First Order Process Electrical Engineering (EE) Notes | EduRev

Electrical Engineering (EE) : First Order Process Electrical Engineering (EE) Notes | EduRev

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First Order Process
A first order process is a process whose output y(t) is modeled by a first order differential equation.
First Order Process Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                                35

where, First Order Process Electrical Engineering (EE) Notes | EduRev  are input and output of the process espectively. If  First Order Process Electrical Engineering (EE) Notes | EduRev , then define the following:  First Order Process Electrical Engineering (EE) Notes | EduRev  and  First Order Process Electrical Engineering (EE) Notes | EduRev

Hence, the first order differential equation takes the following form:

First Order Process Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                           36

At steady state condition  First Order Process Electrical Engineering (EE) Notes | EduRev , the equation can be re-written as
First Order Process Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                                         37

Subtracting eq. (36) from eq. (37), we obtain

First Order Process Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                   38

Alternatively,
First Order Process Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                            39

Where,  First Order Process Electrical Engineering (EE) Notes | EduRev are the deviation forms of the input and output variables of the process around the steady state, whose initial conditions are assumed to be the following:  First Order Process Electrical Engineering (EE) Notes | EduRev .

Taking Laplace Transform of the eq. (39) we obtain,
First Order Process Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                                 40

Rearranging the above we obtain,

First Order Process Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                                     41

Gp(s) is called the transfer function of the process. Kp and τp are called as gain and time constant of the process. The unit of gain is the ratio of the units of output to that of input, whereas the unit of time constant is same as that of time.

 Example of a first order process

The following figure represents a water storage system.
First Order Process Electrical Engineering (EE) Notes | EduRev

It has a cylindrical tank of cross sectional area . Water flows into the tank with a rate of Fi and flows out of the tank with a rate of Fo . Height of water level is represented by h . A control valve, located on the inlet pipe, indicates that Fi can be considered as the manipulated input of the process whereas h can be considered as the controllable output of the process. From the basic knowledge of fluid mechanics (Bernoulli's principle), it is understood that pressure head of the water column provides the necessary kinetic energy for water to eject out of the tank at its bottom. Hence,

First Order Process Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                              42

where, ρ is the density of water, g is the gravitational constant, v is the velocity of water, a is the cross sectional area of the exit pipe. Thus,

First Order Process Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                           43

Where, is a process constant. A simple material balance would yield the following equation:

Accumulation of water inside the tank = Water flow in – Water flow out

or,

First Order Process Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                       44

The steady state (nominal) point of operation is around  First Order Process Electrical Engineering (EE) Notes | EduRev . The above equation is a nonlinear equation and hence this needs to be linearized before a Laplace Transform can be used. Hence, in order to linearize the nonlinear term (√h) of the above equation around the nominal point of operation, Taylor series expansion is carried out on it as the following: 

First Order Process Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                            45

Hence, eq.(44) can be re-written as,
First Order Process Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                          46

At steady state,
First Order Process Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                47

Subtracting eq.(47) from eq. (46), we obtain
First Order Process Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                  48

or

First Order Process Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                                 49

Where,  First Order Process Electrical Engineering (EE) Notes | EduRev  and  First Order Process Electrical Engineering (EE) Notes | EduRev  are the deviation variables of inlet flow rate of water and height of the water level in the tank respectively. Taking the Laplace Transform of eq.(49), we obtain

First Order Process Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                         50

or
First Order Process Electrical Engineering (EE) Notes | EduRev                                                                                                                                                          51

The above equation indicates that the water storage system is a first order process whose
First Order Process Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                                               52

 

 

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