Material Balance Basics

As described here, it is important to consider the process of approaching, formulating, and solving 

If the flow diagram is not provided with the problem statement, then draw a complete one on a whole page. Clarity at this point is critical and will help you get a feel for the problem. If a flow diagram is provided, sometimes it will be better to redo and complete it with all the available information. 

Define all variables (known or unknown). Some variables will be specified, but it is advisable to have a definition for each variable and codify it in such a way that it will be easy for you to match each variable with your flow diagram. As was just mentioned in example 3, in general, avoid using generic variable names such as X, Y, and Z. It is much clearer to codify the variable according its role in the problem. An exception is the use of x for mass fraction composition. 

To formulate a specific material balance, it is vital to first define the system under analysis. But who defines the system You, the person formulating it As shown in example 3, before writing a material balance it is strictly necessary to define the system and its boundaries. Only then can we write a material balance equation and determine which flows are entering and leaving the system. For an example, see Fig. 7.13. 

For nonreactive systems (this chapter) it is better to use mass units for all streams (Fig. 7.13). As was pointed out earlier, mass is conservative. Recall the example of ethanol and water that showed that volume was not necessarily additive. As we will see in the next chapter (material 

---

Equation 235 is the basic expression of material balance for a closed system in which r chemical reactions occur. It asserts that ia such a system... 

When the basic physical laws are expressed in this form, the formulation is greatly facilitated. These expressions are quite often given the names, material balance, energy balance, and so forth. To be a little more specific, one could write the law of conservation of energy in the steady state as... 

Multiple-Effect Evaporators A number of approximate methods have been published for estimating performance and heating-surface requirements of a multiple-effect evaporator [Coates and Pressburg, Chem. Eng., 67(6), 157 (1960) Coates, Chem. Eng. Prog., 45, 25 (1949) and Ray and Carnahan, Trans. Am. Inst. Chem. Eng., 41, 253 (1945)]. However, because of the wide variety of methods of feeding and the added complication of feed heaters and condensate flash systems, the only certain way of determining performance is by detailed heat and material balances. Algebraic soluflons may be used, but if more than a few effects are involved, trial-and-error methods are usually quicker. These frequently involve trial-and-error within trial-and-error solutions. Usually, if condensate flash systems or feed heaters are involved, it is best to start at the first effect. The basic steps in the calculation are then as follows ... 

Although dynamic responses of microbial systems are poorly understood, models with some basic features and some empirical features have been found to correlate with actual data fairly well. Real fermentations take days to run, but many variables can be tried in a few minutes using computer simulation. Optimization of fermentation with models and reaf-time dynamic control is in its early infancy however, bases for such work are advancing steadily. The foundations for all such studies are accurate material Balances. 

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... 

Three basic fluid contacting patterns describe the majority of gas-liquid mixing operations. These are (1) mixed gas/mixed liquid - a stirred tank with continuous in and out gas and liquid flow (2) mixed gas/batch mixed liquid - a stirred tank with continuous in and out gas flow only (3) concurrent plug flow of gas and liquid - an inline mixer with continuous in and out flow. For these cases the material balance/rate expressions and resulting performance equations can be formalized as ... 

Used to present the heat and material balance of a process. This may be in broad block form with specific key points delineated, or in more detailed form identifying essentially every flow, temperature and pressure for each basic piece of process equipment or processing step. This may and usually does include auxiliary services to the process, such as steam, water, air, fuel gas, refrigeration, circulating oil, etc. This type of sheet is not necessarily distributed to the same groups as would receive and need the piping flowsheet described next, because it may contain detailed confidential process data. 

Equilibrium Basic Consideration, 1 Ideal Systems, 2 K-Factor Hydrocarbon Equilibrium Charts, 4 Non-Ideal Systems, 5 Example 8-1 Raoult s Law, 14 Binary System Material Balance Constant Molal Overflow Tray to Tray,... 

In LP steam boiler systems, this problem of uncontrolled or excessive water or condensate return loss also occurs. It may be uncontrolled perhaps because of leaking steam traps or excessive as a result of too frequent or prolonged BD. Irrespective of the basic cause, it is necessary to obtain an accurate assessment of materials balance as a first step in understanding the magnitude of the problem and providing resolution. 

The basic differential equation for mass transfer accompanied by an nth order chemical reaction in a spherical particle is obtained by taking a material balance over a spherical shell of inner radius r and outer radius r + Sr, as shown in Figure 10.12. 

These equations can be solved simultaneously with the material balance equations to obtain x[, x, xf and x1,1. For a multicomponent system, the liquid-liquid equilibrium is illustrated in Figure 4.7. The mass balance is basically the same as that for vapor-liquid equilibrium, but is written for two-liquid phases. Liquid I in the liquid-liquid equilibrium corresponds with the vapor in vapor-liquid equilibrium and Liquid II corresponds with the liquid in vapor-liquid equilibrium. The corresponding mass balance is given by the equivalent to Equation 4.55 ... 

The starting point for the development of the basic design equation for a well-stirred batch reactor is a material balance involving one of the species participating in the chemical reaction. For convenience we will denote this species as A and we will let (— rA) represent the rate of disappearance of this species by reaction. For a well-stirred reactor the reaction mixture will be uniform throughout the effective reactor volume, and the material balance may thus be written over the entire contents of the reactor. For a batch reactor equation 8.0.1 becomes... 

For nonisothermal reactors the key questions that the reactor designer must answer are (1) How can one relate the temperature of the reacting system to the degree of conversion that has been accomplished and (2) How does this temperature influence the subsequent performance of the system In responding to these questions the chemical engineer must use two basic tools—the material balance and the energy balance. The bulk of this chapter deals with these topics. Some stability and selectivity considerations are also treated. 

In addition to the time representation and material balances, scheduling models are based on different concepts or basic ideas that arrange the events of the schedule over time with the main purpose of guaranteeing that the maximum capacity of the shared resources is never exceeded. As can be seen in Figure 8.5 and Table 8.1, we classified these concepts into five different types of event representations, which have been broadly utilized to develop a variety of mathematical formulations for the batch scheduling problem. Although some event representations are more... 

During the process hazards identification and definition phase of a project design, a basic process control system (BPCS) strategy is normally developed in conjunction with heat and material balances for the process. 

The component material balance equation, combined with the reactor energy balance equation and the kinetic rate equation, provide the basic model for the ideal plug-flow tubular reactor. 

In this volume, there is an account of the basic theory underlying the various Unit Operations, and typical items of equipment are described. The equipment items are the essential components of a complete chemical plant, and the way in which such a plant is designed is the subject of Volume 6 of the series which has just appeared. The new volume includes material on flowsheeting, heat and material balances, piping, mechanical construction and costing. It completes the Series and forms an introduction to the very broad subject of Chemical Engineering Design. 

By referring to an elementary section of reactor as shown in Fig. 13.18, the basic material balance for any reaction component... 

The MAM described here is a generalization of the model previously published (10). Hence, only a summary of the derivation will be given here. Details can be found elsewhere (17). The basic equations are the surfactant and counterion material balances and the minimization of the Gibbs free energy of the system with respect to the micelle concentration c , and mole fraction x (11). Equation 4 from Ref. (11) has been changed to... 

Equation 235 is the basic expression of material balance for a closed system in which r chemical reactions occur. It asserts that in such a system there are at most r mole-number-related quantities, S , capable of independent variation. Note the absence of implied restrictions with respect to chemical reaction equilibria the reaction coordinate formalism is merely an accounting scheme, valid for tracking the progress of each reaction to any arbitrary level of conversion. 

Performances of dryers with simple flow patterns can be described with the aid of laboratory drying rate data. In other cases, theoretical principles and correlations of rate data are of value largely for appraisal of the effects of changes in some operating conditions when a basic operation is known. The essential required information is the residence time in the particular kind of dryer under consideration. Along with application of possible available rules for vessel proportions and internals to assure adequate contacting of solids and air, heat and material balances then complete a process design of a dryer. 

The basic set of differential material-balance equations for the various species in the column can also be written in terms of the h instead of Xi and t/i. This new set of differential equations reveals a particular property of the H-function roots no root values other than those appearing in the initial and boundary conditions can arise anywhere in the column. The column behavior thus is completely described by the trajectories of the initial and boundary values of the roots. (The same is not true for concentrations, as the behavior of species 2 in the example in Figure 4 has shown.)... 

Basically, the general material balance for a reactor follows the same pattern as all material and energy balances, namely ... 

The basic equation for a tubular reactor is obtained by applying the general material balance, equation 1.12, with the plug flow assumptions. In steady state operation, which is usually the aim, the Rate of accumulation term (4) is zero. The material balance is taken with respect to a reactant A over a differential element of volume 8V, (Fig. 1.14). The fractional conversion of A in the mixture entering the element is aA and leaving it is (aA + SaA). If FA is the feed rate of A into the reactor (moles per unit time) the material balance over 8V, gives ... 

Rather than proceed by trying to read a reaction factor fA from Fig. 4.3, it is better to set out the basic material balance for mass transfer and reaction as below. Locating the position of 0 on Fig. 4.3 does however confirm that reaction will be occurring in the main bulk of the liquid and that an agitated tank is a suitable type of reactor. 

The modelling of real food webs can be an exceedingly complicated task but, to illustrate the basic technique, a situation may be defined where a continuous stirred-tank biological reactor contains two species, one the predator, the other the prey. The food for the prey is assumed to enter as the sterile feed stream to the reactor, so that the predator may only consume the prey which grows in the reactor. Material balances can be drawn up for the process in much the same way as has... 

The basic items normally considered in a material balance of a cooling tower are the range, T, circulating water rate, C (in U.S. gpm), the ppm (parts per million of any soluble salts in C (Xc), and M, B, E and W. Xc is normally expressed in terms of equivalent chlorides,... 

The basic developments reported herein can be used to predict multicomponent adsorption rates in different types of contacting units, such as slurry adsorbers, moving bed units and columnar operation. The adsorption rate and equilibrium equations would remain the same, but the reactor material balance equation would be different depending on the reactor type and geometry. Thus, the multicomponent adsorption model can be utilized in practical processing applications in the chemical industry, air pollution control, and water and wastewater treatment. 

The deviation from ideal plug flow due to the axial mixing can be described by the dispersion model (Levenspiel, 1972). Let s look at the differential element with a thickness dx in a holding tube as shown in Figure 8.1. The basic material balance for the microorganisms suspended in the medium is... 

---