Exergy Analysis of a PWR Core Heat Transfer

For many decades processes taking place in complex systems such as thermal power plants have been based on a traditional thermodynamic analysis considering energy and mass conservation. The First Law, however, is not sufficient to accurately evaluate the performance of such systems: in fact, during all real processes entropy is produced due to irreversibility of energy transformations. As regards efficiency evaluation of a thermal system doing work it is necessary to deal with all sources of thermodynamic ineffectiveness: to this end, exergy turns out to be the optimal measure system [1].

In thermodynamics, given a certain amount of total energy, exergy is defined as the maximum useful work at the end of the transformation process; as a consequence, exergy allows to evaluate energy degradation associated with the process due to irreversibilities (exergy is conserved only in the case of reversible, ideal process) [21].

One of the most relevant characteristic of the exergy concept is its huge versatility, because it is possible to estimate fluxes and balances of any kind of energy for each element of a complex system, using a simple efficiency criterion [15, 25]. Moreover, the exergy concept can be usefully utilized for technical and economic analyses aimed at optimization of a process (thermo-economic analysis) and/or for a more efficient energy use (exergo-environmental analysis) [17,18, 26].

To date, despite the undisputed usefulness of the concept of exergy in the context of process optimization, energy analyses performed in the design of industrial installations never include evaluation of the exergetic efficiency. It is true that, since the 50s, a significant research activity has been carried out and that, in some applications, also exergy efficiency of a nuclear power plant was analyzed. In such works, however, the nuclear reactor was always modelled as a simple black box, taking into account only coolant inlet and outlet temperatures and reactor core thermal power [9,16].

In this paper, a detailed exergetic analysis of a pressurized water reactor (PWR) is carried out in order to define more accurately its exergetic efficiency. The results are compared with the average data obtained in literature, and used to create an exergetic core model that can support other design tools (thermal design, hydraulic design, mechanical design, neutronic design) in order to optimize the nuclear reactor also in terms of exergy efficiency [8,12,13].

3.1 Thermodynamic theoretical equations

Considering a reactor operating in steady-state as an open system, the exergetic analysis is based on the four balances outlined in Fig. 2, and in the following corresponding equations [2, 7].

figure_2.png

Figure 2. Thermodynamic balances [2, 7 ]

Σ_in ṁ = Σ_out ṁ  (Mass balance)(i)

Σ_in ṁ (h+ gz + w²/2) +$\dot{Q}$= Ĺ+ Σ_out ṁ (h+ gz + w²/2) (Energy balance)(ii)

Σ_in  | $\dot{Q}$ /T|+ Σ_in ṁ s + Ś_gen = Σ_out | $\dot{Q}$ /T| + Σ_out ṁ s (Entropy balance)(iii)

Σ_in ṁ ex + Σ Ėx_q - Ėx_δ = Ĺ + Σ_out ṁ ex (Exergy balance)(iiii)   

To obtain at a glance the definition of exergetic efficiency, it is useful to resort to the Grassmann diagram shown in Fig.3, together with the exergetic efficiency formula [10, 11, 14, 22].

figure_3.jpg

Figure 3. Exergetic efficiency formula [11]

3.2 Mathematical modeling of the reactor

Fig. 4 illustrates the exergetic modeling of a pressurized water reactor, in perfect analogy to Fig. 3.

figure_4.png

Figure 4.  MARS reactor exergy balances

Referring to this figure, the theoretical equations of the thermodynamics are applied as follows:

$\dot{m}_{1}=\dot{m}_{2}$(1)

$\dot{m}_{1} h_{1}+\dot{Q}_{f i s s}-\dot{Q}_{l o s s}=\dot{m}_{1} h_{2}$(2)

$\dot{m}_{1} s_{1}+\frac{\dot{Q}_{f i s s}}{T_{f i s s}}+\frac{\dot{Q}_{l o s s}}{T_{V e s s}}+\dot{S}_{g e n}=\dot{m}_{1} s_{2}$(3)   

$\dot{m}_{1} e x_{1}+\dot{E} x_{f i s s}-\dot{E} x_{l o s s}-\dot{E} x_{\delta R P V}=\dot{m}_{1} e x_{2}$(4)

for the Exergy Efficiency [10,14]:

$\varepsilon_{R P V}=1-\frac{\dot{E} x_{l o s s}+\dot{E} x_{\delta R R V}}{\dot{E} x_{\varrho f s s}}$(5)

in which:

$\dot{E} x_{Q f s s}=\dot{Q}_{f i s s}\left(1-\frac{T_{0}}{T_{f i s s}}\right)$(6) 

$\dot{E} x_{l o s s}=\dot{Q}_{l o s s}\left(1-\frac{T_{0}}{T_{V e s s}}\right)$(7)

$\dot{E} x_{\delta R P V}=\dot{E} x_{\varrho f s s}-\dot{E} x_{P}-\dot{E} x_{l o s s}$(8)

To carry out the exergetic analysis of the reactor, the following simplifying hypotheses are assumed:

1) steady-state operation mode;

2) average constant enrichment of all fuel rods in the core, equal to 2.3%;

3) net energy for each fission  equal to 200 MeV;

4) coolant mass flow homogeneously divided in each fuel rod;

5) constant c_p value of the coolant water around each fuel rod (it increases by about 7% between the top and the bottom of the core) and equal to 4.74kJ/kg K;

6) reference condition for each exergy assessment: P₀=101.3 [kPa]; T₀=298.15 [K]; s₀=0.367 [kJ / kg K]; h₀=104.93 [kJ / kg].

With reference to the fuel exergy as estimated in equation (6), the following steps are performed :

• $\dot{Q}_{\text {fiss}}$  is evaluated from the average value of the core volumetric power q”’ as follows:

${{q}^{'''}}\,={{E}_{fiss}}\,{{\sigma }_{fiss}}\,{{\Phi }_{n}}\,{{\rho }_{UO2}}\,\,\,\frac{{{N}_{A}}}{{{M}_{UO2}}(\bar{X})}\,\bar{X}$ (9)

${{\dot{Q}}_{fiss}}$=${{q}^{'''}}$V_f (10)

• $T_{f i s s}$ is taken to be the average temperature at which heat is generated by fission, assuming that this energy is instantaneously generated at the center of each fuel rod . To obtain $T_{f i s s}$  it is first necessary to simulate the fission temperature profile along the fuel rod axis: to this end, an inverse procedure is followed which requires as preliminary step the simulation of the temperature profile of the coolant flow along the fuel rod. This profile permits, knowing the coefficients of water exchange, the conductivity of the Zircaloy clad, the exchange coefficient of helium in the clad gap and the fuel thermal conductivity, to go back to center temperature profile of  fuel rods.

To simulate the axial temperature shape of the cooling water, the following expression is used:

$T_{H 2 O}(z)=T_{H 2 o i n}+\frac{q_{\max }^{\prime} H_{e}}{\pi \overline{c}_{p H 2 O} \dot{m}_{H 2 O}} \times\left[\operatorname{sen}\left(\frac{\pi z}{H_{e}}\right)+\operatorname{sen}\left(\frac{\pi H a}{2 H_{e}}\right)\right]$(11)

To solve such an expression the linear power density q'_max and the extrapolated fuel rod height, H_e, (at which neutron flux is equal to zero) are required.

To obtain q’_max, the linear heat flux at the center of the rod height, as illustrated in Fig 5, the following expression is used [6]:

${{T}_{H2O}}(z)\,=\,{{T}_{H2O}}_{in\,}+\,\frac{q{{'}_{\max }}\,{{H}_{e}}_{\,}}{\pi \,{{{\bar{c}}}_{pH2O}}\,{{{\dot{m}}}_{H2O}}}\,\times \,\left[ sen\,\ \left( \frac{\pi \,z}{{{H}_{e}}}\, \right)+sen\ \,\left( \frac{\pi \,Ha}{2{{H}_{e}}} \right) \right]\,\,$ (12)

figure_5.png

Figure 5. Linear heat flux for a PWR reactor

Since all the parameters of interest are axially symmetric, as evident from Fig. 5, in all the simulations the origin of the reference axes has always been positioned at the center of the rod at its half height.

To obtain the extrapolated fuel rod height, H_e, the following expression can be used [6]:

$H_{e}=H_{a}+2 \delta=H_{a}+2 \frac{0.71}{\Sigma_{n}}$(13)

in which δ is the adjunctive length necessary to obtain the correct position of the zero neutronic flux boundary condition, and S_tr is the macroscopic transport cross section.

This quantity is equal to the macroscopic total cross section multiplied by the transport factor ($1-\overline{\mu}_{0}$) [3]:

$\Sigma_{\mathrm{tr}}=\Sigma_{\mathrm{tot}}\left(1-\overline{\mu}_{0}\right)$(14)

S_tot can be calculated in terms of the middle temperature of the coolant water, $\overline{T}_{H 2 O}$ , as it follows:

$\Sigma_{t o t}=N_{H 2 O}\left(\overline{T}_{H 2 O}\right)\left[\sigma_{d b s}\left(\overline{T}_{H 2 O}\right)+\sigma_{s c}\left(\overline{T}_{H 2 O}\right)\right]$(15)

in which (T₀ = 293.15 K and T_H2O =  235°C):

$\sigma(\overline{T})=\sigma_{0} \sqrt{\frac{T_{0}}{\overline{T}}}$(16)

figure_6.png

Figure 6. Sub-channel temperature profile for a PWR reactor

With the above relationships it is possible to estimate the cooling water temperature profile along the sub channel around a fuel rod and, referring to a general shape as showed in Fig. 6, it is possible to go back to the central temperature of a fuel rod.

Knowing the temperature of the coolant water, T_H20(z), the convective heat transfer coefficient for water flowing parallel to the clad  can be calculated by means of the Weisman correlation (describing a single-phase forced convection in turbulent regime):

$N u=C R_{e}^{0.8} P_{r}^{1 / 3}$(17)

in which  C is a function of the pitch-to-diameter ratio of a single subchannel [27].

With this correlation the convective heat transfer coefficient for turbulent water can be obtained as follows:

$h_{\text {wall}}=\frac{N u K_{H 2 O}}{D_{e}}$(18)

and the wall temperature, which is the external temperature of the clad, can be obtained from the following equation:

$q^{\prime \prime}=h_{w a l l}\left(T_{w a l l}-T_{H 2 O}\right)$(19)

The internal temperature of the clad, $T_{g i}$ , assuming the  Zircaloy conductivity as a constant along the axis, can be assessed using the Fourier equation:

$q^{\prime}=\frac{2 \pi K_{z i r c}}{\ln \left(R_{e s t} / R_{g}\right)}\left(T_{g i}-T_{w a l l}\right)$(20)

The external temperature of the fuel pellet, $T_{S}$ , depends upon the thermal conductivity of the Helium in the gap. Helium is supposed to be stagnant, at an intermediary temperature among T_S and T_gi. The Zircaloy conductivity correlation used in the work (heat transfer for radiation is neglected) is the following [20]: 

$K_{H e}=0.144\left[1+2.7 \times 10^{-4} p_{H e}\left(T_{H e} / T_{o}\right)\right]^{0.71\left(1-2 \times 10^{-4} p_{B e}\right)}$(21)

in which: T₀ = 273.15 K and  p_He = 11,25 bar;

T_S is calculated by means of an iterative application of the Fourier equation in the following terms [6,27]:

$q^{\prime}(z)=\frac{2 \pi K_{H e}}{\ln \left(R_{p} / R_{g}\right)}\left(T_{S}-T_{g i}\right)$(22)

Knowing T_S, the central temperature of the fuel rod, can be calculated by means of the Integral Conductivity correlation which only depends on the characteristic of the fuel and on the temperature T_c at the center of the rod:

$q^{\prime}(z)=4 \pi I . C .=4 \pi \int_{T_{s}}^{T_{c}} K_{U O 2}(T) d T$(23)

in which [23]:

$K_{U O 2}(T)=\frac{100}{6.548+23.533\left(\frac{T}{10^{3}}\right)}+\frac{6400}{\left(\frac{T}{10^{3}}\right)^{5 / 2}} \exp ^{\left(\frac{-16.35}{T / 0^{3}}\right)}$(24)

T_c is calculated along the fuel rod axis by means of an iterative application of the above mentioned equations along the active rod height thus obtaining the T_c(z) profile.

Finally T_fiss is evaluated as the average value of the T_c(z) curve among the top and the bottom of the fuel rod.

With reference to Exergy losses as estimated in equation (7), $\dot{Q}_{\text {loss}}$ , and therefore $\dot{E} x_{l o s s}$ , are easily found, knowing PRV geometrical dimensions, steal heat thickness and conductivity and reference containment temperature.

At this point of the modeling all data are available to calculate $\dot{E} x_{\delta}$ using equation (8) and, therefore, the reactor Exergy Efficiency using equation (5).

Modeling results confirm the order of magnitude obtained by other researches in which the reactor, analyzed as a black box inside the equipment of a nuclear power plant, is proved to have exergetic efficiencies always between 50 and 60 %.  This efficiency range once again shows that the nuclear reactor presents a second law efficiency approximately equal to that of a traditional combustion chamber.

Next step of the present work – in the frame of a wider ongoing research - is to develop the thermo-economic analysis of a PWR nuclear power plant in order to assess the actual cost of products obtainable downstream of the plant (electric energy and thermal energy useful for industrial and civil users, very different products in terms of exergy contents), and compare them with the costs of similar products obtained from conventional thermal power plants.

The final goal of the research is to create a whole exergetic model of the entire primary PWR loop that can support other design tools (thermal, hydraulic, mechanical and neutronic design) in order to optimize the nuclear reactor loop also in terms of exergy efficiency.