Course code: KKEK 3152 Modeling of Chemical Processes Equations
are required to design equipments
Top left: Distillation Column Top Right: Steam Boiler Bottom Left:
Fluidized Bed Reactor Bottom Right: Natural Gas Compressor Modeling
of Chemical Processes
CHAPTER ONE Introduction and modeling principles 1- Definition of
Model 2- Application of Process Models 3- Mathematical Models and
Terms Definition 4- A Systematic Approach for Developing Dynamic
Models 5- Classifying Mathematical Models 6- Types of Mathematical
Models 7- Process Modeling Fundamentals Introduction and modeling
principles
1- Definition of model: A model of a system is: a representation of
the essential aspects of the system in a suitable (mathematical)
form that can be experimentally verified in order to clarify
questions about the system 2- Application of Process Models:
Applications of models in engineering can be found in: Research and
Development. This type of model is used for the interpretation of
knowledge or measurements. An example is the description of
chemical reaction kinetics from a laboratory set-up. Process
Design. These types of model are frequently used to design and
build (pilot) plants and evaluate safety issues and economical
aspects. Planning and Scheduling. These models are often simple
static linear models in which the required plant capacity, product
type and quality are the independent model variables. Process
Optimization. These models are primarily static physical models
although for smaller process plants they could also be dynamic
models. Prediction and Control. Application of models for
prediction is useful when it is difficult to measure certain
product qualities, such as the properties of polymers, for example
the average molecular weight. Process models are also used in
process control applications, especially since the development of
model-based predictive control. These models are usually empirical
models, they can not be too complex due to the online application
of the models. 3- Mathematical models and terms definition
A mathematical model usually describes a system by a set of
variables and a set of equations that establish relationships
between the variables. The variables represent some properties of
the system, for example, measured system outputs often in the form
of signals, timing data, .etc. One or more assumptions are imposed
on the model. These assumptions limit the functionality of the
model. They are put to simplify the formulation and/or solution of
the model. State the modeling objectives and the end use of the
model. They determine the required levels of model detail and model
accuracy. 3-2 Terms Definition State Variables Input variables
Parameters
A state variable is a variable that arises naturally in the
accumulation term of dynamic material or energy balance.A state
variable is measurable ( at least conceptually) quantity that
indicate the stateof a system. For example, temperature is the
common state variable that arises from a dynamic energy balance.
Concentration is a state variable that arises when dynamic
component balances are written. Input variables An input variable
is a variable that normally must be specified before a problem can
be solved or a process can operated. Input variables typically
include: Flow rates of streams Compositions or temperatures of
streams entering a process. Input variables are often manipulated
(by process controllers) in order to achieve desired performance.
Parameters A parameter is typically a physical or chemical property
value that must be specified or know to mathematically solve a
problem.Examples include density, viscosity, thermal conductivity,
heat transfer coefficient, and mass-transfer coefficient. 4- A
Systematic Approach for Developing Dynamic Models
Draw a schematic diagram of the process and label all process
variables. List all of the assumptions that are involved in
developing the model. The model should be no more complicated than
necessary to meet the modeling objectives. Determine whether
spatial variations of process variables are important. If so, a
partial differential equation model will be required. Write
appropriate conservation equations (mass, component, energy, and so
forth). (continued) Introduce equilibrium relations and other
algebraic equations (from thermodynamics, transport phenomena,
chemical kinetics, equipment geometry, etc.). Perform a degrees of
freedom analysis to ensure that the model equations can be solved.
Simplify the model. It is often possible to arrange the equations
so that the dependent variables (outputs) appear on the left side
and the independent variables (inputs) appear on the right side.
This model form is convenient for computer simulation and
subsequent analysis. Classify inputs as disturbance variables or as
manipulated variables (for process control). 5- Classifying
mathematical models
5-1 Linear vs. nonlinear 5-2 Deterministic vs. probabilistic
(stochastic) 5-3 Static vs. dynamic 5-4 Lumped parameters vs.
distributed parameters 5-1 Linear versus nonlinear models
Mathematical models are usually composed by variables, which are
abstractions of quantities of interest in the described systems,
and operators that act on these variables, which can be algebraic
operators, functions, differential operators, etc. If all the
operators in a mathematical model present linearity, the resulting
mathematical model is defined as linear. If one or more of the
objective functions or constraints are represented with a nonlinear
equation, then the model is known as a nonlinear model. Linear
ordinary differential equations (ODE)
continue Linear ordinary differential equations (ODE) Nonlinear ODE
5-2 Deterministic versus probabilistic (stochastic)
A deterministic model is one in which every set of variable states
is uniquely determined by parameters in the model and by sets of
previous states of these variables. Therefore, deterministic models
perform the same way for a given set of initial conditions.
Conversely, in a stochastic model, randomness is present, and
variable states are not described by unique values, but rather by
probability distributions. Familiar examples of processes modeled
as stochastic time series include: stock market exchange rate
fluctuations, signals such as speech, audio and video, random
movement such as Brownian motion or random walks. 5-3 Static versus
dynamic models
A static (steady-state) model does not account for the element of
time, while a dynamic model does. Static model: Static models are
usually used for determining the final state of the system. Steady
state: No further changes in all variables No dependency in time:
No transient behavior Can be obtained by setting the time
derivative term zero Continue Dynamic model Describes time behavior
of a process due to changes in input, parameters, initial
condition, etc. Described by a set of differential equations (DE),
- ordinary (ODE), partial (PDE) Mostly used in safety, process
control and real time simulation 5-4 Lumped vs. distributed
parameters models
If the model is homogeneous (consistent state throughout the entire
system) the parameters are lumped. If the model is heterogeneous
(varying state within the system), then the parameters are
distributed. Distributed parameters are typically represented with
partial differential equations. When the spatial effects are of
less importance or do not vary considerably, a lumped parameters
model is used. On the other hand a distributed parameter model will
be used to account for these variations Case A. Continuous
Stirred-Tank Reactor
If the tank is well-mixed, the concentrations and density of the
tank contents are uniform throughout. This means that the outlet
stream properties are identical with the tank properties, in this
case concentration CA and density . The balance region can
therefore be taken around the whole tankas in fig below. The total
mass in the system is given by the product of the volume of the
tank contents V (m3) multiplied by the density (kg/m3), thus V
(kg). The mass of any component A in the tank is given in terms of
actual mass or number of moles by the product of volume V times the
concentration of A, CA (kg of A/m3 or kmol of A/m3), thus giving V
CA in kg or kmol. Case B. Tubular Reactor In the case of tubular
reactors, the concentrations of the products and reactants will
vary continuously along the length of the reactor, even when the
reactor is operating at steady state. This type of behavior can be
approximated by choosing the incremental volume of the balance
regions sufficiently small so that the concentration of any
component within the region can be assumed approximately uniform.
The basic concepts of the above lumped parameter and distributed
parameter
systems are shown in Fig. below. 6- Types of Mathametical
Models
Theoretical models (based on physicochemical law) Advantage provide
physical insight into process behavior applicable over wide ranges
of conditions Disadvantage expensive and time consuming to develop
complex processes typically include some model parameters which are
not readily available, such as reaction rate coefficients, physical
properties, or heat transfer coefficients. Empirical models
(obtained by fitting experimental data) Easer to develop than
theoretical models but they have a serious disadvantage which is
typically do not extrapolate well, i.e., should be used with
caution for operating conditions that were not included in the
experimental data used to fit the model. Semi-empirical models
(combined approach) can be extrapolated over a wide range of
operating conditions than empiricalmodels. require less development
effort than theoretical models. Therefore semi-empirical models are
widely used in industry. 7- Process Modeling Fundamentals
7-1 System States. 7-2 Mass Relationship for Liquid and Gas 7-3
Energy Relationship 7-4 Composition Relationship 7-1 System States
Conservation Laws
To describe a process system we need a set of variables that
characterize the system and a set of relationships that describe
how these variables interact and change with time. The variables
that characterize a state, such as concentration, temperature and
flow rate, are called state variables. They can be derived from the
conservation balances for mass, component, energy and momentum.
Open system mass balance
Component balance Energy balance : result of the first law of
thermodynamics. Momentum balance : result of general caseof Newtons
second law 7-2 Mass Relationship for Liquid and Gas
7-2-1 Mass balance This equation relates the rate of change in mass
m to the difference between inlet mass flow (Fm,in) and outlet mass
flow (Fm,out): For N inlet flows and M outlet flows: When
volumetric flow is used instead ofmass flow: If the density i and
volumetric flow Fv,i are measured variables The density in Eqn.
above is defined as the mass per unit volume at a certain pressure,
temperature and composition Accumulation term The rate of change in
the mass of a system can be described by: Liquid Accumulation If
there is only one inlet flow and one outlet flow, the accumulation
in a liquid vessel can be written as: If the temperature and
pressure effects can be neglected If the inlet and outlet density
are the same, the equation becomes: Gas Accumulation For an ideal
gas volume it holds that: in which: n number of moles, M molecular
weight (kg/mole), V volume (m3), P absolute pressure (N/m2), R gas
constant (N.m/mole.K), T absolute temperature (K) 7-2-2 Properties
of Liquid and Gas Mass Transfer
Characterization of mass transport Liquids and gases (generally
fluids) are not capable of passing on static pressures. If a fluid
is subject to shear stress as a result of flow, the shear stress
will lead to a continuous deformation. For gases and so called
Newtonian fluids at constant pressure and temperature, the
viscosity is independent of the shear stress: in which: shear
stress, N/m2 dynamic viscosity, kg/m.s dv/dy velocity gradient, s1
v flow velocity, m/s The flow pattern inside a body or along a body
with diameter d (for example a tube) depends on the flow velocity v
and can be characterized by the Reynolds number: density, kg/m3 d
characteristic flow dimension, m Resistance to flow Figure below
shows a pipe with a flow restriction in the form of an orifice. The
flow through the orifice is turbulent. The velocity will increase
from point A to point B. Ares is the area of the opening of the
restriction, and Ccor is a correction factor between 0 and 1,
depending on the type of opening. As can be seen from above, the
flow (F) can be determined by measuring the pressure drop and
taking the square root. 7-3 Energy Relationship (Energy
Balance)
Energy y Balance:Analogous to the mass balance with N inlet and M
outlet mass flows, the energy balance for a system can be described
as: E is the total energy, which is equal to the sum of
internalenergy U, kinetic energy KE and potential energy PE. E is
the total energy per unit mass The terms on the right-hand side of
the energy balance refer to entering convective energy flows, the
leaving convective energy flows, the net heat flux Q that enters
the system and the net amount of work W that acts upon the system
with: in which: WS applied mechanical work, J/s and WE expansion
energy, J/s. If the pressure is constant, we may write for the
expansion energy: Temperature dependency
In most thermal applications, the energy balance can be further
simplified: KE 0 because the flow velocities are often small, the
contribution of the kinetic energy can be ignored. PE 0 because
differences in height are often small, the contribution of the
potential energy can be ignored. d(P/)/dt 0 For many liquids,
because pressure differences are often small The result of these
simplifications is an enthalpy balance. This balance does not
account for mechanical changes but is valid for most thermal
systems: Temperature dependency The specific enthalpy H i of a
substance i depends on the temperature T with the specific heat
capacity cP: The absolute specific enthalpy at a certain
temperature is related to a reference temperature Tref according:
However, not the absolute enthalpy, but only the contribution of
the enthalpy flux is of interest in the energy balance.: Phase
dependency Example
If a liquid mass flow Fm of a component i with a constant specific
heat cP,iis heated up from a initial Tinit to an operating
temperature T, then the enthalpy flux can be written as: Phase
dependency If in the temperature trajectory a transfer of phase is
included, for instance, from liquid to vapor at boiling temperature
Tbp with a heat of vaporization Hi,vap , then the absolute specific
enthalpy becomes: Only the contribution of the enthalpy change is
of interest in the energy balance. If a liquid mass flow Fm of a
component i with a constant specific heat cP,i is heated up from
Tinit to the boiling point Tbp and evaporated, then the enthalpy
flux can be written as: 7-3 Energy Relationship (Thermal Transfer
Properties)
Convective heat transfer At an interface between gas and liquid or
solid or between liquid and solid, convective heat transfer can
take place when those media have a temperature difference. It can
be in the form of free convection, such as in the case of a central
heating radiator, or it can be forced convection, for example, an
air flow from a blower. The left situation will occur at gas-liquid
or liquid-liquid interfaces. Both media show a gradient. When in a
gas or liquid temperature differences exist, natural convection
flow will raise and eliminate these differences. The right
situation will occur at the interface between gas or liquid and a
conductive solid. The heat flow Q convection per unit area A at a
temperature difference T on the boundary layer can be given by: The
thermal resistance for heat transfer is 1/. Thermal conduction
Conduction takes place within stagnant gas or liquid layers
(layers, which are sufficiently thin that no convection as a result
of temperature gradients can occur), solids or on the boundary
between solids. The model describes the heat transport Q in terms
of the thermal conductivity and the temperature gradient dT/dx
Thermal radiation Heat transport can also take place from one
object to another object through radiation. The wavelength of
electromagnetic heat radiation is in the infrared range. An object
radiates energy proportional to the fourth power of its absolute
temperature: in which A is the surface area of the object and the
Stefan-Boltzmann constant (56.7 109 W/m2K4). 7-4 Composition
Relationship
7-4-1 Component Balance The component balance of component k for a
considered process system or phase can be described as a
concentration balance with concentration Ck [mole.m3] and
volumetric flows Fv: where i = 1, N is the number of inlet flows
and j = 1, M the number of outlet flows. As a partial mass balance
with mass fraction xk [kg.kg1] and mass flows Fm this equation can
be written as: 7-4-2 Component Equilibria
Liquidliquid or liquidgas equilibria The stationary distribution of
a component j between two phases can be described by a distribution
coefficient Kj which is a function of the temperature. xj is the
molar or weight fraction in one of the phases and yj the fraction
in the other. The line which describes the relationship between y
and x is called the equilibrium curve. This relationship is often
linear over a certain range. Vaporliquid equilibria at boiling
point The Antoine equation gives the vapor pressureof the pure
component j. A, B and C are constants. For a binary mixture, it is
sometimes permissible to assume constant relative volatility. When
the pressure P and the liquid compositions x1 and x2 or vapor
compositions y1 and y2 are known, the relative volatility is
defined as: 7-4-3 Component Transfer Properties
Interface component transfer When components are transferred across
a liquid-liquid or a liquid-gas interface A, in each phase, owing
to the resistance to component transfer, a concentration gradient
will occur. In the model that describes this interfacial transfer,
it is assumed, as shown in the figure, that the concentration
gradient restricts itself to a boundary layer. The flow J phase of
component j per unit area A at a composition difference on the
boundary layer can be given by: in which xj and yj are the bulk
concentrations of phase-1 and phase-2, xj interface and yj
interface are the concentrations at the interface. Component
diffusion Diffusion takes place within stagnant gas or liquid
layers (layers in which no convection occurs). The component flow
per unit area as a result of conduction is determined by the Ficks
law: The model describes the component transport J in terms of the
diffusivity D and the composition gradient dCj /dx of component j.
If no steady state is established, no constant concentration
gradient exists. Then the change of the concentration with time has
to be considered as represented by the equation: Reaction kinetics
Chemical reactions produce or consume components. The
transformation is expressed in moles, because of the stoichiometric
relationship between produced and consumed components. The reaction
rate rj for component j is defined as the moles of component j
produced or consumed per unit of time and per volume. This reaction
rate is positive when a component is produced and negative when it
is consumed. When component i is consumed or produced k times
faster than component j, then The reaction rate depends on the
temperature and component concentrations, according to: The order
of the reaction equals the sum of the coefficients (, ,). For a
first-order reaction the reaction rate equation becomes: Example of
a component system: A CSTR
In a stirred reactor as shown in Figure, component A is converted
to component B according the stochiometric relationship: The
reaction is exothermal and the components are dissolved. The
following assumptions are made: the reactor is well mixed and can
be considered as a lumped system in the operating range the density
and the heat capacities do not vary as a function of temperature
and composition Mass balance The general equation for the mass
balance is Since the reactor has one entering and one effluent
flow, the balance can be simplified to: and since the density is
constant (1) Component balances The general equation for the
component balance of component k is Since for the reaction it holds
that: the component equation for component A can be simplified to:
Homework Substitution of the mass balance derived previously Equ
(1):
gives after elimination of equal terms: A similar component
equation can be derived for component B: Homework Energy balance A
simple approach is to consider the specific enthalpy H not as a
function of the temperature and composition H(T,C)but only as a
function of temperature H(T). Then the reaction heat has to be
accounted for as a separate term. For the reactor with one entering
flow and one effluent flow, the basic Eqn. can be written as: which
can be modified to: (2) Rewriting of the left-hand side term gives:
Replacing of dV/dt (according to equ. 1) and remember that dH =cp
dT Therefore dH(T)/dt is equal to (3) substitution of equ 3 in equ
2 results in:
Elimination of equal terms gives: The terms cP Fv,in (Tin T ) and
rAB V HAB are contributions of enthalpy changes. The first indicate
the heating-up of the feed and the second the reaction heat by the
conversion, and for the cooling term (Qcool Example of hierarchy of
the model
As mentioned previously, the real purposes of the modeling effort,
the scope and depth of these decisions will determine the
complexity of the mathematical description of a process. To further
demonstrate the concept of model hierarchy and its importance in
analysis, let us consider a problem of heat removal from a bath of
hot solvent by immersing steel rods into the bath and allowing the
heat to dissipate from the hot solvent bath through the rod and
thence to the atmosphere . For this elementary problem, it is wise
to start with the simplest model first to get some feel about the
system response. Level 1 In this level, let us assume that:
(a) The rod temperature is uniform, that is, from the bath to the
atmosphere. (b) Ignore heat transfer at the two flat ends of the
rod. (c) Overall heat transfer coefficients are known and constant.
(d) No solvent evaporates from the solvent air interface. The many
assumptions listed above are necessary to simplify the analysis
(i.e., to make the model tractable). Let T0 and T1 be the
atmosphere and solvent temperatures, respectively. The steady-state
heat balance (i.e., no accumulation of heat by the rod) shows a
balance between heat collected in the bath and that dissipated by
the upper part of the rod to atmosphere where T is the temperature
of the rod, and L1 and L2 are lengths of rod exposed to solvent and
to atmosphere, respectively. Obviously, the volume elements are
finite (not differential), being composed of the volume above the
liquid of length L2 and the volume below of length L1. Solving for
Tfrom Eq. above yields where Equation above gives us a very quick
estimate of the rod temperature and how it varies with exposure
length. Level 2 Let us relax part of the assumption (a) of the
first model by assuming only that the rod temperament below the
solvent liquid surface is uniform at a value T1. This is a
reasonable proposition, since the liquid has a much higher thermal
conductivity than air. The remaining three assumptions of the level
1 model are retained. Next, choose an upward pointing coordinate x
with the origin at the solvent-air surface. The figure shows the
coordinate system and the elementary control volume. Applying a
heat balance around a thin shell segment with thicknessx gives
where the first and second terms represent heat conducted into and
out of the element and the last term represents heat loss to
atmosphere. We have decided, by writing this, that temperature
gradients are likely to exist in the part of the rod exposed to
air, but are unlikely to exist in the submerged part. Dividing
previousEq.by R2 x and taking the limit as x > 0 yields :
Substitution of the heat flux along the axis is related to the
temperature according to Fourier's law of heat conduction yields:
Equation above is a second order ordinary differential equation,
and to solve this, two conditions must be imposed. One condition
was stipulated earlier: The second condition (heat flux) can also
be specified at x = 0 or at the other end of the rod, i.e., x = L2.
Level 3 In this level of modeling, we relax the assumption (a) of
the first level by allowing for temperature gradients in the rod
for segments above and below the solvent-air interface. Let the
temperature below the solvent-air interface be T1 and that above
the interface be T11. Carrying out the one-dimensional heat
balances for the two segments of the rod, we obtain Level 4 Let us
investigate the fourth level of model where we include radial heat
conduction. This is important if the rod diameter is large relative
to length. Setting up the annular shell shown in Figure and
carrying a heat balance in the radial and axial directions that
leads to following equation: Here we have assumed that the
conductivity of the steel rod is isotropic and constant, that is,
the thermal conductivity k is uniform in both x and r directions,
and does not change with temperature.