# Water Condensation of Passive Dew Condensers

## Introduction

A Passive Dew Condenser (PDC) is a device that collects water from the atmosphere without using a power source [Beysens, 2018]. These devices aim to provide an alternative water source, so the scientific community is looking for ways to optimize the efficiency of PDC devices in collecting water.

The PDE model created here will be compared to an actual device located in Mirleft, Morocco [Lekouch et al., 2012], showing how much water is collected in approximately 24 hours.

A visualization of the device and the experimental setup can be seen in [Lekouch et al., 2012]. This simulation uses the data obtained in this experiment, given in appendix 1 of [Flores-Castillo, 2022], and compares the measured amount of water with our simulation results.

This tutorial explains how to simulate a transient model of a PDC using actual meteorological data in order to estimate the dew yield. The model considers the surface characteristics of the PDC, such as the material and the tilt angle and, and external variables, including meteorological parameters, such as ambient temperature, relative humidity, cloud cover, and solar radiation. This model can be used as a tool to optimize and design better PDC devices. The scientific model presented here is based on [Flores-Castillo, 2022] and is a two-dimensional model of a PDC.

A passive dew collector works under the meteorological phenomenon called dew. The formation of dew corresponds to the process of water vapor from the atmosphere condensing on a surface [Beysens, 2018]. For dew to form on a surface, the surface must have a temperature below the dew point temperature. The dew point temperature is defined as the temperature to which humid air must be cooled to become saturated. In other words, right at the dew point water vapor condenses into a liquid at the same rate it evaporates, assuming a constant pressure. Thus, on a surface with a temperatures below the dew point more liquid water will condensate than evaporate. This excess of water can be collected.

The natural releasing of thermal energy of the collector will allow the surface to cool down below the dew point. Additionally, meteorological effects and the technical conditions will affect how well the PDC surface can cool down.

The PDC device that will be used here is a planar PDC positioned at a 30° angle from the horizontal plane. A PDC is mainly composed of a condenser surface on top of an insulating material. The characteristics of the condenser surface, such as emissivity and heat capacity, will be of great importance as they affect how much water is condensed [Flores-Castillo, 2022].

This model takes into account various physical phenomena. For example, it considers the velocities of the surrounding air, the heat transfer in the air and in the PDC device itself, and different meteorological parameters; all of which are phenomena that change over time. The complex nature of this model makes it computationally expensive in both execution time and memory.

## Overview of the Passive Dew Condenser Model

In this model a fluid dynamics model and a heat transfer model will be run sequentially. The result will then be used to feed a condensate model. In the next sections an overview of the sub-models is given.

We start with a sketch displaying a passive dew collector and its components, which is composed of a condenser and an insulator. Air is surrounding the PDC. An airflow enters from the left boundary and exits at the right boundary . Note that we use a 2D domain with an - coordinate system.

Depiction of a passive dew collector and its components, surrounded by air.

### Fluid Dynamics Model

The fluid dynamics model will simulate the airflow over the passive dew collector. The fluid dynamics model used is a time dependent Navier-Stokes equation. The vector valued Navier-Stokes equation is given by:

where is the density and is the viscosity . is the pressure and is the vector valued velocity field . It's components are , in the - and -direction, respectively. The dependent variables to be solved for are the components of velocity and the relative pressure .

The result of the fluid dynamics model is the velocity field of the air flow over the device. This fluid velocity field solution drives the temperature distribution in the air surrounding the PDC device, something which is necessary to know if we want to estimate how much water is condensed.

### Heat Transfer Model

As the second component to PDC simulation, a heat transfer model will be used to obtain the condenser surface temperature. The surface temperature will allow us to compute the heat transfer coefficient between the condenser surface and the air surrounding it, which is needed to calculate the amount of water condensed at the surface.

The heat equation will be active in both the fluid region and in the PDC device. For a transient heat transfer model the temperature distribution is described by the transient heat equation. The transient heat equation is given by:

where is the constant pressure heat capacity , is the temperature and is the thermal conductivity . The air fluid flow computed with the fluid dynamics model described above is used to model the heat convected over the device. The air flow velocity computed with the fluid flow model with the components and , in the - and -direction, respectively, drives the convective term .

The PDC exchanges heat with its surroundings through various effects [Flores-Castillo, 2022]. The device loses thermal energy though convection , loses or absorbs energy through radiation , absorbs energy through solar irradiance and through condensation .

A summary of the various thermal processes of heat transfer at the condenser surface of the PDC. The convective term models the air flow velocity over the device and removes thermal energy from the PDC. Radiation models in- and outflow of thermal energy of the PDC with the surroundings through radiation. The condensation process , considered as an incoming heat flux, also effects the temperature of the device. The solar irradiance , considered as incoming heat flux, also effect the temperature of the device.

### Condensation model

The fluid flow model and the heat transfer model are accompanied by a condensation model. The condensation model allows us to compute the amount of condensate. The condensation is not modeled as a PDE but modeled through an integral.

The condensation model is based on a single equation which expresses a rate of mass change of condensed water and is described by the following equation [Beysens, 2018]:

where is the surface area of the exposed condenser surface, is the water pressure in the humid air above the condenser as a function of the ambient temperature , and is the water vapor pressure at the condenser temperature . is the water vapor transfer coefficient and is proportional to the heat transfer coefficient , is given by the following equation:

where is the latent heat of evaporation, which is the heat required for water to change phase from liquid to gas, and is the psychrometric constant, which relates the partial pressure of water vapor in air to air temperature, and is commonly used in disciplines like meteorology and climatology.

To solve equation (1) it will be necessary to know all the variables mentioned before, particularly the heat transfer coefficient .

## Physics Domain Coupling

In principle, the amount of condensed water can be calculated in two ways: in a serial and a segregated approach. The main difference between them is that in the serial method, the temperature of the PDC does not depend on the condensation term . Not considering the condensation term allows one to run the model sequentially, meaning that the condensation and heat transfer models are not fully coupled. That makes the serial approach simpler and faster to compute, but on the other hand the segregated method is more accurate.

In the serial approach, the Navier-Stokes equation (2) is solved first, then the heat equation (3). After that, the heat transfer coefficient and the rest of the variables of equation (4) are calculated in order to compute the amount of water.

This serial solution procedure is shown in the following diagram.

Serial method to solve water condensation.

## Meteorological parameters

Using actual weather data as input for the PDE model benefits the analysis. It helps to get a more realistic and accurate result from the model at a specific geolocation. Furthermore using actual data can help to optimize the design of devices for typical or extreme conditions.

The meteorological data used in the simulation includes: the ambient temperature , the relative humidity , the dew temperature , the wind speed , and the cloud cover fraction , which refers to the fraction of the sky obscured by clouds on average from a particular location. All meteorological parameters are time dependent.

The model presented here will use data shown in Appendix I of the thesis [Flores-Castillo, 2022]. It is necessary to convert these data points into a continuous function. To do that, we interpolate the data and generate an interpolating function.

In addition, in the Appendix section it is shown how to use of current and historical weather meteorological data from all standard weather stations worldwide available from WeatherData. With this the performance of a passive dew collector can be estimated at any point in the world.

The meteorological variables obtained are used throughout the entire simulation.

### Meteorological functions

First, meteorological functions to be used are defined. These include the dew temperature , the sky temperature and , and the sky emissivity and with cloud cover fraction .

The dew temperature is dependent on the ambient temperature and the relative humidity , and is given by:

In equation (5), the ambient temperature must be given in .

The sky emissivity , dependents on the dew temperature in and on the site elevation in can be given by:

The sky emissivity including cloud coverage is given by:

The sky temperature is related to the ambient temperature according to the following equation:

So, the sky temperature including cloud coverage can be deduced by replacing the sky emissivity with :

Later on, the ambient temperature , the relative humidity , the wind speed and the cloud cover fraction functions will be obtained using the meteorological data.

### Meteorological data from Mirleft, Morocco

To run the PDC simulation meteorological data from Mirleft, Morocco [Lekouch et al., 2012] is used. These data are from September to September , 2007. Seen that experimental data from this location exists, the model can be compared.

First, we define the geographical parameters, which are a latitude of and longitude of .

Then, we proceed to import the data .

Note, that for the wind speed and the cloud coverage, a GaussianFilter was applied in order to smooth the data.

Unfortunately, real life data might have all sorts of issues, like a negative cloud cover fraction. To accommodate for these shortcomings a clean up function is created to deal with negative values.

For the wind speed data, it is necessary to specify a lower speed limit [Flores-Castillo, 2022], which in this case will be

In general, wind speed is measured at elevation from ground. If one wants to know the velocities at lower heights, an extrapolation is needed. A classical logarithmic variation is typically used for this, which is given by:

where is the height at which the wind speed theoretically becomes zero and is the wind velocity at from the ground. is generally taken to be 0.1 [Beysens, 2018].

So, if we want to know the wind speed at from the ground, we just need to apply equation (11) with .

## Fluid Dynamics Model Setup

### Overview

In this section the fluid dynamics model is set up and solved. The Navier-Stokes equation is used to solve for the transient flow field which surrounds the PDC device.

The vector valued Navier-Stokes equation is given by:

where is the density and is the viscosity . is the pressure and is the vector valued velocity field . It's components are , in the - and -direction, respectively. The dependent variables to be solved for are the components of velocity and the relative pressure .

To simplify matters, no external forces, like gravity, act. In this simulation air is considered to be incompressible, which means that the density is constant. Additionally, to reduce the complexity of the numerical solution we neglected natural convection, and only forced convection is considered [Flores-Castillo, 2022].

The Navier-Stokes equation will be only active in the air domain of the geometry.

Besides the fluid dynamic PDE model, it is necessary to specify boundary conditions:

- A no-slip boundary condition on the PDC walls and the ground level. No-slip means that the velocity in all directions at that surface is zero. This is a common boundary condition in fluid dynamics to model walls, however, choosing a no-slip condition on the PDC surface is essential for heat transfer model shown below and will be explained below.

- An outflow condition at the right-hand side of the domain where the pressure is set to the arbitrarily value of 0.

The relative pressure is the fluid's pressure with respect to a reference pressure. In most cases, the reference pressure is set to the ambient pressure of . When necessary one can calculate the absolute pressure of the system with the following equation:

This equation means that our outflow condition is set to an absolute pressure of 1 atmosphere. Initial conditions and boundary condition should be defined using relative pressure values.

### Domain

The domain of analysis consists of the PDC device in gray and the surrounding fluid area in blue, as shown in the figure below.

Domain of analysis Ω with the PDC device in gray and the surrounding air area in blue.

When simulating fluid flow around the PDC device turbulence will form below the device. The cause of the turbulence is the model's geometry and fluid flow conditions as mentioned in [Flores-Castillo, 2022]. See the Flow Regime subsection for a more theoretical explanation.

To avoid the formation of turbulence below the PDC, a solid region (in red) was added beneath the PDC device, whose primary function is to prevent turbulence from forming beneath the PDC [Flores-Castillo, 2022]. A further step to prevent turbulence from developing is by placing the PDC right corner at the right boundary of the air domain, as shown in the figure below. That is how we got from the domain shown above to the one below.

Modified domain of analysis Ω conformed by the PDC device in gray, a solid region in red and the surrounding air area in blue.

The PDC device has a length of 1 and a thickness of 0.0253 . The PDC thickness is divided by 10% of condenser, , and the rest of insulator, . Moreover, the PDC is tilted with an angle of 30° from the horizontal.

Dimensions of the PDC device. The condenser in white and the insulator in gray.

We start by specifying the PDC region as rectangle, that is orthogonal to the axis. Then, through geometric transformation, the rectangle will be centered at {PDC_{centerX},PDC_{centerZ}} and rotated 30° from the horizontal.

In order to geometrically transform the coordinates, we use RotationTransform and TranslationTransform.

The grey rectangle is the insulator and the one with the red dot is the translated and rotated one. The same rotation and translation applies to the condenser, which can be seen as a small black line at the top of the insulator. The condenser is very small compared to the insulator.

### Mesh Generation

The boundary mesh is created using ToBoundaryMesh. In addition to the coordinates of the PDC, a solid region is defined to avoid simulating the turbulent flow below the PDC. The new solid region is a triangle, placed beneath the PDC device. Two of the triangle coordinates are the two bottom coordinates of the insulator, while the third coordinate is at a specific height at the right boundary of the domain.

The area above the PDC is of great importance. In that area the differential temperature for the heat transfer coefficient computation (eqn. 12) will be obtained. To get an accurate result, a finer mesh will be used in that specif area. The mesh refinement function used for refining the area is given by a semi-circle strictly above the PDC surface.

The above defines a mesh refinement function that produces a mesh elements of an area less than 0.00005 units when inside the refinement region and less than 0.0001 units when outside. Now the full element mesh can be generated.

### Material parameters

The fluid medium is air. We set up the material properties.

### Velocity functions

This section computes the velocity field around the PDC based on inflow wind speeds obtained from weather stations nearby. The wind speeds are on the order of . Using them directly, will result in an unnecessarily long run time of NDSolve, mainly because the high velocities increase the dynamics of the PDE, and NDSolve, or any other FEM solver for that matter, needs to take smaller time steps to time integrate the PDE.

To avoid this, the following approach is used: The Navier-Stokes equation is solved for lower inflow velocities, around and solved up to a time of . Then, the velocity field is extrapolated beyond the velocity field calculated at . This is justified given that the flow regime over of the PDC surface is laminar. The approach assumes that the velocity field increases linearly as the incoming velocity increases. This approach is validated in [Flores-Castillo, 2022] and will be explained and done after solving the fluid model as an additional post-processing step.

For the inflow boundary condition to be consistent with the initial conditions, a helper function that smoothly ramps up the inflow over time is created. The smoothed step function is defined as follows:

Here the minimum value and the maximum value the function can reach are denoted by and . The location of the step is controlled by and the smoothed steepness is controlled by .

The parameters chosen for the ramp-function are taken from [Flores-Castillo, 2022].

The next step is to generate an incoming wind velocity profile along the z-axis. The velocity profile will follow the classical logarithmic variation given in equation (13):

U(z)=U_{10}Log(z/z

_{c})/Log(10/z

_{c})

As the PDC is not exactly on the ground the velocity profile is shifted by in -direction.

Our velocity at is going to have a lower value than the existing meteorological wind velocity at 10 . Remember that the velocity is solved for smaller inflow velocities.

The value of 0.35 was chosen due to its effects on increasing the convergence of the simulation. Different values were tested using this same model, but the value of 0.35 was the best [Flores-Castillo, 2022].

### Flow Regime

The value of the Reynolds number tells us if a flow is laminar or turbulent and is calculated as:

where is the density of the fluid, is the viscosity, is the average velocity, and is the characteristic length.

It is always advisable to calculate the Reynolds number before doing a fluid flow simulation, so one can determine the type of the flow. For an open planar structure, turbulence for air occurs for , corresponding to wind velocities with , meaning that most air flows surrounding PDC devices should correspond to laminar flows [Beysens, 2018]. Nevertheless, this is a theoretical concept and small inhomogeneities in the PDC geometry can make the flow turbulent at much smaller values of . For this reason the model has a solid region beneath the PDC, just to prevent turbulence from forming in the simulation.

In this model the average velocity is given by the velocity profile at , the coordinate of , and the characteristic length is taken as the length of the PDC: .

This value tell us that theoretically the flow is in the laminar regime.

### Boundary Conditions

There are four types of the boundary conditions involved in the fluid dynamics model:

At the inlet the flow velocity is set to , where is the velocity profile.

The ground level of the domain is represented as a wall and also the exterior boundaries of the PDC device, so the flow velocity is set to zero on the boundaries to model a no-slip flow condition.

The no-slip condition specified at the walls of the PDC device are used to fulfil the condition that the velocities on the surface need to be 0. A 0 velocity on the boundary is essential for the computation of the heat transfer coefficient as outlined in the condensation model section.

At the outlet, a pressure outlet boundary conditions is used to model the outgoing flow. Here the outlet pressure is set to an absolute pressure of , . Equivalent to a relative pressure of .

At the top of the domain, an open boundary condition is applied. An open boundary condition describes a boundary in which the fluidic section of the geometry ends, but which is contact with the rest of the fluid domain that is not represented in the geometry. This boundary condition is modeled as a Neumann 0 value. As these are the natural default boundary conditions, these can be left out.

### Solve the PDE Model

First, we build the fluid dynamics PDE to solve for the flow velocity field .

Since the Navier-Stokes equation is a high-index differential algebraic equation, options are specified to time-integrate the equations efficiently. The maximum difference order of the time integrator is reduced to minimize the number of rejected steps. The time dependent boundary conditions are differentiated to create a consistent system of equations that are more efficient to time integrate. Lastly, a stable solution can be found if the velocities are interpolated with a higher order than the pressure. Here the velocities and are set to be interpolated with second order and the pressure with first order.

To read more about the set of options NDSolve has for Finite Elements and how to specify them, check the following tutorial NDSolve Options for Finite Elements.

Also, we specified to NDSolve to use as a linear solver the Multifrontal algorithm instead of the default one. In this case, the default linear solver can not solve the nearly singular matrices given. If the default is used, the following message will appear:

To inspect the fluid field, the following visualizations show a stream plot of the velocity field and a contour plot for the velocity magnitude with .

The extrapolation of the velocity field is done in the following subsection.

### Extrapolation Method

The Navier-Stokes equation was solved for low inflow velocities, around , and solved up to a time of . But, we want a solution that can cover the whole period on which the heat transfer simulation is done, which is approximately hours or , and that has real wind velocity values. So, to accomplish that, the velocity field previously obtained is extrapolated to match the actual wind velocities at all times. We will use the wind velocity function already obtained here to know when is it necessary to modify the velocity values.

To understand how the extrapolation works, it is instructive to look at a hypothetical case. Let's say, the wind velocity at from ground , which is , is compared against the meteorological field velocity at , which is . If the wind speed is higher than , as it is in this case, the solution field from the Navier-Stokes solution at will be multiplied by the factor , and that new velocity field will be the field for . If the velocity is less than in the given case, then the velocity field for will be taken directly from the Navier-Stokes solution at a specific . Let's say for now that it takes the solution at , because the velocity field from the Navier-Stokes equation at this time step already reassembles the winds speeds magnitudes at .

Remember that the solution field was solved until , so in the case that the velocity is less , it will be necessary to develop a way in which, for times larger than , we can still access the solution we have from the Navier-Stokes equation.To fix this time inconsistency, we developed two functions that will help us to accomplish that.

The first function receives as input the ratio between and the wind speed data , and as output, gives a time that lies between the range of the fluid simulation. This function aims to know which time velocity solution to use for longer times, i.e., for a time of ; the function says that you can use the velocity solution at .

The times that we are gonna be obtaining come from solving the following equation:

where is the same ramp-up function used in the fluid simulation and is a constant value.

The ramp-up function is the function to solve for , and the value is going to be or the ratio given as an input. The value of also is the threshold value which will determine the value of and, consequently, the return time value.

Note that Solve always returns a nested list of rules. To use the solution, we must first replace the rule and then extract the value from the list.

So, the function compares the input ratio and . If the velocity ratio is bigger or equal to 0.98, the ramp-up will be solved for given . If the condition is not satisfied, the ramp-up function will be solved for given .

So, as the condition is true, the return time value will be ; otherwise, it will be a lower time value that depends on the ratio of velocities.

After setting this function, we can proceed to make an interpolating function that will relate the fluid simulation to the heat transfer simulation times. The values will range from 0 to , and the values will be from to . This function will allow us to extract the solution for times longer than regardless of the time domain of the heat transfer solution.

Now that we have a way to access the Navier-Stokes solution, we can continue with the implementation of the extrapolation.

The implementation uses If conditional statements to compare when the wind velocity is above or below . Suppose the wind velocity at time is lower than the velocity at () then the velocity field at time will be from the Navier-Stokes solution without any modification. If not, the velocity field at time is multiplied by the ratio between the wind speed at time and the velocity at , .

where ranges from to and time is a function timeFunction that returns a time value between and .

The velocity from eqn. 14 will looks as follows:

Finally, to define the velocity functions, we must consider that the heat transfer model will use a different mesh that includes the PDC region. So, the PDC device should have no velocity field as it is solid. For this purpose, an element marker, to be defined later, is used in the following functions to assign a velocity of zero inside the PDC.

## Heat Transfer Model Setup

### Overview

In this section, the heat transfer model will be discussed and set up, including equations, domain of analysis, material parameters, and boundary conditions.

The heat equation will be active in both the fluid region and in the PDC device. For a transient heat transfer model the temperature distribution is described by the transient heat equation. The transient heat equation is given by:

What follows is just the general heat equation and is not what is used in the simulation. The parameters will be specified in the Heat Transfer Parameter Specification subsection.

Besides the heat transfer PDE model, it is necessary to specify boundary conditions:

- The convective term models the air flow velocity over the device and removes thermal energy from the PDC.

- Radiation models in- and outflow of thermal energy of the PDC with the surroundings through radiation.

- The condensation process , considered as incoming heat flux, also effects the temperature of the device.

In summary, the heat transfer model will simulate the temperature distribution of the fluid and, on the other hand, the temperature distribution of the condenser and insulator, taking into account various boundary conditions.

One should notice that this setup does not consider a support structure for the PDC device itself. It is assumed that the condensing surface is fixed on an adiabatic material [Beysens, 2018] which makes a simulation of a mount unnecessary.

From here, meteorological data is used throughout the whole simulation.

### Mesh generation

In this subsection, the mesh element number is reduced by increasing the mesh element size and this mesh now includes the PDC region not considered in the fluid simulation. The red solid triangle region has also been removed as in the fluid simulation. This area is of no interest for the simulation, and removing this area does not change the heat equation solution [Flores-Castillo, 2022].

As was discussed before, the domain is composed of different materials which have their own properties. To do this, we assign markers to the different subregions. These markers are used to set up the material parameters. Using markers for this is much simpler than specifying the formula for each subregion.

For markers to be attributed to different subregions, coordinates that lie in those subregions need to be specified.

The element markers are going to be: "Air"->10, "Condenser"->20, and "Insulator"->30.

When specifying different subregions it is also possible to refine each of them.

The mesh refinement focuses only on the interface between the PDC surface and the air as it can be seen in the mesh from above.

The blue region represents the air, gray the insulator, and white the condenser. There is no need to refine the condenser surface. What is of interest is the interface between the condenser surface and the air, and the region of above it.

### Heat Transfer Parameter Specification

We need to specify the material properties of each subregion. The fluid medium is air. The condenser is made of low-density polyethylene (LDPE) and the insulator of Styrofoam.

Then, specify the velocities which were defined in the Extrapolation Method subsection.

### Boundary conditions

Besides the heat transfer PDE model, it is necessary to specify boundary conditions:

- The convective term models the air flow velocity over the device and removes thermal energy from the PDC.

- Radiation models in- and outflow of thermal energy of the PDC with the surroundings through radiation.

- The condensation process , considered as incoming heat flux, also effect the temperature of the device.

We proceed to quantify the boundary condition terms of the heat transfer model. We use the abbreviation for ambient, for condenser and for sky.

Special notice that these boundary conditions are applied on inner boundaries and not on the simulation domain boundary.

Due to the no-slip fluid condition specified at the walls of the PDC device, the velocities on the surface are 0. So, a convective boundary condition can not be directly used in the heat transfer model. Nevertheless, the heat transfer coefficient is computed, which is necessary for calculating the amount of water as outlined in the condensation model section. This process ensures that the convective contribution is taken into account.

In the serial approach, the temperature of the PDC does not use the condensation term , so this boundary condition is not used.

is the radiative cooling term, expressed as the difference between absorbed radiation from the sky, and the radiation emitted by the condenser and it is expressed with the following equation [Flores-Castillo, 2022]:

where is the emissivity of the sky, is the emissivity of the condenser, is the Stefan-Boltzmann constant and is a view factor that ranges from 0 to 1.

can be simplified further by considering that the sky temperature is related to the ambient temperature according to equation (15):

T_{sky}=T

_{a}ϵ

_{sky}

^{1/4}

Substituting equation (16) into (17) gives:

The boundary condition expressed in equation (18) given in [Flores-Castillo, 2022] is not conventional in the sense that it does not directly map to the typically used HeatRadiationValue. Never the less the given boundary condition can be implemented. The here given radiative cooling model does not consider the condenser surface absorptivity factor α, typically used when expressing the incoming radiation. More details can be found in Thermal Radiation Boundary Conditions or in [Beysens, 2018, equation 7.20], where a more common form of the equation is shown that can be modeled with HeatRadiationValue.

In this subsection the radiation term will be further refined. Instead of using the above equation, we will now also consider cloud cover fraction effects on the sky temperature , turning into .

For this boundary, we will use the sky temperature with cloud coverage , emissivity , the Stefan-Boltzmann constant , and a view factor which is dependent on the tilt angle of the PDC.

To get a value of the view factor we start with Planck's Law, which describes the spectral radiance and provides the power radiated by a surface element at different wavelength . The power is measured per unit surface area of the body, per unit solid angle , and per unit wavelength [Beysens, 2018]. The Planck radiation law is the following one:

We will be using PlanckRadiationLaw for further calculations. PlanckRadiationLaw uses the same formula as equation (19).

When the spectral radiance is integrated over all the wavelengths and solid angles at a given temperature, one obtains the total black body spectraL radiance:

where we integrate over a half-sphere and integrate from to . The cosine appears because black bodies obey Lambert's cosine law. In this equation represents the zenith angle () and the azimuthal angle ().

The part of the integral which involves is computed with PlanckRadiationLaw, while the rest is computed manually using symbolic integration.

When the PDC is inclined, the solid angle is not integrated over the whole sky, instead the calculation is divided into two integrals. In the first integral spans from 0 to and from 0 to , while in the second integral what changes is the range of , which is now from θ (the tilt angle) to :

The view factor of the PDC tilted at a specific angle is given by the ratio between and .

As was mentioned before, radiative cooling is the essential thermal process for which it is possible to collect water and the emissivity of the device plays a crucial roll on how much energy is radiated from the PDC. The surface emissivity can depend on several factors, like temperature, wavelength, angle, and other variables. Here, we used a constant emissivities to simplify the model.

It is important to realize that the value of cooling power is very sensitive to the value of the condenser emissivity . So, one can adjust the emissivity values in the simulation to fit the data of a reference experiment [Beysens, 2018].

In this model two different emissivities are used depending on the situation of the condenser. If there is water on the condenser, the emissivity of water is used, otherwise emissivity of the condenser is used. In this specific case we are using the same values for both emissivities. These values must always be in the range between zero and one.

An If conditional is used in the definition of to know when the surface temperature is below the dew point temperate and thus select between and . Remember that for dew to form on a surface, the surface must have a temperature below the dew point temperature .

The solar irradiance heat flux and thermal radiation both happen at the condenser's surface.

The condenser surface is specified with the help of a line . The line will give us the -coordinates of the surface. To create this line that overlaps the condenser surface, we define a function that uses the equation of a line , where is the slope of the line and is the intersection with the -axis.

The inputs for are defined below.

The thermal radiation boundary condition cannot be implemented with HeatRadiationValue as does not directly map to what HeatRadiationValue uses. For this reason a NeumannValue is set up.

Another heat contribution to the condenser is the solar irradiance . Solar radiation causes the condenser temperature to rise or decrease depending on the daytime. A simple solar radiation model [Wilkerson et al., 2013] was adapted for this model. This solar radiation model allows for predicting how much heat flux comes from the sun, considering sunset and sunrise times for any location on Earth. Later on, it will be implemented as a Neumann boundary condition.

First, we define the Earth's axis of rotation, represented as a rotation matrix, where the rotation angle will be the tilt angle of the Earth which is .

The following two equations give the Earth's rotation. The first one is again a rotation matrix with an angle of rotation given by , which depends on the time of day and the year's season.

The total hours are given in seconds, starting at zero at midday [Flores-Castillo, 2022].

The season is the number of days from January to the end date, divided by .

The position of the Earth to the sun throughout the year is defined in the following function.

Next, the location on Earth is defined also as a vector .

One can obtain the vector normal to the surface from the two rotation matrices and location vector.

Finally, the incident radiation is calculated from the angle of incidence times a maximum solar irradiance of 1360 .

The solar irradiance was adjusted to fit the experimental data available. The adjustment is made during the initial hour so the sunset and sunrise times of the simulation can match with the real ones [Flores-Castillo, 2022].

The solar irradiance plot shows that sunset and sunrise occur approximately at and , respectively. Notice that the solar irradiance goes to zero at night.

To visualize the accumulated solar irradiance during the period, we need to create a list of points from the solar irradiance function. We then use the function Accumulate to get the total solar irradiation.

The heat flux boundary condition will represent the amount of thermal energy from the sun flowing into the PDC surface.

Here, we deleted the "HeatConvectionVelocity" entry because this boundary condition is for the PDC surface, which is a solid. If the heat convection velocities are included the HeatFluxValue will be modelling a boundary of a fluid that includes a convection velocity term associated with the boundary unit normal , which is not wanted here.

This the only boundary condition that is applied to the simulation domain boundary.

At the flow inlet the temperature is set to the ambient temperature .

### Solution and Visualization

In this subsection, the heat equation will be solved. First, we define the initial condition to denote an ambient temperature at over all the domain, and then we will set up the PDE.

Also, in this model, options are specified to time-integrate the equation. A MaxStepSize of 900 and a StartingStepSize of 1 were chosen. The time-integration method used is the "IDA" method with a maximum difference order of 1, set up through the "TimeIntegration" option. Finally, the time-dependent boundary conditions are differentiated to create a consistent system of equations that are more efficient to time integrate. To read more about the set of options NDSolve has for Finite Elements and how to specify them, check the following tutorial NDSolve Options for Finite Elements.

The previous warning message is generated due to the nature of the model and the mesh used. It does not affect the results of the model.

The plot from above shows the temperature that has the condenser surface and the air temperature above the condenser. Both temperatures are essential for calculating the heat transfer coefficient and consequently the amount of condensed water .

## Condensation Model

### Overview

The condensation model is based on the supersaturation equation (20):

In this equation two process are shown: condensation and evaporation. If the value of is positive, then condensation is taking place and . For a negative value of evaporation is taking place. Since we do not want to consider evaporation in the model the rate of mass change will be set to zero .

Zero here means that the evaporation of the mass of water is not considered [Flores-Castillo, 2022]. So, all water condensed is collected and the equation can be simplified to:

Another way of looking at equation (21) is as an integral:

This integral allows for computing the mass of the condensed water. With the exception of the heat transfer coefficient the values in equation (22) can directly be obtained from formulas that will be shown later. At this stage, the computation of the heat transfer coefficient needs some further explanation.

It is essential to understand that the calculation of heat transfer coefficient is based on two assumptions: The flow over the PDC is laminar and the flow velocity on the surface is 0.

As explained in the heat equation derivation in a fluid we can have conductive and convective components. The convective component is proportional to the fluid velocity . Since we want to establish the heat transfer coefficient right at the interface between the fluid and PDC surface and because we set fluid flow velocity at that boundary to 0 we know that the convective part of the fluid flow must be zero at the boundary. At the PDC surface we remain with Fourier's law of heat conduction applied right at the wall.

At the boundary we can now equate Fourier's law of heat conduction with Newton's law of cooling:

where is the heat transfer coefficient, is the ambient temperature and is the condenser temperature.

Equating gives the heat transfer coefficient :

If the fluid flow is not laminar or the fluid flow velocity is not zero at the boundary a different model for computing the heat transfer coefficient needs to be used.

Knowing the heat transfer coefficient and the surface temperature of the condenser one is able to calculate the condensation rate .

Let us stress once more that this model assumes that all water condensed is collected and no water evaporates.

The serial approach can estimate the amount of condensed water but it will be overestimated as the PDC surface temperature will be lower than if water condensation was present [Flores-Castillo, 2022].

### Water Generation without Latent Heat

The condensed water will be computed by first calculating the heat transfer coefficient using equation (23) and then integrating equation (24).

To calculate the heat transfer coefficient, first, we need to compute the temperature gradient in the normal direction to the PDC.

The temperature gradient is approximated by getting the differences on temperatures and distances between the PDC surface and a line parallel to the surface:

For computing the gradient the PDC surface is divided into 200 sections, and the temperature is taken at the plate directly from the solution. The second temperature is taken from a distance over a line parallel to the PDC surface [Flores-Castillo, 2022].

To create the parallel line and make the evaluations on it, we create a straight line equation and a new intersection to the -axis which is dependent of the position x.

Then, we define a function that is going to do the temperature evaluations at the parallel line normal to the surface and a second function that is going to get the difference in temperatures .

The final step is to generate an interpolating function of as a function of time and PDC position.

The same can be done for and .

One can use the function for a specific time or for a period of time of the simulation and then use the interpolating functions to visualize the behavior of the heat transfer coefficient and other variables.

First, we show how the heat transfer coefficient and temperature gradient behave at .

From the plots, it can be seen that the functions are coarse; this can be improved by making the mesh finer or by filtering the data. The latter one will be applied by using a GaussianFilter.

To obtain the full-time behavior of , we need to generate different interpolating functions representing different times. In this case, the calculation of will be every , and from these interpolating functions, we will create a single one dependent on time and space.

The time domain of the interpolating functions will be from to . The time is omitted to avoid a division by zero.

This plot shows that left side of the surface, where the air touches first, has a higher heat transfer coefficient. This plot is helpful because it tell us how some portions of the PDC surface condense more water based on the localized heat transfer coefficient [Flores-Castillo, 2022].

To know if the computed heat transfer coefficient is valid, we can compare it with a theoretical approximation, which follows the following equation:

where is a corrective factor, is an empirical value and is the typical condenser length-scale.

The velocity will be the wind speed above and near the plate [Beysens, 2018]. So, we will be using the wind speed at 1 m from ground.

One can see that the heat transfer coefficient ranges from 0 to 5 .

Water condensation can be estimated by equation (25). Both pressures needed to solve the equation are calculated by the following formula:

where represents the vapor pressure and can be or depending on , which represents the ambient temperature or the condenser temperature T_{c} [°C].

For calculating the last unknown, the latent heat of evaporation , we use the following equation which only applies for the temperature range of 248 to 313 :

Equation (26) can be simplified if the temperature range is from 273 to 290 :

Equation (26) will be used for the calculations. The input temperature must be given in and the output is given in .

With these functions, we can proceed to calculate according to equation (27).

In addition, here we will include the necessary temperature condition for dew formation, which is the next one: if the surface temperature is higher or equal to the dew temperature the mass rate will be zero.

Note that the surface area is 1 and that the latent of evaporation is multiplied by 1000 to get the SI unit .

Now, to better understand how the different variables behave, one can visualize them at different times.

The surface temperature plot shows us that at water accumulation will happen at of the condenser surface, because is where is below the dew point .

This plot shows that indeed water starts to accumulate where .

The most common rainfall measurement is the total rainfall depth during a given period and is usually expressed in millimeters , the height reached by the water within a container. To know how much volume of water was condensed then, we must know the catchment area. If the catchment area is 1 then 1 is equivalent to 1 [Gires, 2018]. Since 1 of water equals to 1 , so our condensation rate is measured in or .

we apply approximations, explained in the following.

First, the equation is numerically integrated along the length of the PDC at time using NIntegrate. Then this result is multiplied by the time step used. For each time interval, a list of values is generated, so at the end, we can use Accumulate to get the total sum of condensed water.

At the first time steps, NIntegrate will fail to converge because at these time values has a value of zero over the entire surface. NIntegrate will give an error message and it will return a value of zero for these time steps.

In the following computation, we will Quiet the message error generated by NIntegrate.

The plot shows that the calculated water condensation with no latent heat of condensation present is approximately 0.226 , equivalent to 226 .

Although the condensation term was not used in the simulation, it can be calculated according to its equation:

where is the latent heat of condensation and is the mass rate of condensate water. The latent heat of condensation is the heat released by water when it transitions from gas to liquid phase.

We have already calculated , all that is left is the latent heat of condensation , which is computed using the following equation:

where the latent heat of condensation is the negative of the latent heat of evaporation .

The plot shows that heat is effectively released at the condenser surface due to condensation of water on the condenser surface with a maximum value of approximately at . Different amounts of heat will be released at different times.

According to [Lekouch et al., 2012], the observed water condensation for Mirleft, Morocco, was 0.22 , equivalent to 220 . In contrast, the calculated water condensation with no latent heat of condensation present is approximately 0.226 , equivalent to 226 . Apart from comparing the water volume, one can compare the PDC surface temperature with experimental results from [Lekouch et al., 2012].

The surface temperature varies with respect to at the PDC surface. So, three different surface temperatures are shown, each representing the temperature at a specific location: at the surface's left, right, and center. Also, an average surface temperature is included in the plot.

From this plot the effect of the solar irradiance can be observed at the two extremes of the plot when the surface temperatures goes down and up steeply. These high surface temperatures are due to the solar irradiance, so it can be seen that only at night water can condensate on the PDC surface.

Another peculiarity that shows this graph is that the temperature on the left side of the surface is always higher than on the right and this is due to the fact that the heat transfer coefficient has a much higher value on the left side of the surface. Also the cloud coverage effect can be observed around which is when the cloud coverage effect increases and therefore also the surface temperature .

The surface temperature values of the region after are the ones that are closer to the observed experimental values [Flores-Castillo, 2022].

## Appendix

WeatherData is a built-in function that gives access to world wide weather data. When given a time interval the function will return a time series of the weather parameter. In locations where there is no weather station one can use data from a nearby station as a reasonable estimation of the ambient conditions for any model.

We define a function that will allow to access data from different weather stations. WeatherData is returned as a times series, and it is necessary to convert these data points into a continuous function, an interpolating function. For missing data, a MissingDataMethod needs to be specified.

For this case, MissingDataMethod will use an interpolation of order 0 to fill missing data values. On the other hand, the interpolating functions will be generated by using Interpolation with an interpolation order of 2.

Here is an example of how the function WeatherParameter works. The location used is the nearest weather station to Mirleft, Morocco. The location is given as an ICAO code, which is a standardized weather station identifier.

WeatherData sometimes does not have the data for the period you define. For example, here we could only get 21 hours from the original 23.75 hours.

Sometimes the parameters won't cover the same period. To fix this, we define the minimum length of the data, which will be used for plotting the variables and do other computations.

## Nomenclature

## References

1. Beysens, D. (2018). Dew Water. Aalborg: River Publishers.

2. Flores-Castillo, Pedro. (2022). Computational Analysis of Water Condensation for Passive Dew Condensers to Optimize Water Condensation. Master's thesis, Harvard University Division of Continuing Education.

3. Gires, A. (2018) How Do We Measure Rainfall?. Front. Young Minds. 6:38. doi: 10.3389/frym.2018.00038

4. Lekouch, I., Lekouch, K., Muselli, M., Mongruel, A., Kabbachi,B., & Beysens, D. (2012). Rooftop dew, fog and rain collection in southwest Morocco and predictive dew modeling using neural networks. Journal of Hydrology (Amsterdam), 448-449, 60-72. doi: 10.1016j.jhydrol.2012.04.004

5. Wilkerson, S., Wattenberg, F. A., & Park, R. (2013). Solar Energy Incident on Earth's Surface. Wolfram Demonstrations Project. Retrieved from https://demonstrations.wolfram.com/SolarEnergyIncidentOnEarthsSurface/