mech 430: thermal system designs - queen's...
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MECH 430: Thermal System Designs
Psychrometrics Review
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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)
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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
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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)
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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)
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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)
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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)
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We can use these equations to convert back and forth
between relative humidity and the humidity ratio.
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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.
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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)
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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)
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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)
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Enthalpy is measured in a per unit mass of
dry air basis. This gives units of kJ/kga.
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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.
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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)
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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)
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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.
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The condensation process can be roughly shown on a
psychrometric chart
There are two distinct components, the sensible
cooling, and the latent cooling.
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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
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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.
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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
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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)
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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
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See Properties of
Saturated Water Table from
Thermo Text or EES to
estimate the properties of
water vapour.