2.1 The method of energy balances in the analysis of an irreversible process.
The second method for estimating the dissipative function is based on the use of a system of integral equations for a fixed control volume of a thermodynamic system (FTS). The configuration of the FTS boundaries is unchanged, but the boundaries themselves are permeable for convective and non-convective flows of mass and energy. These equations relate the rate of change of extensive quantities (total mass, mass of an individual k-th component, total energy, kinetic and potential energy, entropy, exergy) with the causes causing these changes: flows carried across FTS boundaries and internal sources, if any.
Gross mass balance equation:
where – total mass density, kg/m3; – convective flow of the total mass through the permeable sections of the FTS (sections i = 1.2 ... n), kg/s.
Total energy balance equation:
where – kinetic energy density, J/m3;
– potential energy density, J/m3;
– internal energy density, J/m3;
, , – mass average values of enthalpy, velocity, specific gravitational potential, J/kg;
– flow of external work in the form of kinetic or electrical energy, W;
– convective heat flux, W;
– enthalpy flux supplied due to the mass transfer process at the boundary of the system and the external environment, W.
The calculation of the enthalpy flow is carried out according to the ratio:
where – partial value of the enthalpy of the k component, determined by local values of temperature, pressure, composition at the boundary, J/mol;
– density of the mass flux of k component (k = 1, 2 … m) through the boundaries of the FTS, determined by the conditions of the mass transfer process at the boundary, (mole of component k)/(m2∙s);
– normal to the surface;
– surface area, m2.
Balance equation of kinetic and potential energy:
where – functional that determines the source or sink of kinetic energy during one-dimensional motion of a continuous medium with a finite velocity in the field of pressure forces P, W; – dissipation of kinetic energy, W.
Entropy balance equation:
where – entropy density, J/(m3∙K);
– mass-average value of entropy in the i-th section of the FTS apparatus
(i = 1, 2 … n), J/(kg∙K);
– entropy flux introduced together with the heat flux at equilibrium heat transfer at the boundary, W/K. It is determined by the ratio (1.4);
– entropy flux introduced with the convective flow of matter in an equilibrium mass transfer process at the boundary, W/K. It is determined by relation (1.5);
– rate of entropy production within the FTS due to the irreversibility of the process, W/K.
Mass balance equation for each component k:
where – partial density of component k in the mixture (k = 1, 2 … m), mol/m3;
– mass fraction of component k in the mixture (k = 1, 2 … m),
(kg of k component)/kg of mixture;
– molecular weight of k component, kg/kmol;
– nonconvective flow of the k component through the FTS boundary, mol/s;
– rate of the reaction resulting in the appearance or disappearance of k component, (mol of k component)/(m3∙s).
These relations (2.1 - 2.6) connect the rate of change of extensive quantities with the causes that cause them, both external, due to the convective flows of the substance through the permeable sections of the fixed control volume of the apparatus, and internal sources, if any. The left side of equations (2.1 - 2.6) is the rate of accumulation of extensive characteristics, i.e. changes per unit time of the values of the mass of the component, total mass, entropy, total energy, kinetic and potential energy. The values of these quantities are determined by the distribution of the density of the corresponding substance over the entire volume of the thermodynamic system. On the right side of equations (2.1 - 2.6), the first terms represent the resulting flow of substance due to visible movement due to convective transfer through the open sections of the apparatus. To calculate these quantities, the mass-average values of enthalpy , entropy , velocity , mass fraction of the k component are used. . External flows of substance include flows of heat , substance , due to reversible heat and mass transfer processes at the boundary, the flow of external work, , as well as flows of entropy and , and introduced together with heat and substance flows through the permeable sections of the apparatus under reversible heat and mass transfer conditions at the boundary.
To the inner sources of substance, i.e. generated within the process due to its irreversibility, include: the reaction rate, as a result of which the k component appears or disappears ; the rate of entropy production within the process; ; dissipation of kinetic energy , i.e. the rate at which kinetic energy is converted into internal energy due to overcoming frictional forces.
We illustrate the application of the method using the example of task 1 on page 9.
Calculate the dissipation of kinetic energy during the movement of a 50% aqueous solution of glycerol in a smooth pipe with a radius R = 0.03 m and a length L = 25 m under isothermal conditions. Laminar motion mode
Re = 1800. Glycerin parameters: pressure P = 20 bar, temperature t = 40 °C.
We tackle the task using the differential equation for the balance of kinetic and potential energy (2.4) for the FST of a thermodynamic system:
where ΔРтр can be calculated using the Hagen–Poiseuille equation, since the mode of motion is laminar.
Calculation of pressure loss in straight smooth pipes during isothermal flow can be carried out according to the following formula:
where – dimensionless coefficient of friction, in laminar motion .
We substitute the values into the calculation formula (2.7):
The result shows the equivalence of both approaches to the analysis of dissipation during the motion of viscous media.
2.2 Exergy method for the analysis of a thermodynamic system.
The main tasks that should be solved when analyzing the efficiency of energy consumption in chemical production are as follows:
1) a generalized assessment of material and energy resources based on classical thermodynamics;
2) analysis of the efficiency of energy and substance consumption in non-equilibrium processes;
3) search for ways to improve the efficiency of the process, i.e. development of general principles for the rational use of material and energy resources;
4) the applied part of the energy-chemical-engineering system (ECES), i.e. assessment of technological aspects of production.
Evaluation of the effectiveness of the use of matter is based on the law of conservation of matter. Below we present the mass balance equation for a real process (2.8). ∑_(i=1)^n▒〖νiMi=∑_(j=1÷a)^m▒〖νjMj+∑_(j=1)^a▒〖νjMj= ∑_(j=1)^m▒νjMj〗〗〗 (2.8)
∑_(i=1)^n▒νiMi – total flow of incoming substances, including the main components and auxiliary ones. Catalysts can be an example of the additive.
∑_(j=1)^m▒νjMj – total flow of outgoing substances, including:
∑_(j=1÷a)^m▒νjMj – by-products (semi-products, waste);
∑_(j=1)^a▒〖νjMj 〗– target products.
In chemical production, there are solid wastes, gas emissions, and liquid effluents. The processing of these flows (crushing, filtering, settling) consumes a significant amount of energy, but the cost of these costs is justified by the guarantee of the technospheric safety of the regime. Waste may be negligible, but their harmfulness, i.e. toxicity is very high. This circumstance is taken into account by environmental indicators and sanitary and epidemic standards, i.e. indicators that are directly focused on the cost of the final product.
Based on the value of the mass utilization factor ηM, the actual cost of the target product (2.8) is estimated:
ηM = 1 - (∑_(j=a+1)^m▒νjMj)/(∑_(i=1)^n▒νiMi) (2.9)
Energy consumption is estimated by the energy utilization factor η_E (2.10):
η_E=(∑_(j=1)^a▒〖v_j H ̃_j 〗+Q ̇_k (T ̅_k )+w ̇)/(∑_(i=1)^n▒〖v_i (H_i ) ̃ 〗+Q ̇_k (T_k>T_c )-w ̇ ) (2.10)
Equation (2.9) is derived from the total energy balance equation (2.2):
∑_(i=1)^n▒〖v_i H ̃_i 〗-∑_(j=1)^m▒〖v_j (H_j ) ̃ 〗+ΣQ ̇_k (T ̅_k )-w ̇Э¬ = 0
где:∑_(j=1)^m▒〖v_j H ̃_j 〗 – resulting convective enthalpy fluxes;
∑Q ̇_k (T ̅_k ) – convective heat fluxes that have their own thermal potential;
-w ̇Э – the total flow of electricity that is supplied (removed) to the ECES.
Relationship (2.9) does not take into account the mass transfer processes occurring on the control surface with the environment, since they occur, as a rule, within the system itself, as well as the kinetic and potential (gravitational) components of energy. The contribution of these types of energies is evaluated in specific processes and devices of a given design under certain conditions (for example, a jet apparatus).
η_E≅1-(E ̇_ср^noT)/E ̇^' = (losses to the environment)/(supplied energy input) (2.11)
where: (E_ср ) ̇^noт – loss of energy (heat, matter) to the environment due to poor tightness (thermal insulation), which go away with waste (gas emissions, liquid effluents, solid waste);
E ̇^'-flows of energy that are supplied to the system.
Evaluation of energy efficiency, i.e. a qualitative indicator of energy consumption is carried out using the Gibbs component of energy, i.e. exergy.
In general terms, the integral exergy balance equation is as follows (2.12):
(E') ̇_x=E ̇_(х целевые)^''+ⅇxD ̇+ⅈnD ̇ (2.12)
(E') ̇_x – incoming exergy flows; E ̇_(х целевые)^'' - target exergy flows;
ⅇxD ̇ – external exergy losses that go into the environment. These losses are due to the waste with which they take place;
ⅈnD ̇ – internal exergy losses, which are caused by ⅈnS ̇_12 (the flow of entropy produced within the process itself due to irreversibility.
The model of the analytical apparatus with indication of exergy flows is represented by
in fig. 4.
Fig. 4. Functional diagram of a real energy converter.
We introduce the concept of exergy efficiency of η_ex, i.e. target efficiency in terms of the Gibbs energy (2.13):
η_ex=goal/costs=(E ̇_(x goal)^'')/(E ̇_x^(' costs) )=1-(ⅇxD ̇+ⅈnD ̇)/〖E ̇_x〗^'costs (2.13)
Attention should be paid to the expediency of adding an exergy balance and determining the value of the exergy efficiency of the entire installation, as well as the influence on its value of the contribution of its individual links. Such an analytical approach makes it possible to determine not only quantitative indicators of the consumption of material and energy resources, but also to compare and evaluate the efficiency of converting various types of energy.
The technological process should be divided into stages, since it is important to obtain the values of both the overall efficiency and the efficiency, as well as for each individual stage, so that these quantities (2.14) can be analyzed.
η_(ex whole installation)=1-(∑_(i=1)^f▒D ̇_(i whole installation) )/(E_(x whole installation)^' ) (2.14)
Estimation of the total loss of exergy of the i-th section is determined by the Gouy-Stodola formula
ⅈnD ̇=To.c. ⅈnS ̇_12
D ̇_i/(E_x^' )=(D ̇_i⋅E ̇_(x_i)^')/(E_x^'⋅E ̇_(x_i)^' )=(1-η_(ex_i ) )⋅ⅆ_i (2.15)
a_i=(E ̇_(x_i)^')/(E_x^' )=ⅆ_i – share of exergy introduced in the i-th stage from the total exergy flow introduced into the entire installation as a whole;
D ̇_i/(E ̇_(x_i)^' )=(1-η_(ex_i ) ), where η_(ex_i )- the exergy efficiency of i-th stage
Let's analyze the expression in comparison with the previously obtained formula:
η_(ex уст-ки)=1-(∑_(i=1)^f▒D ̇_i )/(E ̇_(x уст-ки)^' )=1-∑_(i ̇=1)^f▒〖(1-η_(ex_i ) ) d_i 〗 (2.16)
D ̇уст-ки = ∑_(i=1)^f▒D ̇_i - losses of the entire installation as a whole equal to the sum of losses in all stages (with i=1…до f).
We have presented a formula for dividing the installation by space, and perhaps division by stages, i.e. different processes occurring in the same space. There are also variable parameters (eg reflux ratio). If there are many such parameters, then multiparameter optimization is carried out.
External exergy losses ⅇxD ̇ – due to the loss of matter and energy into the environment (imperfection of thermal insulation, waste, effluents, emissions). For the final choice of the technology mode, the economic side of the problem, environmental indicators and hardware design are taken into account. Technospheric security is a unifying problem for all civilized countries and excites the minds of both world-famous scientists and young scientists.