Method of Referred Derivatives

Solving nonlinear simultaneous equations in a process modei the Method of Referred Derivatives... 

Equation (2.98) represents a set of k linear simultaneous equations in the k unknowns, dzifdt, which are calculated by reference to the derivatives of the state variables and the derivatives of the inputs - hence the name Method of Referred Derivatives. 

Using the Method of Referred Derivatives, it is possible to integrate the vector dz/dt in the same way as the vector dx/dt. Thus this method replaces the need to solve a set of nonlinear, simultaneous equations at each timestep by the simpler requirement of solving a set of linear, simultaneous equations, followed by integration of the resultant time-differentials from a feasible initial condition, z(0). 

Thomas, P.J. (1997). The Method of Referred Derivatives a new technique for solving implicit equations in dynamic simulation. Trans. Insi.M.C., 19, 13-21,... 

The last example of Chapter 11 dealt with the case where a liquid and a gas were present together, but the gas was inert with respect to the liquid. However, there are a large number of systems on a process plant where a liquid and its own vapour are present together evaporators, condensers, steam drums, deaerators, refrigeration systems, stills and distillation columns. These systems exist in a state of vapour-liquid equilibrium, and their behaviour is significantly different from the gas-liquid system dealt with in Chapter 11, Section 11.6. The liquid and its vapour will have the same temperature, and it will not be possible to decouple the mass and energy balance equations for the liquid from those of the vapour. The way to obtain the necessary time differentials explicitly is to use the Method of Referred Derivatives. 

We now have in (12.1) and (12.6) equations in our chosen state variables liquid mass and temperature. But the fact that we do not know the boiloff rate prevents us from solving these equations without the additional differential equation (12.2) for the mass of vapour and the algebraic equations for volume and pressure, (12.9) and (12.13). We may now use the Method of Referred Derivatives, outlined in Chapter 2, Section 2.11 (and in Chapter 18, Section 18.7 in further detail), to put these algebraic constraints into the appropriate form to be used in the differential equations defining the state variables. 

An alternative method of solving the implicit equations associated with complex flow networks is to use the Method of Referred Derivatives discussed in Section 2.11 of Chapter 2. The general set of nonlinear, simultaneous equations... 

However, an option more in keeping with the Method of Referred Derivatives is that of Prior Transient Integration . At an assumed time, tp r, prior to the transient start time, we select an artificial set of states, x(t/,rmr)f and inputs, u(tp,jer), chosen solely on the basis that they enable a direct (i.e. non-iterative) solution for the variables z to be found at time tprwr from ... 

Worked example using the Method of Referred Derivatives liquid flow network... 

Table 18.4 Pressures computed by the Method of Referred Derivatives compared with the values found by Iteration...

Table 18.8 Intermediate pressures after 100 s found by applying the Method of Referred Derivatives to the full flow equations and to the modified flow equations. Comparison with exact values...

Given an initial temperature for the node, T, it is possible to find the specific internal energy, u = u(T), and the specific volume, v = v(T), and hence the mass m = V/v. Equation (18.65), taken in conjunction with auxiliary equations (18.63), represents an implicit equation in the nodal pressure, p, which may be solved using the methods already outlined, either iteration or the Method of Referred Derivatives. The upstream and downstream flows, Wyp, and Wj , may then be found, so that it becomes possible to calculate the right-hand side of the temperature differential equation (18.64). Equation (18.64) may then be integrated to find the temperature of the node at the next timestep. The process may then be repeated for the duration of the transient under consideration. 

These pi +k2 + k nonlinear simultaneous equations in Pi -f 2 + unknowns may be solved at each timestep using a nonlinear equation solver of the type discussed in Chapter 2, Section 2.10 or by the Method of Referred Derivatives, discussed in Section 2.11 and in Chapter 18, Sections 18.7 to 18.9. The application of the latter method will be discussed further in the next section. 

Applying the Method of Referred Derivatives, we argue that since G is identically zero at all times, it follows that its derivative will be zero also ... 

Applying the Method of Referred Derivatives to the case where the model has no explicit dependence on time (i.e. 9G/9f = 0) produces the linear equation... 

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