BsysE595 Lecture Basic modeling approaches for engineering systems

– Summary and Review

Shulin ChenJanuary 10, 2013

Topics to be covered

• Review basic terminologies on mathematical modeling

• Steps for model development• Example: modeling a bioreactor

Definition

• Modeling – The process of application of fundamental knowledge or

experience to simulate or describe the performance of a real system to achieve prediction goals.

• Mathematical modeling – Using fundamental theories and principles governing the

systems along with simplifying assumptions to derive mathematical relationships between the variable known of significant.

– The resulting model can be calibrated using historical data from the real system and can be validated using additional data. Predictions can them be made with predefined confidence

Types of models

• Deterministic versus probabilistic – Variables and their changes are well defined with

certainty, the relationship between the variables are fixed, then the model is said to be deterministic; If some unpredictable randomness or probabilities are associated with at least one of the variable of the outcomes, the model is considered probabilistic.

Types of models

• Continuous versus discrete– When the variables in a system are continuous

functions of time, the model is continuous (using differential equations); If the changes in the variables occur randomly or periodically, the modeling is termed discrete (using difference equations).

Static versus Dynamic

• When a system is at a steady state, its inputs and outputs do not vary with passage of time and are average values, the model is known as static or steady-state. The results of the model are obtained by a single computation of all of the equation. When the system behavior is time-dependent, its model is called dynamic. Dynamic models are built with differential equations that yield solutions in the form of functions.

Distributed versus lumped

• When the variations of the variable in a system are continuous functions of time and space, the system has to be modeled by a distributed model; If the variable does not change with space, it is referred as a lumped model. Lumped models are often built of algebraic equations, lumped, static models are often built of ordinary differential equations, distributed models are often built of partial differential equations.

Linear versus nonlinear

• When an equation contains only one variable in each term and each variable appears only to the first power, the equation is termed linear, if not, it is termed non-linear.

Analytical versus numerical

• When all the equations can be solved algebraically to yield a solution in a close form, the can be classified as analytical. If that is not possible, and a numerical procedure is required to solve one or more of the model equations, the model is classified as numerical.

Terminologies

• Variables– The quantitative attributes of the system and of

the surroundings that have significant impact on the system

– Variables include those attributes that change in value during the modeling time span and those that are remain constant during the period. Variables that are constant during the period are referred to as parameters

Terminologies

• Input– Variables that are generated by surroundings and

influence the behavior of the system • Output– Variables that are generated by the system and

influence the behavior of the surroundings

Definition and terminology in mathematical modeling

• System– A collection of one or more relate objects

• Boundary– The system is isolated from its surroundings by the

boundary, which can be physical or imaginary• Close system– When the system does not interact with

surroundings– Neither mass nor energy will cross the boundary

System Modeling

• Systems modeling approach– Definition of systems– System design – Systems modeling • Optimization • Control

Biological modeling approach

– Open system – Growth and transformation – Network – Flux– Regulation and control

Engineering modeling approach

• Principle of conservation– Mass balances • Mass transport

– Flow– Diffusion

• Transformation• Energy balances

Steps of model development

• Problem formulation• Mathematical representation• Checking for validity • Mathematical analysis• Interpretation and evaluation of results

Problem formulation

• Establishing the goal of the modeling effort• Characterize the system– Identifying the system and its boundary– Identify the significant and relevant variables and

parameters– Establish how, when, where, and at what rate the

system interact with its surroundings– Create graphic model

• Simplifying and idealizing the system

Mathematical Representation• Identifying fundamental theories that govern the system

behaviors– Stoichiometry– Conservation of mass and energy– Reaction theory– Reactor theory– Transport mechanisms.

• Deriving relationships– Apply and integrate the theories and principles to derive

relationships between the variables of significance and relevance• Standardizing relationships

– Simplifying, transforming, normalizing, or forming dimensionless groups.

Check for Validity

• Check for units and dimensions– Make sure all terms in both sides of the equation

have the same unit or dimension;• Check for extreme conditions – Assign extreme values to some variables • Give them infinitely large values • Give them infinitely small values • Make them zero

Mathematical Analysis

• Applying standard mathematical techniques ad procedures to “solve” the model to obtain the desirable solutions.

Interpretation and evaluation of results

• Calibrating the model – using observed data from the real system

• Sensitivity analysis– Check system response with changes in selected

variables• Validating the model– Assure mass balance closed– Compare model output with hand calculations– Using a test data set

Example: Modeling a Bioreactor

Step 1. Problem formulation

CCC,V,X, k

Q, Ci Q, Co

Step 2: Mathematical representation

A

∆x

u

C,r

Mass balance

Rate of accumulation of mass within the system boundary = rate of flow of mass into the system boundary - rate of flow of mass out of the boundary + rate of generation within the system boundary

Mass transfer terms - advection

• Advection in which the amount of the mass transported through a flow is the multiplication of the flow rate and the concentration,

• Transport by advection = AuCWhere • A = the cross sectional area, m2• u = average pore velocity, m/d• C = mass concentration of the contaminant, g/m3

Mass transfer terms – dispersion including diffusion

• Dispersion, the longitudinal transport of material due to turbulence and molecular diffusion. Dispersion is driven by the concentration gradient of the substances. There are typically two components that contribute to dispersion. The first is molecular diffusion when the mean flow velocity is very low, and the second component is caused by eddy effect when the liquid particles do not travel at the same speed under a real (non-ideal) flow condition. It becomes a common practice in transport modeling to lump the dispersion and diffusion together, to be represented by a dispersion coefficient. Thus,

• • Transport by dispersion = (5.2.2)• • The reaction rate in the reactor, r, expressed as mass reduction in unit time

and unit volume:• Substrate reaction rate = r (g/m3-d)

x

CDA

Basic Equation

• Substituting all the terms into the mass balance equation, we obtain

xrAx

CADuAC

x

CADuACxA

t

C

xxx

The model

• When x0, the above equation can be reduced to

rx

Cu

x

CD

t

C

2

2

The reaction term• The reaction rate, r, can be represented by kinetics. Kinetics is typically

described by different orders, which are determined by the exponent of the term that represents the constituent involved. If a first order reaction is appropriate. Accordingly:

• •

• • where r = reaction rate (or reaction rate), g/m3.d,• C = concentration of the substrate, g/m3,• t = time, d,• KT= reaction rate constant at a given temperature T, d-1.

CKr T

Step 3: Checking for validity

rx

Cu

x

CD

t

C

2

2

Check for units

Step 4: Mathematical analysis- A steady-state condition

0

2

2

CKx

Cu

x

CD T

)2

exp()1()2

exp()1(

)2

exp(4

22

D

auLa

D

auLa

D

uLa

C

C

o

e

Step 5: Interpretation and evaluation of results

A plug flow reactor

Plug-flow • If the dispersion in a reactor is negligible, equation becomes • •

• • Integration of the above equation • •

• resulting in•

• Where:• Ci = influent concentration, g/m3 (mg/l)• Co = effluent concentration, g/m3 (mg/l)• KT = reaction rate constant, d-1• h = Hydraulic detention time, d.

0 CKx

Cu T

L C

c

Te

dCC

Kudx

0 0

hTT Ku

LK

i

o eeC

C

For a completely mixed reactor

CC

C,V,X, k

Q, Ci Q, Co

rCeCoQdt

dC )(