modelEvaporation

Information

This model computes the water accumulation on the surface of a cooling coil. When the cooling coil operates, water is accumulated on the coil surface up to a maximum amount of water before water starts to drain away from the coil. When the coil is off, the accumulated water evaporates into the air stream.

Physical description

The calculations are based on Shirey et al. (2006).

Parameters

The maximum amount of water that can be accumulated is computed based on the following model, where we used the convention that latent heat removed from the air is negative: The parameter t_wet defines how long it takes for condensate to drip of the coil, assuming the coil starts completely dry and operates at the nominal operating point. Henderson et al. (2003) measured values for t_wet from 16.5 minutes (990 seconds) to 29 minutes (1740 seconds). Thus, we use a default value of t_wet=1400 seconds. The maximum amount of water that can accumulate on the coil is

m_max = -Q̇_L,nom   t_wet ⁄ h_fg

where Q̇_L,nom<0 is the latent capacity at the nominal conditions and h_fg is the latent heat of evaporation.

When the coil is off, the water that has been accumulated on the coil evaporates into the air. The rate of water vapor evaporation at nominal operating conditions is defined by the parameter γ_nom. The definition of γ_nom is

γ_nom = Q̇_e,nom ⁄ Q̇_L,nom,

where Q̇_e,nom<0 is the rate of evaporation from the coil surface into the air stream right after the coil is switched off. The default value is γ_nom = 1.5.

Time dependent calculations

First, we discuss the accumulation of water on the coil. The rate of water accumulation is computed as

dm(t)⁄dt = -ṁ_wat(t)

where ṁ_wat(t) ≤ 0 is the water vapor mass flow rate that is extracted from the air at the current operating conditions. The actual water vapor mass flow rate that is removed from the air stream is as computed by the steady-state cooling coil performance model because for the coil outlet conditions, it does not matter whether the water accumulates on the coil or drips away from the coil. The initial value for the water accumulation is zero at the start of the simulation, and set to whatever water remains after the coil has been switched off and the water partially or completely evaporated into the air stream.

Now, we discuss the evaporation of the water on the coil surface into the air when the coil is off. The model in Shirey et al. (2006) is based on the assumption that the wet coil acts as an evaporative cooler. The change of water on the coil is

dm(t)⁄dt = -ṁ_max(t) η(t),

where ṁ_max(t) > 0 is the maximum water mass flow rate from the coil to the air and η(t) ∈ [0, 1] is the mass transfer effectiveness. For an evaporative cooler,

η(t) = 1-exp(-NTU(t)),

where NTU(t)=(hA)_m/Ċ_a are the number of mass transfer units and Ċ_a is the air capacity flow rate. The mass transfer coefficient (hA)_m is assumed to be proportional to the wet coil area, which is assumed to be equal to the ratio m(t) ⁄ m_max(t). Hence,

(hA)_m(t) ∝ m(t) ⁄ m_max(t).

Furthermore, the mass transfer coefficient depends on the velocity, and hence mass flow rate, as

(hA)_m(t) ∝ ṁ_a(t)^0.8,

where ṁ_a(t) is the current air mass flow rate, from which follows that

NTU(t) ∝ ṁ_a(t)^-0.2.

Therefore, the water mass flow rate from the coil into the air stream is

ṁ_wat(t) = ṁ_max(t) (1-exp(-K (m(t) ⁄ m_tot) (ṁ_a(t) ⁄ ṁ_a,nom)^-0.2 )),

where K > 0 is a constant to be determined below and the rate of change of water on the coil surface is as before

dm(t)⁄dt = -ṁ_wat(t).

The maximum mass transfer is

ṁ_max = ṁ_a (x_wb(t) - x(t)),

where x_wb(t) is the moisture content of air at the wet bulb state and x(t) is the actual moisture content of the air.

The constant K is determined from the nominal conditions as follows: At the nominal condition, we have m(t) ⁄ m_tot=1 and ṁ_a(t) ⁄ ṁ_a,nom=1, and, hence,

ṁ_nom = ṁ_max,nom (1-e^-K).

Because

ṁ_nom = - γ Q̇_L,nom ⁄ h_fg,

it follows that

K = -ln( 1 + γ_nom Q̇_L,nom ⁄ ṁ_a,nom ⁄ h_fg ⁄ (x_nom-x_wb,nom) ),

where x_nom is the humidity ratio at the coil at nominal condition and x_wb,nom is the humidity ratio at the wet bulb condition. Note that the ln(·) in the above equation requires that the argument is positive. See the implementation section below for how this is implemented.

Implementation

Potential for moisture transfer

For the potential that causes the moisture transfer, the difference in mass fraction between the current coil air and the coil air at the wet bulb conditions is used, provided that the air mass flow rate is within 1⁄3 of the nominal mass flow rate. For smaller air mass flow rates, the outlet conditions are used to ensure that the outlet conditions are not supersaturated air. The transition between these two driving potential is continuously differentiable in the mass flow rate.

Computation of mass transfer effectiveness

To evaluate

K = -ln( 1 + γ_nom Q̇_L,nom ⁄ ṁ_a,nom ⁄ h_fg ⁄ (x_nom-x_wb,nom) ),

the argument of the ln(·) function must be positive. However, often the parameter γ_nom is not known, and the default value of γ_nom = 1.5 may yield negative arguments for the function ln(·). We therefore set a lower bound on γ_nom as follows: Note that γ_nom must be such that 0 < ṁ_wat,nom < ṁ_max,nom. This condition is equivalent to

0 < γ_nom < ṁ_a,nom (x_nom-x_wb,nom) h_fg ⁄ Q̇_L,nom

If γ_nom were equal to the right hand side, then the mass transfer effectiveness would be one. Hence, we set the maximum value of γ_nom,max to

γ_nom,max = 0.8 ṁ_a,nom (x_nom-x_wb,nom) h_fg ⁄ Q̇_L,nom,

which corresponds to a mass transfer effectiveness of 0.8. If γ_nom > γ_nom,max, the model sets γ_nom=γ_nom,max and writes a warning message.

Regularization near zero air mass flow rate

To regularize the equations near zero air mass flow rate and zero humidity on the coil, the following conditions have been imposed in such a way that the model is once continuously differentiable with bounded derivatives on compact sets:

• We impose that ṁ_wat(t) → 0 as m → 0 to ensure that there is no evaporation if no water remains on the coil.
• We impose that ṁ_wat(t) ⁄ ṁ_a(t) → 0 as ṁ_a(t) → 0 to ensure that the evaporation mass flow rate remains bounded at zero air flow rate and that it is symmetric near zero without having to trigger an event.

This is implemented by replacing for |ṁ_a(t)| < δ the equation for the evaporation mass flow rate by

ṁ_wat(t) = -C m ṁ_a²(t) (x_nom-x_wb,nom),

where C=K δ^-0.2 which approximates continuity at |ṁ_a|=δ. Note that differentiability is ensured because the two equations are combined using the function Buildings.Utilities.Math.Functions.spliceFunction. Also note that based on physics, we would not have to square ṁ_a, but this was done to avoid an event that would be triggered if |ṁ_a| would have been used. Since the equation is active only at very small air flow rates when the fan is off, the error is negligible for typical applications.

References

Hugh I. Henderson, Jr., Don B. Shirey III and Richard A. Raustad. Understanding the Dehumidification Performance of Air-Conditioning Equipment at Part-Load Conditions. CIBSE/ASHRAE Conference, Edinburgh, Scotland, September 2003.

Don B. Shirey III, Hugh I. Henderson, Jr. and Richard A. Raustad. Understanding the Dehumidification Performance of Air-Conditioning Equipment at Part-Load Conditions. Florida Solar Energy Center, Technical Report FSEC-CR-1537-05, January 2006.

Parameters

Connectors

Components

Contents

Name	Description
Medium	Medium model

Revisions

• September 4, 2026, by Michael Wetter:
  Added missing assignment for XiSatRefOut in branch that is used if the model does not compute evaporation.
  This is for #4715.
• April 5, 2023, by Jianjun Hu:
  Corrected assertion for the condition dX_nominal<0 and the documentation.
  This is for #3322.
• May 7, 2020, by Michael Wetter:
  Corrected limits for dX_nominal.
  This is for #1933.
• April 13, 2017, by Michael Wetter:
  Removed input TWat that is not used.
• March 7, 2017, by Michael Wetter:
  Set start value and removed max attribute for mEva_flow as this can take on zero.
• May 24, 2016, by Filip Jorissen:
  Corrected implementation of wet bulb temperature computation. See Annex60, #474 for a discussion.
• January 21, 2015 by Michael Wetter:
  Converted print statements to assertions with AssertionLevel.warning.
• June 18, 2014 by Michael Wetter:
  Added fixed=true for start value of m to avoid a warning during translation.
• August 21, 2012 by Michael Wetter:
  First implementation.