barcode reader project in c#.net Eq. (9.3) becomes in Software

Maker Code 128B in Software Eq. (9.3) becomes

Eq. (9.3) becomes
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Q + wC(T; - T ) = ,CVg (9.7)
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Taking the Laplace transform of Eq. (9.7) gives Q(s) + wC[T, (s) - T (s)] = pCVsT (s) or
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= - + T;(s)
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(9.8)
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Q(s)
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(9.9)
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*In this analysis, it is assumed that the flow rate of heat q is instantaneously available and independent of the temperature in the tank. In some stirred-tank heaters, such as a jacketed kettle, q depends on both the temperature of the fluid in the jacket and the temperature of the fluid in the kettle. In this introductory chapter, systems (electrically heated tank or direct steam-heated tank) are selected for which this complication can be ignored. In Chap. 21, the analysis of a steam-jacketed kettle is given in which the effect of kettle temperature on q is taken into account.
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This last expression can be written (9.10) where
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If there is a change in Q(t) only, then T, (t) = 0 and the transfer function relating T to Q is
T (s)= l/WC 7s + 1 Q(s)
(9.11)
If there is a change in T;(t) only, then Q(t) = 0 and the transfer function relating T to T/ is (9.12) I;: (s) Equation (9.10) is represented by the block diagram shown in Fig. 9.3~. This diagram is simply an alternate way to express Eq. (9.10) in terms of the transfer functions of Eqs. (9.11) and (9.12). Superposition makes this representation possible. Notice that, in Fig. 9.3, we have indicated summation of signals by the symbol shown in Fig. 9.4, which is called a summing junction. Subtraction can also be indicated with this symbol by placing a minus sign at the appropriate input. The summing junction was used previously as the symbol for the comparator of the controller (see Fig. 9.2). This symbol, which is standard in the control literature, may have several inputs but only one output. A block diagram that is equivalent to Fig. 9.3~ is shown in Fig. 9.3b. That this diagram is correct can be seen by rearranging Eq. (9.10); thus T (s) = [Q(s) + wCT;(s)]~ (9.13)
_ T (s)
1 7s + 1
FIGURE 9-3 Block diagram for process.
THJ3
CONTROL SYSTEM
In Fig. 9.3b, the input variables Q(S) and wCT/(s) am summed before being operated on by the transfer function llwCl(~s + 1). The physical situation that exists for the control system (Fig. 9.1) if steam heating is used requires more careful analysis to show that Fig. 9.3 is an equivalent block diagram. Assume that a supply of steam at constant conditions is available for heating the tauk. One method for introducing heat to the system is to let the steam flow through a control valve and discharge directly into the water in the tank, where it will condense completely and become part of the stream leaving the tank (see Fig. 9.5). If the flow of steam, f (pounds/time), is small compared with the inlet flow w, the total outlet flow is approximately equal to w. When the system is at steady state, the heat balance may be written WC(Tj, - T,) - wC(T, - To) + f,(H, - HI,) = 0 where tering and leaving tank
H, = specific enthalpy of the steam supplied, a constant HI, = specific enthalpy of the condensed steam flowing out at T,, as part
(9.14)
To = reference temperature used to evaluate enthalpy of all streams en-
of the total stream The term HI, may be written in terms of heat capacity and temperature; thus Hr, = C(Ts - To) (9.15)
From this, we see that, if the steady-state temperature changes, HI, changes. In Eq. (9.14), f,(H, - HI,) is equivalent to the steady-state input qs used previously, as can be seen by comparing Eq. (9.2) with (9.14). Now consider an unsteady-state operation in whichfis much less than w and the temperature T of the bath does not deviate significantly from the steady-state
WI Ti -+ L ---. ;-. -tI!F
Steam at constant conditions
w+fYw
FIGURE 9-5 Supplying heat by steam.
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SYSTEMS
temperature T,. For these conditions, we may write the unsteady-state balance approximately; thus
wC(Tj - To) - wC(T - To) + f(Hg - HI,) = pCVs
(9.16)
In a practical situation for steam, H, will be about 1000 Btu/lb,. If the temperature of the bath, T, never deviates from T, by more than loo, the error in using the term f(Hg - HI,) instead of f(Hg - HI) will be no more than 1 percent. Under these conditions, Eq. (9.16) represents the system closely, and by comparing Eq. (9.16) with Eq. (9. l), it is clear that q = fW, -HI,) (9.17)
Therefore, q is proportional to the flow of steam f, which may be varied by means of a control valve. It should be emphasized that the analysis presented here is only approximate. Both f and the deviation in T must be small. The smaller they become, the more closely Eq. (9.16) represents the actual physical system. An exact analysis of the problem leads to a differential equation with time-varying coefficients, and the transfer-function approach does not apply. The problem becomes considerably more difficult. A better approximation will be discussed in Chap. 21, where linearization techniques ate used.
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