Proceedings
of the Tomsk Polytechnic Institute named after S. M. Kirov, awarded the Order of the October Revolution and the Order of the Red Banner of Labor

Volume 205　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1972

On the Optimization of Design Parameters of Boiler Units

I. K. Lebedev, V. V. Kharlamov

(Presented by the scientific seminar of the Department of Boiler Construction and Boiler Installations)

The choice of the most economical version of a boiler unit by the currently used method of variant-by-variant comparison is associated with carrying out, for many variants, cumbersome technical and economic calculations. The proposed method for determining the optimal design parameters of boiler units is intended to reduce the number of variants considered in the comparison; its application makes it possible to find the absolute minimum of the costs of producing a unit of steam.

In what follows, the optimal version of a boiler unit will be understood as the economically most advantageous version satisfying a number of technical requirements determined by the thermotechnical characteristics of the fuel, the parameters of the working heat-transfer medium, the properties of the structural materials used, etc. In analytical form these requirements may be expressed by the following relations:

a) for the temperature of the outgoing flue gases

\[ Q_2 + a_{ukh} I^0_{khv} \geq \vartheta^0_{\omega ukh}, \tag{1} \]

where: \(\omega_{ukh}\) is the water equivalent of the volumes of outgoing flue gases, referred to 1 kg of fuel, kcal/kg deg;

\(\vartheta^0\) is the minimum limiting permissible value of the flue-gas temperature under the conditions of acid corrosion of heating surfaces, \({}^{\circ}\mathrm{C}\).

In this and subsequent formulas, generally accepted notation is not specified.

b) the total heat absorption of all \(k\) stages of the air heater, from the condition of ensuring the normal process of fuel combustion,

\[ \sum_{1}^{k} Q_{VP_k} + \beta'' I^0_{khv} \geq \beta'' \omega_v t^0_{GV}. \tag{2} \]

In relation (2):

\(Q_{VP_k}\) is the heat absorption of the \(k\)-th stage of the air heater, kcal/kg;

\(t^0_{GV}\) is the minimum permissible air-heating temperature from the condition of ensuring the normal process of fuel combustion, \({}^{\circ}\mathrm{C}\);

\(\omega_v\) is the water equivalent of air at \(t^0_{GV}\), referred to 1 kg of fuel, kcal/kg deg.

c) for the temperature of the flue gases at the furnace outlet

\[ \sum_{1}^{n} Q_{\mathrm{HT}_n}+\sum_{1}^{p} Q_{\mathrm{PE}_p}^{\mathrm{rad}} \geqslant \varphi \left[ Q_p^{p}\frac{100-q_3-q_6}{100}-Q_{\mathrm{BBHШ}}+\sum_{1}^{k} Q_{\mathrm{B}\Pi_k}+r I_{\mathrm{comb}}+ \right. \]

\[ \left. +\left(\vartheta''+\Delta \alpha_t+\Delta \alpha_{\mathrm{pl.u}}\right) I_{\mathrm{XB}}^{0} -\omega_m'' t_1 \right]. \tag{3} \]

Here: \(Q_{\mathrm{HT}_n}\) is the heat absorption of the \(n\)-th element of the steam-generating surfaces of the furnace, kcal/kg;

\(Q_{\mathrm{PE}_p}^{\mathrm{rad}}\) is the heat absorption of the \(p\)-th element of the radiant superheater, kcal/kg;

\(t_1\) is the maximum permissible temperature of the flue gases at the furnace outlet under slagging conditions, °C;

\(\omega_m''\) is the water equivalent of the flue gases at the temperature at the furnace outlet, kcal/kg deg.

As in the preceding cases, relation (3) determines the maximum permissible value of the temperature of the flue gases at the furnace outlet, the value of which in the optimal solution may also be lower.

d) the temperature of the flue gases before individual elements of the boiler unit, under the conditions of ensuring reliable operation of these elements,

\[ \vartheta_{ji}\leqslant \vartheta_i \quad \text{or} \quad \vartheta_{ji}\omega_i \leqslant \vartheta_i\omega_i \tag{4} \]

where: \(\vartheta_i\) is the maximum permissible value of the temperature of the flue gases before the \(i\)-th element of the boiler unit, °C;

\(\vartheta_{ji}\) is the temperature of the flue gases before the same element of the boiler unit in its optimal variant, °C;

\(\omega_i\) is the water equivalent of the flue gases within the gas duct of the element under consideration, kcal/kg·deg.

e) Sum of the heat absorptions of individual elements of the boiler unit:

1. Preheating and evaporative surfaces

\[ \sum_{1}^{n} Q_{\mathrm{HT}_n}+\sum_{1}^{m} Q_{\mathrm{ek}_m} = \frac{D_n}{B_p}\left(i_n''-i_{\mathrm{pv}}\right). \tag{5} \]

1. Superheating surfaces (of the first, second . . . \(z\)-th superheats)

\[ \sum_{1}^{p_z} Q_{\mathrm{PE}_{p_z}} = \sum_{1}^{z} \frac{D_z}{B_p}\left(i''-i'\right). \tag{6} \]

In equations (5) and (6) the following notation is used:

\(D_i\) is the flow rate of the heat carrier being heated through the \(i\)-th element of the boiler unit, kg/h;

\(Q_{\mathrm{ek}_m}\) is the heat absorption of the \(m\)-th economizer stage, kcal/kg;

\(i_n''\), \(i_{\mathrm{pv}}\), \(i''\), \(i'\) are, respectively, the heat contents of saturated steam, feed water, and steam at the inlet and outlet of the \(z\)-th superheat, kcal/kg.

Unlike the four preceding ones, relations (5) and (6) are strict equalities, the right-hand sides of which are determined by the amount of the heated heat carrier passed through the element of the boiler unit and by the final and initial parameters of this heat carrier (pressure and temperature).

f) The heat absorption of each element in the optimal variant of the boiler unit cannot be negative, \(Q_i \geqslant 0\) (7).

The set of relations (1)–(7), from the analytical point of view, is a system of constraints within which the economically most advantageous optimal variant of the boiler unit must be found.

The stated constraints may be satisfied by an infinitely large number of variants even of a single type of boiler unit, which will differ from one another in the design characteristics of the elements, the velocities of the heat carriers in them, the magnitudes of heat absorption, and the temperature conditions of the elements. The greatest difficulty in finding the optimal variant of a boiler unit lies in determining the economically most advantageous magnitudes of heat absorption and the corresponding temperature conditions of its elements. The present article is devoted to solving this problem.

In this case, the design characteristics of the elements of the boiler unit, the velocities of the heat carriers in them and, consequently, the heat-transfer coefficients, as well as the order of arrangement of the elements, are assumed to be known. In accordance with this reservation, the unknowns of this variant of the problem of optimizing the design parameters of a boiler unit will be the magnitudes of the heat absorptions and the temperature conditions of its elements. The minimum calculated costs are adopted as the optimality criterion.

For the heating surfaces of a boiler unit in which heat is transferred from the flue gases to the working heat carrier, the costs per unit increase in the energy potential of the heated heat carrier, irrespective of its phase state and parameters, consist of: 1. The cost of the thermal energy of the fuel, delivered to the furnace; 2. The costs of organizing the transfer of thermal energy in an element of the boiler unit from the heating heat carrier to the heated one; 3. Additional costs proportional to the losses of the thermal energy of the fuel accompanying the process of heat transfer from the flue gases to the heated heat carrier.

The magnitude of these costs is determined by the formula:

\[ Z_1 = \frac{B_p}{Q_{ka}} \left[ \sum_{1}^{m} (y + p_{\mathrm{ek}_m}) Q_{\mathrm{ek}_m} + \sum_{1}^{n} (y + p_{\mathrm{nt}_n}) Q_{\mathrm{nt}_n} + \sum_{1}^{p_z} (y + p_{\mathrm{PE}_{p_z}}) Q_{\mathrm{PE}_{p_z}} + y Q_{\mathrm{loss}} \right]. \tag{8} \]

Here \(y\) is the cost of a unit of thermal energy in fuel delivered to the furnace, rub/kcal;

\(p_i\) are the specific calculated costs for the \(i\)-th element of the boiler unit, required to organize the transfer of heat in this element from the heating heat carrier to the heated one, rub/kcal;

The magnitude of the specific calculated costs for an individual element of the boiler unit per hour is expressed by the following relation:

\[ p_i = \frac{\sum a \cdot C_i h_i}{n}, \tag{9} \]

where: \(\sum a\) is the total percentage of deductions, including expenditures for current and capital repairs, physical depreciation, and deductions for expanded reproduction with a standard payback period for additional capital investments \(T_n = 8\) years, \(1/\text{year}\);

\(C_i\) is the capital investment per unit of heating surface of an element of the boiler unit (including the cost of a unit of surface);

heating surfaces, framework, brickwork, casing, ladders, platforms, induced-draft and forced-draft installation, and electric power for in-house needs), rubles/m\(^2\);

\(h_i\) is the value of the specific heating surface for the corresponding element of the boiler unit, m\(^2\) h/kcal;

\(n\) is the number of hours of use of the boiler unit per year.

The specific heating surface in the \(i\)-th element of the boiler unit is determined through the heat-transfer coefficient of this same element, taken as a constant value, and the temperature head, according to the equation

\[ h_i=\frac{1}{k_i\Delta t_i}. \tag{10} \]

Heat recirculation in the boiler unit with air and flue gases entails an additional increase in costs. Heat recirculation with air is associated with costs for organizing the transfer of fuel energy from the gases to the air in the air-heater elements and for return to the furnace chamber. Analogously to the second component of costs for the heating surfaces of the boiler unit in which energy is transferred to the heated working heat carrier, the costs for the stages of the air heater are determined by the equation

\[ Z_{\mathrm{ah}}=B_p\sum_{1}^{k} P_{\mathrm{ah}_k}Q_{\mathrm{ah}_k}, \tag{11} \]

where: \(P_{\mathrm{ah}_k}\) is the specific calculated costs for the \(k\)-th stage of the air heater, rubles/kcal;

The costs for heat recirculation with flue gases are determined by the equation

\[ Z_{\mathrm{rc}}=B_p b_{\mathrm{rc}} r I_{\mathrm{amb}} . \tag{12} \]

Here: \(b_{\mathrm{rc}}\) is the cost per unit of recirculated heat, rubles/kcal.

\(r I_{\mathrm{amb}}\) is the heat of the recirculated gases, kcal/kg.

At the same time, the magnitude of the temperature head at the cold end of the first economizer stage in the direction of water flow has a considerable influence on the value of the costs for the boiler unit.

Taking equations (8), (9), (10), (11), and (12) into account, the hourly costs for the boiler unit are expressed by the following equation:

\[ Z=B_p\left[ \sum_{1}^{m}\left(y+\frac{\sum a C_{\mathrm{ec}_m}}{n k_{\mathrm{ec}_m}\Delta t_{\mathrm{ec}_m}}\right)Q_{\mathrm{ec}_m} +\sum_{1}^{n}\left(y+\frac{\sum a C_{\mathrm{sh}_n}}{n k_{\mathrm{sh}_n}\Delta t_{\mathrm{sh}_n}}\right)Q_{\mathrm{sh}_n} +\right. \]

\[ \left. +\sum_{1}^{p_z}\left(y+\frac{\sum a C_{\mathrm{rh}_{p_z}}}{n k_{\mathrm{rh}_{p_z}}\Delta t_{\mathrm{rh}_{p_z}}}\right)Q_{\mathrm{rh}_{p_z}} +\sum_{1}^{k}\frac{\sum a C_{\mathrm{ah}_k}}{n k_{\mathrm{ah}_k}\Delta t_{\mathrm{ah}_k}}Q_{\mathrm{ah}_k} +yQ_{\mathrm{loss}}+b_{\mathrm{rc}}rI_{\mathrm{amb}} \right]. \tag{13} \]

Knowing the order of arrangement of the elements of the boiler unit, and taking account of constraints (1)--(4), one can approximately specify the boundaries of the flue-gas temperatures within whose interval displacement of the \(i\)-th element is possible in the process of searching for the optimal solution. Taking this into account, from the thermotechnical characteristics of the fuel and the calculated characteristics of the boiler unit, the average values of the water equivalents of the heat-exchanging media within the gas path of each element can be determined with sufficient accuracy. From these characteristics of the heat carriers, the temperature regimes of the elements are expressed as functions of the magnitudes of their heat absorptions.

It follows from the foregoing that the total costs for a boiler unit can be expressed by a nonlinear equation as a function of the heat perceptions of its elements.

The combination of the system of constraints (1)–(7) and the cost equation (13) constitutes a mathematical model of the boiler unit. The nonlinearity of the cost equation (the objective function) determines the affiliation of this problem with one of the types of nonlinear mathematical programming problems, whose solution is possible by the methods of classical analysis.

The continuity of equation (13) over the entire admissible set of values of the heat perceptions of the boiler-unit elements makes it possible to conclude that, at least once on this set, the function attains a maximum and a minimum value. At the extremal points of the function, its first partial derivatives are equal to zero.

Consequently, the problem of optimizing the design parameters of a boiler unit is reduced to solving a system of nonlinear equations obtained by differentiating the cost equation (13) with respect to all linearly independent variables entering into its expression—the heat perceptions of the boiler-unit elements—and to checking whether the solutions obtained satisfy the requirements of the system of linear constraints (1)–(7). Solutions that do not satisfy the requirements of the system of constraints (1)–(7) are inadmissible.

Conclusions

1. This work has shown the possibility of constructing a general mathematical model of a boiler unit for the purpose of subsequently applying methods of quantitative analysis in the process of searching for its optimal design parameters.

2. By purely mathematical techniques it is possible to reveal the tendencies in the behavior of the cost equation (13) with respect to the linearly independent variables entering into its expression—the heat perceptions of the individual elements of the boiler unit.

References

1. Thermal calculation of boiler units (standard method). Gosenergoizdat, 1957.

2. C. Karr, C. Houw. Quantitative Methods of Decision Making in Management and Economics. Mir Publishers, 1966.

3. G. V. Venikov, V. A. Stroev. Application of mathematical methods and computer technology in the design and operation of power systems. Energiya Publishers, 1965.