# Determining the Optimum Inner Diameter of Condenser Tubes Based on Thermodynamic Objective Functions and an Economic Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of Condensers under Consideration

_{1}).

## 3. Mathematical Models for Determining the Optimum Condenser Tube Diameter

#### 3.1. Minimization of the Entropy Generation Rate per Unit Length of a Condenser Tube: SL = f(d_{2i}) → min

#### 3.2. Minimization of the Total Entropy Generation Rate: $\dot{S}$ = f(d_{2i}) → min

#### 3.3. Maximization of the Power Unit’s Output: P = f(p_{1}(d_{2i})) → max

#### 3.4. The Economic Method—Profit Maximization: Z = f(d_{2i}) → max

_{a}:

## 4. Calculation Results

_{1}) are the following: pressure ${p}_{ext1}=0.0299\text{\hspace{0.17em}}\mathrm{MPa}$, temperature ${t}_{ext1}=69{}^{\mathrm{o}}\mathrm{C}$, enthalpy ${h}_{ext1}=2615.7\text{\hspace{0.17em}}\mathrm{kJ}/\mathrm{kg}$, and the steam mass flow rate from the extraction ${\dot{m}}_{ext1}=6.795\text{\hspace{0.17em}}\mathrm{kg}/\mathrm{s}$. The efficiency of the group of stages between the inlet of the LP part and the extraction was 0.86.

#### 4.1. Calculation Results for the Condenser in the 200-MW Power Unit

#### 4.2. Calculation Results for the Condenser in the 500-MW Power Unit

## 5. Conclusions

_{a}) is 29 mm. With most condensers of 200-MW power units, the tube inner diameter is 28 mm, while the outer one is 30 mm. For condenser tube diameters between 27 mm and 33 mm, the graph of the annual surplus profit is rather flat, but has a maximum. In this case, a possibly small change in the diameter is recommended, for the condenser in question to a diameter of 29 or 30 mm.

## Author Contributions

## Conflicts of Interest

## Nomenclature

A | heat transfer area, m^{2} |

${a}_{t}$ | discount factor, - |

${c}_{2}$ | specific heat of water, J/(kg·K) |

${c}_{e}$ | electricity price, zł/MWh |

${c}_{A}$ | price of one square meter of the heat transfer area, zł/m^{2} |

${C}_{m}$ | coefficient of heat transfer intensity for steam and water mixture in a bank of tubes |

d | discount rate, - |

${d}_{2i}$ | tube inner diameter, m |

${d}_{2o}$ | tube outer diameter, m |

f | surface area of steam flow across the section between tubes of a bank in the condenser within its outer circumference, m^{2} |

${f}_{c}$ | fixed cost ratio, 1/year |

${F}_{2}$ | cross-section of the cooling water flow, m^{2} |

g | gravitational acceleration, m/s^{2} |

h | specific enthalpy, J/kg |

k | roughness, mm |

$L$ | tube length, m |

$n$ | number of tubes, - |

N | years of power plant operation, year |

Nu | Nusselt number, - |

$N{u}_{n}$ | Nusselt number for steam condensing on a single tube, - |

$\dot{m}$ | rate of mass flow through one tube, kg/s |

${\dot{m}}_{1}$ | steam mass flow rate to the condenser, kg/s |

${\dot{m}}_{2}$ | mass flow rate of cooling water, kg/s |

${\dot{m}}_{t}$ | steam mass flow rate at the inlet of the low-pressure part of the turbine, kg/s |

$p$ | pressure, Pa (abs) |

$\Delta {p}_{p}$ | pressure rise across the pump, Pa |

${P}_{p}$ | pump power, W |

${P}_{t}$ | power generated in the low-pressure part, W |

$P$ | difference in power between the low-pressure part and the pump, W |

Pr | Prandtl number, - |

$\dot{Q}$ | flow rate of the heat transferred, kW |

$\dot{q}$ | heat flow rate per unit tube length, W/m |

$r$ | phase transition heat of condensing steam, $r=h"({p}_{s})-h\text{\'}({p}_{s})$, kJ/kg |

${r}_{l}$ | thermal resistance per unit length, mK/W |

Re | Reynolds number, - |

s | specific entropy, J/(kg·K) |

S | circumference of steam inflow, measured across a section between tubes outside the bank, m |

$\dot{S}$ | sum of entropy rates, W/K |

${s}_{f}$ | ratio of steam inflow in the area between tubes on the outer circumference of a bank, - |

SL | entropy generation rate per unit length of a condenser tube, W/(mK) |

${\dot{S}}_{2,p}$ | entropy generation rate due to the resistance of cooling-water flow, W/K |

${\dot{S}}_{2,\dot{Q}}$ | entropy generation rate due to heat transfer to water, W/K |

${\dot{S}}_{1}$ | entropy generation rate on the steam side, W/K |

${\dot{S}}_{t}$ | entropy generation rate in the low-pressure part, W/K |

${\dot{S}}_{p}$ | entropy generation rate in the pump, W/K |

${\dot{S}}_{Lp}$ | entropy generation due to flow resistance of cooling water, W/(mK) |

${\dot{S}}_{Lq}$ | entropy generation due to heat flow, W/(mK) |

${T}_{2i}$ | cooling water temperature at the condenser inlet, K |

${T}_{2o}$ | cooling water temperature at the condenser outlet, K |

${T}_{2}$ | average cooling water temperature, $\left({T}_{2o}+{T}_{2i}\right)/2$, K |

${T}_{s}$ | steam saturation temperature, K |

$\Delta {t}_{\mathrm{ln}}$ | logarithmic temperature difference across the steam condenser, °C |

$\Delta {t}_{p}$ | mean temperature difference between steam and the external surface of tubes, °C |

$U$ | overall heat transfer coefficient, kW/(m^{2}·K) |

$w$ | velocity, m/s |

z | number of tube runs in a condenser, - |

${Z}_{a}$ | annual surplusprofit, zł/year |

${Z}_{n}$ | profit over N years of operation, zł |

$\alpha $ | coefficient of heat transfer, kW/(m^{2}·K) |

${\alpha}_{p}$ | heat transfer coefficient for steam condensation on a single horizontal tube, kW/(m^{2}·K) |

${\delta}_{f}$ | thickness of the fouling layer, m |

${\epsilon}_{o}$ | ratio of air content in the condenser to steam flow, - |

${\eta}_{p}$ | pump efficiency, - |

${\lambda}_{1}$ | thermal conductivity of steam, kW/(mK) |

${\lambda}_{2}$ | thermal conductivity of cooling water, kW/(mK) |

${\lambda}_{f}$ | thermal conductivity of the fouling layer, kW/(mK) |

${\lambda}_{fr}$ | flow resistance coefficient, - |

${\lambda}_{k}$ | thermal conductivity of condensate, kW/(mK) |

${\lambda}_{m}$ | thermal conductivity of tube material, kW/(mK) |

${\nu}_{k}$ | kinematic viscosity of condensate, m^{2}/s |

${\mathsf{\Pi}}_{s}$ | similarity number for a bank of tubes, - |

$\rho $ | density, kg/m^{3} |

${\rho}_{k}$ | density of condensate, kg/m^{3} |

$\tau $ | annual power plant operation time, h |

## Subscripts

1 | relates to steam |

2 | relates to cooling water |

ext | relates to parameters at the extraction in steam turbine |

f | fouling |

m | tube material |

ti | relates to parameters at the inlet of the low-pressure part |

to | relates to parameters at the outlet of the low-pressure part |

pi | relates to parameters upstream the pump |

po | relates to parameters downstream the pump |

r | relates to reference (nominal) parameters |

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**Figure 1.**The subsystem under consideration, being a part of the 500-MW power unit: the low-pressure part of the turbine (LP) with extractions ext

_{1}to ext

_{3}; the condenser (C); and the cooling-water pump (CWP).

**Figure 4.**Velocity of cooling water in tubes and pressure drop on the cooling-water side in the condenser as functions of the tube inner diameter.

**Figure 5.**The overall heat transfer coefficient and steam pressure in the condenser as functions of the tube inner diameter.

**Figure 6.**The thermal resistance per unit length and the product of the overall heat transfer coefficient and the heat transfer area as functions of the tube inner diameter.

**Figure 7.**Power generated by the LP part of the turbine and power required to drive the cooling-water pump as functions of the tube inner diameter.

**Figure 8.**The entropy generation rate per unit length of a condenser tube with its two components, and the difference between the power generated by the LP part of the turbine and that used by the cooling-water pump as functions of the tube inner diameter.

**Figure 9.**The total entropy generation rate for the system under consideration, comprising the LP part of the turbine, the condenser, and the cooling-water pump, and the annual surplusprofit as functions of the tube inner diameter.

**Figure 10.**Velocity of cooling water in tubes and pressure drop on the cooling-water side in the condenser as functions of the tube inner diameter.

**Figure 11.**The overall heat transfer coefficient and steam pressure in the condenser as functions of the tube inner diameter.

**Figure 12.**The thermal resistance per unit length and the product of the overall heat transfer coefficient and the heat transfer area as functions of the tube inner diameter.

**Figure 13.**Power generated by the LP part of the turbine and power required to drive the cooling-water pump as functions of the tube inner diameter.

**Figure 14.**The entropy generation rate per unit length of a condenser tube with its two components and the difference between the power generated by the LP part of the turbine and that used by the cooling-water pump as functions of the tube inner diameter.

**Figure 15.**The total entropy generation rate for the system under consideration, comprising the LP part of the turbine, the condenser, and the cooling-water pump, and the annual surplus profit as functions of the tube inner diameter.

Item | Symbol | Unit | Value |
---|---|---|---|

Heat transfer area | A | m^{2} | 2 × 5710 = 11,420 |

Number of tubes | n | - | 2 × 6878 = 13,756 |

Cooling-water mass flow rate | ${\dot{m}}_{2}$ | kg/s | 2 × 4000 = 8000 |

Inlet/outlet cooling water temperature, norm. | t_{2i}/t_{2o} | °C | 17/25.7 |

Rated condensed-steam mass flow rate | ${\dot{m}}_{1}$ | kg/s | 129 |

Tube outer diameter | d_{2o} | mm | 30 |

Tube inner diameter | d_{2i} | mm | 28 |

Tube length | L | mm | 9000 |

Mean water pressure | p_{2} | bar (abs) | 3 |

Number of passes | - | - | 2 |

Item | Symbol | Unit | Value |
---|---|---|---|

Heat transfer area | A | m^{2} | 2 × 9500 = 19,000 |

Number of tubes | n | - | 2 × 16,000 = 32,000 |

Cooling-water mass flow rate | ${\dot{m}}_{2}$ | kg/s | 2 × 6700 = 13,400 |

Inlet/outlet cooling water temperature, norm. | t_{2i}/t_{2o} | ^{o}C | 24/32 |

Rated condensed-steam mass flow rate | ${\dot{m}}_{1}$ | kg/s | 207.5 |

Tube outer diameter | d_{2o} | mm | 24 |

Tube inner diameter | d_{2i} | mm | 22.6 |

Tube length | L | mm | 8000 |

Mean water pressure | p_{2} | bar (abs) | 3 |

Number of passes | - | - | 2 |

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**MDPI and ACS Style**

Laskowski, R.; Smyk, A.; Rusowicz, A.; Grzebielec, A.
Determining the Optimum Inner Diameter of Condenser Tubes Based on Thermodynamic Objective Functions and an Economic Analysis. *Entropy* **2016**, *18*, 444.
https://doi.org/10.3390/e18120444

**AMA Style**

Laskowski R, Smyk A, Rusowicz A, Grzebielec A.
Determining the Optimum Inner Diameter of Condenser Tubes Based on Thermodynamic Objective Functions and an Economic Analysis. *Entropy*. 2016; 18(12):444.
https://doi.org/10.3390/e18120444

**Chicago/Turabian Style**

Laskowski, Rafał, Adam Smyk, Artur Rusowicz, and Andrzej Grzebielec.
2016. "Determining the Optimum Inner Diameter of Condenser Tubes Based on Thermodynamic Objective Functions and an Economic Analysis" *Entropy* 18, no. 12: 444.
https://doi.org/10.3390/e18120444