Plug heat balance

A steady-state heat balance for a plug flow reactor with no radial temperature gradients is given by ... 

For a more detailed analysis of measured transport restrictions and reaction kinetics, a more complex reactor simulation tool developed at Haldor Topsoe was used. The model used for sulphuric acid catalyst assumes plug flow and integrates differential mass and heat balances through the reactor length [16], The bulk effectiveness factor for the catalyst pellets is determined by solution of differential equations for catalytic reaction coupled with mass and heat transport through the porous catalyst pellet and with a film model for external transport restrictions. The model was used both for optimization of particle size and development of intrinsic rate expressions. Even more complex models including radial profiles or dynamic terms may also be used when appropriate. 

In designing and operating a tubular reactor when the heat of reaction is appreciable, strictly isothermal operation is rarely achieved and usually is not economically justifiable, although the aim may be to maintain the local temperatures within fairly narrow limits. On the assumption of plug flow, the rate of temperature rise or fall along the reactor d77dz is determined by a heat balance... 

The heat balance on the bubble phase in Figure 4.24 that is assumed to be in plug flow mode and to have a negligible rate of reaction, as well as a negligible heat transfer with the heating/cooling coil, leads to... 

For suspension-to-gas (or bed-to-gas) heat transfer in a well-mixed bed of particles, the heat balance over the bed under low Biot number (i.e., negligible internal thermal resistance) and, if the gas flow is assumed to be a plug flow, steady temperature conditions can be expressed as... 

Let s look at some examples. First consider the vinyl acetate reactor discussed in Chap. 11. It is a plug-flow system with external cooling. To satisfy the heat balance we have already proposed to close one loop around the reactor, namely between the reactor exit temperature and the coolant temperature (steam pressure). This provides us with one setpoint, Y cf, that we can use to meet economic objectives, YP. provided exit temperature is a dominant variable. 

If the feaction does not occur on the shell side, kf = 0. Likewise, kf = 0 when catalyst is present only on the shell side. Moreover, plug-flow conditions call for the use of Equation (10-36) or (10-44) for the nnacroscopic mass balance and Equation (10-37) or (10-45) for the corresponding heat balance. On the other hand, when perfect mixing prevails in the tube or shell region. Equation (10-54) and Equation (10-55) or (10-56) are used in conjunction with associated heat balance equations. 

Note that this implies that no sweating occurs at or below M - W = 58.15 W/m2. Now, by plugging Ts and Esw into the heat balance equation (first law), we can determine the ambient conditions, i.e. Ta, Va, RHa, MRT and Ici0, necessary for thermal comfort. 

Tubular reactors do not necessarily operate under isothermal conditions in industry, be it for reasons of chemical equilibrium or of selectivity, of profit optimization, or simply because it is not economically or technically feasible. It then becomes necessary to consider also the energy equation, that is, a heat balance on a differential volume element of the reactor. For reasons of analogy with the derivation of Eq. 9.1-1 assume that convection is the only mechanism of heat transfer. Moreover, this convection is considered to occur by plug flow and the temperature is completely uniform in a cross section. If heat is exchanged through the wall the entire temperature difference with the wall is located in a very thin film close to the wall. The energy equation then becomes, in the steady state ... 

It is impossible to create an isothermal process in plug flow reactors as it requires the variation of thermal transfei along the reactor length, according to the kinetics of heat emission. Therefore, plug flow reactors run under adiabatic conditions or at least imder nonisothermal mode conditions with external heat removal. The heat balance equation for steady state conditions for the micro volume of a reactor can be written in the form [4] ... 

In Chapter 2, the design of the so-called ideal reactors was discussed. The reactor ideahty was based on defined hydrodynamic behavior. We had assumedtwo flow patterns plug flow (piston type) where axial dispersion is excluded and completely mixed flow achieved in ideal stirred tank reactors. These flow patterns are often used for reactor design because the mass and heat balances are relatively simple to treat. But real equipment often deviates from that of the ideal flow pattern. In tubular reactors radial velocity and concentration profiles may develop in laminar flow. In turbulent flow, velocity fluctuations can lead to an axial dispersion. In catalytic packed bed reactors, irregular flow with the formation of channels may occur while stagnant fluid zones (dead zones) may develop in other parts of the reactor. Incompletely mixed zones and thus inhomogeneity can also be observed in CSTR, especially in the cases of viscous media. 

The preheater models are based on plug-flow behavior of both gas and the slurry phases, as the L/D ratios in preheater are usually large. Reliable estimates of the fluid properties such as viscosity and density, pressure drop across the preheater and heat transfer coefficient are needed for an optimum design of the preheater and, these have been recently reviewed by Shah (11). Parulekar et al. (17) have proposed a kinetic model for the preheater based on certain fast reactions taking place in the preheater whereas Nunez et al. (18) evaluated the hydrogen mass balance and a heat balance on the preheater. 

Numerical solutions to the coupled heat and mass balance equations have been obtained for both isothermal and adiabatic two- and three-transition systems but for more complex systems only equilibrium theory solutions have so far been obtained. In the application of equilibrium theory a considerable simplification becomes possible if axial dispersion is neglected and the plug flow assumption has therefore been widely adopted. Under plug flow conditions the differential mass and heat balance equations assume the hyperbolic form of the kinematic wave equations and solutions may be obtained in a straightforward manner by the method of characteristics. In a numerical simulation the inclusion of axial dispersion causes no real problem. Indeed, since axial dispersion tends to smooth the concentration profiles the numerical solution may become somewhat easier when the axial dispersion terra is included. Nevertheless, the great majority of numerical solutions obtained so far have assumed plug flow. 

As an example of the application of equilibrium theory to nonisothermal systems we consider here a plug flow system, with one adsorbable component, in which the concentration of adsorbable species and the temperature changes are both small enough to validate the constant velocity approximation. For such a system the differential mass and heat balance equations are... 

Subject to the assumptions of plug flow, thermal equilibrium between fluid and solid and constant linear fluid velocity, the differential mass and heat balance equations for an adiabatic column are given by Eqs. (9.32) and (9.33), while for an irreversible system, the equilibrium isotherm is given by... 

The plugged top tray will prevent the reflux from cascading down to the lower trays. The liquid reflux will just overflow into the condensers and circulate back to the reflux drum. The tip-off to this problem is that neither the reboiler duty nor the bottoms temperature is affected in the normal way by raising reflux. The tower s heat balance appears as if the reflux rate had never been increased. This is not much different from the signs of normal tower flooding, except that the AP on all but the top tray is not excessive. 

The number of parameters is definitely limited to a practical value of two or three because of the limited amount of information available in experimental RTD curves as well as by the accuracy of the numerical technique used to estimate these parameters. This means that the description of the fluid flow remains oversimplified. Actually, these models should not be considered as describing the actual fluid flow hydrodynamics but rather as evaluating the deviations from plug flow or complete mixing in order to write correctly the mass and heat balance equations. 

A heat balance on a volume element of the bed, assuming plug flow, leads to the following differential equation ... 

In this system of coupled differential equations, the mass balance corresponds to the reaction rate and the heat balance is a simplified version showing only the heat production by the reaction and the heat removal by the cooling system, both terms resulting in heat accumulation. This system presents the property of parametric sensitivity, meaning that a small change in one of the parameters may lead to dramatic changes in the solution of the system of equations, that is, in the behavior of the reactor. This is an old [10-12], but always real, problem [13, 14]. This behavior may be observed for batch reactors and for tubular reactors (plug flow reactors) and also for bed reactors [15,16]. Calorimetric methods make it possible to... 

To facilitate the discussion on heat control in microstructured tubular reactors, ideal plug flow behavior will be assumed. Under these conditions, reactant concentrations and temperatures are assumed to he constant over the cross section and a function of only the axial position. The heat balance for a volume element is formulated as follows ... 

Heat Transfer in Rotary Kilns. Heat transfer in rotary kilns occurs by conduction, convection, and radiation. In a highly simplified model, the treatment of radiation can be explained by applying a one-dimensional furnace approximation (19). The gas is assumed to be in plug flow the absorptivity, a, and emissivity, S, of the gas are assumed equal (a = e ) and the presence of water in the soHds is taken into account. Energy balances are performed on both the gas and soHd streams. Parallel or countercurrent kilns can be specified. 

A model of a reaction process is a set of data and equations that is believed to represent the performance of a specific vessel configuration (mixed, plug flow, laminar, dispersed, and so on). The equations include the stoichiometric relations, rate equations, heat and material balances, and auxihaiy relations such as those of mass transfer, pressure variation, contac ting efficiency, residence time distribution, and so on. The data describe physical and thermodynamic properties and, in the ultimate analysis, economic factors. 

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