Molar flow rates mole balances

Here, F, Zf and h are, respectively, the molar flow rate, mole fraction of component of i and total enthalpy, all in cell k their subscripts, ret and perm, refer to retentate and permeate streams. Equations (10.4) and (10.5) are mass balances and mass-transfer equations for each of the components present in the membrane feed. The cross-flow model [Equations (10.3)-(10.7)] was implemented in ACM v8.4 and validated against the experimental data in Pan (1986) and the predicted values of Davis (2002). The Joule-Thompson effect was validated by simulating adiabatic throttling of permeate gas through a valve in Aspen Hysys. Both these validations are described in detail in Appendix lOA. 

To develop E(B) for two CSTRs in series, we use a slightly different, but equivalent, method from that used for a single CSTR in Section 13.4.1.1. Thus, consider a small amount (moles) of tracer M, nMo = F,dt, where Ft is the total steady-state molar flow rate, added to the first vessel at time 0. The initial concentration of M is cMo = nMo/(V/2). We develop a material balance for M around each tank to determine the time-dependent outlet concentration of M from the second vessel, cM2(l). 

Eqs. 1 to 3 relate the rate of production Rj of the balanced reaction component y to the molar amounts or their derivatives with respect to the time variable (reaction time or space time, see above). From the algebraic eq. 2 for the CSTR reactor the rate of production, Rj, may be calculated very simply by introducing the molar flow rates at the inlet and outlet of the reactor these quantities are easily derived from the known flow rate and the analytically determined composition of the reaction mixture. With a plug-flow or with a batch reactor we either have to limit the changes of conversion X or mole amount n7 to very low values so that the derivatives or dAy/d( //y,0) or dn7/d/ could be approximated by differences AXj/ (Q/Fj,0) or An7/A, (differential mode of operation), or to measure experimentally the dependence of Xj or nj on the space or reaction time in a broader region this dependence is then differentiated graphically or numerically. 

After substituting Equations 3.1.2 and 3.1.3 into Equation 3.1.1, the oxygen mole balance reduces to Equation 3.1.4 in Table 3.1.1. Because Equation 3.1.4 is an unsteady-state, first-order differential equation, we need an initial condition to calculate the constant of integration. Initially, the tank contains air, which has an oxygen concentration of approximately 21 % by volume. We could also write the mole balance for nitrogen, but in this case it is more convenient to write the total mole balance, which results in Equation 3.1.5. Once we write Equations 3.1.4 to 3.1.6, the nitrogen mole balance is not an independent equation. Equation 3.1.7 states that the molar flow rate is equal to the product of the molar density and the volmnetric flow rate. 

The mole balance can now be conpleted for one CSTR. The inlet molar flow rate for propylene oxide is calculated above. The inlet molar flow rate of methanol,... 

The design engineer (a) converts the volumetric flow rate of the feed stream to a molar flow rate using the ideal gas equation of state, an approximate relationship between the pressure, temperature, volumetric flow rate, and molar flow rate of a gas (Chapter 5) (b) specifies a condenser temperature of IS C (c) calculates the mole fraction of MEK in the vapor product using Raoult s law—an approximate relationship between the compositions of liquid and vapor phases in equilibrium with each other at a specified temperature and pressure (Chapter 6) and (d) calculates the molar flow rates of the vapor and liquid products from nitrogen and MEK balances (input = output). The results follow. 

The molar flow rate of each species Fj- is obtained from a mole balance on each species. 

Example 3-9 PEE Mole Balances in Terms of Molar Flow Rates Reconsider the elementary gas reaction discussed in Example 3-8. 

The reatsion is to be carried out isothetmally (T = To) and isobaricaliy = P ) in a PFR. Express the rate law and mole balances in terms of the molar flow rates. ... 

There are a number of instances when it is much more convenient to work in terms of the number of moles (iV, N-g) or molar flow rates (Fj, Fg, etc.) rather than conversion. Membrane reactors and multiple reactions taking place in the gas phase are two such cases where molar flow rates rather than conversion are preferred. In Section 3.4 we de.scribed how we can express concentrations in terms of the molar flow rates of the reacting species rather than conversion, We will develop our algorithm using concentrations (liquids) and molar flow rates (gas) as our dependent variables. The main difference is that when conversion is used as our variable to relate one species concentration to that of another species concentration, we needed to write a mole balance on only one species, our basis of calculation. When molar flow rates and concentrations are used as our variables, we must write a mole balance on each species and then relate the mole balances to one another through the relative rates of reaction for... 

Consider again the reaction in Example 6-4. Write the mole balance on a PFR in terms of molar flow rates for each species. 

In combining the mole balance, rate laws, and stoichiometry, we will use our results ftotn Example 6-4. The total molar flow rate of all the gases is... 

We now rewrite mole balances on each species in the total molar flow rate. (1) Mole badance on NO ... 

The unsteady-state energy balance for an open system that has n species, each entering and leaving the system at its respective molar flow rates Fi (moles of i per time) and with its respective energy (joules per mole of i), is... 

This equation is coupled with the mole balances on each species [Equation (8-58)]. Next we express r, as a function of either the concentrations for liquid systems or molar flow rates for gas systems as described in Section 3.4,... 

Now that we have a relationship [Equation (2-10)] between the molar flow rate and conversion, it is possible to express the design equations (i,e., mole balances) in terms of conversion for the flow reactors exantinMi in Chapter 1,... 

Gas I%ase. For gas-phase reactions, the mole balances are given identically in Table 4-6. Consequently, the concentrations in the rate laws need to be expressed in terms of the molar flow rates for example,... 

In complex reaction systems consisting of combinations of parallel and series reactions the availability of software packages (ODE solvers) makes it much easier to solve problems using moles Nj or molar flow rates Fj rather than conversion. For liquid systems, concentration may be the preferred variable used in, the mole balance equations. The resulting coupled differential equations can be easily solved using an ODE solver. In fact, tltis section has been developed to take advantage of the vast number of computational techniques now available on mainframe (e.g., Simulsolv) and personal computers (POLYMATH). 

Rather than combining the concentrations, rate Jaws, and mole balances to write everything in terras of the molar flow rate as we did in the past, it is more convenient here to write our computer solution (either POLYMATH or our own program) using equations fqr F/, and so on. Consequently, we shall write Equations (E6-8.9) through (E6-8.12) and (E6-8.19) through (E6-S.25) as individual lines and let the computer combine them to obtain a solution. 

B. Molar flow rates as the reaction variable ]. Mole balances ... 

For Problem 3-7(b) write the combined PFR mole balance on each species and rate law solely in terms of the molar flow rates and rate law parameters. 

We now insert rate laws written in terms of molar flow rates [e.g., Equation (3-45)] into the mole balances (Table 6-1). After performing this operation for each species we arrive at a coupled set of first-order ordinary differential equations to be solved for the molar flow rates as a function of reactor volume (i.e., distance along the length of the reactor). In liquid-phase reactions, incorporating and solving for total molar flow rate is not necessary at each step along the solution pathway because there is no volume change with reaction. 

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