Modelling the radiant output of orchard heaters

Agricultural Meteorology, 23 (1981)275--286

275

Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

MODELLING THE RADIANT OUTPUT OF ORCHARD HEATERS* J.M. WELLES**, J.M. NORMAN** and J.D. MARTSOLF***

Departments of Horticulture and Meteorology, The Pennsylvania State University, University Park, PA 16802 (U.S.A.) (Received January 2, 1980; revision accepted June 2, 1980)

ABSTRACT Welles, J.M., Norman, J.M. and Martsolf, J.D., 1981. Modelling the radiant output of orchard heaters. Agric. Meteorol., 23: 275--286. The radiant o u t p u t from orchard heaters is an important factor in frost protection, although it is usually neglected in frost protection models. The basic theory for calculating radiant flux incident on a leaf, which is arbitrarily positioned and oriented with respect to a radiation-emitting cylinder, and disc, is presented. A typical heater stack is shown not to emit as a point source; errors of 30% result from applying the "inverse square law". Furthermore, arguments about the importance of heater radiation that are based on pointsource emission from a single orchard heater are invalid, because foliage at any point in an orchard receives radiation from many heaters. Considering the radiant output of only a single heater, instead of the combined effect of all heaters, leads to an underestimate of the importance of heater radiation by a factor of two to ten.

INTRODUCTION

Orchard heaters transfer energy to their surroundings radiatively and convectively. The portion of the fuel's total energy that is converted to radiant energy is known as the radiant fraction, and is typically about one fourth (Perry et al., 1977). Most of the published orchard heating models and energy budgets (Crawford, 1964; Gerber, 1969; Martsolf and Panofsky, 1975) deal only with the convective portion of the heater's output. There is good agreement, however, that the radiant output is an important contributor to orchard protection, especially in windy conditions (Kepner, 1951; Gerber, 1965; Wilson and Jones, 1969; Turrell, 1973; Martsolf, 1974; Perry et al., 1977). Through use of a computer model of orchard foliage temperatures (OFT Model) that takes detailed account of radiation emitted by heaters, Welles et al. (1979) found that the effects of heater radiation and convection were comparable in determining foliage temperatures. In windy conditions with sparse foliage, heater radiation was found to be more important than heater convection. The OFT model also indicates that increases in * Pennsylvania Agricultural Experiment Station Journal Series No. 5809. ** Present address: Agronomy Department, University of Nebraska, Lincoln, NE 68583, U.S.A. *** Present address: Fruit Crops Department, University of Florida, Gainesville, FL 32611, U.S.A.

0002-1571/81/0000--0000/$02.50 © 1981 Elsevier Scientific Publishing Company

276 TABLE I T h e O F T m o d e l c a n b e used t o d e t e r m i n e f r a c t i o n of foliage lost as a f u n c t i o n of b u r n rate. T h e o p t i m u m b u r n r a t e is t h a t n e e d e d t o k e e p losses u n d e r s o m e p e r c e n t a g e (in this e x a m p l e , 30%). See t e x t for m o r e details Time

Unheated air t e m p . (°C)

P e r c e n t loss w i t h b u r n rate of (l/h)

Optimum b u r n rate

(l/h) 0200 --2.0 0300 --2.3 0400 --2.6 0500 --3.0 0600 --2.0 T o t a l fuel use (1) % loss over t h e p e r i o d

3.0

4.0

5.0

30 52 63 72 30 15 72

5 13 28 48 5 20 48

2 3 9 16 2 25 16

3.0 3.5 4.0 4.5 3.0 18.0 30.0

radiant fraction (with resulting decreases in convective output) have the net effect of increasing heater efficiency. The purpose of this paper is to indicate how a heater's radiant o u t p u t can be modeled so that the irradiance of a leaf can be predicted as a function of its orientation and position with respect to one or more fired heaters. This is an essential part of the general OFT model. For an orchard containing 320 trees/hectare with each tree crown centered about 1.5 m above the ground and 1.0 m in radius, the predicted foliage damage as a function of air temperature and heater burn rate is contained in Table I. The canopy foliage density is typical of blossom time (0.17 m -1 ), the wind speed about 1 m s -1 at 4 m above the ground and 100 heaters per hectare (Welles et al., 1979). Not only does this model predict the fraction of foliage lost under frost conditions, but also it can be used to estimate o p t i m u m burn rates under varying environmental conditions to improve the efficiency of oil use. In Table I the optimum burn rate results in 22% less fuel use throughout a typical night (for the same level of protection) than if the heaters were fired at a fixed rate o f 4.51 h -1 . Clearly this model could be used for many applications: (1) to help obtain an optimal heater design; (2) to predict the most economical burn rate that can provide the minimum foliage protection for a successful orchard crop; (3) to assess the level of frost protection and the distribution of protected foliage for various heater arrangements depending on orchard characteristics; and (4) to permit an economic analysis of the return on an investment i n more heaters and indicate advantages of many small heaters versus fewer larger heaters. VIEW FACTORS

The irradiance I (incident energy per time per area) on a leaf from a heater can be expressed as

277 I

=

(W/Ah)Fh

(1)

where W is energy per time emitted by a heater, Ah is heater stack area, and Fh is the view factor for the heater as seen by the leaf. The radiant flux (W/Ah) emitted by a heater can be expressed in terms of burn rate and radiant fraction, or in terms of the mean 4th p o w e r temperature of the stack

W__._= BhkR: = ehOT"~h Ah Ah

(2)

where k -- 1.085 × 104 watt-hours per liter for No. 2 diesel fuel. The view factor Fa is the major hurdle in applying eq. 1. The view factor may be thought of as the fraction of the leaf's hemispheric view (the side of the leaf facing the heater) that is occupied by the heater. Alternatively, it could be thought of as the fraction of radiant flux leaving one side of the leaf that is incident on the heater. A good general discussion of view factors may be found in Eckert and Drake (1959). The view factor for a heater is obtained from the sum of view factors for a hollow cylinder (F c) representing the stack and a flat disc (Fd) which represents the upper end. Fh ----- Fc -{- Fd

(3)

The view factor between an arbitrarily oriented and positioned unit area (leaf) and a hollow cylinder (heater stack) can be calculated from (Fig. 1) F¢ :

H + cos-I --R f f

(R/D)

lr

0 - cos-I

COS~l COS~2 $2 dTdz

(4)

(RID)

where COS ~1 ~--- sin 0 L sin ~ + cos 0 L cos ~ cos (¢' - - e ) COS ~2 : (D cos 7 - - R ) / S

[(Yc -- YT.)/(Xc - - X L ) ] sin-' [(R/f)sin 7] = tan -1 [(Z -- ZL )/f] = + (Z-ZL) 2 = D2 +R 2-2DRcos7 (XL - - X c ) + - - Yc) tan-'

e 5

S2 f2 D2

--eL

Equation 4 is cumbersome and must be evaluated numerically. Fortunately there is a simplified approximate expression for F that is valid for most orchard applications. Hamilton and Morgan (1952) give an analytic solution for the view factor for a special case o f a Cylinder and a unit area. For 0L -~b' = 0, and Z L -~- H then the special case view factor F'c becomes

278

y

•

unit ore~ normal

.

'

,

,,

I

,

.

I

~

I

--~

! ~

I { × L.,',;.,o)

=/~(Xc,Yc,Zc)

IT) t ) ~

I'

~z

~

L-"~''R

( _

I

; "'(Xc,Yc,Z) iI

.....

it

Given: XL,¥L, ZL, eL, ~)t. xc, Yc* H, R

Fig. 1. The geometry for calculating the view factor between an arbitrarily oriented unit area (leaf) and a cylinder (heater stack). 0L is leaf elevation angle (measured posi• t tively up from the horizontal), and ¢ is leaf azimuth angle with respect to the heater.

F : - ~r-~tan

~

+--

lr ~ C%/-A-B

-- c l - - t a n - Z / ( ~ +

~) I

whereA =(C+1)

2 +E 2,B=(C-1)

tan -1

S ( C + 1)

(5) 2 +E 2,C=D/R,andE=H/R.

Note that when the leaf height and the cylinder height are different, eq. 5 must be applied twice: once for a hypothetical cylinder of height ZL, (yielding F'¢ ) and once for a hypothetical cylinder of height Z L -- H [yielding F 'c (for height Z L -- H)]. The difference of these two quantities is the view factor for the real cylinder of height H F'c = F'c(for height ZL ) --F'¢(for height Z L - - H )

(6)

The special case view factor F'c can be generalized for arbitrary leaf orientation by assuming that the view factor is a m a x i m u m when the leaf is facing the cylinder midpoint, and varies as the cosine of the angle between the leaf normal and a line to the cylinder midpoint (Fig. 2). Thus, the maxi-

279

loaf normal

--/

l

X

Fig. 2. The view factor for a unit area and a cylinder as given by Hamilton and Morgan (1952) assumes the unit area normal to be along the X axis. Assuming the view factor to be increased by the secant ~' to its approximate maximum value when the normal points at the cylinder midpoint, the view factor for an arbitrary orientation is this m ax i m u m value times the cosine of the angle a.

mized view factor would be F'c/cos 5'. Stated another way

Fo

QoF'o

(7)

where COS QC

--

cos 5'

sin 0T. sin 5' + cos 0L COS 5' COS ¢' cos 8'

= ( K / D ) sin 0L + COS 0L COS ~b'

(8)

since 5' = t a n - l ( K / D )

where K is the difference in height between the leaf and the cylinder midpoint (positive in Fig. 1, negative in Fig. 2). Comparisons between eqs. 4 and 7 indicate a negligible difference (less than 1%) for cylinders of typical heater dimensions and leaf distances greater than 1.5 m in any direction away from the stack. The view factor for the upper end of the stack F d c a n be represented by (Fig. 3)

Fd =(Z, TZd)! f

R d , 27r

r c o$3s ~

dTdr

0

where cosfll = sin0 L s i n S + c o s 0 L cos ~ cos(~b' -- e) = tan-I [(Z d -- Z L)/f]

(9)

280

(XL'¥L'ZL)~

i !

G~ven:

~ YuZL'eL,¢L Xd,Y,Rdd

Fig. 3. The geometry for the calculation of the view factor between a unit area and a disc. The height of the disc, Z d, is zero in this figure: S f2 ~' D2 e Zd

_ [(Zd _ Z L ) 2 + f 2 ] 1 / 2 = D 2 + r 2 - - 2 D r cos 7 = tan -1 [(Yd - - YL)/(Xd - - X L ) ] --~bL = ( X d _ X L )2 _~ ( Yd - - YL )2 = sin -1 (r sin 7If) = disc h e i g h t (zero in Fig. 3)

E q u a t i o n 9 m u s t be e v a l u a t e d n u m e r i c a l l y , b u t as in t h e case o f the cylinder, an a p p r o x i m a t i o n b a s e d u p o n a special case m a y be used. W h e n 0L = 0 a n d ~b' = 0, t h e view f a c t o r f o r t h e disc is F~ ( H a m i l t o n a n d Morgan, 1952). Fd

=

[(I + G 2 +j2)2

_4G211n

--I

(10)

w h e r e G = R d / D , and J = (Z L - - Z d ) / D . T h e c o r r e c t i o n o f F~ to include a r b i t r a r y l e a f o r i e n t a t i o n is a n a l o g o u s t o t h e c o r r e c t i o n o f F" to m o r e closely approximate F c

Fd ~ QdF~

(11)

and Qd ---- s i n 0 L sin 6 " +COS0L_ COS~"COS~' cos 5 "

(12)

281

Since 5" = tan -1 [(Zd -- ZL )/D], eq. 12 m a y be rewritten as Qd = Za -D ZL sill 0 L -}- COS 0 L COS ~b'

(13)

The error in eq. 11 is smaller than that in eq. 7. Summing the view factors for the side and the top of a heater stack (eq. 3) implies that all the radiation received by the leaf comes directly from the heater with no intermediate reflection from the heater base, the ground, or reflectors on the ground. The a m o u n t of radiation received by a leaf that comes from a reflector placed beneath a heater can be appreciable (Welles et al., 1979), and thus is well worth calculating. The radiant flux Ix reflected by a flat disc underlying a heater that is incident on a leaf is Ir

W paFuf~

(14)

A--d

=

where Ad is the exposed area of the underlying disc, Pa is the disc reflectivity, Fu is the fraction of the radiant energy emitted b y the stack that is incident on the disc, and F~ is the view factor for the underlying disc as seen by the leaf. If eq. 9 is used to calculate F ~ , the blockage o f part of the disc by the stack is accounted for by adjusting the limits of integration. If the approximation (eq. 11) is used, stack blockage m a y be accounted for in t w o ways. The first way reduces the view factor for the entire disc b y an a m o u n t equal to the view factor for a cylinder of height Z * , where

z* =

;

/

(15)

and Z* is never allowed to exceed cylinder height H. The second way is to simply reduce the disc area (for purposes of the view factor calculation) by an a m o u n t equal to the area blocked b y the cylinder. This adjusted disc radius is 1/2 ?r

where A is the area blocked by the cylinder

A--~/~¢ +2ReX

(17)

where

I[~d X =

(ifZ*
g ( R d + D ) / Z L ] --Re

(ifZ*

=H)

282

/"

/

///"

"6 2

/J , - /J

Fig. 4. G e o m e t r y for t h e c a l c u l a t i o n o f t h e view f a c t o r of a disc as seen b y a c y l i n d e r ; i.e., o f t h e f r a c t i o n o f t h e r a d i a n t flux leaving t h e c y l i n d e r w h i c h will be i n t e r c e p t e d b y t h e disc. I n ( 1 8 ) t h e angle 0 h a s already b e e n i n t e g r a t e d (0 t o 27r), yielding 27F.

The view factor F . between the cylinder and the underlying disc is (Fig. 4) H

Rd + ~

1 / ~H 0

-

~ Rc

~zr(rcosO'--Rc) $4

dO'drdz

(18)

-

where S4

=

(Z 2

=

cos-

+ R2c + r 2 -- 2 R c r c o s 0 ' ) 2

1 (R c/r)

Total

(~

__

I)~

I

18(3

- ~ " _ ~L

im,mw.,

E \ 60

~0

,/ "

0

.

. ........122:-..~.~......... : i

,

,

)

2

D

Ref lect t~ disc .

" .....

i ............ ; ......... 3

i 4

(m)

Fig. 5. I r r a d i a n c e I ( W m -2) as c o m p u t e d o n a vertical leaf 2.0 m a b o v e t h e g r o u n d at d i s t a n c e D ( m ) f r o m a h e a t e r e m i t t i n g 104 W m -2 . T h e h e a t e r is sitting o n a disc of reflectivity 0.9 a n d r a d i u s 0.5 m. I = 104 F h .

283

Note t h a t the lower limit of the integration over z is n o t zero if the cylinder does n o t rest on the disc. The total heater view factor Fh (including a surface reflector) is given by Fh = Fc + Fd + (Ah/Ad)PdFuF~

(19)

This model neglects radiation emitted or reflected by bowl covers or bases of heaters. At higher burn rates especially, this assumption will cause underestimates of irradiance, leaving it on the " s a f e " side in a frost protection model. Figure 5 illustrates the relative contributions of the three components of eq. 19 for a specific example.

INVERSE SQUARE LAW

Radiant flux decreases in inverse proportion to the square of the distance from a point source. A heater's stack can behave quite differently than a simple inverse square relation would indicate; however, for leaf positions greater than several meters to the side of a heater, the approximation is much better. A vertical cylinder radiates more in the horizontal than an idealistic point source, and the inverse square relation underestimates itradiance as c o m p u t e d using view factors by 30% or more. For distances less than a few meters from a heater, differences of m u c h more than 30% can arise between the point source assumption and an actual heater. Valli (1970) used the inverse square relation to argue that the radiant effects of a heater are n o t as important as the convective effects. For one heater in an orchard this may be true, but a typical orchard has an array of heaters, and the number of heaters t h a t contribute to heating a particular tree increases as the square of the distance away from that tree. This contribution of far-away heaters tends to balance the decreased contribution per heater with distance. A test was made using the OFT Model (Welles et al., 1979) to determine the overall radiant contribution of an array of heaters to mean foliage temperature at the center of an orchard tree crown (Fig. 6). The heater array used for the test is indicated in the insert in Fig. 6. A sparse foliage density (0.1 m -1 ) typical of spring blossom time was used, along with a burn rate of 4.0 I/h/ heater. The heater array results in 2/3 of the trees (type 1) in the 1 hectare orchard being 2.4 m from the nearest heater, and 1/3 (type 2) 7.4 m from the nearest heater. The numbers n e x t to the plotted points in Fig. 6 indicate the number of heaters within the cut off radius from the particular tree. The radiant o u t p u t of heaters at distances greater than the cut off radius is neglected. The temperature increase plotted is solely due to radiant effects, as the convective effects were ignored for the test. The broken line is an inverse square relation for one heater based on a 1.1°C effect at 2.4 m (tree 1); it is essentially this curve that forms the basis for Valli's (1970) statement

284

7

u

12

2

~

I

s

+~ o %Tree

~

-

5

2 _

~

,0 0 0-0 0 0.0 0 0 .

4

~0.00 o o.o o o,o o o.o

0

I~]

20

CutoTf

30

40

Radius

50

80

(m)

Fig. 6. Effective foliage temperature at the center of two representative trees resulting only from the cumulative radiative output of heaters located at a distance of less than the cutoff radius from the given tree (solid lines). The numbers on each curve represent the number of heaters contributing to heating of the foliage. The dashed line represents the inverse square relation for the heater nearest to tree 1.

that radiant o u t p u t is unimportant. The difference between the dashed curve and solid curves in Fig. 6 illustrates the fallacy in Valli's argument. F o r tree 1, 50% of the temperature increase comes from heaters b e y o n d the nearest one; for tree 2, 90% comes from those b e y o n d the nearest one. Clearly, assessing the importance of radiant heater o u t p u t based on the inverse square loss of p o w e r density from a single heater [as Valli (1970) has done] is a gross error.

COMPARISON

WITH

MEASUREMENTS

Wilson and Jones (1969) measured radiant flux with a net radiometer traversed in a 260 ° arc over a return stack heater. Effective view factors (Fig. 7) were calculated by dividing the mean measured flux at a given angle by the emitted radiant flux (eqs. 1 and 2). These view factors (denoted "measured") are comparable to those obtained using the theoretical equations in this paper. The return stack heater was assumed to be a cylinder (the return stack was neglected) of length 1.05 m and radius 0.096 m. The stack was assumed to sit on a fiat disc of equal temperature of radius 0.2 m; this was to simulate the combined effects of bowl covers, and reflection from the ground. A more elegant treatment of the radiative environment was not done due to lack of information about details of the experiment. At moderate burn rates theory and measurement agree quite well. The cause for the peak of the 1.25 gal/h (4.73 I/h) curve to occur at a lower angle is not known.

285 140 [ 1.2SGal,'~e . . , . ~ . , . ~ ~

I.,.

I / v`,"

8

v.,o.

20

"'

4B

"-'12"

6g

Zenl~h Rngle

80

lilID

12B

(deg.)

Fig. 7. Comparison between " m e a s u r e d " view factors at three burn rates ( 0 , X , * ) f r o m Wilson and Jones (1969) and computed view factors for a unit area traversing an arc over a heater (continuous solid line). ACKNOWLEDGMENT

Support through U.S. DOE Contract EC-77-5-2-4397 is acknowledged. APPENDIX. P A R T I A L LIST OF SYMBOLS Ad

Ah Bh

Fc F"

Fd Fd F~

Fh Fu H I I, k R Rd Rf T. W

Zd (XL, YL, ZL)

area of disc (m 2 ) area of heater stack (m 2 ) heater burn rate (1 h -I ) view factor of cylinder special case cylinder view factor view factor of disc special case disc view factor view factor between a cylinder and an underlying disc view factor for a heater as seen by a leaf view factor for a disc underlying a cylinder heater stack height irradiance on a leaf ( W m -2 ) leaf irradiance that has been reflected from a surface underlying the heater (W m -2 ) 1.085 x 104 watt-hours per liter heater stack radius (m) disc radius (m) radiant fraction heater stack temp (K) Energy per time emitted by a heater (W) disc height (m) coordinates of a leaf

286

(Xc, Yc) (Xd, Yd ) Z* ~h

0L 5'

~H

Pd

horizontal coordinates of a cylinder horizontal coordinates of a disc height of a hypothetical cylinder (m) heater emissivity leaf elevation angle leaf azimuth angle inclination angle from a leaf to a cylinder mid point inclination angle from a leaf to a disc mid point disc reflectivity

REFERENCES Crawford, T.V., 1964. Computing the heating requirement for frost protection. J. Appl. Meteorol., 3: 750--760. Eckert, E.R.G.. and Drake, R.M., Jr., 1959. Heat and Mass Transfer. McGraw-Hill, New York, NY, 530 pp. Gerber, J.F., 1965. Performance characteristics of heating devices. Proc. Fla. State Hortic. Soc., 78: 78--83. Gerber, J.F., 1969. Methods of determining nocturnal heat requirements for citrus orchards. In: H.D. Chapman (Editor), Proc. First Int. Citrus Syrup., 2: 545--550. Hamilton, D.C. and Morgan, W.R., 1952. Radiant interchange configuration factors. Natl. Advisory Committee for Aeronautics, Washington, DC, Tech. Note 2836, pp. 1-86. Kepner, R.A., 1951. Effectiveness of orchard heaters. Calif. Agr. Exp. Sta. Bull., 723. Martsolf, J.D., 1974. Practical frost protection -- frost incidence, site selection, control methods. Pa. Fruit News, 53: 15--30. Martsolf, J.D. and Panofsky, H.A., 1975. A box model approach to frost protection research, Hortic. Sci., 10: 108--111. Perry, K.B., Martsolf, J.D. and Norman, J.M., 1977. Radiant o u t p u t from orchard heaters. J. Am. Soc. Hortic. Sci., 102: 101--105. Turrell, F.M., 1973. The science and technology of frost protection. In: W. Reuther (Editor), The Citrus Industry, Vol. 3. Rev. Edn., University of California Press, Berkeley, pp. 338--446, 505--528. Valli, V.J., 1970. Basic principles of freeze occurrence and the prevention of freeze damage to crops. Proc. Fla. State Hortic. Soc., 83: 98--109. Welles, J.M., Norman, J.M. and Martsolf, J.D., 1979. An orchard foliage temperature model. J. Am. Soc. Hortic. Sci., 104: 602--610. Wilson, E.B. and Jones, A.L., 1969. Orchard heater measurements. Wash. Agr. Exp. Stn. Circ., 511, 18 pp.