 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
barcode reader project in c#.net PHYSICAL EXAMPLES OF FIRSTORDER SYSTEMS in Software
CHAPTER Code128 Reader In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Generate ANSI/AIM Code 128 In None Using Barcode encoder for Software Control to generate, create Code128 image in Software applications. PHYSICAL EXAMPLES OF FIRSTORDER SYSTEMS
Scan Code 128 In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. Paint USS Code 128 In Visual C# Using Barcode creator for VS .NET Control to generate, create Code128 image in Visual Studio .NET applications. In the first part of this chapter, we shall consider several physical systems that can be represented by a firstorder transfer function. In the second part, a method for approximating the dynamic response of a nonlinear system by a linear response will be presented. This approximation is called linearization. Print Code 128 In Visual Studio .NET Using Barcode printer for ASP.NET Control to generate, create Code128 image in ASP.NET applications. Printing Code 128B In .NET Using Barcode generator for Visual Studio .NET Control to generate, create Code 128C image in Visual Studio .NET applications. EXAMPLES OF FIRSTORDER SYSTEMS Liquid Level
Create Code 128 In Visual Basic .NET Using Barcode generator for .NET Control to generate, create USS Code 128 image in .NET framework applications. Printing Data Matrix In None Using Barcode generation for Software Control to generate, create ECC200 image in Software applications. Consider the system shown in Fig. 6.1, which consists of a tank of uniform crosssectional area A to which is attached a flow resistance R such as a valve, a pipe, or a weir. Assume that qO, the volumetric flow rate (volume/time) through the resistance, is related to the head h by the linear relationship Painting GS1  12 In None Using Barcode printer for Software Control to generate, create UPC Symbol image in Software applications. Code39 Creation In None Using Barcode encoder for Software Control to generate, create Code 39 Extended image in Software applications. A resistance that has this linear relationship between flow and head is referred to as a linear resistance. A timevarying volumetric flow 4 of liquid of constant density p enters the tank. Determine the transfer function that relates head to flow. Barcode Generation In None Using Barcode drawer for Software Control to generate, create bar code image in Software applications. Barcode Generation In None Using Barcode encoder for Software Control to generate, create barcode image in Software applications. *A pipe is a linear resistance if the flow is in the laminar range. A specially contoured heir, called a Sutro weir, produces a linear headflow relationship. Formulas used to prepare the shape of Paint UPCE Supplement 5 In None Using Barcode generation for Software Control to generate, create GS1  12 image in Software applications. Draw Bar Code In Java Using Barcode creator for Android Control to generate, create barcode image in Android applications. PIWSICALEXAMPLESOPFIRSTORJXR~Y~TEM~
Printing Data Matrix 2d Barcode In None Using Barcode maker for Microsoft Word Control to generate, create Data Matrix image in Microsoft Word applications. DataMatrix Creation In C#.NET Using Barcode encoder for Visual Studio .NET Control to generate, create DataMatrix image in .NET applications. i&)(t) Creating UPCA Supplement 2 In None Using Barcode encoder for Excel Control to generate, create GTIN  12 image in Excel applications. UPCA Encoder In ObjectiveC Using Barcode printer for iPhone Control to generate, create UPC Symbol image in iPhone applications. tank: Scan Data Matrix ECC200 In VB.NET Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. GS1 RSS Printer In .NET Framework Using Barcode printer for VS .NET Control to generate, create GS1 DataBar Truncated image in Visual Studio .NET applications. FIGURE61
Liquidlevel system.
We can analyze this system by writing a transient mass balance around the Mass flow in  mass flow out = rate of accumulation of mass in the tank In terms of the variables used in this analysis, the mass balance becomes Pm  pqm = y q(t)  q&) = A$
(6.2) Combining Eqs. (6.1) and (6.2) to eliminate qO(t) gives the following linear differential equation: q;=A$ (6.3) We shall introduce deviation variables into the analysis before proceeding to the transfer function. Initially, the process is operating at steady state, which means that dhldt = 0 and we can write Eq. (6.3) as where the subscript s has been used to indicate the steadystate value of the variable. Subtracting Eq. (6.4) from Eq. (6.3) gives (4  qs) = ;(h  h,) + Adchd hs) (6.5) such a weir have been mported in the literature; & Soucek, Howe, and Mavis (1936). tibulent flow through pipes and valves is generally proportional to &. Flow through weirs having simple geometric shapes can be expressed as K/a , where K and n are positive constants. For example, the Bow through a rectangularshaped weir is proportional to h . LINEAR OPENLOOP SYSTEMS
If we define the deviation variables as
e=44s
H = h  h , Eq. (6.5) can be written
Q=;H+Az (6.6) Taking the transform of Eq. (6.6) gives
Q(s) = ;H(s) + ASH(S) (6.7) Notice that H(0) is zero and therefore the transform of dH/dt is simply sH(s). Equation (6.7) can be rearranged into the standard form of the firstorder lag to give R H(s) (6.8) 7s + 1 Q(s) where r = AR. In comparing the transfer function of the tank given by Eq. (6.8) with the transfer function for the thermometer given by Eq. (5.7), we see that Eq. (6.8) contains the factor R. The term R is simply the conversion factor that relates h(t) to q(t) when the system is at steady state. For this reason, a factor K in the transfer function Kl(rs + 1) is often called the steadystate gain. We can readily show this name to be appropriate by applying the finalvalue theorem of Chap. 4 to the determination of the steadystate value of H when the flow rate Q(t) changes according to a unitstep change; thus Q(t) = u(t) where u(t) is the symbol for the unitstep change. The transform of Q(t) is
Q(s) = f
Combining this forcing function with Eq. (6.8) gives
H(s) = f& Applying the finalvalue theorem, proved in Chap. 4, to H(s) gives
ff(t) t*m = liio[sH(s)] = liio & = R
This shows that the ultimate change in H(t) for a unit change in Q(t) is simply R. If the transfer function relating the inlet flow q(t) to the outlet flow is desired, note that we have from Eq. (6.1) PHYSICAL EXAMPLES OF FIRSTORDER SYSTEMS
67 (6.9) Subtracting Eq. (6.9) from Eq. (6.1) and using the deviation variable Q, = q.  qo, gives
Qo = # Taking the transform of Eq. (6.10) gives
(6.10) Combining Eqs. (6.11) and (6.8) to eliminate H(s) gives
1Qo(s> _ 7s + 1 Q(s) (6.12) Notice that the steadystate gain for this transfer function is dimensionless, which is to be expected because the input variable q(t) and the output variable qO(t) have the same units (volume/time). The possibility of approximating an impulse forcing function in the flow rate to the liquidlevel system is quite real. Recall that the unitimpulse function is defined as a pulse of unit area as the duration of the pulse approaches zero, the impulse function can be approximated by suddenly increasing the flow to a large value for a very short time; i.e. we may pour very quickly a volume of liquid into the tank. The nature of the impulse response for a liquidlevel system will be described by the following example. Example 6.1. A tank having a time constant of 1 min and a resistance of i ft/cfm is operating at steady state with an inlet flow of 10 ft3/min. At time t = 0, the flow is suddenly increased to 100 ft3/min for 0.1 min by adding an additional 9 ft3 of water to the tank uniformly over a period of 0.1 min. (See Fig. 6.2 for this input disturbance.) Plot the response in tank level and compare with the impulse response. Before proceeding with the details of the computation, we should observe that, as the time interval over which the 9 ft3 of water is added to the tank is shortened, the input approaches an impulse function having a magnitude of 9. From the data given in this example, the transfer function of the process is H(s) = 1  1 9s+l Q(s) The input may be expressed as the difference in step functions, as was done in Example 4.5. Q(t) = 90[u(t)  u(t  O.l)] The transform of this is

