Heat transfer number Estimation

The final dimensionless group to be evaluated is the interfacial heat-transfer number, and therefore the interfacial heat-transfer coefficient and the interfacial area must be determined. The interface is easily described for this regime, and, with a knowledge of the holdup and the tube geometry, the interfacial area can be calculated. The interfacial heat trasfer coefficient is not readily evaluated, since experimental values for U are not available. A conservative estimate for U is found by treating the interface as a stationary wall and calculating U from the relationship... 

The Nusselt number is basically a dimensionless temperature gradient averaged over the heat transfer surface. The Nusselt number represents the ratio of the heat transfer resistance estimated from the characteristic dimension of the object (L/k) to the real heat transfer resistance (f/h). In many convective heat transfer problems, the Nusselt number is expressed as a function of other dimensionless numbers, e.g., the Reynolds number and the Prandtl number. 

To the extent that radiation contributes to droplet heatup, equation 28 gives a conservative estimate of the time requirements. The parameter ( ) reflects the dependence of the convective heat-transfer coefficient on the Reynolds number ... 

For condensing vapor in vertical downflow, in which the hquid flows as a thin annular film, the frictional contribution to the pressure drop may be estimated based on the gas flow alone, using the friction factor plotted in Fig. 6-31, where Re is the Reynolds number for the gas flowing alone (Bergelin, et al., Proc. Heat Transfer Fluid Mech. Inst., ASME, June 22-24, 1949, pp. 19-28). 

It is assumed that process conditions and physical properties are known and the following are known or specified tube outside diameter D, tube geometrical arrangement (unit cell), shell inside diameter D shell outer tube limit baffle cut 4, baffle spacing and number of sealing strips N,. The effective tube length between tube sheets L may be either specified or calculated after the heat-transfer coefficient has been determined. If additional specific information (e.g., tube-baffle clearance) is available, the exact values (instead of estimates) of certain parameters may be used in the calculation with some improvement in accuracy. To complete the rating, it is necessary to know also the tube material and wall thickness or inside diameter. 

Pressure drop due to hydrostatic head can be calculated from hquid holdup B.]. For nonfoaming dilute aqueous solutions, R] can be estimated from f i = 1/[1 + 2.5(V/E)(pi/pJ ]. Liquid holdup, which represents the ratio of liqmd-only velocity to actual hquid velocity, also appears to be the principal determinant of the convective coefficient in the boiling zone (Dengler, Sc.D. thesis, MIT, 1952). In other words, the convective coefficient is that calciilated from Eq. (5-50) by using the liquid-only velocity divided by in the Reynolds number. Nucleate boiling augments conveclive heat transfer, primarily when AT s are high and the convective coefficient is low [Chen, Ind Eng. Chem. Process Des. Dev., 5, 322 (1966)]. 

Naturally, there are two more Peclet numbers defined for the transverse direction dispersions. In these ranges of Reynolds number, the Peclet number for transverse mass transfer is 11, but the Peclet number for transverse heat transfer is not well agreed upon (121, 122). None of these dispersions numbers is known in the metal screen bed. A special problem is created in the monolith where transverse dispersion of mass must be zero, and the parallel dispersion of mass can be estimated by the Taylor axial dispersion theory (123). The dispersion of heat would depend principally on the properties of the monolith substrate. Often, these Peclet numbers for individual pellets are replaced by the Bodenstein numbers for the entire bed... 

Equation (4.12) indicates the effect of viscous dissipation on heat transfer in micro-channels. In the case when the inlet fluid temperature, To, exceeds the wall temperature, viscous dissipation leads to an increase in the Nusselt number. In contrast, when To < Tv, viscous dissipation leads to a decrease in the temperature gradient on the wall. Equation (4.12) corresponds to a relatively small amount of heat released due to viscous dissipation. Taking this into account, we estimate the lower boundary of the Brinkman number at which the effect of viscous dissipation may be observed experimentally. Assuming that (Nu-Nuo)/Nuo > 10 the follow-... 

Heat transfer in micro-channels occurs under superposition of hydrodynamic and thermal effects, determining the main characteristics of this process. Experimental study of the heat transfer in micro-channels is problematic because of their small size, which makes a direct diagnostics of temperature field in the fluid and the wall difficult. Certain information on mechanisms of this phenomenon can be obtained by analysis of the experimental data, in particular, by comparison of measurements with predictions that are based on several models of heat transfer in circular, rectangular and trapezoidal micro-channels. This approach makes it possible to estimate the applicability of the conventional theory, and the correctness of several hypotheses related to the mechanism of heat transfer. It is possible to reveal the effects of the Reynolds number, axial conduction, energy dissipation, heat losses to the environment, etc., on the heat transfer. 

Below a Reynolds number of about 2000 the flow in pipes will be laminar. Providing the natural convection effects are small, which will normally be so in forced convection, the following equation can be used to estimate the film heat-transfer coefficient ... 

The presence of waves will increase the heat-transfer coefficient, so the use of equation 12.51 above a Reynolds number of 30 will give conservative (safe) estimates. The effect of waves on condensate film on heat transfer is discussed by Kutateladze (1963). 

The heat transfer coefficient to the vessel wall can be estimated using the correlations for forced convection in conduits, such as equation 12.11. The fluid velocity and the path length can be calculated from the geometry of the jacket arrangement. The hydraulic mean diameter (equivalent diameter, de) of the channel or half-pipe should be used as the characteristic dimension in the Reynolds and Nusselt numbers see Section 12.8.1. 

While it is possible to calculate the existing overall heat transfer coefficient from the operating data, it is not possible to calculate the individual film transfer coefficients. The individual film transfer coefficients can be combined in any number of ways to add up to an overall value of 285 W-m 2-K 1. However, the film transfer coefficients can be estimated from the correlations in Appendix C. Given that the tube-side correlations are much more reliable than the shell-side correlations, the best way to determine the individual coefficients is to calculate the coefficient for the tube-side and allocate the shell-side coefficient to add up to U = 285 W-m 2-K 1. Thus, to calculate the tube-side film transfer coefficient, KhT must first be determined. 

If the hydrodynamics of the two-phase systems are understood, and the holdups and phase Reynolds numbers known, the rate of heat transfer can be estimated with about the same confidence as that for single-phase systems. 

Macbeth (M5) has recently written a detailed review on the subject of burn-out. The review contains a number of correlations for predicting the maximum heat flux before burn-out occurs. These correlations include a dependence upon the tube geometry, the fluid being heated, the liquid velocity, and numerous other properties, as well as the method of heating. Sil-vestri (S6) has reviewed the fluid mechanics and heat transfer of two-phase annular dispersed flows with particular emphasis on the critical heat flux that leads to burn-out. Silvestri has stated that phenomena responsible for burn-out, due to the formation of a vapor film between the wall and the liquid, are believed to be substantially different from phenomena causing burn-out due to the formation of dry spots that produce the liquid-deficient heat transfer region. It is known that the value of the liquid holdup at which dry spots first appear is dependent on the heat flux qmi. The correlations presented by Silvestri and Macbeth (S6, M5) can be used to estimate the burn-out conditions. 

The heat-transfer coefficient U can be evaluated by using the gas-phase Reynolds number and the gas-phase Prandtl number in Eq. (33). The term U can also be estimated by using Eq. (33), but the gas-phase Reynolds number used for evaluating U should probably be based on the relative velocity vt, which is defined as... 

The interfacial heat transfer coefficient can be evaluated by using the correlations described by Sideman (S5), and then the dimensionless parameter Ni can be calculated. If the Peclet numbers are assumed to be infinite, Eqs. (30) can be applied to the design of adiabatic cocurrent systems. For nonadiabatic systems, the wall heat flux must also be evaluated. The wall heat flux is described by Eqs. (32) and the wall heat-transfer coefficient can be estimated by Eq. (33) with... 

Now let us consider the mixing time, t. This will be estimated by an order of magnitude estimate for diffusion to occur across the boundary layer thickness, <5Bl- If we have turbulent natural conditions, it is common to represent the heat transfer in terms of the Nusselt number for a vertical plate of height, , as... 

In a supersonic gas flow, the convective heat transfer coefficient is not only a function of the Reynolds and Prandtl numbers, but also depends on the droplet surface temperature and the Mach number (compressibility of gas). 154 156 However, the effects of the surface temperature and the Mach number may be substantially eliminated if all properties are evaluated at a film temperature defined in Ref. 623. Thus, the convective heat transfer coefficient may still be estimated using the experimental correlation proposed by Ranz and Marshall 505 with appropriate modifications to account for various effects such as turbulence,[587] droplet oscillation and distortion,[5851 and droplet vaporization and mass transfer. 555 It has been demonstrated 1561 that using the modified Newton s law of cooling and evaluating the heat transfer coefficient at the film temperature allow numerical calculations of droplet cooling and solidification histories in both subsonic and supersonic gas flows in the spray. 

The designer now needs to make some estimates of mass transfer. These properties are generally well known for commercially available adsorbents, so the job is not difficult. We need to re-introduce the adsorber cross-section area and the gas velocity in order to make the required estimates of the external film contribution to the overall mass transfer. For spherical beads or pellets we can generally employ Eq. (7.12) or (7.15) of Ruthven s text to obtain the Sherwood number. That correlation is the mass transfer analog to the Nusselt number formulation in heat transfer ... 

Ionic liquids (ILs) are, together with water and supercritical fluids, one of the few alternative media for environmentally friendly processes, which seem to have more possibility of industrial application in the next 10 years. The range of demonstrated or proposed applications of ILs is extraordinary, going from their use as nonvolatile, non-flammable solvents in organic synthesis to catalysts, materials for aiding separations and gas capture, advanced heat transfer fluids, lubricants, antistatics, and so on [2 ]. Surpassing in magnitude the number of potential uses is the number of possible IL compositions, estimated to be in the billions [5]. The term ionic liquids includes all compounds composed exclusively by ions that are liquid... 

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