The mathematical model of a TPP cold end system consists of mathematical models for the following components: the cooling tower, the steam condenser, the low-pressure part of the steam turbine, and the circulating water pumps and pipelines. The goal of the mathematical model is to find a mathematical relationship between the process parameters and the dimensional parameters of the system equipment and to establish the mutual dependence of the decision variables based on the general principles of the laws of physics and generally accepted engineering calculation methods.

The mathematical model of a natural draft cooling tower consists of the mathematical model for the thermal calculation of the cooling tower and the mathematical model for aerodynamic calculation of the cooling tower. Water cooling in a wet cooling tower is the result of two physical processes: heat transfer by convection and mass transfer by evaporation. The intensity of these processes is determined by two laws of physics (Newton’s law and Dalton’s law) which form the basis for obtaining Merkel’s differential equations on which the thermal calculation of the cooling tower is based [18] .

$\text{d}Q=G_{\text{cw}}\cdot c_{\text{cw}}\cdot \text{d}{T}_{\text{cw}}={\beta }_{\text{xv}}\cdot \left(i^{″}-i\right)\cdot \text{d}V$ (10)

$\text{d}i=\frac{c_{\text{cw}}}{\lambda }\cdot \text{d}{T}_{\text{cw}}$, where $\lambda =\frac{G_{\text{a}}}{G_{\text{cw}}}$ (11)

In his approach, Merkel neglected the amount of water that evaporates in the cooling process. L. D. Bermam introduced a correction factor that considers the amount of water that evaporates, which is calculated by the formulas [19] :

$k=1-c_{\text{cw}}\cdot \frac{T_{\text{cw2}}}{r_{\text{Tcw2}}}$ (12)

$r_{\text{Tcw2}}=r_0-\left(c_{\text{cw}}-c_{\text{swv}}\right)\cdot T_{\text{cw2}}$ (13)

Considering equation (12), equation (11) can be written in the form:

$\text{d}i=\frac{c_{\text{cw}}}{k\cdot \lambda }\cdot \text{d}{T}_{\text{cw}}$ (14)

The solutions for equations (10) and (14), for the case of counterflow, as shown in Figure 4 , can be written in the form:

$Me={\displaystyle {\int }_{T_{\text{cw2}}}^{T_{\text{cw1}}}\frac{c_{\text{cw}}}{i^{″}-i}\cdot \text{d}{T}_{\text{cw}}}={\displaystyle {\int }_0^V\frac{{\beta }_{\text{xv}}}{G_{\text{cw}}}\cdot \text{d}V}$ (15)

$Me=\frac{c_{\text{cw}}\cdot \left(T_{\text{cw1}}-T_{\text{cw2}}\right)}{6}\cdot \left[\frac{1}{{i^{″}}_1-i_2}+\frac{4}{{i^{″}}_{\text{m}}-i_{\text{m}}}+\frac{1}{{i^{″}}_2-i_1}\right]=A\cdot {\lambda }^n\cdot H_{\text{CTf}}$ (16)

$i_2-i_1=\frac{c_{\text{cw}}\cdot \left(T_{\text{cw1}}\cdot T_{\text{cw2}}\right)}{k\cdot \lambda }$ (17)

Expression (16) is obtained based on the following conditions and assumptions:

- numerical integration of $Me={\displaystyle {\int }_{T_{\text{cw2}}}^{T_{\text{cw1}}}\frac{c_{\text{cw}}}{i^{″}-i}\cdot \text{d}{T}_{\text{cw}}}$ using Simpson’s rule [20] where i_m” is enthalpy of saturated air at temperature $T_{\text{cwm}}=\frac{T_{\text{cw1}}+T_{\text{cw2}}}{2}$, and $i_{\text{m}}=\frac{i_1+i_2}{2}$

- βxv does not depend on the thermodynamic parameters of water and air, which was confirmed by tests,

- mass transfer coefficient in the cooling tower fill has the form:

${\beta }_{\text{xv}}=A\cdot {\left(\frac{G_{\text{a}}}{A_{\text{CTf}}}\right)}^n\cdot {\left(\frac{G_{\text{cw}}}{A_{\text{CTf}}}\right)}^m$ (18)

and considering that for most types of the cooling tower fills, m = 1 ˗ n.

The aerodynamic calculation of a natural draft cooling tower is based on the condition that available draft and total air flow resistance in the cooling tower must be equal [21] [22] .

$H_{\text{CTb}}\cdot \left({\rho }_{\text{a1}}\cdot {\rho }_{\text{a2}}\right)\cdot g={\displaystyle {\sum }_1^n{\zeta }_n\cdot \frac{v_n^2}{2}\cdot {\rho }_n}$ (19)

where, H_CTb is effective tower height of buoyancy, m; ρ_a1 is air density at the CT entrance, kg/m³; ρ_a2 is air density at the CT exit, kg/m³; g is acceleration due to gravity, m/s²; ζ_n is local pressure loss coefficient; v_n is local air velocity, m/s; ρ_n is local air density, kg/m³.

Since dominant airflow resistance is in the cooling tower fill, it is common to express the air flow resistance in the tower as a function of the air velocity in the tower fill (v_CTf) and the average air density in the tower fill (ρ_am) [21] [22] :

${\displaystyle {\sum }_1^n{\zeta }_n\cdot \frac{v_n^2}{2}\cdot {\rho }_n}={\zeta }_{\text{t}}\cdot \frac{v_{\text{CTf}}^2}{2}\cdot {\rho }_{\text{am}}$ (20)

${\rho }_{\text{am}}=\frac{{\rho }_{\text{a1}}+{\rho }_{\text{a2}}}{2}$ (21)

where, ζ_t is the total air flow resistance coefficient reduced to the air velocity in the tower fill.

If the air speed in the tower fill is expressed through the mass flow of air G_a, in kg/h, it can be written:

$v_{\text{CTf}}=\frac{G_{\text{a}}}{3600\cdot A_{\text{CTf}}\cdot {\rho }_{\text{am}}}$ (22)

$v_{\text{CTf}}=\frac{\lambda \cdot G_{\text{cw}}}{3600\cdot A_{\text{CTf}}\cdot {\rho }_{\text{am}}}$ (23)

If the water flow G_cw, in kg/h, is expressed through the parameter that represents the hydraulic water load of the tower fill q_CTf, in m³/m²h, the equation (23) is transformed into the following form:

$v_{\text{CTf}}=\frac{\lambda \cdot q_{\text{CTf}}\cdot A_{\text{CTf}}\cdot {\rho }_{\text{cw}}}{3600\cdot A_{\text{CTf}}\cdot {\rho }_{\text{am}}}$ (24)

$v_{\text{CTf}}=\frac{\lambda \cdot q_{C\text{Tf}}}{3.6\cdot {\rho }_{\text{am}}}$ (25)

From equations (19), (20), (21) and (25), the final expression for the effective tower height of buoyancy is obtained in the form:

$H_{\text{CTb}}={\zeta }_{\text{t}}\cdot {\left(\frac{\lambda \cdot q_{\text{CTf}}}{11.276}\right)}^2\cdot \frac{1}{{\rho }_{\text{a1}}^2-{\rho }_{\text{a2}}^2}$ (26)

The required total tower height can now be calculated [21] :

$H_{\text{CT}}=H_{\text{CTb}}+0.5\cdot \left(H_{\text{CTf}}+0.5\right)+0.75\cdot H_{\text{CTi}}$ (27)

Expression (27) considers that the heat transfer in the cooling tower, in addition to the cooling fill, also occurs in the rain and spray zones, as pointed out by D. Bohn & K. Kusterer in chapter 6 of reference [23] , i.e., these zones are a part of the cooling fill.

The total air flow resistance coefficient ζ_t is calculated according to the methodology given in references [21] [22] [24] [25] , where the local flow resistance coefficients in the cooling fill and drift eliminators are calculated according to the expressions given in references [26] and [27] .

The mathematical model of the steam condenser is based on the heat transfer equation:

$Q_{\text{SC}}=U_{\text{SC}}\cdot A_{\text{SC}}\cdot \Delta T_{\text{LMTD}}$ (28)

$\Delta T_{\text{LMTD}}=\frac{\Delta T_{\text{cw}}}{\ln \frac{\Delta T_{\text{cw}}+\Delta T_{\text{TTD}}}{\Delta T_{\text{TTD}}}}$ (29)

From equation (28), the area of the steam condenser (A_SC) is calculated. The number of condenser tubes (N_SCt) and the length of the condenser tubes (L_SCt) are calculated according to expressions (30) and (31), respectively [28] :

$N_{\text{SCt}}={10}^6\cdot \frac{4}{\pi }\cdot \frac{G_{\text{cw}}\cdot z}{{\rho }_{\text{cw}}\cdot v_{\text{SCt}}\cdot d_{\text{ID}}^2}$ (30)

$L_{\text{SCt}}={10}^3\cdot \frac{A_{\text{SC}}}{N_{\text{SCt}}\cdot \pi \cdot d_{\text{OD}}}$ (31)

The average heat transfer coefficient of a steam condenser (U_SC) can be calculated according to the methodology given in references [28] and [29] .

The purpose of the mathematical model of the LPST is to give the dependence of the turbine power change on the pressure in the SC, on the steam flow through the last stage of the turbine and on the design parameters of the last stage. In the most general case, this dependence can be obtained by a detailed calculation of the turbine, in various modes of operation. This is not usually done in optimization calculations. Either curves provided by the turbine manufacturer are used or a theoretical model is used that gives very good results according to measurements on objects. In this paper, the latter procedure was adopted, the base of which can be found in references [30] - [32] . Three cases of the turbine operating mode are characteristic: the subcritical flow region when ΔP_LPST < 0 (Equation 32), the supercritical flow region when ΔP_LPST > 0 (Equation 33), and the transitional flow region when ΔP_LPST = 0 [32] .

$\Delta P_{\text{LPST}}=G_{\text{s}}\cdot a^{\ast }{}^2\cdot \left[\frac{1}{k-1}\cdot \left(1-{\epsilon }_k^{\frac{k-1}{k}}\right)\cdot {\eta }_{\text{oi}\ast }-\frac{1}{2}\cdot \left({\epsilon }_k^{\frac{-2}{k}}-1\right)+\frac{u\cdot \cos {\beta }_2}{a^{\ast }}\cdot \left({\epsilon }_k^{\frac{-1}{k}}-1\right)\right]$ (32)

$\Delta P_{\text{LPST}}=G_{\text{s}}\cdot u\cdot a^{*}\cdot x\cdot \left\{{\left[\frac{k+1}{k-1}\left(1-\frac{2}{k+1}{\epsilon }_k^{\frac{k-1}{k}}\right)-{\epsilon }_k^{\frac{-2}{k}}\cdot {\sin }^2{\beta }_2\right]}^{\frac{1}{2}}-\cos {\beta }_2\right\}$ (33)

$A_2=D_{\text{m}}\cdot \pi \cdot l_{\text{STb}}\cdot \sin {\beta }_2$ (34)

$p^{*}=\frac{a^{*}\cdot G_{\text{s}}}{{\mu }_2\cdot k\cdot A_2\cdot N_{\text{LPST-ES}}}$ (35)

${\epsilon }_{\text{k}}=\frac{p_{\text{cond}}}{p^{*}}$ (36)

$u=\frac{D_{\text{m}}\cdot \pi \cdot n_{\text{r}}}{60}$ (37)

where, G_s is steam flow rate, kg/s; a* is critical speed of sound, m/s: k is isentropic coefficient of steam in the LPST last stage; ɳ_oi* is internal efficiency coefficient of the LPST last stage; β2 is exit steam velocity angle of the LPST last stage, ˚; x is moisture content of the steam exiting the LPST last stage, %; D_m is mean diameter of the LPST last stage, m; l_STb is blades length of the LPST last stage, m; μ₂ is flow coefficient of the LPST last stage; N_LPST-ES is number of exit sections of the LPST; n_r is number of revolutions of the ST, rpm.

When steam expansion in a turbine reaches the “limit vacuum” (p_cond < p*), further lowering the condenser pressure doesn’t increase power output from the LPST. This limit is determined by the following relationship [32] :

${\epsilon }_{\text{kl}}=\sin {\beta }_2^{\frac{2k}{k+1}}$ (38)

$p_{\text{cond-ΔPLPSTmax}}=p^{*}\cdot {\epsilon }_{\text{kl}}$ (39)

The main purpose of the mathematical model of cooling water pumps and cooling water pipelines is to calculate the power used to drive the cooling water pumps:

$P_{\text{CWP}}=\frac{{\rho }_{\text{cw}}\cdot g\cdot Q_{\text{CWP}}\cdot H_{\text{CWP}}}{{\eta }_{\text{CWP}}}$ (40)

$H_{\text{CWP}}=H_{\text{CWPs}}+H_{\text{CWPd}}$ (41)

The static head of the CWP (H_CWPs) measures the total vertical distance that the cooling water pump raises water in the cooling tower. The dynamic (frictional) head of the CWP (H_CWPd) is spent on overcoming frictional flow resistance in the system and can be expressed as the sum of the pump head spent on overcoming the flow resistance in the cooling water pipeline (ΔH_CWPL) and the flow resistance in the steam condenser (ΔH_SC). For the calculation of ΔH_CWPL, it is most convenient to use the Hazen-Williams empirical formula, which is intended for turbulent water flow. The ΔH_SC can be calculated according to the methodology given in references [28] and [29] .

In the past, the main focus was to make the optimization calculation procedures for an engineering system as efficient as possible to save computer time, but now this approach has little meaning, and more importance is given to simplicity of implementation. Based on this premise, the exhaustive search method [33] [34] was used in this study for optimal design of the recirculating cooling water system of TPPs. The main strength of the exhaustive search method is that it is guaranteed to find the optimal solution from among the domain specified for the decision variables.

For the numerical solution of the system of nonlinear equations defined in the mathematical model and the objective function, the computer program (written in FORTRAN) has been developed. It consists of the following components: a main program, a subroutine for the cooling tower thermal calculation, a subroutine for the cooling tower aerodynamic calculation, a subroutine for calculating the temperature of the cold water in the cooling tower as a function of the plant location and prevailing atmospheric conditions, a subroutine for the steam condenser sizing, a subroutine for calculating the steam condensation pressure, a subroutine for calculating the power of the low-pressure part of the steam turbine, and several subroutines for calculating thermodynamic properties of fluids. Figure A1 in Annex shows the algorithm of the main computer program based on the exhaustive search algorithm for solving the optimization problem.

In the program, the decision variables (T_app, ΔT_cw, q_CTf, H_CTi, H_CTf, ΔT_TTD, and v_SCt) vary between the lower and upper bounds with a variation step fixed for each variable. The AC_CWS is calculated for each variation of the decision variables in an iterative procedure. The AC_CWS calculated in the previous iteration (AC_CWSpi) is compared with the AC_CWS in the next iteration (AC_CWSni), and only the smaller value is memorized so that at the end of the iterative procedure the minimum value of the annual costs of the cooling water system AC_CWSmin is obtained, which corresponds to the optimal values of the decision variables (ΔT_app-opt, ΔT_cw-opt, q_CTf-opt, H_CTi-opt, H_CTf-opt, ΔT_TTD-opt, and v_SCt-opt).

The exhaustive search algorithm has been streamlined (by adjusting the variation step for all the decision variables) to prune the discretized search space with the intention to remediate the time and space complexity of the algorithm.

All the decision variables, at the end of the optimization process, should be between the lower and upper set values, apart from the parameters of ΔT_app and ΔT_TTD, whose minimum values are included in the optimization constraints. If one or more of the decision variables, which are not included in the optimization constraints (ΔT_cw, q_CTf, H_CTi, H_CTf, and v_SCt), at the end of one iteration cycle are at the lower or upper bounds of their given values, then their given bounds are moved until all the decision variables fall within their given intervals. To achieve this, it usually takes several cycles because it often happens that when one of the decision variables “falls” within the limits of the given interval, it “drives” one or more of the other decision variables to their lower or upper limit. For this reason, at the end, the variation step for all the decision variables was reduced to 0.1.

The existing 300 MW TPPs Gacko and Ugljevik with the steam turbine type K-300-240 LMZ and the cold end system components (CT, SC, LPST, and CWPs) of known design parameters and dimensional characteristics were used in the validation process of the proposed optimization model.

The mathematical model and computer program were checked in such a way that the calculation results of the cold end system components were compared with their actual characteristics for the same design and operating conditions. The results are found to be conforming and accurate. In addition, based on the author’s extensive design experience, the optimal values of the decision variables and optimal sizes of the cold end system equipment are within the expected ranges.

In this part (Part I) of the article, Case Study 1 is presented as the base case study.

Input design and operating conditions for the SC are:

Notes:

1) The values of the above-listed input data and constraints are chosen to be as realistic as possible.

2) Constraints of the process and dimensional parameters of the system equipment resulting from the balance of mass and energy and the general laws of physics, as stated in the mathematical models, are implied and are not specifically stated here.