Thermodynamic models and energy distribution of single-phase heated surface in a boiler under unsteady conditions

Abstract

A coal-fired power unit frequently operates under unsteady conditions; thus, in order to acquire scientific energy analysis of the unit, thermodynamic analysis of a single-phase heated surface in a boiler under such conditions requires investigation. Processes are analyzed, and distributions of energy and exergy are qualitatively revealed. Models for energy analysis, entropy analysis, and exergy analysis of control volumes and irreversible heat transfer processes are established. Taking the low-temperature superheater of a 610 t/h-boiler as an example, the distribution of energy, entropy production, and exergy is depicted quantitatively, and the results are analyzed.

Keywords

thermodynamic model / energy distribution / boiler / unsteady conditions

Introduction

Coal-fired power units should operate under steady conditions; however, with the increase in unit scale and peak-valley difference, exceedingly unsteady processes related to startup, shutdown, and other large state variations occur. That is, the unit more frequently operates under unsteady conditions. Despite this situation, correlative research on this issue has been few. Guo [1-3] studied the basic methods, heat distribution in boilers, and coal consumption calculation methods of units under unsteady conditions. Falcetta and Sciubba [4] carried out an exergy-based analysis of the start-up cycle of a combined cycle power plant, and Olslmmer et al. [5] investigated the time-dependent thermoeconomic modeling and optimization of the energy system.

Against this background and taking the single-phase heated surface in a boiler as the study object, models for energy analysis, entropy analysis, and exergy analysis are established, and the corresponding energy distributions are obtained and analyzed. This paper aims to provide scientific energy analysis of the unit under unsteady conditions, and lay the foundation for new energy conservation and emission reduction methods.

Theoretical analyses

In the single-phase heated surface of a boiler, four processes are involved: irreversible heat transfer from gas to metal, energy transfer and conversion in metal, irreversible heat transfer from metal to fluid, and energy transfer and conversion in fluid. According to working principles, these processes can be divided into two kinds. One is energy transfer and conversion in the control volume, and the other is irreversible heat transfer process. In this paper, control volume refers to metal and fluid.

For the irreversible heat transfer process that occurs in the system under unsteady conditions, the thermodynamic principle is the same as that of the system under steady conditions. For energy transfer and conversion in the control volume, however, energy change, entropy change, and exergy change in the volume with respect to time should be considered aside from the factors taken into account under steady conditions. Figures 1 and 2 show the energy as well as exergy transfer and conversion in the single-phase heated surface of the boiler, respectively. Figure 1 indicates that the final uses of the heat released by gas are metal storage, flowing fluid heating and internal energy change in the fluid contained within the volume. Figure 2 indicates that the final uses of the exergy released by gas are metal storage, flowing fluid, and exergy change in the fluid contained within the volume, and exergy destructions of processes that occur between gas and metal, in metal, between metal and fluid, and in fluid. Because entropy is not a kind of energy, entropy analysis is not illustrated here. Entropy production occurs in both irreversible heat transfer processes and control volumes.

Thermodynamic models

Model for control volumes

Energy analysis model

The control volume energy rate balance [6] is calculated by

(1){\displaystyle \frac{\mathrm{d}{E}_{\mathrm{C}\mathrm{V}}}{\mathrm{d}\tau }}={\displaystyle \sum _j\stackrel{\cdot }{Q_j}}-{\dot{W}}_{\mathrm{C}\mathrm{V}}+{\displaystyle \sum _i\stackrel{.}{m_i}\left(h_i+{\displaystyle \frac{v_i^{\mathrm{2}}}{\mathrm{2}}}+g{z}_i\right)}-{\displaystyle \sum _e\stackrel{.}{m_e}\left(h_e+{\displaystyle \frac{v_e^{\mathrm{2}}}{\mathrm{2}}}+g{z}_e\right)}.

For the system in this paper, the only work interaction at the boundary of a control volume is flow work at the areas where matter enters and exits; thus, the {\dot{W}}_{\mathrm{C}\mathrm{V}} of the energy rate balance can be set to zero. In addition, the kinetic and potential energies of the flowing streams can be ignored at the inlets and exits. Thus, the energy rate balance is reduced to

(2){\displaystyle \frac{\mathrm{d}{E}_{\mathrm{C}\mathrm{V}}}{\mathrm{d}\tau }}={\displaystyle \sum _j{\dot{Q}}_j}+{\displaystyle \sum _i{\dot{m}}_i{h}_i}-{\displaystyle \sum _e\stackrel{.}{m_e{h}_e}}.

The control volume rate equation for mass [6] is

(3){\displaystyle \sum _i\stackrel{.}{m_i}}-{\displaystyle \sum _e\stackrel{.}{m_e=}}{\displaystyle \frac{\mathrm{d}{m}_{\mathrm{C}\mathrm{V}}}{\mathrm{d}\tau }}.

Because control volume mass can be expressed in m_{\mathrm{C}\mathrm{V}}=\rho V_{\mathrm{C}\mathrm{V}}, Eq. (3) becomes

(4){\displaystyle \sum _i\stackrel{.}{m_i}}-{\displaystyle \sum _e\stackrel{.}{m_e=}}{V}_{\mathrm{C}\mathrm{V}}{\displaystyle \frac{\mathrm{d}\rho }{\mathrm{d}\tau }}.

m_{\mathrm{C}\mathrm{V}}=\rho V_{\mathrm{C}\mathrm{V}}; thus, the energy contained within the control volume can be calculated by E_{\mathrm{C}\mathrm{V}}=\rho V_{\mathrm{C}\mathrm{V}}e. The specific enthalpy is defined as h=e+pv. Employing Eq. (4), the time rate of energy contained within the control volume can be quantified by

(5){\displaystyle \frac{\mathrm{d}{E}_{\mathrm{C}\mathrm{V}}}{\mathrm{d}\tau }}=m_{\mathrm{C}\mathrm{V}}{\displaystyle \frac{\mathrm{d}h}{\mathrm{d}\tau }}-V_{\mathrm{C}\mathrm{V}}{\displaystyle \frac{\mathrm{d}p}{\mathrm{d}\tau }}+h\left({\displaystyle \sum _i\stackrel{.}{m_i}}-{\displaystyle \sum _e\stackrel{.}{m_e}}\right).

Entropy analysis model

The control volume entropy rate balance [6] is derived by

(6){\displaystyle \frac{\mathrm{d}{S}_{\mathrm{C}\mathrm{V}}}{\mathrm{d}\tau }}={\displaystyle \sum _j{\displaystyle \frac{\stackrel{\cdot }{Q_j}}{T_j}}}+{\displaystyle \sum _i\stackrel{.}{m_i}}{s}_i-{\displaystyle \sum _e\stackrel{.}{m_e}}{s}_e+\stackrel{.}{{\sigma }_{cv}}.

Similar to the time rate of energy, m_{\mathrm{C}\mathrm{V}}=\rho V_{\mathrm{C}\mathrm{V}}, and S_{\mathrm{C}\mathrm{V}}=\rho V_{\mathrm{C}\mathrm{V}}s. Therefore, employing Eq. (4), the time rate of entropy within the control volume can be quantified by

(7){\displaystyle \frac{\mathrm{d}{S}_{\mathrm{C}\mathrm{V}}}{\mathrm{d}\tau }}=m_{\mathrm{C}\mathrm{V}}{\displaystyle \frac{\mathrm{d}s}{\mathrm{d}\tau }}+s\left({\displaystyle \sum _i\stackrel{.}{m_i}}-{\displaystyle \sum _e\stackrel{.}{m_e}}\right).

Exergy analysis model

The control volume exergy rate balance [6] is calculated by

(8){\displaystyle \frac{\mathrm{d}E{x}_{\mathrm{C}\mathrm{V}}}{\mathrm{d}\tau }}={\displaystyle \sum _j\left(\mathrm{1}-{\displaystyle \frac{T_{\mathrm{0}}}{T_j}}\right)\stackrel{.}{Q_j}}-\left(\stackrel{.}{W_{\mathrm{C}\mathrm{V}}}-p_{\mathrm{0}}{\displaystyle \frac{\mathrm{d}{V}_{\mathrm{C}\mathrm{V}}}{\mathrm{d}\tau }}\right)+{\displaystyle \sum _i\stackrel{.}{m_i}e{x}_i}-{\displaystyle \sum _e\stackrel{.}{m_e}e{x}_e}-\stackrel{.}{E_d}.

Similar to the control volume energy rate balance, the {\dot{W}}_{\mathrm{C}\mathrm{V}} of the exergy rate balance can be set to zero. Because the volume remains constant, {\displaystyle \frac{\mathrm{d}{V}_{\mathrm{C}\mathrm{V}}}{\mathrm{d}\tau }}=\mathrm{0}, and the exergy rate balance is reduced to

(9){\displaystyle \frac{\mathrm{d}E{x}_{\mathrm{C}\mathrm{V}}}{\mathrm{d}\tau }}={\displaystyle \sum _j\left(\mathrm{1}-{\displaystyle \frac{T_{\mathrm{0}}}{T_j}}\right)\stackrel{.}{Q_j}}+{\displaystyle \sum _i\stackrel{.}{m_i}e{x}_i}-{\displaystyle \sum _e\stackrel{.}{m_e}e{x}_e}-\stackrel{.}{E_{\mathrm{d}}}.

Similar to the time rate of entropy, the time rate of change in exergy within the control volume can be quantified by

(10){\displaystyle \frac{\mathrm{d}E{x}_{\mathrm{C}\mathrm{V}}}{\mathrm{d}\tau }}=m_{\mathrm{C}\mathrm{V}}{\displaystyle \frac{\mathrm{d}ex}{\mathrm{d}\tau }}+ex\left({\displaystyle \sum _i\stackrel{.}{m_i}}-{\displaystyle \sum _e\stackrel{.}{m_e}}\right).

Model for irreversible heat transfer processes

The irreversible heat transfer processes studied in this paper, including gas-metal and metal-fluid processes, all occur between a solid surface at temperature T_b and an adjacent moving fluid at another temperature, T_f . Absolute value of transferred heat rate \dot{Q} remains constant.

The rate of entropy change in fluid due to process irreversibility is derived by

(11)\mathrm{\Delta }{\dot{S}}_{\mathrm{f}}={\displaystyle \frac{\pm \left|\dot{Q}\right|}{T_{\mathrm{f}}}}.

The rate of entropy change in the solid surface due to process irreversibility is determined by

(12)\mathrm{\Delta }{\dot{S}}_{\mathrm{b}}={\displaystyle \frac{\pm \left|\dot{Q}\right|}{T_{\mathrm{b}}}}.

According to the increase of entropy principle, the time rate of entropy production due to irreversibility is

(13)\dot{\sigma }=\mathrm{\Delta }{\dot{S}}_{\mathrm{f}}+\mathrm{\Delta }{\dot{S}}_{\mathrm{b}}.

The value of “±” in Eqs. (11) and (12) should be defined by this principle: the heat transfer into a system is taken as positive, and the heat transfer from a system is taken as negative. That is, for the gas-metal process “±” in Eq. (11), the heat transfer should be negative, whereas in Eq. (12) the heat transfer should be positive. For the metal-fluid process “±” in Eq. (11), the heat transfer should be positive, whereas negative in Eq. (12).

The time rate of exergy transfer accompanying heat from or into fluid is computed by

(14){\dot{E}}_{Q\mathrm{f}}=\left(\mathrm{1}-{\displaystyle \frac{T_{\mathrm{0}}}{T_{\mathrm{f}}}}\right)\left|\dot{Q}\right|.

The time rate of exergy transfer accompanying heat from or into metal is derived by

(15){\dot{E}}_{Q\mathrm{b}}=\left(\mathrm{1}-{\displaystyle \frac{T_{\mathrm{0}}}{T_{\mathrm{b}}}}\right)\left|\dot{Q}\right|.

The time rate of exergy destruction due to irreversibility is calculated by

(16)\stackrel{.}{E_{\mathrm{d}}}=\left|{\dot{E}}_{Q\mathrm{f}}-{\dot{E}}_{Q\mathrm{b}}\right|.

Methods for heat and property calculation

The rate of energy transfer between a solid surface at temperature T_b and an adjacent moving gas or liquid at temperature T_f can be quantified by

(17)\stackrel{\cdot }{Q}=\alpha F(T_{\mathrm{b}}-T_{\mathrm{f}}).

Heat transfer coefficient α is not a thermodynamic property but an empirical parameter that incorporates into the heat transfer relationship, nature of the flow pattern near the surface, fluid properties, and geometry.

The working fluid of the heated surface in the boiler is water or steam. The properties including ρ, h, s, and ex are evaluated by the water or steam thermodynamic property calculation program.

Energy distribution simulation and discussion

The energy distribution is simulated by applying the models built above to the low-temperature superheater of a 670 t/h-boiler at 70% load condition. In this simulation, the time rate of steam pressure change is 0.012 MPa/s and the time rate of steam temperature change is 6°C/min [7,8]. The temperature of the environment is t₀=0°C. Structure parameters of the low-temperature superheater and initial values of steam and gas parameters are illustrated in Table 1.

Results

The distributions of energy, entropy and exergy in the low-temperature superheater under before-mentioned condition are illustrated in Fig. 3–5, respectively.

Discussion

Figure 3 shows the percentage of each final energy use in the low-temperature superheater. The image indicates that 83.33% of the heat released by gas is absorbed by the flowing steam for enthalpy increase. The remaining 16.68% is stored in the superheater, broken down into 13.34% in metal and 3.34% in the steam contained within the volume for internal energy change.

For a certain system the energy stored in metal is dominated by the time rate of metal temperature. The energy change in the steam contained within the volume is related not only to the time rates of steam parameters, but also to the rate of energy change with respect to the steam parameter, which is ultimately decided by the steam parameter. That is, the energy change in the steam is mainly related to the steam parameters and time rates of steam parameters for a certain system. Figure 1 presents the energy distribution of certain conditions. Once the conditions change—which result in variations in the time rate of metal temperature, steam parameters, and time rates of steam parameters—the energy distribution varies accordingly.

Figure 4 demonstrates the percentage of each entropy production in irreversible heat transfer processes according to the heat transfer direction, which clarifies the processes. The results reveal that the irreversibility of processes between gas and metal, as well as inside the metal, is greater than that of the other two processes. The irreversibility of the metal-steam process is so minimal that it can sometimes be neglected. This is mainly because of the huge variations in the temperature differences among the processes. The entropy production of steam is decided by the irreversible factors within the volume.

Figure 5 depicts the percentage of each final exergy use in the low-temperature superheater, which indicates that 69.87% of the exergy accompanying the heat released by gas enters into the flowing steam accompanying heat absorption. The remaining 15.83% is lost, and 14.30% is stored in the superheater. The percentage stored is found in metal (12.00%) and in the steam (2.30%) contained within the volume for exergy change.

Similar to energy distribution, for a certain system the exergy change in the steam contained within the volume is mainly related to the steam parameters, time rates of steam parameters, and the environment. The exergy stored in metal is dominated by metal temperature, time rate of metal temperature, and the environment. Figure 5 shows the exergy distribution under certain conditions. Once the conditions change, the exergy distribution varies accordingly.

The stacked bar in Fig. 5 illustrates the exergy destruction distribution. The results are in agreement with that in Fig. 4. The Gouy-Stodla equation is validated.

The storage of energy and exergy is related not only to the conditions, but also to the structure and materials of the system. The results also change with system variation.

Conclusions

We studied the energy transfer and conversion of single-phase heated surface in a boiler under unsteady conditions. Processes occurring in the single-phase heated surface were analyzed, and distributions of energy and exergy were qualitatively revealed. Thermodynamic models, including models for energy analysis, entropy analysis, and exergy analysis of control volumes and irreversible heat transfer processes were established. Taking the low-temperature superheater of a 610 t/h-boiler as an example, the thermodynamic analysis of unsteady conditions was completed. The results are summarized as follows:

1) The distributions of energy, entropy production, and exergy under certain conditions can be quantitatively illustrated.

2) The irreversibility of processes between gas and metal, as well as inside the metal is greater than that of the other two processes. The irreversibility of metal-steam process is so minimal that it can sometimes be neglected.

3) The storage of energy and exergy is related to the conditions, time rates of steam parameters, time rate of metal temperature, and the structure and materials of the system.

References

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[4] Falcetta M F, Sciubba E. Unsteady numerical simulation of combined cycle plants. Proc. TAISE’97, Beijing, China, 1997, 224-231

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[8] Zhang B H. Life Time Management and Peak-load Operation of Large Capacity Fossil Fuel Power Units. Beijing: China Water and Electric Power Press, 1988 (in Chinese)

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