Mixing Tank Example

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Mixing Tank Example
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1 Introduction to the Mixing Tank Example

Figure 1: Process Scheme

This article was contributed by Asim Vodencarevic (see the forum topic). I've

There is one mixing tank and two inlet valves. Control valves can control input flow of hot and cold water. There is one outlet pump. The objective is to control both temperature and level.

2 Model of the Mixing Tank

We need to write differential equations which describe the dynamics of the system.

2.1 Water Level

The differential equation for level is a matter of mass balance.

LaTeX: input-output=accumulation



LaTeX: \frac{dV}{dt}=F_{in}-F_{out}



LaTeX: A\frac{dL}{dt}=F_{1}+F_{2}-F_{3}


where

LaTeX: A is the tank cross-sectional area and
LaTeX: L is the tank level.


LaTeX: A\frac{dL}{dt}=\Chi_{hot}\nu_{hot}\left(t\right)+\Chi_{cold}\nu_{cold}\left(t\right)-\Delta_{out}\left(t\right)


where

LaTeX: \Chi is the Pump Capacity,
LaTeX: \nu is the fraction of full capacity (can be getween [0 1]), and
LaTeX: \Delta_{out} is tank output.

Finally this can be reduced to

LaTeX: \frac{dL}{dt}=\frac{1}{A}\left(F_{1}+F_{2}-F_{3}\right) Eqn. 1



2.2 Water Temperature

LaTeX: \frac{d\left(VT\right)}{dt}=F_{1}T_{1}+F_{2}T_{2}-F_{3}T



LaTeX: A\frac{dL}{dt}T+AL\frac{dT}{dt}=F_{1}T_{1}+F_{2}T_{2}-F_{3}T



There is only one fluid (water), so density and specific heat are the same in all terms. Now if we combine the last equation with the level equation (Eqn. 1)

LaTeX: \left(F_{1}+F_{2}-F_{3}\right)T+AL\frac{dT}{dt}=F_{1}T_{1}+F_{2}T_{2}-F_{3}T


This yields

LaTeX: AL\frac{dT}{dt}=F_{1}\left(T_{1}-T\right)+F_{2}\left(T_{2}-T\right) Eqn. 2



3 Simulink Model

LaTeX: \frac{dL}{dt}=\frac{1}{A}\left(F_{1}+F_{2}-F_{3}\right) Eqn. 1



LaTeX: \frac{dT}{dt}=\frac{F_{1}}{AL}\left(T_{1}-T\right)+\frac{F_{2}}{AL}\left(T_{2}-T\right) Eqn. 3



Figure 2: Simulink Model
Table 1: Mixing Tank, Nominal Operational Parameters
Parameters Value Units

LaTeX: \Chi_{hot}=\Chi_{cold}

0.0042

LaTeX: \frac{m^3}{s}

LaTeX: \nu_{hot}\left(0\right)

0.4

N/A

LaTeX: \nu_{cold}\left(0\right)

0.5

N/A

LaTeX: L_{final}

1

m

LaTeX: T_{final}

40

deg C

LaTeX: \Delta_{out}

0.00378

LaTeX: \frac{m^3}{s}

LaTeX: A

1

LaTeX: m^2

3.1 Simulink Model Results

LaTeX: \frac{dL}{dt}=\frac{1}{A}\left(F_{1}+F_{2}-F_{3}\right) Eqn. 1



LaTeX: \frac{dT}{dt}=\frac{F_{1}}{AL}\left(T_{1}-T\right)+\frac{F_{2}}{AL}\left(T_{2}-T\right) Eqn. 3