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

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)

Ice & VaporIce & Vapor TT11

IceIce

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

Ice & VaporIce & Vapor00ooCC

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

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

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

IceIce

Ice & WaterIce & WaterCritical Critical PointPoint

Phase Change (P-T Diagram)

TemperatureTemperature

LiquidLiquid

eessww

GasGaseessii

SolidSolid

Isothermal Compression

LiquidLiquid

eessww

GasGaseessii

SolidSolid

Isobaric Cooling

LiquidLiquid

eessww

GasGas

eessii

SolidSolid

Changes in Atmospheric Pressure

ure LiquidLiquid

GasGas

SolidSolid

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

– Change in Freezing Point

Changes in Atmospheric Pressure

LiquidLiquid

GasGas

SolidSolid

– Change in Boiling Point

Critical Point

LiquidLiquid

Critical Critical PointPoint

GasGaseessii

SolidSolid

Triple Point

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

WaterWater

(hidden)(hidden)

Triple StateTriple StateIc

Temperature

TemperatureSpecific VolumeSpecific Volume

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

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

)(pluu wvvwv

Rearrange

)pu()pu(l wwvvv

Enthalpy is defined as

Substitute

)pu()pu(l wwvvv

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

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

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

ure LiquidLiquid eessww

GasGas

0.010.01ooCC

eessii

SolidSolid

6.11 mb6.11 mb Triple Triple PointPoint

Varies with temperature

VaporVapor

WaterWater

00ooCCTT

IceIce

Variation of Latent HeatVariation of Latent Heat

Let’s examine the latent heat of vaporization

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

First Law of Thermodynamics

pddudq

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

Remember from your early childhood

vwvv R

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!

Back to the First Law

pddudq

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

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

dqc ww

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

duc ww

Substitute into

vwvv Rcc

Another repressed memory ...

vwvv Rcc

vvp Rccvv

wpv cc

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

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

11v kgJK2348dT

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

K%09.Jkg1050.2

kgJK2348

A small factor

1v K%09.dT

SummarySummary

Specific Heat – The amount of heat required to raise the

temperature of a unit mass of a substance by one degree

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

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

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

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

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

Dew Point TemperatureDew Point Temperature

TTatmosphereatmosphere

eesaturationsaturation

RH = 100%RH = 100%

IsobaricIsobaricCoolingCooling

Thermodynamic Wet Bulb Thermodynamic Wet Bulb TemperatureTemperature

TTatmosphereatmosphere

RH = 100%RH = 100%

EvaporationEvaporation

Moisture is added to the atmosphere by evaporation

Heat for evaporation comes from air and water

Heat Balance

VaporizeWaterAir dQdQ

Heat lost by Heat lost by airair

Heat required to Heat required to vaporize watervaporize water==

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(

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(

wwTTw w = wet bulb temperature= wet bulb temperature

TTaa = temperature of the air = temperature of the air

)TT)(cmcmcm( wairwwvpvdpd

)mm(l unsatvsatvv

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

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

)ww(l)TT(c unsatsatvwairdp

)TT)(cmmwcc( wairwdwvpdp

)ww(l unsatsatv

Substitute for mixing ratioTTww

)ww(l)TT(c unsatsatvwairdp

l)TT(c sat

vwairdp

Solve for e

l)TT(c sat

vwairdp

pcee wair

Psychrometric Equation

pcee wair

pc Psychrometric Constant

w’w’

Other Factors– Ventilation– Radiation– Instrumentation

T,wT,wss

w’w’

Measure T & Tw

esat is a Function of Tw via Claussius-Clapeyron

e is a Function of T via Claussius-Clapeyron

pcee wair

Must Be Solved Iteratively or…..

pcee wair

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.

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