LIQUID STORAGE SYSTEMS

Eqn. (*) and are volumetric flow rates. It appears to be a volume balance but it is not since volume is not conserved. This result only follows from a constant density assumption.Three Variations: Inlet or outlet flow rates are constant: This can be achieved by a constant speed, fixed volume pump. If is constant, then the outflow is completely independent of head. Analogous to Ohms LawThe tank exit line may function as a resistance to flow from the tank. This can be down using a valve at the exit line: Flow rate may be assumed to be linearly related to the liquid head.

Resistance

Resistive ProcessEqn. (*) becomes

Eqn. (**)Note that this last equation is similar to charging a capacitor through a resistor and is the time constant. A valve is placed at the exit line and turbulent flow is assumed :The pressure difference driving flow through the valve is

Assumed that flow is discharged at ambient pressureRecall Bernoulli relation

Valve coefficientPressure at the bottom of the tank

Conversion factorAcceleration of gravityAlso, we have

First Order Process Dynamic Characteristics

The general model of a first order lag system is

Process inputProcess outputProcess response time constant

Process steady state gain (output change/input change)

Slope of this line is

Output settles within 2% of final value

Settling time

Example (Resistive Process):Suppose the tank has a cross-sectional area of and operates at when the outflow rate is The resistance and the time constant will be .If we make a small corrective change at of say the resulting change in the level will be and the time to reach 98% of that change will be .

Capcitve Type Processes: Most processes include some form of capacitance or storage capability, either for materials (gas, liquid, or solids) or for energy (thermal, chemical, electrical charge, pressure, fluid hydrostatic head, etc.) Capacitance of a liquid or gas-storage tank, , is expressed in area units. The liquid capacitance equals the cross-sectional area of the tank at the liquid surface. The gas capacitance of a tank is constant and is analogous to electric capacitance.

A purely capacitive process element can be illustrated by a tank with only an inflow connection.

In this case, it is assumed that which implies that Eqn(**) becomes

For an initially empty tank with constant inflow, the level of liquid in the tank is the product of the inflow rate and the time period of charging divided by the capacitance of the tank.

Example (Capacitive Process):Consider what happens if we have a steady state condition, where flow into the tank matches the flow out via an orifice or valve with flow resistance . If we change the inflow slightly by the outflow will rise as the pressure rises until we have a new steady state condition. For a small change we can take the resistance to be a constant value. The pressure and outflow responses with follow the first order lag curve and will be given by the equation)Where is the capacitance of the tank.Second Order Process Dynamic CharacteristicsConsider the second order transfer function

With a unit step input, we obtain

Settling time and the percent overshoot is

Example: Consider the system

The mathematical model for this system is

This is a height signal. The flow is Flow signals

3 inputs and one output process

Example: The water level in a tank is controlled by an open loop system, as shown in Figure. A DC motor controlled by an armature current turns a shaft, opening a valve. The inductance of the DC motor is negligible, that is, . Also, the rotational friction of the motor shaft and valve is negligible, that is, . The height of the water in the tank is

the motor constant is , and the inertia of the motor shaft and valve is kg . Determine (a) the differential equation for and .

Solution:

State Variable MethodObjectives: Understand the concept of state variables, state differential equations, and output equations. Know how to obtain the transfer function model from state variable model and vice versa. Be aware of solution methods for state variable models.The state of a system is a set of variables whose values, together with the input signals and the equations describing the dynamics, will provide the future state and output of the system.

Consider the general system

Assume we have the set of state variables . Knowledge of initial values of the state variables ( and the input signals and for suffices to determine the future values of the outputs and state variables.Example: Consider a spring-mass damper system

Define a set of state variables:, Then the state variable repr. is

Example: Consider the RLC circuit

Let , then the state space model is

The State Differential Equation

...

=

State Differential EquationInput VectorOutput VectorSystem MatrixInput Matrix,,, =

Output Equation

Output MatrixFor the above RLC circuit example we have,, ,,

Consider the first order equation

The Laplace transform is

ConvolutionThe inverse Laplace transform is

In similar fashion the system

Gives

If we call the Fundamental or State Transition Matrix.

The Laplace transform of is

We can also find the transfer function from the state variable model by setting the initial conditions .

Transfer Function

Example: Evaluation of the state transition matrixConsider . Determine .

Characteristic Equation of the System with system matrix

When and we have

MATLAB Software can be used based on transfer function model or state variable model

Example: A single input, single output system has the matrix equation

Determine the transfer function .Answer:

Example: A hovering vehicle control system is represented by two state variables and

(a) Find the roots of the characteristic equation(i.e. the roots of the equation ).(b) Find the state transition matrix .Answer: (a) -3, -2(c)

Example: Determine a state variable representation for the system described by the transfer function

Example: Obtain a state variable matrix for a system with a differential equation

Example: A system has the matrix equation

(a) Find the transition matrix .(b) For the initial conditions , find

Exercise: Consider the block diagram below and the transfer function

(a) Verify that the block diagram is in fact a model of .(b) Show that the state space model that represents the block diagram is Where and

Solution:

Exercise: A system has the following differential equation:

Determine and .Solution:

Exercise: A system has a block diagram as shown in figure below. Determine the state variable differential equation and the state transition matrix .

Solution: