Water Condensation Thermal Model

“All models are wrong, but some are useful.“ - George Box

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Overview

I’ve been intrigued by the notion of creating dew, or passively condensing water out of the atmosphere, for a few years now. Thanks to Gunnar Ristroph, the idea entered my head and has never left.

The general idea behind this is creating a system that emits radiative heat into the sky at night, which could have a high enough heat transfer rate to induce water condensation faster than natural dew creators like shrubs, trees, moss, and lichen.

Many historical and current designs do not thermally isolate their systems from existing heat sources. Radiation sources, like mountains and trees, or conduction sources like the ground are simply ignored. I think the thermal isolation can be improved, which with improved radiative heat transfer could allow for an install-and-forget method to combat desertification and help communities in need.

Assumptions For Model

  1. Air speed near condensation surfaces is low. This means that the heat transfer for condensing water is dominant, not convection.

  2. There is no cloud cover. Clouds reflect infrared radiation, so clouds would radiatively couple the ground to the condenser.

  3. Water vapor instantly replenishes in the air around condensing surfaces.

  4. Humidity and dew point change slowly.

  5. The enthalpy of vaporization is the enthalpy of condensation, and is constant.

  6. No condensation occurs until the condensing surfaces reaches the dew point.

  7. The radiative surfaces are the condensing surfaces, so the area is the same.

  8. Water condensation will not interfere with radiative heat transfer.

  9. Conductive heat transfer to and from the radiative area is negligible.

  10. Heat transfers perfectly with no losses.

  11. The dew point is above freezing, i.e. water will condense as liquid, not solid ice, onto the radiative surfaces.

  12. You, the reader, are using a browser that supports MathML, otherwise the next section won’t make any sense.

Radiative Heat Transfer

q r a d = A r a d ⋅ ϵ ⋅ F A ⋅ σ ⋅ ( T p a n e l 4 − T s k y 4 )

Equations

Convective Heat Transfer

q c o n v = A r a d ⋅ h ⋅ ( T a i r − T p a n e l )

Condensation Heat Transfer

q c o n d = m ˙ ⋅ H v a p

Condensing Surface Heat Transfer

q p a n e l = m ⋅ C ⋅ T ˙

q_rad: radiative heat transfer
A_rad: radiative area
e: emissivity
F_A: view factor from radiative area to sky
s: Boltzmann’s constant
T_panel: temperature of radiative area
T_sky: temperature of sky

q_conv: convective heat transfer
A_rad: radiative area (also exposed to air)
h: convective constant
T_air: temperature of the air
T_panel: temperature of radiative area

q_cond: heat transferred into condensation
m_dot: mass per second of water condensing
H_vap: enthalpy of vaporization

Transient Thermal Model

( ϵ ⋅ σ ⋅ A ⋅ t m ⋅ C ) ⋅ T p a n e l 4 + ( 1 + h ⋅ A ⋅ t m ⋅ C ) ⋅ T p a n e l = ( ϵ ⋅ σ ⋅ A ⋅ t m ⋅ C ) ⋅ T s k y 4 + ( h ⋅ A ⋅ t m ⋅ C ) ⋅ T a i r + T 0

Solving this equation for T_panel at each time t yields the radiative area’s temperature. Similarly, setting T_panel equal to the dew point (or any temperature) yields the amount of time t needed to reach it.

q_panel: heat transferred into radiative area
m: mass that provides the radiative area
C: specific heat capacity of the material
T_dot: change in temperature per second

Steady-State Water Production Model

m ˙ = ( ϵ ⋅ σ ⋅ A r a d H v a p ) ⋅ T d e w 4 + ( h ⋅ A r a d H v a p ) ⋅ T d e w − ( ϵ ⋅ σ ⋅ A r a d H v a p ) ⋅ T s k y 4 − ( h ⋅ A r a d H v a p ) ⋅ T a i r

By plugging in all known values, this equation approximates the mass rate of liquid water production. This calculation is only remotely useful once the radiative surfaces cool to the dew point.