Relative Humidity, Pressure and Temperature – Humidity Academy – Part 1 Chapter 2

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After the basics of humidity, the Humidity Academy takes you to a closer look at relative humidity, pressure and temperature. If you do not know this famous Italian physicist and chemist, then you should definitely read on.

Avogadro

Contents

  1. What is Humidity?
  2. Relative Humidity, Pressure and Temperature
  3. The Capacitive Sensor
  4. The Wet Bulb/Dry Bulb Technology
  5. Summary of Other Technologies
  6. Choosing the Right Humidity Measurement Technology

Relative Humidity, Pressure and Temperature

Reviewing the laws of physics that govern water vapor in a moist gas can help you better understand the properties of what you’re measuring. Understanding these properties will help you make a more accurate measurement and do your job more effectively, whether it’s protecting a product from corrosion or maintaining a precise environment for storage or manufacturing.

Humidity and Laws of Physics

From pressure to temperature, the following ideal gas laws help us understand how humidity levels shift depending on the environment.

Name Definition Law Notes
Boyle’s Law At constant temperature, the product of the volume and pressure of a given amount of gas is a constant. P x V = constant The value of the constant depends on how much gas is in the volume.

Name Definition Law Notes
Charles’s Law At constant pressure, the volume of a given quantity of gas is proportional to absolute temperature (°K). Or at constant volume, the pressure of a given quantity of gas is proportional to absolute temperature. V= q x T Or

P= j x Tq is a proportionality constant that depends on the quantity of gas. j is a proportionality constant that depends on the particular sample of gas and its volume. To convert temperature in °C into absolute temperature in °K, add the constant 273.15.Dalton’s Law of Partial PressuresThe total pressure of a mixture of gases is equal to the sum of the pressures that each gas would exert if it were present alone.Pt = P1+ P2+ P3+…P1, P2, etc., are the partial pressures of gases 1, 2, etc.Avogadro’sEqual volumes ofExample: one literThe temperatureHypothesisgases at the sameof any ideal gasof 0°C andtemperature andat a temp. of 0°Cpressure of 101.3pressure containand a pressurekPa is the standardequal numbers ofof 101.3 kPa,temperature andmolecules.contains 2.688 xpressure condition1022 molecules.or STP.

Avogadro

Amedeo Avogadro

Name Definition Law Notes
Volume of a Mole of Gas at Standard Temperature and Pressure (STP) As one liter of gas at STP contains 2.688 x 1022 molecules (or atoms in the case of a mono atomic gas), it follows that a mole of gas (6.022 x 1023 molecules) occupies a volume of 22.4 l at STP. See definitions of mole and Avogadro’s number below.
Ideal Gas Law The product of volume and pressure of a given amount of gas is proportional to absolute temperature. P x V = n x R x T n is the number of moles of gas and R the molar gas constant. The constant R is equal to: 0.08206 atm x liter/K x mole 8.30928 Pa x m3/K x mole

Mole Fractions and Partial Pressure

The composition of one mole of a gas mixture can be expressed in terms of the mole fractions of its components. The mole fraction of a particular component is defined as the total number of moles of the component divided by the total number of moles of all the components. From this definition, it follows that the sum of all mole fractions is equal to one.

Example: Dry air near sea level

  1. Nitrogen: Mole Fraction: 0.78084
  2. Oxygen: Mole Fraction: 0.20948
  3. Carbon Dioxide: Mole Fraction: 0.0004

If Pt is the total pressure of a gas mixture and n1, n2, etc. the mole fractions of its components, it follows that:

Pt= Pt x (n1+ n2 + …) and

Pt= Pt x n1+ Pt x n2 + …

where Pt x n1, Pt x n2, etc. are the partial pressures of components 1, 2, etc.

Water vapor is one of several gases that makes up air. For example if the total pressure of a system such as air at sea level is 1,013 kPa (or 29.9 inches of mercury), and that air is made up of Nitrogen, Oxygen, water vapor and other trace gases, each of those gases contributes to the total pressure of 1,013 kPa. The portion that is water vapor is called the partial pressure of water vapor. The partial pressure of water vapor is a key metric found as a component in the formulas that define all other humidity parameters.

Effect of a Change in Pressure

Therefore, a change in the total pressure of a gas mixture, at constant composition, results in the same change in the partial pressure of each component. For instance, doubling the total pressure of a gas mixture results in doubling the partial pressure of each component. As the total pressure increases, the partial pressure of water vapor increases proportionately.

This is an important fact to understand, as you will see when we define relative humidity and dew point temperature. An increase in the pressure of a closed system will increase the relative humidity and raise the dew point temperature until saturation is achieved.

Vapor Pressure Above a Liquid

Because molecules in a liquid are closer to one another than they are in a gas, intermolecular forces are stronger than in a gas. For a liquid to vaporize, the intermolecular forces have to be overcome by the kinetic energy of the molecules.

If a liquid is placed in a closed container, the particles entering the vapor phase cannot escape. In their random motion, particles strike the liquid and are recaptured by intermolecular forces. Thus, two processes occur simultaneously: evaporation and condensation.

The rate of evaporation increases as temperature increases. This is because an increase in temperature corresponds to an increase in the kinetic energy of molecules. At the same time, the rate of condensation increases as the number of particles in the vapor
phase increases: more molecules strike the surface of the liquid. When these two processes become equal, the number of particles and, therefore, the pressure in the vapor phase, becomes stabilized.

The value of the equilibrium vapor pressure depends on the attractive forces between particles of the liquid and on the temperature of the liquid. Vapor pressure above a liquid increases with increasing temperature.

Condensation

Vapor Pressure of Water

The vapor pressure of water increases strongly with increasing temperature.

Vapor Pressure of Water

Vapor Pressure Above Ice

When water freezes, the molecules assume a structure that permits the maximum number of hydrogen-bonding interactions between molecules. Because this structure has large hexagonal holes, ice is more open and less dense than liquid water. As hydrogen bonding is stronger in ice than in liquid water, the inter-molecular attraction forces are the strongest in ice. That’s why vapor pressure above ice is less than the vapor pressure above liquid water.

Vapor Pressure Above Ice

Definitions of Humidity

Vapor Concentration (Absolute Humidity)

The vapor concentration or absolute humidity of a mixture of water vapor and dry air is defined as the ratio of the mass of water vapor Mw to the volume V occupied by the mixture.

Dv = Mw / V expressed in grams/m3 or in grains/cu ft

The value of Dv can be derived as follows from the equation PV = nRT

Mw = nw x mw where

nw = number of moles of water vapor present in the volume V

mw = molecular mass of water

Dv = Mw / V = nw x mv / V = mw x p / RT where

mw = 18.016 gram

p = partial pressure of water vapor [Pa]

R = 8.31436 Pa x m3 / K x mole

T = temperature of the gas mixture in K

Dv = p / 0.4615 x T [g / m3]

1 gr (grain) = 0.0648 g (gram)

1 cu ft = 0.0283168 m3

Dv [gr / cu ft] = 0.437 x Dv [g / m3]

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Vapor Concentration (Absolute Humidity)

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Specific Humidity

Specific humidity is the ratio of the mass Mw of water vapor to the mass (Mw + Ma) of moist air.

Q = Mw / (Mw + Ma)

Q = p mw / (p mw + (Pb – p) ma)

Q = 1000 p / (1.6078 Pb – 0.6078 p) [g / kg]

1 gr (grain) = 0.0648 g (gram)

1 lb = 0.4535923 kg

Q [gr / lb] = 7 x Q [g / kg]

Mixing Ratio

The mixing ratio r of moist air is the ratio of the mass Mw of water vapor to the mass Ma of dry air with which the water vapor is associated:

Mw = nw x mw = mw x p V / RT

Ma = na x ma = ma x pa V / RT = ma x (Pb – p) V / RT, where:

nw = number of moles of water vapor present in the volume V

na = number of moles of dry air present in the volume V

mw = 18.016 gram

ma = 28.966 gram

p = partial pressure of water vapor [Pa]

pa = partial pressure of dry air [Pa]

Pb = total or barometric pressure [Pa]

R = 8.31436 Pa x m3 / K x mole

T = temperature of the gas mixture in K

V = volume occupied by the air – water vapor mixture

r = mw p / ma (Pb – p)

r = 621.97 x p / (Pb – p) [g / kg]

1 gr (grain) = 0.0648 g (gram)

1 lb = 0.4535923 kg

r [gr / lb] = 7 x r [g / kg]

masse

r = Mw/ Ma

Volume Mixing Ratio

The volume mixing ratio is the ratio of number of moles of water vapor nw to the number of moles of dry air na with which the water vapor is associated.

This usually expressed in terms of parts per million:

PPMv = 106 x nw / na

nw = p V / RT

na = pa V / RT = ma x (Pb – p) V / RT, where:

p = partial pressure of water vapor [Pa]

pa = partial pressure of dry air [Pa]

Pb = total or barometric pressure [Pa]

R = 8.31436 Pa x m3 / K x mole

T = temperature of the gas mixture in K

V = volume occupied by the air – water vapor mixture

PPMv = 106 x p / (Pb – p)

Relative Humidity

Relative humidity is the ratio of two pressures:

%RH = 100 x p/ps where p is the actual partial pressure of the water vapor present in the ambient and ps the saturation pressure of water at the temperature of the ambient.

Relative humidity sensors are usually calibrated at normal room temperature (well above freezing). Consequently, it is generally accepted that this type of sensor indicates relative humidity with respect to water at all temperatures (including below freezing). As already noted, ice produces a lower vapor pressure than liquid water. Therefore, when ice is present, saturation occurs at a relative humidity of less than 100%. For instance, a humidity reading of 75% RH at a temperature of -30°C, corresponds to saturation above ice.

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Dew Point and Frost Point Temperature

The dew point temperature of moist air at temperature T, pressure Pb and mixing ratio r is the temperature to which the air must be cooled in order to be saturated with respect to water (liquid).

The frost point temperature of moist air at temperature T, pressure Pb and mixing ratio r is the temperature to which the air must be cooled in order to be saturated with respect to ice.

Wet Bulb Temperature

The wet bulb temperature of moist air at pressure Pb, temperature T and mixing ratio r is the temperature that the air assumes when water is introduced gradually by infinitesimal amounts at the current temperature and evaporated into the air by an adiabatic process at constant pressure until saturation is reached.

Effect of Temperature and Total Pressure on Vapor Pressure

One common mistake in taking humidity measurements is failing to distinguish the effects of temperature and pressure on water vapor. When considering the effect of temperature, pressure and space on the partial pressure of vapor, make sure to differentiate between the following situations:

  • saturation (liquid or ice) vs. no saturation (vapor only)
  • closed container of fixed volume vs. open space
Saturation

The partial pressure of vapor is equal to the saturation pressure and its value depends only on temperature. There is no difference between the situation in an open environment and that in a closed container.

No Saturation

Water vapor behaves almost like an ideal gas, and the following equation applies regarding the partial pressure of vapor:

p x V = n x R x T

In an open space, the volume V occupied by vapor is free to expand. Therefore, the partial pressure p is not affected by temperature. The partial pressure p can vary only if n varies (vapor is being added or removed) or if the total pressure varies (Dalton’s law of partial pressures). For instance, total pressure drops with increasing altitude, which results in a decrease of the partial pressure of vapor.

In a closed container of fixed volume, vapor occupies the entire volume of the container and this volume is constant. The partial pressure of water vapor(p) can vary only if there is a change in absolute temperature (degrees K) or a change in the amount of water vapor p. The partial pressure p does not vary with a change in total pressure unless the total pressure change is due to a change in the partial pressure of water vapor p.

Effect of Temperature and Pressure on %RH

Saturation vapor pressure depends only on temperature. There is no effect of total pressure, and there is no difference between the situation in an open space and that in a closed container.

In an open space, at constant moisture level and temperature, %RH is directly proportional to the total pressure. However, the value of %RH is limited to 100% as p cannot be greater than ps.

In a closed container of fixed volume, %RH decreases as temperature increases, but not quite as strongly as in open space.

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Examples

(A) Office building

For practical purposes, an office building can be considered an open environment.

A localized increase in temperature created by a heater or an office machine, for instance, doesn’t modify the value of the partial pressure of water vapor, so the local vapor pressure is the same throughout the building. However, the saturation vapor pressure is locally increased. Consequently, relative humidity in the immediate vicinity of the heat source is lowered.

If we assume that elsewhere in the building the temperature is 25°C and relative humidity is 50%, a localized increase of temperature to 30°C lowers relative humidity as follows:

ps at 25°C = 3.17 kPa

ps at 30°C = 4.24 kPa

p = 0.5 x 3.17 kPa = 1.585 kPa, corresponding to 50 %RH

Localized %RH = 100 x 1.585/4.24 = 37.4%

(B) Dew on a chilled mirror

If the temperature of a mirror is lowered to precisely the value that makes dew appear on the surface, the value of the mirror temperature is called dew point. Using the previous example, the dew point corresponding to a condition of 50 %RH and 25°C can be found as follows:

ps at 25°C = 3.17 kPa

p = 0.5 x 3.17 kPa = 1.585 kPa, corresponding to 50 %RH

If there is equilibrium between the dew on the mirror and the environment, it follows that ps at the temperature of the chilled mirror must be equal to the vapor pressure p. Based on a simple interpolation of the values of the saturation vapor tables, we find that a value of ps of 1.585 kPa corresponds to a temperature of 13.8°C. This temperature is the dew point.

The example above shows that converting relative humidity into dew point and vice versa requires the use of a thermometer and saturation vapor tables.

(C) Compression in a closed chamber

If the total pressure inside a closed chamber is increased from one to one and a half atmospheres and temperature is kept constant, the partial pressure of water vapor is increased 1.5 times. Because temperature is the same, so is the saturation pressure ps. If we assume that we had a condition of 50% RH and 25°C before the compression, the condition afterwards is 75 %RH and 25°C.

(D) Injection of a dry gas in a closed chamber

If dry nitrogen is injected in a closed chamber where there is already air at a condition of 50 %RH and temperature is kept constant, total pressure in the chamber increases. However, the partial water vapor pressure p remains constant because the mole fraction of water vapor in the chamber decreases by an amount that exactly balances the increase in total pressure (see Dalton’s law). Because temperature is maintained constant, the saturation vapor pressure ps is also unchanged. Therefore, relative humidity stays at 50%, despite the fact that a dry gas was injected in the chamber.


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