Dew Point Calculation: Find the Temperature

[RMoen]

Hi

I am currently writing my master's thesis on preparation of a ceramic powder by spray pyrolysis, and have a question I hope you could help me with. I have a sample of water vapor mixed with air, and I know the specific amount of water vapor (grams of vapor per unit volume of the gas). How do I now calculate the temperature where condensation of the vapor starts, i.e. the dew point?

 

[Chestermiller]

Welcome to physics forums.

Use the ideal gas law to calculate the partial pressure of the water vapor in the gas phase. Then, find the temperature at which this partial pressure is equal to the equilibrium vapor pressure of water. (The data for this is in tables and graph, including, of course, the steam tables.) That temperature is the dew point.

Chet

 

[colliflour]

I would recommend using the Antoine equation a-b/(c+T)=Log(Pi) rather than the steam tables. Where Pi is the partial pressure of water vapor at the given total pressure of, I would guess, 1 atm. T is the saturation temperature, at which condensation starts. You'd have to look up the constants a, b, and c and they might not be really 'constant' but vary somewhat with temperature or pressure. Determining Pi exactly involves using activity coefficients but you could probably approximate it with either Henry's law or Raoult's law, depending on whether your water fraction is low or high, respectively. Also if you're dealing with a fixed amount of water, your mole fraction will of course go down over the condensation time. Additionally if you're dealing with short time scales, there will be a temperature increase at the drop surface due to heat of vaporization and you'll have to conduct or convect the heat away before condensation can continue. In the case that the ceramic spray begins at a high temperature relative to the air/water vapor mix, you'd need to Nusselt and Sherwood numbers to get some approximation of heat and mass transfer rates.

 

[Chestermiller]

I would recommend using the Antoine equation a-b/(c+T)=Log(Pi) rather than the steam tables. Where Pi is the partial pressure of water vapor at the given total pressure of, I would guess, 1 atm. T is the saturation temperature, at which condensation starts. You'd have to look up the constants a, b, and c and they might not be really 'constant' but vary somewhat with temperature or pressure. Determining Pi exactly involves using activity coefficients but you could probably approximate it with either Henry's law or Raoult's law, depending on whether your water fraction is low or high, respectively. Also if you're dealing with a fixed amount of water, your mole fraction will of course go down over the condensation time. Additionally if you're dealing with short time scales, there will be a temperature increase at the drop surface due to heat of vaporization and you'll have to conduct or convect the heat away before condensation can continue. In the case that the ceramic spray begins at a high temperature relative to the air/water vapor mix, you'd need to Nusselt and Sherwood numbers to get some approximation of heat and mass transfer rates.

This is one of the worst cases of overkill I have ever seen, and can certainly be expected to confuse the OP (for no reason). Most of the things alluded to don't even come into play, if what you are trying to do is determine the dew point. Since air is virtually a non-condensible, you don't need to apply Henry's Law or Raoult's Law or any other VLE equilibrium equation in predicting the dew point. If you check the Henry's law coefficient, you will find that it is huge, and a negligible amount of air will be dissolved in any water that begins to condense. Determining the activity coefficient or fugacity coefficient for water in the liquid phase will not be necessary, since the condensed liquid will be virtually pure water.

The OP is asking about the thermodynamic equilibrium situation at which liquid water just begins to condense out of the gas phase. So talking about things happening at short time scales cannot be relevant to the OP's interest. And he wasn't asking about what happens as you continue to cool the system.

The Antoine equation does a good job of predicting the relationship between temperature and pressure under saturation conditions for pure water, particularly at water vapor partial pressures below 1 atm. If the total pressure is 1 atm., the combination of water vapor and air in the gas phase should very closely approximate ideal gas behavior, in which case the partial pressure of the water vapor will be equal to the total pressure times the mole fraction of water vapor. At the dew point, this partial pressure will be equal to the saturation vapor pressure of water at the dew point temperature. One can check the accuracy of the Antoine equation by comparing its predictions to the relationship between saturation temperature and pressure listed in the steam tables. Of course, the values in the steam tables represent the actual experimental data (rather than a parametric fit), but almost certainly, the Antoine equation will provide a very accurate fit to the data in the steam tables.

 

[Andre]

If you're allowed to 'cheat', http://www.humidity-calculator.com/index.php