Adiabatic saturation

Adiabatic saturation is a process in which water evaporates into air in a duct in such a way that the air is saturated with water vapour at the outlet [30]. The latent heat necessary for evaporation is extracted from the air. This results in a decrease of the air temperature along the duct. It is assumed that the water surface from which water evaporates is in equilibrium with the exit air. The temperature reached at the outlet where the air is saturated is then called the adiabatic saturation temperature. This adiabatic saturation temperature is a property of the inlet air-water vapour mixture conditions only. For the pressure and temperature range encountered in buildings this adiabatic saturation temperature is often closely approximated by the wet bulb temperature. However in this section it will be shown that assuming the adiabatic saturation temperature equal to the wet bulb temperature should be done with caution. For the adiabatic saturation temperature, the following energy

0.E+00 5.E-04 1.E-03 2.E-03 2.E-03 3.E-03 3.E-03 4.E-03 4.E-03 5.E-03 5.E-03 0 0.02 0.04 0.06 0.08 0.1 x (m) Temperature rise (K)

(a) Temperature increase

0.E+00 1.E-04 2.E-04 3.E-04 4.E-04 5.E-04 6.E-04 7.E-04 0 0.02 0.04 0.06 0.08 0.1 x(m)

water vapour density rise (kg/m³)

(b) Water vapour density increase

Figure 3.8: Verification for transport equations in the porous material when only vapour transport is present. Comparison at different times between the numerical model (−) and

balance can be stated [69]:

L (ωsat(Ts) −ωin) =C (Tin−Ts) (3.40a)

Ts=Tas (3.40b)

In these equations Ts[K] is the surface temperature of the water surface and Tas [K] the adiabatic saturation temperature. ωin [kgvapour/kgdb] is the humidity ratio at the inlet of the duct, ωsatis the saturation humidity at the water surface corresponding with the water surface temperature. The humidity ratio is the ratio of the mass of water vapour to the mass of dry air (expressed in kgdrybasis or kgdb). C[J /kgK] is the heat capacity of the air, L[J /kg] the latent heat of evaporation. The humidity ratio can be expressed as a function of the mass fraction Y [kgvapour/kgair+vapour]:

ω = Y

1 − Y (3.41)

Since Y << 1, a humidity ratio difference can be written as a mass fraction difference.

ωsat(Ts) −ωin≈Ysat(Ts) −Yin (3.42) The adiabatic saturation process is schematically depicted in Figure 3.9.

Figure 3.9: Schematic representation of the adiabatic saturation process: water evaporates until the air at the oulet is saturated with water vapour. Fresh water is supplied

at saturation temperature.Tsat=Tas.

A process close to adiabatic saturation is convective drying. When unsaturated air flows over a wet surface, for example a saturated porous material, water will evaporate into the air. For this evaporation, latent heat is needed which results in a drop of the surface temperature. When a sample of porous material saturated with water is placed into an unsaturated air stream and all but one side of the sample are assumed impermeable and adiabatic, the temperature of the sample will consequently start to drop until the wet bulb temperature is reached. The steady-state heat balance at the wet surface is expressed by Eq. 3.43.

Ts=Twb (3.43b) In other words, at equilibrium, the heat flux entering the wet surface due to convection equals the heat flux leaving the surface due to evaporation. h is the convective heat transfer coefficient [W /m2_{K], h}Y

mthe convective mass transfer coefficient on a mass fraction basis [kg/m2s], A equals the surface area [m2]. At equilibrium the temperature at the surface equals the wet bulb temperature Twb.

Eq. 3.43 can now be used to determine the heat transfer coefficient if the mass transfer coefficients is known or visa versa. Expression of this type are often referred to as heat and mass analogy expressions.

A similar expression for the heat and mass analogy was developed by Chilton and Colburn [70, 71] also referred to as the Chilton-Colburn analogy. Eq. 3.44 shows this analogy when the gradient of vapour density is used as driving force for vapour diffusion.

h hρm

=ρCLe2/3 (3.44)

Talukdar et al. [72] developed a similar analogy based on the evaporation of water from a tray. h hρm = L∆ρv ∆T (3.45)

where ∆T and ∆ρv are respectively the logarithmic mean difference of the temperature and the vapour density.

∆T = (Ts−To) − (Ts−Ti) ln [(Ts−To) / (Ts−Ti)]

(3.46) Ts is the surface temperature, Ti the inlet temperature and To the outlet temperature. ∆ρvcan be defined in a similar way.

Chen et al. [73] used vapour density instead of mass fraction in Eq. 3.43 and found: h hρm = (ρv,sat(Twb) −ρv,in)L Tin−Twb (3.47) In this equation ρv,satis the vapour density at saturation condition corresponding with the wet bulb temperature. Tinis the inlet, the bulk or the ambient temperature. When comparing Eqs. 3.45 and 3.47 it can be seen that these two equations become equivalent if the outlet conditions in Eq. 3.45 are equal to the surface conditions. This is the case for an adiabatic saturation process. The wet bulb temperature will then equal the adiabatic saturation temperature.

Chen et al. compared their analogy with the Chilton-Colburn analogy. They found a significant difference in results when applying both approaches on a drying

case. The equilibrium temperature (wet bulb temperature) when applying the Chilton-Colburn analogy was lower than when their analogy was used. The heat and mass transfer coefficient ratio, obtained based on the heat balance equation at the water surface, was different from that derived from the conventional heat and mass analogy. This can however easily be understood if Eq. 3.47 and Eq. 3.40 are investigated more closely.

Eq. 3.40 is only valid for an adiabatic saturation process. If in Eq. 3.47 the surface conditions are equal to the adiabatic saturation conditions, the following heat and mass transfer coefficient ratio can be derived:

h hY m

=C (3.48)

In other words, If the Lewis number is equal to 1, the adiabatic saturation temperature will be equal to the wet bulb temperature. If the Lewis number is different from 1, the adiabatic saturation temperature will differ from the wet bulb temperature. The discrepancy Chen et al. [73] found was the result of a wrong use of the wet bulb temperature in Eq. 3.47 where they falsely assumed that the wet bulb temperature equals the adiabatic saturation temperature. It can be stated that the expression found by Chilton-Colburn [70], Talukdar et al. [72] and Chen et al. [73] are in fact equivalent.

To illustrate all this and to verify that the boundary conditions of the drying model are correctly implemented, an attempt is made to predict the wet bulb temperature with the drying model. Convective drying of a saturated porous material was simulated. Figure 3.10 shows the details of the simulated case. The moisture content of the porous material was kept constant during the transient simulation. The air flowing over the porous material had a bulk dry bulb temperature of 23.8°C and a relative humidity of 44%. The air flowing over the sample was not simulated but instead a constant heat transfer coefficient of 22.5W /m2_{K at the surface was taken. Air properties at 20°C gives λ}

a = 0.0257W /mK, ρa=1.205kg/m3, Ca=1005J /kgK and Dva=2.625e − 5m2/s. This results in a Lewis number equal to 0.808. Using the Chilton-Colburn analogy the mass transfer coefficient hYmequals 0.0258kg/m2s. For a Lewis number equal to 1 the mass transfer coefficient would be equal to 0.0224kg/m2s. The initial temperature of the brick is 23.8°C, the initial moisture content of the brick is 97%wcap. Material properties of the brick are listed in Appendix A. The adiabatic saturation temperature calculated according to Eq. 3.40 equals 15.91°C. If the Lewis number is lower than 1, the mass transfer coefficient is higher which will result in a lower equilibrium (wet bulb) temperature. The wet bulb temperature calculated according to Eq. 3.47 equals 15.51°C and thus 0.4°C lower than the adiabatic saturation temperature. In Figure 3.11 the simulation results are shown for the convective drying case with Le = 0.808 and Le = 1. The temperature in

the porous material drops to the wet bulb temperature and the adiabatic saturation temperature respectively. A perfect match is found between the predicted and simulated temperatures. a d ia b a ti c q g Tinit = 23.8°C w = 97%wcap Tair = 23.8°C RHair = 44% 3 c m

Figure 3.10: Details of 1D simulated case to illustrate adiabatic saturation.