Diffusion of heat

Sohd Catalysts Processes with solid catalysts are affected by diffusion of heat and mass (1) within the pores of the pellet, (2) between the fluid and the particle, and (3) axially and radially within the packed bed. Criteria in terms of various dimensionless groups have been developed to tell when these effects are appreciable. They are discussed by Mears (Ind. Eng. Chem. Proc. Des. Devel., 10, 541-547 [1971] Jnd. Eng. Chem. Fund., 15, 20-23 [1976]) and Satterfield (Heterogeneous Cataly.sls in Practice, McGraw-Hill, 1991, p. 491). 

A long heating cycle is necessaiy because the size of the solid objects or permissible heating temperature requires a long holdup for internal diffusion of heat or moisture. This case may apply when the cycle will exceed 12 to 24 h. 

The mechanism of flame propagation into a stagnant fuel-air mixture is determined largely by conduction and molecular diffusion of heat and species. Figure 3.1 shows the change in temperature across a laminar flame, whose thickness is on the order of one millimeter. 

Where large samples of reactant are used and/or where C02 withdrawal is not rapid or complete, the rates of calcite decomposition can be controlled by the rate of heat transfer [748] or C02 removal [749], Draper [748] has shown that the shapes of a—time curves can be altered by varying the reactant geometry and supply of heat to the reactant mass. Under the conditions used, heat flow, rather than product escape, was identified as rate-limiting. Using large ( 100 g) samples, Hills [749] concluded that the reaction rate was controlled by both the diffusion of heat to the interface and C02 from it. The proposed models were consistent with independently measured values of the transport parameters [750—752] whether these results are transfenable to small samples is questionable. 

The temperature counterpart of Q>aVR ccj-F/R and if ccj-F/R is low enough, then the reactor will be adiabatic. Since aj 3>a, the situation of an adiabatic, laminar flow reactor is rare. Should it occur, then T i, will be the same in the small and large reactors, and blind scaleup is possible. More commonly, ari/R wiU be so large that radial diffusion of heat will be significant in the small reactor. The extent of radial diffusion will lessen upon scaleup, leading to the possibility of thermal runaway. If model-based scaleup predicts a reasonable outcome, go for it. Otherwise, consider scaling in series or parallel. 

Thermal runaway. Temperature control in a tubular polymerizer depends on convective diffusion of heat. This becomes difficult in a large-diameter tube, and temperatures may rise to a point where a thermal runaway becomes inevitable. 

Using the equation of state (Equation 5.1.12) in the energy Equation 5.1.11, and neglecting the diffusion of heat on the scale of the acoustic wavelength, one finds... 

In dimensionless terms, there is a critical value for S (Damkohler number) that makes ignition possible. From Equation (4.23), this qualitatively means that the reaction time must be smaller than the time needed for the diffusion of heat. The pulse of the spark energy must at least be longer than the reaction time. Also, the time for autoignition at a given temperature T is directly related to the reaction time according to Semenov (as reported in Reference [5]) by... 

Comparison between heat-mode and photon-mode processes is given in Table I. The main differences are the superior resolution and the possibility of multiplex recording in photon-mode systems. Because of the diffusion of heat, the resolution of heat-mode recording is inferior to that of photon-mode systems. Furthermore, photons are rich in information such as energy, polarization and coherency, which can not be rivalled by heat-mode recording. 

Owing to this large concentration of OH relative to O and H in the early part of the reaction zone, OH attack on the fuel is the primary reason for the fuel decay. Since the OH rate constant for abstraction from the fuel is of the same order as those for H and O, its abstraction reaction must dominate. The latter part of the reaction zone forms the region where the intermediate fuel molecules are consumed and where the CO is converted to C02. As discussed in Chapter 3, the CO conversion results in the major heat release in the system and is the reason the rate of heat release curve peaks near the maximum temperature. This curve falls off quickly because of the rapid disappearance of CO and the remaining fuel intermediates. The temperature follows a smoother, exponential-like rise because of the diffusion of heat back to the cooler gases. 

The solution of Eq. (6.137) must be combined with the nonsteady equations for the diffusion of heat and mass. This system can only be solved numerically and the computing time is substantial. Therefore, a simpler alternative model of droplet heating is adopted [26, 27], In this model, the droplet temperature is assumed to be spatially uniform at Ts and temporally varying. With this assumption Eq. (6.136) becomes... 

Figure 6.14, taken from Law [28], is a plot of the nondimensional radius squared as a function of a nondimensional time for an octane droplet initially at 300 K burning under ambient conditions. Shown in this figure are the droplet histories calculated using Eqs. (6.137) and (6.138). Sirignano and Law [27] refer to the result of Eq. (6.137) as the diffusion limit and that of Eq. (6.138) as the distillation limit, respectively. Equation (6.137) allows for diffusion of heat into the droplet, whereas Eq. (6.138) essentially assumes infinite thermal conductivity of the liquid and has vaporization at Ts as a function of time. Thus, one should expect Eq. (6.137) to give a slower burning time. 

If the temperature of a water mass is altered after having been isolated from the sea surfece, deviations from saturation can result. Such alterations are a consequence of heat transport via either conduction (molecular diffusion of heat) or turbulent mixing of adjacent water masses. In the case of the former, hydrothermal systems are important subsurface heat sources. If two water masses of different temperature imdergo turbulent mixing, the temperature of the admixture will differ from that of the source waters. 

In this overview we focus on the elastodynamical aspects of the transformation and intentionally exclude phase changes controlled by diffusion of heat or constituent. To emphasize ideas we use a one dimensional model which reduces to a nonlinear wave equation. Following Ericksen (1975) and James (1980), we interpret the behavior of transforming material as associated with the nonconvexity of elastic energy and demonstrate that a simplest initial value problem for the wave equation with a non-monotone stress-strain relation exhibits massive failure of uniqueness associated with the phenomena of nucleation and growth. 

Nucleation is necessary for the new phase to form, and is often the most difficult step. Because the new phase and old phase have the same composition, mass transport is not necessary. However, for very rapid interface reaction rate, heat transport may play a role. The growth rate may be controlled either by interface reaction or heat transport. Because diffusivity of heat is much greater than chemical diffusivity, crystal growth controlled by heat transport is expected to be much more rapid than crystal growth controlled by mass transport. For vaporization of liquid (e.g., water vapor) in air, because the gas phase is already present (air), nucleation is not necessary except for vaporization (bubbling) beginning in the interior. Similarly, for ice melting (ice water) in nature, nucleation does not seem to be difficult. 

Hint The process of heat exchange across an interface can be treated in the same way as the exchange of a chemical at the interface. To do so, we must express the molecular thermal heat conductivity by a molecular diffusivity of heat in water and air, Z)thw and Z)tha, respectively. At 20°C, we have (see Appendix B) flthw = 1.43 xl(T3 cm2 s-1, Dlh a = 0.216 cm2 s 1. Use the film model for air-water exchange with the typical film thicknesses of Eq. 20-18a. 

Many texts, such as Crank s treatise on diffusion [2], contain solutions in terms of simple functions for a variety of conditions—indeed, the number of worked problems is enormous. As demonstrated in Section 4.1, the differential equation for the diffusion of heat by thermal conduction has the same form as the mass diffusion equation, with the concentration replaced by the temperature and the mass diffusivity replaced by the thermal diffusivity, k. Solutions to many heat-flow... 

A large number of technically important problems involve solution of the equation for diffusion of heat, mass, or some other scalar quantity, subject to the existence of a free boundary. This means that the location of the free boundary is in itself dependent, usually as the result of a phase change, upon the amount of diffusant which has been brought up to it in the past. Common examples are the freezing of the surface of a large... 

The following calculation as made for the Saline Water Project (6) shows the relation between pressure applied and production rate. The dominant factors are (1) the salt solution whose osmotic pressure must be overcome, (2) the pressure, as an energy source, (3) the diffusion of heat and (4) vapor as resistance factors, and (5) viscous losses within the cellophane capillaries. 

Diffusion of Heat. In dynamic equilibrium, a transfer of vapor from liquid through a vapor phase to a second liquid (the two liquids being thermally connected only across the thin gap) will require reverse transfer of the heat of vaporization. This will accompany a temperature difference determined by the ratio of heat flow to the thermal conductance of the two heat paths. These two are the diffusion vapor gap and the series of salt water and plastic films. For the diffusion gap the c.g.s. air value 5.7 x 1(H is chosen for the thermal conductivity (neglecting the separating powder), while for the series polyethylene (50 X 10-4 cm. thick), wet cellophane (50 X 10"4 cm. thick), and water (200 X 10-4 cm. thick) the respective thermal conductivities are 3.5 X 10"4, 4 X 10-4, and 14 X 10 4. 

The diffusion of heat through a liquid or a gas by motion of the substance. Coolant ... 

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