Section 4.4.1: Steady State Model


Material balance:
$\sum$ flows in = $\sum$ flows out

Dynamic Model

Material balance:

rate of change of mass
= $\sum$ flows in - $\sum$ flows out
In a molar or volumetric balance, mass is replaced by moles or volume.

Flow through a tank

Steady state material balance

F1 - F2 = 0

Dynamic material balance


dM/dt = F1 - F2

Dynamic balance:

More equations needed


Dynamic material balance
dM/dt = F1 - F2

Need two constraint equations e.g. for flows

F1 - a1 = 0


F2 - a2 = 0

Now we have a DAE system. Get back to ODE by substituting the AEs into the material balance.

Some DAE systems are hard! Care must be used in formulating DAE systems. (See later:sections 4.4.9 and section 4.4.10).

Solve ODE for holdup

Substitute flow constraints $ \rightarrow $ dynamic material balance

dM/dt = a1 - a2

Solution:

M(t) = (a1 - a2)t + c

or

M(t) = bt + c

Initial Conditions

ODEs need initial conditions

Here : one differential variable, M
Give initial holdup M0 at time t0 = 0

Substitute into ODE:

\begin{displaymath}M_{0} = b \times 0 + c \end{displaymath}

gives

c = M0

Hence particular solution

M(t) = M0 + bt

Next - Section 4.4.2: The Richness of Dynamics
Return to Section 4.4 Index
Return to Section 4 Index

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Last modified: Thu Aug 6 11:34:43 BST