Condenser Boiler Modeling

Condenser boiler modeling

João Barros 1

1. Instituto Superior de Técnico; Technical University of Lisbon; Avenida Rovisco Pais, 1-1049- 001 Lisboa; Email Address: [email protected].

Abstract : In the present work a simplified mathematical model for a condensing boiler, was built using the VBA TM programming language. The model was configured to a Bosch condensing boiler and validated through the analysis of experimental tests already carried out. In particular the influence of the air humidity in the boiler efficiency is addressed. The model was also applied to other boilers without condensation. The model was created based on four zones: the burner, the combustion chamber, the primary heat exchanger and the condensation zone. The energy balance in the combustion zone considers complete conversion with heat losses from the flame. Combustion products are considered a gray gas mixture. The primary heat exchanger was modeled using correlations for the convection coefficient for herringbone corrugated fins. For the condensation zone two models were built. The first model considers pure condensation while the second took into account the existence of non-condensable gases. The mathematical model was solved using the Broyden method improved with Sherman-Morrison. The results obtained from the different boilers, show that the model predicts the boiler efficiency, in null relative humidity, with a standard deviation of 3.63% and with 100% of relative humidity, the standard deviation was of 4.46%. The effect of the relative humidity varies depending on input power. The difference on power loss was of 0.16kW to 2.86 kW for the condensing boiler. For the non-condensing boilers, a difference of 0.72 kW to 0.99 kW was obtained.

Nomenclature – Total Area (m 2) – Mole Number From Specie I (Mole) 2 – Envelope of Comb. Chamber Area (m ) – Nusselt Number (mean) (ad.) 2 – Interior Tube Area (m ) 2 – Tube separation length (m) = – Exterior Area (m ) 2 – Tube length (m) – Radiative Surface Area i (m ) – Lower Heating Value (MJ/Kg comb.) – Ratio of Specific Heat – Saturation Pressure at T (MPa) –Outer Annular Fin Diameter (m) @ – Heat transfer Rate (W) – Interior Diameter (m) – Relative Humidity (%) – Interior Annular Fin Diameter (m) – Average Envelope Temperature (K) – Standard deviation – Average Interface Temperature (K) – Emissive Power From Surface i (W) – Average Combustion Gases Temperature – View Factor Between i and j Surfaces at Point i (K) – Irradiation From Surface i (W) – Average Temperature at surface i (K) – Higher Heating Value (MJ/Kg comb.) – Average Water Temperature (K) – Average Condenser Convection Coefficient 2 – Adiabatic Flame Temperature (K) (W/m K) ℎ – Combustion Gases Temperature at point i – Average Envelope Convection (K) Coefficient (W/m 2K) ℎ – Saturation Temperature (K) – Humid Air Enthalpy (kJ/kg dry air) – Water Temperature at Condenser i exit – Combustion Chamber Heat Transfer ℎ (if i=0 doesn’t count) (K) Coefficient (W/m 2K) ℎ – Water Temperature at One of the Zone – Condensation Heat Transfer Exit (K) Coefficient (W/m 2K) ℎ – Global Convection Coefficient (W/m 2K) – Condensate Convection Coefficient 2 (W/m K) ℎ Greek symbols – Modified Vaporization Enthalpy (kJ/kg) 2 – Surface i Absorptivity (ad.) ℎ – Gases Enthalpy at Position i (W/m K) – Efficiency (ad.) ℎ – Noncondensable Gases Convection 2 – Overall Fin Efficiency (%) Coefficient (W/m K) ℎ – Log Mean Temperature Difference (K) – Liquid Water Enthalpy ( ) (kJ/kg) ∆ – Emissivity From the Envelope (ad.) ℎ – Radiosity from surface i (W)ℎ = ℎ – Surface i Emissivity (ad.) – Fin Thermal Conductivity (W/mK) – Heat Ex. Efficiency (ad.) – Tube Thermal Conductivity (W/mK) λ – Excess of Air Coefficient (ad.) – Air Intake Mass Flow (kg/s) – Specific Humidity (g water vapor/ Kg – Condensate Mass Flow (kg/s) gases) – Fuel Mass Flow (kg/s) – Surface i Reflectivity (ad.) – Flue Gas Mass Flow (kg/s) σ –Stefan-Boltzmann Constant – Water Vapor Mass Flow (kg/s) (5,67040×10 -8Js-1m-2K-4) – Water Mass Flow (kg/s) – Surface i Transmissivity (ad.) – Molar Mass of Specie i (kg/kmol) – Coefficient of ε-NTU-θ Method (ad.) – Number of Transfer Units (ad.)

1 Introduction The high efficiency of condensing boilers (>90%) and the alarming rise on gas prices makes condensing boilers more attractive to consumers even if they still cost more than traditional boilers. On the cost side, governments have been making laws and/or special programs for consumers to opt for condensing boilers rather than non-condensing boilers. (The Warren Report, 2007) Boiler certification requires laboratorial tests to validate their high efficiency and thus being able to apply to tax benefits. One of the standards available is the CFR Part 430, Point 6.2.2.2 – Test Procedure Rule – US. D.O.E., that, even being extensive, it doesn’t predict the reference for relative humidity nor its control on the tests. The standard does have reference conditions for inlet temperatures of air and water and other parameters such carbon dioxide (in molar percentage) on dry basis. The standard also measures the efficiency based on HHV (Higher Heating Value) since the standard was developed in USA rather than LHV (Lower Heating Value), the standard for Europe. Because of this in USA the efficiency does not surpass 100% and in Europe it can. These very high efficiencies are achieved at moderate to low powers since the dew point usually does not go beyond 60°C. The objective of the work is to develop a mathematical model of a condensing boiler, using VBA TM language attached to Excel TM . The resulting non-linear equations are solved using the improved Broyden method. The method makes use of the Sherman-Morrison formula (Kelly, 2003) to reduce the computational work and time and improve results. Other optional conditions could also be used. The goal is to evaluate the overall performance of the condensing boiler when there’s variation on several parameters one of them being the relative humidity ( RH ). Additionally the model is able to give information on the several zones (Combustion Chamber, Primary Heat Exchanger, and Condenser(s)) that a condensing boiler consists of. The validation of the model is based on experimental data already acquired. The work also intends to generalize the model to other condensing boilers and non-condensing boilers. Although geometries vary greatly, a standard geometry has been adopted that represents them with some corrections to the correlation’s parameters to adjust them the various cases. Therefore, this model is somewhat different from Makaire et al. (2010) model that is based on Hanby (2007) model by utilizing a general model for condensing boilers (independent from geometry) with a spiral coil for the condensing area. Makaire et al. (2010) proposes the Hanby (2007) model but the condensing area will use the Morisot (2000) model on which the coil is separated into 5 equal parts. The difference between the proposed model and the Hanby-like models is that, boiler properties such as global convection coefficients from each zone must be known by means of experiments. Anthony, S. (1986) also presented a similar model but proposed some experimental correlations based on the Colburn factor. Thus, this model only needs information that does not depend on experiments, making it more independent from real testing. Miranda et al. (2003) also presented a simple model for boilers that includes a detailed model for combustion that allowed them to calculate the height of the combustion chamber and the production of NOx and CO. The combustion chamber included heat transfer by convection and radiation. It was seen that the convection on the primary heat exchanger was important and that it increased with the water mass flow to be heated. For the relation between efficiency and RH, Kuck (1995) concludes that, at atmospheric pressure the use of pre-heated air with extra water vapor (when needed) increases significantly the efficiency of the boiler.

2 Model

Figure 1 - Complete model illustrating heat transfer and enthalpy flows For modeling the condensing boiler was conceptually divided in 3 zones for the energy balances as illustrated in figure 1. The first zone is the combustion zone plus combustion chamber, the second zone is the primary heat exchanger and the third is the condenser(s). The combustion zone will consider only complete and instantaneous combustion so no CO or NOx are considered. The model is able to handle several gaseous fuels such as methane, propane, etc. Since the percentage of carbon dioxide measured in tests is on dry basis the excess of air is calculated from that value without considering water vapor on the products of combustion. The adiabatic flame temperature can be obtained using the Newton-Raphson method presented e.g. by Kelly (2003) that is applied on the following equation. (1) So, on the first zone a closed radiation ( balance) = is( made. ) The base and top (burners and bottom of the primary heat exchanger) of the combustion chamber are assumed to be black bodies and the walls are considered grey and the emissivity of the gases is determined from Leckner correlation presented e.g. by Modest (2003). Therefore radiosity equations for this zone are the following.

= σT (2) J = εσT + (F J + F J + FJ) + εσT Irradiation equations are required for theJ = energyσT balances to the bottom, top and walls. These equations are defined generally by: (3) And the irradiation equations are: = ( − ) (4) = ( + ) + σT (5) = ( + + ) + σ T (6) In adition to that, heat transfer =by convection( + was) + addedσT using correlation for flat plate present in Incropera and De Witt (2006). For the equation system a virtual surface on the top of the combustion chamber (primary heat exchanger) is considered as a black body according to Azevedo (2000). This way, the heat transfer

4 will occur to the primary heat exchanger and the top of the primary heat exchanger will have negligible heat transfer by radiation since the view factor from the primary heat exchanger and the top is virtually zero. The heat exchanger is thus considered as a black surface at the average temperature of the water T +T T = w1 w2 . The temperature of the burner is calculated from an energy balance to the gases s2 2 considering that the heat transferred from the equivalent surface by radiation is the enthalpy variation of the gases. Heat transfer by convection was also added as stated previously. (7) The variation of the combustion gasesℎ −temperatureℎ = ( is then− )calculated from the heat balance:

(8) ℎ − ℎ = σT − 1 − + ℎ ( − ) where the sum represents the heat exchanged with each surface and considering the grey gas assumption that is the absorption is equal to emissivity. The temperature of the gases Tg3 in the chamber is taken as the geometric mean, Tg3 = Tg2 Tg4 . The energy balance to the water on the combustion chamber zone is then given by: (9) Furthermore, the(ℎ mean − ℎ wall) ≅ temperature( − ) − can , be + obtainedℎ from( − the ) temperature distribution calculated while modeling the wall.

(10) − The losses, q are calculated based= on the natural+ convection plus radiation. The correlations env,total 4 for natural correlation can be taken from Churchill et al. (1975) with a maximum Rayleigh number of 10 13 . The losses are evaluated with the following equation. (11) For the second zone, the, primary= ℎ heat exchanger( − ) con+ siders finned( tubes− ) with herringbone fins (wavy fins) since most boilers geometries for the fins resemble wavy fins. Adaptations on the geometric values must be moderate. For the primary heat exchanger the energy balance to water has the sum of the convection and radiation contributions: (12) For convection the (ℎ − ℎ method) = ( −is used) +as an approximation − since not all the energy is transferred as it passes through − the − exchanger. The heat balance to the gas then becomes: (13) So it is necessary to calculate theℎ global− ℎ convecti= on − coefficient for the primary heat exchanger. The following expression is for finned tubes.

(14) 1 ln (/) 1 = + + The tube convection coefficient canℎ be2 obtained usingℎ correlations from literature. When no turbulator exist the Gnielinski (1995) correlation is used for the water side. When turbulators exist several correlations were identified and for the cases considered in this work the correlation chosen was the Delta-wing Geometry from table 1. Table 1 - Turbulator Correlations Correlation Delta-wing Geometry (Eiamsa-ard and Promvonge, 2011) Twisted Tape and Wire Coil Geometry (Promvonge, 2008) V-Nozzle Geometry (Promvonge and Eiamsa-ard, 2007) Wire coil Geometry (Eiamsa-ard et al., 2011)

For the external convection coefficient Kim et al. (2008) correlation for herringbone finned tubes was considered. The can be determined using expressions from Kays and London (1984) for cross flow with n passes and one row. The first equation is for when is on the gas side and the second for the water side. The C = (mc ) (mc ) and NTU= AU (m c) = ε θ. r p min p max ( ) p min 5

(15) 1 = 1 − (16) The condenser is modeled by a condensate = 1 − film approximation that has a thermal resistance due to the liquid film. The model considered is based on the works of Shi et al (2011), Herranz et al. (2000) and Webb (1983,1985) and considers a layer of non-condensable gases creating additional thermal resistance. Based on the two resistances in series the temperature in the interface is calculated from the following energy balance to be solved where is the convection coefficient from the condensate layer and is the convection coefficientℎ from the non-condensable gases layer. Average temperature whereℎ used to facilitate the calculations. (17) Consider then, condensers ℎwhere ( − subscript) = ℎ represents − the inlet of combustions gases and the subscript the outlet. The same criteria are applied to the water side where subscript denotes the outlet of water and subscript the inlet. The range of the subscripts are, p=1,2,3, 6≤i≤9, 6≤j≤9, 0≤k≤2 e ≤ ≤ 0 l 2. Then, the energy balance for the condensers uses the temperature difference uses the Newton’s cooling law. Nonetheless, for those temperatures, an arithmetic mean is used. (18) = (ℎ − ℎ) = ∆ (19) = ,ℎ − ,ℎ − ℎ = ∆ (20) − − − ∆ = These models will only be made available − to the pro−gram when Ts

(23) ln ln 1 1 = + + + ℎ ℎ 2 2

(24) ln ln 1 1 = + + + ℎ ℎ 2 2 The condenser model is used to calculate the gas temperature and the condensate mass flow after passing the tubes. The mass flow of gases may decrease since it is expected that water vapor condensates on the finned tubes. Therefore the mass flow, enthalpy and the global convection coefficient will vary along the condensers. Initially the specific humidity is known as well as the inlet temperature. Therefore, the next specific humidity can be calculated using the following expression:

(25) ℎ Θ − = ℎ

The water vapor mass flow from the outlet of the condenser can be calculated using the next expression where the water vapor saturation pressure is calculated. (26) @ = @ + 3 Results The set of non-linear equations of the model is solved using the Broyden method with the data provided to validate it. Eight experiments were conducted. The last three of them were on non- condensing boilers. The Broyden method achieved a minimum tolerance of 1x10 -8 and took 17 iterations to achieve it on the first simulation that had a 10 equations system. The losses were about 1% of the input power at full load. The combustion sub-program behaved flawlessly with the adiabatic flame temperature decreasing with increase of relative humidity and achieving a maximum at an excess of air of 1 (since no dissociation is considered). The emissivity of the walls was fixed at 0.8 after several tests and with a qualitative deviation of 10% since no precise temperatures could be obtained. The radiation model adopted has a small sensibility to the emissivity of the wall so other approximations may improve the results. The humidity changed very slightly the emissivity but the combustion chamber temperature difference was not enough to change the outcome. Figure 2 tries to improve Miranda et al. (2003) conclusions that an increase in wall emissivity increases the percentage of radiation share on the combustion chamber. It can be seen that a “heavier” fuel decreases the percentage of radiation share. This happens because, even with the increase in participant gases percentage, the flame temperature is lower than the flame temperature of the “lighter” fuel. This can be confirmed with Miranda et al. (2003) results since for butane the CO and NOx increase regarding to propane for example. The results obtained are in line with Kenna et al. (2007).

80% 20% 78% 22% 76% 24% 74% 26% 72% 28% 70% 30% 68% 32% Radiation (%)

Convection (%) 66% 34% 64% 36% 62% 38% 60% 40% 0 5 10 15 20 25 30 35 Mass flow of water (l/min) for ΔT=41,37°C and ε=0,8

Figure 2 – Comparison of fuel composition on heat transfer mechanisms (triangle for propane and lozenge for methane) Furthermore, when RH was increased, the efficiency increases marginally on test #1, as shown in table 2. Also, for higher loads the RH influence on efficiency drops. This happens because the available energy increases and since tube wall temperature increases, thus increasing interface temperature and reducing the driving force . Table 2 also shows that on test #5 efficiency is below 100% (based on LHV) although condensation occurred. This is because the is no turbulators on the condensers between test #2 and #5, further increasing the interface temperature and somewhat reducing the heat transfer due to latent heat. This means the contribution of latent heat is underestimated. Considering figure 3, test #3 (2 condensers with turbulators and test #4 (3 condensers with turbulators) the increase in efficiency is of about 7.3% for test #3 and of 8.4% for test #4, from null RH to the maximum RH. The efficiency difference between the two tests was of more than 5% for a RH of 100%. These results indicate that condensation exists for the entire range of RH minus when RH is null.

Consequently benefits of condensation are clear and procedures should be implemented to enhance the boiler capability of achieving condensation.

Table 2 - Results for tests #1, #2 and #5 Simulation / Temperature after Flue gas temperature Efficiency Test Experiment primary HEX (°C) (°C) (LHV) Experiment ------52.40 102.16% #1 Simulation. RH=0% 196.00 85.70 101.21% (49.59 kW) Simulation. RH=100% 198.72 93.24 101.43% Experimental 250.00 50.00 103.73% #2 Simulation. RH=0% 220.48 96.17 100.08% (58.28 kW) Simulation. RH=100% 223.41 103.04 100.16% Experimental 255.00 52.00 101.53% #5 Simulation. RH=0% 219.20 154.77 95.52% (58.28 kW) Simulation. RH=100% 220.07 162.44 95.33%

118%

116%

114%

112%

110% Rendimento(%)

108%

106% 0% 20% 40% 60% 80% 100% HR (%)

Figure 3 - Efficiency vs. RH (%) for simulation #3 (red line) and #4 (yellow line)

Table 3 indicates that the differences between the experiments and the calculations were substantial but, when viewing from the power side the differences were small. Table 3 – Enthalpy of the flue gas and the difference obtained in power for different experiments on condensing boiler Test Obtained Value (kJ/Kg) Experimental Value (kJ/Kg) Difference (kW) 1 (49.59kW) 68.53 29.83 0.88 2 (58.28kW) 80.02 30.01 1.23 3 (5.82kW) -6.74 1.03 0.45 4 (5.82kW) -12.79 -2.07 0.16 5 (58.28kW) 144.40 30.01 2.86

Larger differences were encountered when testing non-condensing boilers but the maximum difference obtained was 0.99 kW. Flue gas temperature differences were between 35.1°C and 74.1°C. Notice that all of these differences for flue gas powers were obtained at 50% of RH.

Analyzing the efficiencies from the deviation point of view, an expression for standard deviation is presented:

ͦ 1ʡ (27) ͈ − When analyzing the efficiencies at= 0% of RH standard deviation using the below expression the value obtained for all simulations was 5.55%. For a RH of 100% the standard deviation was higher with a value of 4.62%. Both results make the model suitable for further testing.

120% 120%

110% 110%

100% 100%

90% 90% η pred(%) 80% η pred(%) 80%

70% 70%

60% 60% 60% 80% 100% 120% 60% 80% 100% 120% η exp (%) η exp (%)

Figure 4 - Comparison between experimental efficiencies and calculated efficiencies for +-10% at RH of 0% (left) and for RH of 100% (right)

4 Conclusions From the results obtained it is possible to retain some important aspects. The influence of the RH on the condensing boilers is relevant in a way that increasing the RH increases the efficiency of the boiler when condensation occurred due to latent heat. Marginal decreases were detected when no condensation occurred or when the latent heat was not significant due to high power tests that increased the interface temperature, diminishing the driving force . Increases in boiler efficiency of 7.3% to 8.4% were observed for condensing boilers in line with Kuck (1995). This shows the importance of the conditions used for tests and the influence on the location and installation of the boiler. When boilers act as non-condensing boilers, the efficiency drops marginally with the increase of RH. Therefore at full load there were no benefits from the increase in RH. The model built showed that there are differences on the order of 10% for the combustion chamber and 10% on the primary heat exchanger that can be explained by the simple model adopted for radiation and the uncertainty in the heat transfer correlations. No further fitting was done to the model although this is a possibility to custom the model to the experimental results. Therefore the standard deviation for the efficiencies calculated was 3.63% for a RH of 0% and 4.64% for a RH of 100%. Consequently, the model is capable of predicting the evolution of several parameters with a reasonable deviation.

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