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CONTROL OF A STEAM-JACKETED KETTLE
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The dynamic response and control of the steam-jacketed kettle shown in Fig. 21.1 are to be considered. The system consists of a kettle through which water flows at a variable rate w lb/time. The entering water is at temperature Ti, which may vary with time. The kettle water, which is well agitated, is heated by steam condensing in the jacket at temperature TV and pressure pv. The temperature of the water in the kettle is measured and transmitted to the controller. The output signal from the controller is used to change the stem position of the valve, which adjusts the flow of steam to the jacket. The major problem in this example is to determine
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FIGURE 21-1 C4mtml of a steam-jacketed ke.ttle.
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the dynamic characteristics of the kettle. The kettle is actually a nonlinear system, and in order to obtain a linear model a number of simplifying assumptions are needed. Analysis of Kettle The following assumptions are made for the kettle: 1. The heat loss to the atmosphere is negligible. 2. The holdup volume of water in the kettle is constant. 3. The thermal capacity of the kettle wall, which separates steam from water, is negligible compared with that of the water in the kettle. 4. The thermal capacity of the outer jacket wall, adjacent to the surroundings, is finite, and the temperature of this jacket wall is uniform and equal to the steam temperature at any instant. 5. The kettle water is sufficiently agitated to result in a uniform temperature. 6. The flow of heat from the steam to the water in the kettle is described by the expression 4 = U(Z , - To) where q = flow rate of heat, Btu/(hr)(ft2) U = overall heat-transfer coefficient, Btu/(hr)(ft2)( F) TlJ = steam temperature, T To = water temperature, I The overall heat-transfer coefficient U is constant,
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7. The heat capacities of water and the metal wall am constant. 8. The density of water is constant. 9. The steam in the jacket is saturated. The assumptions listed hem am mote or less arbitrary. For a specific kettle operating under a particular set of conditions, some of these assumptions may require modification. The approach to this problem is to make an energy balance on the water side and another energy balance on the steam side. In order to aid the development of the transfer functions, a schematic diagram of the kettle is shown in Fig. 21.2. The symbols used throughout this analysis am defined as follows: Ti = To = TV = T, = W= w, = w, = m = ml = v = c = Cl = A = t = H, = H, = u, = temperature of inlet water, T temperature of outlet water, F temperature of jacket steam, F temperature of condensate, T flow rate of inlet water, lb/time flow rate of steam, lb/time flow rate of condensate from kettle, lb/time mass of water in kettle, lb mass of jacket wall, lb volume of jacket steam space, ft3 heat capacity of water, Btu/(lb)( F) heat capacity of metal in jacket wall, Btu/(lb)(T) cross-sectional area for heat exchange, ft2 time specific enthalpy of steam entering, BtuAb specific enthalpy of condensate leaving, BtuAb specific internal energy of steam in jacket, BtuAb Pv = density of steam in jacket, lb/ft3
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FlGURE 213
Schematic diagram of kettle.
THEORETICAL
ANALYSIS
OF COMPLEX PROCESSES
An energy balance on the water side gives wC(Ti - To) + UA(T, - To) = rnC% (21.1)
InEq.(21.1),thetermsC,U,A,andmareconstants.ThefirstterminEq.(21.1) is nonlinear, since it contains the product of flow rate and temperature, that is, wTi and w To. In order to obtain a transfer function from Eq. (21. l), these nonlinear terms must be linearized. Before continuing the analysis, we shall digress briefly to discuss the general problem of linearization of a function of several variables. Consider a function of two variables, z(x,y). By means of a Taylor series expansion, the function can be expanded* around an operating point x S,yS as follows: z = Z(XstYs) + g IX& (x - x,) + 2 lx,.y, (Y - Ys) + higher-order terms in (X - x,) and (y - yS) The subscript s stands for steady state. In control problems, the operating point (x s,ys), around which the expansion is to be made, is selected at steady-state values of the variables before any disturbance occurs. Linearization of the function z consists of retaining only the linear terms, on the basis that the deviations (x - x J, etc., will be small. Thus, (21.3) z = zs + z&$(x - x,) + Z,,(Y - Ys) where znS and zY, are the partial derivatives in Eq. (21.2). If z is a function of three or more variables, the linearized form is the same as that of Eq. (21.3) with an additional term for each variable. The linearization expressed by Eq. (21.3) may be applied to the terms WTi and wT,, in Eq. (21.1) to obtain (21.2)
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