Trays hydraulic balance

Refer to the earlier section of this chapter concerning tray hydraulics. Please note that the balance of the tray hydraulics fixes all downcomer areas when one side downcomer area is determined. This is true for any multitray liquid pass 2-pass, 3-pass, or 4-pass and greater. Please refer again to Fig. 3.1, which diagrams these multitray liquid passes. The following equations are therefore presented ... 

The symbols are P for profit, / for equality constraints, g for inequality constraints, x for optimization variables, y for dependent variables, and / (constant) for updated parameters. The objective function is a scalar measure of plant profit it is usually the instantaneous profit ( /hr), because the optimization variables do not involve the time value of money. Typical equality constraints include material and energy balances, heat and mass transfer relationships, and thermodynamic and kinetic models, and typical inequality constraints include equipment limitations limit compressor horsepower, and distillation tray hydraulics. The optimization variables are flow rates, pressures, temperatures, and other variables that can be manipulated directly. The dependent variables involve intermediate values required for the detailed models for example, all distillation tray compositions, flow rates, and temperatures. Because of the fundamental models often used in RTO, the number of dependent variables can be quite large, on the order of hundreds of thousands. 

The pressure P and the vapor flow V are the result of the energy balance and the pressure drop relationship. The liquid fraction x follows from the liquid component balance. Finally, the tray mass A/ and the liquid flow L are determined by the mass balance and the tray hydraulics. This relationship does not determine any intensive variable. The relationships between flie variables are shown in Fig. 2.2. 

The mass balance together with the tray hydraulics determines the liquid flow. 

The model equations are described in the subsequent sections. The sequence of the sections is (i) material balances including equilibrium relations (ii) energy balances (iii) tray hydraulics and (iv) tray pressure drop. 

In the previous sections the various model parts were described material balances including equilibrium relationships, energy balances, tray hydraulics and tray pressme drop. The coherence of these model elements is shown in a simplified tray behavioral model shown in Fig. 16.7. 

Once the weight of liquid on one portion— the lowest area—of a tray deck exceeds the dry tray pressure drop, the hydraulic balance of the entire tray is ruined. Vapor flow through the low area of the tray deck ceases. The aeration of the liquid retained by the weir on the low area of the tray deck stops, and hence the hydraulic tray pressure drop increases even more. As shown in Fig. 4.3, the liquid now drains largely... 

For the ideal chemical cases, a dynamic model is simulated in Matlab. This model consists of ordinary differential equations for tray compositions and algebraic equations for vapor-liquid equilibrium, reaction kinetics, tray hydraulics, and tray energy balances. The dynamic model is used for steady-state design calculations by mnning the simulation out in time until a steady state is achieved. This dynamic relaxation method is quite effective in providing steady-state solutions, and convergence is seldom an issue. 

Suggestion Run a few trials of any tray design you wish on the Trayxx program and observe the downcomer areas. Try 1-, 2-, 3-, and 4-pass trays, noting downcomer areas. You will soon realize the importance of keeping a balanced hydraulic system. 

A detailed model of the pilot-plant MVC was derived and validated against experimental data in a previous study (Barolo et al., 1998 and also see Chapter 4). The model consists of material and energy balances, vapour liquid equilibrium on trays (with Murphree tray efficiency to account for tray nonideal behaviour), liquid hydraulics based on the real tray geometry, reflux subcooling, heat losses, and control-law calculations based on volumetric flows. The model provides a very accurate representation of the real process behaviour, but is computationally expensive for direct use within an optimisation routine. Greaves et al. (2003) used this model as a substitute of the process. 

Bolles (191) correlated the reduction in efficiency in terms of the distribution ratio, i.e., the maximum-pass LfV ratio divided by the minimum-pass LfV ratio. The L and V for each pass are determined from the normal pressure balance and hydraulic relationships, applied to each pass. At high distribution ratios, a substantial drop in tray efficiency occurs. Bolles shows that if this distribution ratio is kept lower than 1.2, the loss in efficiency due to maldistribution is negligible. Bolles recommends designing multipass trays for such low distribution ratios. Detailed guidelines for achieving low distribution ratios (<1.2), thus minimizing the effects of pass maldistribution on efficiency, are contained in a companion book (1) and in Bolles s paper (191). 

The segments of the buildup are (a) the equivalent clear liquid bead on the tray h Ll (b) any hydraulic gradient A caused by resistance to liquid flow across the tray, which usually is not significant for sieve trays, (c) liquid head equivalent to pressure loss due to flow under the downcomer apron. A. and (d) total pressure loss across the tray above, necessarily included to maintain the dynamic pressure balance between point A (just above the floor of tray 3) and point B in the vapor space above tray 2. 

The RD model consists of sets of algebraic and differential equations, which are obtained from the mass, energy and momentum balances performed on each tray, reboiler, condenser, reflux drum and PI controller instances. Additionally, algebraic expressions are included to account for constitutive relations and to estimate physical properties of the components, plate hydraulics and column sizing. Moreover, initial values are included for each state variable. A detailed description of the mathematical model can be found in appendix A. The model is implemented in gPROMS /gOPT and solved using for the DAE a variable time step/variable order Backward Differentiation Formulae (BDF). 

Momentum Balance on Liquid Phase This balance is written in order to consider the hydraulic delay that is usually experienced. The momentum balance written on a tray becomes... 

Flooding is the most common of the hydraulic constraints likely to be encountered. There are two mechanisms that cause it. The first, downcomer flooding, arises if the maximum internal liquid rate is exceeded. Liquid flows through the downcomer under gravity. The level of liquid built up in the downcomer is as result of a balance between the pressure drop across it and the head of liquid held above it on the tray. As the flow increases, the pressure drop increases (with the square of the flow) and so the head must increase. Ultimately the level reaches the tray above and the tray ceases to provide any separation. 

To simulate multistage separation processes with reactions, the reaction equations must be included in the model equations. The component balances. Equation 13.2 or 13.49, will include an additional term that represents the rate of generation or disappearance of components by kinetic reactions. Another equation is included in the model to represent composition changes due to equilibrium reactions. The kinetic reaction rate is calculated by a power law expression. A holdup term is computed from tray geometry and fluid hydraulics. The equilibrium conversion is calculated from the equilibrium constant, which may be determined from experimental data or calculated from Gibbs free energies. 

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