Dynamic Modeling of Transport Process Systems

C. A. Silebi, William E. Schiesser

• Publisher: Academic Press
• Year: 2012
• ISBN:
  9780080925820
• Topic: Technology & Engineering
• Subtopic: Mechanical Engineering
• Index: Technology & Engineering

This book presents a methodology for the development and computer implementation of dynamic models for transport process systems. Rather than developing the general equations of transport phenomena, it develops the equations required specifically for each new example application. These equations are generally of two types: ordinary differential equations (ODEs) and partial differential equations (PDEs) for which time is an independent variable. The computer-based methodology presented is general purpose and can be applied to most applications requiring the numerical integration of initial-value ODEs/PDEs. A set of approximately two hundred applications of ODEs and PDEs developed by the authors are listed in Appendix 8.

1

The Nature of Dynamic Systems

In the analysis of transport process systems, systems in which the transport or transfer of mass, momentum, and energy takes place, the assumption of steady-state operation is commonplace. Specifically, the assumption is made that the input and output flows are exactly balanced so that the internal state variables that describe the system, for example, concentration or temperature, do not change with time. Mathematically, this steady-state condition is expressed simply as

(1.1)

where “input” and “output” denote the rates at which the principal streams flow into and out of the system, respectively, for example, flows of energy and material.

The exact balance of inputs and outputs expressed by eq. (1.1) represents an idealized situation, however, which is rarely realized in practice; more typically an imbalance between inputs and outputs occurs due to, for example, a disturbance that enters through the inputs. Thus, for this situation, eq. (1.1) must be generalized to

(1.2)

Equation (1.2), although quite simple in appearance, serves as the basis for most of the models and associated computer simulations that we will consider in the remainder of this book.

Two special cases of eq. (1.2) can be considered:

for which accumulation will take place within the system since input exceeds output (consider a tank with liquid inlet and outlet lines for which the inlet flow exceeds the outlet flow so liquid accumulates within the tank), and

for which depletion will take place within the system since the output exceeds the input (again, the tank with liquid inlet and outlet streams would have a decreasing liquid level). Thus we can interpret depletion as negative accumulation (from the second case).

1.1 The Origin of Differential Equations

As a specific example of the application of eq. (1.2), consider the holding tank in Figure 1.1. If we apply eq. (1.2) to this tank, the input and output flow rates are simply Q₀ and Q, respectively, which typically would have the units of cm³/sec. The first thing we notice from this example is that the variables usually involve time, for example, the preceding flow rates. Although this also occurs in steady-state analysis, the next term we will consider in eq. (1.2) involves time in a way that is not encountered in steady-state analysis. The accumulation term is dV/dt, that is, the time rate of change of the volume of liquid in the tank, which also has the typical units of cm³/sec.

Fig. 1.1 Dynamics of a holding tank.

Thus eq. (1.2) applied to the holding tank becomes

(1.3)

Equation (1.3) really results from a mass balance, for which we have assumed that the liquid density is constant (generally a good assumption for liquids). If this could not be assumed, eq. (1.3) would be written as

Note that for constant density (ρ = ρ₀), ρ cancels to give eq. (1.3).

Equation (1.3) is an example of the mathematical equations we will be considering throughout this book, that is, differential equations that are in contrast to the algebraic equations that are typically used in steady-state analysis. Technically, eq. (1.3) is a first order ordinary differential equation (ODE). It is “first order” because the highest order derivative in the equation is first order, and it is an “ordinary differential equation” since it has only one independent variable, t (and as a consequence, the derivative, dV/dt, is a total derivative).

1.2 Well-Posed Problems

We might at this point consider solving eq. (1.3). The first question that naturally arises is, what do we mean by a solution? Generally the answer is, the dependent variable as a function of the independent variable, in this case V(t). However, before we obtain this solution we must attend to some details to insure that the problem is well-posed, that is, that all of the mathematical details have been properly stated.

The first thing we note about eq. (1.3) is that it has three variables that could conceivably be functions of time, V(t), Q₀(t) and Q(t). In effect we have three unknowns and one equation, so that two more mathematical relationships are required. For example, if we assume that the outlet line has a valve that regulates the flow, we can use a typical valve equation, which for gravity flow can be written as

(1.4)

where C_v is a constant for a particular valve, and h is the height of liquid in the tank. The problem with eq. (1.4) is that if we substitute it in eq. (1.3) we have introduced another unknown, h, so that we still have three unknowns, V(t), Q₀(t), and h(t). However, we also recognize that V = Ah, where A is the tank cross-sectional area (assumed constant), and therefore substitution in eq. (1.3) gives

(1.5)

We have intentionally not combined eqs. (1.4) and (1.5) since, in general, as the mathematical models (sets of equations) for transport process systems become more complicated, we will find that systems of equations are more convenient to use in computing solutions.

Finally, we will assume that the entering flow rate, Q₀, is a known function of time, for example, a given constant. Thus all three variables in eqs. (1.4) and (1.5) are now defined mathematically, and we might therefore expect that we can obtain a solution, in this case h(t) [remember again that we mean by a solution the dependent variable of the differential equation as a function of the independent variable, or h(t)]. However, ther...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• Dedication
• Preface
• Chapter 1: The Nature of Dynamic Systems
• Chapter 2: Basic Concepts in the Numerical Integration of Ordinary Differential Equations
• Chapter 3: Accuracy in the Numerical Integration of Ordinary Differential Equations
• Chapter 4: Stability in the Numerical Integration of Ordinary Differential Equations
• Chapter 5: Systems Modeled by Ordinary Differential Equations
• Chapter 6: Systems Modeled by First Order Partial Differential Equations
• Chapter 7: Systems Modeled by Second Order Partial Differential Equations
• Chapter 8: Systems Modeled by First/Second Order, Multidimensional and Multidomain Partial Differential Equations
• APPENDIX 1: Integrator INT1
• APPENDIX 2A: Main Program DSS2S
• APPENDIX 2B: Subroutine INTEG Called by Main Program DSS2S
• APPENDIX 3: Integrator INT8
• APPENDIX 4: Integrator INT15
• APPENDIX 5: Integrator INTM15
• APPENDIX 6: Main Program to Call Subroutine LSODE
• APPENDIX 7: Main Program to Call Subroutine DASSL
• APPENDIX 8: Library of ODE and PDE Applications
• APPENDIX 9: Spatial Differentiation Routines
• Index