Análisis del sector

The mole balance equation was developed in Chapter 3 and applied to a variety of ideal reactors. Energy changes were not considered, which led to some constraints on the sys-tems that were analyzed. For the batch reactor case, the contents of the reactor had a uni-form temperature, and this temperature did not change with time. For flow reactors, the temperature was constant with respect to both time and spatial position, and the inlet and outlet process stream temperatures were the same. With these imposed constraints, the mole balance equation, along with a rate expression, is sufficient to determine the conver-sion in a reactor. In the operation of most real reactors, regardless of whether they are labo-ratory units, pilot-scale, or full-size reactors, these constraints on the temperature do not apply. Thus, in a batch reactor, the temperature of the reaction mixture may change with time. In a flow reactor, the temperature may change with time and position, or the feed and effluent streams may have different temperatures. To account for these effects, reactor analysis must include the energy balance.

In this chapter, the energy balance equation is first developed in a general way—then it is applied to the three ideal reactor cases, namely the perfectly mixed batch reactor, the perfectly mixed CSTR, and the PFR. In the case of the CSTR, both transient and steady-state behavior is discussed. The reader may wish to study again the thermodynamics review presented in Chapter 2 prior to reading this chapter.

4.1 Influence of Temperature on Reactor Operation

The effects of temperature on the reaction rate were discussed in Chapters 1 and 2, and are briefly summarized here. Temperature changes influence the reaction rate, and hence the mode of reactor operation in two ways. The first is through the temperature dependence of the rate parameters; the second is through the temperature dependence of the equilib-rium constant.

4.1.1 Temperature Dependence of Reaction Rate Constants

The temperature dependence of the reaction rate constants was mentioned in Chapter 1.

The temperature dependence of each constant in the reaction rate expression has the gen-eral form (where N = 0 for the Arrhenius law)

k AT E

R T

N

g

= ⎛ −

⎝⎜

⎞

exp ⎠⎟ (4.1)

The exponential dependence of the rate constants on the temperature means that the reaction rate can change rapidly as the temperature changes. The larger the value of the

120 Introduction to Chemical Reactor Analysis

activation energy, the greater will be the change in reaction rate for a given change in temperature. This effect can cause operating problems. For example, consider the case of an exothermic chemical reaction. As the reaction proceeds, thermal energy is released. If this energy is not removed from the reactor, the process stream will increase in tempera-ture, causing the reaction rate to increase. Then, as the reaction rate increases, the rate of heat release also increases, which further increases the rate of reaction, and so on. In some cases, this effect may lead to reactor runaway and catastrophic failure, which should be avoided.

4.1.2 Effect of Temperature on Equilibrium Conversion

The equilibrium constant is also a function of temperature; hence, the equilibrium conver-sion will be affected by changes in temperature. The temperature dependence of the equi-librium constant is given by the van’t Hoff equation (see Section 2.3.3):

∂

∂ ln K

T

H R T

R g

( )

₌ ^Δ

2 (4.2)

For an exothermic reaction, an increase in temperature decreases the equilibrium yield, while for an endothermic reaction an increase in temperature increases the equilibrium yield, everything else being equal.

4.1.3 Energy Balance for Ideal Reactors

The mole and energy balance equations are a coupled set of equations that together describe the performance of an ideal reactor. Typically, the energy balance is expressed in terms of the reactor temperature, which may vary with space and time. The nature of the variations depends on the reactor type and mode of operation. In perfectly mixed batch reactors, the temperature is the same at all locations in the reactor, but changes with time as the reaction proceeds. Sometimes the reactor is operated with heat transfer through the reactor walls, or via heating or cooling coils inserted into the reactor. Adiabatic operation is also possible. In a plug flow reactor, the temperature varies with axial position (i.e., in the direction of flow) in the reactor. There may or may not be heat transfer through the reactor walls. In a CSTR, the temperature is uniform everywhere in the reactor and is equal to the outlet temperature. However, the inlet and outlet temperatures are different. There may or may not be heat transfer with the surroundings, either through the vessel walls or via heating or cooling coils within the reactor.

In the following sections, the general energy balance equation is introduced. This intro-duction is followed by a detailed development of the energy balance equation for the three ideal reactors (batch, PFR, and CSTR).

4.2 General Energy Balance

We start the derivation of the energy balance equation by considering the energy changes experienced by a general open system. The energy of the system may change with time as

121 Energy Balances in Ideal Reactors

a result of heat exchange between the system and the surroundings, work done on the system by the surroundings, or by accumulation of mass in the system. Energy crosses the system boundary as mass flows into or out of the system. Energy changes also occur as chemical reactions proceed. The general energy balance equation can thus be written in words as

The reader may recall that in books on heat transfer the general energy balance equation includes a term for thermal energy generation. Such a term does not appear in the above energy balance because this equation represents a total energy balance. Although a chemi-cal reaction may generate or absorb thermal energy, this energy change is balanced by an equal change in energy owing to a compositional change. Strictly speaking, energy is never generated in any system unless there are nuclear reactions where matter is trans-formed into energy.

The energy of a system or a flowing stream consists of potential energy (which depends on the system height relative to a reference point, z), kinetic energy (which depends on the fluid velocity, v), and the internal energy, U (which is the sum of the molecular energies).

The total energy per unit mass is the sum of these three components, and is written as

E U v gz

= + +

2 2

2 (4.3)

In most chemical reactors, changes in kinetic and potential energy are considered to be neg-ligible and are ignored. This is usually a good approximation because the velocities of most process streams are relatively small, and the differences in elevation are minor. The largest energy effect is that owing to chemical reaction. The following approximation is thus made:

E⊕U (4.4)

The heat transfer between the system and the surroundings is denoted as q, and has the units of watts. It has a positive value if heat is transferred to the system from the surround-ings. The work done by the system is given the symbol W and also has units of watts. It has a positive value if the work is done by the system on the surroundings. For a system with multiple species crossing the system boundary with molar flow rates of F_j, the energy bal-ance may be written as

d

Each of the terms in Equation 4.5 has units of power, joules/s (J/s) or watts (W). The work done on or by the system, W, may be divided into flow work, W_f, expansion work (also called pressure/volume work), W_E, and shaft work, W_S.

122 Introduction to Chemical Reactor Analysis

Flow work is the work that a fluid stream does on the system or the surroundings by entering or leaving the system. The flow work depends on the volumetric flow rate, Q, of the stream crossing the system boundary and the pressure of the system, as follows:

Wf = −PQ (4.7)

The volumetric flow rate can be expressed in terms of the molar flow rate by introducing the molar concentration of each species, C_j:

W P F

The volume, V_j, in Equation 4.8 is the molar volume of component j in the process stream.

The shaft work, denoted W_S, is the work done on or by the system by a mechanical device such as a mixer. The expansion work is the work done on or by the system as a result of volume changes of the system. For a constant-volume system, it is equal to zero: substitu-tion of Equasubstitu-tion 4.8 into Equasubstitu-tion 4.5 and rearrangement thus give

d

We now introduce the enthalpy, H. The enthalpy is a defined thermodynamic quantity and is related to the internal energy by the following equation:

H =U+PV (4.10)

Equation 4.10 can be substituted into Equation 4.9 to obtain d energy balance to the batch reactor, the PFR, and the CSTR.

4.3 Batch Reactor

It was seen in Chapter 3 that the batch reactor can be operated in either constant- or variable-volume mode. The form of the energy balance equation for the batch reactor depends on which of these two modes is used. Most batch reactors are operated at constant volume, and this case will be treated first and in more detail, followed by the variable-volume case.

123 Energy Balances in Ideal Reactors

4.3.1 Constant-Volume Batch Reactor

The CVBR has no inlet or outlet streams; thus, it is a closed system. The energy terms in Equation 4.11 for flow streams are thus dropped. The expansion work in a constant-volume system is zero. The internal energy change of the system is the sum of the changes of all of the species in the system. For this analysis, the energy input from work done by mixing devices (shaft work) is ignored. The energy balance for a CVBR becomes

d

The internal energy change of the reactor contents is thus equal to the rate of heat trans-fer with the surroundings. Equation 4.12 can be expanded using the chain rule:

N U

It is not convenient to work directly in terms of internal energy changes: temperature is a much more useful variable. The internal energy change of a species can be expressed in terms of the temperature and the constant-volume heat capacity. For any species j, this substitution gives

Therefore, the first term on the left-hand side of Equation 4.13 becomes

N U

Consider a reaction given by the following overall expression:

aA+bB→cC+dD (4.16)

The reaction can be expressed on the basis of 1 mol of A as A+ bB→ C+ D

Let I denote any inert substances present in the system. For the reaction of Equation 4.16, the right-hand side of Equation 4.15 is equal to

N C T

124 Introduction to Chemical Reactor Analysis

Equation 4.19 contains terms for the reaction rates of each species in the reaction. Each rate can be expressed in terms of the rate of change of the number of moles of A. From the stoichiometry, the relationship between the rates of reaction of the different reactants and products is

q F C+ T⁰ P⁰

(

T⁰−TE

)

⁻

(

F^A0− F^AE

)

^ΔHR^{,A = 0}

Substitution of these terms into Equation 4.19 and simplification gives

U N

The difference between the internal energies of the products and the reactants, multi-plied by the respective stoichiometric coefficients, equals the internal energy change of reaction, which is

In Equation 4.21, ∆UR,A is the internal energy change owing to reaction per mole of A.

The energy balance equation for a CVBR can therefore be written as

N C T expressed in terms of this reference component. Equation 4.22 contains two time deriva-tive terms. The derivaderiva-tive of the moles of A with respect to time can be eliminated by incorporating the mole balance. The general mole balance for a batch reactor is

−_V^{1 d}_d^N_t^A = −

(

^rA

)

^(4.23)

Substituting Equation 4.23 into Equation 4.22 gives the energy balance as

N C T

125 Energy Balances in Ideal Reactors

Equation 4.24 can be rearranged to give an explicit expression in temperature; thus d

The mole balance and energy balance equations between them describe the behavior of a nonisothermal CVBR. The two equations are coupled because the energy balance includes concentration terms and the mole balance includes the temperature. Owing to the exponential temperature dependence of the rate constants, a numerical solution of the two equations is required. In the general case, both C_V and (∆UR,A) depend on the temperature, giving very nonlinear equations. This nonlinearity can lead to difficulties in the numerical solution.

4.3.2 Relationship between ∆UR and ∆HR

The energy balance in the CVBR includes the internal energy of reaction, ∆UR. It is com-mon to express the energy change on reaction in terms of the enthalpy of reaction, ∆HR

(see Chapter 2), and this latter quantity is the one usually encountered in reference texts.

The relationship between ∆UR and ∆HR is given by the following equation:

ΔU ΔH P ΔU

In Equation 4.26, νj is the stoichiometric coefficient, which is negative for a reactant and positive for a product, and V_j is the molar volume of component j. For most liquid reac-tions, the following is usually a good approximation:

νj j

In a gas-phase reacting system, we consider two cases. In the first case, there is no change in moles on reaction, and it follows that

νj ν

and therefore ∆HR= ∆UR. Furthermore, the change in heat capacity is the same for both constant-pressure and constant-volume heat capacity, that is

ΔCP = ΔCV (4.28)

Therefore, either C_P or C_V values may be used in calculating energy changes. In the second case, there is a change in moles on reaction. Thus, Σ νj = ΔN and therefore ΔHR ≠ ΔUR.

126 Introduction to Chemical Reactor Analysis

To determine the difference between ∆HR and ∆UR, consider the case of an ideal gas mix-ture. For an ideal gas, the internal energy depends only on the temperature, and the molar volume is given by the ideal gas law; therefore

∂

∂ ΔU

V V R T

P

R T

j g

⎛

⎝⎜

⎞

⎠⎟ = 0 and = It follows that

νj ν

j n

j g

j j

n

V R T

= P =

∑

₁ ⁼

∑

₁ ^(4.29)

The sum of the stoichiometric coefficients equals the change in moles on reaction, that is, Σ νj= ΔN, where ∆N is the change in moles on reaction when νA is set equal to 1. Therefore

ΔU ΔH P ΔNR T

R R Pg

,A = ,A − ⎛

⎝⎜

⎞

⎠⎟ (4.30)

Simplifying gives the result

ΔUR,A = ΔHR,A −ΔNR Tg (4.31) In this scenario, the value of ∆UR,A is different from the value of ∆HR,A, and error would result if they were assumed to be the same. The magnitude of the difference that can arise between the two is illustrated in Example 4.1.

Example 4.1

The gas-phase hydrogenation of benzene produces cyclohexane. The overall reaction is

C H6 6+3H2→ C H6 12 (4.32)

Compute the enthalpy and internal energy changes of this reaction over the tem-perature range 300–800 K. Assume that the species behave as ideal gases. The following data are available. The temperature dependence of the heat capacity is

CP = +a bT+cT²+dT³J mol K⋅ (4.33) The values of the constants in Equation 4.33 are

Compound a b ∙ 10² c ∙ 10⁵ d ∙ 10⁹

C6H6 −33.92 47.39 −30.17 71.30

H2 27.14 0.9274 −1.381 7.645

C6H12 −54.54 61.13 −25.23 13.21

127 Energy Balances in Ideal Reactors

The enthalpies of formation of benzene and cyclohexane at 298.15 K are

ΔHf,298,_C6H6 =8 298 10. × 4J mol (4.34)

and

ΔHf,298,_C6H12 = −1 232 10. × 5J mol (4.35)

SOLUTION

We start the solution by computing an expression for the enthalpy of reaction at 298.15 K. The enthalpy of reaction as a function of temperature can be computed from the enthalpies of formation and the heat capacity data. The standard enthalpy of forma-tion of hydrogen is zero by definiforma-tion, and therefore, the enthalpy change of reacforma-tion at 298.15 K is

ΔHR,298,_C6H6 = −1 232 10. × 5 −8 298 10. × 4 = −2 062 10. × 5J mol (4.36)

Using this value for the enthalpy of reaction and the heat capacity data, a temperature-dependent expression is developed for the enthalpy of reaction (see Example 2.6 for the methodology):

ΔHR,_C6H6 = −1 813 10. × ⁵ −102 0. T+0 0548. T²+3 024 10. × ⁻⁵T³ −2 026 10. × ⁻⁸⁸T⁴J mol (4.37) The relationship between the internal energy change of reaction and the enthalpy change of reaction for an ideal gas is given by Equation 4.31:

ΔUR,C6H6 =ΔHR,C6H6 −ΔNR Tg (4.38)

For this hydrogenation reaction, there are 4 mol of reactants consumed for every mole of products that is produced. Therefore, it follows that

ΔN = −1 4= −3 (4.39)

Equation 4.38 is written as

ΔUR,_C6H6 =ΔHR,_C6H6 + ×3 8 314. ×T (4.40)

Combining Equations 4.37 and 4.40 gives

ΔUR,_C6H6 = −1 813 10. × ⁵ −77 058. T+0 0548. T²+3 024 10. × ⁻⁵T³ −2 026 10. × ⁻⁻⁸T⁴J mol (4.41) The following table gives some values for the two energy changes at various tempera-tures, computed from Equations 4.37 and 4.41:

128 Introduction to Chemical Reactor Analysis

Note that the percent difference between the two energy values is about 3.8% at 300 K, and rises to about 9.9% at 800 K.

4.3.2.1  Assuming a Constant Value for Heat Capacity

It was seen in Chapter 2 that the heat capacity is usually a function of temperature. This temperature dependence must be included in the solution of the energy balance equation to obtain an exact solution. Furthermore, as the composition of the reaction mixture changes with time, the heat capacity of the mixture will usually change, even if the tem-perature is held constant. It is, however, not uncommon to assume that the heat capacity is either independent of the temperature or composition, or both. The energy balance is then expressed in terms of an average heat capacity value.

When the heat capacity is assumed to be independent of composition, the heat capacity is often based on the mass of the system. With this assumption, the temperature change term in Equation 4.24 becomes

N U

t m C T

j j t

j n

d t V

d

d

= d

∑

=₁₀ ^(4.42)

The units of CV are J/kg ⋅ K. Now, substitute Equation 4.42 into Equation 4.24:

q m C T

t U r V

t V R

= ^d_d +

(

^{Δ ,A}

) (

^−A

)

^(4.43)

Note that the use of an average heat capacity based on the mass of the system does not preclude the temperature dependence of the heat capacity from being used. Writing Equation 4.43 explicitly in terms of the temperature change of the reactor as a function of time gives

d d

A A

T t

q m C

U r V

m C

t V

R t V

= −

(

^{Δ ,}

) (

⁻

)

_(4.44)

The mole balance equation for a CVBR may be written in terms of concentration:

d d

A A

C

t = − −

(

r

)

^(4.45)

T (K) �HR,C H6 6(J/mol) �UR,C H6 6(J/mol) 300 - 2.063 × 10⁵ - 1.988 × 10⁵ 400 - 2.119 × 10⁵ - 2.019 × 10⁵ 500 - 2.161 × 10⁵ - 2.036 × 10⁵ 600 - 2.189 × 10⁵ - 2.039 × 10⁵ 700 - 2.203 × 10⁵ - 2.029 × 10⁵ 800 - 2.206 × 10⁵ - 2.007 × 10⁵

129 Energy Balances in Ideal Reactors

Equations 4.44 and 4.45 are a system of two coupled ordinary differential equations and are initial value problems. The simultaneous numerical solution of these two equations yields the temperature and concentration in the reactor as a function of time.

4.3.3 External Heat Transfer in CVBR

The heat transfer term, q, may have various forms depending on the reactor and surround-ings. The reactor may be heated by a heater that supplies a constant heat flux, in which case the value of q would be a constant. If there is no heat transfer with the surroundings, the reactor is adiabatic and q has a value of zero. Alternatively, q might be varied continuously to achieve a specified heating or cooling rate in the reactor.

A common method used to provide heat exchange with a reactor is to place heating or cooling coils inside the reactor, or to place a jacket containing a heat transfer fluid around the reactor surface. In this case, the rate of heat transfer is governed by the temperature difference between the reactor and the heat transfer fluid, and the value of the overall heat transfer coefficient, U. The heat transfer equation is written as

q=UA T

(

^∞−T

)

^(4.46)

In Equation 4.46, A is the heat transfer area and T_∞ is the temperature of the heat transfer fluid. If the temperature of the heat transfer fluid is held constant during the course of reaction, and the reacting fluid temperature changes, then the rate of heat transfer is a function of time. Refer to Case Study 4.1 in Section 4.3 for an example of a batch reactor with external heat transfer.

4.3.4 Adiabatic Temperature Change for CVBR

When the reactor is adiabatic, Equation 4.22 can be written as

N C T

Eliminating the time derivative term in Equation 4.47 and rearranging give

d

Provided that ∆UR and C_V are known as a function of T, Equation 4.48 can be integrated to give T as a function of N_A. If N_A is assumed to be constant, and furthermore we assume a constant average C_V, the integration is straightforward and we obtain the result

T T U

m C^R N N

t V

− ⁰ = ^{Δ ,A}

(

^A − ^A0

)

^(4.49)

130 Introduction to Chemical Reactor Analysis

Equation 4.49 can be written using fractional conversion, with an initial conversion of zero:

T T U

m C^R N X

t V

− = −

0 Δ ,A 0

A A (4.50)

Equation 4.50 can be used to check the maximum or minimum reactor temperature that can be attained with adiabatic operation by setting the fractional conversion equal to one.

Equation 4.50 can be used to check the maximum or minimum reactor temperature that can be attained with adiabatic operation by setting the fractional conversion equal to one.

In document ESTUDIO DE PREFACTIBILIDAD PARA LA INSTALACIÓN DE UNA PLANTA DE PRODUCCIÓN DE MORINGA OLEIFERA EN POLVO ENRIQUECIDA CON CAMU CAMU PARA EL MERCADO LIMEÑO (página 36-41)