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rR12.9Using this boundary condition, C can be evaluated and equation 12.8 further integrated to obtain cp Tr TrR Rk mV ln 1 4 cp L 12.10 To solve for TrR and xV rR we eliminate mV from equations 12.6 and 12.10 and obtain cpTr TrR L 12.11 Given the transport and thermodynamic properties k, cp , L, and D neglecting variations of these with temperature as well as Tr and , this equation relates the droplet surface mass fraction, xV rR , and temperature TrR . Of course, these two quantities are also connected by the thermodynamic relation ln 1 xV rR pV rR MV V rR p M 12.12 xV r xV rR Dcp ln 1 k xV rR 1where MV and M are the molecular weights of the vapor and the mixture. Equation 12.11 can then be solved given the relation 12.12 and the saturated vapor pressure pV as a function of temperature. Note that since the droplet300size does not occur in equation 12.11, the surface temperature is independent of the droplet size. Once the surface temperature and mass fract!
ion are known, the rate of evaporation can be calculated from equation 12.7 by substituting mV 2 dRdt and integrating to obtain 4LR R2 Rt0 2 2k cp Tr TrR ln 1 cp L cp Tr TrR L1t12.13Thus the time required for complete evaporation, tev , is tev cp R2 t0 2k ln 1 12.14This quantity is important in combustion systems. If it approaches the residence time in the combustor this may lead to incomplete combustion, a failure that is usually avoided by using atomizing nozzles that make the initial droplet size, Rt0 , as small as possible. Having outlined the form of the solution for an evaporating droplet, albeit in the simplest case, we now proceed to consider the combustion of a single droplet. 12.5.2 Single droplet combustion For very small droplets of a volatile fuel, droplet evaporation is completed early in the heating process and the subsequent combustion process is unchanged by the fact that the fuel began in droplet form. On the other hand for larger droplets or le!
ss volatile fuels, droplet evaporation will be a controlFigure 12.13. Schematic of single droplet combustion indicating the radial distributions of fuelvapor mass fraction, xV , oxidant mass fraction, xO , and combustion products mass fraction. 301ling process during combustion. Consequently, analysis of the combustion of a single droplet begins with the single droplet evaporation discussed in the preceding section. Then single droplet combustion consists of the outward diffusion of fuel vapor from the droplet surface and the inward diffusion of oxygen or other oxidant from the far field, with the two reacting in a flame front at a certain radius from the droplet. It is usually adequate to assume that this combustion occurs instantaneously in a thin flame front at a specific radius, rf lame, as indicated in figure 12.13. As in the last section, a steady state process will be assumed in which the mass rates of consumption of fuel and oxidant in the flame are denoted by mV C !
and mOC respectively. For combustion stoichiometry we therefore have !
mV C mOC 12.15where is the massbased stoichiometric coefficient for complete combustion. Moreover the rate of heat release due to combustion will be QmV C where Q is the combustion heat release per unit mass of fuel. Assuming the mass diffusivities for the fuel and oxidant and the thermal diffusivity kcp are all the same a Lewis number of unity and denoted by D, the thermal and mass conservation equations for this process can then be written as mV d dT dr dr d dxV dr dr d dxO dr dr 4r 2D dT dr dxV dr dxO dr 4r 2 QmV C cp 12.16mV 4r 2 D 4r 2 mV C 12.17mV 4r 2 D 4r 2 mOC 12.18where xO is the mass fraction of oxidant. Using equation 12.15 to eliminate the reaction rate terms these become mV d d cp T QxV dr dr 4r 2 D d cpT QxV dr d cpT QxO dr d xV xO dr 12.19mV d d cpT QxO dr dr d d xV xO dr dr1,870 49.3 1,925 50.71,829 49.5 1,864 50.51,965 49.7 1,988 50.31,969 49.6 2,000 50.41635 44.2 1299 35.2 761 20.61378 36.3 1297 34.2 1120 29.51302 35.3 1231 33.!
3 1160 31.41288 32.6 1254 31.7 1411 35.71103 27.8 1132 28.5 1734 42.7The number of participants from 20042005 to 20072008 has been previously reported 16. doi10.1371journal.pntd.0002462.t001percentage of cases caused by DENV2 increased in 2005 through 2008 study Years 24, reaching 95.3 of detected cases in 2007 2008 Year 4 Figure 4. DENV3 entered the study population in 20082009 and was the predominant serotype in 20082009 86.4 and 20092010 82.4. Two cases of coinfection with two different DENV serotypes were also identified a DENV1 and DENV4 coinfection in 20042005 and a DENV1 and DENV2 coinfection in 20052006 Figure 4. A spatiotemporal video of dengue cases in the PDCS from 20042010 can be viewed at httpyoutu.beyQOEdsMhoSk. Using the 1997 WHO definitions, of 351 denguepositive cases, 319 90.9 had classic dengue fever, 21 6.0 had DHF, and 11 3.1 had DSS. The majority of these DHFDSS cases occurred in two years of the study, with five DHF cases 23.8 and three DSS cases 27.3!
in the 20072008 dengue season, and 12 DHF cases 57.1 and seven DSS cas!
es 63.6 in the 20092010 dengue season. Using the 2009 WHO definitions of dengue severity, 182 cases were classified as dengue with warning signs 51.6 of dengue cases and 53 15.0 as severe dengue. There was one death due to DENV infection during the six years of the cohort.DENV infectionsAnalysis of DENV infections was limited to participants who completed each year and contributed a blood sample at the beginning and at the end of the year. In total, 5,073 children contributed 19,708 personyears with 1,778 DENV infections, for an incidence rate of 90.2 infections per 1,000 personyears 95 CI 86.1, 94.5 Table 3. The incidence of DENV infections by study year ranged from 67.0 to 119.7 infections per 1,000 personyears. Excluding 14yearold children due to small sample size, among oneyear age groups, the highest DENV infection incidence was observed in fouryear old children, with an incidence of 100.4 infections per 1,000 personyears, although the incidence was fairly constant acr!
oss age groups Figure 2B. Of the 5,073 children who contributed at least one set of paired samples, 3,570 70.4 did not experience a DENV infection during the study, 1,250 24.6 experienced one documented DENV infection, 231 4.5 experienced two DENV infections, and 22 0.4 experienced three DENV infections, according to serological analysis. Over the six years, 2,779 denguenaive children contributed 8,881 personyears of time. The incidence of primary DENV infections was 78.8 infections per 1,000 personyears 95 CI 73.1, 84.9 Table 4. Yearly incidence rates ranged from 45.2 to 105.3 primary DENV infections per 1,000 personyears. Excluding 12 and 13yearold children due to small sample size, thePLOS Neglected Tropical Diseases www.plosntds.org 4highest incidence rate of primary DENV infection was observed in nineyear old children 109.4 95 CI 81.7, 146.6. Although the highest incidence rates of primary infections were seen in older children, relatively few older children were at !
risk for a primary infection and therefore the majority of DENV infecti!
ons in older children were secondary Figure 5. The 2,813 nondenguenaive children contributed 10,830 personyears. The incidence of secondary DENV infections was 99.5 infections per 1,000 personyears 95 CI 93.8, 105.7 Table 4, with annual secondary infection incidence rates ranging from 59.6 to 115.2 infections per 1,000 personyears. The highest incidence of secondary infections was observed in children aged five and under. However, when the percentage of primary and secondary infections was examined by year of age Figure 5, the majority of DENV infections in younger children were primary, presumably due to the relatively small percentage of young children at risk for a secondary infection. To examine the potential effects of both waning antibody levels and the use of the Inhibition ELISA assay on our estimate of overall DENV infection incidence, we performed a sensitivity analysis by determining how many symptomatic DENV infections reported in this study were correctly iden!
tified as DENV infections by Inhibition ELISA in paired annual samples.. This yielded a sensitivity of 79.7, which we then applied to our observed estimate of incidence to arrive at an incidence rate of 112.5 DENV infections per 1,000 personyears 95 CI 107.7, 117.4. Therefore, our observed incidence is a conservative estimate of the true incidence in the cohort.Repeat DENV infections Of the 700 children who entered the cohort denguenaive and experienced a primary DENV infection, 138 went on to experience a second DENV infection. The 700 children contributed 1,137 personyears of time, yielding an incidence rate of 121.3 second DENV infections per 1,000 personyears 95 CI 102.7, 143.4 Table 3. Of the 138 children who contributed 188.3 personyears of time at risk, 16 experienced third DENV infections, for an incidence rate of 84.9 third DENV infections per 1,000 personyears 95 CI 52.0, 138.7 Table 3. There were no fourth DENV infections observed during the 8.0 personyears of ti!
me at risk for a fourth infection.Dengue cases among DENV infectionsThe !
overall rate of symptomatic cases among DENV infections was 18.2 dengue cases per 100 DENV infections Table 5. This rate varied dramatically by year from 4.9 cases per 100 infectionsSeptember 2013 Volume 7 Issue 9 e2462Incidence of Dengue in a Nicaraguan CohortFigure 1. Flowchart of participants in the Pediatric Dengue Cohort Study, Managua, Nicaragua, 20042010. doi10.1371journal.pntd.0002462.g001PLOS Neglected Tropical Diseases www.plosntds.org5September 2013 Volume 7 Issue 9 e2462Incidence of Dengue in a Nicaraguan CohortTable 2. Incidence of dengue cases, Managua, Nicaragua, 20042010.
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