Examples of reversible reactions are
A Sd B (1–29) A 1 B Sd C 1 D (1–30) As will be discussed subsequently, for both irreversible and reversible reactions, the rate of reaction will be an important consideration in the design of the treatment facilities in which these reactions will be carried out. Special attention must be given to the design of mixing facilities, especially for reactions that are rapid.
Heterogeneous Reactions. Heterogeneous reactions occur between one or more constituents that can be identified with specific sites, such as those on an ion-exchange resin in which one or more ions is replaced by another ion. Reactions that require the pres- ence of a solid-phase catalyst are also classified as heterogeneous. Heterogeneous reactions are usually carried out in packed and fluidized bed reactors [see Fig. 1–4(f ), (g), and (h)].
These reactions are more difficult to study because a number of interrelated steps may be involved. The typical sequence of these steps, as quoted from Smith (1981) is as follows:
1. Transport of reactants from the bulk fluid to the fluid-solid interface (external sur- face of catalyst particle)
2. Intraparticle transport of reactants into the catalyst particle (if it is porous) 3. Adsorption of reactants at interior sites of the catalyst particle
4. Chemical reaction of adsorbed reactants to adsorbed products (surface reaction) 5. Desorption of adsorbed products
6. Transport of products from the interior sites to the outer surface of the catalyst particle
Rate of Reaction
The rate of reaction is the term used to describe the change (decrease or increase) in the number of moles of a reactive substance per unit volume per unit time (for homogeneous reactions), or per unit surface area or mass per unit time (for heterogeneous reactions) (Denbigh and Turner, 1984).
For homogeneous reactions, the rate of reaction r is given by r51
V d[N]
dt 5 moles
(volume)(time) (1–31)
If N is replaced by the term VC, where V is the volume and C is the concentration, Eq. (1–31) becomes
r51 V
d(VC) dt 51
V
VdC1CdV
dt (1–32) If the volume remains constant (i.e., isothermal conditions, no evaporation), Eq. (1–32) reduces to
r5 6dC
dt (1–33)
where the plus sign indicates an increase or accumulation of the substance, and the minus sign indicates a decrease of the substance.
For heterogeneous reactions where S is the surface area, the corresponding expression is r51
S d[N]
dt 5 moles
(area)(time) (1–34)
For reactions involving two or more reactants with unequal stoichiometric coefficients, the rate expressed in terms of one reactant will not be the same as the rate for the other reac- tants. For example, for the reaction
aA 1 bB S cC 1 dD (1–35) the concentration changes for the various reactants are given by
21 a
d[A]
dt 5 21 b
d[B]
dt 51 c
d[C]
dt 51 d
d[D]
dt (1–36) Thus, for reactions in which the stoichiometric coefficients are not equal, the rate of reac- tion is given by
r51 ci
d[Ci]
dt (1–37)
where the coefficient term (1/ci) is negative for reactants and positive for products.
The rate at which a reaction proceeds is an important consideration in wastewater treatment. For example, in some cases the operative reaction may take too long to go to completion. In such cases, treatment processes are designed on the basis of the rate at which the reaction proceeds rather than the equilibrium position of the reaction. Often, quantities of chemicals in excess of the stoichiometric, or exact reacting amount, may be used to accomplish the treatment step in a shorter period of time by driving the reaction to completion.
Specific Reaction Rate
From the law of mass action it can be shown that the rate of reaction for a given reaction is proportional to the remaining concentration of the reactants. Thus, for a reaction involv- ing a single component A, the rate of reaction is given by
r56 kCA (1–38) Where k is a constant of proportionality formally defined as the specific reaction rate (also known as the reaction-rate constant, velocity constant, and the rate coefficient). The spe- cific reaction rate has the units of the specific reaction and concentration. For Eq. (1–38), the units of the specific reaction-rate constant are
k5 r C 5 1
V dN
dt 1
C5 mole
L?s(mole/L)5 1
s (1–39) In application, the rate of reaction, r, takes into account the effects of concentration, and the specific reaction-rate constant, k, takes into account the effects of all the other variables that may affect the reaction. Of the many variables in any given situation, temperature is usually the most important.
Effects of Temperature on Reaction Rate Coefficients
The temperature dependence of the specific reaction rate constants is important because of the need to adjust for other temperatures. The temperature dependence of the rate constant is given by the van’t Hoff-Arrhenius relationship.
d(ln k) dT 5 E
R2 (1–40)
1–10 Introduction to Process Kinetics
31
EXAMPLE 1–1
Solution
Comment
where k 5 reaction rate constant at temperature T T 5 temperature, K 5 273.15 1 °C
E 5 activation energy (a characteristic value for a reaction (e.g., J/mole) R 5 ideal gas constant, 8.314 J/mole?K (1.99 cal/mole?K)
Integration of Eq. (1–40) between the limits T1 and T2 gives lnk2
k1
5 E(T22T1) RT1T2
5 E RT1T2
(T22T1) (1–41) With k1 known for a given temperature and with E known, k2 can be calculated.
Activation Energy. The activation energy, E, can be calculated using Eq. (1–41) by determining the k at two different temperatures as illustrated in Example 1–1. Common values of E for wastewater treatment processes are in the range of 8400 to 84,000 J/mole (2000 to 20,000 cal/mole).
Determination of Activation Energy For a given chemical reaction it has been observed that the rate of reaction doubles for each 10°C increase in temperature. If the initial temperature was 10°C, estimate the activation energy for the reaction.
1. Solve Eq. (1–41) for the activation energy. The required equation is E5 R ln(k2/k1)
(1/T1 2 1/T2)
2. Substitute known values and solve for E:
T15 (273 1 10°C) 5 283 K T25 (273 1 20°C) 5 293 K k25 2k1
R 5 8.314 J/mole?K
E5 (8.314 J/mole?K)(ln 2k1/k1)
(1/283K 2 1/293K) 5 48,024 J/mole
Although used as a constant, the value of the activation energy, E, will vary somewhat with temperature according to the above equation. However, the temperature range in which wastewater treatment process operate is relatively limited. There is much greater variabil- ity in the measured reaction rates.
Temperature Coefficient, U. Because most wastewater treatment processes are carried out over a relatively narrow temperature range, the term, E/RT1T2, in Eq. (1–41) may be assumed to be constant for all practical purposes. If the term E/RT1T2 is designated by C, then Eq. (1–41) can be written as
lnk2
k1
5C(T22T1) (1–42)
k2
k1
5eC(T22T1) (1–43)
Replacing the term, eC, in Eq. (1–43) with a temperature coefficient, u, yields the following expression:
k2
k15u(T22T1) (1–44) Equation (1–44) is used commonly in the sanitary engineering field to adjust the value of the operative rate constant to reflect the effect of temperature. It should be noted, however, that although the value of u is assumed to be constant, it can often vary considerably with temperature. Therefore, caution must be used in selecting appropriate values for u for dif- ferent temperature ranges. Typical values for various processes for different temperature ranges are given, where available, in the sections in which the individual topics are dis- cussed. Values for u for some biological treatment systems vary from about 1.020 to 1.10.
Reaction Order
The rate at which reactions occur is determined usually by measuring the concentration of either a reactant or product as the reaction proceeds to completion. The measured results are then compared to the corresponding results obtained from various standard rate equa- tions by which the reaction under study is expected to proceed.
The order of a reaction with respect to a specified compound is equal to the stoichio- metric coefficient for that compound. For example, in the following reaction, the reaction order for compound A is a, compound B is b, and so on.
aA 1 bB 1 . . . S pP 1 qQ1 . . . (1–45)
If the rate is experimentally found to be proportional to the first power of the concentration of A (i.e., a 5 1), then the reaction is said to be first order with respect to A.
When the mechanism of reaction is not known, the reaction rate for Eq. (1–45) may be approximated with the following expression:
r5kCaACbBCcC pCpP5kCnA (1–46) where a and b are the reaction orders with respect to reactants A and B, and n is the overall reaction order (n 5 a 1 b 1 . . . p). The sum of the exponents to which the concentration(s) are raised is known as the order of the reaction. Several reaction rate expressions with dif- ferent reaction orders are as follows.
r 56k (Zero order) (1–47)
r 56kC (First order) (1–48)
r 56k(C 2 Cs) (First order) (1–49)
r 56kC 2 (Second order) (1–50)
r 56kCACB (Second order) (1–51)
r5 6 kC
K1C (Saturation or mixed order) (1–52)
r5 6 kC
(11rtt)n (First order retarded) (1–53)
1–10 Introduction to Process Kinetics
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The application of various rate expressions in wastewater treatment are described in the following discussion.
Rate Expressions Used in Wastewater Treatment
The physical, chemical, and biological processes that are applied in wastewater treatment for the conversion or separation of constituents are numerous and varied. Important con- stituent treatment processes, along with the constituents affected, are reported in Table 1–10.
The various processes listed in Table 1–10 will be referred to throughout this text.
Table 1–10