Rate equations pressure

Comparing this with equation (A.2.2), it is seen to predict exactly the same dependence of che effusion rate on pressure and temperature. Furthermore, Che ratio of specific heats y depends relatively weakly on che nature of the gas, through its molecularity, so che prediction chat dV/dt 1/M, which follows from equation (A-2.2) and agrees with Graham s results, is not markedly inconsistent with equation (A.2.3) either. 

Equation 39 can often be simplified by adopting the concept of a mass transfer unit. As explained in the film theory discussion eadier, the purpose of selecting equation 27 as a rate equation is that is independent of concentration. This is also tme for the Gj /k aP term in equation 39. In many practical instances, this expression is fairly independent of both pressure and Gj as increases through the tower, increases also, nearly compensating for the variations in Gj. Thus this term is often effectively constant and can be removed from the integral ... 

The scale-up of conventional cake filtration uses the basic filtration equation (eq. 4). Solutions of this equation exist for any kind of operation, eg, constant pressure, constant rate, variable pressure—variable rate operations (2). The problems encountered with scale-up in cake filtration are in estabHshing the effective values of the medium resistance and the specific cake resistance. 

The hydrauhc diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters Dg defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, Dg (l2SQ[LL/ KAPy. Equivalent diameters are not the same as hydraulie diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/(tiDe/4). Equivalent diameter Dg is not to be used in the friction factor and Reynolds number ... 

Any property of a reacting system that changes regularly as the reaction proceeds can be formulated as a rate equation which should be convertible to the fundamental form in terms of concentration, Eq. (7-4). Examples are the rates of change of electrical conductivity, of pH, or of optical rotation. The most common other variables are partial pressure p and mole fraction Ni. The relations between these units... 

TABLE 7-9 Integration of Rate Equations of a PFR at Constant Pressure... 

A reaction A 3B takes place in a tubular flow reactor at constant temperature and an inlet pressure of 5 atm. The rate equation is... 

Variable-Pressure, Variable-Rate Filtration The pattern of this categoiy comphcates the use of the basic rate equation. The method of Tiller and Crump (loc. cit.) can be used to integrate the equation when the characteristic curve of the feed pump is available. 

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. 

The partial pressures in the rate equations are those in the vicinity of the catalyst surface. In the presence of diffusional resistance, in the steady state the rate of diffusion through the stagnant film equals the rate of chemical reaction. For the reaction A -1- B C -1-. . . , with rate of diffusion of A limited. 

Data on Illinois No. 6 and Kentucky No. 9 coals were used by Wen and Han (Prepi Pap.—Am. Chem. Soc., Div. Fuel Chem. 20(1) 216-233, 1975) to obtain a rate equation for coal dissolution under hydrogen pressure. These data included a temperature range of 648 to 773 K (705 to 930°F) and pressures up to 13.8 MPa (2000 psia). An empirical rate expression was proposed as... 

When a unimolecular reaction occurs with an initial product partial pressure of the reactant A, to yield an amount of die product, jc, the first-order reaction rate equation reads... 

We now turn attention towards the ease of eonstant-rate filtration. When sludge is fed to a filter by means of a positive displaeement pump, the rate of filtration is nearly constant, i.e., dV/dx = constant. During constant-rate filtration, pressure increases with cake thickness. As sueh, the principal filtration variables are pressure ind filtrate volume, or pressure and filtration time. Integrating the filtration equation for a constant-rate process, we find that the derivative dV/dx ean simply be replaeed by V/x, and we obtain ... 

As follows for the filtration of incompressible sediment (at a constant rate), the pressure increases in a direct proportion to time. However, the above equation... 

In gaseous reaetions, the eomposition term in the rate equation is often expressed as the partial pressure of the reaeting speeies. These pressures are then transformed to eoneentration. 

Caution must be exereised when the rate equations with gas phase reaetions are expressed in terms of the partial pressure as eompared to eoneentration. This is beeause eaeh rate gives different aetivation energy for the same data and for the same reaetion. Levenspiel [4] suggests that tlie differenee ean be ignored for reaetions with reasonably high aetivation energies as the amount is only a few kJ. 

The average rate eonstant k = 3.85 and die final rate equation is Rate = 3.85 PB,H5PMe,co- The eomputed rate eonstant from PROG3 is 4.68 and die ealeulated value from Table 3-12 is 3.85. The pereentage deviadon between diese values is 17.8%. Eigure 3-22 shows plots of partial pressures of BjHg and MejCO versus die initial rate. 

The data on whieh the rate equations are based were obtained at a total pressure of 1 atm and temperatures of 1,265°F and 1,400°F in a... 

To get the equilibrium sticking coefficient we assume that at an ambient pressure Pq the adsorbate is in equilibrium at a temperature T with partial coverages Hq, m, and Iq. We then increase the pressure slightly to p = Pq- - AP and linearize the rate equations in the increase in the precursor coverages Am = (m) —m and Al = (/) — Iq. If adsorption into and desorption from the precursors is much faster than transitions from the precursors into the adsorbed state, we can ignore terms proportional to An = n) -6 on the right-hand side of Eqs. (70-72) and also assume that the precursors will be in a steady state. It has been shown that the sticking... 

The rate of reaction nearly always depends upon reactant concentrations and (for reversible reactions) product concentrations. The functional relationship between rate of reaction and system concentrations (usually at constant temperature, pressure, and other environmental conditions) is called the rate equation. 

First, the kinetics of the reactions of 0-, m-, and p-xylene as well as of toluene were studied separately (96) at various combinations of initial partial pressures of the hydrocarbon and hydrogen. From a broader set of 23 rate equations, using statistical methods, we selected the best equations for the initial rate and determined the values of their constants. With xylenes and toluenes, these were Eqs. (17a) and (17b). 

In an effort to account for this effect, Spalding and Johnson have considered the effect of radiation losses from the solid surface. Johnson has also included a provision for a volume rate of energy loss, but he does not state the basis for the term. In the case of surface pyrolysis, the resulting equations predict the relations of burning-rate versus pressure as shown in Fig. 16. 

The route from kinetic data to reaction mechanism entails several steps. The first step is to convert the concentration-time measurements to a differential rate equation that gives the rate as a function of one or more concentrations. Chapters 2 through 4 have dealt with this aspect of the problem. Once the concentration dependences are defined, one interprets the rate law to reveal the family of reactions that constitute the reaction scheme. This is the subject of this chapter. Finally, one seeks a chemical interpretation of the steps in the scheme, to understand each contributing step in as much detail as possible. The effects of the solvent and other constituents (Chapter 9) the effects of substituents, isotopic substitution, and others (Chapter 10) and the effects of pressure and temperature (Chapter 7) all aid in the resolution. 

Equilibrium Compositions for Single Reactions. We turn now to the problem of calculating the equilibrium composition for a single, homogeneous reaction. The most direct way of estimating equilibrium compositions is by simulating the reaction. Set the desired initial conditions and simulate an isothermal, constant-pressure, batch reaction. If the simulation is accurate, a real reaction could follow the same trajectory of composition versus time to approach equilibrium, but an accurate simulation is unnecessary. The solution can use the method of false transients. The rate equation must have a functional form consistent with the functional form of K,i,ermo> e.g., Equation (7.38). The time scale is unimportant and even the functional forms for the forward and reverse reactions have some latitude, as will be illustrated in the following example. 

Measurements Using Liquid-Phase Reactions. Liquid-phase reactions, and the oxidation of sodium sulfite to sodium sulfate in particular, are sometimes used to determine kiAi. As for the transient method, the system is batch with respect to the liquid phase. Pure oxygen is sparged into the vessel. A pseudo-steady-state results. There is no gas outlet, and the inlet flow rate is adjusted so that the vessel pressure remains constant. Under these circumstances, the inlet flow rate equals the mass transfer rate. Equations (11.5) and (11.12) are combined to give a particularly simple result ... 

In vapour phase reactions, partial pressure units are often used in place of concentration in the rate equation, for example... 

The simulation emphasises that when reactions start to run away they do so extremely fast. In this example the rate of pressure generation is compounded by the double exponential terms in the Arrhenius and vapour pressure equations. 

Kinetic orders in CO oxidation on M/A1203 can be explained by the classical Langmuir-Hinshelwood expression for the rate equation, as a function of the rate constant k, the adsorption constants K and the partial pressures P ... 

To have a better appreciation of the utility of these representations let us first consider the laws that govern flow rates and pressure drops in a pipeline network. These are the counterparts to KirchofTs laws for electrical circuits, namely, (i) the algebraic sum of flows at each vertex must be zero (ii) the algebraic sum of pressure drops around any cyclic path must be zero. For a connected network with N vertices and P edges there will be (N — 1) independent equations corresponding to the first law (KirchofTs current... 

The second factor is the form of Eq. (36). If it is not explicit in flow rate or pressure, then either formulation C or formulations B and D are infeasible. In this connection we note that since Eqs. (18) and (19) are explicit in both variables, the simulation of water distribution networks using these equations presents no difficulties in this respect. 

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