2006, American Society of Heating, Refrigerating a nd Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in ASHRAE Journal, (Vol. 48, January 2006). For personal use only. Additional distribution in either paper or digital form is not permitted without ASHRAE’s permission.
Psychrometric Spreadsheet By Steve Kavanaugh, Ph.D., Fellow ASHRAE, Barbara Hattemer McCrary and Keith A. Woodbury
M
any engineers use spreadsheet programs for calculations
Review of Psychrometri Psychrometric c Equations umidity Ratio
and graphing because of the variety of relatively easy-to-
Psychrometric charts and equations are convenient methods of dealing with the ® use embedded features. One such feature is the Microsoft Visual hermodynamic properties of mixtures of water vapor and air. Obviously, an ® ® Basic Macro for use in Excel . This tool permits BASIC computer important parameter is the mass of these wo components. The humidity ratio (W (W programming codes to be used to perform computations that are is used to express the mass of water vapor per unit mass of dry air and corresponds o the near right vertical axis of the psycumbersome with conventional spreadsheet equations. chrometric chart. Current practice is to This article describes a series of mac- added benefit of being appropriate appro priate for any use the units of mass of water to mass 1 ros that use psychrometric equations to elevation—not just sea level. The macros compute moist air properties (humidity also can be extended to spreadsheet pro- About the Authors ratio, dew point, enthalpy enthalpy,, specific vol- grams that compute the properties when teve Kavanaugh, Ph.D., is a professor of mehanical engineering at the University of Alabama ume, specific heat, relative humidity) two airstreams are mixed, an airstream by entering the dry-bulb temperature, passes through a cooling coil, a heat re- in Tuscaloosa, Ala. Barbara Hattemer McCrary is an engineer with Johnson, Spellman and Associwet-bulb temperature (RH), and local covery unit, or a heating coil. Engineers tes in Norcross, Ga., and a former grants-in-aid elevation. can cut and paste the necessary macros recipient at the University of Alabama. Keith A. The resulting spreadsheet is essentially from existing public domain programs or Woodbury is an associate professor of mechanical ngineering at the University of Alabama. an electronic psychrometric chart with the develop their own versions. 28
ASHRAE Journal
ashrae.org
January 2006
pw = 0.62198 p – p w
of air (lb (lbw /lba or or g g w /kg kg ). ). Some documents continue the use of grains per pound mass of air, where 7,000 grains = 1 pound (0.45 kg) mass. W = W =
M w M a
lbw lba
7,000
(
(4)
The atmospheric pressure can be cor rected for non-sea level elevations (Z (Z , in feet above sea level) as shown in the 2001 ASHRAE Handbook—Fundamentals 1
grains wlb × lbw lba
(1) The maximum amount of water vapor that can be mixed p(( psia) p psia) = 14.696 14.69 6 (1 – 6.8753 × 10 –6 Z)5.2559 (5) with air increases with temperature. RH i s the mole fraction (or In addition to the easily measured air dry-bulb temperature percent) of water vapor present in the air relative air relative to the mole ( t ), a second indicator is necessary to determine moist air fraction of air that is completely saturated with moisture at a iven temperature. This ratio is also the partial pressure of the properties. Options include the dew-point temperature (t ), (t wb), or RH. water vapor ( w) elative to the partial pressure p ressure of water vapor wet-bulb temperature (t The dew-point temperature can be determined by measuring when the air is saturated ( ws). the temperature of a surface when moisture begins to condense. pw w RH The dew-point temperature also corresponds to the saturation pws at t ws at t temperature or the temperature when RH is 100%. A correlation (2) (t °F) from 32°F to 200°F (0°C In lieu of steam tables to provide the value of p pws as a func- for dew-point temperature (t tion of temperature, Equation 3 is suggested for temperatures to 93°C) as a function of the partial pressure of water vapor ( w (2001 ASHRAE Handbook—Fundamentals): Handbook—Fundamentals):1 psia) is (2001 between 492°R (32°F or 0°C) and 852°R (392°F or 200°C).1 pws = Exp
C 8
( t + C + C
t + t +C
t 2
+
C 12t 3 +
C 13 ln t
(3)
where pws psia t °R C 8 = –1.0440397× 10 C 9 = –1.129465× 10 C 10 = –2.7022355 × 10 –2 C 11 = 1.2890360 × 10 –5 C 12 = –2.4780681 × 10 –9 C 13 = 6.5459673 The partial pressure of water vapor for unsaturated air ( w) can be found by combining Equations 2 and 3 if the relative humidity is known. Equation 4 is used to compute the humidity ratio (W (W from the local atmospheric pressure ( ) and p pw using the relationships of molecular weight MW (MW ), ), mole fractions ( ), and the partial pressures of water and air, M w MW × x w W = W = = M a MW × a January 2006
18.01528 × 28.9645 × x × xa
w
0.62198
pw p
t d 100.45 + 33.193(ln p 33.193(ln pw) + 2.319(ln p 2.319(ln pw)2 + 0.17074 (ln p (ln pw)3 + 1.2063( w)0.1984 (6) The air wet-bulb temperature is determined by placing a thermometer bulb that is i s covered with a completely wetted wick in an airstream. The evaporation rate and corresponding cooling effect noted by the depression of the wet bulb relative to the dry-bulb temperatures provides an indication of the moisture level in the air. When relative humidity is used as the second indicator, the value of p pw can be determined using Equatio n 2 with the value of p pws determined from Equation 3. When the dew-point tem perature is the second indicator, p indicator, pw is the saturation pressure at this dew-point temperature ( ws at t d ), which is also found using Equation 3. In either case, p case, pw is used in Equation 4 to determine the humidity ratio (W (W ). ). If the wet-bulb temperature (t wb) is the second indicator, the humidity ratio is found from 2001 ASHRAE Handbook—Fundamentals: Handbook—Fundamentals:
( )=
W
bw ba
(1,093 – 0.556t 0.556t wb ) W s at t wb 1,093 + 0.444t 0.444t – t wb ASHRAE Journal
– 0.24(t 0.24(t – – t t wb) (7) 29
W s in Equation 7 is determined by inserting the saturation pressure of water vapor at the wet-bulb temperature (t (t wb) into Equation 4. The value in Equation 7 is the thermodynamic wet-bulb temperature (also called the temperature of adiabatic saturation). For moist air, the wet-bulb temperature measured by the proper use of a psychrometer closely approximates the thermodynamic wet-bulb temperature. Equations for Moist Air Properties
The thermodynamic properties of moist air can be determined from the dry bulb temperature and humidity ratio. These include the enthalpy, specific volume (or its inverse, density) and specific heat. In the United States, the current convention is to set base values at 0°F (–18°C) and compute the values at other temperatures. At 0°F (–18°C), hw 1,061 Btu/lb (2468 kJ/kg) and h and ha 0 Btu/lb (0 kJ/kg). The specific heat of air is 0.24 Btu/lb °F [1 kJ/(kg · K)] and water vapor is 0.444 Btu/lb °F [1.9 kJ/(kg · K)]. For moist air at dry bulb temperature (t (t ) and humidity ratio (W (W ,, ·
·
h( Btu/lba) 0.24t + +W Btu/lba) = 0.24t
(1,061 + 0.444t )
(8)
The specific heat of moist air is: p Btu/lba – °F)
0.24 + 0.444W 0.444W
(9)
The specific volume of moist air is: ft 3
( lb ) = 0.37059t 0.37059t + + 459.67[1 + 1.6078W 1.6078W
] / p / p(psia) (10)
function name followed by a set of parenthesis containing the dry-bulb temperature, wet-bulb temperature, and elevation. For example, if a cell contains “=HumRat (80,67,0)”, the displayed value should be 0.0112, which is the humidity ratio in lbw /lba for air with a dry bulb of 80°F (27°C) and a wet bulb of 67°F (19°C) at sea l evel. Once the humidity ratio is known, enthalpy, enthalpy, specific heat, and specific volume can also be calculated using Equations 8, 9, and 10. Since these equations are relatively simple, the values can be computed with a formula in a spreadsheet cell. Function HumRat (db, wb, ElevInFt) onvert the wet bulb in °F to Rankine RT = wb + 459.67 Find the atmospheric pressure in psia from Equation 5 AtmPress = 14.696 × (1 – 0.0000068753 × ElevInFt)5.2559 Use Equation 3 to find the saturation pressure at wet-bulb temperature c8 = –10,440.397 c9 = –11.29465 c10 = –0.027022355 c11 = 0.00001289036 c12 = –0.000000002478068 c13 = 6.5459673 pws = Exp
(c8/RT + c
+ c10 × RT + c11 × RT2 + 12 × RT3 + c13 × Log(RT))
Use Equation 4 to find the saturated humidity ratio at the wet bulb-temperature wsat = (pws × 0.62198) / (AtmPress – pws) Use Equation 7 to find the humidity ratio HumRat = ( 1,093 – 0.556 × wb) × wsat - 0.24 × (db – wb) ) / 1,093 + 0.444 × db – wb ) End Function
Psychrometric Equations in Spreadsheet Macro Format Psychrometric Figure 1: Macro for humidity ratio from dry-bulb dry-bulb temperature, Previous articles in ASHRAE in ASHRAE Journal have Journal have alluded to the use wet-bulb temperature and elevation. of spreadsheet macros for com puting piping pressure drops 2 and solving for friction factors in air ducts.3 These articles emphasized another spreadsheet tool (Goal Seek) that was used to iteratively solve the implicit Colebrook equation for friction factor ( f ( f ). This additional tool is unnecessary since the computation of moist air properties is straightforward once the macro for humidity ratio is developed. Figure 1 is an example macro for computing humidity ratio from the dry-bulb temperature (t a), wet bulb (t (t wb), and elevation (feet above sea level). The macro is stored in an Excel module in the form of a function called HumRat. The function is used just like any other Excel function by clicking on a cell in the main spreadsheet and inserting an “=” sign and the Figure 2: Dew-point spreadsheet. Visual Basic Editor is accessed from a drop-do drop-down wn menu. 30
ASHRAE Journal
ashrae.org
January 2006
For enthalpy (Btu/lb): = 0.24 × 80 + HumRat (80,67,0) × (1,061 + 0.444 × 80) (11) For specific heat (Btu/lb – °F): = 0.24 + 0.444 × HumRat (80,67,0)
(12)
Note that Equation 10 requires the value of atmospheric pressure. This value was found inside the function HumRat, but it is not available since only one value can be passed from a function to the Excel spreadsheet. Therefore, atmospheric pressure must be calculated with Equation 5 in another macro or spreadsheet cell. However, the value of atmospheric pressure rather than elevation must be passed to the macro if it is calculated in a cell. Make Your Own Dew-Point Macro Equation 6 is used to demonstrate the steps required to develop a macro to compute dew-point temperature when the vapor pressure is known. Although this value could also be found with an equation equatio n in a spreadsheet cell, it serves as a simple example. Figure example. Figure 2 shows how the main spreadsheet might look. The value for saturated vapor pressure is entered in Cell B1 and an equation in Cell B2 calls the macro function TDP to compute Figure 3: Screenshot of inserting module in Visual Basic program. the dew-point temperature for the value in Cell B1. Function TDP(pw) Then develop the macro: Select “Tools” on the main toolbar; C 1 =100.45 C = 33.193 Select “Macros” on the first drop-down box; and C = 2.319 Select “Visual Basic Editor” in the second drop down C 4 = 0.17074 C = 1.2063 box ( Figure Figure 2). Alpha = Log(pw) This will bring up a screen as shown in in Figure 3 that has Handbook—Fundamentals s Equation 6—Also Equation 37, Chp. 6, 2001 ASHRAE Handbook—Fundamental a work box on the left side with the heading “Project-VBA TDP = C 1 + C 2 × Alpha + × (Alpha2) + 0.17074 × (Alpha3) + Project.” 1.2063 × pw0.1984 Select “This Workbook”; End Function Select “Insert” on the top toolbar; Figure 4: Macro for dew-point temperature for vapor pressure. Select “Module” on the drop-down box; Type the code (Equation 6) shown in in Figure 4 in the empty box that appears in the right portion of the screen; When the entry is completed, select “File” on the main toolbar; Save the spreadsheet (Note: You You must save the spreadsheet after any changes are made to the macro or the edited macro will not function correctly in the spreadsheet); and Select “File” again, and then “Close and Return to Microsoft Excel. Excel .” • • •
• • • •
•
•
•
Extending Psychrometric Psychrometric Spreadsheets to Mixed Air Processes These spreadsheets are essentially an electronic psychrometric chart and can be combined with the equations for moist air processes that appear in the 2001 ASHRAE Handbook— Fundamentals. Consider Equation 13, which computes the total capacity of a cooling coil (neglecting the small amount of energy in the condensate) as shown in Figure in Figure 5. 5. Known values are typically the airflow rate (Q (Q), the elevation, and the entering Figure 5: Air coil: total cooling. 4 January 2006
ASHRAE Journal
31
air dry-bulb and wet-bulb temperatures. The function HumRat is used to find the humidity ratio for the entering air (Point 1) and Equations 8 and 10 are used to compute the enthalpy (h (h1) and specific volume ( 1). For case where the outlet air dry-bulb and wet-bulb temperatures are known, HumRat is used to find the leaving air humidity ratio and Equation 8 i s used to find the leaving air enthalpy (h (h2).
) Summary Equations are available in the psychrometrics chapter of the 2001 ASHRAE Handbook—Fundame Handbook—Fundamentals ntalsto to determine moist air properties if the dry-bulb temperature, local pressure, and the humidity ratio are known. The pressure can be computed from elevations above sea level. The humidity ratio can be determined from the dry-bulb temperature and one of three other indicators: wet-bulb temperature, relative humidity or dew-point temperature. Visual Basic macros can be used to handle the somewhat cumbersome equations for determining humidity ratio. When completed, complete d, these macros can be combined with equations for moist air properties to develop an electronic psychrometric chart. The usefulness of this tool can be fur ther
extended to solving and analyzing HVAC moist air process problems. Note An objective of this article is to demonstrate the ease of adding functions to spreadsheet programs. The examples are applicable when the dry-bulb, wet-bulb, and dew-point tem peratures are above the freezing point of water. The ASHRAE publicationUnderstanding publication Understanding Psychr Psychrometrics ometrics5 includes additional below freezing equations. Also, the public domain spreadsheet program PsychProcess.xls is available at www.geokiss.com/ software/PsychProcess05.xls. It contains additional macros and moist air programs. References 1. 2001 ASHRAE Handbook—Fundamenta Handbook—Fundamentals, ls, Chapter 6, Psychrometrics. . Lester, T.G. T.G. 2002. “Calculating “Calc ulating pressure drops dro ps in piping systems.” systems .” ASHRAE Journal 44(9):41–43. Journal 44(9):41–43. 3. Lester, T.G. T.G. 2003. “Solving “S olving for friction fr iction factor.” fa ctor.” ASHRAE ASHRAE Journal 5(7):41–44. . Kavanaugh, S.P. 2006. HVAC Simplified. Atlanta: ASHRAE, forthcoming. 5. Gatley, D.P. 2005. Understanding Psychrometrics . Atlanta: ASHRAE.
Advertisement formerly in this space.
32
ASHRAE Journal
ashrae.org
January 2006
