Time delay

Constant delay of input signal

Simple shift

Copies all time points $t$ from the input stream $In$ to the output stream $Out$ at the timepoint $t + \Delta t$ , delaying the signal by a constant value $\Delta t$ .

Norm-based

$\frac{dm}{dt} = \dot{m}_{in}(t-\Delta t) - m$

To correctly take into account the dynamics of the process, norms of each overall parameter (mass flow, temperature, pressure) are maintained as:

$\frac{d||X||}{dt} = (X(t) - X(t-1))^2 - ||X||$

For phase fractions:

$\frac{d||P||}{dt} = \sqrt{\sum_{i}^{N_{P}}{(w_{i}(t) - w_{i}(t-1))^2}} - ||P||$

For compound fractions in each phase:

$\frac{d||C_{i}||}{dt} = \sqrt{\sum_{j}^{N_{C_{i}}}{(w_{i,j}(t) - w_{i, j}(t-1))^2}} - ||C||$

For each distributed parameter:

$\frac{d||D_{i}||}{dt} = \sqrt{\sum_{j}^{N_{D_{i}}}{(w_{i,j}(t) - w_{i,j}(t-1))^2}} - ||D||$

Note

Notations:

${m}$ – current mass

$\dot{m}_{in}$ – input mass flow

$\Delta t$ – time delay

$X(t)$ – value of an overall parameter at time point $t$

$w(t)$ – mass fraction at time point $t$

$N_{P}$ – number of defined phases

$N_{C_{i}}$ – number of defined compounds in phase $i$

$N_{D_{i}}$ – number of classes in distribution $i$

Note

Model parameters:

Name	Symbol	Description	Units	Boundaries
Time delay		Model to use		Norm based, Simple shift
Time delay	$\Delta t$	Time delay	[s]	>=0
Relative tolerance		Relative tolerance for DAE solver	[-]	>0 (0 for flowsheet-wide value)
Absolute tolerance		Absolute tolerance for DAE solver	[-]	>0 (0 for flowsheet-wide value)