MECH 430: Thermal System Designs

Psychrometrics Review

Moist Air is considered an ideal gas and thus follows the

ideal gas law, for a closed system occupying a volume V

at a mixture pressure P and Temperature T:

( )m

RTnRT MPV V

M – Molar mass (kg/kmol)

m – mass (kg)

n – number of moles = m/M

R – Universal Gas Constant (8.314 kJ/kmol K), =RM

T – Temperature (K)

V – Volume (m3)

P - Total Pressure (101325 Pa)

(eqn 12.40 in your thermodynamics book)

The total pressure is made up of the partial pressure of

each component in the system

a vP p p

a a

v v

p y P

p y P

pa - partial pressure of air

pv – partial pressure of water vapour

ya – mole fraction of air

yv – mole fraction of water vapour

(eqn 12.41b)

(eqn 12.41b)

(eqn 12.13)

/ /,

a a v va vm R M T m R M Tn RT n RT

Pa PvV V V V

The humidity ratio (ω) is an absolute method of

indicating the water content in the air

v

a

m

m

/

/

v v v v

a a a a

M p V RT M p

M p V RT M p

Using the ideal gas law

(eqn 12.42)

An important measure of humidity is the relative humidity

(RH or ϕ). It is dependant on temperature as well as water

content.

v

g

p

p (eqn 12.44)

ϕ – Relative Humidity (%)

pg – The saturated vapour partial pressure at the air temperature (Pa)

Introducing the molar mass of air and water and noting

that pa = P-pv (from eqn 12.13)

0.622 v

v

p

P p

(eqn 12.43)

ω – Humidity Ratio (kgv/kga)

The saturation pressure can be found from EES,

psychrometric charts or tables, or calculated from this

equation from the ASHRAE Handbook of Fundamentals

Psychrometric chapter.

2 3

8 9 10 11 12 13exp( / ln )gp C T C C T C T C T C T

T – The air temperature (must be in Kelvins for this equation)

Cn – Constant outlined in table below

C8 -5.8002206E+03 C11 4.1764768E-05

C9 1.3914993 C12 -1.4452093E-08

C10 -4.8640239E-02 C13 6.5459673

(eqn 6 in ASHRAE)

We can use these equations to convert back and forth

between relative humidity and the humidity ratio.

When the relative humidity is at 100%, the air is fully

saturated.

This is the dew point. When air at 100% RH is cooled, the

moisture drops out of the air as condensate.

In other words, the dew point temperature is the

saturation temperature corresponding to the partial

pressure of water in the air.

This value can be found on psychrometric charts

and tables, or by using the equations from the

ASHRAE Handbook of Fundamentals

Psychrometrics chapter shown below. The first is

just a re-arrangement of equation 12.43.

0.622g v

Pp p

pv – Partial vapour pressure of water in air (Pa)

pg – The saturated vapour partial pressure at the air temperature (Pa)

P – Total pressure

ω – Humidity Ratio (kgv/kga)

(eqn 38 from ASHRAE)

2 3 0.1984

14 15 16 17 18(ln ) (ln ) (ln ) ( )d v v v vT C C p C p C p C p

Td – Dew Point Temperature (C)

Cn – Constant shown below

C14 6.54 C17 0.09486C15 14.526 C18 0.4569C16 0.7389

This partial pressure can be used with the rearranged eqn 6

from ASHRAE shown below

(eqn 39 from ASHRAE)

The amount of moisture in the air and the

temperature of the air can then be used to

determine the amount of enthalpy (h) in the air.

As always with enthalpy, when cross referencing

different sources, ensure that the reference

temperatures for and absolute h values are the

same (not needed for enthalpy changes (i.e. h)

Enthalpy is measured in a per unit mass of

dry air basis. This gives units of kJ/kga.

a v a a v vH H H m h m h

va v a v

a a

mHh h h h

m m

(eqn 12.45)

(eqn 12.46)

H – enthalpy (kJ)

ha – enthalpy of dry air (kJ/kga)

hv – enthalpy of water vapour (kJ/kgw) approx. = hg(T)

Values of ha and hv can be read from psychrometric

tables, or they can be roughly calculated using the

equations from ASHRAE Fundamentals, EES or

Steam tables for hg.

1.006ah T

2501 1.86vh T

T - Air Temperature (oC)

1.006 –Average cp of air

2501 –Average heat of vapourization of water

1.86 –Average cp of water vapour

(eqn 31 from ASHRAE)

(eqn 30 from ASHRAE)

( , ) ( ) [ / ]

( , ) 1.006 (2501 1.86 ) [ / ]

( ) [ / ]

1.006 (2501 1.86 ) [ / ]

a fg v a

a

a fg v a

a

h T Cp T h Cp T kJ kg

h T T T kJ kg

h Cp T h Cp T kJ kg

or

h T T kJ kg

(eqn 32 from ASHRAE)

1.006ah T

2501 1.86vh T

T - Air Temperature (oC)

1.006 – Average cp of air

2501 – Average heat of vapourization of water

1.86 – Average cp of water vapour

(eqn 31 from ASHRAE)

(eqn 30 from ASHRAE)

1.006 (2501 1.86 )h T T (eqn 32 from ASHRAE)

An energy balance can be applied to get the enthalpy

change across two points.

1 1 1 2 2 2( ) ( )a a v v a a v vh m h m h m h m h

This assumes that no condensate is created, so the

inlet and outlet humidity ratios would be the same.

If condensation occurs (and inlet and outlet humidity

ratios are no longer the same) additional steps must be

accounted for.

The condensation process can be roughly shown on a

psychrometric chart

There are two distinct components, the sensible

cooling, and the latent cooling.

1 2a am m

1 2 rate of condesation, kg/sw v vm m m

Mass Balance

Since, we can equatev

a

m

m

1 2w

a

m

m

1 1v am m

2 2v am m

Energy Balance

1 1 1 2 2 20 ( ) ( ) ( )r i e a a v v w w a a v vm h h m h m h m h m h m h

Using equations from mass balance gives

1 2 1 1 2 2 1 2 20 [( ) ( ) ] ( )a a a g g f r i em h h h h h m h h

( )r i em h hThe term, is the enthalpy change of the

coolant. This is the amount of energy required to be

removed by the condenser.

This can be simplified to create a rough estimate. The

sensible cooling is the drop in air temperature to the dew

point. It can be simplified to the equation below. The total

enthalpy (h) can be found using an effectiveness model

for heat exchangers.

1 intsensible dewpoh h h

hsensible – The change of enthalpy required to bring to dew point (kJ/kga)

Tdew – The dew point temperature(kJ/kga)

cp,air – Specific heat capacity of air

h1- inlet air enthalpy

hdewpoint- Enthalpy of air at dew point temperature

sensible latenth h h

The calculation for latent heat is shown here. This assumes

the sensible heat change of air is negligible during

condensation.

1 2( )( )latent fgh h

hlatent – The change of enthalpy during condensation (kJ/kga)

ω1/2 – Humidity ratio at inlet/outlet (kgw/kga)

hfg – Heat of vapourization (kJ/kgw)

If the total enthalpy transferred (h) is less then the

energy required to reach the dew point, no

condensation will occur (i.e., humidity ratio remains

constant) and you can just find the outlet temperature

using

, 1 2( )p airh c T T

See Properties of

Saturated Water Table from

Thermo Text or EES to

estimate the properties of

water vapour.