Nusselt number limiting values

TABLE 5-4 Values of Limiting Nusselt Number in Laminar Flow in Closed Ducts... 

Thus, M decreases when % (and the Peclet number) increases. Accordingly, the Nusselt number decreases when the Peclet number increases, and approaches its limiting value Nuoo that corresponds to Pe oo. 

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. 

For a single sphere in a stagnant environment, i.e. where there is no convection, the limiting value of the Nusselt number can be shown (see, for example, Kay and Nedderman, 1985) to be... 

The analytical solution to this problem provides an asymptotic value of Nu = 3.66. Notice that far downstream (i.e., at large values of z), both Tm and dT/dr approach zero. Thus at some sufficiently long downstream position the numerical solution is unable to compute the Nusselt number. The analytic solution can be used to find the limiting result of Nu = 3.66. The solution presented in Fig. 4.17 was computed on auniformly spaced mesh of 16 points, and returned an asymptotic value of Nu = 3.7, which represents about a 1% error. It returned the Nu = 3.7 result until about z = 1.0, before the zero-over-zero situation caused to evaluation to lose accuracy and eventually fail. 

As the diameter is decreased, the heat transfer from a unit volume intensifies because of the increase in the ratio of surface to volume (d-1) heat exchange per unit surface also intensifies. For a constant value of the Nusselt number the heat exchange coefficient is proportional to d l. Under the rough assumption that Tc and E change little from one case to the next, we come to the conclusion that at the limit the Peclet number (numerically equal for gases to the Reynolds number), based on the flame velocity (or adiabatic flame velocity u0) has a specific value... 

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 ... 

At low Reynolds numbers, the Nusselt number will tend to the constant value that would exist in purely free convection, this being designated as Afa.v, whereas at high Reynolds numbers, when the effects of the buoyancy forces are small, the Nusselt numbers will tend to the values that would exist in purely forced convection at the same Reynolds number as that being considered. These forced convection Nusselt numbers are here designated as Nur- In the combined convection regions between these two limits, the Nusselt number variation can be approximately... 

It will be seen from Fig. 10.29 that for small Raw, when the convective motion is weak, the Nusselt number tends to the conduction limit value of 1. However, when Raw is large, the flow will consist mainly of a boundary layer flow up the hot wall and a boundary layer flow down the cold wall. It is to be expected, therefore, that in... 

Data are most frequently correlated by the Nusselt number (Nn )i or (NNu)om, the Graetz number Nc = (NReNpi-D/L), and the Grashof (natural-convection effects) number Ng,. Some correlations consider only the variation of viscosity with temperature, while others also consider density variation. Theoretical analyses indicate that for very long tubes approaches a limiting value. Limiting Nusselt numbers... 

The model was developed to show that if the proper boundary conditions are used,one should not expect at low Reynolds numbers that the Nusselt and Sherwood numbers approach the limiting value of two, which is valid for a sphere in an infinite static medium. Since the particles are members of an assemblage, they assume in their model that there is a concentric spherical shell of radius... 

The discussions above on the local heat transfer coefficients arc insightful however, they are of limited value in heal transfer calculations since the calculation of heat transfer requires the average heat transfer coefficient over the entire. surface. Of the several such relations available in the literature for the average Nusselt number for cross flow over a cylinder, we present the one proposed by Churchill and Bernstein ... 

The first five correlations predict limiting values of the Sherwood or Nusselt number as 2 and are mainly valid for single-particle or very dilute flows. The last two correlations predict Nusselt and Sherwood numbers lower than 2 for dense flows and should be used to simulate dense granular flows. 

For small drop sizes or for stagnant conditions, the Nusselt number has a limiting value of 2. 

Table 17 also shows that the exhaust gas flow is laminar inside the channels of a ceramic monolith with 62 cells cm at all engine operation conditions. With a Reynolds number in the range of about 10 up to about 300, it is no surprise that both the limiting Sherwood and Nusselt numbers assume low values as well, which means that there is only a limited contribution of convection to the transfer of heat and mass from the gas phase to the catalyst surface. 

It is conventional to define the length of the thermal initial region as the distance from the inlet cross-section to the point at which the Nusselt number differs from its limit value (3.5.27) by 1%. Calculations show that the dimensional length of the thermal initial region is l = 0.1 la Pe --... 

Figure 10. Comparison of measured (dots) local Nusselt numbers for parallel-plates micro-channel against limiting fully developed values for prescribed wall temperature (solid line) and heat flux (dashed line) (rearranging from 10 to 242).

Using Eqs. (12.22) and (12.16), theoretical values of the Nusselt number can be obtained, and these values are shown in Fig. 12.2. At low Graetz numbers, only the first term of Eq. (12.16) is significant, and the Nusselt number approaches a limiting value of 3.66. It is difficult to get an accurate measurement of the heat-transfer coefficient at low Graetz numbers, since the final temperature difference is very small. For example, at Nq = I.O, the ratio of exit to inlet driving forces is only 8.3 x 10" . 

FLOW INSIDE PIPES. Correlations for mass transfer to the inside wall of a pipe are of the same form as those for heat transfer, since the basic equations for diffusion and conduction are similar. For laminar flow, the Sherwood number shows the same trends as the Nusselt number, with a limiting value of 3.66 for very long tubes and a one-third-power dependence on flow rate for short tubes. The solution shown in Figure 12.2 can be used for Agh if the Graetz number is based on the dilfusivity or on the Schmidt number as follows ... 

Parallel plate ducts, also referred to as flat ducts or parallel plates, possess the simplest duct geometry. This is also the limiting geometry for the family of rectangular ducts and concentric annular ducts. For most cases, the friction factor and Nusselt number for parallel plate ducts are the maximum values for the friction factor and the Nusselt number for rectangular ducts and concentric annular ducts. 

Turning next to the HI boundary condition, Fig. 10.9 presents the fully developed Nusselt number predictions for the plane parallel plates case covering the power-law index range from 0 to 3. The available predictions for the square duct, with n varying from 0.5 to 1, are also shown. As in the case of the T boundary condition, the slug flow and newtonian flow limits are also available for the HI condition for all aspect ratios. As in the constant-temperature case, a large decrease in the Nusselt number occurs for any aspect ratio when n increases from 0 to 0.5, and the subsequent decrease from 0.5 to 1.0 is more gentle. The dashed lines represent estimates of the fully established Nusselt values for intermediate values of the aspect ratio and power-law index. 

This type of flow is characterised by the uniform velocity across the cross-section of the tube, i.e. V r) = Vq, the constant value. This condition applies near the tube entrance, and is also the limiting condition of n = 0 with power-law model, i.e. infinite pseudoplasticity. In view of its limited practical utility, though this case is not discussed here, but detailed solutions are given in several books, e.g. see Skelland [1967]. However, Metzner et al. [1957] put forward the following expression for Nusselt number under these conditions (for Gz > 100) ... 

As the quality increases (X o), the ratio of Nusselt numbers approaches unity, which means the predicted Nusselt number would approach the gaseous value. The experimental heat transfer of gaseous hydrogen has been correlated using a Nusselt number based upon film conditions [7]. Thus the correlations (2) can be applied even into the completely gaseous phase. However, it is more difficult to assign the limit of applic Dility to small values of quality where Xtt is large. 

In general, values for Sh and Nu under reaction conditions are different from those observed at these limits (Hayes and Kolaczkowski [121]). Correlations which attempt to describe the variation of Sherwood (or Nusselt) numbers with the Damkohler number have been proposed. Tronconi et al. [104] and Groppi and Tronconi [122] used the result from Brauer and Petting [123] for mass/heat transfer correlations when modeling monolith reactors with circular, square, and triangular shape ... 

It is further suggested that the estimated actual value be used in tandem with the theoretical Nusselt number for process decisions. Both Nusselt numbers can be used to calculate the upper and lower limits of the expected temperature rise. Such calculations could then be used in making the most conservative engineering decisions with respect to a process. 

---