Component continuity equation

B. COMPONENT CONTINUITY EQUATIONS (COMPONENT BALANCES). Unlike mass, chemical components are not conserved. If a reaction occurs inside a system, the number of moles of an individual component will increase if it is a... 

The flows in and out can be both convective (due to bulk flow) and molecular (due to dithision). We can write one eomponent continuity equation for each component in the system. If there are NC components, there are NC component continuity equations for any one system. However, the am total mass balance and these NC component balances are not all independent, since the sum of all the moles times their respective molecular weights equals the total mass. Therefore a given system has only NC independent continuity equations. We usually use the total mass balance and NC — 1 component balances. For example, in a binary (two-component) system, there would be one total mass balance and one component balance. 

We have used an ordinary derivative since t is the only independent variable in this lumped system. The units of this component continuity equation are moles of A per unit time. The left-hand side of the equation is the dynamic term. The first two terms on the right-hand side are the convective terms. The last term is the generation term. 

Since the system is binary (components A and B), we could write another component continuity equation for component B. Let be the concentration of B in moles of B per unit volume. 

Assuming first-order reactions, the component continuity equations for components A, B, and C are... 

We now want to apply the component continuity equation for reactant A to a small differential slice of width dz, as shown in Fig. 2.4. The inflow terms can be split into two types bulk flow and diffusion. Diffusion can occur because of the concentration gradient in the axial direction. It is usually much less important than bulk flow in most practical systems, but we include it here to see what it contributes to the model. We will say that the diffusive flux of A, (moles of A per unit time per unit area), is given by a Pick s law type of relationship... 

Write the component continuity equations for a tubular reactor as in Example 2.5 with consecutive reactions occurring ... 

We want to keep track of the amounts of reactant A and product B in each tank, so component continuity equations are needed. However, since the system is binary and we know the total mass of material in each tank, only one component continuity equation is required. Either B or A can be used. If we arbitrarily choose A, the equations describing the dynamic changes in the amounts of... 

If the previous example is modified slightly to permit the volumes in each reactor to vary with time, both total and component continuity equations are required for each reactor. To show the effects of higher-order kinetics, assume the reaction is now nth-order in reactant A. 

In the reactors studied so far, we have shown the effects of variable holdups, variable densities, and higher-order kinetics on the total and component continuity equations. Energy equations were not needed because we assumed isothermal operations. Let us now consider a system in which temperature can change with time. An irreversible, exothermic reaction is carried out in a single perfectly mixed CSTR as shown in Fig. 3.3. 

An equilibrium-flash calculation (using the same equations as in case A above) is made at each point in time to find the vapor and liquid flow rates and properties immediately after the pressure letdown valve (the variables with the primes F , F l, y], x j,.. . shown in Fig. 3.8). These two streams are then fed into the vapor and liquid phases. The equations describing the two phases will be similar to Eqs. (3.40) to (3.42) and (3.44) to (3.46) with the addition of (1) a multi-component vapor-liquid equilibrium equation to calculate Pi and (2) NC — 1 component continuity equations for each phase. Controller equations relating 1 to Fi and P to F complete the model. 

The digital simulation of a distillation column is fairly straightforward. The main complication is the large number of ODEs and algebraic equations that must be solved. We will illustrate the procedure first with the simplified binary distillation column for which we developed the equations in Chap. 3 (Sec. 3.11). Equimolal overflow, constant relative volatility, and theoretical plates have been assumed. There are two ODEs per tray (a total continuity equation and a light component continuity equation) and two algebraic equations per tray (a vapor-liquid phase equilibrium relationship and a liquid-hydraulic relationship). 

Evaluate all derivatives of the component continuity equations for all SC components on all NT trays plus the reflux drum and the column base. 

Exampk (L5. The component continuity equation for an irreversible nth-order, nonisothermal reaction occurring in a constant-volume, variable-throughput CSTR is... 

Example 6A An isothennal, constant-holdup, constant-throughput CSTR with a first-order irreveraible reaction is described by a component continuity equation that is a first-order linear ODE ... 

This type of mass balance is known as a component continuity equation. It can be set up for each component. This means that there are as many of these equations as there are components. The summation over all the components leads to a continuity equation for the total mass, due to J2 k = 0, H Qk = 0 and 5Z 0KwKi = 0wi In place of the N component continuity equations for a system of N components, N — 1 component continuity equations along with the continuity equation for the total mass can be used. 

The term in the square brackets disappears due to the continuity equation (3.20) for the total mass. The component continuity equation is then... 

Example 3.2 Show that, under the presumption of film theory — steady-state mass transfer only in the direction of the coordinate axis adjacent to the wall, vanishing production density — that the component continuity equation (3.25) transforms into (1.186) of film theory. 

The same reasoning as before, whereby the mass transfer in the direction of flow is negligible in comparison to that through the boundary layer, leads to the equations for the concentration boundary layer. These are found from the component continuity equations (3.25) by neglecting the relevant expressions, to be... 

The equations (3.109), (3.117) or (3.118) and (3.120) for the velocity, thermal and concentration boundary layers show some noticeable similarities. On the left hand side they contain convective terms , which describe the momentum, heat or mass exchange by convection, whilst on the right hand side a diffusive term for the momentum, heat and mass exchange exists. In addition to this the energy equation for multicomponent mixtures (3.118) and the component continuity equation (3.25) also contain terms for the influence of chemical reactions. The remaining expressions for pressure drop in the momentum equation and mass transport in the energy equation for multicomponent mixtures cannot be compared with each other because they describe two completely different physical phenomena. 

As the material properties have been presumed to be independent of the temperature and composition, the velocity field is independent of the temperature and concentration fields, so the continuity and momentum equations can be solved independently of the energy and component continuity equations. 

In the calculation of the temperature and concentration fields the velocities are replaced by the stream function in the energy equation (3.162) and the component continuity equation (3.163). In addition a dimensionless temperature... 

Component continuity equations (component mass balances) ... 

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