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FIGURE 5-l Cross-sectional view
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thermometer.
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The following assumptions* will be used in this analysis: 1. All the resistance to heat transfer resides in the film surrounding the bulb (i.e., the resistance offered by the glass and mercury is neglected). 2. All the thermal capacity is in the mercury. Furthermore, at any instant the mercury assumes a uniform temperature throughout. 3. The glass wall containing the mercury does not expand or contract during the transient response. (In an actual thermometer, the expansion of the wall has an additional effect on the response of the thermometer reading. (See Iinoya and Altpeter (1962) .) It is assumed that the thermometer is initially at steady state. This means that, before time zero, there is no change in temperature with time. At time zero the thermometer will be subjected to some change in the surrounding temperature x(t). By applying the unsteady-state energy balance Input rate - output rate = rate of accumulation we get the result dy h A ( x - y ) - 0 = mC, where A = surface area of bulb for heat transfer, ft2 C = heat capacity of mercury, Btu/(lb,)( F) m = mass of mercury in bulb, lb, t = time, hr h = film coefficient of heat transfer, Btu/(hr)(ft2)(T) For illustrative purposes, typical engineering units have been used.
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(5.1)
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*Making the first two assumptions is often referred to as the lumping of p a r a m e t e r s because all the resistance is lumped into one location and all the capacitance into another. As shown in the analysis, these assumptions make it possible to represent the dynamics of the system by an ordinary differential equation. If such assumptions were not ma&, the analysis would lead to a partial differential equation, and the representation would be referred to as a distributed-parumeter system. In Chap. 21, distributed-parameter systems will be considered in detail.
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Equation (5.1) states that the rate of flow of heat through the film resistance surrounding the bulb causes the internal energy of the mercury to increase at the same rate. The increase in internal energy is manifested by a change in temperature and a corresponding expansion of mercury, which causes the mercury column, or reading of the thermometer, to rise. The coefficient h will depend on the flow rate and properties of the surrounding fluid and the dimensions of the bulb. We shall assume that h is constant for a particular installation of the thermometer. Our analysis has resulted in Eq. (5. l), which is a first-order differential equation. Before solving this equation by means of the Laplace transform, deviation variables will be introduced into Eq. (5.1). The reason for these new variables will soon become apparent. Prior to the change in x, the thermometer is at steady state and the derivative dyldt is zero. For the steady-state condition, Eq. (5.1) may be written t<O hA(x, - ys) = 0 (5.2) The subscript s is used to indicate that the variable is the steady-state value. Equation (5.2) simply states that yS = n $, or the thermometer reads the true, bath temperature. Subtracting Eq. (5.2) from Eq. (5.1) gives
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hA[tx - xd - (Y - ~~11 = mC
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d(y - ys) dt
(5.3)
Notice that d(y - ys)ldt = dyldt because y, is a constant. If we define the deviation variables to be the differences between the variables and their steady-state values x=x-xxs
y=y-Ys
Eq. (5.3) becomes hA(X - Y) = rnC% If we let mClhA = T, Eq. (5.4) becomes x-y=g!r dt Taking the Laplace transform of Eq. (5.5) gives X(s) - Y(s) = TSY(S) Rearranging Eq. (5.6) as a ratio of Y( S) to X(S) gives 1 Y(s) -=7s + 1 X(s) (5.7) (5.6) (5.5) (5.4)
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