Dimensionless groups heat transfer

Example Buckingham Pi Method—Heat-Transfer Film Coefficient It is desired to determine a complete set of dimensionless groups with which to correlate experimental data on the film coefficient of heat transfer between the walls of a straight conduit with circular cross section and a fluid flowing in that conduit. The variables and the dimensional constant believed to be involved and their dimensions in the engineering system are given below ... 

The dimensionless relations are usually indicated in either of two forms, each yielding identical resiilts. The preferred form is that suggested by Colburn ran.s. Am. In.st. Chem. Eng., 29, 174—210 (1933)]. It relates, primarily, three dimensionless groups the Stanton number h/cQ, the Prandtl number c Jk, and the Reynolds number DG/[L. For more accurate correlation of data (at Reynolds number <10,000), two additional dimensionless groups are used ratio of length to diameter L/D and ratio of viscosity at wall (or surface) temperature to viscosity at bulk temperature. Colburn showed that the product of the Stanton number and the two-thirds power of the Prandtl number (and, in addition, power functions of L/D and for Reynolds number <10,000) is approximately equal to half of the Fanning friction fac tor//2. This produc t is called the Colburn j factor. Since the Colburn type of equation relates heat transfer and fluid friction, it has greater utility than other expressions for the heat-transfer coefficient. 

Not only is the type of flow related to the impeller Reynolds number, but also such process performance characteristics as mixing time, impeller pumping rate, impeller power consumption, and heat- and mass-transfer coefficients can be correlated with this dimensionless group. 

Heat Transfer In general, the fluid mechanics of the film on the mixer side of the heat transfer surface is a function of what happens at that surface rather than the fluid mechanics going on around the impeller zone. The impeller largely provides flow across and adjacent to the heat-transfer surface and that is the major consideration of the heat-transfer result obtained. Many of the correlations are in terms of traditional dimensionless groups in heat transfer, while the impeller performance is often expressed as the impeller Reynolds number. 

From the slope and intereept of the heat absorption line, it is possible to manipulate Equation 6-117 by ehanging operating variables sueh as u, Tq, and T or design variables sueh as the dimensionless heat transfer group UA/puCp. It is also possible to alter the magnitude of the reaetion exotherm by ehanging the inlet reaetant eoneentrations. Any of tliese manipulations ean be used to vary tlie number of loeations of the possible steady states. 

Empirieal dimensionless group eorrelations have been used in the seale-up proeess. In partieular, the eorrelation for the inside film heat transfer eoeffieient for agitated, jaeketed vessels has been employed for the seale-up to a larger vessel. Reaetion ealorimeters are often used to give some indieation of heat transfer eoeffieients eompared to water in the same unit. Conelation for plant heat transfer is of the general form... 

Useful dimensionless groups for heat transfer calculation are ... 

Forced convection heat transfer has been measured under widely differing conditions, and using the dimensionless groups makes correlation of the experimental... 

Klinkenberg, A. and Mqqy, H. H. Chem. Eng. Prog. 44 (1948) 17. Dimensionless groups in fluid friction, heat and material transfer. 

It is convenient to define other dimensionless groups which are also used in the analysis of heat transfer. These are ... 

It may be noted that many of these dimensionless groups are ratios. For example, the Nusselt group h/(k/l) is the ratio of the actual heat transfer to that by conduction over a thickness l, whilst the Prandtl group, (p/p)/(k/Cpp) is the ratio of the kinematic viscosity to the thermal diffusivity. 

The problem of axial conduction in the wall was considered by Petukhov (1967). The parameter used to characterize the effect of axial conduction is P = (l - dyd k2/k ). The numerical calculations performed for q = const, and neglecting the wall thermal resistance in radial direction, showed that axial thermal conduction in the wall does not affect the Nusselt number Nuco. Davis and Gill (1970) considered the problem of axial conduction in the wall with reference to laminar flow between parallel plates with finite conductivity. It was found that the Peclet number, the ratio of thickness of the plates to their length are important dimensionless groups that determine the process of heat transfer. 

A general case of heat transfer under the conditions of combined action of electro-osmotic forces and imposed pressure gradient was considered by Chakra-borty (2006). The analysis showed that in this case the Nusselt number depends not only on parameters z and S, but also on an additional dimensionless group, which is a measure of the relative significance of the pressure gradient and osmotic forces. 

Convective heat transfer to fluid inside circular tubes depends on three dimensionless groups the Reynolds number. Re = pdtu/ii, the Prandtl number, Pr = Cpiilk where k is the thermal conductivity, and the length-to-diameter ratio, L/D. These groups can be combined into the Graetz number, Gz = RePr4/L. The most commonly used correlations for the inside heat transfer coefficient are... 

The heat transfer group, UAexdlVpCp, is dimensionless. Assume its value is 0.02. A controller is needed to regulate Text- The industrial choice would be a two-term controller, proportional plus reset. We skirt the formal control issues and use a simple controller of the form... 

Some of the important dimensionless groups pertinent to heat and mass transfer problems are listed in Table 3.5. 

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... 

For gas-liquid flows in Regime I, the Lockhart and Martinelli analysis described in Section I,B can be used to calculate the pressure drop, phase holdups, hydraulic diameters, and phase Reynolds numbers. Once these quantities are known, the liquid phase may be treated as a single-phase fluid flowing in an open channel, and the liquid-phase wall heat-transfer coefficient and Peclet number may be calculated in the same manner as in Section lI,B,l,a. The gas-phase Reynolds number is always larger than the liquid-phase Reynolds number, and it is probable that the gas phase is well mixed at any axial position therefore, Pei is assumed to be infinite. The dimensionless group M is easily evaluated from the operating conditions and physical properties. 

The dimensionless group hD/k is called the Nusselt number, /VNu, and the group Cp i/k is the Prandtl number, NPl. The group DVp/p is the familiar Reynolds number, NEe, encountered in fluid-friction problems. These three dimensionless groups are frequently used in heat-transfer-film-coefficient correlations. Functionally, their relation may be expressed as... 

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. 

Many of the results and correlations in heat and mass transfer are expressed in terms of dimensionless groups such as the Nusselt, Reynolds and Prandtl numbers. The definitions of those dimensionless groups referred to in this chapter are given in Appendix 2. 

In addition to correlations of the above type, that is, where the ratio of a two-phase heat-transfer coefficient is compared to a fictitious singlephase coefficient, relationships in terms of dimensionless groups have also been presented. In general, these are of the form. 

For heat transfer the Fourier number is cct/a. The heat transfer analogs of the mass transfer dimensionless groups can be found by making the substitutions described in Chapter 1. 

---