Tubular flow reactor velocity profile

Mixing in static mixers considered as chemical reactors was essentially studied by Nauman (165, 166). This author proposed a model which consists of a tubular reactor comprising N zones in laminar flow (parabolic velocity profile). Mixing between each zone is achieved accross a plane by a permutation of the radial position of fluid particles (r — , in this way the flowrate... 

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]... 

The ideal tubular reactor is one in which elements of the homogeneous fluid reactant stream move through a tube as plugs moving parallel to the tube axis. This flow pattern is referred to as plug flow or piston flow. The velocity profile at a given cross section is flat and it is assumed that there is no axial diffusion or backmixing of fluid elements. 

Tubular polymerization reactors frequently show large deviations from the parabolic velocity profile of constant viscosity laminar flow. The velocity profile of a polymerizing mixture can be calculated by combining the equations of motion with the convective diffusion equations for heat and mass, but direct experimental verification of the calculations is difficult. One way of testing the results is to compare an experimental residence time distribution to the calculated distribution. There is a one-to-one correspondence between velocity profile and RTD for well-developed diffusion-free flows in tubes. See Nauman and Buffham (1983) for details. 

A tubular plug flow (Figure 5-28) reactor assumes that mixing of fluid does not take place, the velocity profile is flat, and both temperature and composition are uniform at any cross-section in the reactor. 

Example 8.9 Find the temperature distribution in a laminar flow, tubular heat exchanger having a uniform inlet temperature and constant wall temperature Twall- Ignore the temperature dependence of viscosity so that the velocity profile is parabolic everywhere in the reactor. Use art/P = 0.4 and report your results in terms of the dimensionless temperature... 

The final idealized flow situation that we will consider is laminar flow in a tubular reactor in the absence of either radial or longitudinal diffusion. The velocity profile in such a reactor is given by... 

The PFR model assumes a flat velocity profile across the whole of the reactor cross-section in reality, this is impossible to achieve although in practice certain combinations of physical conditions are closely described by this assumption. If the Reynolds number, dupln, in a tubular reactor is less than about 2100, then the flow therein will be laminar and where the flow is fully developed, the velocity profile across the reactor will be parabolic in form. If one assumes that diffusion is negligible between adjacent radial layers of fluid, then it is relatively straightforward to derive the forms of E(t), E(0) and F(0) associated with this type of reactor [42]. These are given in the equations... 

There are some fundamental investigations devoted to analysis of the flow in tubular polymerization reactors where the viscosity of the final product has a limit (viscosity < >) i.e., the reactive mass is fluid up to the end of the process. As a zero approximation, flow can be considered to be one-dimensional, for which it is assumed that the velocity is constant across the tube cross-section. This is a model of an ideal plug reactor, and it is very far from reality. A model with a Poiseuille velocity profile (parabolic for a Newtonian liquid) at each cross-section is a first approximation, but again this is a very rough model, which does not reflect the inherent interactions between the kinetics of the chemical reaction, the changes in viscosity of the reactive liquid, and the changes in temperature and velocity profiles along the reactor. 

The flow patterns, composition profiles, and temperature profiles in a real tubular reactor can often be quite complex. Temperature and composition gradients can exist in both the axial and radial dimensions. Flow can be laminar or turbulent. Axial diffusion and conduction can occur. All of these potential complexities are eliminated when the plug flow assumption is made. A plug flow tubular reactor (PFR) assumes that the process fluid moves with a uniform velocity profile over the entire cross-sectional area of the reactor and no radial gradients exist. This assumption is fairly reasonable for adiabatic reactors. But for nonadiabatic reactors, radial temperature gradients are inherent features. If tube diameters are kept small, the plug flow assumption in more correct. Nevertheless the PFR can be used for many systems, and this idealized tubular reactor will be assumed in the examples considered in this book. We also assume that there is no axial conduction or diffusion. 

The laminar velocity profile in Figure 8-2la is approximated by a series of annuli, within each of which the velocity is constant as illustrated in Figure 8-2lb. Each annulus is considered to be a plug flow tubular reactor having its own space velocity. The velocities of the fluid elements at different radii are given by the parabolic velocity profile for fully developed laminar flow. The velocity is expressed as... 

Fick s diffusion law is used to describe dispersion. In a tubular reactor, either empty or packed, the depletion of the reactant and non-uniform flow velocity profiles result in concentration gradients, and thus dispersion in both axial and radial directions. Fick s law for molecular diffusion in the x-direction is defined by... 

Plug Flow Reactor. A PFR is a continuous flow reactor. It is an ideal tubular type reactor. The assumption we make is that the reaction mixture stream has the same velocity across the reactor cross-sectional area. In other words, the velocity profile across the reactor is a flat one. In a PFR there is no axial mixing along the reactor. The condition of plug flow is met in highly turbulent flows, as is usually the case in chemical reactors. 

Thirty years later, Gerhard Damkohler (1937) in his historic paper, summarized various reactor models and formulated the two-dimensional CDR model for tubular reactors in complete generality, allowing for finite mixing both in the radial and axial directions. In this paper, Damkohler used the flux-type boundary condition at the inlet and also replaced the assumption of plug flow with parabolic velocity profile, which is typical of laminar flow in tubes. 

Before proceeding to show how the RTD can be used to estimate conversion in a reactor, we shall derive E t) for a laminar flow reactor. For laminar flow in a tubular reactor, the velocity profile is parabolic, with the fluid in the center of the tube spending the shortest time in the reactor. A schematic diagram of the fluid movement after a time t is shown in Figure 13-9. The figure at the left shows how far down the reactor each concentric fluid element has traveled after a time t. 

We first consider nonideal tubular reactors. Tubular reactors may be empty, or they may be packed with some material that acts as a catalyst, heat-transfer medium, or means of promoting interphase contact. Until now when analyzing ideal tubular reactors, it usually has been assumed that the fluid moved through the reactor in piston-like flow (PFR), and every atom spends an identical length of time in the reaction environment. Here, the velocity profile... 

The gas flow through tubular reactors is of particular importance because the composition at any point is influenced by the linear velocity of the gas, the size of the reactor and the size of the catalyst particles. When gas flows through a pipe at low linear velocity (low Reynolds number), the radial velocity is not uniform. As the linear velocity increases, turbulence increases and the velocity profile approaches what is called plug flow. However, in a packed bed, plug flow can never be completely attained because of the high voidage near the reactor wall. 

Under plug flow conditions the convective transport is completely dominant over the diffusive mass transport term. The fluid moves like a plug and the diffusive term can be neglected. The conditions for plug flow are closely satisfied for narrow and long tubular reactors when the viscosity is low. However, this approximation is clearly best for fully developed turbulent flow, for which the velocity profiles are relatively fiat. For dynamic conditions, the species mass balance is a PDF with z and t as the independent variables. The Eulerian species mass balance (1.301) reduces further to ... 

In particular cases simplified reactor models can be obtained neglecting the insignificant terms in the governing microscopic equations (without averaging in space) [9]. For axisymmetrical tubular reactors, the species mass and heat balances are written in cylindrical coordinates. Himelblau and Bischoff [9] give a list of simplified models that might be used to describe tubular reactors with steady-state turbulent flow. A representative model, with radially variable velocity profile, and axial- and radial dispersion coefficients, is given below ... 

Similar balance equations with purely laminar diffusivities can be used for a fully developed laminar flow in tubular reactors. The velocity profile is then parabolic, so the Hagen Poiseuille law (1.353) might suffice. 

A tubular reactor with laminar flow has been mentioned as a good approximation to segregated flow. If the dispersion due to molecular diffusion is neglected, the approximation is exact. Since the flow is segregated and the velocity profile is known, the RTD can be calculated. It is instruc- tive to compare the calculated results with those for the ideal forms given in Fig. 6-5. The velocity in the axial direction for laminar flow is parabolic,... 

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