Thermal energy balance differential reactor

This problem requires an analysis of coupled thermal energy and mass transport in a differential tubular reactor. In other words, the mass and energy balances should be expressed as coupled ordinary differential equations (ODEs). Since 3 mol of reactants produces 1 mol of product, the total number of moles is not conserved. Hence, this problem corresponds to a variable-volume gas-phase flow reactor and it is important to use reactor volume as the independent variable. Don t introduce average residence time because the gas-phase volumetric flow rate is not constant. If heat transfer across the wall of the reactor is neglected in the thermal energy balance for adiabatic operation, it... 

Now, the coupled mass and thermal energy balances can be combined and integrated analytically to obtain a linear relation between temperature and conversion under nonequilibrium (i.e., kinetic) conditions because it is not necessary to consider the temperature and conversion dependence of (Cp mixture)- At high-mass-transfer Peclet numbers, axial diffusion can be neglected relative to convective mass transfer, and the mass balance is expressed in terms of molar flow rate F, and differential volume dV for a gas-phase tubular reactor with one chemical reaction ... 

The final form of the differential thermal energy balance for a generic plug-flow reactor that operates at high-mass and high-heat-transfer Peclet numbers allows one to predict temperature as a function of reactor volume ... 

Coupled mass and thermal energy balances are required to analyze the nonisother-mal response of a well-mixed continuous-stirred tank reactor. These balances can be obtained by employing a macroscopic control volume that includes the entire contents of the CSTR, or by integrating plug-flow balances for a differential reactor under the assumption that temperature and concentrations are not a function of spatial coordinates in the macroscopic CSTR. The macroscopic approach is used for the mass balance, and the differential approach is employed for the thermal energy balance. At high-mass-transfer Peclet numbers, the steady-state macroscopic mass balance on reactant A with axial convection and one chemical reaction, and units of moles per time, is... 

Write differential thermal energy balances on both the solid phase and the gas phase in the porous reactor. 

A complete description of the reactor bed involves the six differential equations that describe the catalyst, gas, and thermal well temperatures, CO and C02 concentrations, and gas velocity. These are the continuity equation, three energy balances, and two component mass balances. The following equations are written in dimensional quantities and are general for packed bed analyses. Systems without a thermal well can be treated simply by letting hts, hlg, and R0 equal zero and by eliminating the thermal well energy equation. Adiabatic conditions are simulated by setting hm and hvg equal to zero. 

Another potential solution technique appropriate for the packed bed reactor model is the method of characteristics. This procedure is suitable for hyperbolic partial differential equations of the form obtained from the energy balance for the gas and catalyst and the mass balances if axial dispersion is neglected and if the radial dimension is first discretized by a technique such as orthogonal collocation. The thermal well energy balance would still require a numerical technique that is not limited to hyperbolic systems since axial conduction in the well is expected to be significant. 

Equation 5.2.55 is a sinoplified dimensionless differential energy balance equation of steady-flow reactors, where each term is divided hy [(Ftot)oCpo ol, the reference thermal energy rate. The first term in the bracket of Eq. 5.2.55 represents the dimensionless heat-transfer rate and its relationship to the dimensionless driving force (6ir — 6), where... 

In this system a reactant enters a chemical reactor at = 0 with a superficial velocity of v (cm/sec). The reactor is packed with catalyst particles. The heat of reaction is known as AH (cal/g mol). The reaction rate is zero-order in the reactant concentrtion and therefore is a constant down the reactor R (g mol/sec cm ). Because of the effect of the catalyst packing, both convective and dispersion thermal effects are present in this reactor. We want to compute the temperature profile down the reactor. We write a steady-state energy balance for the differential reactor volume, AAz, as... 

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