Lecture 10 Thermodynamics of humid air; Part two

What is the surface temperature twb of the material at equilibrium? Wet bulb temperature What is the surface temperature twb of the material at equilibrium? Heat from the air Uniform temperature and moisture distribution Wet material Vapor from the wet material At equilibrium, the energy balance is Humid air (t,x,w) The surface temperature is called the wet bulb temperature twb w = air velocity, m/s = Heat transfer coefficient, W/m2K lv = vaporization heat of water, J/kg

Wet bulb temperature: evaporation rate k = mass transfer coefficient [m/s] T = temperature of the boundary layer [K] Usually, Sometimes T may also be twb+273.15 pv = partial vapor pressure in the air [Pa] pv’(twb) = saturated vapor pressure at the wet bulb temperature [Pa] MH2O = molar mass of water [kg/mol] ptot = total pressure of humid air [Pa] R = gas constant 8.314 J/molK

Wet bulb temperature: mass transfer coefficient Le is Lewis number. In practical calculations, we can usually assume that Le  1 =>

Wet bulb temperature: vaporization heat lv (twb)  2501000 – (cpliq – cpv)twb  2501000 – 2340twb cpliq = specific heat of water [J/kgK] cpv = specific heat of vapor [J/kgK]

Wet bulb temperature: calculation equation where twb is the only unknown term that can be solved iteratively using some suitable software, e.g. Excel Solver tool.

Dew point and wet bulb temperatures on the Mollier diagram NOTE: ptot must be ca 100kPa State of air: 70oC, 0.005 kg/kgda Wet bulb temperature c. 27oC Dew point temperature c. 4oC

Examples Example 1: Heating of humid air Example 2: Moisturizing of air Example 3: Mixing of two air flows

Example 1: Heating of humid air Qin Air, 10oC, 80%, 0.7kgda/s Air, 30oC What is the heat demand in heating? + pv’(10oC) = 1227.5 Pa pv = 0.81227.5 = 982 Pa

Example 2: Moisturizing of air Water 10oC What is the mass flow rate of water into the moisturizing chamber? Air, 60oC, 0.001kg/kgda, 1.2kgda/s Moisturizing chamber Air, 25oC Remeber to check for the outlet air that pv < pv’(25oC)

Heating and moisturizing on the Mollier diagram Moisturizing hin  hout Example 2 Heating xin = xout Example 1

Example 3: Mixing of two air flows 33oC, 0.036 kg/kgda , 20 kgda/s What is the outlet temperature and humidity of the air after mixing? Flow 1 8oC, 0.005 kg/kgda , 18 kgda/s Flow 3 tout , xout Mixing chamber Remeber to check that the state of air flow 3 is above the saturation curve

Example 3: Mixing of two air flows 33 oC, 0.036 kg/kgda , 20 kgda/s Flow 1 8 oC, 0.005 kg/kgda , 18 kgda/s Let’s chech whether the mixing point is below or above the saturation curve Flow 3 tout , xout Mixing chamber Condensate REMEBER ALWAYS TO CHECK THAT NO CONDENSING OCCURS AFTER MIXING.

Example 3: Mixing of two air flows, graphical solution The real mixing point t3  22.5oC x3  0.017kg/kgda Calculated mixing point

Example 3: Mixing of two air flows, numerical solution Energy and water balance after mixing

Condensing of water from humid air t = 40oC,  = 0.5 t = 35 oC,  = ? Air Does water condense from the air flow? t = 15oC At inlet: p’(40oC) = 7381 Pa => pv = 0.57381 = 3691Pa => tdew point  27.5oC If we assume that no condensing occurs, vapor pressure pv remains the same => At outlet: p’(35oC) = 5627 Pa =>  = pv/pv’(35o) = 3691/5627 = 0.66 < 1 Even though the relative humidity at outlet is lower than 1, condensing occurs, becuase the dew point temperature of the inlet air (27.5oC) is greater than the surface temperature (15oC). General condition for condensing. Condensing occurs if some surface temperature is lower than the dew point temperature of the air. - Dew point temperture: p’(27.5oC)  3690Pa - To define the relative humidity at the outlet we should calculate the condensing rate using theories of heat and mass transfer.

Cooling tower at a power plant Humid air Live steam Cooling tower Warm water Vapor Cold water Dry air Make-up water Condensate

Cooling of water in a cooling tower When warm water dopplets enter the cooling tower they begin to cool down due to evaporation and convective heat transfer. Humid air Evaporation m’’lv(tp)aAcdL dL Warm water Cooling tower Air Heat transfer (tparticle - tair)aAcdL Droplets cool down until they reach the equilibrium. At equilibrium Cold water Dry air Evaporation m’’lv(twb)aAcdL Make-up water Air Heat transfer (tair - twb)aAcdL At equilibrium, temperature of the droplets is the same as the wet bulb temperature of the inlet air. => Wet bulb tempearture of the inlet air is the lowest possible temperature of the cold water. m’’ evaporation rate [kg/m2s] lv vaporization heat [J/kg] Ac = cross-sectional area of the tower [m2] a = evaporation surcace of droplets per volume of the towern [m2/m3]  = transfer coefficient [W/m2]

Cooling of water in a cooling tower Humid air What is the lowest possible temperature of the cold water? twb (30oC, 40%)  24oC the lowest possible temperature of water is c. 24oC In reality, the outlet temperature depends on the design parameters of the tower and can be higher than 24oC. Warm water 35oC Cooling tower Cold water Dry air, 30oC, 40%