The mass balance

If there is no change of density during the reaction, the concentration can be suitably measured in moles per unit volume and the linear velocity will be denoted by if reactor is packed, the velocity v is taken to be the volume flow rate divided by the total cross-sectional area of the reactor. (The mean velocity through the interstices of the packed bed is v/e, where is the void fraction.) Also, let c/z) be the concentration of at a distance z from the inlet and r(ej, T) be the pseudo-homogeneous reaction rate in moles per unit reactor volume per unit time. If is the cross-sectional area of the reactor then in a unit of time 

This is a differential equation for c/z) and is subject to the initial conditions that Specify the feed composition  

But since the reactions are independent, the only set of multipliers that make 2 0 = 0 for J is the trivial set X = 0. Hence the S equations 

If the density is not constant, the linear velocity will not be constant, for 

This equation can be put into a number of different forms. First, notice that it is not particularly useful to substitute from Eq. (9.2.3) for then 

The balance equation cannot be applied to uncountable individuals (units) or to intensive properties. Intensive properties are independent of the amount of matter present. Some examples are temperature, pressure, viscosity, hardness, color, honesty, electric voltage, beauty, and density. An example of uncountable individuals is all the decimal fractions between 0 and 1. 

Our example of a balance equation in Sec. 3.1 would be of interest to demographers but not necessarily to engineers. Let us consider the most important engineering balance, the mass balance. Mass obeys the general balance equation the creation and destruction terms are zero. Thus, the mass balance is 

The careful application of this equation is necessary to most fluid-mechanics problems. Naturally we can divide by time and find 

The mass balance cannot be derived from any prior principle. Like all the other basic laws of nature, it rests on its ability to explain observed facts. Every careful experiment which has been run to test it indicates that it is correct. We will see in Chap. 4 that mass and energy can be converted from one to the other. In most engineering problems, we can neglect this fact and use the simple formulation in Eq. 3.5 (but we cannot neglect it in dealing with atomic bombs or the energy source of the sun).  

Example 3.1. Consider the simple pot-bellied stove, burning natural gas, shown in Fig. 3.1. Applying Eq. 3.6 to this stove, we choose as our system 

---

How does the result of this calculation change if we require our assumptions to have an error of less than 1%. In this case we can no longer assume that [HF] >> [F ]. Solving the mass balance equation (6.37) for [HF]... 

From the ladder diagram it appears that we may safely assume that the concentrations of H2L+ and L are significantly smaller than that for HL, allowing us to simplify the mass balance equation to... 

Using these assumptions allows us to simplify several equations. The mass balance for N1T3 is now... 

Equimolar Counterdiffusion. Just as unidirectional diffusion through stagnant films represents the situation in an ideally simple gas absorption process, equimolar counterdiffusion prevails as another special case in ideal distillation columns. In this case, the total molar flows and are constant, and the mass balance is given by equation 35. As shown eadier, noj/g factors have to be included in the derivation and the height of the packing is... 

The mass balance for A is best represented as a straight line in hypothetical coordinates and X ... 

The momentum terms can be rewritten using the mass balance to yield ... 

Design and operation of recirculation systems can be compHcated. Problems are avoided by using a sludge-blanket clarifier, in which feed enters below a blanket of accumulated and flocculated soflds which become fluidized in the zone-settling regime by the upflowing feed. Feed soflds are trapped in the blanket. The soflds content of the blanket continuously increases and part must be bled off in order to maintain the mass balance. 

The flow of hydrothermal solutions iato the oceans from hydrothermal vents, ie, springs coming from the sea floor ia areas of active volcanism, and the chemical reactions occurring there by high temperature alteration of basalts ate of significance ia the mass balance of and. Eurthermore,... 

Phase II Mass Balance. (/) Determine raw material iaputs. 2) Record water usage. 3) Assess present practice and procedures. (4) Quantify process outputs. (5) Account for emissions to atmosphere, to wastewater, and to off-site disposal. (6) Assemble iaput and output information. (7) Derive a preliminary mass balance. (8) Evaluate and refine the mass balance. 

The mass balance across stage n is (/) vapor ) from the stage below (n — 1) flows up to stage n , (2) Hquid ) from the stage above (n + 1)... 

The mass-balance restrictions are the C balances written for the C components present in the system. (Since we will only deal with non-reactive mixtures, each chemical compound present is a phase-rule component.) An alternative is to write (C — 1) component balances and one overall mass balance. 

Since residue cur ves do not by definition cross separatrices, the distillate and bottoms compositions must be in the same distillation region with the mass balance line intersecting a residue cur ve in two places. Mass balance lines for mixing and for other separations not involving vapor-hquid equihbria, such as extraction and decantation, are of course not hmited by distiUation boundaries. 

Under steady state, the mass balance on both solids and hquid yield, respectively ... 

Kinetics The capacity and efficiency of biofilter operation is a function of active surface area, filter void space, target removal efficiency, gas species, gas concentration, and gas flow rate. A simphfied theoretical model described by S.P.P. Ottengraf et al. is schematically represented by in Fig. 25-18. The mass balance made around the hq-uid-phase biolayer can be described as follows ... 

To estimate the current yield of magnesium alloys, the weight loss is determined indirectly over the volumetric measurement of the evolved hydrogen in the apparatus in Fig. 6-9 in Section 6.2.4. The mass balance for oxygen-free media follows from Eqs. (6-1) and (6-5a,b) or (6-14) ... 

This is accomplished by measuring the rate at constant temperature and at various concentrations by varying the feed rate. Calculating 0, multiplying by the measured slope at the calculated 0, and then adding one gives the derivative of the mass balance rate with regard to concentration. 

Often in plant operations condensate at high pressures are let down to lower pressures. In such situations some low-pressure flash steam is produced, and the low-pressure condensate is either sent to a power plant or is cascaded to a lower pressure level. The following analysis solves the mass and heat balances that describe such a system, and can be used as an approximate calculation procedure. Refer to Figure 2 for a simplified view of the system and the basis for developing the mass and energy balances. We consider the condensate to be at pressure Pj and temperature tj, from whence it is let down to pressure 2. The saturation temperature at pressure Pj is tj. The vapor flow is defined as V Ibs/hr, and the condensate quality is defined as L Ibs/hr. The mass balance derived from Figure 2 is ... 

From the mass balance equation for a batch reactor... 

The first order reaction is represented by (-r ) = kC, and applying the mass balance Equation 6-120 and the heat balance Equation 6-121, respectively, gives the fractional conversion in terms of the mass balance equation ... 

The fractional conversions in terms of both the mass balance and heat balance equations were calculated at effluent temperatures of 300, 325, 350, 375, 400, 425, 450, and 475 K, respectively. A Microsoft Excel Spreadsheet (Example6-ll.xls) was used to calculate the fractional conversions at varying temperature. Table 6-7 gives the results of the spreadsheet calculation and Eigure 6-24 shows profiles of the conversions at varying effluent temperature. The figure shows that die steady state values are (X, T) = (0.02,300), (0.5,362), and (0.95,410). The middle point is unstable and die last point is die most desirable because of die high conversion. 

Expression of the mass balances around the bioreactor (process). 

The Right-Hand Side of the Mass Balance Equation... 

We will consider flow through a solid element. Introducing the notations for molar flow density, partial density, and the reaction rate gives an equation for the mass balance ... 

For the mass balance of component A, diffusion velocity and the corresponding diffusion factor are defined with regard to the mean molar velocity V, defined by the equation... 

The airflow rate infiltrating and exfiltrating through each air leakage pass, Q , due to the combined effect of wind, stack, and mechanical ventilation system perfotmance can be calculated ftom the mass balance equation... 

The mass balance equation for the airflow through the gate can be described as... 

---