Energy balance concentration profile

For an isothermal system the simultaneous solution of equations 30 and 31, subject to the boundary conditions imposed on the column, provides the expressions for the concentration profiles in both phases. If the system is nonisotherm a1, an energy balance is also required and since, in... 

For isothermal systems this equation, together with an appropriate expression for rv, is sufficient to predict the concentration profiles through the reactor. For nonisothermal systems, this equation is coupled to an energy balance equation (e.g., the steady-state form of equation 12.7.16) by the dependence of the reaction rate on temperature. 

The energy balance equation was solved for the most exothermic case (Run 5), (Tables I and II) together with the mass balance equation (1). Thus, the r. were deduced from a well-fitted but with respect to the kinetic expression still arbitrary description of the experimental concentration profile along the reactor. Since the AHj are known, it remains to choose hw and Xeff so that the experimentally measured temperature gradient is correctly described. For this,... 

Numerous reactions are performed by feeding the reactants continuously to cylindrical tubes, either empty or packed with catalyst, with a length which is 10 to 1000 times larger than the diameter. The mixture of unconverted reactants and reaction products is continuously withdrawn at the reactor exit. Hence, constant concentration profiles of reactants and products, as well as a temperature profile are established between the inlet and the outlet of the tubular reactor, see Fig. 7.1. This requires, in contrast to the batch reactor, the application of the law of conservation of mass over an infinitesimal volume element, dV, of the reactor. In contrast to a batch reactor the existence of a temperature profile does not allow us to consider the mass balances for the reacting components and the energy balance separately. Such a separation can only be performed for isothermal tubular reactors. 

Solid concentration profiles are produced from a balance of gravitational with buoyancy and kinetic energy transfer forces. For a single particle in a stagnant liquid, the settling velocity,... 

The development of mathematical models to describe the thermochemical process occurring in a fluidized bed involves setting up the material and energy balance equations. The total process is represented in terms of a set of independent equations which are solved simultaneously to obtain such quantities as combustion efficiency, sulfur retention, oxygen utilization, oxygen and sulfur dioxide concentration profiles in the bed, etc. 

The dimensionless material and energy balances, with associated boundary conditions, must be solved simultaneously to get the concentration and temperature profiles through the stagnant film and into the catalyst particle. Those relationships are given below ... 

Solution of Equation (10.2.1) provides the pressure, temperature, and concentration profiles along the axial dimension of the reactor. The solution of Equation (10.2.1) requires the use of numerical techniques. If the linear velocity is not a function of z [as illustrated in Equation (10.2.1)], then the momentum balance can be solved independently of the mass and energy balances. If such is not the case (e.g., large mole change with reaction), then all three balances must be solved simultaneously. 

Real flows are not frictionless. As a result of friction, a flow field is formed with local and time related changes in the velocity. The temperature and concentration fields will not only be determined by conduction and diffusion but also by the flow itself. The form and profiles of flow, temperature and concentration fields are found by solving the mass, momentum and energy balances, which are the subject of the next section. 

Pg-lle Derive the energy balance for a packed bed membrane reactor. Apply th ance to the reaction in Problem PS-Bg for the case when it is reversibk Kr = 0.01 moi/dm at 300 K. Species C diffuses out of the membrane. (b) Plot the concentration profiles or different values of k,. when the re is carried out adiabaticaliy. 

The mass and energy balance equations developed in Secs. 14.1 and 14.2 are the basic equations used in reactor design and analysis. In many cases, however, our needs are much more modest than in engineering design. In particular, we may not be interested in such details as the type of reactor used and the concentration and temperature profiles or time history in the reactor, but merely in the species mass and total energy balances for the reactor. In such situations one can use the general black-box equations of Table 8.4-1 ... 

Temperature Effects During the condensation/evaporation of a particle latent heat is released/absorbed at the particle surface. This heat can be released either toward the particle or toward the exterior gas phase. As mass transfer continues, the particle surface temperature changes until the rate of heat transfer balances the rate of heat generation/ consumption. The formation of the external temperature and vapor concentration profiles must be related by a steady-state energy balance to determine the steady-state surface temperature at all times during the particle growth. 

In a fixed-bed reactor the catalyst pellets are held in place and do not move with respect to a fixed reference frame. Material and energy balances are required for both the fluid, which occupies the interstitial region between catalyst particles, and the catalyst particles, in which the reactions occur. For heterogeneously catalyzed reactions, the effects of intraparticle transport on the rate of reaction must be considered. Catalytic systems operate somewhere between two extremes kinetic control, in which mass and energy transfer are very rapid and intra-partide transport control, in which the reaction is very rapid. Separate material and energy balances are needed to describe the concentration and temperature profile inside the catalyst pellet. The concentrations... 

A set of equilibrium data, determined experimentally, in combination with mass and energy balances enables the calculation concentration profiles along a separation device operated in the countercurrent mode for multicomponent mixtures. ASPEN-h is a commercially available process simulation program. 

The way these equations have been set up, the values of the coefficients Dp depend upon the values of fp and t, that is, on what the temperature and concentration profiles are. In addition, remember that equation (xviii) is for the material balance only. Thus we have two tasks left at this point 1) what of the energy balance and, 2) what of the coefficients D ... 

If T terface and Tbuik replace Ca, equilibrium and Ca, bulks respectively, in the definition of the dimensionless profile P, and the thermal diffusiv-ity replaces a. mix. then the preceding equation represents the thermal energy balance from which temperature profiles can be obtained. The tangential velocity component within the mass transfer boundary layer is calculated from the potential flow solution for vg if the interface is characterized by zero shear and the Reynolds number is in the laminar flow regime. Since the concentration and thermal boundary layers are thin for large values of the Schmidt and Prandtl... 

In conclusion, one solves two coupled first-order ODEs for the molar density profile of reactant A under isothermal conditions, without considering the thermal energy balance. Then, a volumetric average of the rate of conversion of reactant A to products due to multiple chemical reactions is obtained by focusing on the reactant concentration gradient at the external surface of the catalyst ... 

For endothermic reactions, fi<0, and the p curves at each value clearly indicate that the internal effectiveness factor ri will always be less than unity, since both the temperature and the reactant concentration decline toward the center of the particle. In this case, the impact of heat transfer decreases, but the effect of mass transfer becomes almost negligible. An approximate solution can be obtained by ignoring the concentration profile and solving the differential energy balance (Eq. 2.69) by assuming that the reactant concentration is equal to Cas within the particle ... 

In the model considered below, the role of both grain boundary and bulk diffusion in the transformation front and close to it, respectively, is analyzed within the problem of unambiguous determination of the discontinuous precipitation parameters in the binary Pb-Sn system at room temperature [9]. In order to complete this, we use the principle of maximum rate of free energy release and balance of entropy fluxes for the description of discontinuous precipitation kinetics for binary polycrystaUine alloys and independent determination of three basic parameters interlamellar distance, rate of phase transformation front, and concentration profile close to the transformation front. While solving the problem, we also find the optimal concentration distribution of components both along the precipitation lamella behind the transformation front and close to it, as well as the degree of the components separation. 

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