Molar flow rates balance equation

A material balance analysis taking into account inputs and outputs by flow and reaction, and accumulation, as appropriate. This results in a proper number of continuity equations expressing, fa- example, molar flow rates of species in terms of process parameters (volumetric flow rate, rate constants, volume, initial concentrations, etc.). These are differential equations or algebraic equations. 

We must develop a differential mass balance of composition versus position and then solve the resulting differential equation for Ca(z) and Ca(L) (Figure 3-2). We consider a tube of length L with position z going fiom 0 to L. The molar flow rate of species j is Fjo at the inlet (z = 0), Fj (z) at position z, and Fj (L) at the exit L. 

The solution to the reactor model differential equations (7.166) and (7.180) to (7.182) simulates the molar flow rates and the pressure drop and energy balance of the reactor. The solution of the catalyst pellet boundary value differential equations (7.172) and (7.173) provides the effectiveness factors r]j for each reaction labeled j = 1,..., 6 for use inside the differential equations (7.180) to (7.182). 

Evaluate molar flow rates and integrate the mass balance. Once the known data on molar flow rates are substituted into the mass balance, it can be solved for reactor volume. From step 1, the equation to be solved is... 

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. 

For any component i the Lewis-Sorel material balance equations (Section 11.5) and equilibrium relationship can be written in terms of the individual component molar flow rates in place of the component composition ... 

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. 

Density is a frequently needed physical property of a process fluid. For example, engineers often know volumetric flow rates (F) of process streams from flowmeter readings but need to know mass flow rates m) or molar flow rates (h) for material balance calculations. The factor needed to calculate m or h from V is the density of the stream. This chapter illustrates the uses of both tabulated data and estimation formulas for calculating densities. Section 5.1 concerns solids and liquids Section 5.2 covers ideal gases, gases for which the ideal gas equation of state (PV - nRT) is a good approximation and Section 5.3 extends the discussion to nonideal gases. 

In this equation, is the extent of reaction (determined from Equation 9.1-3) n, and H,- are respectively the molar flow rate and specific enthalpy of a process species in an inlet or outlet stream and the summations are taken over all species in all of their inlet and outlet stales. Once calculated, AH is substituted in the open-system energy balance, which is solved for Q or whichever other variable is unknown. 

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,... 

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. 

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. 

So far we have examined single units without a reaction occurring in the unit. How is the count for Nd affected by the presence of a reaction in the unit The way Nv is calculated does not change. As to Nr, all restrictions and constraints are deducted from N that represent independent restrictions on the unit. Thus the number of material balances is not necessarily equal to the number of species (H2O, O2, CO2, etc.) but instead is the number of independent material balances that exist determined in the same way as we did in Secs. 2.2 to 2.4, usually (but not always) equal to the number of elemental balances (H, O, C, etc.). Fixed ratios of materials such as the O2/N2 ratio in air or the CO/CO2 ratio in a product gas would be a restriction, as would be a specified conversion fraction or a known molar flow rate of a material. If some degrees of freedom exist still to be specified, improper specification of a variable may disrupt the independence of equations and/or specifications previously enumerated in the unit of Nr, so be carefiil. 

The material balance from (1.228) is once again the starting point for investigating the concentration profile in a packed column being used to rectify a gaseous mixture. In this case the mole fractions will be used instead of the mole ratios. The amount transferred from the gas to the liquid phase is —NGdy, wherein NG is now the molar flow rate of the gas mixture rather than that of the carrier gas as in (1.228). Equation (1.228) is therefore replaced by... 

Consider a plug flow reactor - a cylindrical tube with reactants flowing in one end and reacting as they flow to the outlet. A mole balance is made here for the case when the operation is steady. Let F be the molar flow rate of a chemical, which changes dF in a small part of the tube take the volume of that small part to be d V. The rate of reaction is expressed as the moles produced per unit of time per unit of volume. If a chemical specie reacts, then the r for that specie is negative. Thus, the overall equation for a smaU section of the tube is... 

Here Fj, Vj, and Lj are the feed, vapor product, and liquid product molar flow rates at stage j, and Zj, Yj, and X- are the feed, vapor, and liquid mole fractions of component i at stage j. Summation of Equations 3.7 and 3.8 gives the total material balance,... 

Next, Equations 17.32 through 17.34 are used to update the vapor and liquid flow rates, and Equations 17.35 and 17.36 are used to calculate the condenser and reboiler duties. The reboiler molar holdup is calculated by over all material balance ... 

Since the pressure of the reactor is typically known, along with the molar flow rates of the components of the inlet stream. Equations (4.572) and (4.573) provide two equations with two unknowns, which can be solved for 4 and hi other words, the energy balance and the chemical equilibrium relations generate a complete set of equations to determine the adiabatic reaction temperature. Whether is greater than or less than 7 depends upon whether the reaction is exothermic or endothermic. 

We see tliiu conversion is not used in this sum. The molar flow rates, Fj, are found by solving the mole balance etjuations. Equation (.1-42) will be u.seci for measures other than ctmversion when we discuss nienibrane reactors (Chapter 4 Part 2) and multiple reactions (Chapter 6). We will use this form of the concentration equation for multiple gas-phase reactions and for membrane reactors. 

The Polymath Program will combine the mole balance, net rates, and stoichiometric equations to solve for the molar flow rate and selectivity profiles for both the con-ventional PFR and the MR and also the selectivity profile. 

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