Linearisation using Taylors series

The linearisation method employed in Section 7.5.2 considers the dependent variable (volumetric flowrate of liquid leaving a tank Q ) being perturbed about its steady-state value Q in response to changes in the level of liquid z in the tank about its steady-state level z. Thus, Qx can be expressed as a function of z in terms of a Taylor s series, viz.  

Neglecting higher order terms (assuming that (z - z ) is small) yields the corresponding linear approximation  

This procedure can be extended to functions of any number of variables(,). For example, suppose that the mass flowrate w and temperature 8 of a particular process stream both vary about their steady-state values w and 8 respectively. The rate of heat flow in the stream at any instant QH (which is assumed also to fluctuate about its steady-state value Q due to the changes in w and 8) can be written in terms of Taylor s series  

For small variations in w and 6, terms involving powers higher than unity in equation 7.177 can be neglected, i.e.  

Water is heated by passing it through a steam-heated kettle (Fig. 7.78) at a mass flowrate w. The inlet and outlet temperatures of the water are 9, and 9 respectively. Steam condenses in the jacket of the kettle at a temperature 0, and a pressure f. It is intended to control the temperature of the water by placing a temperature sensor in the water in the kettle and using this measurement to manipulate the flow of steam to the kettle jacket. In order to tune the controller it is necessary to derive the transfer functions relating 0o to 0j, 0, and w. 

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This is a non-linear relationship. Linearising using Taylor s series, as given in equation 7.24, Volume 3 ... 

The time derivative of Y (t) can be calculated in two ways. For the first method (a linearisation), a Taylor series expansion is used with higher-order terms neglected ... 

Linearising equations 7.39 and 7.40 using Taylor s series (Section 7.5.2), expressing them in deviation form and transforming leads to ... 

Further simplification can be achieved by linearising the water gas-shift reaction rate, and using Taylor s series expansion the flowing expression for the shift reaction can be obtained... 

In order to establish suitable relationships between k and F it is necessary to consider small changes in the variables and by so doing linearise the equations using a two-term Taylor s series. Transpose and square the positive root of (2.52). 

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