Steady tubular flow

Consider a steady unidimensional flow in a tubular reaetor as shown in Figure 8-21 in the absenee of either radial or longitudinal diffusion. The veloeity u(r) is the parabolie distribution for a Newtonian fluid at eonstant viseosity, with the fluid in the eenter of the tube spending the shortest time in the reaetor. 

Steady-State Tubular Flow with Heat Loss... 

There are a variety of limiting forms of equation 8.0.3 that are appropriate for use with different types of reactors and different modes of operation. For stirred tanks the reactor contents are uniform in temperature and composition throughout, and it is possible to write the energy balance over the entire reactor. In the case of a batch reactor, only the first two terms need be retained. For continuous flow systems operating at steady state, the accumulation term disappears. For adiabatic operation in the absence of shaft work effects the energy transfer term is omitted. For the case of semibatch operation it may be necessary to retain all four terms. For tubular flow reactors neither the composition nor the temperature need be independent of position, and the energy balance must be written on a differential element of reactor volume. The resultant differential equation must then be solved in conjunction with the differential equation describing the material balance on the differential element. 

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. 

All chemical reactions are accompanied by some heat effects so that the temperature will tend to change, a serious result in view of the sensitivity of most reaction rates to temperature. Factors of equipment size, controllability, and possibly unfavorable product distribution of complex reactions often necessitate provision of means of heat transfer to keep the temperature within bounds. In practical operation of nonflow or tubular flow reactors, truly isothermal conditions are not feasible even if they were desirable. Individual continuous stirred tanks, however, do maintain substantially uniform temperatures at steady state when the mixing is intense enough the level is determined by the heat of reaction as well as the rate of heat transfer provided. 

Reaction processes may be conducted under nonflow or steady flow conditions. One mode of the latter is tubular flow or, in the limiting case, plug flow, in which all molecules have substantially the same residence time. The rate equation for a plug flow reactor (PFR) is... 

There are a number of drawbacks to using continuous processes. Resources are needed to develop the process the appropriate residence time to reach a level of suitable reaction completion must be determined under the desired conditions of temperature, flow rate, and any other critical parameters. The reaction system may have limited flexibility for running other reactions. Pressure drops occur when using tubular flow reactors, and these can be calculated [18]. Once the conditions have been developed, time is necessary to reach steady-state conditions. What happens to material produced while the conditions are approaching steady state Such material is not produced under the desired conditions and hence is atypical of the majority of the batch. Effective control equipment is mandatory for large-scale operations otherwise expensive material is at risk and may need to be reworked. 

Tubular flow reactors are usually operated under steady conditions so that, at any point, physical and chemical properties do not vary with time. Unlike the batch and tank flow reactors, there is no mechanical mixing. Thus, the state of the reacting fluid will vary from point to point in the system, and this variation may be in both the radial and axial direction. The describing equations are then differential, with position as the independent variable. 

In Sec. 6-5 a non-steady-state mass balance for a tubular-flow reactor (plug flow except for axial dispersion) was used to evaluate an effective diffusivity. Now we consider the problem of calculating the conversion when a reaction occurs in a dispersion-model reactor operated at steady-state conditions. Again a mass balance is written, this time for steady state and including reaction and axial-dispersion terms. It is considered now that the axial diffusivity is known. 

An ideal stirred bioreactor is assumed to be well mixed so that the contents are uniform in composition at all times. The plug-flow bioreactor (PFB) is an ideal tubular-flow bioreactor without radial concentration variations. The nutrient concentration of an ideal batch bioreactor after time t will be the same as that of a steady-state PFB at the longitudinal location of the residence time. Therefore, the following analysis applies for both the ideal batch bioreactor and the steady-state PFB. 

Steady-state flow is never achieved with foamed fluids rather, the flow is dynamic. Foams flow dynamically because the pressure, which is continually changing, affects the viscosity, flow rate, and density of the foam at any given interval in the tubular. This problem can be accounted for by numerically integrating the mechanical energy balance equation from bottomhole to surface conditions. 

Calculate the residence-time distribution (RTD) for a tubular reactor undergoing steady, laminar flow (Hagen-Poiseuille flow). The velocity profile for Hagen-Poiseuille flow is 2, p. 51]... 

In the analysis of batch reactors, the two flow terms in equation (8.0.1) are omitted. For continuous flow reactors operating at steady state, the accumulation term is omitted. However, for the analysis of continuous flow reactors under transient conditions and for semibatch reactors, it may be necessary to retain all four terms. For ideal well-stirred reactors, the composition and temperature are uniform throughout the reactor and all volume elements are identical. Hence, the material balance may be written over the entire reactor in the analysis of an individual stirred tank. For tubular flow reactors the composition is not independent of position and the balance must be written on a differential element of reactor volume and then integrated over the entire reactor using appropriate flow conditions and concentration and temperature profiles. When non-steady-state conditions are involved, it will be necessary to integrate over time as well as over volume to determine the performance characteristics of the reactor. 

Tubular flow reactors are characterized by a continuous and decreasing concen-tiadon of reactants in the direction of flow. This is in contrast to the discontinuous characteristic of the CSTR reactor. Most of these units consist of one or several pipes or tubes in parallel. Either horizontal or vertical orientation is common. The reactants are charged continuously at one end, and the products are removed continuously at the other end. The unit almost always operates in a steady-state mode. This greatly simplifies design and predictive calculations. It is a unit that is amendable to automatic control and to experimental work. When heat transfer is required, a jacketed tube or a construction similar to that of a shell-and-tube heat exchanger is employed. In the latter case the reactants may be on either the tube or shell side. 

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