Assistance creating a new ????VPD formula solving for relative humidity

[Marshy]

I am trying to create a formula to solve for the dependent variable of relative humidity (RH) given the known independent variables of air temperature, leaf surface temperature, and _leafVPD.

I recently created three formulas to calculate:

1. VPD (vapor pressure deficit)
2. RH of VPD
3. _leafVPD (i.e., VPD accounting for leaf temperature assuming 100% relative humidity inside leaves).

The formulas I created to solve for RH of VPD (when the air temperature is known) and _leafVPD (when air temperature, RH, leaf surface temperature, and intra-leaf RH are known) can be used as a model for the formula I am trying to create.

Here are my three formulas and a blank fourth formula I'm trying to create:

(1) Solve for VPD:

exp^{(34.494-4924.99/(T+237.15))}/(T+105)^1.57/1000*(1-RH/100)

Where:

• exp = spreadsheet function returning the constant e (Euler's number), approximately equal to 2.71828
• T = air temperature in degrees Celcius; e.g., 25
• RH = relative humidity in percent; e.g., 70

(2) Solve for RH of VPD:

100*(1-_airVPD/(exp^{(34.494-4924.99/(T+237.15))}/(T+105)^1.57/1000))

Where:

• exp = spreadsheet function returning the constant e (Euler's number), approximately equal to 2.71828
• T = air temperature in degrees Celcius; e.g., 25
• _airVPD = target VPD in kPa; e.g., 1.0

(3) Solve for _leafVPD:

(exp(^{34.494-4924.99/(_lsT+237.15))}/(_lsT+105)^1.57-RH/100*exp^{(34.494-4924.99/(T+237.15))}/(T+105)^1.57)/1000

Where:

• exp = spreadsheet function returning the constant e (Euler's number), approximately equal to 2.71828
• T = air temperature in degrees Celcius; e.g., 25
• _lsT = leaf surface temperature in degrees Celcius; e.g., 23
• RH = relative humidity in percent; e.g., 70

(4) Solve for RH of _leafVPD:

???

Where:

• exp = spreadsheet function returning the constant e (Euler's number), approximately equal to 2.71828
• T = air temperature in degrees Celcius; e.g., 25
• _lsT = leaf surface temperature in degrees Celcius; e.g., 23
• _leafVPD = target _leafVPD in kPa; e.g., 1.0

 

[Deleted member 4993]

exp(34.494-4924.99/(T+237.15))/(T+105)1.57/1000*(1-RH/100)

What does this↑ equal to? How did you derive it?

100*(1-airVPD/(exp(34.494-4924.99/(T+237.15))/(T+105)1.57/1000))

What does this↑ equal to? How did you derive it?

(exp(34.494-4924.99/(lsT+237.15))/(lsT+105)1.57-RH/100*exp(34.494-4924.99/(T+237.15))/(T+105)1.57)/1000

What does this↑ equal to? How did you derive it?

As such your expressions do not make sense to me.

 

[Dr.Peterson]

I am trying to create a formula to solve for the dependent variable of relative humidity (RH) given the known independent variables of air temperature, leaf surface temperature, and _leafVPD.

I, too, would like to see where the formulas come from, in order to make sure we understand them, as there appear to be a couple things you don't understand (or at least aren't communicating in our language).

The spreadsheet function EXP doesn't "return the constant e (Euler's number)"; it raises e to the power of its argument. So your superscript notation is inappropriate.

I tried putting your first couple formulas into a spreadsheet like this (with provided RH in A3, T in B3, and VPD in C3):

(1) VPD: =EXP(34.494-4924.99/(B3+237.15))/(B3+105)^1.57/1000*(1-A3/100)​

(2) RH: =100*(1-C3/(EXP(34.494-4924.99/(B3+237.15))/(B3+105)^1.57/1000))​

Putting in the values of 70, 25, and 0.954, the results agreed. So those at least are consistent. Written normally, they are (I think)

[math]VPD=\frac{e^{34.494-\frac{4924.99}{T+237.15}}}{(T+105)^{1.57}}\cdot\frac{1}{1000}\cdot\left(1-\frac{RH}{100}\right)[/math][math]RH=100\left(1-\frac{VPD}{\frac{e^{34.494-\frac{4924.99}{T+237.15}}}{(T+105)^{1.57}}\cdot\frac{1}{1000}}\right)[/math]
And that looks about right, though I could simplify them.

This suggests that the third formula should look like this (with RH in A3, lsT in B5):

(3) leafVPD: =(EXP(34.494-4924.99/(B5+237.15))/(B5+105)^1.57-A3/100*EXP(34.494-4924.99/(B3+237.15))/(B3+105)^1.57)/1000​

I'll hold off on trying to write this normally until I have confirmation of what you want it to mean. This was a big enough struggle!

 

[Dr.Peterson]

Here are simplified versions of the first two formulas:

[math]airVPD=\frac{e^{34.494-\frac{4924.99}{T+237.15}}}{1000(T+105)^{1.57}}\cdot\left(1-\frac{RH}{100}\right)[/math][math]RH=100\left(1-\frac{1000airVPD(T+105)^{1.57}}{e^{34.494-\frac{4924.99}{T+237.15}}}\right)[/math]
Here is the third formula:
[math]leafVPD=\left(\frac{e^{34.494-\frac{4924.99}{lsT+237.15}}}{(lsT+105)^{1.57}}-\frac{RH}{100}\cdot \frac{e^{34.494-\frac{4924.99}{T+237.15}}}{(T+105)^{1.57}}\right)\cdot\frac{1}{1000}[/math]
If this is correct, it is not hard to solve for RH.

 

[Marshy]

Hello,

Thank you for your response! Please let me know if this answers your questions.

Please note that in my formulas, all instances of the value "237.15" should be "237.1". I have corrected that error below.

As such your expressions do not make sense to me.

My apologies. As Dr. Peterson kindly pointed out, I shouldn't have used the superscript notation for the values after EXP. Dr. Peterson rightly noted in spreadsheets EXP returns Euler's number, i.e., the constant e (2.71828182845904), "raised to the power of number" (1,2).

The corrected formulas are:

(#1) e(34.494-4924.99/(T+237.1))/(T+105)^1.57/1000*(1-RH/100)

(#2) 100*(1-_airVPD/(EXP(34.494-4924.99/(T+237.1))/(T+105)^1.57/1000))

(#3) (e(34.494-4924.99/(_lsT+237.1))/(_lsT+105)^1.57-RH/100*e(34.494-4924.99/(T+237.1))/(T+105)^1.57)/1000

exp^{(34.494-4924.99/(T+237.1))}/(T+105)^1.57/1000*(1-RH/100)
What does this↑ equal to? How did you derive it?

That formula solves for vapor pressure deficit (VPD) of air (kPa) using the variables of air temperature (°C) and RH:

In a spreadsheet cell, it looks like this:

• =EXP(34.494-4924.99/(T+237.1))/(T+105)^1.57/1000*(1-RH/100)

The formula equals a VPD value of ≈0.95099 kPa when replacing the variables of T and RH, where T is 25°C is RH is 70%:

• =EXP(34.494-4924.99/(25+237.1))/(25+105)^1.57/1000*(1-70/100)

I dervied the formula as so:

(1) I used the satured vapor pressure (SVP) over water from formula 17 given by Huang (2018), A Simple Accurate Formula for Calculating Saturation Vapor Pressure of Water and Ice:


Where:

• Ps is the saturation vapor pressure as pascal (Pa).
• t is the air temperature (>0°C).

(2) I then converted the calculated SVP from PA to kilopascal (kPa) by dividing by 1000.

(3) Finally, to find the VPD of air, I modified the _airVPD calculation presented by Stasiak et al. (2022). Replacing Tetens’s (1930) formula for SVP over water formula used in plant science before my current efforts with the SVP over water from formula 17 by Huang (2018). And simplifed the actual vapor pressure (AVP) formula given by Stasiak et al. (2022).

That is, I multiplied the SVP in kPa (from step #2) by the AVP. Where AVP is the difference of the measured relative humidity (RH) divided by 100 from the maximum possible SVP (i.e., 100% relative humidity divided by 100).

Stasiak et al., (2022), Growing facilities and environmental control:

100*(1-airVPD/(exp(^{34.494-4924.99/(T+237.15))}/(T+105)^1.57/1000))
What does this↑ equal to? How did you derive it?

That formula solves for the RH of _airVPD using the known variables of _airVPD (kPa) and air temperature (°C):

In a spreadsheet cell, it looks like this:

• =100*(1-airVPD/(EXP(34.494-4924.99/(T+237.1))/(T+105)^1.57/1000))

The formula equals an RH value of ≈70% when replacing the variables of _airVPD and T, where _airVPD is 0.95099 kPa and T is 25°C:

• =100*(1-0.95099/(EXP(34.494-4924.99/(25+237.1))/(25+105)^1.57/1000))

I derived the formula as so:

I used the HTML code from this webpage as a template, which solves for RH of _airVPD using Tetens's (1930) formula for SVP over water. By modifying my formula solving for VPD of air described above to solve for the RH of _airVPD using Huang's (2018) #17 formula for SVP over water in place of Tetens's (1930) formula for SVP over water.

(exp^{(34.494-4924.99/(lsT+237.15))}/(lsT+105)^1.57-RH/100*exp^{(34.494-4924.99/(T+237.15))}/(T+105)^1.57)/1000
What does this↑ equal to? How did you derive it?

That formula solves for _leafVPD using the known variables of air temperature (as T in °C), leaf-surface temperature (as _lsT in °C), and RH:

In a spreadsheet cell, it looks like this:

• =(EXP(34.494-4924.99/(lsT+237.1))/(lsT+105)^1.57-RH/100*EXP(34.494-4924.99/(T+237.1))/(T+105)^1.57)/1000

The formula equals a _leafVPD value of ≈0.59212 kPa when replacing the variables of _lsT, RH, and T, where _lsT 23°C, is RH is 70%, and T is 25°C:

• =(EXP(34.494-4924.99/(23+237.1))/(23+105)^1.57-70/100*EXP(34.494-4924.99/(25+237.1))/(25+105)^1.57)/1000

I derived the formula as so:

(1) I used the SVP over water from formula 17 given by Huang (2018) to find _leafSVP by swapping the air temperature (°C) in his formula with leaf-surface temperature (°C).

(2) Then, I subtracted the quotient of RH divided by APV (100% RH) from _leafSVP.

(3) Next, I used the SVP over water from formula 17 given by Huang (2018) to find SVP using the air temperature (°C). Then I multiplied SVP by the difference of RH divided by AVP from _leafSVP.

(4) Finally, I divided by 1000 to convert from Pa to kPa.

 

[Marshy]

I, too, would like to see where the formulas come from, in order to make sure we understand them, as there appear to be a couple things you don't understand (or at least aren't communicating in our language).

The spreadsheet function EXP doesn't "return the constant e (Euler's number)"; it raises e to the power of its argument. So your superscript notation is inappropriate.

I tried putting your first couple formulas into a spreadsheet like this (with provided RH in A3, T in B3, and VPD in C3):

(1) VPD: =EXP(34.494-4924.99/(B3+237.15))/(B3+105)^1.57/1000*(1-A3/100)​

(2) RH: =100*(1-C3/(EXP(34.494-4924.99/(B3+237.15))/(B3+105)^1.57/1000))​

Putting in the values of 70, 25, and 0.954, the results agreed. So those at least are consistent. Written normally, they are (I think)

[math]VPD=\frac{e^{34.494-\frac{4924.99}{T+237.15}}}{(T+105)^{1.57}}\cdot\frac{1}{1000}\cdot\left(1-\frac{RH}{100}\right)[/math][math]RH=100\left(1-\frac{VPD}{\frac{e^{34.494-\frac{4924.99}{T+237.15}}}{(T+105)^{1.57}}\cdot\frac{1}{1000}}\right)[/math]
And that looks about right, though I could simplify them.

This suggests that the third formula should look like this (with RH in A3, lsT in B5):

(3) leafVPD: =(EXP(34.494-4924.99/(B5+237.15))/(B5+105)^1.57-A3/100*EXP(34.494-4924.99/(B3+237.15))/(B3+105)^1.57)/1000​

I'll hold off on trying to write this normally until I have confirmation of what you want it to mean. This was a big enough struggle!

Thank you for your time and effort!

I am very appreciative and thankful. I would not have been able to convert my spreadsheet formulas to the normal format you have presented without your and Subhotosh Khan's willingness to assist me. Rest assured that your efforts are not in vain because my formulas in the near future should become the standard method to calculate VPD and leafVPD in plant science and horticulture.

If you don't mind, once you and Subhotosh Khan are satisfied with the results, I would like to include the normal format of the formulas you're presenting (giving full credit) in a white paper I am preparing to publish.

It's past 2 AM here, so I will more fully respond to your posts later today.

For now, I thought you might like to see the formulas when I include the enhancement factor to obtain the saturation vapor pressure for moist air. Including the enhancement factor is the most accurate way to calculate SVP and, therefore _leafVPD and _airVPD. But the difference is in the megapascal (MPa) range (0.001-0.009 kPa). So, if you're not keen to include the enhancement factor in the normal format when writing the formulas, I understand.

According to Huang (2018),

​

Moist air does not, in strict terms, satisfy the ideal gas law. The saturation vapor pressure over water or ice should be multiplied by an enhancement factor to obtain the saturation vapor pressure for moist air. The enhancement factor, which is a weak function of temperature and pressure, is defined as the ratio of the saturation vapor pressure for moist air to that of pure water vapor over a plane of water (Buck 1981). The temperature effect is negligible. Thus, the enhancement factor is determined by (Alduchov and Eskridge 1996).​

​

Enhancement factor:​

​

f = 1.00071e(0.000000045P*3386.39)​

Where f is the enhancement factor and P is the atmospheric pressure (inHg).​

Here are my three formulas with the enhancement factor. I converted the standard atmosphere (atm) used in the enhancement factor from Pa to inches of mercury because inHG is a commonly used unit for weather stations and smartphone barometers:

(#1) _airVPD = e(34.494-4924.99/(T+237.1))/(T+105)^1.57/1000*(1-RH/100)*1.00071e(0.000000045*P*3386.39)

The formula equals _airVPD value of ≈0.956 kPa when replacing the variables of T, RH, and P, where T is 25°C, is RH is 70%, and P is 30 inHg:​

​

Spreadsheet formula:​

=EXP(34.494-4924.99/(25+237.1))/(25+105)^1.57/1000*(1-70/100)*1.00071*EXP(0.000000045*30*3386.39)​

​

(#2) RH of _airVPD = 100*(1-_airVPD/(EXP(34.494-4924.99/(T+237.1))/(T+105)^1.57/1000))*1.00071e(0.000000045*P*3386.39)

The formula equals RH value of ≈70.2 when replacing the variables of _airVPD, T, and P, where _airVPD is 0.956 kPa, T is 25°C, and P is 30 inHg:​

​

Spreadsheet formula:​

=100*(1-0.956/(EXP(34.494-4924.99/(25+237.1))/(25+105)^1.57/1000))*1.00071*EXP(0.000000045*30*3386.39)​

​

(#3) _leafVPD = (e(34.494-4924.99/(_lsT+237.1))/(_lsT+105)^1.57-RH/100*EXP(34.494-4924.99/(T+237.1))/(T+105)^1.57)/1000*1.00071e(0.000000045*P*3386.39)

The formula equals _leafVPD value of ≈0.5953 kPa when replacing the variables of _lstVPD, RH, T, and P, where _lstVPD is 23°C, RH is 70%, T is 25°C, and P is 30 inHg:​

​

Spreadsheet formula:​

=(EXP(34.494-4924.99/(23+237.1))/(23+105)^1.57-70/100*EXP(34.494-4924.99/(25+237.1))/(25+105)^1.57)/1000*1.00071*EXP(0.000000045*30*3386.39)​

 

[Marshy]

Here are simplified versions of the first two formulas:

[math]airVPD=\frac{e^{34.494-\frac{4924.99}{T+237.15}}}{1000(T+105)^{1.57}}\cdot\left(1-\frac{RH}{100}\right)[/math][math]RH=100\left(1-\frac{1000airVPD(T+105)^{1.57}}{e^{34.494-\frac{4924.99}{T+237.15}}}\right)[/math]
Here is the third formula:
[math]leafVPD=\left(\frac{e^{34.494-\frac{4924.99}{lsT+237.15}}}{(lsT+105)^{1.57}}-\frac{RH}{100}\cdot \frac{e^{34.494-\frac{4924.99}{T+237.15}}}{(T+105)^{1.57}}\right)\cdot\frac{1}{1000}[/math]
If this is correct, it is not hard to solve for RH.

Hello Dr. Peterson,

These look great, thank you! When comparing the three formulas written normally, it is easier to visualize how to solve for RH of _leafVPD. Later today, based on what you provided, I will attempt to create the fourth formula (although I may fail).

Regarding my previous post, I noticed I made two errors when posting last night. My apologies:

• The first error is I shouldn't have rounded the inHg to Pa conversion factor, which is 3386.38864034.
  
  
• The third error is in my description for _leafVPD, where I wrote "_lstVPD" when I intended to write "_lsT".

Also, I decided it's best to use hectopascals (hPa) as the unit for atmospheric pressure. The World Meteorological Organization recommends hPa for meteorological purposes, "largely because one hectopascal equals one millibar (mbar), the formerly used unit." In the future, I will modify these formulas to use imperial units (degrees Fareigneht and inches of mercury) for the USA.

Enhancement factor:​

f = 1.00071e(0.0000045P_hPa)​

​

Where f is the enhancement factor and P_hPa is the atmospheric pressure as hectopascal.​

Here are the spreadsheet formulas with the enhancement factor in hPa:

(#1) _airVPD = (e(34.494-4924.99/(T+237.1))/(T+105)^1.57/1000*(1-RH/100)*1.00071e(0.0000045P_hPa))

The formula equals _airVPD value of ≈0.956 kPa when replacing the variables of T, RH, and P, where T is 25°C, is RH is 70%, and P_hPa is 1015.9 hPa:​

​

=EXP(34.494-4924.99/(25+237.1))/(25+105)^1.57/1000*(1-70/100)*1.00071*EXP(0.0000045*1015.9)​

(#2) RH of _airVPD = 100*(1-_airVPD/(e(34.494-4924.99/(T+237.1))/(T+105)^1.57/1000*1.00071e(0.0000045P_hPa)))

The formula equals RH value of ≈70.00 when replacing the variables of _airVPD, T, and P_hPa, where _airVPD is 0.956 kPa, T is 25°C, and P_hPa is 1015.9 hPa:​

​

=100*(1-0.956/(EXP(34.494-4924.99/(25+237.1))/(25+105)^1.57/1000*1.00071*EXP(0.0000045*1015.9)))​

(#3) _leafVPD = (e(34.494-4924.99/(_lsT+237.1))/(_lsT+105)^1.57-RH/100e(34.494-4924.99/(T+237.1))/(T+105)^1.57)/1000*1.00071e(0.0000045P_hPa)

The formula equals _leafVPD value of ≈0.595 kPa when replacing the variables of _lsT, RH, T, and P_hPa, where _lsT is 23°C, RH is 70%, T is 25°C, and P_hPa is 1015.9 hPa:​

​

=(EXP(34.494-4924.99/(23+237.1))/(23+105)^1.57-70/100*EXP(34.494-4924.99/(25+237.1))/(25+105)^1.57)/1000*1.00071*EXP(0.0000045*1015.9)​

​

(#4) RH of _leafVPD = ???

The formula equals RH value of ≈??? when replacing the variables of _leafVPD, _lsT, T, and P_hPa, where _leafVPD is 0.595 kPa, _lsT is 23°C, T is 25°C, and P_hPa is 1015.9 hPa:​

​

​

 

[Marshy]

Hello again,

I attempted to rewrite the _leafVPD formula to solve for RH, but I was unsuccessful. However, I adjusted the formulas Dr. Peterson wrote to include the enhancement factor. Below is the current state of the formulas:

How should this formula that is solving for _leafVPD be rewritten to solve for RH?

[math]_{leaf}VPD=\left(\dfrac{e^{34.494-\tfrac{4924.99}{(_{ls}T+237.1)}}}{(_{ls}T+105)^{1.57}}-\dfrac{RH}{100}\cdot \dfrac{e^{34.494-\tfrac{4924.99}{(T+237.1)}}}{(T+105)^{1.57}}\right)\cdot0.00100071e^{0.0000045P_{hPa}}[/math]​

Where, for example:

• _lsT = 23
• RH = 70
• T = 25
• P_hPa = 1015.9

Spreadsheet forumula:

=(EXP(34.494-4924.99/(_lsT+237.1))/(_lsT+105)^1.57-RH/100*EXP(34.494-4924.99/(T+237.1))/(T+105)^1.57)*1/1000*1.00071*EXP(0.0000045*P_hPa)​

Here are the other formuals:

(1) Solving for _airVPD:

[math]_{air}VPD=\dfrac{e^{34.494-\tfrac{4924.99}{(T+237.1)}}}{1000(T+105)^{1.57}}\cdot\left(1-\dfrac{RH}{100}\right)\cdot1.00071e^{0.0000045P_{hPa}}[/math]​

Where, for example:

• T = 25
• RH = 70
• P_hPa = 1015.9

Spreadsheet formula:

=EXP(34.494-4924.99/(T+237.1))/(1000*(T+105)^1.57)*(1-RH/100)*1.00071*EXP(0.0000045*P_hPa)​

(2) The formula solving for _airVPD was rewritten so to solve for RH:

[math]RH=100\left(1-\dfrac{1000_{air}VPD(T+105)^{1.57}}{e^{34.494-\tfrac{4924.99}{(T+237.1)}}} \middle/ {(1.00071e^{0.0000045P_{hPa}}})\right)[/math]​

Where, for example:

• _airVPD = 0.95602073
• T = 25
• P_hPa = 1015.9

Spreadsheet formula:

=100*(1-1000*_airVPD*(T+105)^1.57/EXP(34.494-4924.99/(T+237.1))/(1.00071*EXP(0.0000045*P_hPa)))​

 

[Deleted member 4993]

leafVPD=(e34.494−4924.99(lsT+237.1)(lsT+105)1.57−RH100⋅e34.494−4924.99(T+237.1)(T+105)1.57)⋅0.00100071e0.0000045PhPa_{leaf}VPD=\left(\dfrac{e^{34.494-\tfrac{4924.99}{(_{ls}T+237.1)}}}{(_{ls}T+105)^{1.57}}-\dfrac{RH}{100}\cdot \dfrac{e^{34.494-\tfrac{4924.99}{(T+237.1)}}}{(T+105)^{1.57}}\right)\cdot0.00100071e^{0.0000045P_{hPa}}leafVPD=((lsT+105)1.57e34.494−(lsT+237.1)4924.99−100RH⋅(T+105)1.57e34.494−(T+237.1)4924.99)⋅0.00100071e0.0000045PhPa

This equation can be re-written as:

_leafVPD = (A - RH * B) * C →

RH =[A - (_leafVPD/C)]/B