Heat-transfer coefficients

One-Dimensional Conduction Semi-infinite Plate Consider a semi-infinite plate with an initial uniform temperature T,. Suppose that the temperature of the surface is suddenly raised to T that is, the heat-transfer coefficient is infinite. The unsteady temperature of the plate is 

Two- and Three-Dimensional Conduction The one-dimensional solutions discussed above can be used to construct solutions to multidimensional problems. The unsteady temperature of a rectangular, solid box of height, length, and width 2H, 2L, and 2 W, respectively, with governing equations in each direction as in (5-18), is 

Similar products apply for solids with other geometries, e.g., semiinfinite, cylindrical rods. 

Convection is the transfer of energy by conduction and radiation in moving, fluid media. The motion of the fluid is an essential part of convective heat transfer. A key step in calculating the rate of heat transfer by convection is the calculation of the heat-transfer coefficient. This section focuses on the estimation of heat-transfer coefficients for natural and forced convection. The conservation equations for mass, momentum, and energy, as presented in Sec. 6, can be used to calculate the rate of convective heat transfer. Our approach in this section is to rely on correlations. 

In many cases of industrial importance, heat is transferred from one fluid, through a solid wall, to another fluid. The transfer occurs in a heat exchanger. Section 11 introduces several types of heat exchangers, design procedures, overall heat-transfer coefficients, and mean temperature differences. Section 3 introduces dimensional analysis and the dimensionless groups associated with the heat-transfer coefficient. 

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The problem with Eq. (7.5) is that the overall heat transfer coefficient is not constant throughout the process. Is there some way to extend this model to deal with the individual heat transfer coefficients ... 

By constrast, Fig. 7.46 shows a diflFerent arrangement. Hot stream A with a low coefficient is matched with cold stream D, which also has a low coefficient but uses temperature diflferences greater than vertical separation. Hot stream B is matched with cold stream C, both with high heat transfer coefficients but with temperature differences less than vertical. This arrangement requires 1250 m of area overall, less than the vertical arrangement. 

Stream Supply temp. T, rc) Target temp. Tr rC) AH (MW) Heat capacity flow rate CP (WN C- ) Heat transfer coefficient h(MW... 

Thus, for a given exchanger duty and overall heat transfer coefficient, the 1-2 design needs a larger area than the 1-1 design. However, the 1-2 design offers many practical advantages. These... 

These small positive and negative errors partially cancel each other. The result is that capital cost targets predicted by the methods described in this chapter are usually within 5 percent of the final design, providing heat transfer coefficients vary by less than one order of magnitude. If heat transfer coefficients vary by more than one order of magnitude, then a more sophisticated approach can sometimes be justified. ... 

Now consider the heat transfer area required by enthalpy interval k, in which the overall heat transfer coefficient is allowed to vary... 

Uij = overall heat transfer coefficient between hot stream i and cold stream j... 

Equation (F.l) shows that each stream makes a contribution to total heat transfer area defined only by its duty, position in the composite curves, and its h value. This contribution to area means also a contribution to capital cost. If, for example, a corrosive stream requires special materials of construction, it will have a greater contribution to capital cost than a similar noncorrosive stream. If only one cost law is to be used for a network comprising mixed materials of construction, the area contribution of streams requiring special materials must somehow increase. One way this may be done is by weighting the heat transfer coefficients to reflect the cost of the material the stream requires. 

The relationship between heat exchanger area and overall heat transfer coefficient U is given by... 

Liquid viscosity is one of the most difficult properties to calculate with accuracy, yet it has an important role in the calculation of heat transfer coefficients and pressure drop. No single method is satisfactory for all temperature and viscosity ranges. We will distinguish three cases for pure hydrocarbons and petroleum fractions ... 

If solvent recovery is maximized by minimizing the temperature approach, the overall heat-transfer coefficient in the condenser will be reduced. This is due to the fact that a large fraction of the heat transfer area is now utilized for cooling a gas rather than condensing a Hquid. Depending on the desired temperature approach the overall heat-transfer coefficients in vent condensers usually range between 85 and 170 W/m K (ca 15 and 30 Btu/h-ft. °F). 

Bed-to-Surface Heat Transfer. Bed-to-surface heat-transfer coefficients in fluidized beds are high. In a fast-fluidized bed combustor containing mostly Group B limestone particles, the dense bed-to-boiling water heat-transfer coefficient is on the order of 250 W/(m -K). For an FCC catalyst cooler (Group A particles), this heat-transfer coefficient is around 600 W/(600 -K). 

The heat-transfer coefficient of most interest is that between the bed and a wall or tube. This heat-transfer coefficient, is made up of three components. To obtain the overall dense bed-to-boiling water heat-transfer coefficient, the additional resistances of the tube wall and inside-tube-waH-to-boiling-water must be added. Generally, the conductive heat transfer from particles to the surface, the convective heat transfer... 

The heat-transfer coefficient depends on particle size distribution, bed voidage, tube size, etc. Thus a universal correlation to predict heat-transfer coefficients is not available. However, the correlation of Andeen and Ghcksman (22) is adequate for approximate predictions ... 

Fundamental models correctly predict that for Group A particles, the conductive heat transfer is much greater than the convective heat transfer. For Group B and D particles, the gas convective heat transfer predominates as the particle surface area decreases. Figure 11 demonstrates how heat transfer varies with pressure and velocity for the different types of particles (23). As superficial velocity increases, there is a sudden jump in the heat-transfer coefficient as gas velocity exceeds and the bed becomes fluidized. 

Fig. 11. Variation of heat-transfer coefficient, where O represents experimental results at 100 kPa , 500 kPa 0, 1000 kPa and , 2000 kPa, of pressure (23) for (a) a 0.061-mm glass—CO2 system (Group A particles) and (b) a 0.475-mm glass—N2 system (Group B and D particles). To convert kPa to psi,...

Traditionally, production of metallic glasses requites rapid heat removal from the material (Fig. 2) which normally involves a combination of a cooling process that has a high heat-transfer coefficient at the interface of the Hquid and quenching medium, and a thin cross section in at least one-dimension. Besides rapid cooling, a variety of techniques are available to produce metallic glasses. Processes not dependent on rapid solidification include plastic deformation (38), mechanical alloying (7,8), and diffusional transformations (10). 

I ewton s Cooling L w of Heat Convection. The heat-transfer rate per unit area by convection is directly proportional to the temperature difference between the soHd and the fluid which, using a proportionaUty constant called the heat-transfer coefficient, becomes... 

Table 2. Values of the Convective Heat-Transfer Coefficient ...

In the forced convection heat transfer, the heat-transfer coefficient, mainly depends on the fluid velocity because the contribution from natural convection is negligibly small. The dependence of the heat-transfer coefficient, on fluid velocity, which has been observed empirically (1—3), for laminar flow inside tubes, is h for turbulent flow inside tubes, h and for flow outside tubes, h. Flow may be classified as laminar or... 

Nusse/t Number. Empidcal correlations can be obtained for a particular size of tube diameter and particular flow conditions. To generalize such results and to apply the correlations to different sizes of equipment and different flow conditions, the heat-transfer coefficient, Z, is traditionally nondimensionalized by the use of the Nusselt number, Nu named after Wilhelm Nusselt,... 

Correlations for Convective Heat Transfer. In the design or sizing of a heat exchanger, the heat-transfer coefficients on the inner and outer walls of the tube and the friction coefficient in the tube must be calculated. Summaries of the various correlations for convective heat-transfer coefficients for internal and external flows are given in Tables 3 and 4, respectively, in terms of the Nusselt number. In addition, the friction coefficient is given for the deterrnination of the pumping requirement. 

The convective heat-transfer coefficient and friction factor for laminar flow in noncircular ducts can be calculated from empirically or analytically determined Nusselt numbers, as given in Table 5. For turbulent flow, the circular duct data with the use of the hydrauhc diameter, defined in equation 10, may be used. 

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