Effective wall heat transfer

Crider, J.E. and Foss, A.S. (1965) Effective wall heat transfer coefficients and thermal resistances in mathematical models of packed beds. AIChE J., 11,1012. 

Dimensionless effective radial thermal conductivity X and effective wall heat transfer coefficient h as function of Peclet-number (a measured under reacting conditions, b calculated according to /42/for nonreacting conditions)... 

Perhaps the simplest to determine is the effective wall heat transfer coefficient (based on tube wall temperature) used in conjunction with the one-dimensional model. A heat balance for a bed packed with inert pellets is given by ... 

Figure 14.5 Dependence of effective wall heat transfer coefficient on reactor length for various conditions. (De-wasch and Froment 1972 reprinted with permission from Chemical Engineering Science. Copyright by Pergamon Press, Inc.)...

Work in connection with desahnation of seawater has shown that specially modified surfaces can have a profound effect on heat-transfer coefficients in evaporators. Figure 11-26 (Alexander and Hoffman, Oak Ridge National Laboratory TM-2203) compares overall coefficients for some of these surfaces when boiling fresh water in 0.051-m (2-in) tubes 2.44-m (8-ft) long at atmospheric pressure in both upflow and downflow. The area basis used was the nominal outside area. Tube 20 was a smooth 0.0016-m- (0.062-in-) wall aluminum brass tube that had accumulated about 6 years of fouhng in seawater service and exhibited a fouling resistance of about (2.6)(10 ) (m s K)/ J [0.00015 (fF -h-°F)/Btu]. Tube 23 was a clean aluminum tube with 20 spiral corrugations of 0.0032-m (lA-in) radius on a 0.254-m (10 -in)... 

The baffle cut determines the fluid velocity between the baffle and the shell wall, and the baffle spacing determines the parallel and cross-flow velocities that affect heat transfer and pressure drop. Often the shell side of an exchanger is subject to low-pressure drop limitations, and the baffle patterns must be arranged to meet these specified conditions and at the same time provide maximum effectiveness for heat transfer. The plate material used for these supports and baffles should not be too thin and is usually minimum thick-... 

Obtain by dimensional analysis a functional relationship for the wall heat transfer coefficient for a fluid flowing through a straight pipe of circular cross-section. Assume that the effects of natural convection can be neglected in comparison with those of forced convection. 

Equations (8) are based on the assumption of plug flow in each phase but one may take account of any axial mixing in each liquid phase by replacing the molecular thermal conductivities fc, and ku with the effective thermal conductivities /c, eff and kn eff in the definition of the Peclet numbers. The evaluation of these conductivity terms is discussed in Section II,B,1. The wall heat-transfer terms may be defined as... 

From this illustration we can see that the added detail of the radial temperature profile near the wall that could be provided by CFD simulations does not help in obtaining better estimates for the standard heat transfer parameters. It also implies that experimental efforts to measure temperatures closer to the wall are, in fact, counter-productive. Finally, it is clear that the standard model with plug flow and constant effective transport parameters does not fit satisfactorily to temperature profiles in low-Abeds. These considerations have led us to look for improved approaches to near-wall heat transfer. 

De Wasch and Froment (1971) and Hoiberg et. al. (1971) published the first two-dimensional packed bed reactor models that distinguished between conditions in the fluid and on the solid. The basic emphasis of the work by De Wasch and Froment (1971) was the comparison of simple homogeneous and heterogeneous models and the relationships between lumped heat transfer parameters (wall heat transfer coefficient and thermal conductivity) and the effective parameters in the gas and solid phases. Hoiberg et al. (1971)... 

Figure 1736. Effective thermal conductivity and wall heat transfer coefficient of packed beds. Re = dpG/fi, dp = 6Vp/Ap, s -porosity, (a) Effective thermal conductivity in terms of particle Reynolds number. Most of the investigations were with air of approx. kf = 0.026, so that in general k elk f = 38.5k [Froment, Adv. Chem. Ser. 109, (1970)]. (b) Heat transfer coefficient at the wall. Recommendations for L/dp above 50 by Doraiswamy and Sharma are line H for cylinders, line J for spheres, (c) Correlation of Gnielinski (cited by Schlilnder, 1978) of coefficient of heat transfer between particle and fluid. The wall coefficient may be taken as hw = 0.8hp.

Two runs at high CO2 concentrations (9.8 mole percent CO2/ N2/5A 1/4" and 13.2 mole percent C02/air/5A 1/8" LMS pellets), for which it was determined that effects of heat transfer could be very important, were run in a special column designed by F. W. Leavitt (developer of the MASC program) to simulate essentially adiabatic behavior. The column was constructed of thin-walled sheet metal and was 24.8 cm in diameter. Electric heating jackets placed in sections along the wall of the column and controlled by thermocouples placed at corresponding intervals along the centerline of the bed were used to maintain the wall at essentially the same temperature as the bed interior. 

However, the circumferential wall heat transfer area increases as the aspect ratio increases. So, from a heat transfer, dynamic stability perspective, a large aspect ratio is desirable. Figure 2.8 shows the effect of aspect ratio on the various design and operating parameters for a design conversion of 80%. Increasing L/D increases heat transfer area, which decreases the required AT driving force and raises jacket temperature. The reactor stability index improves substantially. 

The static contribution l. , incorporates heat transfer by conduction and radiation in the fluid present in the pores, conduction through particles, at the particle contact points and through stagnant fluid zones in the particles, and radiation from particle to particle. Figure 19-20 compares various literature correlations for the effective thermal conductivity and wall heat-transfer coefficient in fixed beds [Yagi and Kunii, AlC hE J. 3 373(1957)]. 

FIG. 19-20 Thermal conductivity and wall heat transfer in fixed beds, (a) Effective thermal conductivity, (h) Nusselt number for wall heat transfer. (Figs. 11.7.1-2 and 11.7.1-3 in Froment and Bischoff. Chemical Reactor Analysis and Design, Wiley. 1990.)... 

Eqns. (3) - (9) enable the effective radial thermal conductivity (kr>eff), the apparent wall heat transfer coefficient (hw>eff) and the overall heat transfer coefficient (U) to be predicted in terms of the physical properties p, kg and Cp of air, together with measurable parameters such as , 6, dt, kp, dp (A), dp>(v), dp>(V/A), e and T, the mean bed temperature. The predicted and observed values of U are compared in Figure 6. The averaged normalised standard error... 

For y > 5 the turbulence stress and heat transfer rate become important However, near the wall the total shear stress and total heat transfer rate will remain effectively constant and equal to the wall shear stress and wall heat transfer rate, respectively. 

For Rep < 100 and 0.05 < rp/r, < 0.2, wall Biot numbers range between 0.8 and 10 [28], so this means that wall effects cannot be neglected a priori [38]. Also this criterion contains procurable parameters. For the wall heat transfer coefficient hw and the effective heat conductivity in the bed Abc(r, the correlations in Table 2, eqs. 44-47 can be used [8, 39]. These variables are assumed to be composed of a static and a dynamic (i.e. dependent on the flow conditions) contribution. Thermal heat conductivities of gases at 1 bar range from 0.01 to 0.5 Js m l K l, depending on the nature of the gas and temperature. 

Liquid bismuth flows at a rate of 4.5 kg/s through a 5.0-cm-diameter stainless-steel tube. The bismuth enters at 415°C and is heated to 440°C as it passes through the tube. If a constant heat flux is maintained along the tube and the tube wall is at a temperature 20°C higher than the bismuth bulk temperature, calculate the length of tube required to effect the heat transfer. 

Air enters a small duct having a cross section of an equilateral triangle, 3.0 mm on a side. The entering temperature is 27°C and the exit temperature is 77°C. If the flow rate is 5 x 10 3 kg/s and the tube length is 30 cm, calculate the tube wall temperature necessary to effect the heat transfer. Also calculate the pressure drop. The pressure is 1 atm. ... 

For mass transfer outside particles, the transfer coefficient can be calculated by using Equation (7). Inside the particles, the diffusion coefficient can be modified by means of the effective porosity (see Section I.B., or determined by another method such as that of Lou et al., 1991). For bed-to-wall heat transfer, the transfer coefficient can be computed by using relevant correlations presented in Chapter 5. 

Along with wall heat transfer coefficient in non-adiabatic reactors, another effect frequently added in models is that of thermal conduction in the solid phase (27, 28, 29). One should be particularly careful here, since most of the correlations available in the literature (9 y 32) are for effective... 

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