PRESENTATION OF EXPERIMENTAL DATA

CHAPTER 2 CHAPTER 2

5.2 PRESENTATION OF EXPERIMENTAL DATA

reasonable temperature difference between the fl.owing materials and the copper plate. Then the power output of the guard heater was adjusted with great care to make the temperature difference across the phenolic plate as small as possible.

The temperature difference between the fl.ow and the copper plate was usually about 20°C, and the temperature difference across the phenolic plate was less than 0.2°C. When a steady state was achieved, the mass fl.ow rate and the depth of fl.ow were measured. Following this, all the temperatures were quickly recorded. Finally, the mean solid fraction was measured. The whole procedure took 15 ,..., 25 minutes.

Following the works of Sullivan and Sabersky [1975] and Patton et al. [1986], the present data are plotted in terms of the modified Nusselt number, Nu*, and the modified Peclet number, Pe*, where

Nu*= hd k ' _g

Pe*= UL ( d

)2 (k

⁰

)2.

a0 L kg

(5.1)

(5.2) In the above, h is the convection heat transfer coefficient, d the particle diameter, L the length of the heating plate, U the mean velocity. And kg is the thermal conductivity of gas, and k₀ and a0 are the thermal conductivity and diffusivity of the granular material measured at the critical solid fraction. Note that air was used as the interstitial gas in all of the present experiments. In Figure 5.4, for a given material, Nu* increases with increasing Pe*, and after it reaches a maximum, Nu* decreases with further increase in Pe*. This phenomenon is closely associated with that of Figure 5.3, in that the data for increasing Nu* with increasing Pe* correspond to the subcritical fl.ow, and the data for decreasing Nu*

with high Pe* a.re in the supercritical regime. In other words, the heat transfer rate increases with increasing velocity since the solid fraction remains near the critical solid fraction. Once the flows become supercritical, Nu* decreases because of the decrease of solid fraction as Pe* increases. All the granular materials show similar trends, and all the data in the subcritical regime form a single curve regardless of particle size or material. This is the same phenomenon observed by Patton et al. [1986], and the single curve has been predicted from the model developed by Sullivan and Sabersky [1975]:

Nu*

=

- - - , = = 1

x+ v:a'

^(5.3)

where

x

is an experimental constant. For the present data

x

= 0.025 is chosen while Sullivan and Sabersky used

x

^{= 0.085. In} Figure 5.4, the equation (5.3) with

x

= 0.025 is also presented.

The above model of Sullivan and Sabersky predicts the data quite satisfac- torily in the subcritical regime where granular materials fl.ow slowly with high solid fraction, but it fails to predict the behavior of the heat transfer as the ft ow becomes supercritical. This incompleteness of the model leads to the introduc- tion of two new parameters; the effective Nusselt number, Nu:ff, and the effective Peclet number, Pe:ff.

The effective N usselt number and the effective Peclet number are defined as follows:

{5.4) (5.5) where ke and O!e are the effective thermal conductivity and diffusivity of granular materials at a given mean solid fraction. The effective thermal conductivity is obtained from the results ofGelperin and Einstein [1971] as follows:

ke v(l - k,.,/ka)

kQ

=

^l

+

^k,,/ka

+

0.28(1 - 1.1)0.6S{leg/1c.)-O.l&' (5.6) where v is the solid fraction, and kg/ k. is the ratio of the thermal conductivity of gas to the thermal conductivity of the solid particle. The effective thermal diffusivity is obtained as

O ! e = - - , ke

Pp//Cp

where Pp is the particle density, and Cp is the specific heat capacity of the solid particle. The effective Nusselt number and the effective Peclet number simply take into account the variation of solid fraction which was neglected in the Sul- livan and Sabersky formula. Note that when the mean solid fraction is equal to the critical solid fraction, the effective Nusselt and Peclet numbers are identical with the modified Nusselt and Peclet numbers.

The data for glass beads are plotted in Figure 5.5 using the effective coordi- nates. All the data points follow the shape of a single curve fairly well regardless of the flow regime. This may be explained by considering that when the effective

thermal conductivity and diffusivity are used for the data with solid fraction less than the critical value, the effective Nusselt number becomes larger than the modified one, and the effective Pedet number becomes smaller. AE. a. result, the data in the supercritical regime in Figure 5.4 are shifted to the predicted curve for the subcritical regime. Therefore, when the effective Nusselt number and the effective Peclet number are introduced, all the data in both fl.ow regimes can be represented reasonably well by a single equation which is of the same form as that developed by Sullivan and Sabersky (equation (5.3)). The following expression using the effective Nusselt and Peclet numbers is plotted in Figure 5.5.

(5.7)

where the value for

x

is chosen as 0.025.

The data with materials different from glass beads are also plotted in Figure 5.6. The data for polystyrene beads and mustard seeds in the subcritical regime are well correlated with glass beads. But the data. for polystyrene beads and mustard seeds in the supercritical regime are deviated from equation (5.7) for reasons that are not yet fully understood.