Page 131 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

P1: ZCK/FLS P2: FYK/FLS QC: FYK Revised Pages
Encyclopedia of Physical Science and Technology EN002G-100 May 19, 2001 18:49

Chemical Process Design, Simulation, Optimization, and Operation 753

assumes that a variable of interest (temperature, for ex- f (x) = 0
ample) changes with only one independent variable (time,
which can be solved using numerical techniques, as shown
but not space, for a dynamic systems example). A typical
in Section III.
lumped parameter system is a perfectly mixed (stirred)
tank, where the temperature throughout the tank is uni-
form (there is no spatial gradient). A distributed param- B. Distributed Parameter Models
eter system has more than one independent variable; for
Consider a tubular reactor where a chemical reaction
example, temperature may vary with both spatial position
changes the concentration of the fluid as it moves down the
and time.
tube. Assuming first-order chemical reaction, isothermal
reactor, and constant density, the modeling equation is
A. Lumped Parameter Models
2
∂C A ∂C A ∂ C A
Consider the dynamic behavior of a process that can be =−v z + D Az 2 − kC A (3)
∂t ∂z ∂z
considered perfectly mixed. The lumped parameter model
where z is the spatial coordinate, C A is the concentration
has the following form:
of component A, D Az is the diffusion coefficient, k is
dx the reaction rate constant, v z is the velocity, and t is
xY= = f (x, p, u) (1)
dt time. Since this is a second-order partial differential
where x = states, p = parameters, u = inputs, t = time. equation, the initial condition (C A as a function of z) and
To illustrate the basic principles, consider a perfectly two boundary conditions must be specified. Notice that,
mixed, isothermal chemical reactor, with a series reaction at steady state, this results in a second-order ordinary
of the form: differential equation:
2
A B C d C A dC A
D Az 2 − v z − kC A = 0 (4)
dz dz
Assuming constant density and volume, the following
modeling equations can be written, where it is assumed which can be solved, given the boundary conditions.
that each reaction is a first-order decomposition:
F
dC A
= (C Ain − C A ) − k 1 C A III. PROCESS SIMULATION
dt V
F
dC B Process simulation refers to the numerical solution of pro-
= (C Bin − C B ) + k 1 C A − k 2 C B (2)
dt V cess models. At the process design stage, process simu-
F lation often refers to the numerical solution of an entire
dC C
= (C Cin − C C ) + k 2 C B
dt V process flowsheet.
where F = volumetric flowrate; V = reactor volume; C A ,
C B , C C = concentrations of components A, B, and C, A. Algebraic Equations
respectively; k 1 , k 2 = reaction rate constants; and C Ain ,
The steady-state behavior of lumped parameter systems is
C Bin , C Cin = feedstream concentrations of components
characterized by a set of algebraic equations that have the
A, B, and C, respectively.
form:
In terms of the state variable notation of Eq. (1), there
are three states, three parameters, and four inputs (often f (x) = 0 (5)
the feedstream compositions of components B and C will
be zero): obtained from Eq. (1) with a fixed p and u.
The most commonly used numerical techniques are re-
C C ] T
x = [C A C B lated to Newton–Raphson iteration. The “guess” for it-
V ] T eration k + 1 is determined from the value at iteration k,
p = [k 1 k 2
using:
u = [F C Ain C Bin C Cin ] T −1
x(k + 1) = x(k) − J(k) f (x(k)) (6)
Notice that steady-state models can be obtained by set-
where f (x(k)) is the vector of function evaluations at it-
ting the derivative (accumulation) terms to zero. In this
eration k, and J(k) is the Jacobian matrix:
case, the ordinary differential equations become algebraic
equations. For a fixed set of values of the parameters and ∂ f i
J ij (k) = (k) (7)
inputs, the algebraic equations have the form: ∂x j