Profile flow, steady state temperature

Here a steady-state formulation of heat transfer is considered (Pollard, 1978). A hot fluid flows with linear velocity v, through a tube of length L, and diameter D, such that heat is lost via the tube wall to the surrounding atmosphere. It is required to find the steady-state temperature profile along the tube length. 

Derive the equation for the steady state temperature profile in a simple shear flow with viscous dissipation. Assume a Newtonian viscosity model. 

A flow controller is installed on the feedstream to the reactor. Figure 6.49 shows that the action is reverse and the typical flow controller tuning constants are used (Kc = 0.5 and Tj = 0.3 min). Figure 6.50 shows the final flowsheet, the three controller faceplates, and the steady-state temperature profile. The setpoint of the peak temperature controller is 441 K, and the coolant temperature is 400 K. 

Under more realistic reaction parameters, only a spatially one-dimensional solution could be obtained. Here it was assumed that the gas flow over the whole cross sectional area was reduced uniformly to 1/4 at time zero. The resulting dynamics are given in Figure 8. In this case the transient temperature is so high that the fluid concentration is completely consumed and the reaction zone moves like a front through the whole reactor. Finally a flat steady state temperature profile is established again. 

In the transition range, spreading with the stirrer speed appears, and indeed all the more strongly, the higher the rotation speed. There is a satisfactory explanation for this circumstance, after Rautenbach and BoUenrath [451] calculated the temperature profile for a stirred tank with a low wall clearance for the laminar flow range according to their modified penetration theory. This theory assumes, that a parallel layered flow exists between the tank wall and the stirrer (similar to Couette flow), but that the liquid occluded by the stirrer is ideally mixed. For the temperature profile this means, that the layer near to the wall is at rest, but the layer near the stirrer at every stirrer blade pass is included and is mixed with the liquid bulk and acquires its temperature. After a certain number of stirrer blade passes a steady-state temperature profile is realized. 

By modifying Boussinesq s transformation to allow for unsteady state, the time-dependent temperature distribution in potential flow past a spherical bubble was recently obtained (B6) numerically. For low Peclet numbers (2 < (Npf)c < 100), where the conduction contribution cannot be neglected, graphical integration of the steady-state temperature profiles yields approximately... 

The heat capacity of the catalyst and the reactor wall are not included in the heat balance, since once the steady-state temperature profile is established, the solids cannot store any more energy, and all the heat released must be absorbed by the flowing gas. From Eqs. (3.55) and (3.36),... 

Fig. 1.28. Integrated, autothermal methane reformer with cocurrent flow in the reaction zone and countercurrent heat recovery. Simulated steady-state temperature and conversion profiles.

The governing equation to determine the temperature distribution in the tube is the thermal energy equation, (2-110). To solve this equation, we need to know the form of the velocity distribution in the tube. We have already seen that the steady-state velocity profile for an isothermal fluid, far downstream from the entrance to the tube, is the Poiseuille flow solution given by (3-44). In the present problem, however, the temperature must depend on both r and z, and hence the viscosity (which depends on the temperature) will also depend on position. The dependence on z is due to the fact that heat is added for all z > 0, and thus the temperature must continue to increase with the increase of z. The dependence on r is due to the fact that there must be a nonzero conductive heat flux in the fluid at the tube wall to match the prescribed heat flux through the wall, and thus the temperature must have a nonzero r derivative. It follows that the velocity field will generally differ from Poiseuille flow. 

By systematic variation of reactor geometry it has been conclusively demonstrated that methane added through a peripheral slot to a nitrogen jet titrates, apparently in the true meaning of the word, some potential of the nitrogen jet to react with the thermal decomposition products of methane to form HCN. It is further demonstrated that the potential to react is simply related to heat flow even far down the reactor and is therefore probably the consequence of some steady-state temperature and/or composition profile in a flow with local thermal equilibrium. 

For this case it will be necessary to calculate the steady-state temperature and fractional conversion profiles along the length of the tubular reactor. For a plug flow reactor the appropriate differential material balance for the reaction at hand is... 

The ID pseudo-homogeneous model is the most used model to describe packed bed membrane reactors, especially for laboratory-scale applications. In its simplest form, namely the plug flow steady state model, the model describes only axial profiles of radially averaged temperatures and concentrations. 

After a large number of flow reversals periodic steady state temperature and concentration profiles are established as shown in fig. 2.4. The maximum temperature will reach a limit, which is shown in fig. 2.5, where the development of the maximum temperature in the reactor beginning from the startup to the periodic steady state is given. 

Figure 5.14 Steady-state axial profiles of the volumetric gas flow rate (a) and temperature (b) for aqueous-phase glycerol reforming and DME synthesis without in situ H2O removal.

These authors employed an identical technique to that used previously by Yagi and Kunii [l4]. Steady state heat flow and radial temperature profiles were measured in an annular bed, the walls of which were maintained at different known temperatures, the fluid (air) being stagnant. This technique permits the simultaneous estimation of wall heat transfer coefficients and the effective conductivity in the bed centre whereas... 

The introduction of more liquid feed means that more liquid benzene is cascading down the trays. As liquid reaches the bottom tray, the bubble point of the liquid on that tray decreases. The temperature controller reduces the valve opening on the bottoms stream to conqtensate. The reboiler level rises, resulting in more steam being introduced by the action of the level controller. The benzene and pentane are vaporized and move back up the column. As the column adjusts to the increased feed flow rate, the temperature profile in the column rises. The bottoms flow rate is then increased and settles back down to its new steady-state value. 

Continuous Polymerizations As previously mentioned, fifteen continuous polymerizations in the tubular reactor were performed at different flow rates (i.e. (Nj g) ) with twelve runs using identical formulations and three runs having different emulsifier and initiator concentrations. A summary of the experimental runs is presented in Table IV and the styrene conversion vs reaction time data are presented graphically in Figures 7 to 9. It is important to note that the measurements of pressure and temperature profiles, flow rate and the latex properties indicated that steady state operation was reached after a period corresponding to twice the residence time in the tubular reactor. This agrees with Ghosh s results ). 

The model is described in Sec. 4.3.3. The steady-state balances are written in terms of moles. The Ideal Gas Law is used to calculate the volumetric flow rate from the molar flow at each point in the reactor. This gives also the possibility of considering the influence and temperature or pressure profiles along the tube. 

Bunimovich et al. (1984) point out that if the period of flow reversal, t, is very small relative to the time required for the temperature front to creep through the bed and the high-temperature zone occupies most of the bed a relaxed steady state is achieved in which the temperature profile is constant through most of the bed. This profile can be calculated and leads to a steady-state model for this extreme variant of flow reversal. 

Tubular flow units, like the CSTR, usually are operated at steady state. It is not always easy to measure the temperature profile accurately. In some high temperature operations, the coil is immersed in a fluidized sand bed or lead bath so there is fairly good temperature control. Sometimes it is felt desirable to do the laboratory work in a tubular unit if the commercial unit is to be of that type, but rate data from any kind of equipment are adaptable to the design of PFR. 

The coefficient of thermal conductivity can be defined in reference to the experiment shown schematically in Fig. 12.2. In this example the lower wall (at z = 0) is held at a fixed temperature T and the upper wall (at z = a) is held at some higher temperature T + AT. At steady state there will be a linear temperature profile across the gap, with temperature gradient dT/dz = AT/a. Heat will flow from the hot wall toward the colder wall, and the heat flux q is proportional to the areas of the plates, proportional to the temperature... 

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