Info

I I max

I max 1

I I max where, 1^1 is the maximum absolute eigenvalue of the Jacobian matrix of the ODEs.

With an example of batch distillation, Seader and Henley showed that the time step needed in implicit Euler's method was 200 times of that needed for explicit Euler's method.

4.5.2.3. Stiffness Ratio

Quite often chemical engineering systems are encountered with widely different time constants, which give rise to both long-term and short-term effects. The corresponding ordinary differential equations have widely different eigenvalues. Differential equations of this type are known as stiff systems. Seader and Henley (1988) derived the expressions for maximum and minimum eigenvalues for the differential component mass balance equations related to intermediate plates and reboiler respectively.

Lj +kijVj ~ 2- for intermediate plates

for reboiler

The stiffness ratio, SR, is defined as:

Seader and Henley (1998) noted that for SR = 20, the system is not stiff, SR = 1000, the system is stiff and SR = 1,000,000, the system is very stiff.

4.5.3. Initialisation of the DAE System

The initialisation of variables in a system of equations is important. While, in systems of ODEs all of the state variables must be initialised, in DAE systems only some of the variables need to be initialised, which is the same as the number of differential variables for index one system. The other variables can be determined using the algebraic equations. It is inconvenient for the user to be required to initialise all of the variables as this might require the solution of a set of nonlinear algebraic equations. Pantelides (1988) developed a procedure for consistent initialisation of DAE systems. Readers are directed to this reference for further details.

Nomenclature

D, Ld distillate flow rate (kmol/hr)

F feed flow rate in a continuous column or flow rate out of feed tank

accumulator and condenser holdup respectively (kmol) plate, reboiler and feed tank holdup respectively (kmol) Liquid, vapour enthalpy (kJ/kmol)

liquid flow rate into the feed tank in MVC columns (kmol/hr) liquid, vapour flow rate, vapour boilup rate (kmol/hr) bottom product flow rate (kmol/hr)

number of plates, feed tank location, minimum number of plates nc no of components

Qc, Qr condenser or reboiler duty (kJ/hr)

K, k vapour-liquid equilibrium constant kr array of reaction rate constants r internal reflux ratio [0, 1]

rb internal reboil ratio [0, 1]

R, Rmin external reflux ratio, minimum reflux ratio

Rmin,u minimum reflux ratio (Underwood)

Rmin.g minimum reflux ratio (Gilliland)

t batch time (hr)

x, y liquid or vapour composition (molefraction)

x, y array of liquid or vapour composition (molefraction)

xa accumulated distillate composition (molefraction)

xD instant distillate composition (molefraction)

r reaction rate (1/hr)

An change in moles due to chemical reaction

Superscripts and subscripts i component number (1, nc)

j stage number (1, AO

k column section in MultiBD columns m vessel no. in MultiBD columns

Greek letters a relative volatility

<j> common root

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