Teaching CHE 696

Distributed Parameter Systems Modelling:

We consider several models present in the process and mechanical systems control:

Transport-reaction systems:

1.) Tubular reactor

<br />
\frac{\partial x}{\partial t}=-v(\zeta)\frac{\partial x}{\partial \zeta}+\psi(\zeta) x

with boundary conditions x(0,t)=u(t), and initial condition x(\zeta,0)=x_0(\zeta)

2.) Tubular reactor with recycle

\frac{\partial x}{\partial t}=-v(\zeta)\frac{\partial x}{\partial \zeta}+\psi(\zeta)x

with boundary conditions x(0,t)=Rx(1,t)+(1-R)u(t)  and initial conditions  x(\zeta,0)=x_0(\zeta)

3.) Countercurrent heater

\frac{\partial x_1}{\partial t} =-\frac{\partial x_1}{\partial \zeta}+k(x_2-x_1),x_1(t,0)=u_1(t)

\frac{\partial x_2}{\partial t} =\frac{\partial x_2}{\partial \zeta}-k(x_2-x_1),x_2(t,1)=u_2(t)




4.) Equation of heat conduction (parabolic PDE)

<br />
\frac{\partial x}{\partial t}=\frac{\partial^2 x}{\partial \zeta^2}

with x(0,t)=0=x(1,t), x(\zeta,0)=x_0(\zeta).

5.) Reaction-diffusion-convection models of dispersion reactor (parabolic PDE)

\frac{\partial x}{\partial t}=\frac{\partial^2 x}{\partial \zeta^2}-v\frac{\partial x}{\partial \zeta}+\psi x

with \frac{\partial x}{\partial \zeta}(0,t)=Pe(x(0,t)-u(t)) and \frac{\partial x}{\partial \zeta}(1,t)=0x(\zeta,0)=x_0(\zeta).

Strings & Beams

1.) String Equation (2nd Order Hyperbolic PDE)

Let us consider:
\frac{\partial ^2 x}{\partial t^2}=c^2\frac{\partial ^2 x}{\partial \zeta ^2}
with Derichlet boundary conditions x(0,t)=0=x(1,t), and  x(\zeta,0)=\phi(\zeta) and \dot{x}(\zeta,0)=\psi(\zeta). The change of variable leads to \frac{\partial x}{\partial \zeta}=x_1, and \frac{\partial x}{\partial t}=x_2, which leads to the following \frac{\partial }{\partial t}\frac{\partial x}{\partial \zeta}=\frac{\partial x_1}{\partial t} and also \frac{\partial }{\partial \zeta}\frac{\partial x}{\partial t}=\frac{\partial x_2}{\partial \zeta}, from this we obtain \frac{\partial x_1}{\partial t}=\frac{\partial x_2}{\partial \zeta} and \frac{\partial }{\partial t}[x_2]=c^2\frac{\partial x_1}{\partial \zeta}, and therefore:



2.) Euler-Bernoulli Beam Equation 
Let us consider following version of the beam equation which is a simply supported undamped beam:
\mu \frac{\partial^2 x}{\partial t^2}+EI\frac{\partial^4 x}{\partial \zeta^4}=q

where \mu in the mass per unit length, E is the elastic modulus and I is the second moment of area of the beam's cross-section.The x represents deflection of the beam in the z direction at some point \zeta

3.) Rayleigh Beam Equation

\frac{\partial ^2 x}{\partial t^2}-\alpha \frac{\partial^4 x}{\partial\zeta^2 \partial t^2}-a \frac{\partial^3 x}{\partial\zeta^2 \partial t}+\frac{\partial^4 x}{\partial\zeta^4}=0

with boundary conditions:

\frac{\partial^2 x}{\partial \zeta^2}(0,t)=u(t)
\frac{\partial^2 x}{\partial \zeta^2}(1,t)=0


3.) Shear Beam Equation

\frac{\partial ^2 x}{\partial t^2}-\alpha \frac{\partial^4 x}{\partial\zeta^2 \partial t^2}+\frac{\partial^4 x}{\partial\zeta^4}=0


\alpha\frac{\partial ^2 x}{\partial t^2}=\frac{\partial^2 x}{\partial \zeta^2}-\frac{\partial \psi}{\partial \zeta}

0=\alpha \frac{\partial^2 \psi}{\partial \zeta^2}-\psi+\frac{\partial x}{\partial \zeta}


4.) Timoshenko Beam Equation

\rho A\frac{\partial ^2 x}{\partial t^2}-q(\zeta,t)=\frac{\partial }{\partial \zeta}[\kappa AG (\frac{\partial x}{\partial \zeta}-\phi)]
\rho I \frac{\partial^2 \phi}{\partial t^2}=\frac{\partial }{\partial \zeta} (EI\frac{\partial \phi}{\partial \zeta})+\kappa AG (\frac{\partial x}{\partial \zeta}-\phi)

where the x is translational displacement of the beam and \phi is angular velocity.

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