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An 'n' stages system

The classical 2nd order system

n 9 :

T 1 :

α 1.25 :

β 0.5 :

K.p 1 :

==================== XFR [Transfer Function] =======================

XFR of 'n' multistage system of
=  Unit Time Constant [1] t nth 1 t - T / exp 1 k! / t T / (k^* (k 0 n sum * - :

XFR of the 2nd order classical system:
Kp [Process gain], α,β [Dynamics] t XFR K.p 1 α β α - (/ t - α / exp * + β β α - (/ t - β / exp * - (* :

==================== Reaction Rate =======================

Reaction rate of the nth system t Laplace t n 1 - (^n 1 - (! / t - T / exp T n^/ * :

Reaction rate 2nd order x XFR x diff K.p x β / - exp x α / - exp - (* β α - / ≡

x nth x Laplace 2 1 sys x XFR x XFR x diff 2 1 sys

Source: Inst_PID Discrete Schema

The XFR transfer function of an 'nth order system'

... is the graphical representation of the open loop response

in time of a system under the Laplace unit step response [1/s]

... i.e: the response from the steady state 0 to the stady state 1

... i.e: natural response w/o oscillation(s) .

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Laplace algebra of 2nd order under 
unit step ... Gp => Process gain s G.P α β G3 G.P s 1 α s * + (* 1 β s * + (* / :

The XFR(t) is the InverseLaplace  of open loop Laplace algebra... in Smath from the "maple companion" t G.P α β XFR s G.P α β G3 s t invlaplace maple : G.P β α 1 - t α / - exp + (* + β t β / - exp * - (* β α - /

G.P 1 : α 0.5 : β 1.25 : 31 line

x xfr 0 x ≤ x G.P α β XFR invalid if :

x xfr 1 2 1 sys

You can copy the RHS maple result and export. Always recommanded for isolation purpose.

============================================================================

n 9 :

T 1 :

XFR of 'n' multistage system of
=  Unit Time C0onstant 'T' [1] t nth 1 t - T / exp 1 k! / t T / (k^* (k 0 n sum * - :

x nth

n T 21 mat 9121 mat

Working more Maple/Smath

This suite is not related to DE solvers, just something interesting, Edu. Observe: invlaplace, s, t laplace, t, s

μ 5 :

T μ s d 1 T s * 1 + (μ^/ :

t xfr T μ s d s t invlaplace maple : t 4^t - exp * 24 /

# t n 1 - (^n 1 - (! / t - T / exp T n^/ * ≡

EQ.0

n T x xfr x n 1 - (^n 1 - (! / x - T / exp T n^/ * : (reaction rate ≡

n T x XFR 1 x - T / exp 1 k! / x T / (k^* (k 0 n sum * - :

# 1 t - T / exp 1 k! / t T / (k^* (k 0 n sum * - ≡

EQ.1

n 4 :

T 1 :

'n' time constant 'T' system

0 x ≤ n T x xfr x < 0 invalid if 0 x ≤ n T x XFR x < 0 invalid if 2 1 sys

0 x ≤ n T x xfr x < 0 invalid if 0 x ≤ n T x XFR x < 0 invalid if 21 sys

1 t - T / exp 1 k! / t T / (k^* (k 1 n sum * - 1 t - T / exp t T / exp 1 n 1 + (! / t T / (n 1 +^* n 1 + (* t T / (1 - n -^* t T / exp * n 1 + Γ n 1 + t T / Γ - (* - (* - s / ≡

XFR multistage 'n' time constant 'T' 1 t - T / exp 1 k! / t T / (k^* (k 1 n sum * - n 1 + Γ n 1 + t T / Γ - s n 1 + Γ * / ≡

http://sites.tufts.edu/andrewrosen/files/2013/09/reactor_design_guide1.pdf