where zheater is the length of the heated portion of the system and Dzheater represent lengths of 1 mm segments that were used to fit the temperature profile of the heated section. As shown in Table 12, velocities as high as 4 m/min could be obtained for fairly modest (5 to 10 oC) temperature differences between the heater and cooler.

Table 12. Velocities obtained for various loading densities, pressures, and cooler/heater temperatures. Effective density difference is described in the text.

The steady-state solution of the continuity, momentum, and energy equations is the main interest in this work:

First, the initial Tin, Pin and rin were set at a point in the loop and at the system pressure. A velocity was assumed and Dx was incremented. Then, eqs. (2)-(4) were used to determine u1, P1, and H1. Knowledge of P1 and H1 allowed determination of a new density, rnew. This iteration was repeated until the densities converged. Once the density for a single grid, Dx, was converged, the process was repeated around the loop to provide Tout and Pout. For a correct solution, Tout and Pout were determined to match Tin and Pin within a given tolerance. Over the range of densities and pressures shown in Table 12, one-dimensional finite-difference simulation could predict the velocities to within about 35%. A correlation in terms of dimensionless groups was determined to be as follows:

where rini is the initial density, and Gr and Pr are the Grashof and Prandtl numbers averaged according to the heated and cooled lengths of the apparatus, and Dreff is the effective density difference determined from eq. (1). Correlation of the data with Eqn (2) gave an R2 value of about 0.90. Empirical correlation of the data gave the following equation:

where u is in m/min, P is in MPa and Dr is in kg/m3. Eq. (6) could represent the data to within about 10% as shown in Figure 29.

Figure 29. Correlation of velocities with eq. (6)