Particle Nusselt number

The Nusselt number for the heat transfer between a gas and a solid particle of radius d, is given by the Ranz-Marshall equation... 

The Biot number is essentially the ratio of the resistance to heat transfer within the particle to that within the external fluid. At first sight, it appears to be similar in form to the Nusselt Number Nu where ... 

The second approach assigns thermal resistance to a gaseous boundary layer at the heat transfer surface. The enhancement of heat transfer found in fluidized beds is then attributed to the scouring action of solid particles on the gas film, decreasing the effective film thickness. The early works of Leva et al. (1949), Dow and Jacob (1951), and Levenspiel and Walton (1954) utilized this approach. Models following this approach generally attempt to correlate a heat transfer Nusselt number in terms of the fluid Prandtl number and a modified Reynolds number with either the particle diameter or the tube diameter as the characteristic length scale. Examples are ... 

Nwp = Nusselt number for wall-particle heat transfer... 

Particle Nusselt number Nu — V - kf Peclet number Pe — pv°fpdp kf Prandtl number Pr — r kf Turbulent Prandtl number Prt — ... 

It may be noted that in this expression the Nusselt number with respect to the tube wall Nu is related to the Reynolds number with respect to the particle Re c. 

The experimental results obtained for a wide range of systems(96-99) are correlated by equation 6.58, in terms of the Nusselt number (Nu = hd/k) for the particle expressed as a function of the Reynolds number (Re c = ucdp/fx) for the particle, the Prandtl number Pr for the liquid, and the voidage of the bed. This takes the form ... 

The Nusselt number with respect to the tube Nu(= hdt/k) is expressed as a function of four dimensionless groups the ratio of tube diameter to length, the ratio of tube to particle diameter, the ratio of the heat capacity per unit volume of the solid to that of the fluid, and the tube Reynolds number, Rec = (ucdtp/p,). However, equation 6.59 and other equations quoted in the literature should be used with extreme caution, as the value of the heat transfer coefficient will be highly dependent on the flow patterns of gas and solid and the precise geometry of the system. 

Calculate the coefficient for heat transfer between the gas and the particles, and the corresponding values of the particle Reynolds and Nusselt numbers. Comment on the results and on any assumptions made. 

Cornish(128) considered the minimum possible value of the Nusselt number in a multiple particle system. By regarding an individual particle as a source and the remote fluid as a sink, it was shown that values of Nusselt number less than 2 may then be obtained. In a fluidised system, however, the inter-particle fluid is usually regarded as the sink and under these circumstances the theoretical lower limit of 2 for the Nusselt number applies. Zabrodsky 1 291 has also discussed the fallacy of Cornish s argument. 

Here d is the model particle diameter Nu is the Nusselt number. The Nusselt number for the th particle is determined as... 

Experimental data are available for large particles at Re greater than that required for wake shedding. Turbulence increases the rate of transfer at all Reynolds numbers. Early experimental work on cylinders (VI) disclosed an effect of turbulence scale with a particular scale being optimal, i.e., for a given turbulence intensity the Nusselt number achieved a maximum value for a certain ratio of scale to diameter. This led to speculation on the existence of a similar effect for spheres. However, more recent work (Rl, R2) has failed to support the existence of an optimal scale for either cylinders or spheres. A weak scale effect has been found for spheres (R2) amounting to less than a 2% increase in Nusselt number as the ratio of sphere diameter to turbulence macroscale increased from zero to five. There has also been some indication (M15, S21) that the spectral distribution of the turbulence affects the transfer rate, but additional data are required to confirm this. The major variable is the intensity of turbulence. Early experimental work has been reviewed by several authors (G3, G4, K3). 

So far we have considered an infinite value of the gas-to-particle heat and mass transfer coefficients. One may encounter, however, an imperfect access of heat and mass by convection to the outer geometrical surface of a catalyst. Stated in other terms, the surface conditions differ from those in the bulk flow because external temperature and concentration gradients are established. In consequence, the multiple steady-state phenomena as well as oscillatory activity depend also on the Sherwood and Nusselt numbers. The magnitudes of the Nusselt and Sherwood numbers for some strongly exothermic reactions are reported in Table III (77). We may infer from this table that the range of Sh/Nu is roughly Sh/Nu (1.0, 104). 

Since the particles are so small in most cases, the Nusselt number based on the diameter reaches its lower limiting value of 2, i.e., Nu = (h2rs)/kg = 2 when k is the thermal conductivity of the gas and h is the heat transfer coefficient. Then the preceding equation becomes ... 

Re = wsdpPg/T g is the Reynolds number in which dp refers to particle diameter, us superficial velocity, pg gas density and r g dynamic viscosity. The exponential term accounts for the enhancement effect due to the catalytic bed. in the Nusselt number the characteristic length is the tube diameter, such that Nu = awdt/7tg. For hydrogen-rich mixtures the thermal conductivity is remarkably high. 

While the particle is undergoing the accelerative motion as described above, heat is being transfered between it and the surrounding gas stream, also in an unsteady state. By using the Nusselt number Nu for evaluating the heat-transfer coefficient h from the slip velocity between the particles and the gas (Kramers, 1964),... 

There are many correlations available for heat and mass transfer to particles. These are all have the Nusselt number Nu = adjk (or the Sherwood number Sh = kgd/D) as a function of the Reynolds number Re = udjv and the Prandtl number Pr = vl (Al pcp) (or the Schmidt number Sc = v/D). 

Because of the fact that the Nusselt number and the Peclet number are not simply expressed in terms of the particle diameter it is convenient to use them as intermediate variables in the calculation. Since the mass velocity is fixed, the Reynolds number and the Nusselt number are determined by the particle diameter. The calculation proceeds by finding the values of r2 from selected values of dp, as determined from ki, and fc2 separately. By inverse interpolation, the value of dp that makes the two values of r2 equal is found. 

Kato and Wen (5) found, for the case of packed beds,that there was a dependency of the Sherwood and Nusselt numbers with the ratio dp/L. They proposed that the fall of the heat and mass transfer coefficients at low Reynolds numbers is due to an overlapping of the boundary layers surrounding the particles which produces a reduction of the available effective area for transfer of mass and heat. Nelson and Galloway W proposed a new model in terms of the Frossling number, to explain the fall of the heat and mass transfer coefficients in the zone of low Reynolds numbers. 

Here Dab is the binary diffusion coefficient, dp is the particle diameter, and Nu is the Nusselt number. Major data used in this work have been taken from both experimental data [7] and beech wood data [12]. 

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