Prof. Fred Remer University of North Dakota Phase Changes and Latent Heat Where’s the heat? Solid...

Prof. Fred RemerUniversity of North Dakota

Phase ChangesPhase Changesand Latent Heatand Latent Heat

Where’s the heat?Where’s the heat?

SolidSolidLiquidLiquid

GasGas


ReadingReading Hess

– Phase Diagram pp 49 – 51

– Dew Point, Wet Bulb Temperature and Wet Bulb Potential Temperature

pp 60 – 63

Bohren & Albrecht– pp 218-223

Wallace & Hobbs– p. 84


ObjectivesObjectives

Be able to describe the changes in temperature, equilibrium pressure, volume and heat during various phase changes



Be able to recall from memory the definition of critical point

Be able to recall from memory the definition of triple point



Be able recall from memory the values of temperature and pressure for the triple point of water

Be able to recall from memory the values of temperature and pressure at the critical point of water



Be able to show isobaric, isochoric and isothermal changes on phase diagrams

Be able to determine changes of boiling and melting temperatures with changes in atmospheric pressure



Be able to recall from memory the definition of latent heat

Be able to determine whether latent heat is released or absorbed during a phase change

Be able to provide the name given to each type of phase change



Be able to describe how enthalpy and latent heat are related

Be able to perform calculations to determine the amount of latent heat released during a phase change

Be able to perform calculations to determine the change in latent heat with temperature


ObjectiveObjective

Be able to recall from memory the definition of wet bulb temperature

Be able to compare the differences between wet bulb temperature and dew point temperature


SolidSolidLiquidLiquidGasGas

Phase ChangesPhase Changes

Phase change results in a transformation of the molecular structure


TT

Phase ChangePhase Change

Temperature of substance does not change during transformation



Equilibrium (or saturation) pressure does not change during phase change



Can Occur at Various Temperatures and Equilibrium Pressures

VaporVapor

WaterWater

Water &Water &VaporVapor

Volume (V)Volume (V)

Pre

ssu

re (

e)P

ress

ure

(e)

Ice & VaporIce & Vapor TT11

IceIce

TT22

TT33

TT44

TT55



Volume changes significantly during phase change

CondensationCondensation



Entropy also changes

SolidSolid LiquidLiquid GasGas

Increasing EntropyIncreasing Entropy



– Vapor to Ice– Water to Ice– Triple Line

The thermodynamic state at which three phases of a substance exist in equilibrium.

VaporVapor

WaterWater

Water Water &&

VaporVapor


Pre

ssu

re (

e)P

ress

ure

(e)

Ice & VaporIce & Vapor00ooCC

TT

IceIce

Ice & WaterIce & Water

Phase Change (P-V Diagram)

Triple LineTriple Line



– Triple Line T = 273.16K es = 6.107 mb

VaporVapor

WaterWater

Water Water &&

VaporVapor


Pre

ssu

re (

e)P

ress

ure

(e)


TT

IceIce



Triple LineTriple Line



– Vapor to Water Critical Point (Pc)

– The thermodynamic state in which liquid and gas phases of a substance coexist in equilibrium at the highest possible temperature.


VaporVapor

WaterWater

Water Water &&

VaporVapor


Pre

ssu

re (

e)P

ress

ure

(e)


TT

IceIce

Ice & WaterIce & WaterCritical Critical PointPoint



– Vapor to Water Critical Point (Pc)

– No liquid phase can exist at temperatures higher than the critical temperature

– Tc = 647 K

– Pc = 222,000 mb


VaporVapor

WaterWater

Water Water &&

VaporVapor


Pre

ssu

re (

e)P

ress

ure

(e)


TT

IceIce

Ice & WaterIce & WaterCritical Critical PointPoint



Phase Change (P-T Diagram)

TemperatureTemperature

Pre

ssu

reP

ress

ure

LiquidLiquid

eessww

GasGaseessii

SolidSolid



Isothermal Compression


Pre

ssu

reP

ress

ure

LiquidLiquid

eessww

GasGaseessii

SolidSolid



Isobaric Cooling


Pre

ssu

reP

ress

ure

LiquidLiquid

eessww

GasGas

eessii

SolidSolid



Changes in Atmospheric Pressure


Pre

ssu

reP

ress

ure LiquidLiquid

GasGas

SolidSolid

-.-.007007ooC C atmatm-1-1

– Change in Freezing Point



Changes in Atmospheric Pressure


Pre

ssu

reP

ress

ure

LiquidLiquid

GasGas

SolidSolid

– Change in Boiling Point



Critical Point


Pre

ssu

reP

ress

ure

LiquidLiquid

Critical Critical PointPoint

GasGaseessii

SolidSolid



Triple Point


Pre

ssu

reP

ress

ure

LiquidLiquid

Critical Critical PointPoint

GasGas

0.010.01ooCC

eessii

6.11 mb6.11 mb Triple Triple PointPoint

SolidSolid


Three Dimensional Phase Three Dimensional Phase DiagramDiagram

Ice & waterIce & water

WaterWater&&

VaporVapor

Vapor

Vapor

WaterWater


(hidden)(hidden)

Triple StateTriple StateIc

e

Temperature

TemperatureSpecific VolumeSpecific Volume

Vap

or P

ress

ure

Vap

or P

ress

ure

Critical PointCritical Point


Three Dimensional Phase Three Dimensional Phase DiagramDiagram



Liquid Water Molecule– Hydrogen Bonds– Shearing Energy too great



Ice– Volume Increases



GasGas


Heat is absorbed or released during the phase changes



GasGas

MeltingMelting

SublimationSublimationEvaporationEvaporation


Heat Absorbed



Heat Released


GasGas

FreezingFreezing

DepositionDepositionCondensationCondensation



Latent Heat– The heat required to change the

molecular configuration of a substance




Latent Heat


GasGas

FusionFusion(l(lff))

SublimationSublimation(l(lss))

Vaporization Vaporization (l(lvv))



Latent Heat– Increase in internal energy results from

the change in molecular configuration



Latent HeatLatent Heat

First Law of Thermodynamics– Internal Energy changes– Temperature is constant!– Pressure is constant– Volume changes

Work is done

pddudq



Rearrange

pddudq

pddqdu

vv = specific volume of vapor = specific volume of vapor

ww = specific volume of liquid = specific volume of liquid

)(ppd wv

For a phase change from liquid to vapor


pddqdu

uuvv = internal energy of vapor = internal energy of vapor

uuww = internal energy of liquid = internal energy of liquid

)(ppd wv

Define the change in Internal Energy


Substitute

Into )(pdqdu wv

)(pdquu wvwv


)(pdquu wvwv


Latent Heat (lv) = Change in Heat (dq)

)(pluu wvvwv

Rearrange

)pu()pu(l wwvvv



Enthalpy is defined as

Substitute

)pu()pu(l wwvvv

puh

wvv hhl dhlv oror

Latent Heat is a change in Enthalpy!



Latent Heat of Transformation (l)– ratio of the heat absorbed (Q) to the

mass undergoing a phase change

m

dQdhl

dqdhl



The amount of heat absorbed (or released) during a phase change is

lmdQ

m

dQl



Representative Values at 0oC

– Latent Heat of Fusion (lf) 3.34x105 J kg-1

– Latent Heat of Vaporization (lv)

2.500x106 J kg-1



Latent Heat of Sublimation (ls) at 0oC

ls = lf + lv

ls = 2.834x106 J kg-1


Pre

ssu

reP

ress

ure LiquidLiquid eessww

GasGas

0.010.01ooCC

eessii

SolidSolid

6.11 mb6.11 mb Triple Triple PointPoint



Varies with temperature

VaporVapor

WaterWater


Pre

ssu

re (

e)P

ress

ure

(e)

00ooCCTT

IceIce

dQdQ

dQdQ

dQdQ

dQdQ


Variation of Latent HeatVariation of Latent Heat

Let’s examine the latent heat of vaporization

dqm

dQlv

It’s easier to show the variation using entropy, but we’ll follow Hess



First Law of Thermodynamics

pddudq

dqm

dQlv

Substitute

pddulv



Expand

And since w << v

pddulv

)(e)uu(l wvswvv

vswvv e)uu(l



The Ideal Gas Law (or Equation of State)

Substitute

vswvv e)uu(l

vvs TRe

vwvv TR)uu(l



Differentiate with respect to temperature

vwvv TR)uu(l

vwvv R

dT

du

dT

du

dT

dl



Remember from your early childhood

vwvv R

dT

du

dT

du

dT

dl

v

vv dT

duc

v

ccvvvv = specific heat of vapor at a constant volume = specific heat of vapor at a constant volume



The internal energy of water is a little more tricky!

vw

vvv R

dT

duc

dT

dl

?dT

duW



Back to the First Law

pddudq

Differentiate with respect to temperature for water (remembering es is constant)

dT

de

dT

du

dT

dq ws

w



But the change in specific volume of water with temperature is very small

dT

de

dT

du

dT

dq ws

w

dT

du

dT

dqc ww

dT

de

dT

du

dT

dq ws

w

ccww = specific heat of liquid water = specific heat of liquid water


vw

vv R

dT

duc

dT

dlv

dT

duc ww


Substitute into

vwvv Rcc

dT

dlv



Another repressed memory ...

vwvv Rcc

dT

dlv

vvp Rccvv

wpv cc

dT

dlv


wpv cc

dT

dlv


Change in the Latent Heat of Vaporization with Temperature – Difference between

Specific Heat of Vapor (at constant pressure) Specific Heat of Liquid Water



Evaluate wpv cc

dT

dlv

ccpvpv = specific heat of vapor = 1952 J K = specific heat of vapor = 1952 J K-1-1 kg kg-1-1

ccww = specific heat of liquid water = 4218 J K = specific heat of liquid water = 4218 J K-1-1 kg kg-1-1

1111v kgJK4218kgJK1870dT

dl

11v kgJK2348dT

dl



Is this a factor to be considered?

llvv = latent heat of vaporization @ 273.16K = 2.5 x 10 = latent heat of vaporization @ 273.16K = 2.5 x 1066 J kg J kg-1-1

11v kgJK2348dT

dl

116

11

v

v

K%09.Jkg1050.2

kgJK2348

ldTdl



A small factor

1v K%09.dT

dl


SummarySummary

Specific Heat – The amount of heat required to raise the

temperature of a unit mass of a substance by one degree

dT

dqc


SummarySummary

Specific Heat– Dry Air

Constant Volumecv = 717 J K-1 kg-1

Constant Pressurecp = 1004 J K-1 kg-1


SummarySummary

Specific Heat– Water Vapor

Constant Volume– cvv

= 1463 J K-1 kg-1

Constant Pressure– cpv

= 1870 J K-1 kg-1


SummarySummary

Specific Heat– Liquid Water (0oC)

cw = 4218 J K-1 kg-1

cw = 1 cal g-1 K-1

– Ice (0oC) ci = 2106 J K-1 kg-1


SummarySummary

Latent Heat– The heat required to change the

molecular configuration of a substance



SummarySummary

Latent Heat– The change in enthalpy between states

dhl


SummarySummary

Latent Heat– The amount of heat absorbed (or

released) during a phase change


GasGas

FusionFusion(l(lff))

SublimationSublimation(l(lss))

Vaporization Vaporization (l(lvv))


SummarySummary

Latent Heat– The ratio of the heat absorbed (Q) to the

mass undergoing a phase change

dqm

dQl

dqm

dQl


SummarySummary

Latent Heat– Vaporization

lv = 2.50 x 106 J kg-1

– Fusion lf = 3.34 x 105 J kg-1

– Sublimation ls = 2.834 x 106 J kg-1


Moisture VariablesMoisture Variables

Wet-Bulb Temperature (Tw)

– The temperature to which air is cooled by evaporating water into it at constant pressure until the air is saturated

TTww



Wet Bulb Temperature (Tw)

– Two methods to compute Thermodynamic (or Isobaric) Method Adiabatic Method

vp mldTc


Themodynamic Wet Bulb Themodynamic Wet Bulb TemperatureTemperature

Different than Dew Point Temperature

TTww

TTdd



Dew Point (Td)

– Temperature to which air must be cooled at constant pressure in order for it to become saturated with respect to liquid water

TTdd


Dew Point TemperatureDew Point Temperature


Pre

ssu

reP

ress

ure

eess

TTatmosphereatmosphere

eesaturationsaturation

RH = 100%RH = 100%

TTdd

IsobaricIsobaricCoolingCooling


Thermodynamic Wet Bulb Thermodynamic Wet Bulb TemperatureTemperature


Pre

ssu

reP

ress

ure

eess

TTatmosphereatmosphere

ee

RH = 100%RH = 100%

TTdd

EvaporationEvaporation

TTww



Moisture is added to the atmosphere by evaporation

Heat for evaporation comes from air and water

TTww



Heat Balance

VaporizeWaterAir dQdQ

Heat lost by Heat lost by airair

Heat required to Heat required to vaporize watervaporize water==

TTww

dQdQ

dQdQccp p = specific heat of air = specific heat of air

ccw w = specific heat of water = specific heat of water

mmvv = mass of water that evaporates = mass of water that evaporates

llv v = latent heat of vaporization = latent heat of vaporization

vvwp dmldT)cc(



TTww

ccppdd = specific heat of dry air= specific heat of dry air

ccppvv = specific heat of water vapor= specific heat of water vapor

ccw w = specific heat of liquid water = specific heat of liquid water

mmdd = mass of dry air = mass of dry air

mmv v = mass of water vapor = mass of water vapor

vvwwvpvdpd dmldT)cmcmcm(

vvwp dmldT)cc(



TTww

TT

TTww

wwTTw w = wet bulb temperature= wet bulb temperature

TTaa = temperature of the air = temperature of the air

)TT)(cmcmcm( wairwwvpvdpd

)mm(l unsatvsatvv




TTww

TT

TTww

wwmmvvsatsat

= mass of water vapor of saturated air = mass of water vapor of saturated air

mmvunvunsatsat = mass of water vapor of unsaturated air = mass of water vapor of unsaturated air

mmvunvunsatsat-- mmvunvunsat sat

= amount of water vapor = amount of water vapor

evaporated into airevaporated into air


)mm(l unsatvsatvv




Divide both sides by md

TTww

TT

TTww

ww

wwsatsat

= mixing ratio of saturated air = mixing ratio of saturated air

wwunsatunsat

= mixing ratio of unsaturated air = mixing ratio of unsaturated air

)TT)(cmmwcc( wairwdwvpdp

)ww(l unsatsatv


)mm(l unsatvsatvv



As md increases, wcpv

and mw/md decreases

– Can be neglected

TTww

TT

TTww

ww

)ww(l)TT(c unsatsatvwairdp

)TT)(cmmwcc( wairwdwvpdp

)ww(l unsatsatv



Substitute for mixing ratioTTww

TT

TTww

ww

)ww(l)TT(c unsatsatvwairdp

p

e

ep

ew

)ee(p

l)TT(c sat

vwairdp



Solve for e

TTww

TT

TTww

ww

)ee(p

l)TT(c sat

vwairdp

)TT(l

pcee wair

v

dpsat



Psychrometric Equation

TTww

TT

TTww

ww

)TT(l

pcee wair

v

dpsat

vdp

l

pc Psychrometric Constant


TTww

TT

TTww

ww

w’w’


Other Factors– Ventilation– Radiation– Instrumentation


TTww

T,wT,wss

TTww

ww

w’w’


Measure T & Tw

esat is a Function of Tw via Claussius-Clapeyron

e is a Function of T via Claussius-Clapeyron

)TT(l

pcee wair

v

dpsat



Must Be Solved Iteratively or…..

)TT(l

pcee wair

v

dpsat



Psychrometric Charts– Equation Solved for Various Temperatures

Relative Humidity %

Dry Bulb oF



Psychrometric Tables


Adiabatic Wet Bulb Adiabatic Wet Bulb TemperatureTemperature

The temperature an air parcel would have if cooled to saturation and then compressed adiabatically to the original pressure in a moist adiabatic process


Adiabatic Wet Bulb Adiabatic Wet Bulb TemperatureTemperature

TTTTdd TTww


Wet Bulb Potential Wet Bulb Potential Temperature (Temperature (w w ))

The wet bulb temperature the air would have if it were expanded or compressed adiabatically from its existing pressure and wet bulb temperature to a standard pressure of 1000 mb.

286.

ww P

1000T

Prof. Fred Remer University of North Dakota Phase Changes and Latent Heat Where’s the heat? Solid...

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Transcript of Prof. Fred Remer University of North Dakota Phase Changes and Latent Heat Where’s the heat? Solid...