Besides earlier described quasistatic processes of free evaporation, which are stationary and reverse in their limit, in nature and in engineering are widely spread the irreversible processes, that proceed in thermodynamics systems and connect with establishment and maintenance in them of stationary states. As a rule those states develop during long time and can exist in the systems as long as conditions at the system's boundaries are constant. The stable equilibrium in such systems cannot be attained either owing to continuous energy and mass exchange with its environment or owing to limitations and links imposed on their degrees of freedom. However variables defining thermodynamic state of the system become time-independent character since certain moment. This permits one to say the processes running in such systems are also stationary and therefore must be described only by autonomous systems of ordinary differential equations, which do not involve time as independent variable in their right-hand sides.
Further it will be analyzed for convenience only stationary processes of continuous rectification of multi-component mixtures including the simplest ones running in packed or film columns with adiabatic walls and phase counter-flow. This means that in each infinite small element of volume of the rectifying column during the average residence time dt takes place the change liquid composition - dxi that does not perturb the time-independent sequence of average states of liquid, which is established in the whole column. These states are described with a complex function that has taken into account both the driving forces of the process and the corresponding kinetic coefficients and define a rate of the process.
On mathematic description of mentioned processes one has to disregard the actual distribution of vapor and liquid composition and temperature over the column cross-section and to make use of values averaged over the cross-section. It is also assumed that the process runs at constant pressure, which allows one to exclude the hydrodynamics processes associated with the drop in pressure along the column and having different tensor dimensions (Curie's principle). As a result energy dissipation and entropy production related to those processes appear also excluded from consideration.
Simultaneously the vapor of average composition is assumed to be at its dew-point while the liquid is at its boiling point. The thermophysical and thermodynamic properties can be determined by assumption of these conditions.
All said above permits one to formulate within the approximations made a closed mathematical model of the process in a column with adiabatic walls, which contain quite realistic limitations imposed on the system's degrees of freedom, such as integral and differential equations of material balance as well as equations of energy (enthalpy) and entropy balance. During more then 60 years that passed since its first publication in the classic monograph by F.Boshnyakovich [1] this model stood against the test by time and became generally accepted in theory of rectification.
In stationary rectification processes of binary mixture this system of limiting equations and restrictions can be clearly interpreted in terms of doubled diagram enthalpy - entropy - composition [1]; in rectification of multi-component mixtures the same limitations, say, for rectifying part of a column may be formulated as the following system of analytic equations, for classification of which can be used the classification accepted in theoretical mechanics [2].
The geometrical or finite connections. For rectification column with adiabatic walls, working in stationary regime, these connections relate to boundary sections of column. They are deduced from condition, that the maintenance of stationary regime in column is possible in that and only in that event if the loss or increase of mass of separate components will not happen in any its parts, as these change connected with some disruptions of constant composition of vapor and liquid and also its properties and state.
These conditions are mathematical reduced to necessity of obvious observance of general material balance, material balance of each component, as well as balance of energy (enthalpy).
| (1) |
| (2) |
The constant flow of energy (enthalpy) through rectifying part
of column is determined usually by assignment reflux numbers
RD = LD
/ D in top section of column in which
yDi = xD
i are followed from the organization of process.
| (3) | |||||||||||||||||||||||||||||||||||
Differential integrating connection. Equations of material and thermal balances (balance of enthalpy), recorded in differential form, play the paramount role at mathematical description kinetics of process, as establish the quantitative correlation between changes of component flows sizes of vapor and liquid and changes of their composition and thermodynamics properties. These correlation follow from the laws of preservation and should be executed at maintenance of stationary state in each infinitely small volume of column, limited by its intermediate sections [3,4].
| (4) |
| (5) |
| (6) |
| (7) | ||||||||||||||||||||||||||||||||||||
which can be brought to view:
| (8) | ||||||||||||||||||||||||||||||||||||||||
In contrast to differential equations (4)¸(8) integrated equations of material and thermal balance (balance of enthalpy) are determined the absolute size and composition of flows of vapors and liquids, intersecting any intermediate sections of column. They include the constants of integration - D, xDi, qDC, which size is determined by geometrical connections - equations (1)¸(3); the constant of integration in equation (11) equal to unit, is determined by accepted method of concentration's determination of components in moles fraction.
| (9) |
| (10) |
| (11) |
| (12) | |||||||||||||||||||||||||||||||||||||||||
In equations (3) and (12) hold-up, constantly presented in column, is described for clearness as it were consisting from two subsystems:
The first is formed by recycling flows of vapor and liquid, size and composition of which in each section of column are identical equal one another, i.e. y° miº xmi;
The second is formed by flow of vapor having constant size and composition, that come into feed entry section of the rectifying part of column and leave it in quality of distillate.
It may be observed in passing that namely on this subsystem is being made the work of dividing. At other identical condition in columns with adiabatic walls the more size of this work the less is rate of entropies production.
Characteristic feature of equations (9)¸(11) is the fact that they do not reduce (in difference from similar equations, describing laws of preservation in closed thermodynamic systems) alternative change of liquid phase, i.e. do not impose any restrictions on its degrees of freedom.
For example, from analysis of differential equations follows, that the sequence of composition of liquid phase, realizing through height rectifying part of column or, that the same, form of trajectory of process in phase space of concentration, should not depend on the size of draw off distillate and its composition, i.e. from quantity, which on relation to considered system of differential equations are obvious constants of integration.
From consideration of integrated equations the most result is that
it permit to put in conformity to composition of liquid phase in each
section of column the composition of vapor phase, keeping
n - 1 of degrees of freedom for change of composition
of liquid phase.
Note:
For comparison, according to rule of Djugem variance of
closed multi-phase multi-componet system equal 2,
by p = const
variance equal 1.
The restrictions, following from second principle of thermodynamiks. It is known, that at constant pressure on system are imposed the connections, stipulated by existence of one-to-one dependencies between compositions of liquid and its temperatures of boiling, as well as between composition of saturated vapor and its temperature of dew-point.
| (13) |
| (14) |
In turn these quantity determine one of major laws of rectification process so-called rule of Fr.Boshnjakovic [1], directly following from condition of necessity of summary increase entropy in process, in which is produced the work of dividing, i.e. from second principle of thermodynamics. Pursuant to these rules the division of mixes by rectification is possible only in that range of compositions, within the limits of which the temperature of the vapor in any section of column exceeds the temperature of liquid, i.e. the condition is executed only if:
| (15) |
The obvious observance of this rule underlie the choosing during minimum of reflux number, limiting consumption of energy on given dividing; the infringement of rule makes the rectification's process thermodynamic impracticable one.
It is simultaneously to notice, that the requirement, following from disparity (15), is incompatible with conditions, determining the achievement and maintenance of steady thermodynamic equilibrium, major from which assumes the equality of temperatures the vapor and liquid. At number of components of mixture more 2-nd this predetermines obvious erroneous of method of calculating of multi-component rectification processes based on use only representation about theoretical plates or steps of dividing. At dividing of binary mixes the thermodynamic inconsistent mathematical description is excluded only owing to correct choosing of reflux number, as well as because off the dividing system has for its change only one degree of freedom.
Thermal plate as the basis of step calculation rectification columns. As it was mentioned above, the mathematic theory allow give purely kinematic interpretation of a kinetic of stationary rectification process in the concentration phase space, so that the rates of heat and mass exchange is associated with the motion of figurative point along the trajectory of the process [5].
The trajectories of the stationary rectification processes to be solutions autonomous systems of ordinary differential equations according to the classification followed from mathematical theory can be ascribed to semi-stable, which have not any limiting equilibrium positions.
This conclusion is based on the fact that the whole trajectory lose its stability and the separation become thermodynamically impossible if thermal equilibrium between vapor and liquid has established only at one point of the trajectory. This is the main difference between rectification and isothermal absorption processes, which take place along stable trajectories and are completed when the system attains a stable thermodynamic equilibrium.
Another important property of the trajectories is their ability for dividing by thermal equilibrium positions into a calculable number of intervals that can be called thermal plates by analogy to the theoretical plates.
The calculation of number thermal plate practically repeats the calculation theoretical plate, as for binary mixtures both these plate concepts completely coincide.
Knowing the composition distillate, xDi in top section (m = 1) of column find the composition of vapor in top section, using equalities yDi = xDi, and on equation (14) his temperature of dew point, TG1;
Further by means of numerical integration differential equation (16) [6]:
| (16) |
find on trajectory the point having composition of liquid phase x2i, and temperature of boiling - TL2 = TG1 in accordance with equation (13). The interval of trajectory from xm, i to xm+1, i correspond one thermal plate.
On calculating the separation of multi-component mixtures the thermal plate conception can be used to determine the number of real plates and the size of column required in practice. An obvious advantage of this concept is that unlike the theoretical plate concept it does not contain any wholly hypothetical and unrealizable conditions. To latter belong such as ideal mixing of the liquid and infinitely long contact's time of phases within a single plate to be necessary before the system reaches thermodynamic equilibrium.
Using thermal plates conceptions for calculation multi-component separation by analogy with using theoretical plates for calculation binary mixture separation one can formulate the following efficiencies: the local or point efficiency - hO; the efficiency of plate by analogy to Murphree coefficient - hT; the column efficiency - hK. Also one can establish the relationship between those coefficients and the coefficients of heat and mass exchange, thermal units of transfer, etc. [7].
It is important that the deflection of phases from equilibrium during rectification processes cannot be infinitely large. At infinitely large reflux ratio it is limited from above with the condition compositions equality of liquid and vapor (the deflection from equilibrium can be quantitatively described by a dimensionless ratio of difference in temperature of vapor and liquid to their absolute temperature). Under such conditions the kinetics of the process discussed can be described with sufficient accuracy by linear phenomenological equations. Therefore in the first approximation the extent of deflection from equilibrium of the binary mixtures separated in the real plates may be assumed to be the same, if it is determinated with using thermal plate efficiency, hT, or with using Murphree coefficient.
To carry out consideration show the possibility of using the conception of thermal plates for the design of rectifying columns for the separation not only binary bat also multi-component mixtures.
D, Di - distillate flow rate and flow of component with
distillate, respectively, mol/s;
F - feed flow rate, mol/s;
G, Gi - vapour flow and flow of component with vapour
phase flow mol/s;
H - enthalpy and specific mole enthalpy,
J, J/mol;
H
i
,
h
i
- partial mole enthalpies of vapour and liquid
components, J/mol;
K - total mass tranfer coefficient,
mol/(m2.s.(mol/mol));
qDC - constant flow heat through section of rectifying
part of column, J/s;
p - pressure, N/m2;
R - reflux ratio: R = L / D;
T - absolute temperature, °K;
W - reboiler waste output, mol/s;
x, y - component concentrations in liquid and vapour,
respectively, mol/mol;
h0, hp, hk - efficiency: local, of plate, of column;
t - time, s;
| D | - distillate product and top section of column; |
| F | - feed; |
| G | - vapour phase; |
| i, j, k | - component's numbers; |
| L | - liquid phase; |
| m | - number of plate or section of column; |
| n | - number of component , equal to number of components; |
| W | - bottom product. |