Material balances integral

To derive the concentration profile for progressive freezing, a material balance is employed for solidification of a small fraction dg of melt, as shown in Figure 1. Integration from the beginning of solidification gives (1,4,8) ... 

Tubular flow reaclors operate at nearly constant pressure. How the differential material balance is integrated for a number of second-order reactions will be explained. When n is the molal flow rate of reactant A the flow reactor equation is... 

Replace the holdup derivatives in Eqs. (13-149) to (13-151) by total-stage material-balance equations (e.g., dMj/dt = Vj + i + Ej- — Vj — Lj) and solve the resulting equations one at a time by the predictor step of an explicit integration method for a time increment that is determined by stability and truncation considerations. If the mole fraclions for a particular stage do not sum to 1, normalize them. 

For fast irreversible chemical reactions, therefore, the principles of rigorous absorber design can be applied by first estabhshing the effects of the chemical reaction on /cl and then employing the appropriate material-balance and rate equations in Eq. (14-71) to perform the integration to compute the required height of packing. 

Evaluation of the integral in Eq. (14-86) requires a knowledge of the liquid-phase bulk concentration of B as a function of y. This relationship is obtained by means of a material balance around the tower, as shown in Eq. (14-73). Numerical integration by a quadrature method such as Simpson s nrle normally will be required for this calculation. 

The integration of Eq. (16-140) or (16-141) as an indefinite integral will give an integration constant that must be evaluated to center the transition properly. The material balance depicted in Fig. 16-26 is used. The two shaded regions must be of equal area if the stoichiometric center of the transition is located where the throughput parameter is unity. Thus, we have... 

As can be seen for infinite recycle ratio where C = Cl, all reactions will occur at a constant C. The resulting expression is simply the basic material balance statement for a CSTR, divided here by the catalyst quantity of W. On the other side, for no recycle at all, the integrated expression reverts to the usual and well known expression of tubular reactors. The two small graphs at the bottom show that the results should be illustrated for the CSTR case differently than for tubular reactor results. In CSTRs, rates are measured directly and this must be plotted against the driving force of... 

Equation 1 is normally integrated by graphical or numerical means utilizing the overall material balance and the saturated air enthalpy curve. 

When the Freeman and Lewis rate constants are applied to an experimental situation and integrated. Fig. 7 results. This figure shows the same fundamental trends seen in the data. There are some differences, however. The Freeman and Lewis measurements, as presented in their Fig. 2, appear to exceed the available phenol by about 39%. This is probably one reason why Zavitsas et al. state that the Freeman rate constants do not fit the data [80], Flowever, the calculations made using their rate constants do maintain the overall material balance. As presented here, they are not as precise as they could be because the calculation interval has been set at 1 h. Flowever, they are as good as the data at this level. 

Differential and Integral Balances. Two types of material balances, differential and integral, are applied in analyzing chemical processes. The differential mass balance is valid at any instant in time, with each term representing a rate (i.e., mass per unit time). A general differential material balance may be written on any material involved in any transient process, including semibatch and unsteady-state continuous flow processes ... 

For any transient process that begins at time t and is terminated at a later time tp, the general integral material balance equation has the form... 

A special case of Equation 2-218 is directly applicable to batch processes for which the mass flowrate terms are zero. The integral material balance for any component i in such a process is... 

Perform an overall material balance and the necessary component material balances so as to provide the maximum number of independent equations. In the event the balance is written in differential form, appropriate integration must be carried out over time, and the set of equations solved for the unknowns. 

In integral analysis concentration-versus-time (or equivalently concentration-versus-distance from the inlet of the integral flow reactor) data are known. Kinetic expressions to be determined are incorporated into the differential material balance equations ... 

From the rate law and the material-balance equation 2.2-10, the equation to be integrated is... 

PFR OS integral reactor. In Figure 3.8, the entire vessel indicated from sampling points Sjn to Sout, over which a considerable change in fA or cA would normally occur, could be called an integral PFR. It is possible to obtain values of kinetics parameters by means of such a reactor from the material balance equation 2.4-4 rearranged as... 

Repeat problem 6-4 using an integral method For this purpose, substitute the rate law into the material balance for a constant-volume BR, and integrate the resulting expression to relate /a and t. Then, with cAo as a parameter (corresponding to P0 in problem 6-4), show that, for a... 

The time f required to achieve fractional conversion fA is obtained by integration of the material balance, equation 2.2-4 ... 

It is not possible to predict a priori which of the possible stationary-states is actually attained, based on steady-state operating considerations. It may be done by integrating the material-balance equation in unsteady-state form, equation 14.3-2 or equivalent, with the given rate law incorporated. For this, the initial concentration of A in the reactor cA(f) at t= 0 must be known this is not necessarily the same as cAg. 

To evaluate the effect of pressure drop on performance, differential equations for the pressure drop (15.2-11), material balance (15.2-4), and energy balance (15.2-10) must be integrated simultaneously to solve for P,fA, and T as functions of axial position, x ... 

The E-Z Solve software can be used to integrate numerically the differential equation resulting from the combination of the material-balance equation (15.2-17) and the rate law [equation (A)]—see file exl5-3.msp. The same results are obtained for V and t. 

Suppose the liquid-phase reaction A products is second-order, with ( rA) = kAcA, and takes place in a PFR. Show that the SFM gives the same result for 1 - fA = cAlcAo as does the integration of equation 15.2-16, the material balance. 

The reaction is A - B + H2. The reactor length corresponding to the reactor volume and diameter specified is 110 m. The fA(x) and P(x) profiles can be determined by simultaneously integrating the material-balance equation (15.2-4) and the pressure-drop equation (15.2-11) ... 

Since the material-balance equations, 16.2-1 and -2, derived above refer to a particular radial position, we must integrate radially to obtain an average concentration cA at any axial position x, including at the outlet, where x = L. The latter is the observed outlet concentration that corresponds to the outlet fractional conversion. We develop the expression in terms of cA,... 

The interpretation of cA(t) comes from the realization that each cylindrical shell passes through the vessel as an independent batch. Thus, cA(/) is obtained by integration of the material balance for a batch reactor (BR). Accordingly, we may rewrite equation 16.2-11, in terms of either cA(x) or fA(x), as... 

Component D can be found by material balance after the three different equations have been integrated. 

Method (b). Numerical integration of the four equations, (3) to (6) is accomplished with ODE, either Constantinides or POLYMATH. Alternately, the material balances (11 and (2) and only the two differential equations (3) and (4) can be solved together. This procedure is better carried out with POLYMATH ODE. 

Tubular flow reactors usually operate at nearly constant pressure. For a reactant A, the differential material balance is -dna = na0dx - -V dCa = V Ca0dx = radVr One form of the integration is,... 

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