1. Lecture 10 Thermodynamics of humid air; Part two

2. 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

3. 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
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 J/molK

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

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

6. 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.

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

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

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

10. 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)

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

12. Example 3: Mixing of two air flows
33oC, kg/kgda ,
20 kgda/s
What is the outlet temperature
and humidity of the air after mixing?
Flow 1
8oC, 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

13. Example 3: Mixing of two air flows
33 oC, kg/kgda ,
20 kgda/s
Flow 1
8 oC, 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.

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

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

16. 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.

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

18. 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]

19. 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%