CHE334 Instrumentation and Process Control Lecture 8 Chapter 4 & 5 Mathematical Modeling of behavior of Chemical Process

By Dr. Maria Mustafa

Department of Chemical Engineering

• Total Number of Equations

Type of relationship Number

Equilibrium relationships N+1

Liquid flow relationships N

Mass balance around ith tray (where i= 2, …….., N-1, i≠f) Total mass balance Total component mass balance

N-3 N-3

Balance around feed tray 2

Balances around top tray 2

Balances around bottom tray 2

Balances around column base 2

Balance around reflux accumulation 2

4N+5

Independent Variables for an N stage Binary distillation Pocess

Independent Variables Numbers

Liquid composition (xi) i= 1,2,……f,…N,D,B N+2

Vapor Composition (yi) i= 1,2,……f,…N,B N+1

Liquid Holdups Mi i= 1,2,……f,…N,RD,B N+2

Liquid Flows Li i= 1,2,……f,…N N

Other Variables ( Ff, cf,FD, FB, FR, V)

6

Total= 4N+11

Modeling Considerations for Chemical Process

• Development of Input-Output model from control point of view.

• Direct relationship Model b/w input and output (cause and effect )

Output = f(input variables )

yi=f(m1,m2,……,mk; d1,d2,…..dl) for i= 1,2,…..,m

Process

Outputs Manipulated Variables

Disturbances d1 d2 dl

m1

m2

m3

y1

y2

ym

…………..

….

….

Example Q1: Consider the tank heater system. A liquid enters the tank with flow rate Fi and temperature Ti and leaves the tank with flowrate F and T. The liquid in tank is well mixed. The tank is heated with steam ( having flow rate of Fs). Assume tank do not move and momentum of heater is negligible.

Part 1 1. Construct the process diagram for the above system. 2. What are the relevant balances of fundamental

quantities for the system? 3. Identify the appropriate state variables to describe

the system. 4. Develop Mathematical model for above system.

1. Process Diagram

• Assumptions

– The momentum of the heater remain constant.

– Tank does not move so kinetic and potential energy is zero.

– For liquid systen

• dU/dt =dH/dt Fs

Fi, Ti

h

F, T

T

Q

2. What are the relevant balances of fundamental quantities for the

system? • Total Mass Balance

• Total energy balance

2. Identify the appropriate state variables

to describe the system.? • Height of the liquid in the tank

• Temperature of the liquid in the tank

Applying mass and energy balance around the heater tank system

For total mass balance 𝑑(𝜌𝐴ℎ)

𝑑𝑡= 𝜌𝑖𝐹𝑖 − 𝜌 𝐹

𝐴𝑑(ℎ)

𝑑𝑡= 𝐹𝑖 − 𝐹

Total energy balance 𝑑(𝐸)

𝑑𝑡=𝑑(𝑈 + 𝐾 + 𝑃)

𝑑𝑡= 𝜌𝑖𝐹 ℎ𝑖 − 𝜌 𝐹𝑖ℎ + 𝑄 ±𝑊𝑠

2. Develop Mathematical model for above system

𝑑(𝐻)

𝑑𝑡= 𝜌𝑖𝐹𝑖ℎ𝑖 − 𝜌 𝐹 ℎ + 𝑄

Rate of change of Enthalpy =𝐻 = 𝑚 𝑐𝑝∆𝑇 =

𝜌𝐹𝑐𝑝 𝑇 − 𝑇𝑟𝑒𝑓

Putting the H value in above equation we have

𝑑[𝜌𝐴ℎ𝑐𝑝 𝑇 − 𝑇𝑟𝑒𝑓 ]

𝑑𝑡=

= 𝜌 𝐹𝑖𝑐𝑝 𝑇𝑖 − 𝑇𝑟𝑒𝑓 − 𝜌 𝐹𝑖𝑐𝑝 𝑇 − 𝑇𝑟𝑒𝑓+ 𝑄

𝐴ℎ𝑑(𝑇)

𝑑𝑡= 𝐹𝑖(𝑇𝑖 − 𝑇) +

𝑄

𝜌𝑐𝑝

Summarize

State Equation 𝑨𝒅(𝒉)

𝒅𝒕= 𝑭𝒊 − 𝑭

𝑨𝒉𝒅(𝑻)

𝒅𝒕= 𝑭𝒊(𝑻𝒊 − 𝑻) +

𝑸

𝝆𝒄𝒑

State Variables : h,T

Output Variables : h,T

Input Variables

Disturbances : Ti, Fi

Manipulated Variables = Q,F, Fi

Part 2

1. Consider the flow rate at inlet of the heated tank is equal to the flow rate at the outlet.

a) What are the relevant balances of fundamental quantities for the system?

b) How many state model equations would be in this case?

c) Develop Input-out model for the above system

c) Develop input – Out put model

𝑨𝒉𝒅(𝑻)

𝒅𝒕= 𝑭𝒊(𝑻𝒊 − 𝑻) +

𝑸

𝝆𝒄𝒑

Amount of heat supplied by the tank is given by 𝑄 = 𝑈𝐴𝑡(𝑇𝑠𝑡 − 𝑇)

Replacing in first equation by its formula 𝑨𝒉𝒅(𝑻)

𝒅𝒕= 𝑭𝒊(𝑻𝒊 − 𝑻) +

𝑈𝐴𝑡(𝑇𝑠𝑡 − 𝑇)

𝝆𝒄𝒑

𝑨𝒉𝒅(𝑻)

𝒅𝒕= 𝑭𝒊𝑻𝒊 − 𝑭𝑻 +

𝑈𝐴𝑡 𝑇𝑠𝑡

𝝆𝒄𝒑−𝑈𝐴𝑡𝑇

𝝆𝒄𝒑

𝑽𝒅(𝑻)

𝒅𝒕= 𝑭𝒊𝑻𝒊 − (𝑭 +

𝑈𝐴𝑡

𝝆𝒄𝒑)𝑻 +

𝑈𝐴𝑡 𝑇𝑠𝑡

𝝆𝒄𝒑

𝑽𝒅(𝑻)

𝒅𝒕+ (𝑭 +

𝑈𝐴𝑡

𝝆𝒄𝒑)𝑻 = 𝑭𝒊𝑻𝒊 +

𝑈𝐴𝑡 𝑇𝑠𝑡

𝝆𝒄𝒑

Dividing by V and putting F=Fi 𝒅(𝑻)

𝒅𝒕+ (

𝑭𝒊𝑽+𝑈𝐴𝑡

𝑉𝝆𝒄𝒑)𝑻 =

𝑭𝒊𝑽𝑻𝒊 +

𝑈𝐴𝑡 𝑇𝑠𝑡

𝑉𝝆𝒄𝒑

𝒅(𝑻)

𝒅𝒕+ 𝒂𝑻 =

𝟏

𝝉𝑻𝒊 +𝑲𝑻𝒔𝒕

𝟏

𝝉=𝑭𝒊𝑽,𝑲 =

𝑈𝐴𝑡𝑇

𝝆𝒄𝒑, 𝒂 =

𝑭𝒊𝑽+𝑈𝐴𝑡

𝑉𝝆𝒄𝒑=𝟏

𝝉+ 𝑲

• At steady state dT/dt = 0 𝒅𝑻𝒔𝒅𝒕

+ 𝒂𝑻𝒔 =𝟏

𝝉𝑻𝒊,𝒔 +𝑲𝑻𝒔𝒕,𝒔

Subtract steady state equation with un-steady equation ,

𝒅(𝑻−𝑻𝒔)

𝒅𝒕+ 𝒂 𝑻 − 𝑻𝒔 =

𝟏

𝝉𝑻𝒊 − 𝑻𝒊,𝒔 +𝑲(𝑻𝒔𝒕−𝑻𝒔𝒕,𝒔)

𝒅(𝑻′)

𝒅𝒕+ 𝒂𝑻′ =

𝟏

𝝉𝑻′𝒊 +𝑲𝑻′𝒔𝒕

𝑇′ = 𝑇 − 𝑇𝑠, 𝑇

′𝑖 = 𝑇𝑖 − 𝑇𝑖,𝑠 𝑎𝑛𝑑 𝑇

′𝑠𝑡 = 𝑇𝑠𝑡 − 𝑇𝑠𝑡,𝑠

are

0

𝒊𝒏𝒅𝒊𝒄𝒂𝒕𝒆𝒔 𝒕𝒉𝒆 𝒅𝒆𝒗𝒊𝒂𝒕𝒊𝒐𝒏 𝒇𝒓𝒐𝒎 𝒕𝒉𝒆 𝒄𝒐𝒓𝒓𝒆𝒔𝒑𝒐𝒏𝒅𝒊𝒏𝒈 𝒔𝒕𝒂𝒕𝒆 𝒔𝒕𝒂𝒕𝒆 𝒗𝒂𝒍𝒖𝒆

• The solution of the above equation is

𝑇′(𝑡) = 𝑐1𝑒−𝑎𝑡 + 𝑒−𝑎𝑡 𝑒𝑎𝑡

𝑡

0

1

𝜏𝑇′𝑖 + 𝐾𝑇′𝑠𝑡 𝑑𝑡

Initially the heater is at steady state that T’(t) =0

Therefore c1 = 0 and above equation gives

𝑇′(𝑡) = 𝑒−𝑎𝑡 𝑒𝑎𝑡𝑡

0

1

𝜏𝑇′𝑖 + 𝐾𝑇′𝑠𝑡 𝑑𝑡

The above relationship b/w input and o

ut put variables constitutes the input-output model for the tank heater

Pictorial Diagram

𝟏

𝝉

𝑲

𝑒−𝑎𝑡 𝑒𝑎𝑡𝑡

0

. dt (𝟏

𝝉𝑻′𝒊 +𝑲𝑻′𝒔𝒕)

𝟏

𝝉𝑻′𝒊

𝑲𝑻′𝒔𝒕

𝑻′𝒊

𝑻′𝒔𝒕

𝑻′

Inp

uts

Ou

tpu

t

Next Lecture

• Modeling Considerations for Chemical Process