The starts as soon as there is a

The single channel natural convection cases (fig. 1 and 2) may be solved in a much simpler way by 1-D-model, considering their nature as direct contact balanced counter current heat exchangers. this is shown for the parallel plates duct in fig. 3. The 1-D model in this case leads to simple explicit formulae for the flow rate as a linear function of temperature difference (Rayleigh number) and predicts the critical Rayleigh number about 3% higher than the harder 2-D approach. looking at many parallel plate ducts, the situation is different. No circulation will take place within each one of the single parallel tubes, but in pairs of tubes in that caseWhile the circulating flow in case of the horizontal channel starts as soon as there is a temperature difference greater zero, this is not true for the case of a vertical channel as shown in fig.2. Only if a critical value Rac of the (dimensionless) temperature difference Ra is surpassed, the circulating flow will start. for Ra < Rac the critical Rayleigh numbers (temperature differences) have been calculated from a stability analysis for a cylindrical vertical tube by Taylor (1954) and for a vertical parallel plates duct by Unger (1980). The definition of Ra, as well as the values of RacLength and diameter is used in the Rayleigh number as: d2L, the Rayleigh numbers of the single channel cases, containing d4/L must be multiplied by (L/d)2 to approach the correct form of Rayleigh for the bundle so we can find, that Rac single tube (L/d)2 had to be less than Rac tube bundle for circulation within a single tube to occur in bundle. using values from Fig. 2 and 4, namely Ra c single tube =1087 and Rac tube bundle= 384, we find (L/d2) < 384/1087 or L/d <0.59. so in the case of tube with lengths that is less than 0.6 diameter, the single tube internal convection will never happen in a bundle. using a 1-D approximate solution, which is certainly good for engineering approach, the temperature profiles in the tubes of the bundle as a function of the individual flow rates are shown in fig.5


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