Energy balances equilibrium temperature

The calculational base consists of equilibrium relations and material and energy balances. Equilibrium data for many binary systems are available as tabulations of x vs. y at constant temperature or pressure or in graphical form as on Figure 13.4. Often they can be extended to other pressures or temperatures or expressed in mathematical form as explained in Section 13.1. Sources of equilibrium data are listed in the references. Graphical calculation of distillation problems often is the most convenient... 

Here x represents a vector of n continuous variables (e.g., flows, pressures, compositions, temperatures, sizes of units), and y is a vector of integer variables (e.g., alternative solvents or materials) h(x,y) = 0 denote the to equality constraints (e.g., mass, energy balances, equilibrium relationships) g(x,y) < 0 are the p inequality constraints (e.g., specifications on purity of distillation products, environmental regulations, feasibility constraints in heat recovery systems, logical constraints) f(x,y) is the objective function (e.g., annualized total cost, profit, thermodynamic criteria). 

Some examples of equations are mass balances, energy balances, equilibrium relationships, and equalities (identities) of the temperature and/or pressure of two phases. If the system contains C components, then there are C independent mass balances. One example would be a mass balance for each component. In this case, the total mass balance is not independent since it is just the sum of the component balances. 

Vibrational distributions in non-equilibrium plasma are mostly controlled by W-exchange and VT-relaxation processes, while excitation by electron impact, chemical reactions, radiation, and so on determine averaged energy balance and temperatures. At steady state, the Fokker-Planck kinetic equation (3-116) gives J(E) = const. At E oo = 0,... 

From your flowchart, determine the number of unknowns in the process. What qualifies as an unknown depends on what you re looking for, but in a material balance calculation, masses and concentrations are the most common. In equilibrium and energy balance calculations, temperature and pressure also become important unknowns. In a reactor, you should include the conversion as an unknown unless it is given OR you are doing an atom balance. 

Try the following problem to sharpen your skills in working with material and energy balances. Crude oil is heated to 525° K and then charged at a rate of 0.06 m /hr to the flash zone of a pilot-scale distillation tower. The flash zone is maintained at an absolute temperature of 115 kPa. Calculate the percent vaporized and the amounts of the overhead and bottoms streams. Assume that the vapor and liquid are in equilibrium. 

With high concentrations, heat effects in the chromatographic column may be important. This would require the simultaneous application of an energy balance and the introduction of a term reflecting the influence of temperature on the adsorption equilibrium. 

The procedure developed by Joris and Kalitventzeff (1987) aims to classify the variables and measurements involved in any type of plant model. The system of equations that represents plant operation involves state variables (temperature, pressure, partial molar flowrates of components, extents of reactions), measurements, and link variables (those that relate certain measurements to state variables). This system is made up of material and energy balances, liquid-vapor equilibrium relationships, pressure equality equations, link equations, etc. 

When considering heat transfer in a multiphase system, one can either treat each phase separately or the phases can be assumed to be in local equilibrium. In general, it is more complicated to treat each phase separately hence, we will use the local equilibrium approach that assumes all the phases have the same local average temperature. By performing an energy balance over a differential resin element one obtains,... 

Van der Waals molecular volume is the volume contained by van der Waals surface of a molecule which is defined as the surface of the intersection of spheres each of which is centered at the equilibrium position of the atomic nucleus with van der Waals radius of each atom 62). Since the van der Waals radius of an atom is the distance at which the repulsive force balances the attraction forces between two non-bonded atoms, van der Waals molecular volume is regarded as the volume impenetrable for other molecules with thermal energies at ordinary temperatures. 

In order for a process to be controllable by machine, it must represented by a mathematical model. Ideally, each element of a dynamic process, for example, a reflux drum or an individual tray of a fractionator, is represented by differential equations based on material and energy balances, transfer rates, stage efficiencies, phase equilibrium relations, etc., as well as the parameters of sensing devices, control valves, and control instruments. The process as a whole then is equivalent to a system of ordinary and partial differential equations involving certain independent and dependent variables. When the values of the independent variables are specified or measured, corresponding values of the others are found by computation, and the information is transmitted to the control instruments. For example, if the temperature, composition, and flow rate of the feed to a fractionator are perturbed, the computer will determine the other flows and the heat balance required to maintain constant overhead purity. Economic factors also can be incorporated in process models then the computer can be made to optimize the operation continually. 

The specific enthalpies in the above equation can be determined as described earlier, provided the temperatures of the product streams are known. Evaporative cooling crystallizers (described more completely in Section V) operate at reduced pressure and may be considered adiabatic. In such circumstances, Eq. (9) is modified by setting Q = 0. As with many problems involving equilibrium relationships and mass and energy balances, trial-and-error computations are often involved in solving Eqs. (7) through (9). 

From this energy balance we see that the temperature indicated by the thermometer is not the true gas temperature but some radiation-convection equilibrium temperature. Very large errors can result in temperature measurements if this energy balance is not properly taken into account. Radiation shields are frequently employed to alleviate this difficulty. 

We now employ the energy balance indicated in Eq. (12-60) to calculate the radiation equilibrium temperature, taking the radiation surroundings at - 70°C ... 

If the bounding walls are mostly sink-type surfaces of area Ai and temperature Ti, but in small part refractory surfaces of area A in radiative equilibrium at unknown temperature T, an energy balance on Ar is in principle necessary to determine Tr and the effect on energy flux. However, the total heat transfer to the sink may be visualized as corresponding to its having an effective area equal to its own plus a fraction X of that of the refractory, with the only temperatures involved being those of the gas and the heat sink. The fraction x varies from zero when the ratio of refractory to heat-sink surface is very high to unity when the ratio is very low and the value of g is low. If Ar is small compared with Ai, a value for x of 0.7 may be used in the approximate method. 

The number of unknown variables for a single unit is the sum of the unknown component amounts or flow rates for ail inlet and outlet streams, plus all unknown stream temperatures and pressures, plus the rates of energy transfer as heat and work. The equations available to determine these unknowns include material balances for each independent species, an energy balance, phase and chemical equilibrium relations, and additional specified relationships among the process variables. 

Exothermic Reactions. Figures 8-6 and 8-7 show typical plots of equilibrium conversion as a function of temperature for an exotiiermic reaction. To determine the maximum conversion that can be achieved in an exothermic reaction carried out adiabaticaUy, we find the intersection of the equilibrium conversion as a function of temperature with temperature-conversion relationships from the energy balance (Figure 8-7). For Tjo = Tq, ... 

We now consider an adiabatic reactor of fixed sire or catalyst weight and investigate what happens as the feed temperature is varied. ITie reaction is reversible and exothermic. At one temperature extreme, using a very high feed temperature, the specific reaction rate will be large and the reaction will proceed rapidly, but the equilibrium conversion will be close to zero. Consequently, very little product will be formed. A plot of the equilibrium conversion and the conversion calculated from the adiabatic energy balance,... 

Figure 8-7 Graphical solution of equilibrium and energy balance equations to obtain adiabatic temperature and equilibrium conversion.

If the entering temperature is increased from Tq to Toi> the energy balance line will be shifted to the right and will be parallel to the original line, as shown by the dashed line. Note that as the inlet temperature increases, the adiabatic equilibrium conversion decreases. 

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