculation for our problem at a grid-size of 60 x 120, in a P4 workstation with 512 MB RAM have a ratio of experiment time and numerical simulation time of 20:1.

NUMERICAL EXPERIMENTATION

For a better insight in the natural convection in phase change heat transfer in an annulus, the effects of the eccentricity as well as the angle of inclination of eccentricity are considered to be of great importance. It is already observed that the effects of Rayleigh number and Prandtl number on the natural convective heat transfer is well established. In this study the eccentricity is varied from 0 to 0.5. Similar such positions of eccentricity is varied among the different angle of inclination ([psi]) between [pi]/2 to [pi]/2 (e.g. -90[degrees], -60[degrees], -30[degrees], 0[degrees], 30[degrees], 60[degrees], 90[degrees]) as shown in Figure 3.