Latent Heat of Vaporization – Delta Hvap – of Water calculated by corresponding states correlation in a one cell excel formula

updated 14 May 2019

The latent heat of vaporization , the ‘DeltaHvap’ , of water is one of the key thermophysical properties of theoretical and practical interest. How much energy in the form of heat has to be added to one kilogram of water -at its boiling point- to convert it in one kilogram of steam ? A lot! The latent heat of evaporation for water is very high compared to many other substances due to the hydrogen bonding between the water molecules in addition to the van der Waals attraction forces binding the water molecules tightly together. To escape the liquid phase and free itself from these bonds a water molecule needs to ‘pick up’ sufficient (kinetic) energy. The high latent heat of vaporization makes water a preferred medium to transport and transfer energy.

In this post I want to share a compact, closed equation for the latent heat of vaporization / condensation that covers the entire temperature range from the triple to the critical point and still predicts the ‘DeltaHvap’ with good accuracy. I developed this equation in the form of a corresponding states correlation. It can generate latent heat of evaporation data with an average percentage error of 0.15 % compared to the Steam Table Data! If this amount of remaining error still makes you cringe, it must mean you likely will be concerned with what edition , what update of the International Steam Tables are involved. If that is the case then you will have to resort to equations like the series expansion type of equations with many constants, terms , and exponents to squeeze out these last inaccuracies. My preference is for compact, explicit equations that can be used directly in a worksheet not requiring iterative calculations to solve or forcing me to set up a Visual Basic program to deal with lengthy formula’s which requires more time to implement and verify.

This post has three sections. In the first I will present the new equation for the heat of evaporation of (saturated) water, i.e. steam condensate, as a function of reduced temperature. In the second section applications of this equation will be discussed. And finally in the third section a short description of how this new compact equation was developed.

SECTION 1. A new explicit equation for the Latent Heat of Evaporation of saturated Water in the form of a corresponding states formula.

A new equation for the relation of the Latent Heat of Evaporation ‘DeltaHvap’ was developed with the following goal in mind: good accuracy over the entire temperature range from triple point of water to the critical point!  I arrived at the following equation in which the ‘DeltaHvap’ is expressed in a corresponding states type formula. Please note the formula has been written here in ‘excel format style where the symbol ‘*’ represents ‘multiply’ and the ‘/’ stands for ‘dividing’ :

DHv * MW / ( R * Tcrit) = A * (1-Tr)^n / Tr * Ln(1 – Tr) + B  ; Validity range 0.4221 <= Tr < 1.00

in which the following symbols are used:  ‘DHv’ is the latent heat of vaporization of water , the ‘DeltaHvap’ in  kJ/kg ; ‘MW’ is the Molecular Weight of water 18 ; ‘R’ is the Universal Gas Constant equal to 8.31451 kJ/kmol/oK ; ‘Tcrit’ the critical absolute temperature of water of 647.15 degrees Kelvin ; he constant ‘A’ is equal to -9.11 ; the exponent ‘n’ is equal to 0.785 ; the symbol ‘Ln’ stands for the natural logarithm and ‘B’ is a constant  equal to 0.646. This equation covers the entire range of temperatures from the triple point up to but not including the critical temperature itself :  validity range     0.4221  <=  Tr   <  1  ; This equation predicts  the latent heat of evaporation with an average percentage error of 0.15%! The highest error are found at the ‘edges’ of the temperature range. The error at zero degrees Celsius is 0.29% while the largest error is 0.51% at 365 degrees Celsius. In the next Chart I have plotted the values calculated with this corresponding states correlation as function of the reduced temperature ‘Tr‘ alongside the Steam Table Data as follows: ( click on image to enlarge)

Latent Heat of Vaporization of Water versus Reduced Temperature – New Corresponding State Correlation

In this Chart you can observe that the predicted values tightly match the Steam Table Data. You can use this equation in your spreadsheet simulation calculations with confidence without having to worry whether your ‘working’ temperature range is outside of the validity range of any other equation for ‘DHvap’ you may have employed earlier! No error flags or error messages need to be involved or raised when working with the above equation!  To further demonstrate the goodness of fit of this equation I have created two additional Charts. The first one shows the calculation results plotted against the absolute temperature and the second one show the ‘DHv’ values plotted against the temperature expressed in degrees Celsius. (click on the image to enlarge):

Latent Heat of Vaporization of Water versus Absolute Temperature

 

In the second Chart the new correlation has been plotted against the temperaature in degrees Celsius together with the Steam Table Data. (click on Chart to enlarge)

Note: in this Chart a full grid is shown so this Chart can be used as a quick (manual) reference to the ‘DeltaHvap’ value for a given temperature in degrees Celsius.

SECTION 2.  Two application examples.

First application example: Calculating  the Enthalpy of Saturated Steam in one Chart , creating a single Chart of the phase envelope of the entire Steam/Condensate region. The new equation for the latent heat of vaporization for saturated water (condensate) can be combined with the new correlation for the enthalpy of saturated steam condensate presented in the previous post to calculate the saturated (vapor) Steam Enthalpy. The following Chart shows the enthalpy of saturated steam and condensate in a single graph. (click in image to enlarge):

Enthalpy of Saturated steam Hv and Condensate hl combined

Second example: Determination of the amount of (Flash) Steam that can be generated from ‘hot condensate’. 

I discussed in the previous post the efficient use of hot condensate discharged from a steam condenser. Hot condensate can , depending on its temperature, still hold an appreciable amount of usable energy.  One option is to generate ‘Flash Steam’ by letting the hot condensate down in pressure and using that ‘flash steam’ at a lower pressure/temperature level. The question then is how much steam can be produced from that hot condensate? This can be determined by making a material- and enthalpy balances over the letdown (control) valve.   The flash evaporation process over the letdown valve is ‘isenthalpic‘ because no heat is lost or added , no additional work is done other than the expansion of steam against the letdown pressure which the thermodynamical property of enthalpy already takes into account. The let down process is portrayed in the following sketch. The hot condensate flow ‘F’ fed into the valve is at pressure level ‘P1’ whereas emerging on the outlet side are the (vapor) steam flow ‘V’ and a remaining (liquid) condensate flow ‘L” at pressure ‘P2’.

at level ‘P1’  :       F ———>  lXl ——–> V   and L         at level ‘P2’

Material balance:  F = V + L ;   Enthalpy Balance:  F * hl,1 = V * Hv,2 + L * hl,2     Combining gives for the Flash fraction V / F  = ( hl1 – hl2) / Delta Hvap 2 

The above discussed new equations/correlations can each be easily entered an excel spreadsheet to calculate the flash fraction ‘V/F”.  The outcome of these calculations are demonstrated in the next Chart where the flash fraction has been plotted as function of the upstream or source pressure ‘P1’ for various  letdown pressures ‘P2’.  Such Chart can be handy for the quick ‘scoping’of  a design-or revamp change to evaluate the economic benefit of this heat recovery through flash steam generation can be.  This Chart shows the weight percentage of Steam generated from the hot condensate stream ( at source pressure ‘P1’) let down over the flash valve to letdown pressure ‘P2’. (Click on image to enlarge):

Flash Steam generated from hot saturated Condensate letdown to lower pressure

The highest Up stream Pressure ‘P1’ has been chosen as 90 Bar absolute. Each curve represents a letdown pressure level ‘P2’. The vertical scale is percentage of hot condensate flashed to steam at pressure level ‘P2’. The advantage of flash steam generation as an energy recovery route is that the heat is in a form that can be used efficiently and effectively as condensing steam has a very high heat transfer coefficient. The disadvantage is, it is steam at a lower temperature  being the saturation temperature at the letdown pressure ‘P2’. A cost/ benefit analysis can tell whether this is the best route for energy recovery from the hot condensate under study in the revamp project. 

SECTION 3. How this new ‘DHv’ equation was developed. 

The oldest equation – at least that I am aware of – for the heat of vaporization as function of reduced temperature was the  one by K.M. Watson published in 1943. ( see the original paper “Thermodynamics of the liquid state” on the web ). It reads as follows :

DHv1 / DHv2 = ( ( 1-Tr1) / (1- Tr2 ) )^0.38

in which ‘DHv1′ is the difference in (saturated) vapor an liquid Enthalpy – ‘DeltaHvap’ – at reduced temperature ‘Tr1′ and ‘DHv2′ that for reduced temperature ‘Tr2′. A remarkable equation that he found to be valid for a wide range of different substances. How does this equation apply to Water and Steam ?  Actually, as I will show in the following text,  it gives good, rough estimate, however not good enough for my purpose that is: I need a good accuracy over as wide a range as possible. Taking the original Watson equation and casting it in the following form:

DHv * MW/(R*Tcrit) = A * ( 1 – Tr ) ^ n              (Eq. 1)

and regressing it against  41 data points of ‘DeltaHvap’  taken from the Steam Table – each at temperatures 10 Degrees apart, except near the triple point and near the critical point, taken 5 degrees apart. This yielded  for the constants ‘A’ and exponent ‘n’ the following values:

 DHv * MW /(R *Tcrit)  = 10.538 * (1 – Tr) ^ 0.3809         (Eq. 2)

confirming the “Watson exponent” of 0.38. However, when calculating the average percentage error between predicted and actual Steam Table Data I found this to be 1.35 % over the whole range from 0 degrees  to 370 degrees Celsius. The maximum error though can be as high as 2.3% at zero degrees and 2.2% at 260 oC. Comparing the Steam Table Data against the points calculated with this correlation, by plotting them side by side in the same graph. You can see for yourselves what the ‘goodness of fit’ is as shown in the  following Chart (click on the image to enlarge):

Latent Heat of Vaporization of Water versus Reduced Temp- Watson type correlation

In this graph you can see that the exponent 0.38 catches the general trend over the whole range of reduced temperatures but apparently this exponent is not exactly constant over that range! If I plot the logarithm of the Steam Table Data for the latent heat of vaporization -DeltaHvap – against the  logarithm of the reduced temperature ( this plot and Chart not shown) an interesting observation can be made. It looks like the points roughly fall on a straight line as would be expected from a constant exponent ‘ruling’ the relationship with temperature. However, closer inspection shows that there is a ‘bend’ in the apparent straight line around a reduced temperature of ~ 0.73  corresponding to ~ 200 degrees Celsius. The lower temperature points tend to curve downwards toward the temperature axis. Each of the  line sections in themselves appear to be straight from which we may conclude that each is ruled by a different exponent that I found to be 0.33 for the lower temperature part and 0.39 for the higher temperature part . The existence of this ‘bend’ in the Log – Log plot of the data reminds of the similar behavior of vapor pressures against temperature , leading Pitzer in 1955 to define his acentric factor omega as another constant in characterizing a substance in addition to the critical data of Tc, Pc and Vc. And this similarity should be no surprise as the fundamental Clausius Clapeyron equation connects the slope of the vapor pressure curve with the heat of vaporization! Pitzer defined the acentric factor omega  as ” -10 baseLog( Pr)  -1 ” taken at Tr = 0.70 with Pr being the saturated reduced vapor pressure. For noble gases Argon , Krypton , where only the van der Waals forces play a role and whose gas molecules are spherical and apolar and with no hydrogen bonds present, the value of omega equals zero. For water the acentric factor omega = 0.344. Looking at the performance of the above Watson type equation from a useful application perspective you can see that in the range often used in practice – the temperature range from 200 to 300 degrees Celsius -, this equation happens to show it’s highest errors!  In short if I want to develop and arrive at an equally wide ranging but more accurate correlation then a more elaborate  equation than the above Watson type equation with two constants ‘A’ and ‘n’ will be needed, which has been described above.