7.2 Calculation procedure
7.2.2 Calculation procedure for vertical tube bank heat exchangers
Consider then how flue gas in tubes air preheaters and boiler banks with many passes are treated. Like horizontal tube bank heat exchangers, also vertical tube bank heat exchangers are modeled so that they consist of tube banks and small cavities. This is illustrated in figure 58.
Figure 58. Vertical tube bank heat exchangers with small cavities.
In the case of the boiler bank, there are small cavities at the inlet and outlet of the boiler bank. Imagined turning cavities between passes are not cavities and thus all passes are as-sumed to belong to the same tube bank. That is, in the case of convection heat transfer the row number correction is applied only for the first tube rows of the first pass.
In the case of flue gas in tubes air preheaters, there are small cavities at the inlet and outlet of the air heater on the air side because the calculation model assumes that small cavities are always on the outside fluid side. Thus, the small cavity at the inlet of the air heater on the flue gas side is not supported by the calculation model. Small cavities on the air side are basically not calculated because there is not any heat transfer surface in those cavities and because non-radiant air does not release any radiation for the tube banks. Turning cavities between passes are not assumed to be cavities because there is no need for the radiation calculations of those cavities. Thus, all passes are assumed to belong to the same tube bank.
This is probably a good assumption also from the convection point of view because flow conditions may be quite turbulent in those cavities which may lead to that the flow is not stabilized between passes. So, the row number correction is applied only for the first tube rows of the first pass.
In the case of flue gas in tubes air preheaters, the outside fluid is colder than tube tempera-tures. Thus, for example, the surface temperature of the outside fouling layer is calculated as follows:
𝑇𝑓,𝑜 = 𝑇𝑜+ (𝜙𝑐− 𝜙𝑒𝑥) ⋅ 𝑅𝑜,𝑡𝑜𝑡 (7.18) 7.2.3 Stabilization of the calculation procedure
The method where new calculated values are used as new initial guesses works usually well in situations where the temperature difference between fluids is big. However, when the temperature difference is smaller, instability of the iteration process may exist, and thus the calculation model does not find the solution. This problem can be solved using a so-called under-relaxation method, that is, the newly calculated value is not used entirely as a new initial guess, but the newly calculated value and the earlier value are taken for the new guess at some weights. For example, a new guess for some of the tube temperatures can be calcu-lated as follows:
𝑇𝑛𝑒𝑤 = (1 − 𝑓𝑢)𝑇𝑛−1+ 𝑓𝑢𝑇𝑛 (7.19) where 𝑇𝑛𝑒𝑤 is the new guess for the temperature (K), 𝑓𝑢 is the under-relaxation factor (-), 𝑇𝑛−1 is the earlier temperature value (K) and 𝑇𝑛 is the newly calculated value (K)
When the value of the under-relaxation factor is reduced, the stability of the iteration process increases, but the drawback is that also the computation time increases. For that reason, it would be useful to find the optimal value for the under-relaxation factor which gives the stable solution in as short computation time as possible. However, the optimal value seems to vary a lot between different cases, so the common optimal value is not existing. Thus, in the calculation model, the value 0,5 for the under-relaxation factor is used which offers at least a stable solution process in most situations.
8 VALIDATION OF THE CALCULATION MODEL
The main purpose of the calculation model was to provide information about what is hap-pening in the different parts of the heat exchanger. In figure 59 there is an example where temperatures and heat rates of each row are shown in the case of a 22 tube row counter flow superheater which has 1 parallel row in a pass.
Figure 59. Temperature and heat rate profiles for superheater.
In the above figure, there are curves for flue gas temperature, tube outer surface temperature, steam temperature, and heat rate to steam. Also, the measured steam side outlet temperature is shown. Tube row indexes are defined in direction of the flue gas flow. As can be seen, the heat rate decreases when the temperature difference between the flue gas and steam de-creases. The effect of external radiation from the cavities is high due to high flue gas tem-perature and it can be noticed from the outermost tube rows where heat rates are high. Con-sider then also a 12 tube row counter flow economizer which also has just 1 parallel row in a pass.
Figure 60. Temperature and heat rate profiles for economizer.
From the above figure, it can be seen that the heat rate remains now quite constant due to the quite constant temperature difference between flue gas and water. In this case, the flue gas temperature is much lower than in the earlier superheater case so the effect of external radiation for the outermost tube rows is much smaller. The effect of turbulence increasing to the heat transfer can now be seen more clearly than in the earlier case because of the more constant temperature difference and the smaller amount of radiation. The turbulence of the flue gas flow increases through few first tube rows so the convection heat transfer and then also total heat transfer rate to the water increases as it can be seen from the figure.
Results in the earlier superheater and economizer cases were calculated without an outside fouling factor. As can be seen from figures 59 and 60, measured and calculated outlet tem-peratures do not match due to the effect of fouling and other uncertainties. In order to get the calculation to match better to the measurements, the fouling factor needs to be defined. In the following subchapter 8.1, the procedure for that is introduced. In subchapter 8.2, results from the new matrix calculation model are compared to the results from the earlier conven-tional calculation model to see that can the new model offer better overall accuracy than the old model.
8.1 Determination of fouling factors
Consider first how the results from the calculation model correspond with the real experi-mental data gathered from the power plants. In the validation process, only superheaters and economizers of CFB boilers were investigated. For BFB boilers, separate validation would need to be done because the working principle is slightly different and thus probably also the resulting fouling behavior.
Inlet and outlet temperatures of the heat exchangers are usually measured in power plants so the accuracy of the overall performance prediction of the calculation model can be investi-gated. Usually, water and steam side measurements are more reliable than flue gas and air side measurements, so water and steam side measurements are used as a comparing point, as was shown in figures 59 and 60. In the validation process, the outside fouling factor is adjusted so that the calculated total heat rate matches the measurements. In figure 61, there are outside fouling factors for all investigated superheater cases as a function of boiler load.
Figure 61. Outside fouling factors for superheaters as a function of boiler load.
As can be seen from the earlier figure, fouling factors vary quite a lot and clear fuel or load dependence cannot be seen. Consider then which kind of fouling factors were needed in the case of economizers. Those are shown in figure 62.
Figure 62. Outside fouling factors for economizers as a function of boiler load.
Also, in the case of economizers, load dependence cannot be seen but fuel dependence seems to exist. Especially, the difference between coal and biomass is clearly visible since fouling factors for coal are close to one and fouling factors for biomass are close to 0,7. From deter-mined fouling factors, the average fouling factor can be calculated for different cases. Those are shown in table 3 with an accuracy of 0.05.
Table 3. Outside fouling factors for different fuel-heat exchanger combinations.
Superheaters Economizers
Coal 0,80 1,05
Peat 0,85 0,80
Biomass 0,85 0,70
As it can be seen from table 3, economizers need to be corrected more than superheaters in the case of peat and biomass. The reason is probably that in the economizer section of the back-pass the flue gas and tube temperatures are enough low for the condensation fouling mechanism that was discussed in chapter 5.2.4. If condensation fouling occurs and the foul-ing layer begins to form on the tube surface, it probably enhances the inertial impaction fouling mechanism as well. In the case of coal, similar behavior cannot be seen because inorganic species cannot vaporize so easily and thus there are not condensable vapors that could condensate and cause the fouling. Actually, table 3 shows that economizers need to be corrected less than superheaters in the case of coal but because the amount of coal validation cases was very low, it is not worth drawing any conclusions from it yet. If more coal cases were investigated, fouling factors for superheaters and economizers would probably be fairly the same. What comes to the effect of fuel on the fouling factor, it affects in the case of economizers where condensation fouling can occur. There, biomass cases need to be cor-rected more than coal cases. In the case of superheaters, condensation fouling does not occur so much and thus fuel has probably no effect on the fouling factor, although the fouling factor of coal differs from fouling factors of peat and coal. However, if more coal cases were investigated, the value of 0,8 would probably also approach the value 0,85.
After average fouling factors for different cases have been determined, those can be applied in the calculations. Consider again the earlier 22 tube row superheater case with the differ-ence that now the outside fouling factor is in use. Temperature and heat rate profiles are shown in figure 63.
Figure 63. Corrected temperature and heat rate profiles for the superheater.
As can be seen, calculated and measured outlet temperatures match much better than earlier.
Outlet temperatures do not correspond to each other perfectly because the demanded outside fouling factor to get a perfect match varies between different cases. However, using the de-termined average fouling factor, many cases can now be got to match quite well with the measurements. When the outside fouling factor is in use, calculation provides also surface temperatures of the outside fouling layer in each row which can be seen in figure 63.
8.2 Accuracy of the new model compared to the conventional model
Consider then how the results of the matrix calculation model differ from the results of the earlier conventional calculation model to see that can the new model offer any extra accuracy for the overall performance prediction. Results are shown in figure 64.
Figure 64. Comparison between new and old calculation models.
In the above figure, there are fouling factors that are needed to get the results to match the measurements. Red spots show the fouling factors needed in a certain case so that the fouling factor used in the earlier conventional calculation model can be seen from the x-axis and the fouling factor used in the new matrix calculation model can be seen from the y-axis. If a red spot is above the blue line and both fouling factors are less than one, results from the matrix calculation model have been corrected less and thus the matrix method has been more accu-rate. If both fouling factors are greater than one, the situation is vice versa. If only one of the fouling factors is greater than one, the place of the spot does not tell which of the models were more accurate. However, there are just a few of these kinds of spots. As can be seen from the figure, both models give quite similar results. However, the new matrix model seems to offer a small improvement to the calculation accuracy. That is also the truth since the average absolute difference between fouling factor and value of one was 0,183 in the case of the matrix calculation model and 0,200 in the case of the conventional calculation model.
9 SUMMARY AND CONCLUSIONS
The purpose of this Master’s thesis was to develop a heat transfer calculation model for back-pass tube bank heat exchangers of fluidized bed steam boilers so that intermediate tempera-tures of each tube row inside the heat exchanger can be obtained. This goal was achieved using a so-called matrix method where tube banks of the heat exchangers were divided into many small parts, in this case to tube rows. The development of the calculation model re-quired, among other things, an understanding of different heat transfer mechanisms which are conduction, convection, and thermal radiation. Also, different heat exchanger configura-tions and how matrix equaconfigura-tions differ between all those situaconfigura-tions were covered. In addition, energy balance and pressure loss calculations of the heat exchangers were discussed.
In the calculation model, heat exchangers consisted of tube banks and cavities between them.
The calculation procedure proceeded so that cavities were calculated first in order to solve the radiation that they sent for the tube banks. Then tube banks were solved, and this alter-nation was continued until the solution was found. Pressure losses were calculated conven-tionally for the whole heat exchanger because pressure does not have a big effect on the heat transfer and because this way computational costs can be reduced.
In the validation process of the calculations model, results from experimental measurements were utilized to find fouling factors for the calculations. As a result of that process, correction factors for different fuel-heat exchanger combinations were found. It was noticed that cal-culations must be corrected more in the case of economizers than superheaters probably due to the bigger effect of condensation fouling mechanism which occurs in lower temperatures.
Also, it was found that there where condensation fouling can occur, that is in economizer section, calculations must be corrected more in case of biomass fuels than coal because of the larger amount of condensable inorganic species in the biomass fuels. In the case of su-perheaters, fuel seems to not have an effect on the fouling factor. In the validation process, the total amount of investigated cases was quite low and only heat exchangers of CFB boilers were investigated. Thus, many more cases need to be validated in the future to obtain more reliable validation results, and also separate validation needs to be done for BFB boilers.
Results of the new matrix calculation model were also compared to the results from the earlier conventional calculation model and it was found that the matrix calculation model offers slightly more accurate results. The main reason for that is probably the fact that fluid properties and thus also all heat transfer parameters are calculated more accurately in the new model than in the conventional model where just overall average fluid properties are used. In addition, the new model makes it possible to take into account changes in tube material and tube thickness through the heat exchanger that is not possible in the earlier conventional calculation model where averaged values need to be used for the whole heat exchanger.
In the future, it would be good to compare the results from the calculation model to the more accurate measurements where also some intermediate temperatures from the real heat ex-changers are known in addition to overall inlet and outlet temperatures. This is important because even if the outlet temperature calculated by the model corresponds to the measure-ments, the temperature profile given by the model may still be different from the real tem-perature profile. In addition to experimental measurements, the results from the computa-tional fluid dynamics (CFD) simulations could also provide good reference data for the val-idation process.
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Matrix equations where only outside of the matrix A and outside variables are shown
Outside inlet is known 1 row
Outside outlet is known 1 row
Matrix equations where only inside of the matrix A and inside variables are shown
Parallel flow, not mixed, 1 row per pass
Parallel flow, not mixed, 1 row per pass