Isothermal tube wall

The catalyst in an isothermal tube-wall reactor (experiment TWR-6 in Ref. 2) deactivated much more slowly than did the catalyst in the best test (experiment HGR-14) in an adiabatic HGR reactor (0.009 vs. 0.0291 %/mscf/lb), and it also produced much more methane (177 vs. 32 mscf/lb catalyst). This indicates that adiabatic operation of a metha-nation catalyst between 300° and 400°C is not as efficient as isothermal operation at higher temperature ( 400°C). 

The enhancement of heat transfer at a isothermal tube wall due to temperature-dependent viscosity has been correlated by Kwant et al. [1973] as follows ... 

Example 15.4 A reboiler is required to supply 0.1 krnol-s 1 of vapor to a distillation column. The column bottom product is almost pure butane. The column operates with a pressure at the bottom of the column of 19.25 bar. At this pressure, the butane vaporizes at a temperature of 112°C. The vaporization can be assumed to be essentially isothermal and is to be carried out using steam with a condensing temperature of 140°C. The heat of vaporization for butane is 233,000 Jkg, its critical pressure 38 bar, critical temperature 425.2 K and molar mass 58 kg krnol Steel tubes with 30 mm outside diameter, 2 mm wall thickness and length 3.95 m are to be used. The thermal conductivity of the tube wall can be taken to be 45 W-m 1-K 1. The film coefficient (including fouling) for the condensing steam can be assumed to be 5700 W m 2-K 1. Estimate the heat transfer area for... 

Several simplifying assumptions are adopted (1) the flow in the tube is considered to be one dimensional without turbulence or mixing (2) the processes in the tube are adiabatic. There is no heat conduction in the tube and no heat exchange between the gas and the tube walls (3) each heat exchanger is isothermal. 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

An important class of non-isothermal fixed-bed reactors is that for which the tube wall temperature is not constant, but varies along the reactor length. Such would be the case when the cooling tubes and reactor... 

The solution of Eq. (173) poses a rather formidable task in general. Thus the dispersed plug-flow model has not been as extensively studied as the axial-dispersed plug-flow model. Actually, if there are no initial radial gradients in C, the radial terms will be identically zero, and Eq. (173) will reduce to the simpler Eq. (167). Thus for a simple isothermal reactor, the dispersed plug flow model is not useful. Its greatest use is for either nonisothermal reactions with radial temperature gradients or tube wall catalysed reactions. Of course, if the reactants were not introduced uniformly across a plane the model could be used, but this would not be a common practice. Paneth and Herzfeld (P2) have used this model for a first order wall catalysed reaction. The boundary conditions used were the same as those discussed for tracer measurements for radial dispersion coefficients in Section II,C,3,b, except that at the wall. 

An incompressible fluid flows into one end of an isothermal porous tube that is blocked on the far end (Fig. 4.31). The mass flux through the tube walls is proportional to the local pressure difference across the tube wall,... 

On the other hand, the tubular reactor is a simple and inexpensive apparatus. It s small inner diameter requires a low thickness of the tube-wall to resist high pressure, and facilitates the removal or heating of the feed in order to operate the reactor under isothermal conditions. Solid catalyst can easily be placed in the tubular reactor. 

Non-isothermal and non-adiabatic conditions. A useful approach to the preliminary design of a non-isothermal fixed bed reactor is to assume that all the resistance to heat transfer is in a thin layer near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the approximate design of reactors. Neglecting diffusion and conduction in the direction of flow, the mass and energy balances for a single component of the reacting mixture are ... 

A tube-wall reactor, in which the catalyst is coated on the tube wall, is conceptually ideally suited for highly exothermic and equilibrium-limited reactions because the heat generated at the wall can be rapidly taken away by the coolant. Previous work (1) has numerically demonstrated that for highly exothermic selectivity reactions, the optimized tube-wall reactor is superior from both steady state production and dynamic points of view to the fixed-bed reactor. Also, the tube-wall reactor is being advanced as a possible reactor for carrying out methanation in coal gasification plants (2). From a reaction engineering point of view, it therefore seems appropriate to analyze the reactor for the analytically resolvable case of complex first-order isothermal reactions. 

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. 

For calculating the concentration profile as function of channel length and radius the model of the isothermic tube reactor with catalytically coated wall and laminar flow (ref. 5) was used with the following simplifications ... 

In the early stages of development, the graphite atomizers suffered from relatively low analytical precision, severe chemical interferences and from the intense background signals. With the introduction of the L vov platform and Zeeman-effect background correction considerable improvements have been made with respect to these deficiencies. The L vov platform placed inside the graphite tube (first proposed by L vov in 1978) is mainly heated by atomization from the tube wall as the temperature increases. Atomization of the sample from the platform delays atomization until near isothermal gas-phase conditions are present... 

Figure 5.7 Spatiotemporal distribution of the degree of conversion. The front is initiated at the test tube wall (r = 8) and moves toward the tube axis (r = 0). The profiles are shown over equal time intervals. Densely drawn profiles propagating to the left from the test tube wall toward the axis correspond to the isothermal mechanism of propagation,...

The calculations apply to an isothermal reactor, a condition which generally is satisfied in case of low concentration of the waste gas. Further it is assumed that the kinetics of the chemical reaction taking place at the tube wall can be described by... 

For the fully developed isothermal flow of an incompressible fluid in a pipe of radius R and length L where the pressure drop along the length of the tube is AP, the shear stress distribution is given by (AF/L)(r/2) hence, the shear stress at the tube wall is given by this equation with the radial position r = R. Also, the apparent shear rate, y, is given by 4Q/(ttR ), where Q is the volumetric flow rate, which can be calculated from the fluidity data from... 

Air and water flow at 8 x 10 3 kg/s and 0.4 kg/s upwards in a vertical, smooth-wall tube of internal diameter dt = 20 mm and length L = 1.3 m. Using the homogeneous flow model, calculate the pressure drop across the tube (neglecting end effects). The fluids are at a temperature of 20 °C and the expansion of the air may be assumed to be isothermal. The exit pressure is 1 bar. 

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