Solar Engineering of Thermal Processes, 5th ed.

Index by Chapter then by equation or table number

Nomenclature

Chapter 1, Solar Radiation
Solar Time (E1.5.2)
Equation of time (E1.5.3)
Average day of the month (T1.6.1)
Number of days in a month (T1.6.1)
Day of the year given month and day of the month (T1.6.1)
Declination (E1.6.1)
Cosine of the incidence angle (E1.6.2)
Cosine of the zenith angle (E1.6.5)
Solar Azimuth angle (E1.6.6)
Sunset or sunrise hour angle (E1.6.10)
Cosine of the incidence angle for an east-west one axis tracker (E1.7.2a)
Cosine of the incidence angle for a north-south one axis tracker (E1.7.3a)
Cosine of the incidence angle for a vertical one axis tracker (E1.7.4a)
Ratio of beam radiation on an inclined surface to horizontal surface (E 1.8.1)
Instantaneous extraterrestrial radiation on a horizontal surface (E1.10.1)
Extraterrestrial radiation on a horizontal surface for one day (E1.10.3)
Extraterrestrial radiation on a horizontal surface in a time interval (E1.10.4)
Angles for solar radiation

Chapter 2, Available Solar Radiation
Ratio of hourly diffuse radiation to hourly total radiation using Erbs' correlation (E2.10.1)
Ratio of daily diffuse radiation to daily total radiation using Erbs' correlation (E2.11.1)
Ratio of monthly average diffuse radiation to monthly average total radiation using Erbs' correlation (E2.12.1)
Fraction of day's total radiation in a particular hour (E 2.13.2)
Fraction of day's diffuse radiation in a particular hour (E 2.13.4)
Hour's radiation on tilted surface using LJ (E 2.15.1), HD( E2.16.4) or HDKR (Eqn 2.16.7) methods
Monthly average ratio of beam radiation on a tilted surface to beam on a horizontal surface. (E2.19.3, 2.19.4 and 2.20.3)
Monthly average daily radiation on tilted surface using LJ (E2.19.1), KT (Eqn 2.20.4) or the K (Eqn 2.20.5) methods
Monthly average hourly radiation on sloped surface from monthly average daily radiation using the LJ model (E2.23.4)
Daily utilizability on a south facing surface (E2.23.5)
Monthly average daily utilizability on an equator facing tilted surface (E2.24.4)

Chapter 3, Selected Heat Transfer Topics
Fraction of black body radiation in wavelength interval (E3.6.3)
Sky temperature (E3.9.2)
Radiation heat transfer coefficient (E3.10.2)
Nusselt number for convection heat transfer between two flat plates (E3.11.4)
Heat transfer coefficient for flow inside a tube (E3.14.1-3.14.4)
Nusselt number for flow inside a tube (E3.14.1-3.14.4)
Heat transfer coefficient for free standing collectors due to wind (E3.15.1)
Heat transfer coefficient for flush mounted collectors (E3.15.10)
Pressure drop of air flowing across a packed bed of pebbles using Dunkle and Ellul (E3.16.5)
Effectiveness or Ntu of various heat exchanger configurations (E3.17.6)

Chapter 4, Radiation Characteristics of Opaque Materials
Ratio of solar absorptance at any incidence angle to absorptance at zero incidence (E4.11.1)
Effective absorptance of a cavity (E4.12.1)

Chapter 5, Radiation Transmission Through Glazing; Absorbed Radiation.
Transmittance of a "half" cover considering reflection at one interface and absorption in the cover (E5.1.1-4 & 5.2.2)
Transmittance of N covers for off-normal incident radiation (E5.3.4)
Incidence angle for ground reflected radiation (E 5.4.1)
Incidence angle for diffuse sky radiation (E 5.4.2)
Transmittance-absorptance product (E 5.5.1)
Absorbed hourly solar radiation on a sloped collector (Section 5.9)
Absorbed monthly average solar radiation on a sloped collector Section 5.10)

Chapter 6, Flat-Plate Collectors
Top loss coefficient using exact equations (Section 6.4)
Top Loss coefficient using exact equations for two covers.
Top loss coefficient using Klein Equation (E6.4.9)
Collector plate efficiency factor (E6.5.18)
Collector heat removal factor (E6.7.5)
Collector useful gain (E6.7.6)
Collector mean plate and fluid temperatures (E6.9.2 and E6.9.4)
Serpentine Collector heat removal factor (E6.13.1)
Flow rate correction factor (E6.20.3)

Chapter 10
Collector-tank heat exchanger correction factor (10.2.3)
Collector duct and pipe losses - absorbed radiation correction (E10.3.9 and E10.3.10)

Chapter 11
Present worth factor (E11.5.1)
Life cycle savings (E 11.8.1)
Economic factor P1 (E11.8.2)
Economic factor P2 (E11.8.3)
Partial derivative of present worth factor wrt period of analysis (E11.9.12)
Partial derivative of present worth factor wrt inflation rate (E11.9.13)
Partial derivative of present worth factor wrt discount rate (E11.9.14)

Chapter 20
fchart X and Y (Eqn 20.2.3 and 20.2.4)
Solar fraction for a liquid based solar heating system (E20.3.1)
Solar fraction for an air based solar heating system (E20.4.1)
Solar fraction for a solar domestic hot water system (E 20.5.1)

Chapter 21
PhiBarFChart (Equation 21.3.5)

Chapter 23
FindPV_ref_par (Section 23.2)

Chapter 24
Find c and k (E 24.2.9 and 24.2.10)













Nomenclature and conventions

Equations are found in the order in which they appear in the text:
Solar Engineering of Thermal Processes, 5th. Ed.
J. A. Duffie (deceased) and W. A. Beckman
J. Wiley and Sons, NY (2020)
ISBN 978-0-471-87366-3

All functions end with the underscore, "_" and have the form:
value = FunctionName_(par1, par2, .... , parN)

and all procedures end with P underscore, "P_" and have the form:
call ProcedureNameP_(par1, par2, .... , parN : result1, result2, ...., resultN)

Units:
All angles are in degrees.  Be sure you set EES units to one of the following:

$UnitSystem SI Mass Deg PA C or K
or
$UnitSystem ENG Mass Deg PSIA F or R

In a few instances, due to the dimensional nature of some empirical equations, temperature units may be other than expected from the EES settings.  Instantaneous solar energy quantities are in W, W/m^2, BTU/hr or Btu/hr-ft^2.  Integrated solar energy quantities (e.g., the energy per unit area for an hour or a day) are expressed without the time basis.  That is, the solar energy incident on a unit area surface for an hour or a day are both expressed as either MJ/m^2 or Btu/ft^2.  The time interval is built into the wording, such as the daily solar radiation is 1.23 MJ/m^2.

Names:
Function, procedure and variable names have been chosen to represent the physical quantity and are usually the same as the name used in the text. Slope is an exception; instead of using beta, we use slope. For variables that represent a ratio, the forward slash (divisor) is replaced by the reverse slash "\", so that I_d/I is replaced by I_d\I.  EES does not distinguish between upper and lower case names so delta and DELTA are the same variable.  However, delta will appear as d and DELTA will appear as D in the EES formatted equations window.

Errors:
Please report errors to wbeckman@wisc.edu.

Index


EqnTime_(nDay)

Usage
"Find the equation of time correction (in minutes) for the last day of the year"
nDay=365
DELTAtime=EqnTime_(nDay)


Reference
Equation 1.5.3: calculates the Equation of Time in minutes from the day number, nDay.  The equation of time is used to convert standard time to solar time.

Index


SolarTime_(LocalStdTime, GMT, Long_local, nDay)

Solar time is the time used in all of the sun-angle relationships; it does not coincide with local clock time.  It is necessary to convert standard time to solar time by applying two corrections.  First, there is a constant correction for the difference in longitude (also called meridian) between the observer's longitude and the longitude on which the local standard time is based. The sun takes 4 minutes to transverse 1 degree of longitude so each hour corresponds to 15 degrees of rotation.  The local standard longitude (L_st) can be found by multiplying GMT, the time difference between Greenwich Mean TIme and local standard time by 15 (GMT is between 0 and 24 hours for all locations).  The longitude is measured as the angle west from Greenwich, ranging from 0° at Greenwich to +360° westward so New York City (NYC) is at +40.8°.  Two other longitude conventions exist, all measured from Greenwich:   -180°  to +180° with minus measured towards the west so NYC is at -40.8° and 0 to +180 with E or W appended so that NYC is at 40.8°W.  A similar situation exists for Latitude where a location latitude could be between -90° and +90° or between 0° and 90° with N or S appended.

The second correction is E the equation of time (in minutes), which takes into account the perturbations in the earth's rate of rotation which affects the time the sun crosses the observer's meridian.  

The difference in minutes between solar time and standard time is found from Equation 1.5.2:

Solar time - Standard time = 4 (L_st - L_loc) + E

Usage
Example 1.5.1
LocalStdTime=10:30[hr]
GMT=6[hr]
Long_local=89.4[deg]
Long_std=15[deg/hr]*GMT
Month=2
day=13
nDay=nDay_(month, day)
Time_solar=SolarTime_(LocalStdTime, GMT, Long_local, nDay)

The solar time could also be found from:
E=EqnTime_(nDay)    
SolarTime=LocalStdTime+(4[min/deg]*(Long_std-Long_local)+E)*Convert(min, hr)


Reference
Equation  1.5.2

Index


AveDay_(month)

Usage
"Generate an array of average days for all 12 months."
Duplicate i=1,12
AveDay[i]=AveDay_(i)
end

Reference
Table 1.6.1:  Returns n for the average day of the month from the month number, 1-12.

The average day is the day that has the same extraterrestrial radiation as the monthly average extraterrestrial radiation.  

These average days do not hold at latitudes above the Arctic circle or below the Antarctic circle.

Index


NumDay_(month)

Usage
"Generate an array containing the number of days in each month."
Duplicate i=1,12
DayNumber[i]=NumDay_(i)
end

Reference
Table 1.6.1:  Returns the number of days in a month given the month number, 1-12.

Index


nDay_(month, day)

Usage
"Find the day of the year for July 4."
month=7
day=4
nDay=nDay_(month, day)

Reference
Table 1.6.1: Returns the day of the year (1<nDay<365) given the month and the day of the month.

Index


dec_(nDay)

Usage
"Find the declination for July 4."
month=7
day=4
nDay=nDay_(month, day)
delta=dec_(nDay)


Reference
Equation 1.6.1:  Calculates declination dec in degrees from day number n.

If n is positive Equation 1.6.1b is used and if nDay is negative Equation 1.6.1a is used.  The solutions in the text use Equation 1.6.1a but the solutions manual for the 4th edition uses Equation 1.6.1b.  All of the functions that call dec_ use a positive value for nDay.

Index


cosIncAng_(Lat, nDay, HrAng, Slope, SurfAzAng)

Usage
!Example 1.6.1
Month=2
day=13
nDay=nDay_(month, day)
Lat=43[deg]
slope=45[deg]
SurfAzAng=15[deg]
Time_Solar=10.5[hr]
hrAng=(Time_Solar-12[hr])*15[deg/hr]
Cos(theta)= CosIncAng_(Lat, nDay, HrAng, Slope, SurfAzAng)  

Reference
Equation 1.6.2:  Calculates the cosine of the angle of incidence of beam radiation on a surface. 

If the angle of incidence is greater than 90 degrees, i.e., if the beam radiation is not incident on the surface, the function will return CosIncAng = 0 (i.e., IncAng = 90).  

All angles are in degrees and nDay is the day number.

Index


Latitude

The angular location north or south of the equator, north positive and south negative.  The angle must be between -90 and + 90.

Index


hour angle

The angular displacement of the sun east or west of the local meridian due to rotation of the earth on its axis at 15 degrees per hour, mornings negative and afternoons positive.   The hour angle can be calculated from:

HrAng=15*(time-12.00)

where time is solar time expressed as the decimal portion of a 24 hour clock (i.e., 2:15pm is 14.25 hrs).

Index


slope

the angle between the plane of the surface in question and the horizontal. The slope is between 0 and 180 degrees.  For angles greater than 90, the surface is facing downward.

Index


surface azimuth angle

The deviation of the projection on a horizontal plane of the normal to the surface from the local meridian, with zero due south, east negative and west positive.  

Index


day of the year

The day of the year is a number between 1 and 365.  See function nDay

Index


declination

Declination is the angular position of the sun at solar noon (i.e., when the sun is on the local meridian) with respect to the plane of the equator, north positive; -23.45 ?d ?+23.45.  See function dec_.

Index


zenith angle

the angle between the vertical and the line to the sun, i.e., the angle of incidence of beam radiation on a horizontal surface.  See function CosZenAng.

Index


solar altitude angle

the angle between the horizontal and the line to the sun, i.e., the complement of the zenith angle.

Index


day length

the number of hours between sunrise and sunset.  The day length is found from

Day_length=(2/15)* SunsetHrAng(lat, nDat)

Index



solar azimuth angle

The angular displacement from south of the projection of beam radiation on the horizontal plane, shown in Figure 1.6.1. Displacements east of south are negative and west of south are positive.  See function SolAzAng.

Index



CosZenAng_(Lat, nDay, HrAng)

The zenith angle is the angle between the vertical and the line to the sun and is equal to the incidence angle for a horizontal surface.

Usage
"Find the zenith angle at 14:15 for July 4 at a latitude of 43 degrees."
Lat=43
month=7
day=4
time=14+15/60 "time must be decimal time"
nDay=nDay_(month, day)
HrAng=15*(time-12.00)
Cos(THETA_z)=CosZenAng_(Lat, nDay, HrAng)


Reference
Equation 1.6.5:  Calculates the cosine of the zenith angle.  Uses Eq. 1.6.1b for declination.  If the sun is below the horizon, i.e., if ZenAng is greater than 90 degrees, the function will return CosZenAng = 0 (or ZenAng = 90).  

Index


SolAzAng_(Lat, nDay, HrAng,)

The solar azimuth angle is the angular displacement from south of the projection of beam radiation on the horizontal plane, shown in Figure 1.6.1. Displacements east of south are negative and west of south are positive. 

Usage
"!Example 1.6.2a"
Lat=43
month_a=2
day_a=13
n_a=nDay_(month_a, day_a)
Time=9.5[hr]
hrAng_a=(Time-12[hr])*15[deg/hr]
gamma_a=SolAzAng_(Lat,n_a, HrAng_a)  

"!Example 1.6.2b"
month_b=7
day_b=1
n_b=nDay_(month_b, day_b)
Time_b=18.5[hr]
hrAng_b=(Time_b-12[hr])*15[deg/hr]
gamma_b=SolAzAng_(Lat,n_b, HrAng_b)


Reference
Equation 1.6.6

All angles are in degrees and n is the day number.

Index







SunsetHrAng_(Lat, nDay)

The sunset hour angle is the angular displacement of the sun east or west of the local meridian at the time of sunset.  The sunrise hour angle is the negative of the sunset hour angle.
Usage
"Determine sunrise on July 4 at a latitude of 43o."
Lat=43
Month=7
day=4
nDay=nDay_(month, day)
SunRise= -SunsetHrAng_(Lat, nDay)

Reference
Equation 1.6.10:  Calculates sunset hour angle (and thus the negative of sunrise hour angle).  

All angles are in degrees.

Index


CosIncAngEW_(nDay, HrAng)

Usage
"!Example 1.7.1"
Lat=40[deg]
HrAng_a=+30[deg]
delta=21[deg]
delta=dec_(nDay) "Since the declination is given we need to find nDay, the day of the year.  Set nDay limits 0<n<366."
Cos(Theta_a)=CosIncANgEW_(Lat, nDay, HrAng_a)

HrAng_b=100[deg]
Cos(Theta_b)=CosIncANgEW_(Lat, nDay, HrAng_b)

Reference
Equation 1.7.2a:  Calculates the cosine of the angle of incidence of beam radiation on a surface tracking the sun on a horizontal east-west axis.  

If beam radiation is not incident on the surface, the function will return CosIncAngEW = 0 (or IncAngEW = 90 degrees).  

All angles are in degrees and n is the day number.

Index


CosIncAngNS_(Lat, nDay, HrAng)

Usage
"Determine solar incidence angle at 2:15 pm on July 4 at a latitude 43o  of a north-south mounted tracking solar collector."
Lat=43[deg]
Month=7
day=4
nDay=nDay_(month, day)
time=14.25[hr]
HrAng=15*(time-12.00)
Cos(Theta)= CosIncAngNS_(Lat, nDay, HrAng)

Reference
Equation 1.7.3a:  Calculate the cosine of the angle if incidence of beam radiation on a surface tracking the sun on a horizontal north-south axis.  

If beam radiation is not incident on the surface, the function will return CosIncAngNS = 0 (or IncAngNS = 90 degrees).  

All angles are in degrees and nDay is the day number.

Index


CosIncAngVert_(Lat, nDay, HrAng, Slope)

Usage
"Determine the solar incidence angle at 2:15 pm on July 4 at a latitude 43o  for a surface with a slope of 45o that tracks on a vertical axis."
Lat=43[deg]
slope=45[deg]
Month=7
day=4
nDay=nDay_(month, day)
time=14.25[hr]
HrAng=15*(time-12.00)
Cos(Theta)= CosIncAngVert_(Lat, nDay, HrAng, slope)

Reference
Equation 1.7.4a:  Calculates angle of incidence of beam radiation on a surface with fixed slope tracking on a vertical axis.  Uses Fctn. Dec_ for Declination and  Fctn. cosZenAng_ for Zenith Angle.  

If beam radiation is not incident on the surface, the function will return CosIncAngNS = 0 (or IncAngNS = 90 degrees).  

All angles are in degrees and nDay is the day number.

Index


R_beam_(Lat,nDay,HrAng,Slope,SurfAzAng)

R_beam_p_(Lat,nDay,HrAng,Slope,SurfAzAng: IncAng,ZenAng,R_beam)

Usage
"!Example 1.8.1"
Month=2
day=13
nDay=nDay_(month, day)
Lat=43[deg]
slope=45[deg]
SurfAzAng=15[deg]
Time_Solar=10.5[hr] 
hrAng=(Time_Solar-12[hr])*15[deg/hr]
R_b=R_Beam_(Lat, nDay, HrAng, Slope, SurfAzAng)   

Call R_beam_P_(Lat, nDay, HrAng, Slope, SurfAzAng: IncAng, ZenAng, R_beam)

Reference
Equations, 1.8.2 and 1.8.3:  R_beam is  the ratio of beam radiation on a tilted surface to that on a horizontal surface.  Also see Example 2.14.1 for conditions around sunrise and sunset.

All angles are in degrees and nDat is the day number.


Index


G_zero_(nDay, Lat, HrAng)

Usage
"Compute the instantaneous extraterrestrial radiation at solar noon on a horizontal surface on July 12 at a latitude of 45 degrees.  "
Lat=45
month=7
day=12
omega=0
nDay=nDay_(month, day)
G_o=G_zero_(nDay, lat, omega)

Reference

Equation 1.10.1:  Calculates the extraterrestrial radiation G_Zero.  

All angles are in degrees, nDat is the day number and G_Zero is in W/m^2 or BTU/hr-ft^2.

Index


H_zero_(nDay, Lat)

Usage
"Determine the monthly average daily extraterrestrial radiation on an horizontal surface for all 12 months at a latitude of 45 degrees in W-hr/m^2-day"

Lat=45[deg]
Duplicate i=1,12
     n[i]=AveDay_(i)
     H_zero[i]=H_zero_(n[i], Lat)*Convert(MJ/m^2, W-hr/m^2)
end


Reference
Equation 1.10.3:  Calculates the day's extraterrestrial radiation H_Zero.  

All angles are in degrees, nDay is the day number and H_Zero is in MJ/m^2 or Btu/ft^2.

Index


H_bar_zero_(month, Lat)

Usage
"Determine the monthly average daily extraterrestrial radiation on an horizontal surface for all 12 months at a latitude of 45 degrees in W-hr/m^2-day."

Lat=45[deg]
Duplicate month=1,12
     H_bar_zero[month]=H_bar_zero_(month, Lat)*Convert(MJ/m^2, W-hr/m^2)
end

"See also Usage for function H_zero_"

Reference
Equation 1.10.3:  Calculates the day's extraterrestrial radiation for the average day of the month from Table 1.6.1 (and function AveDay_) .  H_zero evaluated at the monthly average day is equal to H_bar_zero.

All angles are in degrees and H_bar_Zero is in MJ/m^2 or Btu/m^2.  Note that the time interval of one day is implied.

Index



I_zero_(nDay,Lat,HrAng1,HrAng2)

Usage
"Determine the hourly extraterrestrial radiation for each hour on July 12 at a latitude of 45 degrees."
Lat=45
Month=7
Day=4
nDay=nDay_(month, Day)
Duplicate i=1, 24
     HrAng[i]=15*(i-12)-7.5
     I_o[i]=I_Zero_(nDay, Lat, HrAng[i]-7.5[deg], HrAng[i]+7.5[deg])
End

Reference
Equation 1.10.4:  Calculates an hour's extraterrestrial radiation I_Zero.   Hour Angles 1 and 2 are the beginning and end of the time of integration (usually one hour).  

If the hour includes sunrise or sunset, or if the whole hour is before sunrise or after sunset, the appropriate limits of integration are set to sunrise or sunset.

If I_zero is zero, it is set at an arbitrary value of 0.01.  This avoids "divide by zero" problems when using the function in simulation problems where hours before sunrise or after sunset are treated.

All angles are in degrees and nDay is the day number.  I_Zero is in MJ/m^2 or Btu/ft^2.  The time interval defined by the two hour angles is implied.

Index


Angles_p_(Lat,nDay,HrAng,Slope,SurfAzAng: Dec,CosIncAng,CosZenAng,R_beam,SunSetHrAng,DayLength)

Usage
"!Example 1.8.2."
Month=2
day=16
nDay=nDay_(month, day)
Lat=40[deg]
Slope=30[deg]
SurfAzAng=0[Deg]
HrAng=-15*2.5

CALL ANGLES_p_(Lat, nDay, HrAng, Slope, SurfAzAng : Dec, CosIncAng, CosZenAng, R_beam, SunSetHrAng, DayLength)


Reference
This procedure returns values of angles or cosines that are useful in checking intermediate results and providing the basis for radiation property determinations. 

Index

All angles are in degrees.


I_diff\I_(k_T)

Usage
"Estimate the hourly beam and diffuse solar radiation for Madison, WI (Lat=43) at 11:30 on July 4 when the total radiation for the hour was 3.1 MJ/m^2"

lat=43[deg]
month=7
day=4
nDay=nDay_(month,day)
HrAng1=-15[deg]
HrAng2=0[deg]
I=3.1[MJ/hr]
I_o=I_zero_(nDay,lat,HrAng1, HrAng2)
k_T=I/I_o
I_d=I*I_diff\I_(k_T)
I_b+I_d=I


Reference
Equation 2.10.1: Calculates the fraction of an hour`s total radiation that is diffuse, based on the Erbs et al. correlation.

Index


clearness index, K`_t, K_bar_t and k_t

The monthly average clearness index is the ratio of monthly average daily radiation on a horizontal surface to the monthly average daily extraterrestrial radiation.  In equation form,

   K_bar_T = H_bar/H_bar_zero

We can also define a daily clearness index as the ratio of a particular day's radiation to the extraterrestrial radiation for that day.  In equation form,

   K_T = H/H_zero  (Note K_T and k_T are the same variables in EES, consequently the grave accent, `,  is added to daily K_T` if it is necessary to distinguish it from the hourly k_T.)

An hourly clearness index can also be defined:

   k_t=I/I_zero

The data H_bar, H and I, are from measurements of total solar radiation on a horizontal surface, that is,  the commonly available pyranometer measurements.  Values of H_bar_zero, H_zero and I_zero can be calculated by the functions H_zero_ and I_zero_.  H_bar_zero is found from function H_zero_ with the day calculated from the function AveDay_.






H_diff\H_(Lat, n, K_T)

Usage
"!Example 2.11.1"
Lat=38.6[deg]
Month=9
Day=3
nDay=nDay_(month,day)
H=23[MJ/m^2]
H_o=H_zero_(n,Lat)
K_T=H/H_o
Hdiff\H=H_diff\H_(Lat, nDay, K_T)
H_d=H*Hdiff\H
H_b+H_d=H

Reference
Equation 2.11.1:  Calculates the fraction of a day`s radiation that is diffuse.  

Lat is in degrees.

Index



H_DIFFBAR\HBAR_(Lat, month, K_bar_T) 

Usage
"!Example 2.12.1"
Lat=43[deg]
H_bar=23.0[MJ/m^2]
Month=6
n=AveDay_(Month)
H_bar_o=H_bar_Zero_(month,Lat)  
K_bar_T=H_bar/H_bar_o   
HbarDiff\Hbar=H_DIFFBAR\HBAR_(Lat, month, K_bar_T)    

Reference
Equation 2.12.1:   Calculates a month`s diffuse fraction given the monthly average clearness index.  Lat is in degrees.

Index


r_t_(HrAng,Lat, nDay)


Usage

"!Example 2.13.2"  
lat=43[deg]
month=8
day=23
nDay=nDay_(month,day)
time=13.5[hr]
hrAng=15[deg/hr]*(time-12[hr])
H=31.4[MJ/m^2]
r_T=r_T_(hrAng, lat, nDay)
I=r_T*H

Reference
Equation 2.13.2a-c:   Calculates the fraction of a day`s total radiation that occurs in an hour centered at HrAng.  If the hour angle entered is not between sunrise and sunset, the function returns r_t=0.  

Lat is in degrees.  nDay is the day number.

Index


r_d_(HrAng,Lat, nDay)

Usage

"!Example 2.13.3" 
month=6
Lat=43.1
H_bar=23
time=10.5
nDay=AveDay_(month)
HrAng=15[deg/hr]*(time-12[hr])  
"The afternoon hour has the same hour angle only of different sign. The results for the two hours are exactly the same - that is, the day is symmetrical about solar noon."
H_bar_o=H_zero_(n, Lat)
K_bar_T=H_bar/H_bar_o
H_bar_d/H_bar=H_DIFFBAR\HBAR_(Lat, month, K_bar_T)
r_d=r_d_(hrAng, Lat, n)
r_t=r_t_(hrAng, Lat, n)
I_bar=r_t*H_bar
I_bar_d=r_d*H_bar_d
I_bar=I_bar_b+I_bar_d

Reference

Equation 2.13.4:  Calculates the fraction of a day`s diffuse radiation that occurs in an hour centered at HrAng.  If the hour angle entered is not between sunrise and sunset, the function returns r_d=0.

Lat is in degrees.  nDay is the day number.

Index


I_T_(Type$, mode, I_h, I_bn, Lat, nDay, Slope, SurfAzAng, GrRef, HrAng)

I_T_p_(Type$, mode, I_h, I_bn, Lat, nDay, Slope, SurfAzAng, GrRef, HrAng :    I_T, I_b , I_d,  I_g,  CosIncAng)

This function calculates the hour`s radiation on a sloped surface using one of three methods: the Liu and Jordan (Type$=`LJ`) , Hay and Davies (Type$=`HD`), or the Hay, Davies, Klucher, Reindl  (Type$=`HDKR`) method.  Type$ must be 'LJ', 'HD', or 'HDKR'. 

I_h and I_bn are the total horizontal and bean normal solar radiation.  Most data sets provide both of these values but sometimes only the total horizontal radiation is known.  If mode=1 then only the total horizontal solar radiation I_h is used and the beam and diffuse components are estimated from Equation 2.10.1, the Erbs correlation using function I_diff\I.  I_bn can be set to anything.  
If mode=2 then both he total horizontal solar radiation I_h and bean normal solar radiation I_bn are used.

HrAng is the hour angle for the mid-point of the hour.   All angles are in degrees. Radiation is in MJ/m^2 or Btu/t^2.  The time interval of one hour is implied.

Usage

"!Example 2.15.1 and 2.16.1 (using the Liu and Jordan model, the Hay-Davies model and again using the Hay-Davies-Klucher-Reindl model model"
Type1$='LJ'
Type2$='HD'
Type3$='HDKR''
month=2
day=20
nDay=nDay_(month, day)
Time=9.5[hr]
HrAng=15[deg/hr]*(time-12[hr])
Lat=40[deg]
slope=60[deg]
rho_g=0.6
SurfAzAng=0[deg]
I_h=1.04[MJ/m^2]
mode=1
I_bn=1000[MJ/m^2]
I_T_LJ=I_T_(Type1$, mode, I_h, I_bn, Lat, nDay,Slope, SurfAzAng, rho_g, HrAng)    
I_T_HD=I_T_(Type2$, mode, I_h, I_bn, Lat, nDay,Slope, SurfAzAng, rho_g, HrAng)   
I_T_HDKR=I_T_(Type3$, mode, I_h, I_bn, Lat, nDay,Slope, SurfAzAng, rho_g, HrAng)   

"Alternatively, we can call the procedure"
{Call I_T_p_(Type1$, mode, I_h, I_bn, Lat, nDay, Slope, SurfAzAng, rho_g, HrAng :  I_T`_LJ,  I_b_LJ , I_d_LJ,  I_g_LJ,  CosIncAng_LJ)
Call I_T_p_(Type2$, mode, I_h, I_bn, Lat, nDay, Slope, SurfAzAng, rho_g, HrAng :  I_T`_HD,  I_b_HD , I_d_HD,  I_g_HD,  CosIncAng_HD)
Call I_T_p_(Type3$, mode, I_h, I_bn, Lat, nDay, Slope, SurfAzAng, rho_g, HrAng :  I_T`_HDKR,  I_b_HDKR , I_d_HDKR,  I_g_HDKR,  CosIncAng_HDKR) 
}

Reference
Equation 2.15.1 (Liu and Jordan), 2.16.4 (Hay and Davies) and 2.16.7 (Hay, Davies, Klucher and Reindl). HrAng is the hour angle for the mid-point of the hour.  

All angles are in degrees.  I is in MJ/m^2.  nDay is the day number.


Index


Ground reflectance

The average diffuse reflectance of solar radiation incident on the surface in front of the receiver.  Typical values range from 0.7 for snow cover to 0.2 for grass.


Solar radiation

Solar radiation is short wave radiation that originates from the sun and is in the wavelength range of 0.3 to 3 mm.  The solar radiation that reaches the ground consists of beam radiation (the portion that casts a shadow) and diffuse radiation (the portion that is scattered in the atmosphere by dust, water vapor, clouds etc.).  

H is the daily total radiation in MJ/m^2 or Btu/ft^2.
H_bar is the monthly average daily total radiation in MJ/m^2 or Btu/^2.

I is the total radiation in a time period (almost always 1 hour) in MJ/m^2 or Btu/ft^2.
I_bar is the monthly average radiation in a time period (almost always 1 hour) in MJ/m^2 or Btu/ft^2..
I and I_bar are sometimes considered to be energy rates per unit area (power density) in MJ/m^2-hr or Btu/ft^2..

G and G_bar are the solar power density in (usually) W/m^2 or Btu/hr-ft^2.

Subscript T represents radiation on a tilted surface, H_bar_b_T is thus the monthly average beam solar radiation on a tilted surface..  

Subscripts b, d and g represent beam, diffuse and ground reflected radiation.  The subscript o (zero) represents extraterrestrial radiation.

The time units for I (per hour) and H (per day) are not provided but are implied.  The reason is that if all hourly radiation I (in MJ/m^2-hr) were added up in EES to obtain the daily radiation H, the units of H would be MJ/m^2-hr and not MJ/m^2-day. To avoid units inconsistency time is implied for H and I.


I_bar_T_(H_bar, Lat, Month, HrAng, Slope, SurfAzAng, GrRef)

I_bar_T_p_(H_bar, Lat, Month, HrAng, Slope, SurfAzAng, GrRef : I_bar_T, I_bar_Tbeam I_bar_Tdiff, I_bar_Tgrref)

These functions estimate the monthly average hourly radiation on a sloped surface at a location where the monthly average daily radiation is H_bar.

H_bar is the hour's total global horizontal solar radiation.   I_bar_T, the total hour's radiation on the sloped surface.  The Erbs correlation for monthly average diffuse radiation (Eqn 2.12.1) is used to estimate the monthly diffuse fractlion.   All angles are in degrees.  H_bar and I_bar_T are in either MJ/m^2 or  BTU/ft^2.  The time interval of one hour is implied.

Usage

"Example 2.23.2 (radiation only)"
Lat=40[deg]
slope=40[deg]
SurfAzAng=0[deg]
K_bar_T=0.5
GrRef=0.7
HrAng=7.5[deg]
month=2
I_bar_T1=I_bar_T_(H_bar, Lat, month, HrAng, slope, SurfAzAng, GrRef)

Call I_bar_T_p_(H_bar, Lat, month, HrAng, slope, SurfAzAng, GrRef: I_bar_T2,  I_bar_T_b,  I_bar_T_d,  I_bar_T_g)

Reference
Equation 2.23.4 calculates monthly average hourly radiation on a sloped surface at a location where the monthly average daily radiation is H_bar.

I_bar_T is in MJ/m^2 or Btu.ft^2.  The time interval of one hour is implied. 

Index


R_bar_b_(Type$, month, Lat, slope, azimuth, K_bar_T)

If Type$ is equal to either  'LJ'  or 'KT' R_bar_beam finds the monthly average ratio of beam radiation on a horizontal surface to that on a south facing surface (in the northern hemisphere) or a north facing surface (in the southern hemisphere).  

If Type$='K' then the surface does not need to face the equator and R_bar_beam is calculated numerically.    

Usage
"Examples 2.19.1 and 2.19.2"

{Array H_bar} "TMY2"
H_bar[1]=6.44[MJ/m^2];  H_bar[2]=9.89[MJ/m^2]
H_bar[3]=12.86[MJ/m^2];  H_bar[4]=16.05[MJ/m^2]
H_bar[5]=21.36[MJ/m^2];  H_bar[6]=23.041[MJ/m^2]
H_bar[7]=22.58[MJ/m^2];  H_bar[8]=20.33[MJ/m^2]
H_bar[9]=14.59[MJ/m^2];  H_bar[10]=10.48[MJ/m^2]
H_bar[11]=6.37[MJ/m^2];  H_bar[12]=5.74[MJ/m^2]
{Array H_bar end}

Type1$='LJ'
Type2$='KT'
Type3$='K'
Azimuth=0[deg]
Lat=43[deg]
slope=60[deg]

Duplicate i=1,12
month[i]=i
H_bar_o[i]=H_bar_Zero_(month[i],Lat)     
K_bar_T[i]=H_bar[i]/H_bar_o[i]
R_bar_b_LJ[i]=R_BAR_b_(Type1$, month[i], Lat, Slope, Azimuth, K_bar_T[i]) 
R_bar_b_KT[i]=R_BAR_b_(Type2$, month[i], Lat, Slope, Azimuth, K_bar_T[i])
R_bar_b_K[i]=R_BAR_b_(Type3$, month[i], Lat, Slope, Azimuth, K_bar_T[i])
End


Reference
R_bar_b is the monthly average ratio of beam radiation on the sloped surface to that on a horizontal surface.   When Type$ is equal to 'LJ' Equations 2.19.3 and 2.19.4 are used. If Type$ is equal to 'KT' then Equations 2.20.4a-d are used  If Type$='K' then the surface can face any direction and a numerical method is used to find R_bar_b.  For surfaces facing the equator choosing Type$='KT' or Type$='K' will produce exactly the same results and will be equivalent to using Equation 2.20.5a-i.

All angles are in degrees.  

Index


H_bar_T_(Type$, Month, H_bar, Lat,  Slope, Azimuth, GrRef)

H_bar_T_p_(Type$, Month, H_bar, Lat,  Slope, Azimuth, GrRef, : H_bar_T, H_bar_T_b, H_bar_T_d, H_bar_T_g)

Given the monthly average daily radiation on a horizontal surface, H_bar, these functions calculate the monthly average radiation on a sloped surface using the  Liu-Jordan method (Equation 2.19.1, Type$='LJ') or the  Klein, Theilacker  (Equation 2.20.4, Type$=`KT`) method or a numerical method (Type$='K').  If Type$ is 'LJ' or 'KT' then this function only works for surfaces sloped toward the equator.  If Type$=`K` then R_bar_beam is calculated numerically and works for all orientations.  

H_bar is the total global horizontal solar radiation in either MJ/m^2 or  BTU/ft^2 and H_bar_T, the total radiation on the sloped surface.  The time interval of one day is implied.  The Erbs correlation for monthly average diffuse radiation (Eqn 2.12.1) is used to estimate the diffuse component.   All angles are in degrees.  

Usage

"!Examples 2.19.1and 2.20.1"
{Array H_bar} "TMY2"
H_bar[1]=6.44[MJ/m^2];  H_bar[2]=9.89[MJ/m^2]
H_bar[3]=12.86[MJ/m^2];  H_bar[4]=16.05[MJ/m^2]
H_bar[5]=21.36[MJ/m^2];  H_bar[6]=23.041[MJ/m^2]
H_bar[7]=22.58[MJ/m^2];  H_bar[8]=20.33[MJ/m^2]
H_bar[9]=14.59[MJ/m^2];  H_bar[10]=10.48[MJ/m^2]
H_bar[11]=6.37[MJ/m^2];  H_bar[12]=5.74[MJ/m^2]
{Array H_bar end}

{Array rho_g}
rho_g[1]=.7;  rho_g[2]=.7
rho_g[3]=.4;  rho_g[4]=.2
rho_g[5]=.2;  rho_g[6]=.2
rho_g[7]=.2; rho_g[8]=.2
rho_g[9]=.2;  rho_g[10]=.2
rho_g[11]=.2;  rho_g[12]=.4
{Array rho_g end}

Type1$='LJ'
Type2$='KT'
Type3$='K'
Azimuth=0[deg]
Lat=43[deg]
slope=60[deg]

Duplicate i=1,12
month[i]=i
H_bar_o[i]=H_bar_Zero_(month[i],Lat)     
K_bar_T[i]=H_bar[i]/H_bar_o[i]
H_bar_T_LJ[i]=H_BAR_T_(Type1$, month[i], H_bar[i], Lat, Slope, Azimuth, rho_g[i])
H_bar_T_KT[i]=H_BAR_T_(Type2$, month[i], H_bar[i], Lat, Slope, Azimuth, rho_g[i])
Call H_BAR_T_p_(Type3$, month[i], H_bar[i], Lat, Slope, Azimuth, rho_g[i] : H_bar_T_K[i] H_bar_T_b_K[i],  H_bar_T_d_K[i], H_bar_T_g_K[i])
End



Reference
Equation 2.19.1 and 2.20.1:  Calculates monthly average daily radiation on a south facing sloped surface using the Liu and Jordan or the Klein and Theilacker method.  If the surface were not facing the equator then the 'K' method must be used.  Note that for this south facing surface the 'KT' method and the 'K' method produce exactly the same results.

All angles are in degrees.  H is in MJ/m^2 or Btu/ft^2.  THe time interval of one day is implied.

Index




Phi_(H_bar, I_T_c, Lat, month, HrAng, slope, SurfAzAng, rho_g)

Usage
"!Example 2.23.3"
Lat=40
SurfAzAng=0[deg]
rho_g=0.7
slope=40
month=2
K_bar_t=0.5
hr=1[hr]
I_T_c=1.28[MJ/m^2]/hr*Convert(MJ/m^2-hr, W/m^2)
HrAng=-7.5[deg]
H_bar_zero=H_bar_zero_(month, lat)
K_bar_T=H_bar/H_bar_zero
phi=phi_(H_bar, I_T_c,  Lat, Month, HrAng, Slope, SurfAzAng, rho_g)

Result: phi=0.52


Reference
Equations 2.23.5 a-c are used to estimate the monthly average hourly utilizability on a sloped surface for an hour given the monthly average horizontal radiation.

All angles are in degrees, H_bar is in MJ/m^2 or Btu/ft^2.  The time interval of one day is implied.  The critical radiation level I_T_c is in W/m^2 or Btu/hr-ft^2.

Index


Phi_bar_(H_bar, month, Lat, Slope, GrRef, I_Tc)

Phi_bar_p_(H_bar, month, Lat, Slope, GrRef, I_Tc : K_bar_T, R_bar, R_beam_n, R_noon, X_bar_c, Phi_bar)

Usage
""!Example 2.24.1"

Lat=43[deg]
slope=60[deg]
SurfAzAng=0[deg]
K_bar_T=0.49
rho_g=0.4
HrAng=7.5[deg]
month=3
n=AveDay_(month)
I_Tc=145[W/m^2]
H_bar=12.86[MJ/m^2]
H_o=H_zero_(n, Lat)
phi_bar=PHI_BAR_(H_bar, month, Lat, Slope, rho_g, I_Tc)
H_bar_T=H_BAR_T_('LJ', month, H_bar, Lat, Slope, Azimuth, rho_g)
Ndom=numDay_(month)
UE=H_bar_T*Ndom*phi_bar

Call PHI_BAR_p_(H_bar, month, Lat, Slope, rho_g, I_Tc : K`_bar_T, R_bar, R_beam_n, R_noon, X_bar_c, Phi`_bar)


Reference
Equation 2.24.4a:  Returns Phi_Bar, the daily utilizability (i.e., the fraction of the monthly average daily energy above the critical level, I_Tc).  The procedure also returns values of intermediate variables used in the calculation.  R_noon is calculated from Eq. 2.24.2, and X_bar_c from Eq. 2.24.3. 

The monthly radiation on a tilted surface H_bar_T is calculated using the Liu & Jordan method and so is restricted to surfaces sloped toward the equator.  If the surface did not face the equator then the 'K' method should be used to find H_bar_T.

All angles are in degrees.  H is in MJ/m2 and I is in W/m2.


Index


Critical Radiation

Solar radiation above a specified level, called the critical radiation level,  is called "utilizable" energy.  The energy below the critical radiation level is called the "unutilizable" energy.  The concept is useful in the evaluation of solar systems (collectors, passive houses, photovoltaics, etc.).  The fraction of the monthly average energy above the critical level is phi for an hour and phi_bar for a day. 

The dimensionless critical level (Eqn 2.24.3) used in the calculation of daily utilizability is the ratio of the critical radiation to the monthly average noontime radiation.


BlBodyFract_(Wavelength*Temperature)

Usage

"!Example 3.6.1"
"Solve (F4) by maximizing E_lambda_b wrt lambda"
T=5777[K]
E_lambda_b=C1#/(lambda^5*(exp(C2#/(lambda*T))-1))
LT=Lambda*T
Lambda_2=0.78[micrometer]
Lambda_1=0.38[micrometer]

f_2=BlBodyFract_(Lambda_2*T)
f_1=BlBodyFract_(Lambda_1*T)
DELTAf=f_2-f_1

Reference
Equation 3.6.3:   Calculates the fraction of blackbody radiation at wavelengths less than Wavelength at a surface Temperature  based on the polynomial approximations to Equation 3.6.3, i.e.,  3.6.4 and 3.6.5.

The product, Wavelength*Temperature, has units of micrometers-K.

Index


SkyTemp_(T_amb, T_dp, HrAng)

Usage

"Example of sky temperature at noon"
T_hot=35[C]
T_dp_hot=30[C]

T_cold=-20[C]
T_dp_cold=-10[C]
HrAng=0[deg]

T_sky_hot=SkyTemp_(T_hot,T_dp_hot, HrAng) 
T_sky_cold=SkyTemp_(T_cold,T_dp_cold, HrAng) 

Reference
Equation 3.9.2:  Calculates an equivalent black-body clear-sky  temperature for use in radiation calculations given the ambient temperature, T_amb  and the dew point temperature, T_dp.  

SKYTEMP will be returned in C, K, F or R depending upon the units chosen for EES.  The ambient and dew point temperatures used as inputs must be in  K, C, F or R, depending upon the EES units settings.    

Index


RadHtTrCoef_(T1, T2, epsilon_1, epsilon_2)

Usage
"!Example 3.10.1"
{The EES units setting must be temperature in K}
$UnitSystem K
T_c=323[K]
T_p=343[K]
epsilon_p=0.15
epsilon_c=0.88
sigma=sigma#
q=sigma*(T_p^4-T_c^4)/(1/epsilon_p+1/epsilon_c-1)
h_r=RadHtTrCoeff_(T_p, T_c, epsilon_p, epsilon_c)
q_linear=h_r*(T_p-T_C)
{Note that the calculated values of q are exactly the same.}

Reference
Equation 3.10.2:  Calculates radiation heat transfer coefficient from surface 2 to surface 1 with F12=1.  For large parallel plates (i.e., flat-plate collectors), temperatures and emittances are those of plate and covers.  

For radiation from a body to a cavity (i.e., collector to surroundings), set Emitt1 to unity.

Heat transfer Q is RadHtTrCoeff*Area*(T2-T1). 

T1 and T2 are in C or K, depending upon the units chosen.   RadHtTrCoeff is in W/m^2-C.

This function is of little use in EES since EES can easily handle equation 3.8.3.


Index


NuFlatPlate_(Fluid$, Slope, Pressure, T_lower,T_upper, PlateSpace, PlateLength, PlateWidth )

Usage

"!Example 3.11.1"
$UnitSystem C
PlateWidth=2[m]; PlateSpace=0.025[m]; FLuid$='Air'; PlateLength=10[m]; Pressure=po#
Slope=45[deg]
T_h=70[C]; T_c=50[C]; T_bar=(T_h+T_c)/2
k=Conductivity(Fluid$, T=T_bar)
Nusselt=NuFlatPlate_(Fluid$, Slope, Pressure, T_h,T_c, PlateSpace, PlateLength, PlateWidth )
htc=Nusselt*k/PlateSpace


Reference
Equation 3.11.4:  Temperatures can be C, K, F or R, depending upon the EES unit settings.  
 
For angles >70 deg, the correlation is modified according to  Catton, Proc. 6th Int. Heat Transfer Conf., 1978, Vol. 6, pp.13-31 and Arnold, Catton and Edwards, ASME Paper 75-HT-62, 1975 as described in section 6.2.6 of Nellis and Klein.  
 
The ratio PlateWidth to PlateSpace is assumed to be > 12.  Also, PlateLength/PlateSpace should be >12 for the correlation to be applicable..  

Slope is in degrees.  Plate dimensions are in meters or feet.

This function calls the function Tilted_Rect_Enclosure in the EES library FreeConvection.lib.

Index


hfi_(m_dot, D, L, Fluid$, T, P)

Usage
hfi is the fluid heat transfer coefficient inside a constant temperature tube.  
"!Example 3.14.1"
T=80 [C]
p=101.3 [kPa]
W=1.5[m]
L=3[m]
D=0.010[m]
spacing=0.1[m]
n_tubes=W/spacing
m_dot=0.075[kg/s]
m_dot_tube=m_dot/n_tubes 
h_fi=hfi_(m_dot_tube, D, L, 'water', T, P)  

Reference
This procedure calls Procedure PipeFlow in the EES convection library and reports the constant wall temperature Nusselt number.  Procedure PipeFlow( Fluid$, T, P, m_dot, D, L,RelRough: h_T, h_H, DELTAP, Nusselt_T, f, Re) returns lower and upper bounds on the average heat transfer coefficient and the pressure drop for a specified mass flow rate (m_dot) through a circular tube of diameter D and length L. The procedure assumes simultaneous hydrodynamic and thermally developing flow. Properties are evaluated at the bulk temperature T and pressure P. Units of the inputs and outputs depend upon the unit settings in EES. The tube is assumed to be smooth with a relative roughness of 0.00001. 
Inputs:
Fluid$ can be any fluid in the EES database. The fluid can be an ideal gas or a real fluid. 
T - the bulk temperature of the fluid in [C], [K], [F], or [R]. 
P - pressure can be in [Pa], [kPa], [bar], [MPa], [atm], or [psia]. 
m_dot - mass flow rate in [kg/s], or [lbm/hr]
D - diameter of the tube in [m] or [ft]
L - length of the tube in [m] or [ft]
Output:
hfi_ - heat transfer coefficient in [W/m^2-K] or [Btu/hr-ft^2-R] assuming that the pipe wall is at constant temperature

See the function PipeFlow for more options.


Index


PipeFlow_(Fluid$, T_bulk, P, m_dot, D, L, RelRough : h_T, h_H, DELTAP, Nusselt_T,  f, Re)


Usage

"!Example 3.14.1"
T=80 [C]
p=101.3 [kPa]
W=1.5[ m]
L=3 [m]
D=0.010 [m]
spacing=0.1 [m]
n_tubes=W/spacing
m_dot=0.075 [kg/s]
m_dot_tube=m_dot/n_tubes 
RelRough=0.00001
call PipeFlow(Fluid$, T, P, m_dot_tube, D, L, RelRough : h_T, h_H, DELTAP, Nusselt_T,  f, Re)


Reference

The built-in EES Heat Transfer library must be loaded for this procedure to work.

Procedure PipeFlow( Fluid$, T, P, m_dot, D, L,RelRough: h_T, h_H, DELTAP, Nusselt_T,  f, Re) returns lower and upper bounds on the average heat transfer coefficient and the pressure drop for a specified mass flow rate (m_dot) through a circular tube of diameter D and length L.  The procedure assumes simultaneous hydrodynamic and thermally developing flow as reported in Section 5.2.3 of Nellis and Klein.  Properties are evaluated at the bulk temperature T and pressure P.  Units of the inputs and outputs depend upon the unit settings in EES. 

Inputs:
Fluid$ can be any fluid in the EES database.  The fluid can be an ideal gas or a real fluid.  
T - the bulk temperature of the fluid  in  [C], [K], [F], or [R]. 
P -  pressure can be in [Pa], [kPa], [bar], [MPa], [atm], or [psia].  
m_dot  -  mass flow rate in [kg/s], or [lbm/hr]
D -  diameter of the tube in [m] or [ft]
L - length of the tube in [m] or [ft]
RelRough - the ratio of the dispersions on the wall of the tube to the tube diameter (must be between 0 and 0.05)

Outputs (all but the first output are optional):

h_T -  heat transfer coefficient in [W/m^2-K] or [Btu/hr-ft^2-R] assuming that the pipe wall is at constant temperature (lower bound)
h_H -  heat transfer coefficient in [W/m^2-K] or [Btu/hr-ft^2-R] assuming a constant heat flux at the pipe wall (upper bound)
DELTAP - pressure difference between the inlet and outlet of the pipe in the pressure units set in the EES Unit System dialog
Nusselt_T - Nusselt number (determined for a constant temperature wall) [-]
f -  friction factor [-] 
Re - Reynolds number [-]

Two values of the heat transfer coefficient are returned.  The first, h_T is determined assuming that the wall is at constant temperature.  The second is determined assuming that the wall is subjected to a constant heat flux.  For laminar flow, these values should provide lower and upper bounds on the heat transfer coefficient.  For turbulent flow, these values are identical.

The procedure will determine if the flow is laminar or turbulent.  Transitional flow is assumed to occur for Reynold's numbers between 2300 and 3000 and interpolation is applied between the laminar and turbulent correlations.  The ratio of L/D is used to apply a developing flow correction based on simultaneous hydrodynamic and thermal development; set L to a large number if developing flow corrections are not applicable.

Index


Reynolds number

The Reynolds number is the ratio of momentum forces to viscous forces.  For flow in a tube it is defined as rVD/m, where r is the fluid density, V is the average velocity, D is the hydraulic diameter and m is the viscosity.  


Prandtl number

The Prandtl number is cp m/k where cp is the constant pressure specific heat, m is the viscosity and k is the thermal conductivity.


WindCoeff`FS_(WindSp, Length, Width, T_amb)

Usage
"Estimate the wind heat transfer coefficient for a 10 mph wind blowing over a collector of length 6 feet and width 2.5 feet.  The ambient temperature is 70 F."
T=ConvertTemp('F', 'C', 70)
V=10*convert(mph, m/s)
L=6*Convert(ft,m)
w=2.5*Convert(ft,m)
h_w=WindCoeff`FS_(V, L, W, T)


Reference
Equation 3.15.1:  Calculates a wind (convection) coefficient for Free Standing collectors, using the Sparrow correlation.   Note that for some circumstances, the Reynolds number exceeds that for which the correlation was developed, but there may be no better alternative (except possibly Equation 3.15.3).

T_amb is in C, K, F or R.  Length and width are in meters or feet.  WindSp is in m/s or miles/hour.  The minimum value of the wind coefficient is 5 W/m^2-K or 0.8806 Btu/hr-ft^2-R.


Index


Nusselt number

The Nusselt number is a dimensionless heat transfer coefficient, defined as hL/k where h is the heat transfer coefficient, L is the characteristic length (hydraulic diameter for the inside of a tube) and k is the fluid thermal conductivity.


WindCoeff`FM_(WindSp, BldgVol)

Usage
"Estimate the heat transfer coefficient for a building with length, width and height of 30, 20, 15 meters in a 5.5 m/s wind."
V=5.5[m/s]
L=30[m]
W=20
H=15
Vol_bldg=L*W*H
h_w=WindCoeff`FM_(V, Vol_bldg)

Reference
Equation 3.15.10:  Calculates a wind (convection) coefficient for collectors Flush Mounted on a building surface.

WindSp is in m/s or mph.  BldgVol is in m^3 or ft^3.  The minimum value of the wind coefficient is 5 W/m^2-K or 0.88 Btu/hr-ft^2-R.}


Index


PresDrop_(Length, MassVel, PartDia, T)


Usage
"!Example 3.16.1."

L=4[m]
W=3.7[m]
A=L*W
D=2.1[m]
Dia=0.235[m]
Vel=0.143[m/s]
void=0.41
T=40[C]
rho=density('air', T=T, p=101.3)
G_o=rho*Vel
DELATp=PresDrop(D, G_o, Dia, T)

Reference
Equation 3.16.5:  Calculates the pressure drop in Pa across a packed bed energy store with the Dunkle and Ellul correlation.  

Length is the bed length in the flow direction in feet or meters.  MassVel is the mass flow rate per unit area in lb_m/hr-ft^2 or kg/s-m^2.  PartDiam is the particle diameter in feet or meters.  T is the average bed temperature in R, F, K or C, depending upon the EES units setting.


Index


HX_(Type$, P, C_1, C_2, Return$

Usage

"!Example 3.17.1"
Type$='CounterFlow'
UA=6500[W/C]
T_h_in=62[C]
T_c_in=35[C]
m_dot_glycol=1.25[kg/s]
m_dot_water=0.864[kg/s]
c_p_glycol=3850[J/kg-C]
c_p_water=4190[J/kg-C]
C_h=m_dot_glycol*c_p_glycol
C_c=m_dot_water*c_p_water
Ntu=UA/min(C_h, C_C)
Return$='epsilon'
epsilon=HX_(Type$, Ntu, C_h, C_c, Return$)
Q=epsilon*Q_max
Q_max=min(C_h, C_c)*(T_h_in-T_c_in)
Q=C_h*(T_h_in-T_h_out)
Q=C_c*(T_c_out-T_c_in)


Reference

Equations from Introduction to Heat Transfer by Incropera and DeWitt.  When Return$='epsilon' and Type$ = 'counterflow' then the function solves Equation 3.17.7.
HX_ returns either the heat exchanger effectiveness (epsilon) given the number of transfer units (Ntu) or it returns Ntu given the effectiveness.

The general form is either:
Return$='epsilon'
epsilon=HX_(TypeHX$, Ntu, C_1, C_2 , Return$)
or
Return$='NTU'
Ntu=HX_(TypeHX$, epsilon, C_1, C_2 , Return$)
 
If epsilon is known, then it is more efficient to set Return$='Ntu'.  If Ntu is known, then it is more efficient to set Return$='epsilon'.  There is always a solution for epsilon given any valid Ntu but there may not be a solution for Ntu given any valid epsilon.
 
Capacitance rates, C1 and C2 :
The product of the mass flow rate (e.g., kg/s or lbm/hr) and the fluid specific heat (e.g., J/kg-K or Btu/lbm-R) is the capacitance rate. The two capacitance rates are C_1 and C_2.  In most cases it does not matter which fluid is C_1 or C_2.  However, for a cross-flow heat exchanger with one fluid mixed, C_1 must be the unmixed fluid.  If one fluid is at a constant temperature (e.g., an evaporator or condenser), that fluid capacitance rate should be set to a large number compared to the other fluid (e.g., 1000 times bigger is usually sufficient).  
 
Heat Exchanger types:
The heat exchanger flow arrangement is specified with the parameter TypeHX$.  The following are acceptable values for TypeHX$:
                  'parallelflow'
                  'counterflow'
                  'crossflow_both_unmixed'        {HX_ cannot directly solve for Ntu given epsilon for this case.}
                  'crossflow_one_unmixed'         {C_1 must be the unmixed fluid.}
                  'shell&tube_N' {where N is an integer between 1 and 9, specifying the number of shell pass.  The number of tube passes is then, 2N, 4N, 6N, .... . }


Index


abs\abs_n_(IncAng)


Usage
"The solar absorptance of a surface at normal incidence is 0.91.  Estimate the absorptance at an angle of 45 degrees."
alpha_n=0.91
theta=45
alpha/alpha_n=abs\abs_n_(theta)

Reference
Modified Equation 4.11.1:  Calculates the ratio of absorptance at IncAng to absorptance at normal incidence for a "standard" black surface.  The curve fit used is good to IncAng = 90 deg.

IncAng is in deg.

Index


abs_eff_(alpha_i, Area_aperture, Area_in)


Usage

"!Example 4.12.1"
alpha_n=0.60
alpha_i/alpha_n=abs\abs_n_(theta)
D=1
L=1
Area_aperture=pi*D^2/4
Area_in=pi*D^2/4+pi*D*L
alpha_eff=Abs_eff_(alpha_i, Area_aperture, Area_in) 

Reference
Equation 4.12.1 with alpha_i from 4.11.1 with an incidence angle of 60 degrees:  Calculates absorptance of a cavity from the absorptance at normal incidence and area of the inside surface of the cavity and the area of aperture. 

Abs_In is calculated at IncAng = 60 with Eq. 4.11.1.  

Areas must both be in the same units.


Index


TransIncAng_(N_cov, IncAng, KL, RefrInd)

TransIncAng_p_(N_cov, IncAng, KL, RefrInd : tau, alpha, rho, RefrAng)

Usage
"!Example 5.3.1"
n_cov=1
K=32[1/m]
L=0.0023[m]
THETA=60[deg]
RefrInd=1.526
tau_1=TransIncAng_(N_cov, Theta, K*L, RefrInd)
Call TransIncAng_p_(N_cov, Theta, K*L, RefrInd : tau_2, alpha, rho, theta_ref)


Reference
Equation 5.3.4:  Estimates the transmittance of Ncov glass covers for radiation incident at IncAng, using the approximate method of Equation 5.3.4, and Equations 5.1.4, 5.1.1, 5.1.2, 5.1.9, and 5.2.2.  KL is the product of the cover extinction coefficient and the cover thickness.  RefrInd is the refractive index of the cover material, averaged over the radiation spectrum of interest (e.g., the solar spectrum).

The procedure returns the cover transmittance cover absorptance and cover reflectance along with the refraction angle . 

Index


TransHalfCov_(IncAng, KL, RefrInd)

TransHalfCov_p_(IncAng, KL, RefrInd : TransHalfCov, SurfRef, TransCover_abs, RefrAng)

Usage
"Calculate the transmittance of one surface of a 2 mm thick (L) glass cover at angle of 60 degrees with an index of refraction (RefrInd) of 1.526 and an extinction coefficient (K) of 32 reciprocal meters. "

RefrInd=1.526
L=0.002[m]
K=32[1/m]
theta=60[deg]
tau=TransHalfCov_(theta, K*L, RefrInd)

Call TransHalfCov_p_(theta, K*L, RefrInd : TransHalfCov, SurfRef, TransCover_abs)

Reference
Equations 5.1.1-4 and 5.2.2:  Calculates the fraction of solar energy incident on a cover that is transmitted to the back surface of a cover.  It includes reflection from one air-cover interface and absorption in the cover.

The procedure returns the cover transmittance, cover absorptance, cover reflectance and the refraction angle. 

Index


IncAngEffGrRef_(Slope)


Usage
"Calculate the effective incidence angle for ground reflected radiation for a collector sloped at 45 degrees."
slope=45
theta_gr=IncAngEffGrRef_(slope)

Reference
Equation 5.4.1:  Estimates effective angle of incidence of ground-reflected radiation on a sloped flat surface.  

Slope is in degrees.

Index


IncAngEffDiff_(slope)


Usage
"Calculate the effective incidence angle for diffuse sky radiation for a collector sloped at 45 degrees."
slope=45
theta_sky=IncAngEffDiff_(slope)

Reference
Equation 5.4.2:  Estimates effective angle of incidence of diffuse radiation on a sloped flat surface.  

Slope is in degrees.

Index


TauAlphaProd_(N_cov, IncANg, KL, RefrInd, Alpha_n)

Usage

"!Example 5.5.1"
"Note that Example 5.5.1 assumes the absorber plate's absorptance is independent of direction so that the results here are a bit smaller."
Alpha_n=0.90
theta=50
KL=0.0370
N_cov=2
RefrInd=1.526
TauAlpha=TauAlphaProd_(n_cov, theta, KL, RefrInd, Alpha_n)

Reference
Equations 5.5.1:  Calculates the product of the cover transmittance and the absorber plate absorptance from the properties of the cover and absorber and the angle of incidence.  Uses Fctn Abs\Abs_n_, which is based on a modified Eq. 4.11.1.

IncAng is in degrees.

Index


S_T_(Type$, mode, I_h, I_bn, Lat, nDay, slope, SurfAzAng, GrRef, HrAng, N_cov, KL, RefrInd, Abs_n)

S_T_p_(Type$, mode, I_h, I_bn, Lat, nDay, slope, SurfAzAng, GrRef, HrAng, N_cov, KL, RefrInd, Abs_n : S_T, S_T_beam, S_T_diffuse, S_T_GrRef)


Usage

"!Examples 5.9.1 (L&J), 5.9.2 (HDKR) and HD methods"
Type1$='LJ'
Type2$='HDKR' 
Type3$='HD'
Mode1=1
Mode2=2
RefrInd=1.526
Ncov=1
Abs_n=0.93
SurfAzAng=0[deg]
Slope=60[deg]
KL=0.037
rho_g=0.6
HrAng=-7.5[deg]
nDay=13 "Day and month not given - on day 13 the incidence angle is 7 degrees as stated in problem."
Lat=40[deg]
I_h=1.79[MJ/m^2]
I_bn=3.128[MJ/m^2]
S_T=S_T_(Type1$, mode1, I_h, I_bn, Lat,nDay,Slope,SurfAzAng,rho_g,HrAng,Ncov,KL,RefrInd,ABS_n)

Call S_T_P_(Type1$, mode2, I_h, I_bn,Lat,nDay,Slope,SurfAzAng,rho_g,HrAng,Ncov,KL,RefrInd,ABS_n : S_T_P, S_T_beam, S_T_diff, S_T_grref) 

Reference
Equation 5.9.1, or  5.9.2 or the Hay Davies equivalent  :  This function calculates the hour`s absorbed radiation on a sloped surface using the Liu and Jordan (Type$=`LJ`) , Hay and Davies (Type$=`HD`), or the Hay, Davies, Klucher, Reindl  (Type$=`HDKR`) method.  Type$ must be 'LJ', 'HD', or 'HDKR'. 

I_h and I_bn are the total horizontal and bean normal solar radiation.  Most data sets provide both of these values but sometimes only the total horizontal radiation is known.  If mode=1 then only the total horizontal solar radiation I_h is used and the beam and diffuse components are estimated from Equation 2.10.1, the Erbs correlation.  I_bn can then be set to anything.  If mode=2 then both the total horizontal solar radiation I_h and bean normal solar radiation I_bn are used.  Radiation for the hour is in MJ/m^2 or Btu/ft^2.

HrAng is the hour angle for the mid-point of the hour.   All angles are in degrees.  

Allows use of Ncov=0.5, a situation encountered in PV modules where there is reflection loss from an air-cover interface and a cover-cell interface, and absorption in the cover.

If Ncov=0.5 then the angle of incidence on the cell surface is the angle of refraction for the air-cover interface.

Inside the function, I_zero is set to a finite value when it is zero.  (This still gives S_T_ = 0.)  This avoids divide-by-zero difficulties in simulation calculations. 

Includes calculation of I_zero with Fctn I_zero_ and Idiff\I with Fctn I_diff\I_. Transmittances are calculated with Fctn TransIncAng_ and absorptances with Fctn Abs\Abs_n_.    Transmittance-absorptance products are 1.01*Trans*Abs.

S_T is the total absorbed radiation on the tilted surface and S_T_beam, S_T_diff and S_T_grref are the three components of S_T.

Effective angles of incidence are from Fctn  IncAngEffGrRef_and Fctn IncAngEffDiff_

The procedure S_T_p_ also returns the beam, diffuse and ground-reflected components of S_T

Index


TauAlpha_bar_b_(month, Lat, slope, SurfAzAng, RefrInd, KL, N_covers, alpha_n)

TauAlpha_bar_b_p_(month, Lat, slope, SurfAzAng, RefrInd, KL, N_covers, alpha_n : TauAlpha_bar_b, R_bar_b, Theta_bar_b)



Usage
"!Example 5.10.1"
Slope=90
Lat=40
month=1  "Use Parametric Table"
SurfAzAng=0
Refrind=1.526
KL=0.0125
N_covers=2
alpha_n=0.9
H_bar=6.63[MJ/m^2]
rho_g=0.3

Call TauAlpha_bar_b_p_(month, Lat, slope, SurfAzAng, RefrInd, KL, N_covers, alpha_n : TauAlpha_bar_b, R_bar_b, Theta_bar_b)

"Result: theta_bar_b=0.73"


Reference
This function replicates figure 5.10.1 (a-f)  and Figure 5.6.1 to estimate the monthly average transmittance-absorptance product of the cover system and R_bar_b.  The surface can have any orientation of latitude, slope and surface azimuth angle.  Numerical integration is used rather than analytical equations.  The assumption is that the monthly average hourly beam radiation can be estimated from:

I_bar_b=(r_t*H_bar -r_d*H_bar_d)

which assumes that the day is symmetric about solar noon.

All angles are in degrees.

Index


S_bar_T_(Type$, month, H_bar, Lat, Slope,  SurfAzAng, rho_g, RefrInd, KL, N_cov, alpha_n)

S_bar_T_p_(Type$, month, H_bar, Lat, Slope,  SurfAzAng, rho_g, RefrInd, KL, N_cov, alpha_n : S_bar_T, S_bar_T_beam, S_bar_T_diff, S_bar_T_GrRef, H_bar_T, taualpha_bar)
  
Type$ directs the solution method and can have three values, 'LJ', 'KT' and 'K'.  The 'LJ' and 'KT' methods are discussed in Section 5.10 and problem 5.10.1 used the 'LJ' method.  For surfaces facing the equator the 'KT' method and the 'K' methods are nearly identical (the 'K' method uses a numerical method to find R_bar_b).  For surfaces not facing the equator only the 'K' method can be used.  It is recommended that the 'K' method be used.

Usage

"!Example 5.10.1" 
 
$UnitSystem SI
 
gamma=0[deg]; beta=90[deg]; phi=40[deg]
N_cov=2; RefrInd=1.526; KL=0.0125
alpha_n=0.9; rho_g=0.3
 
{Array H_bar}
H_bar[1..12]=[6.63,9.77,12.97,17.20,21.17,23.80,23.36,20.50,16.50,12.13,7.68,5.57]
{Array H_bar end}
 
Type$='LJ'
{Type$='K'}
{Type$='KT'}
 
Duplicate i=1,12
month[i]=i
Call S_bar_T_p_(Type$,  month[i], H_bar[i],  phi, beta,  gamma,  rho_g, RefrInd, KL, N_cov, alpha_n : S_bar_T[i], S_bar_T_b[i], S_bar_T_d[i], S_bar_T_g[i], H_bar_T[i], ta_bar[i])
Call H_bar_T_p_(Type$,  month[i], H_bar[i], PHI,  beta,  gamma,  rho_g : H_bar_T2[i], H_bar_T_b[i], H_bar_T_d[i], H_bar_T_g[i])
end


Reference
S_bar is estimated using one of three methods: Equation 5.10.2 (Liu and Jordan), the Klein-Theilacker equivalent or Klein (a numerical method discussed in Chapter 5 reference, Klein(1979)) .  Type$ must be either LJ,  KT or K.  Only surfaces facing the equator are supported for the LJ method as R_bar_b is found from Equation 2.19.3.  For the KT or K methods the estimate is for surfaces of any latitude, slope and azimuth.  For the KT method, R_bar_b is found from Equation 2.20.5.  For the K method R_bar_b is found numerically.  For both the KT and K methods the monthly average effective beam incidence angle (IncAngEffBeam) is found numerically and is equivalent to that from Figure 5.10.1 or 5.10.2.   The basic assumption is that the monthly average hourly beam radiation can be estimated from:

I_bar_b=(r_t*H_bar -r_d*H_bar_d)

which assumes that the day is symmetric about solar noon.

Allows use of Ncov=0.5, a situation encountered in PV modules where there is reflection loss from an air-cover interface and a cover-cell interface, and absorption in the cover.  If Ncov=0.5 then the angle of incidence on the cell surface is the angle of refraction for the air-cover interface.

Transmittances are calculated with Fctn. TransIncAng_ and absorptances with Fctn. Abs\Abs_n_.  Transmittance-absorptance products are 1.01*Trans*Abs.

S_bar and H_bar are MJ/m^2 or Btu/hr with the time interval of one day implied.  Angles are in degrees.

Index


U_top_Klein_(T_pl, T_amb, Slope, WindCoeff, epsilon, N_cov)


Usage
"!Example 6.4.2"
T_amb=10[C]
T_pl=100[C]
slope=45[deg]
WindCoeff=10[W/m^2-C]
epsilon=0.95
N_cov=1
U_t=U_top_Klein_(T_pl, T_amb, Slope, WindCoeff, epsilon, N_cov)


Reference
Equation 6.4.9:  Calculates the top loss coefficient for a collector using the Klein approximation to the solution of the detailed loss Equations.  

Epsilon is the plate emittance.  The glass emittance is assumed to be 0.88.  T_pl and T_amb are the plate and ambient temperatures in C or K, depending upon the EES units setting.  Slope is in degrees.  WindCoeff is in W/m2-C.


Index


<U_T_2>



U_T_1_(T_pl, T_amb, T_sky, Slope, WindCoeff, epsilon, PlateSpace) 

Usage

"!Example 6.4.1"
T_amb=10[C]
T_pl=100[C]
T_sky=T_amb  
slope=45[deg]
WindCoeff=10[W/m^2-K]
epsilon=0.95
Space=0.025[m]
N_cov=1
U_t=U_T_1_(T_pl, T_amb, T_sky, Slope, WindCoeff, epsilon, Space) 


Reference
Example 6.4.1 calculates the top loss coefficient for a single cover collector using the detailed loss equations of this section.   For two cover collectors use Function U_T_2_  

Epsilon is the plate emittance.  The glass emittance is assumed to be 0.88.  T_pl, T_sky and T_amb are the plate, sky and ambient temperatures in C or K, depending upon the EES units setting.  Slope is in degrees.  WindCoeff is in W/m^2-C or Btu/hr-ft^2-F.  Space is the plate spacing in m or ft.  



Index


U_T_2_(T_pl, T_amb, T_sky, Slope, WindCoeff, epsilon, PlateSpace) 

Usage

"!Example 6.4.1 extended to 2 covers."
T_amb=10[C]
T_pl=100[C]
T_sky=T_amb  
slope=45[deg]
WindCoeff=10[W/m^2-K]
epsilon=0.95
Space=0.025[m]

U_t=U_T_2_(T_pl, T_amb, T_sky, Slope, WindCoeff, epsilon, Space) 


Reference
Section 6.4:  Example 6.4.1 calculates the top loss coefficient for a single cover collector using the detailed loss equations of this section.   For two cover collectors use Function U_T_2_  

Epsilon is the plate emittance.  The glass emittance is assumed to be 0.88.  T_pl, T_sky and T_amb are the plate, sky and ambient temperatures in C or K, depending upon the EES units setting.  Slope is in degrees.  WindCoeff is in W/m^2-C or Btu/hr-ft^2-F.  Space is the plate spacing in m or ft.  The spacing from plate to cover 1 is assumed to be equal to the spacing between the two covers.



Index




F`_(U_L, Cond_pl, Thick_pl, TubeSp, TubeDia, BondCond, h_fi)


Usage

"!Example 6.5.1"
U_L=8[W/m^2-K]
Sp_tube=0.150[m]
Dia_tube=0.010[m]
Thick_plate=0.0005[m]
k_plate=385[W/m-K]
h_fi=300[W/m^2-K]
C_b=1e6 [W/m-K]"note the bond conductance was set to a large value"
F`=F`_(U_L, k_plate, Thick_plate, Sp_tube, Dia_tube, C_b, h_fi)

Reference
Equation 6.5.18:  Calculates collector efficiency factor for a flat-plate collector.  

Linear dimensions are in meters or feet.  U_L, the top loss coefficient and h_fi, the heat transfer coefficient inside the tubes, and are in W/m^2-C or Btu/hr-ft^2-F.  Cond_Pl is the plate thermal conductivity in W/m-C or Btu/hr-ft-F.  BondCond is the conductance between the plate and the tubes in W/m-C or Btu/hr-ft-F.

Index


F``

F`` is the collector flow factor (Equation 6.7.5) and is defined as the ratio F_R / F`.



F_R_(m_dot, c_p, A_c, U_L, CollEffFact)


Usage
"!For the collector of Example 6.5.1 with a mass flow rate of 0.03 kg/s and an area of 2 m^2 (1 by 2), calculate the collector heat removal factor.  "
U_L=8[W/m^2-K]
Sp_tube=0.150[m]
Dia_tube=0.010[m]
Thick_plate=0.0005[m]
k_plate=385[W/m-K]
h_fi=300[W/m^2-K]
C_b=1e6 [W/m-K]"note the bond conductance was set to a large value"
F`=F`_(U_L, k_plate, Thick_plate, Sp_tube, Dia_tube, C_b, h_fi)

m_dot=0.03[kg/s]
A_c=2[m^2]
C_p=4190[J/kg-K]
F_R=F_R_(m_dot, c_p, A_c, U_L, F`)

Reference
Equation 6.7.5:  Calculates heat removal factor for collector.  

m_dot, the mass flow rate is in kg/s or Lbm/hr.  The collector area,A_c is in m^2 or ft^2.  c_p  is in J/kgC or Btu/lbm-F.  U_L is in W/m^2C or Btu/hr-ft^2-F.

Index


Q_u_(A_c, F_R, S, U_L, T_in, T_amb)

Q_u is the useful gain of a collector accounting for transmission losses and thermal losses.

Usage

"!Example 6.7.1"
{Array T_amb}
T_amb[1]=-11;  T_amb[2]=-8;  T_amb[3]=-2
T_amb[4]=2;  T_amb[5]=3;  T_amb[6]=6
T_amb[7]=7;  T_amb[8]=8;  T_amb[9]=9
T_amb[10]=7
{Array T_amb end}
{Array I_T}
I_T[1]=0.02;  I_T[2]=0.43;  I_T[3]=0.99
I_T[4]=3.92;  I_T[5]=3.36;  I_T[6]=4.01
I_T[7]=3.84;  I_T[8]=1.96;  I_T[9]=1.21
I_T[10]=0.05
{Array I_T end}
{Array S}
S[1]=0;  S[2]=0.35;  S[3]=0.82;  S[4]=3.29
S[5]=2.84;  S[6]=3.39;  S[7]=3.21
S[8]=1.63;  S[9]=0.99;  S[10]=0
{Array S end}

U_L=8
A_panel=2
N_c=10
A_C=A_panel*N_c
F`=0.841
m_dot=0.03*N_c
c_p=4190
T_in=40

F_R=F_R_(m_dot, c_p, A_c, U_L, F`)

Duplicate i=1,10
Q_u[i]=Q_U_(A_c, F_R, S[i], U_L, T_in, T_amb[i])
eta[i]=Q_u[i]/(I_T[i]*A_c)
end

Q_day=sum(Q_u[i], i=1,10)
I_day=sum(I_T[i], i=1,10)
eta_day=Q_day/(A_c*I_day)

Reference
"Equation 6.7.6:  Calculates collector useful gain for an hour.  If Q_u is negative, the function returns Q_u = 0 (implying a controller).  

Q_u is in MJ or Btu for the hour.  A_c is in m^2 or ft^2.  U_L is in W/m^2-C or Btu/hr-ft^2-F.  S is in MJ/m^2 or Btu/ft^2 for the hour.  T_in and T_amb must be in the same units (C or K or F or R), depending upon the EES units settings.

Index


T_bar_(T_in, Q_u, A_c, F_R, F``, U_L : T_bar_f, T_bar_p)

The procedure T_bar calculates the mean collector plate and fluid temperature.

Usage
"!Example 6.9.1"

U_L=8[W/m^2-K]
T_amb=3[C]
T_in=40[C]
S=2.84[MJ/m^2]
A_c=20[m^2]
F_R=0.797:
F``=0.948
Q_u=Q_U_(A_c, F_R, S , U_L, T_in, T_amb)
Call T_bar_(T_in, Q_u, A_c, F_R, F``, U_L : T_bar_f, T_bar_p)

Reference
"Equations 6.9.2 and 6.9.4:  Calculates the mean fluid and mean plate temperatures in an operating collector.  Useful in fluid property determination. 

Q_u is in MJ or Btu for the hour.  A_c is in m^2 or ft^2. T_bar_f and T_bar_p are in the same units (C or K) or (F or R)  as T_in. U_L is in W/m^2- C or Btu/hr-ft^2-F.

Index


F_R_serpentine_(k, delta, U_L, W, D, N, L, m_dot, c_p, C_b, D_i, h_fi)

This function calculates F_R for a serpentine collector.

Usage

"!Example 6.13.1"
"knowns"
L=1.2   [m] "the length of one serpentine segment"
W=0.1   [m] "the distance between tubes"
N=6   "the number of tube segments"
delta=0.0015 [m] "the plate thickness"
D=0.0075  [m]" the tube outside diameter"
D_i=0.0065 [m] "the tube inside diameter"
k=211    [W/m-K] "the plate thermal conductivity"
U_L=5   [W/m^2-K]" the overall loss coefficient"
m_dot=.014 [kg/s] "the fluid mass flow rate"
C_p=3352  [J/kg-K] "the fluid specific heat"
h_fi=1500  [W/m^2-K] "the fluid-to-tube heat transfer coefficient"
C_b=1e6  [W/m-K] "the bond conductance"

A_c=N*W*L "[m^2] the collector area"
F_R=F_R_serpentine_(k, delta, U_L, W, D, N, L, m_dot, c_p, C_b, D_i, h_fi)


Reference
"Equation 6.13.1 calculates the ratio of F_R to parameter F_1.  There are three dimensionless variables, F_1, F_2 and F_3.  The parameters F_4, F_5 and F_6 are functions of F_2 only.  The function returns F_R for the serpentine flow arrangement.

Index



FlowRateCorr_(FRUL_test, m_dot_test, m_dot_use, C_p_test, C_p_use, A_c)

Usage

"!Example 6.20.1"
c_p=4187[J/kg-C]  "Both use and test conditions use water."
A_c=4.1[m^2]
m_dot_use=0.020[kg/s]
m_dot_test=0.040[kg/s]
FRUL_test=7.62[W/m^2-C]
FRta_n_test=0.78
F_corr=FlowRateCorr_(FRUL_test, m_dot_test, m_dot_use, C_p, C_p, A_c)
FRUL=FRUL_test*F_corr
FRta_n=FRta_n_test*F_corr


Reference
Equation 6.20.2 and 6.20.4:  Calculates the ratio r of FrUl and Frta at use and test conditions.  This assumes that F`Ul does not change significantly with flow rate, i.e., that effects of a change in h_fi on F` are small.  

m_dot  is in kg/s or Lbm/hr.  FrUl is in W/m^2-C or Btu/hr-ft^2-F.  A_c is in m^2 or ft^2.  C_p is in J/kg-C, kJ/kg-C or Btu/lbm-F.

Index


HXPenalty_(A_c, FrUL, m_dot_col, c_p_col, m_dot_tk, c_p_tk, epsilon_HX)

This function calculates the penalty (increase in collector area) when a heat exchanger is added to a solar system.


Usage

"!Example 10.2.1"
FrUl=3.75[W/m^2-K]
epsilon_HX=0.7
c_p_col=3350{J/kg-K]
c_p_tk=4190{J/kg-K]
A_c=1[m^2]
m_dot_col=0.0139 [kg/s]
m_dot_tk=0.0139 [kg/s]
Fr`\Fr=HXPenalty_(A_c, FrUl, m_dot_col, c_p_col, m_dot_tk, c_p_tk, epsilon_HX)

Reference
Equation 10.2.3:  Calculates the collector heat exchanger penalty.  By multiplying F_R by F`_R/F_R, a collector and heat exchanger system can be modeled as a collector system without a heat exchanger. 

A_c is in m^2 or ft^2.  FrUl is in W/m^2-C or  Btu/hr-ft^2-F.  MassFR is in kg/s or Lbm/hr.  C_p is in J/kg-C or Btu/lbm-F.}

Index


DuctLoss_(U_d, A_in, A_out, m_dot, C_p, A_c, FRta, FRUL : F`Rta, F`RUL)

Usage

"!Example 10.3.1"
A_c=50[m^2]
FRUL=3[W/m^2-C]
FRta=0.60
A_inlet=10[m^2]
A_outlet=10[m^2]
delta_ins=0.033[m]
h_outside=10[W/m^2-C]
U_duct_ins=1[W/m^2-C]
U_duct_unins=10[W/m^2-C]
c_p=1000[J/kg-C]
m_dot=0.5[kg/s]
Call DuctLoss_(U_duct_ins, A_inlet, A_outlet, m_dot, c_p, A_c, FRta, FRUL : FRta_ins, FRUL_ins)
Call DuctLoss_(U_duct_unins, A_inlet, A_outlet, m_dot, c_p, A_c, FRta, FRUL : FRta_unins, FRUL_unins)

"Solving this set of equations results in:  FRta_ins=0.5882, FRta_unins=0.0.500, FRUL_ins=3.275 and FRUL_unins=5.333 so that the Q_u equations become

Q_u_ins=50*[0.588* G_T - 3.275*(T_in-T_amb)]
Q_u_unins=50*[0.500* G_T - 5.333*(T_in-T_amb)]

Which will lead to a very significant reduction in performance without duct insulation."


Reference
Equation 10.3.9:  Calculates the correction factor to be applied to the absorbed radiation term (FRta in Equation 6.7.6) to account for pipe/duct losses.
Equation 10.3.10:  Calculates the correction factor to be applied to the loss term (FRUL in Equation 6.7.6) to account for pipe/duct losses.  

U_d, FRUL and F`RUL are in W/m^2-C or Btu/hr-ft^2-F.  Areas are in m^2 or ft^2.  m_dot  is in kg/s or Lbm/hr.  C_p is in J/kg-C, kJ/kg-C or Btu/lbm-F.

Index


PWF_(n, inf, dis)


Usage

"!Examples 11.5.1"
n=20
dis=0.10
inf=0.0.08
Cost_firstyear=500[$'
PW=Cost_firstyear*PWF_(n, inf, dis)

Reference
Equations 11.5.1:  Calculates the present worth factor for a series of n payments (usually years but could be months or even days) inflating at rate inf and discounted at rate dis.  

Inf and dis are decimals (e.g., a discount rate of 6% per period means dis = 0.06 per period).

Index


PrTx is the property taxes that must be paid on the investment. and Tx_bar is the average income tax that must be paid on any profit in a commercial business.  

Index



inf is the inflation rate per period, usually years or months.  It may have a subscript such as F for fuel 

Index


N_D is the number of periods in the depreciation schedule.

Index


dis is the discount rate per period, usually years or months.

Index


n or N or N_e in economic calculations is the number of periods to be considered in the economic analysis, usually years or months.

Index


Invest is the investment in solar equipment excluding any investment that is also part of the non-solar investment. 

Index


C is a switch to distinguish commercial property (C=1) from non-commercial property (C=0) for tax purposes.

Index


<N_L is the number of periods of the system N_L
Index


m is the mortgage interest rate as a decimal (i.e., 6% is entered as 0.06)

Index


LCS_(P_1, CfuelxLoad, Fract_sol, P_2, Invest)

LCS is the life cycle savings of a solar system over a conventual energy system.

Usage

"!Example 11.8.2"
{Array Area}
Area[1..6]=[0.01,25,39,50,75,150]
{Array Area end}
{Array F}
F[1..6]=[0,.37,.49,.56,.71,.92]
{Array F end}


dis=0.08
inf_fuel=0.10
inf=0.06
C_fuel_1=10[$/GJ]
Load=125.5[GJ]
Down=0.10
m=0.09
N_L=20
N_e=20
C=0
TX_bar=0.45
M_s=0.01
N_D=0 
R_v=0.40
Val=1
PrTx=0.02
P_1=P1_(N_e, inf_fuel, dis, TX_bar,C)
P_2=P2_(m, inf, dis, N_e, N_L, N_D, PrTx, TX_bar, Down, M_s, Val, R_v, C)

"Linear regression under the EES table menu was used to find the following curve fit."
$CheckUnits off
F=4.19687335E-04+1.97007052E-02*Area-2.32793369E-04*Area^2+1.27803131E-06*Area^3
$CheckUnits on

C_area=200[$/m^2]
C_fix=1000[$]
Invest=C_Area*Area+C_FIX
CFxLoad=C_fuel_1*Load

"We will maximize LCS wrt the Area using the EES min/max solution."
LCS=LCS_(P_1, CFxLoad, F, P_2, Invest)

"The result is LCS=$4681 at an Area= 45.1 [m^2]"

Reference
Equation 11.8.1:  Calculates Life Cycle Savings.  P1 and P2 must be supplied or calculated by the appropriate functions.  

CFxLoad is the first year fuel cost times the Load in $.  Invest is in $.

Index


Val is the ratio of the assessed valuation of the system in the first year to the initial investment.

Index


M_s is the ratio of the first year miscellaneous costs to the initial investment,

Index


R_v is the ratio of the resale value at the end of the period of analysis to the initial investment.

Index


Down is the down payment (as a decimal fraction) when the system is purchased over time. 

Index


P1_(N_e, inf_fuel, dis, TX_bar, C)


Usage

"!Example 11.8.2"
N_e=20
Inf_fuel=0.10
dis=0.08
TX_bar=0
C=0
P_1=P1_(N_e, inf_fuel, dis, TX_bar, C)



Reference
Equation 11.8.2:  Calculates P_1 using the function PWF_.  N_e is the years of the economic analysis, inf_fuel is the fuel inflation rate, dis is the discount rate, TX_bar is the average tax rate for income producing equipment and C should be set to 1 for an income producing commercial enterprise and to zero for a non-income producing enterprise. If C=0, then there are no tax implications.
Inf_fuel and dis are decimals (e.g., a discount rate of 6% per period means dis = 0.06 per period).

Index


P2_(m, inf, dis, N_e, N_L, N_D, PrTx, TX_bar, Down, M_s, Val, R_v, C)


Usage

"!Example 11.6.3"
m=0.09
inf=0.06
dis=0.08
N_e=20
N_L=20
N_D=0 
PrTx=0.02
TX_bar=0.45
Down=0.10
M_s=0.01
Val=1
R_v=0.40
C=0
P_2=P2_(m, inf, dis, N_e, N_L, N_D, PrTx, TX_bar, Down, M_s, Val, R_v, C)

inf_fuel=0.10
P_1=P1_(N_e, inf_fuel, dis, TX_bar, C)


C_fuel_1=10
Fract_solar=0.56
Load=125.5
Invest=11000









P_1=P1_(N_e, inf_fuel, dis, TX_bar,C)


"See LCS for using P_1 and P_2"



Reference
"Equation 11.8.3:  Calculates P2 using Function PWF_.  

Note that N_D cannot be set to 0.  

Note that additional terms may have to be added in the evaluation of P2.  Note also that parameters in terms not included in an analysis will have to be assigned values such that those terms are zero.  

m is the mortgage interest rate, inf is the general inflation rate, dis is the discount rate, N_e is the years of the economic analysis, N_L is the years for the loan, N_D is the years depreciation schedule, PrTx is the effective property tax, tx_bar is the average income tax rate, Down is the fraction paid as a down payment on the loan, M_s is the ratio of the first year miscellaneous costs to the initial investment, Val is the ratio of the assessed valuation of the system in the first year to the initial investment, R_v is the ratio of the resale value at the end of the period of analysis to the initial investment and C is a flag that should be set to 1 for a commercial (income producing) property and to zero for a non-income producing property.

Inf, dis, m, and _bar are decimals (e.g., a discount rate of 6% per period means dis = .06 per period).  PrTax, Down, R_v, and Val are fractions.}

Index


PartialPWF\N_(N, inf, dis)

Usage
"Numerically check the function for calculating the partial derivative of the PWF with respect to the number of years of the analysis."

inf=0.10
dis=0.08
N=20
delta=1
partial_function=PartialPWF\N_(N, inf, dis)
Partial_numerical=(PWF(N+delta, inf, dis)-PWF(N-delta, inf, dis))/(2*delta)

Reference
Equations 11.9.12 and 11.9.15:  The function returns values of the partial derivative of the Present Worth Factor with respect to the period of economic analysis, N.  The function works for any combination of inflation rate, inf, and discount rate, dis.  

Inf and dis are decimals (e.g., a discount rate of 6% per period means dis = .06 per period).

Index


PartialPWF\inf_(N, inf, dis)


Usage
"Numerically check the function for calculating the partial derivative of the PWF with respect to the inflation rate."

inf=0.10
dis=0.08
N=20
delta=0.001
partial_function=PartialPWF\inf_(N, inf, dis)
Partial_numerical=(PWF(N, inf+delta, dis)-PWF(N, inf-delta, dis))/(2*delta)

Reference
Equations 11.9.13 and 11.9.16:  Calculates the partial derivative of the present worth factor with respect to the inflation rate.

The function works for any combination of inflation rate and discount rate.  


Inf and dis are decimals (e.g., a discount rate of 6% per period means dis = .06 per period).}

Index


PartialPWF\dis_(N, Inf, dis)


Usage
"Numerically check the function for calculating the partial derivative of the PWF with respect to the discount rate."

inf=0.10
dis=0.08
N=20
delta=0.001
partial_function=PartialPWF\dis_(N, inf, dis)
Partial_numerical=(PWF(N, inf, dis+delta)-PWF(N, inf, dis-delta))/(2*delta)


Reference
Equations 11.9.14 and 11.9.16:  Calculates values of the partial of the present worth factor with respect to the discount rate.  

The function works for any combination of inflation rate and discount rate.  

Inf and dis are decimals (e.g., a discount rate of 6% per period means dis = .06 per period).

Index


XXYY_p_(Frta_n, FRUL, FR`\FR, tabar\ta, H_bar_T, T_bar_a, month, A_c, Load : X, Y)

Usage
"!Example 20.2.1 "
Frta_n=0.74
FrUl=4[W/m^2-C]
lat=43[deg]
Fr`\Fr=0.97
Load=36e9[J]
H_bar_T=13.7e6[J/m^2]
T_bar_a=-8[C]
tabar\ta=0.96
A_c=25[m^2] {A_c=50}
month=1

Call XXYY_p_(Frta_n, FRUL, FR`\FR, tabar\ta, H_bar_T, T_bar_a, month, A_c, Load : X, Y)

Reference
Calculates the dimensionless "loss ratio" X in the f-Chart method using Equation 20.2.3.
Calculates the dimensionless "absorbed radiation ratio" Y in the f-Chart method using Equation 20.2.4..  

Any set of dimensions can be used, with the restriction that the result must be dimensionless.  Thus, in SI units H_bar_T should be in Joules/m^2 for the month and Load should be in Joules for the month.





SolFract`L_p_(month, FrUl, FR`\FR, T_bar_a, A_c, Load, Frta_n, ta_bar\ta, H_bar_T, StoreCap, LoadHXEff, C_min, UA_h : X, Y, f)


Usage

"!Example 20.3.1"
{Array T_bar_a}
T_bar_a[1]=-8[C];  T_bar_a[2]=-5[C]
T_bar_a[3]=1[C];  T_bar_a[4]=9[C];  
T_bar_a[5]=14[C]; T_bar_a[6]=19[C]
T_bar_a[7]=22[C]; T_bar_a[8]=20[C]
T_bar_a[9]=15[C]; T_bar_a[10]=11[C]
T_bar_a[11]=2[C]; T_bar_a[12]=-5[C]
{Array T_bar_a end}

{Array Load}
Load[1]=36[GJ];  Load[2]=30.4[GJ];  Load[3]=26.7[GJ]
Load[4]=15.7[GJ];  Load[5]=9.2[GJ];  Load[6]=4.1[GJ]
Load[7]=2.9[GJ];  Load[8]=3.4[GJ];  Load[9]=6.3[GJ]
Load[10]=13.2[GJ];  Load[11]=22.8[GJ]; Load[12]=32.5[GJ]
{Array Load end}

{Array H_bar} "TMY2"
H_bar_T[1]=13.7[MJ/m^2];  H_bar_T[2]=18.8[MJ/m^2]
H_bar_T[3]=15.8[MJ/m^2];  H_bar_T[4]=14.7[MJ/m^2]
H_bar_T[5]=16.6[MJ/m^2];  H_bar_T[6]=16.51[MJ/m^2]
H_bar_T[7]=16.8[MJ/m^2];  H_bar_T[8]=17.5[MJ/m^2]
H_bar_T[9]=15.6[MJ/m^2];  H_bar_T[10]=15.2[MJ/m^2]
H_bar_T[11]=11.4[MJ/m^2];  H_bar_T[12]=12.7[MJ/m^2]
{Array H_bar end}

"Liquid System"
Lat=43 [deg]
slope=60[deg]
A_c=50[m^2]
Frta_n=0.74
FrUl=4[W/m^2-K]
Fr`\Fr=0.97
ta_bar\ta=0.96

StoreCap=75[l/m^2]  "The standard value"
"The following are chosen so that LoadHXeff*C_min/UA_h=2, the standard value."
LoadHXEff=1
C_min=2*UA_h
UA_h=463[W/C]

Duplicate i=1,12
     Call SolFract`L_p_(i, FrUl, FR`\FR, T_bar_a[i], A_c, Load[i]*Convert(GJ, J), Frta_n, ta_bar\ta, H_bar_T[i]*convert(MJ, J), StoreCap, LoadHXEff, C_min, UA_h : X[i], Y[i], f[i])
end
Load=sum(load[i], i=1,12)
F=Sum(f[i]*Load[i], i=1,12)/Load

Reference
Equations 20.3.1, 20.2.3, 20.2.4, 20.3.2, and 20.3.3:  Calculates month`s solar fraction for a LIQUID SYSTEM from X and Y using the correction factors for store capacity and load heat exchanger.  

If the storage ratio is outside of the allowable range of Eq. 20.3.2, or if the ratio (LoadHXEff*C_min/UA_h) is outside of the allowable range of Eq. 20.3.3, the function returns a solar fraction of 0.

Depending on EES units setting, In SI units T_ref must be either 100 C  or 373 K and In Eng units T_ref must be either 212 F or 672R.  T_bar_a is the monthly average ambient temperature.  Time is the number of seconds (SI) or hours (Eng) in the month.  A_c is the collector area.  Load is the monthly load in J or Btu.  H_bar_T is the incident solar radiation on the collector in J or Btu.  Ndom is the number of days in the month.  Storage capacity is in liters/m2 of collector area.  LoadHXEff is the effectiveness of the load heat exchanger, C_min is the minimum of the two capacitance rates through the load HX.  UA_h is the house loss coefficient.  All dimensions must be such that X and Y are dimensionless.

Index


SolFract`A_p_(month, FrUl, F_R`\F_R, T_bar_a, A_c, Load, Frta_n, tabar\ta_n, H_bar_T, StoreCap, AirFLRate: X, Y, f)


Usage
"!Example 20.4.1"

{Array H_bar} "TMY2"
H_bar_T[1]=13.7[MJ/m^2];  H_bar_T[2]=18.8[MJ/m^2]
H_bar_T[3]=15.8[MJ/m^2];  H_bar_T[4]=14.7[MJ/m^2]
H_bar_T[5]=16.6[MJ/m^2];  H_bar_T[6]=16.51[MJ/m^2]
H_bar_T[7]=16.8[MJ/m^2];  H_bar_T[8]=17.5[MJ/m^2]
H_bar_T[9]=15.6[MJ/m^2];  H_bar_T[10]=15.2[MJ/m^2]
H_bar_T[11]=11.4[MJ/m^2];  H_bar_T[12]=12.7[MJ/m^2]
{Array H_bar end}

{Array T_bar_a}
T_bar_a[1]=-8[C];  T_bar_a[2]=-5[C]
T_bar_a[3]=1[C];  T_bar_a[4]=9[C];  
T_bar_a[5]=14[C]; T_bar_a[6]=19[C]
T_bar_a[7]=22[C]; T_bar_a[8]=20[C]
T_bar_a[9]=15[C]; T_bar_a[10]=11[C]
T_bar_a[11]=2[C]; T_bar_a[12]=-5[C]
{Array T_bar_a end}

{Array Load}
Load[1]=36;  Load[2]=30.4;  Load[3]=26.7
Load[4]=15.7;  Load[5]=9.2;  Load[6]=4.1
Load[7]=2.9;  Load[8]=3.4;  Load[9]=6.3
Load[10]=13.2;  Load[11]=22.8
Load[12]=32.5
{Array Load end}

Frta_n=0.49
FrUl=2.84
lat=43
Fr`\Fr=1.0
T_bar_a=-7
ta_bar\ta=0.93
A_c=50
StoreCap=0.25     
AirFlRate=10

"The standard value"
"The following are chosen so that LoadHXeff*C_min/UA_h=2, the standard value."
LoadHXEff=1
C_min=2*UA_h
UA_h=463

Duplicate i=1,12
month=i
     Call SolFract`A_p_(month, FrUl, FR`\FR, T_bar_a[i], A_c, Load[i]*1e9, Frta_n, ta_bar\ta, H_bar_T[i]*1e6, StoreCap, AirFlRate : X[i], Y[i]. f[i])
end

F_annual=sum(F[i]*Load[i], i=1,12)/sum(Load[i], i=1,12)  {This is to find the annual solar fraction}

Reference

Equations 20.4.1, 20.2.3, 20.2.4, 20.4.2, and 20.4.3:  Calculates month`s solar fraction for an AIR SYSTEM from X and Y using the correction factors for store capacity and load heat exchanger.  If X>15 the function uses X=15, and if Y>3, it uses Y=3.  

If the storage ratio is outside of the allowable range of Eq. 20.4.3, or if the air flow rate ratio is outside of the allowable range of Eq. 20.4.2, the function returns a solar fraction of 0.

Depending on EES units setting, In SI units T_ref must be either 100 C or 373 K and In Eng units T_ref must be either 212 F or 672R.    T_bar_a is the monthly average ambient temperature.  Time is the number of seconds (SI) or hours (Eng) in the month.  A_c is the collector area.  Load is the monthly load in J or Btu.  H_bar_T is the incident solar radiation on the collector in J or Btu.  Ndom is the number of days in the month.  Storage capacity is in m^3 of pebbles per m^2 of collector area.   Air flow rate is in liters/*m2.  All dimensions must be such that X and Y are dimensionless.


Index


SolFract`HW_p_(month, FrUl, Fr`\Fr, T_bar_a, A_c, Load, Frta_n, ta_bar\ta_n, H_bar_T, Store_Cap, T_set, T_main : X, Y, f)


Usage
"!Example 20.5.1"

FrUl=3.64
Frta_n=0.64
ta_bar\ta_n=0.94
Fr`\Fr=1
H_bar_T=12.7e6[J/m^2]
T_bar_a=-8[C]
V_tank=225
D_tank=0.50
L_tank=1.16
Store_cap=75
A_c=10
U_tank=0.62
Draw=400
T_main=11
T_set=60
T_room=20
T_ref=100
C_p=4190
month=1
Time=NumDay_(month)*24*3600

Load_HW=Draw*NumDay_(month)*C_p*(T_set-T_main)
A_tank=pi*D_tank*L_tank+2*pi*D_tank^2/4
Loss_tank=U_tank*A_tank*(T_set-T_room)*NumDay_(month)*CONVERT(1/s, 1/day)
Load=Load_HW+Loss_tank

Call SolFract`HW_p_(month, FrUl, Fr`\Fr, T_bar_a, A_c, Load, Frta_n, ta_bar\ta_n, H_bar_T, Store_Cap, T_set, T_main : X, Y, f)

Reference
Equations 20.5.1, 20.3.1, 20.2.3, 20.3.2, and 20.2.4:  Calculates month`s solar fraction for DOMESTIC HOT WATER SYSTEM from X and Y and the correction factors for store capacity and set and mains water temperatures.  

If the storage ratio is outside of the allowable range of Equation 20.3.2, the function returns a solar fraction of 0. 

Depending on EES units setting, In SI units T_ref must be either 100 C  or 373 K and In Eng units T_ref must be either 212 F or 672R.    T_bar_a is the monthly average ambient temperature.  Time is the number of seconds (SI) or hours (Eng) in the month.  A_c is the collector area.  Load is the monthly load in J or Btu.  H_bar_T is the incident solar radiation on the collector in J or Btu.  Ndom is the number of days in the month.  Storage capacity is in liters/m2 of collector area.  T_set is the hot water delivery temperature, T_main is the mains water temperature.  All dimensions must be such that X and Y are dimensionless.

Index


PhiBarFChart_p_(A_c, FRUL, FRta x tabar\tan, L_total, H_bar, i, Lat, Slope, rho_g, I_Tc, Store\A : X, Y, f)

The units must be SI with A_c in m^2, FRUL in W/m^2-C, L_total (the useful load plus the tank losses) in GJ, H_bar in MJ/m^2, the critical level I_Tc in W/m^2-C and Store\A_c = m_store x c_p/A_c in kJ/K-m^2.  The function returns X (Eqn 21.3.4), Y (Eqn 21.3.4) and the solar fraction f (Eqn 21.3.5).  Note that FRUL and FRta both should be corrected for duct losses, collector-storage heat exchanger and array plumbing (see Chapter 10).

Usage
"!Example 21.3.2"
Lat=41[deg]
slope=40[deg]
SurfAzAng=0[deg]
T_min=60[C]
tabar\tan=0.94
FRta=0.72
FRUL=2.63[W/m^2-C]
A_c=50
C_p=Cp(Water,T=T_min, P=Po#)
Store\A=4180[kg]*C_p/A_c
rho_g=0.2
UA_s=5.9 [W/c]
hr=1[hr]

"T_st=60"  "The average tank temperature was set to a value during the development of the following equation set."

"The unknown storage temperature T_st and quantities dependent n T_st are included in some of the EES Library functions so that a direct solution of the equations fails without good guesses.  Here we show an alternative method  for solving for T_st in which initial guesses are less important.  We define an ERROR as the absolute value of the difference in phi_bar calculated in two different ways.  The Min\Max Table solution option (F5) minimizes ERROR with respect to the tank temperature, T_st. This is similar to the hand trial-and-error solution shown in Example 21.3.2"

Q_st=UA_s*(T_st-20[C])*3600[s/hr]*24[hr]*NumDay*Convert(J, GJ)
NumDay=NumDay_(i)
L=12[kW]*12[hr]*Convert(kWh, GJ)*numDay
L_total=L+Q_st
H_bar_o=H_bar_zero_(i, Lat)
K_bar_T=H_bar/H_bar_o
I_Tc=FRUL*(T_min-T_bar_a)/(FRta*tabar\tan)
call PhiBarFChart_(A_c, FRUL, FRta*tabar\tan, L_total, H_bar, i, Lat, Slope, rho_g, I_Tc, Store\A : X, Y, f_TL)
Phi_bar_1=f_TL/Y
I`_Tc=FRUL*(T_st-T_bar_a)/(FRta*tabar\tan)
Call PHI_BAR_p_(H_bar, i, Lat, Slope, rho_g, I`_Tc : K`_bar_T, R_bar, R_beam_n, R_noon, X_bar_c, Phi_bar_2)
Error=Abs(Phi_bar_1-Phi_bar_2)

   "Error=0"
"Solving the above equations (with Error=0) will work if the following initial guesses and bounds are set.: 
T_min < T_st < 200 with initial guess of T_min
0 < I`_Tc < 1000 with initial guess equal to I_Tc
"
"The annual solar fraction is found from:"
f_annual=sum(TableValue(i, 'F_TL')*TableValue(i, 'L_total'), i=1,12)/Sum(TableValue(i, 'L_total'), i=1,12)

The solution for January is f=0.50 and for the year f=0.65.



Reference
Equation 21.3.5 and Figure 21.3.2 represent the phi_bar f-charts.  This is a solar system design tool that can replace the original f-chart method of Chapter 20.  These correlations are more convenient to use than the f-chart correlations which require limits on the variables X and Y.

Index


FindPV_RefPar_p_(I_o_ref, I_sc_ref,  V_oc_ref, mu_Voc, V_mp_ref, I_mp_ref, mu_Isc, G_ref, epsilon_ref, N_s, C, TC_c_ref : E5, R_sh_ref, R_s_ref, I_L_ref, a_ref )

This Subprogram will find the five PV model reference parameters (I_o_ref, R_sh_ref, R_s_ref, I_L_ref and a_ref; see Example 23.2.1).  Note that four of the parameters are in the call outputs and one is an input.  The variable E5 in the output is an error that we wish to minimize by finding the correct value of the fifth parameter, I_o_ref.  


Usage
The process for finding all five parameters is a two step process and is best illustrated using the data from Example 23.2.1.

I_sc_ref=4.5[A]
V_oc_ref=21.4[V]
mu_Voc=-0.085[V/K]
V_mp_ref=16.5[V]
I_mp_ref=3.95[A]
mu_Isc=0.00026[A/K]
G_ref=1000[W/m^2]
epsilon_ref=1.12[eV]
N_s=36
C=0.0002677[1/K]
TC_c_ref=25[C]

Call FindPV_RefPar_(I_o_ref, I_sc_ref,  V_oc_ref, mu_Voc, V_mp_ref, I_mp_ref, mu_Isc, G_ref, epsilon_ref, N_s, C, TC_c_ref : E5, R_sh_ref, R_s_ref, I_L_ref, a_ref )
E5E5=E5^2  

"Set up a 10-row parametric table with the six variables I_o_ref, E5, R_sh_ret, R_s_ref, I_L_ref and a_ref (actually only the first two columns are needed but the others provide interesting information). Set the 10 values of I_o_ref in increments of 10 from 1E-5 to 1E-14.  Run the table (make sure `Stop if error occurs` is not checked).  All rows in the table may not be solved but this is OK as long as the rows where E5 changes sign are solved (in this example E5 crosses zero between 1E-11 and 1E-10).  When E5 is zero a solution has been found.  Run the table again but stop at the row where the absolute value of E5 is the smallest (row 7 in this example) and under the calculate menu choose to update guess values to prepare for the next step.  At this point you have three options.  The first option is to allow EES to solve the equations with the new guess values; this ofter works. The second solution alternative is to use the EES min/max option.   Minimize E5E5 (the square of E5; we could have minimized the absolute value of E5) with respect to I_o_ref.  Choose an appropriate lower and upper bound on I_o_ref from the parametric table results (1E-11 to 1E-10 for this example).   Generally a solution is found very quickly.  The last option is to change the values of I_o_ref in the table to range between 1E-11 and 1E-10 and run the table again to find a new range of I_o_ref.  Continue the process until a satisfactory solution accuracy has been found or use the first or second option with the new guess values. "


Reference
Equations 23.2.3 through 23.2.12 define the PV model.  The equations are highly nonlinear and consequently difficult to solve.  This EES function is constructed to continually improve the guess values until EES can solve the set of simultaneous equations.  

Index


Find_ck_(U_bar, Sigma: c, k)

The units of U_bar and sigma are either m/s or mph.

Usage
Find the Weibull shape factor (k) and the Weibull scale factor (c) for a wind data set with an average wind velocity of 10 m/s and a standard deviation of 2 m/s.  

U_bar=10[m/s]
sigma=2[m/s]
Call Find_ck_(U_bar, sigma : c, k)

Solution: c=10.80 and k=5.797

Reference
Equations 24.2.9 and 24.2.10 are solved simultaneously for c and k given the average wind speed U_bar and the wind speed standard deviation, sigma.



Index