Steady-state enthalpy balances for

The steady-state enthalpy balance for a slab with thickness 2 L is then given by ... 

Each term in the preceding equations has units of energy/time. Note the signs on each term indicating that heat is removed or added to the reactor. We preserve the minus sign on A Hji because we are more interested in exothermic reactions for which A Hr < 0. The student can recognize each term on the right side from the steady-state enthalpy balance we derived in the previous section from the thermodynamics of a steady-state flow system. 

The streams incident to the node are oriented according to Fig. 6-1. Thus oriented mass flowrate F > 0 for inlets, Fj < 0 for outlets, W is mechanical power supply, and Q is heat supply per unit time we consider the heat supply through the walls divided into several streams 2h If compared with the notation in (5.1.1) we have Fj = and W = -W. Electromagnetic phenomena are not considered, hence the steady-state enthalpy balance reads... 

In developing the enthalpy balance for a PFR, we consider only steady-state operation, so that the rate of accumulation vanishes. The rates of input and output of enthalpy by (1) flow, (2) heat transfer, and (3) reaction may be developed based on the differential control volume dV in Figure 15.3 ... 

In calculating temperature distribution in the dilute phase, solid motion must be accounted for. Solid motion in the dilute phase is shown in Sections II and VI. Laboratory-scale fluid beds exhibit a circulating flow of solid particles with ascending central core and descending peripheral region. The enthalpy balances for both ascending and descending zcHies for the steady state are... 

By combining (12.4.48)-(12.4.50) and (12.4.52)-(12.4.53) with the energy balance (12.4.47), we obtain a form of the steady-state energy balance that allows us to compute heat effects for chemical reactors. To obtain numerical values for quantities on the rhs in (12.4.48)-(12.4.53), we need an equation of state to obtain residual enthalpies, along with ideal-gas heat capacities and ideal-gas heats of reaction. 

We have chosen, in Section 5.3, a classical technology where available thermodynamic data allow us to compute the balance according to the general scheme (5.2.11). The formulation (5.2.11) of the enthalpy (approximate steady-state energy) balance is, however, not the only possible, and not always the most convenient one. In practice, one must frequently put up with a number of empirical data specific to the system and not found in thermodynamical tables. The data can even be more reliable than those found by a hypothetical thermodynamic path involving not precisely known items. For example the standard enthalpies of the components are, in fact, certain extrapolations computed backwards from reaction heats under realistic experimental conditions for instance [see (5.3.9)] sulphur is certainly not burnt at, say, 273 K. Because the theoretical items are subtracted in the input - otput node balance, a relatively small error in the data may cause a significant relative error in the result. 

Mathematical model of three-way catalytic converter (TWC) has been developed. It includes mass balances in the bulk gas, mass transfer to the porous catalyst, diffusion in the porous structure and simultaneous reactions described by a complex microkinetic scheme of 31 reaction steps for 8 gas components (CO, O2, C2H4, C2H2, NO, NO2, N2O and CO2) and a number of surface reaction intermediates. Enthalpy balances for the gas and solid phase are also included. The method of lines has been used for the transformation of a set of partial differential equations (PDEs) to a large and stiff system of ordinary differential equations (ODEs . Multiple steady and oscillatory states (simple and doubly-periodic) and complex spatiotemporal patterns have been found for a certain range of operation parameters. The methodology of studies of such systems with complex dynamic patterns is briefly introduced and the undesired behaviour of the used integrator is discussed. 

In unsteady states the situation is less satisfactory, since stoichiometric constraints need no longer be satisfied by the flux vectors. Consequently differential equations representing material balances can be constructed only for binary mixtures, where the flux relations can be solved explicitly for the flux vectors. This severely limits the scope of work on the dynamical equations and their principal field of applicacion--Che theory of stability of steady states. The formulation of unsteady material and enthalpy balances is discussed in Chapter 12, which also includes a brief digression on stability problems. 

The time derivative is zero at steady state, but it is included so that the method of false transients can be used. The computational procedure in Section 4.3.2 applies directly when the energy balance is given by Equation (5.28). The same basic procedure can be used for Equation (5.25). The enthalpy rather than the temperature is marched ahead as the dependent variable, and then Tout is calculated from Hout after each time step. 

A dynamic model should be consistent with the steady-state model. Thus, Eqs (1) and (4) should be extended to dynamic form. For the better convergence and computational efficiency, some assumption can be introduced the total amounts of mass and enthalpy at each plate are maintained constant. Then, the internal flow can be determined by total mass balance and total energy balance and the number of differential equations is reduced. Therefore, the dynamic model can be established by replacing component material balance in Eq. (1) with the following equation. 

Here, h = u + P/p is the enthalpy per unit mass of fluid. Note that the inlet and exit streams include enthalpy (i.e., both internal energy, u, and flow work, P/p), whereas the system energy includes only the internal energy but no P/p flow work (for obvious reasons). If there are only one inlet stream and one exit stream (m, =m0 = m) and the system is at steady state, the energy balance becomes... 

The enthalpy change, AH, can be calculated for a steady-state process, using H°f, which is the enthalpy of formation of the various output and input components. Under the assumption that the inputs and outputs are at ambient conditions, the enthalpy of the components corresponds to the standard enthalpy of formation of each component. The kinetic and potential energy terms are neglected from the energy balance. It is also assumed that water enters the process as a liquid and hydrocarbon products leave the process as a liquid. All other components are in the gas phase. 

The input and output terms of equation 1.5-1 may each have more than one contribution. The input of a species may be by convective (bulk) flow, by diffusion of some kind across the entry point(s), and by formation by chemical reaction(s) within the control volume. The output of a species may include consumption by reaction(s) within the control volume. There are also corresponding terms in the energy balance (e.g., generation or consumption of enthalpy by reaction), and in addition there is heat transfer (2), which does not involve material flow. The accumulation term on the right side of equation 1.5-1 is the net result of the inputs and outputs for steady-state operation, it is zero, and for unsteady-state operation, it is nonzero. 

The steady state material and energy balances for the evaporator are listed in Table VI and VII, and the notation in Table VIII. Table IX lists the enthalpy relationships for the various streams as well as the boiling point versus pressure and concentration relationships in functional form for NaOH solutions and pure water. The list of unknown variables and the numbers assigned to each is given in Table X. At this stage in the analysis there are 25 equations and 27 unknown variables. Another pair of equations comes from the problem statement in which the following is given... 

For the energy balance we make a similar differential balance on the enthalpy flow between z and z + dz for a tube of length L and diameter D as sketched in Figure 5-2. In steady state this ener balance is... 

This forms the basis of constructing an enthalpy budget in which the total enthalpy flux is compared with the scalar heat flux, 7q(W m-3), obtained from dividing heat flow by size (volume or mass) of the living matter. If account is made of all the reactions and side reactions in metabolism, the ratio of heat flux to enthalpy flux, the so-called energy recovery ( Yq/H = Jq/Jh) will equal 1. If it is more than 1, then the chemical analysis has failed fully to account for heat flux and if it is less than 1, then there are undetected endothermic reactions. Account for all reactions may seem a formidable task, but it should be borne in mind that anabolic processes dissipate insignificant amounts of heat compared with those of catabolism and that ATP production and utilization are balanced in cells at steady-state. Catabolism is generally limited to a relatively few well-known pathways with established overall molar enthalpies. So, as will be seen later, the task is by no means mission impossible. ... 

Equation 7.4-15 states that the net rate at which energy is transferred to a system as heat and/or shaft work (Q- W,) equals the difference between the rates at which the quantity (enthalpy + kinetic energy potential energy) is transported into and out of the system (A// + AE + A p). We will use this equation as the starting point for most energy balance calculations on open systems at steady state. 

The key step in the derivation by Reuter et al. of their lattice model is the use of detailed balance to determine the sticking coefficients for each species on each type of site.31 The total adsorption rate at a particular site can be expressed as Tad = SI(p, T), where S is the local sticking coefficient and I(p,T) is the impingement rate of the species of interest from a gas phase with partial pressure p and temperature T. At steady state, the total adsorption and desorption rates must satisfy the detailed balance condition TdesjTad = exp[(Fb—/j,(T, p))/kT, where Fb is the free energy of the adsorbed species and fi(T, p) is the chemical potential of the gas phase species. The adsorption free energy is well approximated by the adsorption enthalpy, which is simply the adsorption energy calculated by a DFT calculation. This approach provides a direct link between the adsorption and desorption rates and the pressure and temperature of the bulk gas phase. 

Nonisothermal stirred tanks are governed by an enthalpy balance that contains the heat of reaction as a significant term. If the heat of reaction is unimportant so that a desired Tout can be imposed on the system regardless of the extent of reaction, then the reactor dynamics can be analyzed by the methods of the previous section. This section focuses on situations where Equation 14.3 must be considered as part of the design. Even for these situations, it is usually possible to control a steady-state CSTR at a desired temperature. If temperature control can be achieved rapidly, then isothermal design techniques again become applicable. Rapid means on a time scale that is fast compared to reaction times and composition changes. 

Finally, we often assume that the diffusivity, thermal conductivity and partial molar enthalpies are independent of temperature and composition to produce the following coupled mass and energy balances for the steady-state problem... 

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