Proposal Method for Assessment Evaluation of Air Cooled Condenser Used in a Biomass Power Plant of Cuba

2.1 Basic methodology for the selection of the condenser more adequate

For the selection of the most adequate condenser, the methodology given in the Figure 1 was applied. For this purpose is required to have a group of primary parameters, which are summarized in the Table 1. The diagram of utilization (see Figure 2) is a fast application method, very used to verify the effectiveness use of the ACC in pre-established operating conditions.

In the diagram of utilization an interaction between the initial temperature difference (ITD) and the temperature of dry bulb (T_TBS) are shown, in terms of the steam pressure in the turbine outlet (P_Cond). The zone recommended of favorable operation was obtained in recent investigations, being shady in pink in the Figure 2 [9].

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Figure 1. Selection of the more adequate condenser for BPP

The initial temperatures difference (ITD) is obtained as, [10]:

$\mathrm{ITD}=\mathrm{T}_{\mathrm{EntVapor}}-\mathrm{T}_{\mathrm{TBS}}$    (℃)        (1)

In Eq. (1) T_EntVapor is the fluid temperature in the condenser outlet, in ℃.

Table 1. Initial parameters required for the ACC evaluation

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Figure 2. Diagram of utilization

2.2 Basic methodology for the selection of the condenser more adequate

The proposal methodology follows the same logical order shown in the Kröger's method, since it is the one with the greatest acceptance and dissemination among researchers and specialists working in this field. In this new method are considered news procedures for the estimation of the average heat transfer coefficient, pressure drop and thermal assessment of the ACC. The proposed procedure is described step to step of the following manner:

1- Define the initial required variables (see Table 1)

2- Determine the ideal pressure of the vapor in the turbine outlet, (see Table 2).

Table 2. Ideal pressure of the vapor in the turbine outlet

3- Define the correcting factors for the effect of the wind (C_rec) and relative humidity (CChr), see Figures 3 and 4.

4- Determine the real pressure of the vapor in the turbine outlet.

$P_{S T}=P_{B a c k} \cdot C_{r e c} \cdot C C h r \quad(k P a)$      (8)

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Figure 3. Correcting factors for the effect of the wind

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Figure 4. Correcting factors for the effect of the relative humidity

5- Define the vapor production of the steam boiler $\mathrm{m}_{\mathrm{I}}$ , the rate of vapor in the intermediate extraction $\mathrm{m}_{\mathrm{E}}$ and the flow at the turbine outlet m_agua.

6- Obtain the thermodynamic properties (enthalpy and entropy) in the turbine inlet (h_I; s_I) and intermediate extraction in the turbine, (h_E; s_E).

7- Establish the required initial variables of the combined process, (see Table 3).

Table 3. Initial required variables of the combined process

8- Calculate the useful power in the turbine installation, using the following Equation:

$W_{\text {Elec}}=\left[\begin{array}{c}

m_{\text {agua}} \cdot\left(h_{I}-h_{E}\right)+ \\

+\left(m_{\text {agua}}-m_{E}\right) \cdot\left(h_{I}-h_{E}\right)

\end{array}\right] \cdot \eta_{\text {em}} \cdot \eta_{r i}$     (9)

In Eq. (9) η_em and η_ri is the electromechanical and relative internal performance of the turbine respectively.

9- Obtain the thermodynamic properties of the vapor in the turbine outlet, (see Table 4).

Table 4. Thermodynamic properties required, turbine outlet

10- Define the tube arrangements and the transverse pitch S_T and longitudinal pitch S_L measured between tube centers, in m.

11- Define characteristics of the fin tubes (all the dimensions will be given in meters), which include, d_i and d_e is the inner and outer diameter of the bare tube (without fins), fins height h_a, fins thicknesse_a, wall thickness of the tube e_T, tube length $1$ and number of fins per linear meter of tube F_a.

12- Assume a global coefficient of heat transfer K, in the range of values K₁=30 to 100 W⁄(m²∙℃).

13- Define in a first approximation the condensed outlet temperature in the ACC by means of the following Equation:

$\mathrm{T}_{\mathrm{scond}}=\left(\mathrm{T}_{\mathrm{EntVapor}}+\mathrm{T}_{\mathrm{TBS}}\right) / 2\left({ }^{\circ} \mathrm{C}\right)$      (10)

14- Calculate the initial temperatures difference (ITD) with the use of the Equation (1).

15- Check in the Figure 2 if ITD values are located in the recommended zone, otherwise taking a value between ITD=22 to 28 ℃.

16- Determine the Logarithmic Mean Temperature Difference (LMTD) in the ACC, by means of the following Equation:

$\mathrm{LMTD}=\frac{\mathrm{ITD}-\mathrm{TTD}}{\ln \left(\frac{1 \mathrm{TD}}{\mathrm{TTD}}\right)}$ (℃)     (11)

In Equation (11) $TTD$  is the temperature difference of the flow in the the condensed side, in ℃.

17- Obtain the thermodynamic properties in the ACC outlet, enthalpy h_fluid (in kJ/kg) and entropy s_fluid, in kJ⁄(kg∙℃).

18- Determine the transferred heat in the ACC by the following Equation:

$\mathrm{Q}=\mathrm{m}_{\text {agua }} \cdot\left(\mathrm{h}_{\text {cond }}-\mathrm{h}_{\text {fluid }}\right)(\mathrm{kW})$       (12)

19- Obtain the heat transfer surface of a single fin (in $\mathrm{m}^{2}$ by the following Equation:

$A_{\mathrm{a}}=\pi \mathrm{e}_{\mathrm{a}}\left(\mathrm{d}_{\mathrm{e}}+2 \mathrm{h}_{\mathrm{a}}\right)+2 \pi\left[\left(\frac{\mathrm{d}_{\mathrm{e}}}{2}+\mathrm{h}_{\mathrm{a}}\right)^{2}-\left(\frac{\mathrm{d}_{\mathrm{i}}}{2}\right)^{2}\right]$    (13)

20- Define the fins number per tube.

$\mathrm{n}_{\mathrm{aT}}=\mathrm{F}_{\mathrm{a}} \cdot \mathrm{l}$    (14)

21- Determine the heat transfer surface of and finned tube. by the following Equation:

$A_{T}=\pi d_{e} \cdot\left(l-e_{a} n_{a T}\right)+A_{a} n_{a T} \quad\left(m^{2}\right)$       (15)

22- Determine the inner surface of the fiined tube:

$A_{I}=\pi \cdot l \cdot\left(d_{e}-2 e_{T}\right) \quad\left(m^{2}\right)$      (16)

23- Obtain  the  necessary heat transfer for  the ACC, using the following Equation:

$\mathrm{F}=\mathrm{Q} /\left(\mathrm{LMTD} \cdot \mathrm{K}_{1}\right) \quad\left(\mathrm{m}^{2}\right)$      (17)

23- Determine the number of tubes that would have the ACC in a first approximation, using the following Equation:

$n_{\text {tubos }}=F /\left(A_{T} \cdot A_{I}\right)$      (18)

23- Calculate the liquid and vapor Reynolds number, using the following Equations [11, 12]:

$\mathrm{Re}_{\mathrm{L}}=\frac{\mathrm{m}_{\mathrm{agua}} \cdot(1-\mathrm{x}) \cdot \mathrm{d}_{\mathrm{i}}}{\mu_{\mathrm{L}} \cdot 0.785 \cdot \mathrm{d}_{\mathrm{i}}^{2} \cdot \mathrm{n}_{\mathrm{tubos}}}$     (19)

$\operatorname{Re}_{\mathrm{V}}=\frac{\mathrm{m}_{\mathrm{agua}} \cdot \mathrm{x} \cdot \mathrm{d}_{\mathrm{i}}}{\mu_{\mathrm{V}} \cdot 0.785 \cdot \mathrm{d}_{\mathrm{j}}^{2} \cdot \mathrm{n}_{\mathrm{tubos}}}$       (20)

In Eqns. (19) and (20), X is the thermodynamic vapor quality.

24- Calculate the heat transfer coefficient in the inner portion of the tube and the drop pressure by means of the procedure gived in [6, 7].

25- Determine the thermodynamic properties of the air at T_TBS, (see Table 5).

Table 4. Thermodynamic properties required, ACC inlet in the air side

26- Determine the enthalpy of the air at ACC outlet h_as by air side, in kJ⁄kg.

27- Calculate the airflow necessary, as:

$\mathrm{m}_{\mathrm{aire}}=\mathrm{Cp}_{\mathrm{a}}\left(\mathrm{T}_{\mathrm{SACC}}-\mathrm{T}_{\mathrm{TBS}}\right) \quad(\mathrm{kg} / \mathrm{s})$     (21)

In Eq. (21) $\mathrm{T}_{\mathrm{SACC}}$ is the air temperature at ACC outlet.

28- Calculate the initial air velocity as:

$\mathrm{V}_{\mathrm{n}}=\mathrm{m}_{\text {aire }} /\left(75.6 \cdot \mathrm{n}_{\text {vent }}\right) \quad(\mathrm{m} / \mathrm{s})$     (22)

In Eq. (22) n_vent is the number of fans used in the ACC, for a first aproximation take n_vent=1.

29- Calculate the maximum fluid velocity of the air in the tube bank.

30- Calculate the heat transfer coefficient in the air side portion and the drop pressure by means of the procedure gived in [5].

31- Determine the thermal resistances in the ACC.

32- Determine the thermal efficiency of an single fin, η_A.

33- Determine the performance of the extended surface as:

$\eta_{\mathrm{W}}=1-\mathrm{n}_{\mathrm{aT}} \mathrm{A}_{\mathrm{a}}\left(1-\mathrm{n}_{\mathrm{aT}}\right) /\left(\mathrm{A}_{\mathrm{T}} \cdot \mathrm{n}_{\mathrm{tubos}}\right)$      (23)

34- Determine the global heat transfer K₂.

35- Compare the global coefficients K₂ and K₁. If the average error E_E is lower than 5% the values K₂.are acepted, otherwise, is necessary to repeat the study from the point 12 with the new value of the global heat transfer coefficient K₂. The average error E_E is obtained as:

$\mathrm{E}_{\mathrm{E}}=100 \cdot\left|\frac{K_{2}-K_{1}}{K_{2}}\right|$   (%)      (24)

36- Obtain the real heat transfer surface $\mathrm{F}^{\prime}$ required by the ACC as:

$\mathrm{F}^{\prime}=\mathrm{Q} /\left(\mathrm{LMTD} \cdot \mathrm{K}_{2}\right) \quad\left(\mathrm{m}^{2}\right)$       (25)

37- Determine the cost of ecological trace, in USD⁄(GJ∙year), for the estimated emissions associated to the use of AAC as:

$G_{E m i s}=(6.02-0.434 B) \cdot A \cdot e^{0.226 B}$      (26)

In Eq. (26) e is de Euler constant value, while A and B are constants values, conditioned to the emission of greenhouse effect gases. Values of constant A and B are obtained as [13]:

$A=\ln \left[\left(C O_{2}\right)^{0.1} \cdot S O_{2}\right]^{0.1}+0.252$     (27)

$B=\log \left[\frac{\left(C H_{4} \cdot N O_{X} \cdot(C O)^{0.04}-(S O)^{2}\right)^{2}}{N_{2} O}\right]$      (28)

In Eqns. (27) and (28) the volumes of polluting gases are given in Gg.