LECTURE – 1
THE CONTENTS OF THIS LECTURE ARE AS FOLLOWS:
1.0 INTRODUCTION
2.0 VARIOUS TERMINOLOGY USED IN PSYCHROMETRY
2.1 Vapour Pressure
2.2 Saturation Vapour Pressure (SVP)
2.3 Gas Constant of Unsaturated Air
2.4 Specific Heat of Moist Air
2.5 Latent Heat
2.6 Sensible Heat
2.7 Sigma Heat and Total Heat
2.8 Density of Humid Air
REFERENCES
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1.0 INTRODUCTION
You might have experienced that in rainy season, clothes take longer time to dry
compared to summer days or even moderate winter season. Do you find any
reason behind such phenomenon? If we ask this question to a lay man, he may say
that, it is because of higher temperature during summer. But, what about the
moderate winter days when temperature is lower compared to rainy season? Let
me tell you that this phenomenon is governed by psychrometric properties of air. In
psychrometry, we learn about the thermodynamic behavior of air-water vapour
mixture. In rainy season, air has relatively more water vapour content (air has
more relative humidity) compared to those of summer days or moderate winter.
Thus, it is the water-vapour content of air (relative humidity) in combination with
temperature which decides the time required for drying of clothes. Again, one may
ask that, how can we say that air has more water vapour (relative humidity) during
rainy season? Obviously we have many apparatus using which we can say about
the water-vapor content of the air. But, to prove this, let us take example from day
to day life. There are many substances around us which has the property of
absorbing water content from atmosphere. One such substance is “table salt”used
in our home for cooking purpose. You might have noticed that table salt become
wet during rainy season if kept open to atmosphere. While it does not happen in
summer days or moderate winter days because air has less water-vapour content.
It is very interesting to know that chemically water-vapour is not a constituent of
air. We have already learnt about the composition of dry air earlier. But,
atmospheric air is not dry and has water-vapour mixed homogeneously with it. This
water-vapour component of atmospheric air keeps on varying. It may vary from 0
to 4 % in composition. There are various terminologies that we need to discuss to
understand the concepts involved in psychrometry. Let us discuss them.
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2.0 VARIOUS TERMINOLOGIES USED IN PSYCHROMETRY
2.1 Vapour Pressure
Dalton’s law states that the total pressure exerted by a gaseous mixture is equal to
the sum of the partial pressures exerted by the constituent gases of the mixture.
Atmospheric air can be taken as mixture of dry air and water vapour. Thus, the
total atmospheric pressure at a place can be written as:
𝑃𝑏 = 𝑃𝑑 + 𝑒…..(1)
Where,
Pb = total barometric pressure (kPa)
Pd= partial pressure due to dry air (kPa)
e = Vapour pressure or Partial pressure due to water vapour (kPa)
Hence, vapour pressure is defined as the partial pressure of water vapour present
in a certain volume of air. It can be calculated using the following equation.
𝑒 = 𝑒𝑠𝑤 − 0.000644 𝑃𝑏 (𝑡𝑑 − 𝑡𝑤 ) 𝑘𝑃𝑎 …(2)
Where,
e = vapour pressure (kPa)
esw = saturation vapour pressure at wet bulb temperature (kPa)
Pb = barometric pressure (kPa)
td = dry bulb temperature (℃)
tw= wet bulb temperature (℃)
It is also called actual vapour pressure(AVP).
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2.2 Saturation Vapour Pressure (SVP)
At a specific or particular temperature, the air can hold a certain maximum amount
of water vapour. Further, addition of water vapour in the air will lead to
condensation of water vapour. In this situation, air is said to be fully saturated at
the specific temperature. We are talking of specific temperature, because air
holding capacity of water vapour is temperature dependent. We know that if more
molecules of a gas are present, it will exerts higher pressure in the surrounding.
Hence, the partial pressure due to water vapour will be maximum at a temperature
when air will be in saturation. The SVP at a temperature can be calculated using the
following formula.
𝑒𝑠 = 610.6 𝑒𝑥𝑝 (
17.27∗𝑡
237.3+𝑡
)Pa
……(3)
Where,
es =saturation vapour pressure (Pa)
t = temperature (°C)
At this point let me tell you that SVP is independent of pressure but SVP increases
with increase in temperature. This is because water-vapour holding capacity of air
increases with increase in temperature.
2.3 Gas Constant of Unsaturated Air
From ideal gas equation we have
PV = mRT
Where,
P = pressure (Pa)
V= volume (m3)
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m = mass of the gas (kg)
R= gas constant(J/kgK)
T= temperature (K)
Suppose, we have a mixture of dry air and water-vapour. Let the amount of water
vapour per kg of dry air be X kg. In other words, let the specific humidity of the
moist air be ‘X’ kg/kg of dry air. Let pressure due to dry air be Pd(Pa) and due to
vapour be e (Pa). Also, let total pressure due to the moist air will be P (Pa). Using
Dalton’s law we can write P as the sum of the individual pressures.
P = Pd + e
We can write individual pressure due to dry air and water vapour in the form of
ideal gas equation.
𝑃𝑑 𝑉 = 𝑚𝑅𝑑 𝑇
where,
Rd = gas constant of dry air(J/(kgK))
and
𝑒𝑉 = 𝑚𝑋𝑅𝑣 𝑇
where,
Rv = gas constant of water vapour (J/(kg)K)
It should be kept in mind that, masses of the individual gases are different but,
they occupy the same volume. Total mass of moist air is m+mX (kg). Therefore,
the pressure due to moist air can be written as
𝑃𝑑 𝑉 + 𝑒𝑉 = 𝑚𝑅𝑑 𝑇 + 𝑚𝑋𝑅𝑣 𝑇
=
𝑚(1 + 𝑋)𝑅𝑚 𝑇
Where,
Rm = gas constant of the moist air
Hence, we can write the gas constant of moist air as
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𝑅𝑑 +𝑋𝑅𝑣
𝑅𝑚 =
𝐽/(𝑘𝑔𝐾)
1+𝑋
………..(4)
Equation (4) indicates that for moist air, gas constant is the sum of gas constants
of dry air and water vapour in proportion to their relative masses in the moist air.
This is true for any homogeneous mixture of gases.
2.4 Specific Heat of Moist Air
Like gas constant of moist air, we can apply the same formula (Eqn. 4), for
evaluating the specific heat of moist air. In other words, we can write the specific
heat of moist air as the sum of the specific heat of dry air and specific heat of water
vapour in proportion to their relative masses in the moist air.
We know that, for gases two types of specific heat exist- specific heat at constant
pressure and specific heat at constant volume. The expressions for two types of
specific heats of moist air are given below.
a. Specific heat at constant pressure
Cpm =
Cpd +XCpv
1+X
…….(5)
where,
(J/(kg K)
Cpm = Specific heat of moist air at constant pressure
Cpd = constant heat of dry air at constant pressure (J/(kg K)
Cpv = constant heat of water vapour at constant pressure (J/(kg K)
X = specific humidity of the moist air (kg/kg dry air)
b. Specific heat at constant volume
Cvm =
Cvd +XCvv
1+X
……(6)
where,
Cvm = specific heat of moist air at constant volume
(J/(kg K))
Cvd = specific heat of dry air at constant volume (J/(kg K))
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Cvv = specific heat of water vapour at constant volume (J/(kg K))
X = specific humidity of the moist air (kg/kg dry air)
2.5 Latent Heat
You might have experienced that steam is more dangerous than boiling water,
though both are at 100°C. Can you find any reason behind this phenomenon? Isn’t
it surprising that they are at same temperature but their effect is different.
Temperature is associated with degree of hotness and not the heat content. Steam
at 100°C has more heat content than boiling water at 100°C. This difference in the
heat content is due to ‘latent heat’. Latent heat is defined as the heat energy that
has to be supplied or removed in order to change the state of the substance
without having any effect on its temperature. Similarly, 2260 kJ of heat per kg of
water has to be added as latent heat to boiling water for its conversion to water
vapour. Latent heat is usually expressed in kJ/kg.
Depending upon the change in state which a substance undergoes, different names
have been given to this latent heat. They are as given in Table 1.
Table 1 Different types of latent heat
S. No.
1
2
3
Change of state
Solid to liquid
Liquid to solid
Liquid to vapour
Heat added/removed
Added
Removed
Added
4
Vapour to liquid
Removed
Type of latent heat
Latent heat of fusion
Latent heat of solidification
Latent heat of
evaporation/vaporization
Latent heat of condensation
Note:
In general while solving problem, Latent heat of evaporation or vaporization is
taken numerically equal to Latent heat of condensation. Also, latent heat of fusion
is taken numerically equal to latent heat of solidification. Note that they are taken
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equal in general, but they are not the same. The reason is that, they change to
some extent by temperature or pressure.
They are not constant for a particular
substance also. For example, the latent heat of condensation of water vapour is
2260 kJ/kg at 100°C, but it is equal to 2460 kJ/kg at normal air temperature. You
might be surprised by this example, especially when we talk of latent heat of
condensation at normal temperature.
Evaporation of water takes place at all
temperatures. Similarly, condensation may also take place at all temperatures.
The higher is the initial temperature of the liquid, the lesser is latent heat required
to evaporate it. For water, latent heat of evaporation is linearly related to
temperature by the equation:
𝐿 = (2502.5 − 2.386𝑡) × 1000
𝐽/𝑘𝑔
….(7)
Where,
𝐿 = latent heat of vaporization / evaporation (J/kg)
t = temperature (°C)
2.6 Sensible Heat
The word ‘sensible’ itself suggest that, it is the heat which can be sensed. As
opposed to latent heat, it is the heat responsible for changing the temperature of a
substance. Change in temperature can be sensed/perceived and hence it got the
name as sensible heat.
2.7 Sigma Heat and Total Heat
Let us take an example where we have a mixture of dry air and water vapour. Let
its specific humidity be X kg/kg dry air. Let the mass of dry air in the mixture be m
kg. Then, mass of water vapour is mX kg. Assume that the temperature of moist air
is 𝑇𝑑 ℃. Let us assume that at 0°C, the water vapour content of the mixture is in
liquid phase. Our interest is to calculate heat to be added to dry air and water
(liquid) so as to heat them up to 𝑇𝑑 ℃. It is easy to calculate heat added separately
to dry air and water.
For air, we can heat it directly to 𝑇𝑑 ℃.
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Therefore
𝒒𝒂𝒊𝒓 = 𝒎 × 𝑪𝒑𝒅 × 𝑻𝒅 𝑱𝒐𝒖𝒍𝒆
Cpd in the above equation is the specific heat of dry air (J/kg˚C)
For water, it involves change of state. One of the way is to heat water is shown in
Fig. 1
Heat the water from 0℃ to wet bulb temperature 𝑇𝑤 <𝑇𝑑 .
𝑞𝑎 = 𝑚𝑋 × 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 ℎ𝑒𝑎𝑡 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟(𝐶𝑤 ) × 𝑇𝑤 𝐽𝑜𝑢𝑙𝑒
Heat the water at wet bulb temperature 𝑇𝑤 ℃ to evaporate it.
𝑞𝑏 = 𝑚𝑋 × 𝐿𝑤
𝑗𝑜𝑢𝑙𝑒𝑠
Superheat the water vapour to 𝑇𝑑 ℃ 𝑓𝑟𝑜𝑚 𝑇𝑤 ℃.
𝑞𝑐 = 𝑚𝑋 × 𝐶𝑝𝑣 × (𝑇𝑑 − 𝑇𝑤 )
𝑗𝑜𝑢𝑙𝑒𝑠
Fig. 1 Different steps involved for conversion of water to water
vapour
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Thus , the total heat required to heat the mixture upto 𝑇𝑑 ℃ taking 0℃ as the datum
line is given by
(𝑞𝑎𝑖𝑟 + 𝑞𝑎 + 𝑞𝑏 + 𝑞𝑐 ) 𝑗𝑜𝑢𝑙𝑒𝑠
In terms of enthalpy ’H’, the above equation can be rewritten as
𝐻=
𝑡𝑜𝑡𝑎𝑙 ℎ𝑒𝑎𝑡
𝑗𝑜𝑢𝑙𝑒
= 𝐶𝑝𝑑 𝑇𝑑 + 𝑋𝐶𝑤 𝑇𝑤 + 𝑋 𝐿𝑤 + 𝑋𝐶𝑝𝑣 (𝑇𝑑 − 𝑇𝑤 )
𝑚𝑎𝑠𝑠 𝑜𝑓 𝑑𝑟𝑦 𝑎𝑖𝑟
𝑘𝑔 𝑑𝑟𝑦 𝑎𝑖𝑟
This is called the total heat of moist air taking 0℃ as the datum.
Sigma heat, S is given by total heat of moist air less the heat of the liquid.
Numerically it is given by
𝑆 = 𝐻 − 𝑋𝐶𝑤 𝑇𝑤
𝑗𝑜𝑢𝑙𝑒
𝑘𝑔 𝑑𝑟𝑦 𝑎𝑖𝑟
For air at saturation, sigma heat is given by
𝑆 = 𝐿𝑤 𝑋𝑠 + 𝐶𝑝𝑑 𝑇𝑤
= 𝐿𝑤 𝑋𝑠 + 1005𝑇𝑤
𝐽
𝑘𝑔 𝑑𝑟𝑦 𝑎𝑖𝑟
2.8 Density of Humid Air
From ideal gas equation we have
PV = mRT
Therefore, m/V =P /(RT); m/V = density
Hence, for humid air if we substitute the value of gas constant in the above
equation, it results in
𝑚
𝑉
=
(1+𝑋)𝑃
(𝑅𝑑 +𝑋𝑅𝑣 )𝑇
kg moist air/m3
We know that for any gas, at constant temperature, volume and pressure
Gas constant ∝
Therefore,
𝑅𝑑
𝑅𝑣
=
1
𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑚𝑎𝑠𝑠
1
𝑑𝑒𝑛𝑠𝑖𝑡𝑦
𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑣𝑎𝑝𝑜𝑢𝑟
𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑑𝑟𝑦 𝑎𝑖𝑟
Thus, we have 𝑅𝑑 = 0.622𝑅𝑣 .
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𝑜𝑟 ∝
=
18
28.97
= 0.622
Substituting for ‘X’ and ‘Rv’, we have
(𝑃−𝑒+ 0.622 𝑒)𝑃
𝑚
(𝑃−𝑒)
𝜌 = = 𝑃×𝑅 −𝑒×𝑅 +0.622𝑒𝑅
𝑑
𝑑
𝑣
𝑣
(
)𝑇
(𝑃−𝑒)
=
=
(𝑃 − 0.378𝑒)
𝑅𝑑 𝑇
(𝑃 − 0.378𝑒)
𝑘𝑔 𝑚𝑜𝑖𝑠𝑡 𝑎𝑖𝑟/𝑚3
287.04𝑇
We can also find apparent density of moist air in terms of dry air, given by
𝜌𝑎𝑝𝑝 =
(𝑃 − 𝑒)
287.04𝑇
𝑘𝑔 𝑜𝑓 𝑑𝑟𝑦 𝑎𝑖𝑟/𝑚3
Some of the constants to be used in solving numerical problems are listed in Table
2.
Table 2 Value of different constants
S. No.
1
2
3
4
5
6
7
8
9
10
11
11.
Constants
Specific heat of air at constant pressure
Specific heat of air at constant volume
Specific heat of water vapour at constant
pressure
Specific heat of water vapour at constant
volume
Specific heat of water
Specific heat of ice
Latent heat of fusion of ice
Latent heat of condensation of water vapour
at 100℃
Latent heat of condensation of water vapour
at normal air temperature
Gas constant of dry air
Gas constant of water vapour
Atmospheric pressure, 1 atm
Values in SI
1005 J/kg℃
718 J/kg℃
1884 J/kg℃
1422 J/kg℃
4187 J/kg℃ *
2094 J/kg℃
335 kJ/kg
2260 kJ/kg
2460 kJ/kg
287.04 J/kg K
461.50 J/kg K
101. 325 kPa
REFERENCES
Banerjee S.P. (2003); “Mine Ventilation”; Lovely Prakashan, Dhanbad, India.
Hartman, H. L., Mutmansky, J. M. & Wang, Y. J. (1982); “Mine Ventilation and Air
Conditioning”; John Wiley & Sons, New York.
Le Roux, W. L. (1972); Mine Ventilation Notes for Beginners”; The Mine Ventilation
Society of South Africa.
Page 11 of 12
McPherson, M. J. (1993); Subsurface Ventilation and Environmental Engineering”;
Chapman & Hall, London.
Misra G.B.
(1986); “Mine Environment and Ventilation”; Oxford University Press,
Calcutta, India.
Vutukuri, V. S. & Lama, R. D. (1986); “Environmental Engineering in Mines”;
Cambridge University Press, Cambridge.
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