NUMERICAL MATLAB ROUTINE EVALUATION OF MASS AND HEAT OF EVAPORATION IN AN AIR-WATER SURFACE INTERFACE

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1 2013 International Nuclear Atlantic Conference - INAC 2013 Recife, PE, Brazil, November 24-29, 2013 ASSOCIAÇÃO BRASILEIRA DE ENERGIA NUCLEAR - ABEN ISBN: NUMERICAL MATLAB ROUTINE EVALUATION OF MASS AND HEAT OF EVAPORATION IN AN AIR-WATER SURFACE INTERFACE João Jachic 1, Maria L. Moreira 2 1 Comissão Nacional de Energia Nuclear, Instituto de Engenharia Nuclear. Av. Brigadeiro Trompowski s/n - Ilha do Fundão - C.P Cidade Universitária Rio de Janeiro, RJ - Brasil - Caixa-postal: jachic@ien.gov.br 2 Comissão Nacional de Energia Nuclear, Instituto de Engenharia Nuclear. Av. Brigadeiro Trompowski s/n - Ilha do Fundão - C.P Cidade Universitária Rio de Janeiro, RJ - Brasil - Caixa-postal: malu@ien.gov.br ABSTRACT A psychrometric chart data can be used to estimate some air-water mixture properties but it requires reading graphs. Assuming the humid air as ideal gas, this work applies some analytical equations together with experimental database and semi empirical correlations relating humidity ratio, specific mass, dew point, vapor mass, total air-vapor mixture pressure, air speed and water temperature, to incorporate in a numerical simulation routine using MATLAB technical computing language. The simulation routine calculates the dew point temperature, the specific humidity or the relative humidity, the partial pressure of the vapor, the saturation pressure as well as the evaporation mass rate per unit area of the pool surface and also the corresponding evaporation heat rate per unit area. The results show fairly good agreement of the simulated data with reference and experimental values. 1. INTRODUCTION A psychrometric chart can and usually is utilized to read graphically some air-water mixture properties. However, in a step by step computation routine, it is required to employ an evaluation that can be automatically updated with the last known dependent input variable. Then, assuming the humid air as ideal gas, in this work, it is applied some analytical equations [1] together with experimental database [1], [2] and semi empirical correlations [3] relating humidity ratio, specific mass, enthalpy, dew point, vapor mass, total air-vapor mixture pressure, air speed and water temperature to incorporate in a computation routine using MATLAB technical language. The saturation vapor pressure is updated with the temperature by applying a spline interpolating routine fed with adequate database. A small cooling water pool is simulated along with an exhausting air system above the pool surface in order to remove heat from a spent fuel type plate inserted in the bottom of the pool. The amount of vapor in the air entering the exhausting system will be increased due to some water evaporation from the water surface. The air-vapor mixture can be specified by its temperature (T), specific humidity (UE) or yet by its relative humidity (UR).

2 2. DEVELOPMENT The air-vapor mixture is assumed to be an ideal gas, with some water molecules inserted in a matrix of dry air composed mainly with nitrogen and oxygen. Let the subscript a stand for dry air and v labeling the vapor in a volume V of air at temperature T with total pressure P t. Then P a plus P v adds to P t. Let M be the molecular weight, then M v and M a are 18g and g/mol, respectively. Applying Clapeyron s equation for each gas, results: m v = P v M v (1) m a = (P t - P v )M a (2) where R is universal gas constant. Let P vs be the saturated steam pressure. Then, the vapor pressure P v tends to P vs at saturation condition. It is common to label the ratio m v /m a as specific humidity (UE) and P v /P vs as relative humidity (UR). Most literature refers to UE and UR as ω and φ, respectively. The equations 1 and 2 above then become: ω = UE = = (3) The specific humidity ω assumes its maximum value (ω s ) at air saturation condition (φ=100%). Let A denote the surface area of the interface water-air. While the saturation has not yet been reached, there will be some vapor released from the water surface (A) to the air mixture. A semi empirical correlation for the corresponding evaporation rate has been presented [4], [5] as follows g = θa(ω s -ω) (4) where g = evaporation rate (kg/h) θ = 25+19v = evaporation coefficient (kg/h/m²) v = air velocity above the water surface (m/s) A = interface water-air surface (m²) ω = specific humidity at temperature T (kg of water / kg of dry air) ω s = specific humidity in saturated air at the same temperature as the water surface (kg of water/ kg of dry air). The mass of vapor leaving the water surface carries along with it a corresponding amount of heat given by: where q = heat carried (kj/h) h e = evaporation heat of water = 2270 kj/kg. q = h e g (5)

3 It can be noted that the exchange of vapor from water surface to the air depend on atmospheric conditions. The temperature (T d ) at which the air becomes saturated is called dew point. At this temperature, the net evaporation rate (g) tends to zero and the corresponding specific humidity reaches it maximum value (ω s ). There are some correlations for the relative humidity at air condition away from the saturation limit. Let us present one of such correlation [6] that has been incorporated in our MATLAB computation routine. UR = φ = e C (6) where C = 5417 (7) T = water temperature (K) T d = dew pint temperature (K) In many cases, however, the dew point temperature is not readily known. Then, it is used yet another correlation [7] for the partial vapor pressure (P v ), namely: where P vs = x 10 D (8) P v = P vs 6.7 x 10-4 P(T - T u ) (9) D = (10) P v = vapor pressure away from saturation (mmhg) P vs = vapor pressure at saturation condition (mmhg) T u = bulk humid (wet-bulk) temperature (K) P = atmospheric pressure (mmhg) Combining those correlations, the dew point temperature (T d ) can be calculated and then used in eq.7 to determine the relative humidity (φ) in eq.6, for any air-vapor condition. 3. RESULTS The correlations exposed in the last section were incorporated in a software using MATLAB s technical computation language that we named SubPTDArVapor. This program allows the determination of UR, UE and the corresponding mass evaporation rate as well as the heat rate removed from the water pool through evaporation for the air exhausting system. This computation routine reproduces correctly the psychrometric chart data. For example, if the water temperature were T = 20ºC and 1 atm total atmospheric pressure, then: P vs (T = 20 o C) = 2333Pa ω s = kg of water / kg of dry air

4 In table 1 it is reproduced such calculation for few water-air mixture temperature T, using the computation software. Table 1: Variation of saturated vapor pressure and maximum specific humidity with temperature Water air temperature T (ºC) Vapor saturation pressure P vs (Pa) Maximum specific humidity ω s (kg of water/ kg of dry air) It is assumed in the above application example that the air in the exhausting system has inlet relative humidity of 50% and is flowing with average horizontal speed of 50 cm/s. Furthermore, it is assumed that the cooling pool dimensions are 5x5x5 m³. Then, it follows: UR = ϕ = 0.5 P v = 0.5 x 2333 = 1166 Pa ω = kg of water / kg of dry air g = (25 x 19 x 0.5) x 5 x 5 x ( ) = 6.47 kg of evaporated water /hour q = 2270 kj/kg x Data on the relative humidity were achieved accessing In 22/ at 8h 14 min, it showed T = 22ºC, UR 99%. Using the simulated 5x5x5 m 3 cooling pool, the amount of evaporated mass rate were obtained by feeding the computation code with relative humidity and air speed values as exposed in the table 2 below. Table 2: Mass evaporation rate variation with air exhausting conditions for T = 22ºC. Evaporated mass Inlet air relative humidity UR(%) Air horizontal speed v air (cm/s) rate for 5x5x5 pool (kg of water evaporated/ month) Furthermore, it is presented in table 3, the water surface pool level decrement rate estimation when inlet air-vapor temperature is 22 o C at various relative humidity and speed, used in the exhausting air system.

5 Table 3: Water surface level decrement rate variation with air velocity and relative humidity for air-vapor temperature t = 22ºC, units of cm/month. UR(%) v air (m/s) The data presented in table 1 can better be interpreted when displayed in a graph such as in figures 1 below when the inlet water-air temperature is 22 o C. 4 x 104 Vapor pressure at saturation (pvs) Vapor pressure at saturation, pvs (kpa) Experimental dada Spline interpolated Temperature of the Air-Vapor mixture, Ta = 22 ºC pvs(t=22ºc) = kpa hvs(t=22ºc) = kJ/kg hl(t=22ºc) = kJ/kg ug(t=22ºc) = kJ/kg uf(t=22ºc) = kJ/kg g= kg/hour/m 2 q= kw/m Temperature, T (ºC) Figure 1: Vapor pressure at saturation temperature and temperature of air-vapor mixture of 22 o C. The data presented in tables 2 and 3 can be visualized in figure 2. It can be noted that the pool surface level decrement increases with air speed of the exhausting system but decreases with the air-vapor relative humidity. It can also be noted that the decrement level increases with the average value of the water temperature in the cooling pool. Figure 3 shows the thermodynamic properties when the average pool water temperature reaches 42 o C.

6 Surface level decrement (cm/month) Water surface level decrement due to evaporation, T=22ºC Relative humidity, UR=10% Relative humidity, UR=30% Relative humidity, UR=60% Relative humidity, UR=90% Air-vapor temperature,t=22ºc Horizontal air velocity, vair (m/s) Figure2: Water surface level decrement due to evaporation. 4 x 104 Vapor pressure at saturation (pvs) Vapor pressure at saturation, pvs (kpa) Experimental dada Spline interpolated Temperature of the Air-Vapor mixture, Ta = 42 ºC pvs(t=42ºc) = kpa hvs(t=42ºc) = kJ/kg hl(t=42ºc) = kJ/kg ug(t=42ºc) = kJ/kg uf(t=42ºc) = kJ/kg g=1.3711kg/hour/m 2 q= kw/m Temperature, T (ºC) Figure3: Vapor pressure at saturation temperature and temperature of air-vapor mixture of 42 o C.

7 4. CONCLUSIONS There are additional results, using different input simulator that were generated just to delimit the acceptable range of meaningful data, mainly those that can be readily applied to cooling system evaluations. From the results presented in tables 1, 2 and 3 and figures 1, 2 and 3, it can be concluded that the computation software does properly calculate most of all important air-vapor mixture thermodynamic properties. It can also be noted that the simulated result do agree with those obtained directly through graph reading diagrams of psychrometric charts. It was, however, observed that the computation software does not give good results when is inputted with UR values below 20%. This does not truly represent a drawback since in Brazil the observable relative humidity of the atmosphere ranges from approximately 30% to around 90%. Nonetheless, in warm tropical coastal cities like Rio de Janeiro, the relative humidity may well reach values very close to even 100%. The presented results also show that there will be a significant water surface level decrement when the inlet exhausting air has low relative humidity and high horizontal speed. It is, however, important to note that the presented result were compiled assuming that the pool water stable temperature was low, around 22 o C. If it is allowed to use higher values of stabilized water temperature, then, there will be a significant higher evaporation rate. Consequently, the surface level decrement rate will also increase. It can be concluded that the presented computation software fits fairly well as an online auxiliary thermodynamic calculator. Therefore, it can and it is actually used as a subroutine in another heat removal by natural convection cooling system simulator, which should be used in the near future to aid some experiments with cooling pool for spent nuclear reactor fuel. ACKNOWLEDGMENTS CNEN Comissão Nacional de Energia Nuclear IEN Instituto de Engenharia Nuclear REFERENCES 1. R. W. FOX; A.T. Mcdonald. Introdução à /a Mecânica dos Fluidos, LTC-Livros Técnicos e Científicos, Editora SA, Rio de Janeiro & Brasil (1996). 2. M. C. ROTTER; D. C. WIGGERT. Mecânica dos Fluidos, Thomson, São Paulo & Brasil (2004). 3. R. B. SILVA. Manual da Termodinâmica e Transmissão de Calor, DLP - Departamento de Livros e Publicações do Grêmio Politécnico, São Paulo & Brasil (1968) AirPsychrometrics. 5. internet, psicrometric.pdf-princípiosbásicosdepsicometria. Roberto Prece e Lopes, Juarez de Sousa e Silva Ricardo Caetano Rezende. 6. D. F. YOUNG; B. R. MUNDSON; T. H. OKIIHI. Uma Introdução à Mecânica dos Fluidos, Edgard Blicher, São Paulo & Brasil (2005). 7. R. E. SONNTAG; C. BORGNAKKE; G.I.V. NYLEN. Fundamentals of Thermodynamics, Inc, New York & United States of America (1998).