Axial variation of temperature

Figure 14. The axial variation of temperature in a sooting flame with height above the tube burner...

Kubec el al. (1974) developed a model for a radial flow quench type converter. Kjaer (1985) and Michael and Filippo (1982) formulated model equations accounting for radial as well as axial variation of temperature and concentration. They found that radial variations have insignificant influence on the model predictions. 

The gate corresponds to Z/L = 0.0 in tfie plot. Minimal conversion has occurred (plot not shown) thus these profiles are almost entirely due to conduction and not heat of reaction. Only mild axial variation of temperature is evident here, and even less would occur with more moderate molding temperatures. Under such circumstances, a one rather than two dimensional solution of the curing stage should suffice, simplifying the modeling procedure. 

We also account for density, heat capacity, and molecular weight variations due to temperature, pressure, and mole changes, along with temperature-induced variations in equilibrium constants, reaction rate constants, and heats of reaction. Axial variations of the fluid velocity arising from axial temperature changes and the change in the number of moles due to the reaction are accounted for by using the overall mass conservation or continuity equation. 

Homogeneous one-dimensional model This is the simplest description of a packed bed, with an overall heat-transfer coefficient U. The particle and gas temperatures are identical, and only axial variation in temperature is considered, giving the following mass and energy balance equations for any species C, ... 

When the radial variation of temperature must be taken into account, the problem assumes an entirely different character. Each of the equations is now a partial differential equation, and both radial and axial profiles must be calculated a mesh or network of radial and axial lines is set up, and the temperature and composition are calculated for each intersection. A great deal of work has been done on the formulation of difference equations for solving the related diffusion or heat-conduction equations most of this has been directed towards the case in which there is only one dependent variable and in which the source is a linear function of that variable. Although the results obtained for one dependent variable are only partially applicable to the multiple-variable problem,... 

SOLUTION This heat transfer problem is similar to the problem in Example 2-17, except that we need to obtain a relation for the variation of temperature within the wire with r. Differential equations are v/ell suited for this purpose. Assumptions t Heat transfer is steady since there is no change with time. 2 Heat transfer is one-dimensional since there is no thermal symmetry about the centerline arid no change m the axial direction. 3 Thermal conductivity is constant. 4 Heat generation in the wire is uniform. 

In vertical pneumatic transport the radial particle concentration distribution is almost uniform, but some particle strands may still be identified near the wall. Little or no axial variation of solids concentration except in the bottom acceleration section is observed [58]. The flow associated with transport bed reactors tends to be dilute (typically 1 to 5 % by volume solids) and uniform. By virtue of the smaller reflux and density of the suspension within the dilute pneumatic conveying regime, there might be larger temperature gradients than within the fast fluidization regime [56]. 

We reflect on the more involved case of a tubular reactor described in Section 2.2.4.2, where we will consider the radial variation of temperature in addition to its axial variation. Thus T = T(kz). Furthermore, we also assume that the diffusion coefficient is constant. Consequently, the mass and energy balance are given by the following partial differential equations ... 

The analysis solves the fundamental heat propagation equations using a two-dimensional computational model in which values of the radial and axial variations of the temperature and the refractive index are calculated. The calculated temperature gradients for the two cases are shown in Fig. 18. Figure 19 shows the radial dependence of the average (across the thickness) temperature. The temperature in CVD diamond shows a maximum variation (center to edge) of 3.4°C compared to 23.4°C for the ZnSe case. The temperature variations over the effective diameter of the beam are 1.7°C and 12.4°C for the CVD diamond and ZnSe windows respectively. 

Startup after catalyst replacement, Ap < expected and conversion < standard [maldistribution] and axial variation in temperature/larger size catalyst. Startup after catalyst replacement, conversion < standard and Ap increasing [maldistribution and axial temperatures drfferent] /feed precursors present for polymerization or coking. Startup after catalyst replacement, Apfor this batch of catalyst > previous batch catalyst fines produced during loading/poor loading. Startup after catalyst replacement, conversion < specifications per unit mass of catalyst and more side reactions [maldistribution] /faulty inlet distributor/faulty exit distributor. 

Fruther investigations are required in order to highlight the roles of wall-fluid conjugate heat transfer, temperature-dependent fluid properties (i.e., viscosity), rarefaction, compressibility, and axial variation of the zeta potential in electroosmotic flows in developing flows for the most common microchannel cross-sectional geometries. 

It is interesting to note that for ideal gases for which jS = /Th, the flow work and the viscous dissipation are exactly the same in terms of absolute value and they compensate each other. Eor this reason, for ideal gases, the axial variation of the bulk temperature in a channel is independent of viscous heating and flow work even for large values of the Mach number (highly compressible gases). 

Axial variation of coolant, cladding surface, and fuel element center temperatures. 

In this section, we formulate a ID model with interphase mass and heat transfer coefficients. These lumped models [103] describe the axial variation of concentration and temperature (which are averaged over the channel cross section). The diffusion processes in the transverse directions (represented by differential terms) are replaced by a transfer term, associated with a given driving force. The use of ID models is widespread throughout the literature on monoUth reactor modeling. Chen et al. [3] reviewed some specific appUcations including simulation of simultaneous heat transfer in monofith catalysts... 

Figure 8.5 2D nonisothermal solution without axial dispersion. The axi-symmetric solution plane is rotated 210° along the z-axis. (a) The variation of temperature along with the reactor length at a low inlet partial pressure (b) the effect of the feed amount on the level of hot spot at the inlet of the reactor. 

Atmospheric air enters a 3-m-long, 0.05-m-diameter tube at 0.005 kg/sec. The h is 25 W/m °K, there is a uniform heat flux at the surface of 1000 W/m. Find the outlet and inlet air temperatures and sketch the axial variation of surface temperature. 

Axial Variation of Fuel, Clad, and Coolant Temperatures... 

In order to derive an expression for the axial variation of the central fuel temperature, we start by equating the rate of heat production in the section dz of the fuel to the rate of transport of heat to the coolant. This is given, by a variation of equation (6.44), as... 

Fig. 6.9. Axial variation of heat generation rate per unit volume qZ ), bulk coolant temperature (Tf,), cladding temperature (J ), and center-line fuel temperature (7. ).

There is no variation of temperature or concentration normal to the direction of flow. For a tubular reactor, this means that there is no radial or angular variation of temperature or of any species concentration at a given axial position z. As a consequence, the reaction rate r,-does not vary normal to the direction of flow, at any cross section in the direction of flow. 

Figure 7 Evolution of the axial variation of the temperature on the mean radius during start-up.

This figure also illustrates the strong variation of temperature in the radial direction over the axial length covering the rear half of the reactor. Flow field simulations of the single monolith channels for this case revealed that fuel is already consumed up to 99% at half the reactor length, which in turn renders the rest of the reactor volume a heat sink towards the environment. Reduced radial heat transfer (due to the spatially non-uniform thermal properties of the monolith) reduces the... 

The design equations for a chemical reactor contain several parameters that are functions of temperature. Equation (7.17) applies to a nonisothermal batch reactor and is exemplary of the physical property variations that can be important even for ideal reactors. Note that the word ideal has three uses in this chapter. In connection with reactors, ideal refers to the quality of mixing in the vessel. Ideal batch reactors and CSTRs have perfect internal mixing. Ideal PFRs are perfectly mixed in the radial direction and have no mixing in the axial direction. These ideal reactors may be nonisothermal and may have physical properties that vary with temperature, pressure, and composition. 

---