Is Fick's Law related to the mass transfer coefficient in water evaporation?

[InTuoVultu]

I guess this would go under chemistry. At least I leared what thermodynamics I know from my chem class.

Here's the situation: My dad wants to know if the decreacing water level in the jacuzzi is due to a leak or evaporation. The pool guy told us to fill up a bucket with water and compare the distance the waterline fell in the pool and the bucket.

Now, the pool (jacuzzi really) is about 2-3 times deeper than the bucket and it's one of thoes burried in the ground types. Since both the bottom of the pool and bucket are both white I'll assume they absorb the same amount of solar energy. The heater and jets that are normally on are turned off for this experiment.

My bet is that the bucket test will work because both the bucket and the pool both absorb the same energy per unit surface area. My dad is thinking the bucket will evaporate faster because it is at a hotter temperature during the day. I dunno, maybe I'm missing something.

Thanks,
-Matt

 

[mrjeffy321]

If we assume that all the energy the water in the bucket and the water in the Jacuzzi absorb is from sun light, then I would think that the "bucket experiment" is inherently flawed.

Both the Jacuzzi and bucket will receive the same amount of solar energy per square meter. A larger surface area means that much more solar energy will hit the water.
But the surface area does not just determine the amount of light absorbed but also the rate of evaporation. Since only water molecules at the surface of the water will evaporate, a larger surface area means that it is possible for the water to evaporate much faster.
I bet the Jacuzzi has a much larger surface area than the bucket, does it not?

Also remember the volumes of water the Jacuzzi and bucket contain. The bucket might only hold 1 gallon of water but the Jacuzzi might hold hundreds of gallons of water. The Jacuzzi’s volume is likely to be a couple orders of magnitude greater than the buckets and will thus require much more energy to cause an equal change in temperature.
The rate of evaporation is also dependent on the temperature of the liquid and the Jacuzzi will take longer (more energy) to heat up.
Also, the Jacuzzi is insulated within the ground and will receive the most of its energy from the sun on the surface of the water, but the bucket above ground able to receive a lot more of the ambient energy in the air and the ground underneath it.

If you conduct this experiment and compare the change in height of the water in the bucket to the change in height of water in the Jacuzzi a change in (for example) 1 inch in the bucket does not equal a drop of 1 inch in the Jacuzzi. 1" of water in a bucket might only be a Liter or so, but 1" in a Jacuzzi might be several gallons.


Lets say, for argument sake, that your Jacuzzi is 2 meters in diameter and a constant 1 meter deep (a perfectly cylindrical Jacuzzi). That would amount to a volume of 37.7 m^3 (37700 Liters) with a surface area of 12.6 m^2.
Lets say that you have a bucket which the water level measures 20 cm across (surface area of .12566 m^2) and 15 cm deep (18.85 Liters, just under 5 gallons).
In this case, the Jacuzzi has a surface area which is about 100 times greater, but a volume which is 2000 times greater than the bucket of water. The bucket is going to get hot a lot faster than the Jacuzzi.

A 1" drop in the water level in the bucket means a 3.2 Liter loss in water. A 1" drop in the water level in the Jacuzzi means a 320 Liter loss in water.

 

[TuviaDaCat]

i think that if assuming that both tempertures on both containers is equal, then the water would be lowered in a propotional way, proportional to the surface size of each container.

do make sure that the bucket will not make a shadow on it water as the sun moves...

 

[lalbatros]

Hello,

The evaporation of water from the bucket or from the jacuzzi can be analysed with the same laws of physics.

The water evaporation can be predicted from a "[URL law.
This can be simply written as follows:

$$J_{vapor} = h S \left( p_{sat}(T_{water}) - p_{vapor} \right)$$​

where

$$J_{vapor}$$ is the flux of vapor from the surface say in mole/s
$$h$$ is the mass transfer coefficient for water evaporation
$$S$$ is the surface of water exposed
$$p_{sat}(T_{water})$$ is the saturation of vapor at the temperature of the water surface
$$p_{vapor}^{ }$$ is the partial pressure of vapor in the air
$$T_{water}$$ is the temperature of the water surface
​

This law describes a lot of physics:

note1: the saturation pressure increases very fast with temprature

note2: the exchange coefficient depends on the air flow at the surface of water, better flow means better exchange

note3: the water surface temperature may differ from the bulk water temperature

note4: evaporation cools the bath (surface) because in requires energy

note5: evaporation from the bath increases the partial pressure in the surrounding air, but this stops when this air is saturated somewhere like on the wall of the room where ther will be condensation

note6: in very cold air (jacuzzi in open-air in winter!) the evaporated vapor will quikly condensate outside (cloud), therefore the partial pressure of vapor will always remain very low and evaporation will be fast all the time

If the water surface is so cool that the saturation pressure is lower than the vapor pressure, then no evaporation occurs, but instead condensation occurs. The flux of vapor in the Fick law could be positive (evaporation) or negative (condensation) depending on the conditions.

If the bath is mixed by water jets, then the water surface will be warmer than if it is not stirred (like the bucket), this increases evaporation.

If the bath is stirred with air bubbles, a lot of additional water vapor can be produced. Assume each air bubble enters the bad at a low temperature (25°C). This air warms up. The surface between water and the bubble will evaporate water inside the bubble until the air inside the bubble is saturated (higher temperature means higher saturation pressure). The bubble will leave the bath and release the vapor in the athmosphere. Bubbles jets can increase dramatically the evaporation: just try to evaluate the total surface of the bubbles and the surface of the bath and compare them. In addition, the motion of the bubbles and the continuous apparition of fresh bubbles make this mecanism probably more efficient than the free surface evaopration (usually you don't use a fan to refresh the air when you are in a jacuzzi).

...

We could continue discussing that a lot.
The best could be to make now a mathematical model!
​

Michel

 

[LURCH]

If you're going to use a bucket to compare evaporation rates, you should probably dig a small hole to set the bucket in up to its rim, and push the dirt in from the sides, like the dirt in contact with the sides of the jacuzzi. Heat transfer between the water and the ground sarounding it probably will be non-negligable.

 

[lalbatros]

InTuoVultu,

Please note that the solar energy radiating on the pool or the bucket has usually a totally negligible effect if the water is heated artificially, say to 35°C. By usually I mean with the usual white or light blue colors for example that reflect much of the solar power. Thinks might be different with a suitable design: the sun could heat the water to ebullition or even further theoretically. But this is not -I think- the subject to be discussed.

If solar energy is the only source of energy, then the comparison between bucket and jacuzzi must ensure that the solar irradiation conditions are comparable.

To make sure that this comparison is correct, you could simply check that the water temperatures are the same. If I understood correctly what is discussed here is the evaporation for a given water temperature. Right?

Michel

 

[mpemba]

Hello.

I am a french student and I am working on a subject which deals with evaporation rate.
I would like to know how the Fick law can lead to the formula given by "lalbatros", and if "the mass transfer coefficient for water evaporation "(h) is linked to the diffusion coefiicient (D) which appears in the Fick law.

(I need this to be able to calculate the mass of water evaported during a given period.)

Edouard