Chapter 2: Theoretical Models of Chemical Processes

Introduction to Process Control Romagnoli & Palazoglu

Chapter 2: Theoretical Models of Chemical Processes (cont’d)

Inputs ),( uxfdtdx

=

The knowledge of the process is contained in the model of

the process.

Introduction

Principles of modeling:

Basis AssumptionsMathematical consistencySolution of the modelVerification


A model will help us create an abstraction of the process and, in turn, will be the source of additional insight into its behavior.

Basis

The basis for mathematical models is the fundamental physical and chemical laws, such as:

Law of conservation of massLaw of conservation of energyLaw of conservation of momentum.


Assumptions

Probably the most vital role that the engineer plays in modeling is in exercising his/her judgment as to what valid assumptions can be made.

“An engineering compromise between a rigorous description and

getting an answer that is good enough is always required.”


Consistency

Once all the equations of the mathematical model have been written, we need to make sure that the number of variables equals the number of equations.

degrees of freedom of the system must be zero in order to

obtain a solution.


Solution

The available solution techniques andtools must be kept in mind as themathematical model is developed.

“An equation without a way to solve it is not worth much.”


Verification

An important but often neglected part of developing a mathematical model is proving that the model describes the actual process behavior.

“Model validation is performed using plant

data..”


Analytical Models

To characterize a processing system and its behavior, we need:

A set of fundamental dependent quantities whose values describe the natural state of the system State Variables

A set of equations which will describe how the natural state of the given system changes over time State Equations


Analytical Models

The state equations are derived from application of the balance equations:

TOTAL MASS BALANCECOMPONENT MASS BALANCEENERGY BALANCE MOMENTUM BALANCE

The balance equation for a quantity S states that:

ACCUMULATION = IN - OUT + AMOUNT OF S GENERATED

- AMOUNT OF S CONSUMED


Additional Elements

In addition to the state variables, there are other type of variables that can be classified into two groups:

Parameters: For example cp, ρ, (- ΔH), α, etc.Input Variables: Manipulated and disturbance variables

To solve the system of equations, we must specify both the values of the parameters as well as the values of the input variables.


Additional Elements

In the same way, in addition to balance equations, we need other relationships such as:

Transport equations: Transport of energy, mass and momentum. For example,

Kinetic rate equations: For example,

Reaction and phase equilibrium relationships: For example, phase equilibrium, chemical equilibrium, etc.

Equations of state: Relationships between the intensive variables. For example, ideal gas law, Van der Waals equation, etc.


ARTE

oA cekr /−=

)( TTUAQ s −=

Example - Tank Level

Fin

FV=A h

A : Cross sectional areah: Height of the liquid levelV : Volume of the tank


The fundamental quantity that provides theinformation about the dynamics of the tank is:

The total mass of the liquid in the tank

ρ : density of the liquidV: volume of the liquidA: cross sectional area of the tankh: height of the liquid level

AhVMassTotal ρρ == AhVMassTotal ρρ ==



State Variable: hConstant parameters: ρ and A

Note: It is assumed that the density ρ isconstant and independent of temperature.Conservation principle on the fundamentalquantity: the total mass

[ ] [ ] [ ]timetimetime

mass total of Outputmass total of Inputmass total of onAccumulati−=



( ) FFdtAhd

i ρρρ−=

Fi : inlet volumetric flow rate (m3/min)

F : outlet volumetric flow rate (m3/min)

FFdtdhA i −=

ρ: constant

Additional Equation:

hF α=h : State Variable and Output



Steady-State Condition

A very important concept associated with the model for control is the notion or condition of steady-state of the system.

Steady-State is the state at which the state variables no longer change with time. In this case, the accumulation term is equal to zero.

The application of the steady-state condition leads to a set of algebraic equations

The solution of this set of equations gives us the values of the state variables at steady-state, also called the equilibrium state.



Steady-State Equation

FFdtdhA i −== 0

hFi α−=0

or


It may happen that a system has more than one steady-state solution. In this case, we say

that the system has multiplicity or multiple steady-states.


Steady-State Condition


Multiple Steady-State Condition


Multiple Steady-State ConditionO’all Mass Balance:

At Steady state and assuming constant density,

Component A mass balance:

Energy Balance equation:

First-order reaction:


Multiple Steady-State ConditionAt steady state, solving for CA:

is the steady state concentration for any given reactor temperature Ts

Solving for Ts from the steady-state energy balance equation:

Energy Removed by flow and heat exchange

= Energy generated by reaction


Multiple Steady-State ConditionNote the form of the Energy removal term:

Notice that this is an equation for a line, where the independent variable is reactor tempeature (Ts). The slope of the line is and the intercept is . Changes in jacket or feed temperature shift the intercept, but not the slope. Changes in UA or F effect both the slope and intercept. Now, consider the Energy generation term:

substituting for


Multiple Steady-State Condition

A steady-state solution exists when there is an intersection of the Qrem [or

R(T)] and Qgen [or G(T)] curves.

increasingTfs

Steady-StateReactor Temperature

s

Ste

ady-

Sta

teR

eact

or T

empe

ratu

re

Inlet-feed TemperatureTfs

Ignitionpoint

Extinctionpoint

Role of Process Simulators

Process simulation packages like Dynsim, HYSYS, ASPEN PLUS, gProms and PRO/II provide a valuable basis for the design and overall evaluation of advanced process control applications. This comes with obvious economical, environmental and operational benefits.





Dynamic simulation offers many benefits over steady-state simulation where it allows the design and process engineers to investigate the time-dependent behavior of a process.

All variables can be observed regardless of instrumentation, The process model can be taken well beyond the safe limits of actual operation.

Using a dynamic model, individual and plantwide control strategies can be designed and tested.

Even the control loops can be tuned before choosing one that may be suitable for final implementation.


Open Modeling Architectures

There are many modeling problems which may need the concurrent use of two or more software packages to arrive at a meaningful solution. Each program offers unique advantages and functions, but there is no single program that incorporates all of the data and methods, which may possibly be used to solve every problem. Nor will there ever be an all-encompassing process solverthat would be a solution to all engineering modeling situations. A recent trend is to develop techniques to interconnect different software packages, enabling data communication across established links.


The vision is to create an open system whereby different programs can import and export data and communicate with each other freely.

Open Modeling Architectures

The open software vision


Example - Pilot-Scale Crystallization

HYSYS is responsible for the modeling of the utilities section, including all heating and cooling, temperatures and pressures.

gPROMS provides solutions for the crystallizer temperature and more significantly it provides the crystal size distribution (CSD).

Excel, while acting as a data link and used for data management and visualization, also allows the execution of the crystallizer jacket heat transfer model.


HYSYS flowsheet for the utilities section of the crystallization facility.



The Graphical User Interface (GUI) where connection and simulation execution can be undertaken.



Summary

Dynamic process models are developed by paying close attention to the modeling principles.The starting point is the fundamental laws of conservation for the quantity of interest.Steady-state condition arises when the state variables no longer change with time (equilibrium).The algebraic equations describing the steady-state condition may have multiple solutions (multiplicity).Process simulators like HYSYS, Dynsim and Aspen Plus help process designers and control engineers visualize the dynamic behavior of complete plants with control systems.Open modeling architectures enable the use of multiple modeling technologies to work together.


Documents

Chapter 2: Theoretical Models of Chemical Processes