9 Macroscopic Balance Equations Paul K. Andersen and Sarah W. Harcum New Mexico State University, Las Cruces, New Mexico The prevention of waste and pollution requires an understanding of numerous technical disciplines, including thermodynamics, heat and mass transfer, fluid mechanics, and chemical kinetics. This chapter summarizes the basic equations and concepts underlying these seemingly disparate fields. 1 MACROSCOPIC BALANCE EQUATIONS A balance equation accounts for changes in an extensive quantity (such as mass or energy) that occur in a well-defined region of space, called the control volume (CV). The control volume is set off from its surroundings by boundaries, called control surfaces (CS). These surfaces may coincide with real surfaces, or they may be mathematical abstractions, chosen for convenience of analysis. If matter can cross the control surfaces, the system is said to be open; if not, it is said to be closed. 1.1 The General Macroscopic Balance Balance equations have the following general form: Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. dX dt = ∑ (X CS,i ⋅ ) i +(X ⋅ ) gen (1) where X is some extensive quantity. A dot placed over a variable denotes a rate; for example, (X ⋅ ) i is the flow rate of X across control surface i. The terms of Eq. (1) can be interpreted as follows: dX dt = rate of change of X inside the control volume ∑ (X CS,i ⋅ ) i = sum of flow rates of X across the control surfaces (X ⋅ ) gen = rate of generation of X inside the control volume Flows into the control volume are considered positive, while flows out of the control volume are negative. Likewise, a positive generation rate indicates that X is being a created within the control volume; a negative generation rate indicates that X is being consumed in the control volume. The variable X in Eq. (1) represents any extensive property, such as those listed in Table 1. Extensive properties are additive: if the control volume is TABLE 1 Extensive Quantities Quantity Flow rate (CS i) Generation rate Total mass, mm ⋅ i m ⋅ gen = 0 (conservation of mass) Total moles , NN ⋅ i N ⋅ gen Species mass, m A (m ⋅ A ) i (m ⋅ A ) gen Species moles, N A (N ⋅ A ) i (N ⋅ A ) gen Energy, EE ⋅ i = Q ⋅ i + W ⋅ i + m ⋅ i E ^ i E ⋅ i = Q ⋅ i + W ⋅ i + N ⋅ i E ~ i E ⋅ gen = 0 (conservation of energy) Entropy, SS ⋅ i = Q ⋅ i /T i + m ⋅ i S ^ i S ⋅ i = Q ⋅ i /T i + N ⋅ i S ~ i S ⋅ gen ≥ 0 (second law of thermodynamics) Momentum, p = mvp ⋅ i = m ⋅ i v i p ⋅ gen = F (Newton’s second law of motion) Notes: Q ⋅ i ≡ heat transfer rate through CS i W ⋅ i ≡ work rate (power) at CS i E ^ i ≡ energy per unit mass of stream i; E ~ i ≡ energy per unit mole of stream i S ^ i ≡ entropy per unit mass of stream i; S ~ i ≡ entropy per unit mole of stream i T i ≡ absolute temperature of CS i F ≡ net force acting on control volume Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. subdivided into smaller volumes, the total quantity of X in the control volume is just the sum of the quantities in each of the smaller volumes. Balance equations are not appropriate for intensive properties such as temperature and pressure, which may be specified from point to point in the control volume but are not additive. It is important to note that Eq. (1) accounts for overall or gross changes in the quantity of X that is contained in a system; it gives no information about the distribution of X within the control volume. A differential balance equation may be used to describe the distribution of X (see Section 2.2). Equation (1) may be integrated from time t 1 to time t 2 to show the change in X during that time period: ∆X = ∑ (X) i CS,i +(X) gen (2) 1.2 Total Mass Balance Material is conveniently measured in terms of the mass m. According to Einstein’s special theory of relativity, mass varies with the energy of the system: m ⋅ gen = 1 c 2 dE dt (special relativity) (3) where c = 3.0 × 10 8 m/s is the speed of light in a vacuum. In most problems of practical interest, the variation of mass with changes in energy is not detectable, and mass is assumed to be conserved—that is, the mass-generation rate is taken to be zero: m ⋅ gen = 0 (conservation of mass) (4) Hence, the mass balance becomes dm dt = ∑ ( CS,i m ⋅ ) i (5) 1.3 Total Material Balance The quantity of material in the CV can be measured in moles N, a mole being 6.02 × 10 23 elementary particles (atoms or molecules). The rate of change of moles in the control volume is given by dN dt = ∑ (N CS,i ⋅ ) i +(N ⋅ ) gen (6) where (N ⋅ ) i is the molar flow rate through control surface i and (N ⋅ ) gen is the molar Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. generation rate. In general, the molar generation rate is not zero; the determina- tion of its value is the object of the science of chemical kinetics (Section 4.2). 1.4 Macroscopic Species Mass Balance A solution is a homogenous mixture of two or more chemical species. Solutions usually cannot be separated into their components by mechanical means. Con- sider a solution consisting of chemical species A, B, . . . . For each of the components of the solution, a mass balance may be written: dm A dt = ∑ ( CS,i m ⋅ A ) i +(m ⋅ A ) gen dm B dt = ∑ ( CS,i m ⋅ B ) i +(m ⋅ B ) gen (7) . . . Conservation of mass requires that the sum of the constituent mass generation rates be zero: (m ⋅ A ) gen +(m ⋅ B ) gen + . . . = 0 (conservation of mass) (8) 1.5 Macroscopic Species Mole Balance The macroscopic species mole balances for a solution are dN A dt = ∑ (N CS,i ⋅ A ) i +(N ⋅ A ) gen dN B dt = ∑ (N CS,i ⋅ B ) i +(N ⋅ B ) gen (9) . . . In general, moles are not conserved in chemical or nuclear reactions. Hence, (N ⋅ A ) gen +(N ⋅ B ) gen + . . . = N ⋅ gen (10) 1.6 Macroscopic Energy Balance Energy may be defined as the capacity of a system to do work or exchange heat with its surroundings. In general, the total energy E can expressed as the sum of three contributions: Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. E = K +Φ+U (11) where K is the kinetic energy, Φ is the potential energy, and U is the internal energy. Energy can be transported across the control surfaces by heat, by work, and by the flow of material. Thus, the rate of energy transport across control surface i is the sum of three terms: E ⋅ i = Q ⋅ i + W ⋅ i + m ⋅ i E ^ i (12) where Q ⋅ i is the heat transfer rate, W ⋅ i is the working rate (or power), m ⋅ i is the mass flow rate, and E ^ i is the specific energy (energy per unit mass). Energy is conserved, meaning that the energy generation rate is zero: E ⋅ gen = 0 (conservation of energy) (13) Therefore, the energy of the control volume varies according to dE dt = ∑ ( CS,i Q ⋅ + W ⋅ + m ⋅ E ^ ) i (14) The energy flow rate can also be written in terms of the molar flow rate N ⋅ i and the molar energy E ~ i . Hence, the energy balance can be written in the equivalent form dE dt = ∑ (Q CS,i ⋅ + W ⋅ + N ⋅ E ~ ) i (15) 1.7 Entropy Balance Entropy is a measure of the unavailability of energy for performing useful work. Entropy may be transported across the system boundaries by heat and by the flow of material. Thus, the rate of entropy transport across control surface i is given by S ⋅ i = Q ⋅ i T i + m ⋅ i S ^ i (16) where Q ⋅ i is the heat transfer rate through control surface i, T i is the absolute temperature of the control surface, m ⋅ i is the mass flow rate through the control surface, and S ^ i is the specific entropy or entropy per unit mass. In terms of the molar flow rate and the molar entropy S ~ i , the entropy transport rate is S ⋅ i = Q ⋅ i T i + N ⋅ i S ~ i (17) Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. According to the second law of thermodynamics, entropy may be created— but not destroyed—in the control volume. The entropy generation rate therefore must be non-negative: S ⋅ gen ≥ 0 (second law of thermodynamics) (18) Processes for which the entropy generation rate vanishes are said to be reversible. Most real processes are more or less irreversible. In terms of mass flow rates, the entropy balance is dS dt = ∑ CS,i    Q ⋅ T + m ⋅ S ^    i + S ⋅ gen (19) If molar flow rates are used instead, the entropy balance is dS dt = ∑ CS,i    Q ⋅ T + N ⋅ S ~    i + S ⋅ gen (20) 1.8 Macroscopic Momentum Balance The momentum p is defined as the product of mass and velocity. Because velocity is a vector—a quantity having both magnitude and direction—momentum is also a vector. Momentum can be transported across the system boundaries by the flow of mass into or out of the control volume: p ⋅ i =(m ⋅ v ) i (21) According to Newton’s second law of motion, momentum is generated by the net force F that acts on the control volume: p ⋅ gen = F (Newton’s second law) (22) Hence, the momentum balance takes the form dmv dt = ∑ CS,i    m ⋅ v    i + F (23) Because this is a vector equation, it can be written as three component equations. In Cartesian coordinates, the momentum balance becomes x momentum: dmv x dt = ∑ CS,i    m ⋅ v x    i + F x Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. y momentum: dmv y dt = ∑ CS,i    m ⋅ v y    i + F y (24) z momentum: dmv z dt = ∑ CS,i    m ⋅ v z    i + F z 2 DIFFERENTIAL BALANCE EQUATIONS As noted previously, macroscopic balance equations account only for overall or gross changes that occur within a control volume. To obtain more detailed information, a macroscopic control volume can be subdivided into smaller control volumes. In the limit, this process of subdivision creates infinitesimal control volumes described by differential balance equations. 2.1 General Differential Balance Equation The general macroscopic balance for the extensive property X is Eq. (1): dX dt = ∑ CS,i (X ⋅ ) i +(X ⋅ ) gen (1) Division by the system’s volume V yields d dt    X V    = ∑ CS,i X ⋅ i V + X ⋅ gen V The differential or microscopic balance equation results from taking the limit as V → 0: ∂[X] ∂ =−∇⋅(X)+ [X ⋅ ] gen (25) where [X] is read as “the concentration of X” and X is “the flux of X.” The terms of this equation can be interpreted as follows: ∂ ∂t [X] = rate of change of the concentration of X −∇ ⋅ (X) = net influx of X [X] gen = generation rate of X per unit volume The flux X is the rate of transport per unit area, where the area is oriented perpendicular to the direction of transport. In Cartesian (x, y, z) coordinates, X may be defined as Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. X = X ⋅ x A x i + X ⋅ y A y j + X ⋅ z A z k Here, A x , A y , and A z are the areas perpendicular to the x, y, and z directions, respectively; (i, j, k) are the (x, y, z) unit vectors. In Cartesian coordinates, the del operator ∇ takes the form ∇= ∂ ∂x i + ∂ ∂y j + ∂ ∂z k The form of the del operator in other coordinate systems may be found in texts on fluid mechanics and transport phenomena (1–4). Table 2 shows the concentrations, fluxes, and volumetric generation terms for the extensive quantities considered in this chapter. 2.2 Differential Total Mass Balance Assuming conservation of mass (ρ gen = 0), the differential mass balance can be written as TABLE 2 Concentrations, Fluxes, and Volumetric Generation Quantity Flux Volumetric generation rate Total mass, [m] =ρ m =ρv ρ ⋅ gen = 0 (conservation of mass) Total moles , [N] = c N = cv c ⋅ gen Species mass, [m A ] =ρ A m A =ρ A v + j A (ρ ⋅ A ) gen Species moles, [N A ] = c A N A = c A v + J A (c ⋅ A ) gen Energy, [E] = e E = q +σ⋅v + mE ^ E = q +σ⋅v + NE ~ e ⋅ gen = 0 (conservation of energy) Entropy, [S] = s S = q/T + mS ^ S = q/T + NS ~ s ⋅ gen ≥ 0 (second law of thermodynamics) Momentum, [p] =ρvP= mv =ρvv f =−∇⋅σ+b (second law of motion) Notes: b ≡ body force per unit volume j A ≡ diffusive mass flux of species A J A ≡ diffusive molar flux of species A f ≡ total force per unit volume q ≡ heat flux σ≡ material stress v ≡ fluid velocity Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. ∂ρ ∂t =−∇⋅(m) (26) where m is the mass flux. It is more common to write the mass flux in terms of the density and velocity, m = ρv. Hence, ∂ρ ∂t =−∇⋅(ρv) (27) Equation (27) is called the continuity equation; it is one of the basic equations of fluid mechanics. 2.3 Differential Total Material Balance The differential material balance is ∂c ∂t =−∇⋅(N)+c ⋅ gen (28) where N is the molar flux. In the absence of chemical or nuclear reactions, c ⋅ gen = 0. 2.4 Differential Species Balances Consider a solution consisting of component species A, B, . . . . In general, a chemical species in such a solution may be transported by convection and by diffusion. Convection is transport by the bulk motion of the solution. The convective flux of species A is the product of the mass concentration ρ A and the solution velocity v: ρ A v = convective (mass) flux of A Diffusion is the transport of a species resulting from gradients of concentration, electrical potential, temperature, pressure, and so on. The diffusive flux of species A is denoted by j A : j A = diffusive (mass) flux of A The overall material flux is the sum of the convective and diffusive fluxes: m A =ρ A v + j A (29) A differential mass balance may be written for each of the components of the solution: ∂ρ A ∂t =−∇⋅(ρ A v + j A )+(ρ ⋅ A ) gen Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. ∂ρ B ∂t =−∇⋅(ρ B v + j B )+(ρ ⋅ B ) gen (30) . . . Conservation of mass requires that the constituent mass generation rates sum to zero: (ρ ⋅ A ) gen +(ρ ⋅ B ) gen + . . . = 0 (conservation of mass ) (31) 2.5 Differential Species Material Balance The total molar flux of species A is the sum of the convective and diffusive molar fluxes: N A = c A v + J A (32) where c A v is the convective molar flux and J A is the diffusive molar flux: c A v = convective (molar) flux of A J A = diffusive (molar) flux of A The differential material balances for a solution consisting of species A, B, . . . are ∂c A ∂t =−∇⋅(c A v + J A )+(c ⋅ A ) gen ∂c B ∂t =−∇⋅(c B v + J B )+(c ⋅ B ) gen (33) . . . The sum of the constituent molar generation terms is the total molar volumetric generation rate: (c ⋅ A ) gen +(c ⋅ B ) gen + . . . = c ⋅ gen (34) 2.6 Differential Energy Balance Energy may be transported by heat, by work, and by convection. Thus, the energy flux can be written as the sum of three terms: (35) where q is the heat flux, is the stress tensor (defined as the force per unit area), and mE ^ is the convective energy flux. E = q + ␴ ⋅ v + m E ^ ␴ Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. [...]... NJ: PTR Prentice-Hall, 199 2 8 L D Schmidt, The Engineering of Chemical Reactions New York: Oxford University Press, 199 8 9 F P Incropera and D P DeWitt, Fundamentals of Heat and Mass Transfer, 3rd ed New York: Wiley, 199 0 10 D R Lide (ed.), CRC Handbook of Chemistry and Physics, 80th ed Cleveland, OH: CRC Press, 199 9 11 I M Klotz and R M Rosenberg Chemical Thermodynamics: Basic Theory and Methods, 3rd... (82) 5.5 Fugacity and Activity The chemical potential is generally a function of temperature, pressure, and composition It is common practice to write µA = µ˚A(T) + RT ln aA (83) where µ˚A(T) is the standard chemical potential and aA is the activity of species A The activity is defined as the ratio of the fugacity ƒA to a standard-state fugacity ƒ˚A: aA = ƒA ƒ˚A (84) Activities and standard states are... is found to be a function of the Reynolds number Re and the Prandtl number Pr: Nu = ƒ(Re,Pr) (63) The equivalent relation for mass transfer is Sh = ƒ(Re,Sc) (64) where Sh is the Sherwood number and Sc is the Schmidt number (see Table 3) Many standard handbooks and heat transfer textbooks list dimensionless transport relationships for engineering systems (9, 10) 4 STOICHIOMETRY AND CHEMICAL KINETICS Stoichiometry... W/m2 K4); and ε is the emissivity, a material property Values of the emissivity for many different materials are tabulated in standard handbooks (9, 10) 3.4 Fluid Stress The stress tensor ␴ appears in both the differential energy balance and the differential momentum balance In general, stress is defined as force per unit area The force and the area can be represented as vectors having magnitude and direction:... expressed in the shorthand notation n 0 = ∑ νkIk k=1 The criterion for chemical reaction equilibrium is n ∆G = ∑ νkµk = 0 ( 89) k=1 Reaction equilibrium may also be expressed in terms of the standard Gibbs free-energy change: ∆G0 = RT ln Ka (90 ) Here, Ka is the equilibrium constant, defined by n Ka = ∏ akν k (91 ) k=1 6 ENGINEERING FLUID MECHANICS Fluid mechanics deals with the flow of liquids and gases For most... Refs 4 and 5) 3.2 Heat Conduction Heat conduction (also called heat diffusion) is the transport of thermal energy by random molecular motion Conduction is driven by a temperature gradient In most cases, the heat flux can be described by Fourier’s law: q = −k∇T (51) where k is the conductivity of the material Conductivity values may be found in many standard handbooks and heat transfer textbooks (9, 10)... Methods, 3rd ed Menlo Park, CA: W A Benjamin, 197 2 12 K Denbigh, The Principles of Chemical Equilibrium, 3rd ed Cambridge, U.K.: Cambridge University Press, 197 1 13 N De Nevers, Fluid Mechanics for Chemical Engineers, 2nd ed New York: McGraw-Hill, 199 1 14 Flow of Fluids Through Valves, Fittings, and Pipe, Crane Technical Paper 410 Chicago: The Crane Company, 198 8 Copyright 2002 by Marcel Dekker, Inc All... in species B, and so on The overall order of the reaction n is the sum of the individual reaction orders: n = α + β + (71) More information on the determination of rate constants and their applications can be found in standard texts on kinetics and reactor design (6–8) Copyright 2002 by Marcel Dekker, Inc All Rights Reserved 5 EQUILIBRIUM THERMODYNAMICS Thermodynamics is the study of the relationship... exerted on the surface of the differential control volume by the surrounding material Multiplying the stress ␴ by the material velocity v gives the rate of work done (per unit area) on the surface of the control volume: Rate of work by material stresses (per unit area) = σ ⋅ v Conservation of energy requires that the energy generation rate be zero: ⋅ e gen = 0 (conservation of energy) (36) The differential... momentum concentration is the product of density and velocity: [p] = ρv The momentum flux is the product of the momentum concentration and the velocity: P = ρvv The differential momentum balance is ∂ρv = −∇ ⋅ (ρvv) + f ∂t (44) where f is the net force (per unit volume) acting on the control volume In most fluid systems, f is the sum of a stress term and a body-force term: f = −∇ ⋅ σ + b (45) The most . 9 Macroscopic Balance Equations Paul K. Andersen and Sarah W. Harcum New Mexico State University, Las Cruces, New Mexico The prevention of waste and pollution requires an understanding of. flow rate of X across control surface i. The terms of Eq. (1) can be interpreted as follows: dX dt = rate of change of X inside the control volume ∑ (X CS,i ⋅ ) i = sum of flow rates of X across. concentration of X” and X is “the flux of X.” The terms of this equation can be interpreted as follows: ∂ ∂t [X] = rate of change of the concentration of X −∇ ⋅ (X) = net influx of X [X] gen