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The heat balance of apple buds and blossoms. Part III. The water requirements for evaporative cooling by overhead sprinkler irrigation
Agricultural and Forest Meteorology, 37 (1986) 175--188 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands
175
THE HEAT BALANCE OF APPLE BUDS AND BLOSSOMS P a r t III. T h e w a t e r r e q u i r e m e n t s f o r e v a p o r a t i v e c o o l i n g b y o v e r h e a d s p r i n k l e r irrigation P.J.C. HAMER Pomology Department, East Mailing Research Station, Maidstone, Kent (Gt. Britain)
(Received November 1, 1985; revision accepted January 16, 1986) ABSTRACT Hamer, P.J.C., 1986. The heat balance of apple buds and blossoms. Part III. The water requirements for evaporative cooling by overhead sprinkler irrigation. Agric. For. Meteorol., 37:175--188. The heat balance equations for wetted and unwetted buds were solved in order to estimate the effect of water sprinkling on bud temperature. This involved determination of the heat transfer coefficients of buds at different stages of development. The fluxes of sensible and latent heat were found to be twice or three times as great as that of net radiation. The estimates of the degree of cooling for a given sprinkler irrigation rate, based on these heat balance studies, agreed well with those found by field experimentation. INTRODUCTION T h e use o f w a t e r sprinkling to c o o l f r u i t b u d s and d e l a y t h e i r d e v e l o p m e n t w a s first suggested as a m e t h o d o f f r o s t p r o t e c t i o n b y A n d e r s o n et al. (1973). T h e b u d s m a y be p r o t e c t e d f r o m f r o s t d a m a g e b y delaying t h e frostsensitive l a t e r stages o f d e v e l o p m e n t . T h e d e v e l o p m e n t o f t h e w e t t e d buds c a n be e i t h e r slowed d o w n w h e n t h e s p r i n k l e d b u d t e m p e r a t u r e is g r e a t e r t h a n 4.5°C, or s t o p p e d w h e n it is less t h a n 4.5°C. A p p l e b u d s are initiated in late s u m m e r and early a u t u m n . S u b s e q u e n t l y , a winter-chilling r e q u i r e m e n t m u s t be m e t b e f o r e t h e b u d s e m e r g e f r o m d o r m a n c y . In S o u t h - E a s t England, b u d d e v e l o p m e n t c o m m e n c e s at t h e end o f a c o l d spell in late J a n u a r y or early F e b r u a r y ( H a m e r , 1980). T h e r a t e o f b u d d e v e l o p m e n t , o n c e s t a r t e d in spring, is a linear f u n c t i o n o f t h e r m a l t i m e m e a s u r e d in d a y - d e g r e e s a b o v e a t h r e s h o l d value o f 4.5°C. T h e a c h i e v e m e n t o f t h e m a x i m u m p o s s i b l e d e l a y b y e v a p o r a t i v e cooling requires t h e c o n t i n u a l a p p l i c a t i o n o f w a t e r w h e n e v e r t h e t e m p e r a t u r e is a b o v e 4.5°C, o v e r a p e r i o d f r o m t h e beginning o f F e b r u a r y until full b l o o m . In this p a p e r , t h e h e a t balances f o r u n w e t t e d and w e t t e d b u d s are established a n d t h e " c o o l i n g " e f f e c t r e l a t e d t o e n v i r o n m e n t a l f a c t o r s and t h e p h y s i c a l c h a r a c t e r i s t i c s o f t h e bud. T h e p r e c i p i t a t i o n r a t e s r e q u i r e d f o r e f f e c t i v e e v a p o r a t i v e cooling are t h e n c a l c u l a t e d . Finally, e s t i m a t e s o f t h e w a t e r c o n s u m p t i o n r e q u i r e d are r e l a t e d t o t h e " c o o l i n g " a c h i e v e d in t h e field w i t h a small-scale t e s t s y s t e m . 0168-1923/86/$03.50
© 1986 Elsevier Science Publishers B.V.
METHODOLOG Y
Estimation o f the "cooling" effect When w a t e r is a p p h e d evaporative cooling o c c u r s and the heat balance e q u a t i o n f o r fully wett, ed buds can be w r i t t e n as: /~w = Cw +1-7,,
(t)
w h e r e / ~ w is t h e n e t gain o f h e a t b y radiation, Cw is the loss o f sensible heat by c o n v e c t i o n and I-7, is t h e loss o f l a t e n t heat o f e v a p o r a t i o n . Each t e r m is a m e a n e n e r g y flux per unit surface area in W m -2 . E q u a t i o n 1 assumes t h a t the rate o f change o f s t o r e d heat and t h a t latent heat loss t h r o u g h the cuticle and s t o m a t a are small (Hamer, 1984). F o r the w e t t e d b u d at t e m p e r a t u r e Tw, m e a n radiative, sensible and l a t e n t heat fluxes can be w r i t t e n as:
Rw
= Rn~-- hr(T~ -- Ta)
(21
Cw = hw(T w -Ta)
14,,: = h v ( e s ( T w ) -
(3)
e~)/'7
(4)
w h e r e Rr~ is the i s o t h e r m a l net radiation, Ta is air t e m p e r a t u r e , e,(Tw ) is the s a t u r a t e d v a p o u r pressure over w a t e r at the t e m p e r a t u r e o f the bud and e~ ( m b a r ) is the v a p o u r pressure in the s u r r o u n d i n g air. Here, hr, hw and h v are the h e a t t r a n s f e r c o e f f i c i e n t s for r a d i a t i o n exchange, heat and l a t e n t heat, respectively. F o r d a y t i m e spring t e m p e r a t u r e , h r ~ 5 W m -2 K -1 , h v ~- 1.08 h w (Hamer, 1 9 8 4 ) and 7 the p s y c h r o m e t e r c o n s t a n t = 0 . 6 5 m b a r K --L. E q u a t i o n 4 enables t h e rates o f e v a p o r a t i o n for a free w a t e r surface to be e s t i m a t e d . T o p r o d u c e t h e m a x i m u m e f f e c t o f evaporative cooling, w a t e r s h o u l d be applied at the same or greater rate as w a t e r evaporates f r o m t h e bud surface. Excess w a t e r will r u n off. F r o m eqs. 1--4, the t e m p e r a t u r e o f w e t t e d buds can be given by:
Tw = Ta + (Rnil(hr + hw ) ) - - (es(Tw)
-
-
ea)/7*
{5t
where 3'* = 7(hr + h w ) / h v
(6)
F o r u n w e t t e d buds, t h e l a t e n t heat flux t e r m in t h e h e a t balance eq. 1 is zero and assuming the u n w e t t e d and w e t t e d buds are b o t h e x p o s e d t o t h e same r a d i a t i o n e n v i r o n m e n t , t h e t e m p e r a t u r e o f the u n w e t t e d buds, Tb, can be given by:
Tb = Ta + Rni/(hr + he)
(7)
w h e r e h e is the c o e f f i c i e n t o f h e a t transfer o f u n w e t t e d buds. At the end o f d o r m a n c y w h e n w a t e r sprinkling is c o m m e n c e d , t h e coefficients hc and h w will be equal because the buds are at t h e same p h e n o l o g i c a l
177 stage o f d e v e l o p m e n t . In this case, based o n eqs. 5 and 7, " c o o l i n g " , being d e f i n e d as the t e m p e r a t u r e d i f f e r e n c e b e t w e e n u n w e t t e d and w e t t e d buds, simplifies to: Tb -- Tw = ( e s ( T w ) - - ea)/'Y $
(8)
H a m e r {1985) s h o w e d t h a t t h e r e is a seasonal change in he f o r the same wind speed c o n d i t i o n s due t o changes in b u d size and shape. Because cooling delays the d e v e l o p m e n t o f w e t t e d buds, as t h e season progresses t h e differe n c e b e t w e e n the d e v e l o p m e n t o f the w e t t e d and u n w e t t e d buds increases so t h a t h e ve hw. As the u n k n o w n i s o t h e r m a l net r a d i a t i o n is c o m m o n in eqs. 5 and 7 it can be eliminated; this results in a c o m p l e x e q u a t i o n f o r b u d " c o o l i n g " , w h i c h includes air t e m p e r a t u r e , Ta, in t h e expression. By introducing a t e r m hc~, w h i c h is the m e a n o f t h e c o e f f i c i e n t s h e and h w , t h e air t e m p e r a t u r e t e r m s can be eliminated. Bud cooling can t h e n be w r i t t e n (see A p p e n d i x ) in t h e same general f o r m as eq. 8 as: Tb -- Tw = ( e s ( T w ) - - ea)/~{ +
(9)
where 7 + = ~/(hr + hcw)/hv
(10)
E q u a t i o n 9 is valid if Tu -- Ta ~ Ta -- Tw, w h e n the w e t t e d buds are below air t e m p e r a t u r e b y a margin similar t o the t e m p e r a t u r e d i f f e r e n c e o f the u n w e t t e d buds above air t e m p e r a t u r e . Even if this c o n d i t i o n is n o t satisfied, errors in estimating "),+ are small f o r U.K. c o n d i t i o n s (see A p p e n d i x ) .
Estimation o f the precipitation rate required to maximise the "cooling" effect Water d r o p l e t s applied t o buds f r o m sprinklers have a relatively large h o r i z o n t a l c o m p o n e n t o f v e l o c i t y so t h a t the buds i n t e r c e p t m o r e w a t e r t h a n w o u l d be d e p o s i t e d o n a h o r i z o n t a l area equal t o t h e i r cross-section. Businger ( 1 9 6 5 ) i n t r o d u c e d an i n t e r c e p t i o n factor, I, w h i c h d e p e n d s o n the ratio o f surface area t o cross-sectional area o f t h e bud and t h e d i r e c t i o n o f w a t e r d r o p l e t s falling o n the bud. N o r m a l l y , irrigation systems are designed to a p p l y a given d e p t h o f w a t e r per unit t i m e o n a h o r i z o n t a l surface, r a t h e r t h a n a p a r t i c u l a r flux o f heat. T h e c o r r e s p o n d i n g p r e c i p i t a t i o n rate, P, can be e x p r e s s e d in m m h -1 by:
P = I~qw/pwLe = I H w / 6 8 8
(11)
Pw is t h e d e n s i t y ( ~ 103 k g m -3) and L e is the l a t e n t heat o f v a p o r i s a t i o n (24.8 x 10 s J kg -1 at 10°C) o f water. T h e r e is a simple relationship b e t w e e n the a m o u n t o f w a t e r e v a p o r a t e d f r o m the surface o f a w e t t e d bud and the " c o o l i n g " effect. P r e p r e s e n t s t h e m i n i m u m p r e c i p i t a t i o n rate r e q u i r e d t o p r o d u c e the m a x i m u m " c o o l i n g " e f f e c t and can be e s t i m a t e d b y c o m b i n i n g eqs. 4 and 9--11. T h e p r e c i p i t a t i o n rate r e q u i r e d can be w r i t t e n as:
17~
P = I(h~ + hcw)(T b ..... !',,. i/688 By i n t e r m i t t e n t l y w e t t i n g is restricted i n d i c a t e d by eq. 12. cooling e f f e c t (Tb --
~12
applying w a t e r so t h a t excess w a t e r is n o t applied, buci and t.he t e m p e r a t u r e depression will n o t be as large a:~ In these circumstances, the p r e c i p i t a t i o n rate P ' and the, rl':v ) are related by:
P' = I(hr + hc~)(Tb -- T w ) / 6 8 8
t13
In t e r m s o f " c o o l i n g " in units o f day-degree C (D} and d e p t h o f w a t e r applied (W) in m m , eq. 13 becomes: D -
28.7 W
I(h~ + h~.)
~14)
Determination o f coefficients o[ heat transfer and isothermal net radiation and validation o f the relationship between precipitation rate and "cooling" T h e c o e f f i c i e n t s o f heat t r a n s f e r for u n w e t t e d and w e t t e d buds hc and hw, respectively, d e p e n d o n the n a t u r e o f the c o n v e c t i o n regime and the develo p m e n t a l stage o f the bud or blossom. T h e n o r m a l m o d e o f heat t r a n s f e r for apple buds in t h e field is by f o r c e d c o n v e c t i o n (Hamer, 1 9 8 5 ) and the c o e f f i c i e n t s hw and he, w h i c h vary w i t h w i n d speed, u ( m s - l ) , can be d e s c r i b e d by the e q u a t i o n s :
hw = C1 + C 2 u ~s ~ Cwu°'S;hc "~ Cb u°'s
(15)
w h e r e C1 and C2 are c o n s t a n t s at a p a r t i c u l a r stage o f d e v e l o p m e n t , w h i c h c h a n g e as bud g r o w t h proceeds. It m a y be assumed t h a t the w e t t e d buds have t h e same heat t r a n s f e r c o e f f i c i e n t s as t h e u n w e t t e d buds at the same phenological stage. P a p e r I r e p o r t s d e t e r m i n a t i o n s o f C 1 and C 2 f o r early d e v e l o p m e n t a l stages f r o m " ' d o r m a n c y " to " g r e e n c l u s t e r " , using r a d i a t i o n and t e m p e r a t u r e m e a s u r e m e n t s m a d e at night in low wind speed c o n d i t i o n s . Values o f C 1 and C2 r a n g e d f r o m 3 t o 11 and 46 t o 62, respectively. However, in d a y t i m e , wind speeds in an o r c h a r d c a n o p y are higher. T h e usual range is f r o m 1 to 4 m s -l , with an average o f ca. 2 m s -1 (see H a m e r , 1984}. In this range, the single c o n s t a n t Cw can be used in eq. 15 with little loss o f accuracy. F o r the average wind speed and d o r m a n t buds, C1 = 11 and C2 = 46 while Cw = 54. T h e a p p r o x i m a t i o n gives values o f h w w i t h i n + 5% o f t h o s e f r o m the full e q u a t i o n o v e r the wind speed range 1--4 m s -1 . T h e u n k n o w n h e a t t r a n s f e r c o e f f i c i e n t s a p p r o p r i a t e to later stages o f d e v e l o p m e n t can be e s t i m a t e d f r o m m e a s u r e m e n t s o f t e m p e r a t u r e , T b and T,, and v a p o u r pressure e a. T h e b u d c o n s t a n t s C b and Cw are o b t a i n e d f r o m eqs. 10 and 15 using 7 = 0 . 6 5 m b a r K -1 (at 10°C), hr = 5 W m -2 K -1 and hv = 1.08 hw. H e n c e : C b
---~
2 ( 1 . 6 6 C w 7 + - 5/u °'s) -- Cw
(16)
In the e x p e r i m e n t to be described, at o n e site { " c o n t i n u o u s " } apple buds
179 were wetted to run-off and 7+ determined from the relationship between the "cooling" effect T b - - T w and the vapour pressure difference es(Tw)--e~ (eq. 9). These measurements enabled the seasonal changes of the bud constant, C b, to be determined (eq. 17) because the constant Cw for wetted buds was k n o w n t h r o u g h o u t the period of water sprinkling. The estimation of isothermal net radiation appropriate to the bud is difficult (Hamer, 1984) due to the complex bud geometry and is further complicated by shading by adjacent leaves, flowers and twigs. However, in the field, R,~ for the unwetted buds can be estimated from measurements of the coefficient of heat transfer hc (~ Cb u °'s) and the temperature difference Tb -- Ta (eq. 7). Rm is likely to be closely related to the net radiation balance ( R ~ ) of the underlying orchard, which can be readily measured. The ratio, ~, of isothermal net radiation and the measured radiation balance of the orchard can be written simply as ~ = R m / R n ~ t. At other sites, two levels of intermittent watering were applied, referred to as " i n f r e q u e n t " and " f r e q u e n t " . These minimised run-off and enabled validation of the relationship (eq. 14) between day-degree cooling and depth of water applied. Experimental procedure
Full details of the experimental orchard and instrumentation were given in Hamer (1985). Three pairs of Cox's Orange Pippin trees on M.26 rootsock (Sites 2, 4 and 6) were watered using minisprinklers. The water supplies to the sprinklers were controlled independently using solenoid valves. Two similar control sites (1 and 5) were unwatered. In all cases, water was applied only when air temperature measured at the control Site 1 was greater than 4.5°C. The criteria for irrigation of the three sites differed. At Sites 2 and 4, water was applied intermittently. The application was controlled using signals from four wetness sensors at each site, being turned on when t h e y were dry. The sensors were taped to the branches of watered trees at about 1 m height. The wetness sensors were wired in parallel at Sites 2 and in series at Site 4. Set upper and lower voltage limits (Hamer, 1984) initiated the shutting and opening of the valves. Because the voltage o u t p u t depends on the most wet sensors in the case of parallel-wired sets of sensors and the least wet sensors for series-wired sensors, the watering regime was " i n f r e q u e n t " and " f r e q u e n t " , respectively. At Site 6, water was applied continuously whenever the net radiation was positive. This regime was interrupted when the o u t p u t control facility of the data-logger failed from 23 March to 4 April 1982, inclusive. During this period, Sites 2 and 4 were unwatered, whilst at Site 6 the sprinkler operated continuously. At each site, the temperatures of buds in an upward-facing exposed position at mid-tree height were monitored by inserting fine-gauge (36 s.w.g.) copper-constantan thermocouples into four separate buds. The thermocouples were wired in series and referenced to the air temperature at 1.1 m
18C} 600 ,,
," ~: 4 0 0
200
•D cE
y
St
.,
. .....
Rnet'
, "/
0
-200
08:00
,0:00
,2200
,,;00
,eoo
,800
20.00
22.00
Time (hi
10. C 2.0
v
u E
4.
1.0
2>
x ¸¸
....'r-
0
08~00
10100
12100
14100
16100
18100
20,00
22.00
T i m e (h)
Fig. 1. Diurnal variation of environmental parameters on 27 March 1982. using well-ventilated psychrometer units. The units also housed wet and dry bulb thermometers using 100 ~ platinum resistance elements. These were calibrated to an accuracy of +0.1°C. At Sites 1 and 6, there were two measurements of bud temperature and one at the other sites. Measurements were made at 10-s intervals using a computer-controlled data-logger. Mean values for 10-min periods were retained by the logger for analysis. The depth of water applied per day was measured by 65-ram-diameter plastic funnels with their mouths horizontal, placed adjacent to the buds with inserted thermocouples. At each site, four funnels were connected to a single water bottle. These "gauges" agreed within 3% with a standard rain gauge (Hamer, 1986). Any rain collected at the unwatered control site was deducted from the daily measurements at the other sites. The precipitation rates were estimated from daily measurements of water depth and the recorded time the valve was open at each site (Hamer, 1984). On days following frost it was impossible to measure the applied water directly and it was estimated from the precipitation rates and the length of time the valves were open.
181 20,
15
!
o F-
08:00
10:00
12:00
14:00
16:00
18:00
20:00
22.00
Time (h)
Fig. 2. M e a n h o u r l y values o f t h e w e t ( " ) a n d d r y ([]) b u l b t e m p e r a t u r e s . T e m p e r a t u r e s o f u n t r e a t e d (o) a n d " c o n t i n u o u s " (o) t r e a t m e n t b u d s o n 27 M a r c h 1 9 8 2 (* i n d i c a t e s t h e p e r i o d over w h i c h " u n w e t t e d b u d s " w e r e dry, as i n d i c a t e d b y a w e t n e s s sensor in t h e u n s p r i n k l e d trees).
The stages of bud development were determined by visual examination of the trees according to an established classification (Table I; Hamer, 1980). Observations were made at weekly intervals from the beginning of March until mid-April and two to three times per week subsequently. RESULTS
Sprinkling was commenced on 11 February 1982 and continued until 6 May 1982, when the unwatered control trees blossomed. The " i n f r e q u e n t " and " f r e q u e n t " treatment trees blossomed 5 and 10 days later, respectively. Most of the " s p u r " buds that received the " c o n t i n u o u s " treatment failed to develop and " s e c o n d a r y " buds blossomed 19 days after the control. Data from the " c o n t i n u o u s " t r e a t m e n t for 17 March demonstrate many features of evaporative cooling of apple buds; at this time wetting was continued at night due to equipment failure p. 179. The diurnal courses of solar (St) and net radiation, wind speed (measured at 1.2 m above ground level) and vapour pressure are shown in Fig. 1. At 06.00 h there was thick fog and consequently the wet and dry bulb air temperatures and the temperatures of both the u n w e t t e d and wetted buds were all the same (Fig. 2). With light winds and high solar radiation, bud temperature exceeded air temperature by 4.5°C at l l . 3 0 h (the highest
400
2OO ./
m
j
/ \ w •, J
200
\
E
w @ -r"
61111
0s:0o
10:0o
,2:00
1,:0o
,6:00
,s:00
o:oo
2:0o
Tlme (h) Fig. 3. H o u r l y e s t i m a t e s o f t h e c o n t r i b u t i o n s o f fluxes o f l a t e n t (o), sensible h e a t (m) a n d n e t r a d i a t i o n (x) t o t h e h e a t b a l a n c e o f a c o n t i n u o u s l y w e t t e d b u d o n 27 M a r c h 1 9 8 2 . T h e c o e f f i c i e n t s o f h e a t t r a n s f e r for t h e s e w e t t e d b u d s were e s t i m a t e d as h w --- 11 ÷ 4 6 u °'s ( a p p r o p r i a t e f o r a d o r m a n t b u d , H a m e r , 1 9 8 6 ) w h e r e u is t h e m e a n w i n d speed. T h e fluxes o f sensible h e a t , t h e l a t e n t h e a t o f e v a p o r a t i o n a n d n e t r a d i a t i o n were calculated f r o m eqs. 3, 4 ( w h e r e h v = 1.08 hw) a n d 1, respectively.
recorded in 1982). In the evening when R ~ was negative, bud temperature was lower than air temperature by 0.9°C. When net radiation was positive, the temperature of the watered buds exceeded the measured wet bulb temperature, but at night these temperatures were similar. The m a x i m u m temperature difference between unwetted and wetted buds (i.e., m a x i m u m "cooling") was 9.2°C, the highest value recorded by the author during the years 1981--1983. Hourly values of heat gains and losses for the continuously sprinkled buds are shown in Fig. 3. At the start of the period all fluxes were approximately zero. Thereafter until 11:00 h, Tw was greater than Ta and, therefore, there was a loss of sensible heat. For the remainder of the period, Tw was lower than air temperature and, therefore, Cw was a gain. At 14.30 h when T w -Ta = - 6.4°C, Cw peaked a t - 370 W m -2 . During early afternoon the loss of latent heat peaked at 55OW m -2 , subsequently declining gradually to zero by 21.00 h. Calculated values of isothermal net radiation, R ~ , for the wetted buds are plotted in Fig. 4 against calculated values of R~a for unwetted buds. The isothermal net radiation was positive t h r o u g h o u t the daylight hours, with a peak value o f 2 2 0 W i n -2 , but a loss at night. As indicated by the 1:1 line in the figure, the estimated values of isothermal net radiation for the u n w e t t e d and w e t t e d buds were not significantly different.
183 250
OQ
200 E 150
lOO
50'
'E
•
-50 -50
0
50
100
150
R ni for unwatered buds
200
250
(W m-=)
Fig. 4. T h e values o f i s o t h e r m a l n e t r a d i a t i o n , _Rni, c a l c u l a t e d for t h e " c o n t i n u o u s " treatm e n t b u d s f r o m Rni = h r ( T w - - Ta) 4- Cw ~- Hw ( f r o m eqs. 1 a n d 2) p l o t t e d against c o r r e s p o n d i n g values for t h e u n w e t t e d c o n t r o l buds, Rni ---- (hr + h c ) ( T b - - Ta) ( f r o m e~. 7). T h e c o e f f i c i e n t s o f h e a t t r a n s f e r for t h e u n w e t t e d b u d s were t a k e n as 11 -t- 55 u , t h e value a p p r o p r i a t e f o r 27 M a r c h ( H a m e r , 1 9 8 5 ) . H o u r s w h e n t h e c o n t r o l b u d s were w e t t e d b y d e w w e r e e x c l u d e d f r o m t h e analysis.
T h e seasonal variation o f the b u d c o n s t a n t Cb and t h e ratio o f i s o t h e r m a l n e t r a d i a t i o n to o r c h a r d net r a d i a t i o n (R~/R~ot) w e r e d e t e r m i n e d at w e e k l y intervals. T h e d a t a w e r e regarded as suitable f o r analysis w h e n : (i) T h e wetness sensor at the c o n t r o l (Site 1) was dry. (ii) T h e t w o m e a s u r e m e n t s o f u n w a t e r e d b u d t e m p e r a t u r e agreed within 0.3°C, t h e m e a n o f t h e t w o readings being T b . T h e same t e m p e r a t u r e c r i t e r i o n was used f o r t h e " c o n t i n u o u s " t r e a t m e n t , the m e a n o f t h e t w o readings being T w . (iii) M e a s u r e m e n t s o f v a p o u r pressure at t h e u n w a t e r e d Sites 1 and 5 agreed w i t h i n 0.2 mbar, t h e m e a s u r e m e n t at Site 1 being e a. (iv) T h e wind speed w i t h i n the o r c h a r d c a n o p y was less t h a n 4 m s -1 (it was f o u n d t h a t wind speeds greater t h a n this resulted in some o f t h e buds remaining d r y ) . (v) T h e sprinkler was o p e r a t i n g c o n t i n u o u s l y t h r o u g h o u t t h e p e r i o d o f measurement. L i n e a r regression analyses (with the lines c o n s t r a i n e d t o pass t h r o u g h t h e origin) w e r e carried o u t using m e a n h o u r l y values for data c o l l e c t e d at w e e k l y intervals. T h e reciprocals o f t h e slopes (7 +} d e t e r m i n e d are given in T a b l e I t o g e t h e r w i t h t h e n u m b e r o f d a t a p o i n t s used and the values o f r 2. T h e r e were
TABLE I The values of ~ obtained from tile regression of 7"b- T w against e s ( l w ] ca ;~ measurements collected over weekly periods during the spring of 1982. The data are t'r,m~ the "continuous" t r e a t m e m in which excess water was applied to I he buds Inclusive dates
N umber of data points
"~ mbar K -1 ~
Percem ~ariance
i~ 20 4~ 32
0.64 0.64 0.63 0.68
92 97 ~.)2 95
5--i I April
32
0.67
97
12--18 April 19--25 April 26 April--2 May 3--7 May
40 60 29 !7
0.72 0.71 0.74 0.85
84 89 94 92
8 - 1 4 March 15--21 March 22--18 March 29 March--4 April
I/ 2 )
i n s u f f i c i e n t d a t a f o r analysis p r i o r to 8 March. As t h e b u d s d e v e l o p e d f r o m d o r m a n c y t o full b l o o m , t h e slopes d e c r e a s e d and 3,÷ c h a n g e d f r o m 0 . 6 4 to 0.85 m b a r K - ' , respectively. F o r t h e " c o n t i n u o u s l y " w e t t e d buds, b u d d e v e l o p m e n t was d e l a y e d s u f f i c i e n t l y t o e n a b l e t h e b u d c o n s t a n t s t o be c a l c u l a t e d f r o m m e a s u r e m e n t s o f m e a n w i n d s p e e d and t h e b u d c o n s t a n t s C 1 and C 2 f o r d i f f e r e n t develo p m e n t a l stages. C1 a n d C2 w e r e k n o w n f o r b u d d e v e l o p m e n t a l stages f r o m d o r m a n c y to green c l u s t e r ( H a m e r , 1 9 8 5 ) a n d Cw was d e t e r m i n e d at w e e k l y intervals f r o m eq. 15. T h e c o n s t a n t s f o r t h e u n w e t t e d buds, Cb, w e r e estim a t e d f r o m ~'÷, w i n d s p e e d and Cw (eq. 16). T h e changes o f Cw a n d C b t h r o u g h t h e p e r i o d o f b u d g r o w t h are s h o w n in Fig. 5. T h e b u d c o n s t a n t Cw (in units o f W m -2 K -l ( m s -1)-°'5) f o r t h e " c o n t i n u o u s l y " w e t t e d b u d s r a n g e d f r o m 52 t o 65. Cb increased f r o m 54 w h e n d o r m a n t to 101 at full b l o o m . The bud constants were calculated for the unwatered buds from " d o r m a n c y " t o " g r e e n c l u s t e r " f r o m t h e d a t a given in P a p e r I ( H a m e r , 1 9 8 5 ) and t h e m e a n w i n d speed. T h e c o n s t a n t s are in r e a s o n a b l e a g r e e m e n t w i t h t h e c a l c u l a t e d values o f C b . Figure 6 s h o w s values o f R , , I / R , ~ , f o r t h e d i f f e r e n t p e r i o d s o f b u d g r o w t h derived b y regression w i t h a f i x e d i n t e r c e p t t h r o u g h t h e origin. T h e values r a n g e d f r o m 0 . 3 0 t o 0 . 4 4 w i t h a m e a n o f 0.38. T h e r e l a t i o n s h i p b e t w e e n w a t e r use and cooling was f o u n d t o be a p p r o x i m a t e l y linear (Fig. 7), in a g r e e m e n t w i t h e x p e c t a t i o n s f r o m eq. 14. DISCUSSION AND CONCLUSIONS Values of T+ ranged from 0.63 to 0.85 m b a r K -1 . Early in the season, the values of 7 + agreed with the theoretical estimated (eq. 7) of 0.64 mbar K -I
185 100
8O ma o
E 60
? E v
-- 4 0 c
C 0 O "D
m 2(1
Continuous
,o
20
;o I
ME
o
March
,;
;
A p r il
May
Fig. 5. S e a s o n a l v a r i a t i o n o f " b u d c o n s t a n t " C b f o r t h e " c o n t i n u o u s " t r e a t m e n t b u d s (©) a n d t h e u n w a t e r e d c o n t r o l b u d s (o), c a l c u l a t e d f r o m eq. 16. V a l u e s o f C b (G) w e r e e s t i m a t e d f r o m C1 a n d C2 in P a p e r I. T h e r e c o r d e d stages o f b u d d e v e l o p m e n t are i n d i c a t e d as M E --- m o u s e ear, GC -- g r e e n cluster, PB = p i n k b u d , IB - first b l o o m , F B = full b l o o m . 0.6
0.4
t-
tr
-~ 0.2 tr
10 '
~0 March
3'o
1'9
9
April
2'~
9'
May
Fig. 6. S e a s o n a l v a r i a t i o n o f t h e ratio o f i s o t h e r m a l n e t r a d i a t i o n t o t h e n e t r a d i a t i o n measured above the orchard canopy.
188
'5
(a)
2.5
1.5
15
-/
/
/
eee
v
-==
"== p
.lj •
20
20
1.0
1.0
e/
/ 0.5
05
:
•
/ 5
10
Rate of application of water, W(mm d a y
15 -~)
o
/
; t'o Rate of application of water, W ( m m
d a y °~)
Fig. 7. The daily measured mean cooling (T b -- Tw) for (a) "infrequent" and (b) "frequent" treatments plotted against depth o f water collected in "rain" ganges. The regression lines (forced throught the origins) have slopes o f 0.14 day-degree C mm -1 (r 2 = 40.8) and 0.15 day-degree C mm -1 (r 2 ~ 7616), respectively.
(hw typically 80 W m -~ K -1 ), for buds at the same phenological d e v e l o p m e n t stage. In general, the estimated values of bud constant Cb and hence the coefficient of heat transfer for a given wind speed, increased as bud development proceeded. Hamer (1985) showed a similar seasonal trend in the coefficient of heat transfer estimated from measurements on unwetted buds only. The ratio of the estimated net radiation, Rm, to the measured net radiation to the orchard, R ~ , ranged from 0.30 to 0.44. This range is consistent with geometry. If the buds were assumed to be horizontal plates the ratio would be about 0.5. It was shown in Paper I that the net flux of short-wave radiation for a sphere is complex and is less than for horizontal plates (Gates, 1980). As buds expand, their shapes change and they become more difficult :to characterise. Furthermore, as the season progresses, the buds are more likely to be shaded by adjacent leaves and buds {or flowers). In the conditions when sprinkling is most likely to produce effective cooling, the flux of net radiation (normally regarded as the driving term in the heat balance equation) is much smaller than the fluxes o f latent and sensible heat (Fig. 3). The observed mean ratio of R n i / R ~ = 0.38 can,
187 therefore, be used t h r o u g h o u t the period of bud growth with little loss of accuracy. The validity of the value of the interception factor (I = 2) employed by Hamer (1986) can be assessed from the results for " i n f r e q u e n t " and "'freq u e n t " wetting presented in Fig. 7. From theory (eq. 14), the slopes of the graphs should be 28.7/(I(hr + hc~)). If the mean seasonal values of hw and hc are estimated from the data in Fig. 5, for the mean wind speed of 1.9 m s -1 , C w and C b are equal to 60 and 70W m -2 K -1 (m s-1 )-0.s, respectively and the coefficient h ~ = 90 W m - : K -1 . The corresponding slope in Fig. 7 expected from t h e o r y for I = 2 is 0.15 day-degree C mm -1 . This is remarkably close to the measured values of 0.14 and 0.15 day-degree C mm -1 found for the " i n f r e q u e n t " and "'frequent" treatments, respectively. The data in Fig. 7 for the " i n f r e q u e n t " treatment offer further evidence for the seasonal increase in bud constant. Extrapolation of the data to zero depth of water, indicates a negative value for the "cooling". For the hypothetical condition of no watering, the temperature of these buds (whose blossoming was delayed by 5 days) would be greater than the control, as would be expected because Cw ~ CD. The precipitation rate requirement for daytime cooling, which follows the potential flux of latent heat, will on average vary sinusoidally with time of day (see Fig. 3). The area under the curve represents the mean daily requirement, which is typically 10 mm. A sensible criterion for the design of an irrigation system for daytime cooling is the mean daily m a x i m u m precipitation rate. From eq. 12, a precipitation rate of 1.4 mm h -1 would give T b -T w = 5°C, which was the mean daily m a x i m u m temperature depression in 1982. An effective control system would be desirable to enable intermittent application. The seasonal water requirements for evaporative cooling can be estimated as follows. Using 10 years' climatologcial and phenological data, Hamer (1980) showed t h a t for the m a x i m u m cooling of 122 day-degree C, the wetted buds would be at "green cluster" when u n w e t t e d buds were at "full bloom". The average seasonal water requirement for such a delay is ca. 800 mm (calculated from eq. 14 with a value of hew = 90 W m -2 K -1). This is a vast q u a n t i t y of water and on most farms would be prohibitively expensive (Hamer, 1984). This paper has established a sound theoretical understanding of the technique of evaporative cooling. However, it also demonstrates the need for very substantial quantities of water to delay blossoming effectively. It is unlikely, therefore, that this technique will be adopted by the U.K. fruit industry, although the sprinkler technique in itself may be useful in other climates where trees are regularly submitted to high-temperature stress. ACKNOWLEDGEMENTS I am grateful to Dr. J.A. Clark of the Department of Physiology and Environmental Science, Nottingham University School of Agriculture, for his
constructive c o m m e n t s o n tJ~is paper and to P. N e w m a n for hi~ general assistance. APPENDIX
Derivation o f the relationship between bud "'cooling" and the coefficient o f heat transfer, h c'w
The isothermal net radiation terms can be eliminated from eqs. 5 and 7 to give h r T b w + (hcTba - h w T w a + hcTba -- hwTwa)/2
= h v ( e s ( T w ) - - Ca)/7
(17)
w h e r e Tbw = T b -- T w , Tba = T b - T a a n d Twa = T w -- T a. T h e c o m b i n e d h e a t t r a n s f e r c o e f f i c i e n t hcw = (h e + hw)/2 a n d t h e t e r m hcwTbw c a n be e x p a n d e d t o : hcwTbw
_-- ( h c T b a - - h c T w a 4- h w T b a .... h w T w a ) / 2
118)
F r o m eqs. 17 a n d 18: (h r + h c w ) T b w + (h c -- h w ) T w a / 2 +- ( h c - hw)Tba/2
= h v ( e s ( T w ) - - ea)/~,
(19)
E q u a t i o n 19 s i m p l i f i e s if Twa + Tba =: 0, w h e n b u d " c o o l i n g " , Tbw , c a n be w r i t t e n as eqs. 9 a n d 10 in t h e t e x t : Tbw =
T b-
(es(Tw)--ea)/7 +
Tw =
I9)
where
7 + ~= "),(h r -b hcw)/h v
(101
If t h e c o n d i t i o n Twa A Tba :/= 0 t h e n eq. 19 c a n be w r i t t e n as: -- T b -- T w + ( e s ( T w ) .... Ca)fly + 4- e T
Tbw
(20)
where eT
(Tba + T w a ) ( h w + hc)/(2(hr + h e w ) )
=
(21)
r e p r e s e n t s a n e r r o r t e r m . I n S o u t h - E a s t E n g l a n d , t h e largest v a l u e o f e T t h a t is likely t o o c c u r is f o r t h e c o n & t i o n s Tba = 0 a n d Twa = - - 5 C. F o r t y p m a l v a l u e s o f h w a n d h e o f •
.
-2
-1
,
o
'
o
.
.
o
80 and 1 0 0 W i n K , respectlvely, eT = 0.5 C and compares wlth Tbw = 5 C. Under these extreme conditions, there would be an error of 1 0 % in the estimation of 7 +. REFERENCES Anderson, J.L., Richardson, E.A., Ashcroft, G.L., Keller, J., Alfaro, J., Hanson, G. and Griffin, R.E., 1973. Reducing freeze damage to fruit by overhead sprinkling. Utah Sci., December, 108--109• Businger, J.A., 1965. Protection from the Cold. Meterok Monogr., 6:74--80. Gates, D.M., 1980. Biophysical Ecology• Springer-Verlag, N e w York, 161 pp. Hamer, P.J.C., 1980. A model to evaluate evaporative cooling of apple buds as a frost protection technique• J. Hortic. Sci., 55:157--163. Hamer, P.J.C., 1984. The heat balance of apple buds and blossoms in frost protection. Ph.D. Thesis, University of Nottingham. Hamer, P.J.C., 1985. The heat balance of apple buds and blossoms• Part I. Heat transfer in the outdoor environment. Agric. For. Meteorol, 35: 339--352. Hamer, P.J•C., 1986• The heat balance of apple buds and blossoms. Part II. The water requirements for frost protection by overhead sprinkler irrigation. Agric. For. Meteorol., 37 :159--174
175
THE HEAT BALANCE OF APPLE BUDS AND BLOSSOMS P a r t III. T h e w a t e r r e q u i r e m e n t s f o r e v a p o r a t i v e c o o l i n g b y o v e r h e a d s p r i n k l e r irrigation P.J.C. HAMER Pomology Department, East Mailing Research Station, Maidstone, Kent (Gt. Britain)
(Received November 1, 1985; revision accepted January 16, 1986) ABSTRACT Hamer, P.J.C., 1986. The heat balance of apple buds and blossoms. Part III. The water requirements for evaporative cooling by overhead sprinkler irrigation. Agric. For. Meteorol., 37:175--188. The heat balance equations for wetted and unwetted buds were solved in order to estimate the effect of water sprinkling on bud temperature. This involved determination of the heat transfer coefficients of buds at different stages of development. The fluxes of sensible and latent heat were found to be twice or three times as great as that of net radiation. The estimates of the degree of cooling for a given sprinkler irrigation rate, based on these heat balance studies, agreed well with those found by field experimentation. INTRODUCTION T h e use o f w a t e r sprinkling to c o o l f r u i t b u d s and d e l a y t h e i r d e v e l o p m e n t w a s first suggested as a m e t h o d o f f r o s t p r o t e c t i o n b y A n d e r s o n et al. (1973). T h e b u d s m a y be p r o t e c t e d f r o m f r o s t d a m a g e b y delaying t h e frostsensitive l a t e r stages o f d e v e l o p m e n t . T h e d e v e l o p m e n t o f t h e w e t t e d buds c a n be e i t h e r slowed d o w n w h e n t h e s p r i n k l e d b u d t e m p e r a t u r e is g r e a t e r t h a n 4.5°C, or s t o p p e d w h e n it is less t h a n 4.5°C. A p p l e b u d s are initiated in late s u m m e r and early a u t u m n . S u b s e q u e n t l y , a winter-chilling r e q u i r e m e n t m u s t be m e t b e f o r e t h e b u d s e m e r g e f r o m d o r m a n c y . In S o u t h - E a s t England, b u d d e v e l o p m e n t c o m m e n c e s at t h e end o f a c o l d spell in late J a n u a r y or early F e b r u a r y ( H a m e r , 1980). T h e r a t e o f b u d d e v e l o p m e n t , o n c e s t a r t e d in spring, is a linear f u n c t i o n o f t h e r m a l t i m e m e a s u r e d in d a y - d e g r e e s a b o v e a t h r e s h o l d value o f 4.5°C. T h e a c h i e v e m e n t o f t h e m a x i m u m p o s s i b l e d e l a y b y e v a p o r a t i v e cooling requires t h e c o n t i n u a l a p p l i c a t i o n o f w a t e r w h e n e v e r t h e t e m p e r a t u r e is a b o v e 4.5°C, o v e r a p e r i o d f r o m t h e beginning o f F e b r u a r y until full b l o o m . In this p a p e r , t h e h e a t balances f o r u n w e t t e d and w e t t e d b u d s are established a n d t h e " c o o l i n g " e f f e c t r e l a t e d t o e n v i r o n m e n t a l f a c t o r s and t h e p h y s i c a l c h a r a c t e r i s t i c s o f t h e bud. T h e p r e c i p i t a t i o n r a t e s r e q u i r e d f o r e f f e c t i v e e v a p o r a t i v e cooling are t h e n c a l c u l a t e d . Finally, e s t i m a t e s o f t h e w a t e r c o n s u m p t i o n r e q u i r e d are r e l a t e d t o t h e " c o o l i n g " a c h i e v e d in t h e field w i t h a small-scale t e s t s y s t e m . 0168-1923/86/$03.50
© 1986 Elsevier Science Publishers B.V.
METHODOLOG Y
Estimation o f the "cooling" effect When w a t e r is a p p h e d evaporative cooling o c c u r s and the heat balance e q u a t i o n f o r fully wett, ed buds can be w r i t t e n as: /~w = Cw +1-7,,
(t)
w h e r e / ~ w is t h e n e t gain o f h e a t b y radiation, Cw is the loss o f sensible heat by c o n v e c t i o n and I-7, is t h e loss o f l a t e n t heat o f e v a p o r a t i o n . Each t e r m is a m e a n e n e r g y flux per unit surface area in W m -2 . E q u a t i o n 1 assumes t h a t the rate o f change o f s t o r e d heat and t h a t latent heat loss t h r o u g h the cuticle and s t o m a t a are small (Hamer, 1984). F o r the w e t t e d b u d at t e m p e r a t u r e Tw, m e a n radiative, sensible and l a t e n t heat fluxes can be w r i t t e n as:
Rw
= Rn~-- hr(T~ -- Ta)
(21
Cw = hw(T w -Ta)
14,,: = h v ( e s ( T w ) -
(3)
e~)/'7
(4)
w h e r e Rr~ is the i s o t h e r m a l net radiation, Ta is air t e m p e r a t u r e , e,(Tw ) is the s a t u r a t e d v a p o u r pressure over w a t e r at the t e m p e r a t u r e o f the bud and e~ ( m b a r ) is the v a p o u r pressure in the s u r r o u n d i n g air. Here, hr, hw and h v are the h e a t t r a n s f e r c o e f f i c i e n t s for r a d i a t i o n exchange, heat and l a t e n t heat, respectively. F o r d a y t i m e spring t e m p e r a t u r e , h r ~ 5 W m -2 K -1 , h v ~- 1.08 h w (Hamer, 1 9 8 4 ) and 7 the p s y c h r o m e t e r c o n s t a n t = 0 . 6 5 m b a r K --L. E q u a t i o n 4 enables t h e rates o f e v a p o r a t i o n for a free w a t e r surface to be e s t i m a t e d . T o p r o d u c e t h e m a x i m u m e f f e c t o f evaporative cooling, w a t e r s h o u l d be applied at the same or greater rate as w a t e r evaporates f r o m t h e bud surface. Excess w a t e r will r u n off. F r o m eqs. 1--4, the t e m p e r a t u r e o f w e t t e d buds can be given by:
Tw = Ta + (Rnil(hr + hw ) ) - - (es(Tw)
-
-
ea)/7*
{5t
where 3'* = 7(hr + h w ) / h v
(6)
F o r u n w e t t e d buds, t h e l a t e n t heat flux t e r m in t h e h e a t balance eq. 1 is zero and assuming the u n w e t t e d and w e t t e d buds are b o t h e x p o s e d t o t h e same r a d i a t i o n e n v i r o n m e n t , t h e t e m p e r a t u r e o f the u n w e t t e d buds, Tb, can be given by:
Tb = Ta + Rni/(hr + he)
(7)
w h e r e h e is the c o e f f i c i e n t o f h e a t transfer o f u n w e t t e d buds. At the end o f d o r m a n c y w h e n w a t e r sprinkling is c o m m e n c e d , t h e coefficients hc and h w will be equal because the buds are at t h e same p h e n o l o g i c a l
177 stage o f d e v e l o p m e n t . In this case, based o n eqs. 5 and 7, " c o o l i n g " , being d e f i n e d as the t e m p e r a t u r e d i f f e r e n c e b e t w e e n u n w e t t e d and w e t t e d buds, simplifies to: Tb -- Tw = ( e s ( T w ) - - ea)/'Y $
(8)
H a m e r {1985) s h o w e d t h a t t h e r e is a seasonal change in he f o r the same wind speed c o n d i t i o n s due t o changes in b u d size and shape. Because cooling delays the d e v e l o p m e n t o f w e t t e d buds, as t h e season progresses t h e differe n c e b e t w e e n the d e v e l o p m e n t o f the w e t t e d and u n w e t t e d buds increases so t h a t h e ve hw. As the u n k n o w n i s o t h e r m a l net r a d i a t i o n is c o m m o n in eqs. 5 and 7 it can be eliminated; this results in a c o m p l e x e q u a t i o n f o r b u d " c o o l i n g " , w h i c h includes air t e m p e r a t u r e , Ta, in t h e expression. By introducing a t e r m hc~, w h i c h is the m e a n o f t h e c o e f f i c i e n t s h e and h w , t h e air t e m p e r a t u r e t e r m s can be eliminated. Bud cooling can t h e n be w r i t t e n (see A p p e n d i x ) in t h e same general f o r m as eq. 8 as: Tb -- Tw = ( e s ( T w ) - - ea)/~{ +
(9)
where 7 + = ~/(hr + hcw)/hv
(10)
E q u a t i o n 9 is valid if Tu -- Ta ~ Ta -- Tw, w h e n the w e t t e d buds are below air t e m p e r a t u r e b y a margin similar t o the t e m p e r a t u r e d i f f e r e n c e o f the u n w e t t e d buds above air t e m p e r a t u r e . Even if this c o n d i t i o n is n o t satisfied, errors in estimating "),+ are small f o r U.K. c o n d i t i o n s (see A p p e n d i x ) .
Estimation o f the precipitation rate required to maximise the "cooling" effect Water d r o p l e t s applied t o buds f r o m sprinklers have a relatively large h o r i z o n t a l c o m p o n e n t o f v e l o c i t y so t h a t the buds i n t e r c e p t m o r e w a t e r t h a n w o u l d be d e p o s i t e d o n a h o r i z o n t a l area equal t o t h e i r cross-section. Businger ( 1 9 6 5 ) i n t r o d u c e d an i n t e r c e p t i o n factor, I, w h i c h d e p e n d s o n the ratio o f surface area t o cross-sectional area o f t h e bud and t h e d i r e c t i o n o f w a t e r d r o p l e t s falling o n the bud. N o r m a l l y , irrigation systems are designed to a p p l y a given d e p t h o f w a t e r per unit t i m e o n a h o r i z o n t a l surface, r a t h e r t h a n a p a r t i c u l a r flux o f heat. T h e c o r r e s p o n d i n g p r e c i p i t a t i o n rate, P, can be e x p r e s s e d in m m h -1 by:
P = I~qw/pwLe = I H w / 6 8 8
(11)
Pw is t h e d e n s i t y ( ~ 103 k g m -3) and L e is the l a t e n t heat o f v a p o r i s a t i o n (24.8 x 10 s J kg -1 at 10°C) o f water. T h e r e is a simple relationship b e t w e e n the a m o u n t o f w a t e r e v a p o r a t e d f r o m the surface o f a w e t t e d bud and the " c o o l i n g " effect. P r e p r e s e n t s t h e m i n i m u m p r e c i p i t a t i o n rate r e q u i r e d t o p r o d u c e the m a x i m u m " c o o l i n g " e f f e c t and can be e s t i m a t e d b y c o m b i n i n g eqs. 4 and 9--11. T h e p r e c i p i t a t i o n rate r e q u i r e d can be w r i t t e n as:
17~
P = I(h~ + hcw)(T b ..... !',,. i/688 By i n t e r m i t t e n t l y w e t t i n g is restricted i n d i c a t e d by eq. 12. cooling e f f e c t (Tb --
~12
applying w a t e r so t h a t excess w a t e r is n o t applied, buci and t.he t e m p e r a t u r e depression will n o t be as large a:~ In these circumstances, the p r e c i p i t a t i o n rate P ' and the, rl':v ) are related by:
P' = I(hr + hc~)(Tb -- T w ) / 6 8 8
t13
In t e r m s o f " c o o l i n g " in units o f day-degree C (D} and d e p t h o f w a t e r applied (W) in m m , eq. 13 becomes: D -
28.7 W
I(h~ + h~.)
~14)
Determination o f coefficients o[ heat transfer and isothermal net radiation and validation o f the relationship between precipitation rate and "cooling" T h e c o e f f i c i e n t s o f heat t r a n s f e r for u n w e t t e d and w e t t e d buds hc and hw, respectively, d e p e n d o n the n a t u r e o f the c o n v e c t i o n regime and the develo p m e n t a l stage o f the bud or blossom. T h e n o r m a l m o d e o f heat t r a n s f e r for apple buds in t h e field is by f o r c e d c o n v e c t i o n (Hamer, 1 9 8 5 ) and the c o e f f i c i e n t s hw and he, w h i c h vary w i t h w i n d speed, u ( m s - l ) , can be d e s c r i b e d by the e q u a t i o n s :
hw = C1 + C 2 u ~s ~ Cwu°'S;hc "~ Cb u°'s
(15)
w h e r e C1 and C2 are c o n s t a n t s at a p a r t i c u l a r stage o f d e v e l o p m e n t , w h i c h c h a n g e as bud g r o w t h proceeds. It m a y be assumed t h a t the w e t t e d buds have t h e same heat t r a n s f e r c o e f f i c i e n t s as t h e u n w e t t e d buds at the same phenological stage. P a p e r I r e p o r t s d e t e r m i n a t i o n s o f C 1 and C 2 f o r early d e v e l o p m e n t a l stages f r o m " ' d o r m a n c y " to " g r e e n c l u s t e r " , using r a d i a t i o n and t e m p e r a t u r e m e a s u r e m e n t s m a d e at night in low wind speed c o n d i t i o n s . Values o f C 1 and C2 r a n g e d f r o m 3 t o 11 and 46 t o 62, respectively. However, in d a y t i m e , wind speeds in an o r c h a r d c a n o p y are higher. T h e usual range is f r o m 1 to 4 m s -l , with an average o f ca. 2 m s -1 (see H a m e r , 1984}. In this range, the single c o n s t a n t Cw can be used in eq. 15 with little loss o f accuracy. F o r the average wind speed and d o r m a n t buds, C1 = 11 and C2 = 46 while Cw = 54. T h e a p p r o x i m a t i o n gives values o f h w w i t h i n + 5% o f t h o s e f r o m the full e q u a t i o n o v e r the wind speed range 1--4 m s -1 . T h e u n k n o w n h e a t t r a n s f e r c o e f f i c i e n t s a p p r o p r i a t e to later stages o f d e v e l o p m e n t can be e s t i m a t e d f r o m m e a s u r e m e n t s o f t e m p e r a t u r e , T b and T,, and v a p o u r pressure e a. T h e b u d c o n s t a n t s C b and Cw are o b t a i n e d f r o m eqs. 10 and 15 using 7 = 0 . 6 5 m b a r K -1 (at 10°C), hr = 5 W m -2 K -1 and hv = 1.08 hw. H e n c e : C b
---~
2 ( 1 . 6 6 C w 7 + - 5/u °'s) -- Cw
(16)
In the e x p e r i m e n t to be described, at o n e site { " c o n t i n u o u s " } apple buds
179 were wetted to run-off and 7+ determined from the relationship between the "cooling" effect T b - - T w and the vapour pressure difference es(Tw)--e~ (eq. 9). These measurements enabled the seasonal changes of the bud constant, C b, to be determined (eq. 17) because the constant Cw for wetted buds was k n o w n t h r o u g h o u t the period of water sprinkling. The estimation of isothermal net radiation appropriate to the bud is difficult (Hamer, 1984) due to the complex bud geometry and is further complicated by shading by adjacent leaves, flowers and twigs. However, in the field, R,~ for the unwetted buds can be estimated from measurements of the coefficient of heat transfer hc (~ Cb u °'s) and the temperature difference Tb -- Ta (eq. 7). Rm is likely to be closely related to the net radiation balance ( R ~ ) of the underlying orchard, which can be readily measured. The ratio, ~, of isothermal net radiation and the measured radiation balance of the orchard can be written simply as ~ = R m / R n ~ t. At other sites, two levels of intermittent watering were applied, referred to as " i n f r e q u e n t " and " f r e q u e n t " . These minimised run-off and enabled validation of the relationship (eq. 14) between day-degree cooling and depth of water applied. Experimental procedure
Full details of the experimental orchard and instrumentation were given in Hamer (1985). Three pairs of Cox's Orange Pippin trees on M.26 rootsock (Sites 2, 4 and 6) were watered using minisprinklers. The water supplies to the sprinklers were controlled independently using solenoid valves. Two similar control sites (1 and 5) were unwatered. In all cases, water was applied only when air temperature measured at the control Site 1 was greater than 4.5°C. The criteria for irrigation of the three sites differed. At Sites 2 and 4, water was applied intermittently. The application was controlled using signals from four wetness sensors at each site, being turned on when t h e y were dry. The sensors were taped to the branches of watered trees at about 1 m height. The wetness sensors were wired in parallel at Sites 2 and in series at Site 4. Set upper and lower voltage limits (Hamer, 1984) initiated the shutting and opening of the valves. Because the voltage o u t p u t depends on the most wet sensors in the case of parallel-wired sets of sensors and the least wet sensors for series-wired sensors, the watering regime was " i n f r e q u e n t " and " f r e q u e n t " , respectively. At Site 6, water was applied continuously whenever the net radiation was positive. This regime was interrupted when the o u t p u t control facility of the data-logger failed from 23 March to 4 April 1982, inclusive. During this period, Sites 2 and 4 were unwatered, whilst at Site 6 the sprinkler operated continuously. At each site, the temperatures of buds in an upward-facing exposed position at mid-tree height were monitored by inserting fine-gauge (36 s.w.g.) copper-constantan thermocouples into four separate buds. The thermocouples were wired in series and referenced to the air temperature at 1.1 m
18C} 600 ,,
," ~: 4 0 0
200
•D cE
y
St
.,
. .....
Rnet'
, "/
0
-200
08:00
,0:00
,2200
,,;00
,eoo
,800
20.00
22.00
Time (hi
10. C 2.0
v
u E
4.
1.0
2>
x ¸¸
....'r-
0
08~00
10100
12100
14100
16100
18100
20,00
22.00
T i m e (h)
Fig. 1. Diurnal variation of environmental parameters on 27 March 1982. using well-ventilated psychrometer units. The units also housed wet and dry bulb thermometers using 100 ~ platinum resistance elements. These were calibrated to an accuracy of +0.1°C. At Sites 1 and 6, there were two measurements of bud temperature and one at the other sites. Measurements were made at 10-s intervals using a computer-controlled data-logger. Mean values for 10-min periods were retained by the logger for analysis. The depth of water applied per day was measured by 65-ram-diameter plastic funnels with their mouths horizontal, placed adjacent to the buds with inserted thermocouples. At each site, four funnels were connected to a single water bottle. These "gauges" agreed within 3% with a standard rain gauge (Hamer, 1986). Any rain collected at the unwatered control site was deducted from the daily measurements at the other sites. The precipitation rates were estimated from daily measurements of water depth and the recorded time the valve was open at each site (Hamer, 1984). On days following frost it was impossible to measure the applied water directly and it was estimated from the precipitation rates and the length of time the valves were open.
181 20,
15
!
o F-
08:00
10:00
12:00
14:00
16:00
18:00
20:00
22.00
Time (h)
Fig. 2. M e a n h o u r l y values o f t h e w e t ( " ) a n d d r y ([]) b u l b t e m p e r a t u r e s . T e m p e r a t u r e s o f u n t r e a t e d (o) a n d " c o n t i n u o u s " (o) t r e a t m e n t b u d s o n 27 M a r c h 1 9 8 2 (* i n d i c a t e s t h e p e r i o d over w h i c h " u n w e t t e d b u d s " w e r e dry, as i n d i c a t e d b y a w e t n e s s sensor in t h e u n s p r i n k l e d trees).
The stages of bud development were determined by visual examination of the trees according to an established classification (Table I; Hamer, 1980). Observations were made at weekly intervals from the beginning of March until mid-April and two to three times per week subsequently. RESULTS
Sprinkling was commenced on 11 February 1982 and continued until 6 May 1982, when the unwatered control trees blossomed. The " i n f r e q u e n t " and " f r e q u e n t " treatment trees blossomed 5 and 10 days later, respectively. Most of the " s p u r " buds that received the " c o n t i n u o u s " treatment failed to develop and " s e c o n d a r y " buds blossomed 19 days after the control. Data from the " c o n t i n u o u s " t r e a t m e n t for 17 March demonstrate many features of evaporative cooling of apple buds; at this time wetting was continued at night due to equipment failure p. 179. The diurnal courses of solar (St) and net radiation, wind speed (measured at 1.2 m above ground level) and vapour pressure are shown in Fig. 1. At 06.00 h there was thick fog and consequently the wet and dry bulb air temperatures and the temperatures of both the u n w e t t e d and wetted buds were all the same (Fig. 2). With light winds and high solar radiation, bud temperature exceeded air temperature by 4.5°C at l l . 3 0 h (the highest
400
2OO ./
m
j
/ \ w •, J
200
\
E
w @ -r"
61111
0s:0o
10:0o
,2:00
1,:0o
,6:00
,s:00
o:oo
2:0o
Tlme (h) Fig. 3. H o u r l y e s t i m a t e s o f t h e c o n t r i b u t i o n s o f fluxes o f l a t e n t (o), sensible h e a t (m) a n d n e t r a d i a t i o n (x) t o t h e h e a t b a l a n c e o f a c o n t i n u o u s l y w e t t e d b u d o n 27 M a r c h 1 9 8 2 . T h e c o e f f i c i e n t s o f h e a t t r a n s f e r for t h e s e w e t t e d b u d s were e s t i m a t e d as h w --- 11 ÷ 4 6 u °'s ( a p p r o p r i a t e f o r a d o r m a n t b u d , H a m e r , 1 9 8 6 ) w h e r e u is t h e m e a n w i n d speed. T h e fluxes o f sensible h e a t , t h e l a t e n t h e a t o f e v a p o r a t i o n a n d n e t r a d i a t i o n were calculated f r o m eqs. 3, 4 ( w h e r e h v = 1.08 hw) a n d 1, respectively.
recorded in 1982). In the evening when R ~ was negative, bud temperature was lower than air temperature by 0.9°C. When net radiation was positive, the temperature of the watered buds exceeded the measured wet bulb temperature, but at night these temperatures were similar. The m a x i m u m temperature difference between unwetted and wetted buds (i.e., m a x i m u m "cooling") was 9.2°C, the highest value recorded by the author during the years 1981--1983. Hourly values of heat gains and losses for the continuously sprinkled buds are shown in Fig. 3. At the start of the period all fluxes were approximately zero. Thereafter until 11:00 h, Tw was greater than Ta and, therefore, there was a loss of sensible heat. For the remainder of the period, Tw was lower than air temperature and, therefore, Cw was a gain. At 14.30 h when T w -Ta = - 6.4°C, Cw peaked a t - 370 W m -2 . During early afternoon the loss of latent heat peaked at 55OW m -2 , subsequently declining gradually to zero by 21.00 h. Calculated values of isothermal net radiation, R ~ , for the wetted buds are plotted in Fig. 4 against calculated values of R~a for unwetted buds. The isothermal net radiation was positive t h r o u g h o u t the daylight hours, with a peak value o f 2 2 0 W i n -2 , but a loss at night. As indicated by the 1:1 line in the figure, the estimated values of isothermal net radiation for the u n w e t t e d and w e t t e d buds were not significantly different.
183 250
OQ
200 E 150
lOO
50'
'E
•
-50 -50
0
50
100
150
R ni for unwatered buds
200
250
(W m-=)
Fig. 4. T h e values o f i s o t h e r m a l n e t r a d i a t i o n , _Rni, c a l c u l a t e d for t h e " c o n t i n u o u s " treatm e n t b u d s f r o m Rni = h r ( T w - - Ta) 4- Cw ~- Hw ( f r o m eqs. 1 a n d 2) p l o t t e d against c o r r e s p o n d i n g values for t h e u n w e t t e d c o n t r o l buds, Rni ---- (hr + h c ) ( T b - - Ta) ( f r o m e~. 7). T h e c o e f f i c i e n t s o f h e a t t r a n s f e r for t h e u n w e t t e d b u d s were t a k e n as 11 -t- 55 u , t h e value a p p r o p r i a t e f o r 27 M a r c h ( H a m e r , 1 9 8 5 ) . H o u r s w h e n t h e c o n t r o l b u d s were w e t t e d b y d e w w e r e e x c l u d e d f r o m t h e analysis.
T h e seasonal variation o f the b u d c o n s t a n t Cb and t h e ratio o f i s o t h e r m a l n e t r a d i a t i o n to o r c h a r d net r a d i a t i o n (R~/R~ot) w e r e d e t e r m i n e d at w e e k l y intervals. T h e d a t a w e r e regarded as suitable f o r analysis w h e n : (i) T h e wetness sensor at the c o n t r o l (Site 1) was dry. (ii) T h e t w o m e a s u r e m e n t s o f u n w a t e r e d b u d t e m p e r a t u r e agreed within 0.3°C, t h e m e a n o f t h e t w o readings being T b . T h e same t e m p e r a t u r e c r i t e r i o n was used f o r t h e " c o n t i n u o u s " t r e a t m e n t , the m e a n o f t h e t w o readings being T w . (iii) M e a s u r e m e n t s o f v a p o u r pressure at t h e u n w a t e r e d Sites 1 and 5 agreed w i t h i n 0.2 mbar, t h e m e a s u r e m e n t at Site 1 being e a. (iv) T h e wind speed w i t h i n the o r c h a r d c a n o p y was less t h a n 4 m s -1 (it was f o u n d t h a t wind speeds greater t h a n this resulted in some o f t h e buds remaining d r y ) . (v) T h e sprinkler was o p e r a t i n g c o n t i n u o u s l y t h r o u g h o u t t h e p e r i o d o f measurement. L i n e a r regression analyses (with the lines c o n s t r a i n e d t o pass t h r o u g h t h e origin) w e r e carried o u t using m e a n h o u r l y values for data c o l l e c t e d at w e e k l y intervals. T h e reciprocals o f t h e slopes (7 +} d e t e r m i n e d are given in T a b l e I t o g e t h e r w i t h t h e n u m b e r o f d a t a p o i n t s used and the values o f r 2. T h e r e were
TABLE I The values of ~ obtained from tile regression of 7"b- T w against e s ( l w ] ca ;~ measurements collected over weekly periods during the spring of 1982. The data are t'r,m~ the "continuous" t r e a t m e m in which excess water was applied to I he buds Inclusive dates
N umber of data points
"~ mbar K -1 ~
Percem ~ariance
i~ 20 4~ 32
0.64 0.64 0.63 0.68
92 97 ~.)2 95
5--i I April
32
0.67
97
12--18 April 19--25 April 26 April--2 May 3--7 May
40 60 29 !7
0.72 0.71 0.74 0.85
84 89 94 92
8 - 1 4 March 15--21 March 22--18 March 29 March--4 April
I/ 2 )
i n s u f f i c i e n t d a t a f o r analysis p r i o r to 8 March. As t h e b u d s d e v e l o p e d f r o m d o r m a n c y t o full b l o o m , t h e slopes d e c r e a s e d and 3,÷ c h a n g e d f r o m 0 . 6 4 to 0.85 m b a r K - ' , respectively. F o r t h e " c o n t i n u o u s l y " w e t t e d buds, b u d d e v e l o p m e n t was d e l a y e d s u f f i c i e n t l y t o e n a b l e t h e b u d c o n s t a n t s t o be c a l c u l a t e d f r o m m e a s u r e m e n t s o f m e a n w i n d s p e e d and t h e b u d c o n s t a n t s C 1 and C 2 f o r d i f f e r e n t develo p m e n t a l stages. C1 a n d C2 w e r e k n o w n f o r b u d d e v e l o p m e n t a l stages f r o m d o r m a n c y to green c l u s t e r ( H a m e r , 1 9 8 5 ) a n d Cw was d e t e r m i n e d at w e e k l y intervals f r o m eq. 15. T h e c o n s t a n t s f o r t h e u n w e t t e d buds, Cb, w e r e estim a t e d f r o m ~'÷, w i n d s p e e d and Cw (eq. 16). T h e changes o f Cw a n d C b t h r o u g h t h e p e r i o d o f b u d g r o w t h are s h o w n in Fig. 5. T h e b u d c o n s t a n t Cw (in units o f W m -2 K -l ( m s -1)-°'5) f o r t h e " c o n t i n u o u s l y " w e t t e d b u d s r a n g e d f r o m 52 t o 65. Cb increased f r o m 54 w h e n d o r m a n t to 101 at full b l o o m . The bud constants were calculated for the unwatered buds from " d o r m a n c y " t o " g r e e n c l u s t e r " f r o m t h e d a t a given in P a p e r I ( H a m e r , 1 9 8 5 ) and t h e m e a n w i n d speed. T h e c o n s t a n t s are in r e a s o n a b l e a g r e e m e n t w i t h t h e c a l c u l a t e d values o f C b . Figure 6 s h o w s values o f R , , I / R , ~ , f o r t h e d i f f e r e n t p e r i o d s o f b u d g r o w t h derived b y regression w i t h a f i x e d i n t e r c e p t t h r o u g h t h e origin. T h e values r a n g e d f r o m 0 . 3 0 t o 0 . 4 4 w i t h a m e a n o f 0.38. T h e r e l a t i o n s h i p b e t w e e n w a t e r use and cooling was f o u n d t o be a p p r o x i m a t e l y linear (Fig. 7), in a g r e e m e n t w i t h e x p e c t a t i o n s f r o m eq. 14. DISCUSSION AND CONCLUSIONS Values of T+ ranged from 0.63 to 0.85 m b a r K -1 . Early in the season, the values of 7 + agreed with the theoretical estimated (eq. 7) of 0.64 mbar K -I
185 100
8O ma o
E 60
? E v
-- 4 0 c
C 0 O "D
m 2(1
Continuous
,o
20
;o I
ME
o
March
,;
;
A p r il
May
Fig. 5. S e a s o n a l v a r i a t i o n o f " b u d c o n s t a n t " C b f o r t h e " c o n t i n u o u s " t r e a t m e n t b u d s (©) a n d t h e u n w a t e r e d c o n t r o l b u d s (o), c a l c u l a t e d f r o m eq. 16. V a l u e s o f C b (G) w e r e e s t i m a t e d f r o m C1 a n d C2 in P a p e r I. T h e r e c o r d e d stages o f b u d d e v e l o p m e n t are i n d i c a t e d as M E --- m o u s e ear, GC -- g r e e n cluster, PB = p i n k b u d , IB - first b l o o m , F B = full b l o o m . 0.6
0.4
t-
tr
-~ 0.2 tr
10 '
~0 March
3'o
1'9
9
April
2'~
9'
May
Fig. 6. S e a s o n a l v a r i a t i o n o f t h e ratio o f i s o t h e r m a l n e t r a d i a t i o n t o t h e n e t r a d i a t i o n measured above the orchard canopy.
188
'5
(a)
2.5
1.5
15
-/
/
/
eee
v
-==
"== p
.lj •
20
20
1.0
1.0
e/
/ 0.5
05
:
•
/ 5
10
Rate of application of water, W(mm d a y
15 -~)
o
/
; t'o Rate of application of water, W ( m m
d a y °~)
Fig. 7. The daily measured mean cooling (T b -- Tw) for (a) "infrequent" and (b) "frequent" treatments plotted against depth o f water collected in "rain" ganges. The regression lines (forced throught the origins) have slopes o f 0.14 day-degree C mm -1 (r 2 = 40.8) and 0.15 day-degree C mm -1 (r 2 ~ 7616), respectively.
(hw typically 80 W m -~ K -1 ), for buds at the same phenological d e v e l o p m e n t stage. In general, the estimated values of bud constant Cb and hence the coefficient of heat transfer for a given wind speed, increased as bud development proceeded. Hamer (1985) showed a similar seasonal trend in the coefficient of heat transfer estimated from measurements on unwetted buds only. The ratio of the estimated net radiation, Rm, to the measured net radiation to the orchard, R ~ , ranged from 0.30 to 0.44. This range is consistent with geometry. If the buds were assumed to be horizontal plates the ratio would be about 0.5. It was shown in Paper I that the net flux of short-wave radiation for a sphere is complex and is less than for horizontal plates (Gates, 1980). As buds expand, their shapes change and they become more difficult :to characterise. Furthermore, as the season progresses, the buds are more likely to be shaded by adjacent leaves and buds {or flowers). In the conditions when sprinkling is most likely to produce effective cooling, the flux of net radiation (normally regarded as the driving term in the heat balance equation) is much smaller than the fluxes o f latent and sensible heat (Fig. 3). The observed mean ratio of R n i / R ~ = 0.38 can,
187 therefore, be used t h r o u g h o u t the period of bud growth with little loss of accuracy. The validity of the value of the interception factor (I = 2) employed by Hamer (1986) can be assessed from the results for " i n f r e q u e n t " and "'freq u e n t " wetting presented in Fig. 7. From theory (eq. 14), the slopes of the graphs should be 28.7/(I(hr + hc~)). If the mean seasonal values of hw and hc are estimated from the data in Fig. 5, for the mean wind speed of 1.9 m s -1 , C w and C b are equal to 60 and 70W m -2 K -1 (m s-1 )-0.s, respectively and the coefficient h ~ = 90 W m - : K -1 . The corresponding slope in Fig. 7 expected from t h e o r y for I = 2 is 0.15 day-degree C mm -1 . This is remarkably close to the measured values of 0.14 and 0.15 day-degree C mm -1 found for the " i n f r e q u e n t " and "'frequent" treatments, respectively. The data in Fig. 7 for the " i n f r e q u e n t " treatment offer further evidence for the seasonal increase in bud constant. Extrapolation of the data to zero depth of water, indicates a negative value for the "cooling". For the hypothetical condition of no watering, the temperature of these buds (whose blossoming was delayed by 5 days) would be greater than the control, as would be expected because Cw ~ CD. The precipitation rate requirement for daytime cooling, which follows the potential flux of latent heat, will on average vary sinusoidally with time of day (see Fig. 3). The area under the curve represents the mean daily requirement, which is typically 10 mm. A sensible criterion for the design of an irrigation system for daytime cooling is the mean daily m a x i m u m precipitation rate. From eq. 12, a precipitation rate of 1.4 mm h -1 would give T b -T w = 5°C, which was the mean daily m a x i m u m temperature depression in 1982. An effective control system would be desirable to enable intermittent application. The seasonal water requirements for evaporative cooling can be estimated as follows. Using 10 years' climatologcial and phenological data, Hamer (1980) showed t h a t for the m a x i m u m cooling of 122 day-degree C, the wetted buds would be at "green cluster" when u n w e t t e d buds were at "full bloom". The average seasonal water requirement for such a delay is ca. 800 mm (calculated from eq. 14 with a value of hew = 90 W m -2 K -1). This is a vast q u a n t i t y of water and on most farms would be prohibitively expensive (Hamer, 1984). This paper has established a sound theoretical understanding of the technique of evaporative cooling. However, it also demonstrates the need for very substantial quantities of water to delay blossoming effectively. It is unlikely, therefore, that this technique will be adopted by the U.K. fruit industry, although the sprinkler technique in itself may be useful in other climates where trees are regularly submitted to high-temperature stress. ACKNOWLEDGEMENTS I am grateful to Dr. J.A. Clark of the Department of Physiology and Environmental Science, Nottingham University School of Agriculture, for his
constructive c o m m e n t s o n tJ~is paper and to P. N e w m a n for hi~ general assistance. APPENDIX
Derivation o f the relationship between bud "'cooling" and the coefficient o f heat transfer, h c'w
The isothermal net radiation terms can be eliminated from eqs. 5 and 7 to give h r T b w + (hcTba - h w T w a + hcTba -- hwTwa)/2
= h v ( e s ( T w ) - - Ca)/7
(17)
w h e r e Tbw = T b -- T w , Tba = T b - T a a n d Twa = T w -- T a. T h e c o m b i n e d h e a t t r a n s f e r c o e f f i c i e n t hcw = (h e + hw)/2 a n d t h e t e r m hcwTbw c a n be e x p a n d e d t o : hcwTbw
_-- ( h c T b a - - h c T w a 4- h w T b a .... h w T w a ) / 2
118)
F r o m eqs. 17 a n d 18: (h r + h c w ) T b w + (h c -- h w ) T w a / 2 +- ( h c - hw)Tba/2
= h v ( e s ( T w ) - - ea)/~,
(19)
E q u a t i o n 19 s i m p l i f i e s if Twa + Tba =: 0, w h e n b u d " c o o l i n g " , Tbw , c a n be w r i t t e n as eqs. 9 a n d 10 in t h e t e x t : Tbw =
T b-
(es(Tw)--ea)/7 +
Tw =
I9)
where
7 + ~= "),(h r -b hcw)/h v
(101
If t h e c o n d i t i o n Twa A Tba :/= 0 t h e n eq. 19 c a n be w r i t t e n as: -- T b -- T w + ( e s ( T w ) .... Ca)fly + 4- e T
Tbw
(20)
where eT
(Tba + T w a ) ( h w + hc)/(2(hr + h e w ) )
=
(21)
r e p r e s e n t s a n e r r o r t e r m . I n S o u t h - E a s t E n g l a n d , t h e largest v a l u e o f e T t h a t is likely t o o c c u r is f o r t h e c o n & t i o n s Tba = 0 a n d Twa = - - 5 C. F o r t y p m a l v a l u e s o f h w a n d h e o f •
.
-2
-1
,
o
'
o
.
.
o
80 and 1 0 0 W i n K , respectlvely, eT = 0.5 C and compares wlth Tbw = 5 C. Under these extreme conditions, there would be an error of 1 0 % in the estimation of 7 +. REFERENCES Anderson, J.L., Richardson, E.A., Ashcroft, G.L., Keller, J., Alfaro, J., Hanson, G. and Griffin, R.E., 1973. Reducing freeze damage to fruit by overhead sprinkling. Utah Sci., December, 108--109• Businger, J.A., 1965. Protection from the Cold. Meterok Monogr., 6:74--80. Gates, D.M., 1980. Biophysical Ecology• Springer-Verlag, N e w York, 161 pp. Hamer, P.J.C., 1980. A model to evaluate evaporative cooling of apple buds as a frost protection technique• J. Hortic. Sci., 55:157--163. Hamer, P.J.C., 1984. The heat balance of apple buds and blossoms in frost protection. Ph.D. Thesis, University of Nottingham. Hamer, P.J.C., 1985. The heat balance of apple buds and blossoms• Part I. Heat transfer in the outdoor environment. Agric. For. Meteorol, 35: 339--352. Hamer, P.J•C., 1986• The heat balance of apple buds and blossoms. Part II. The water requirements for frost protection by overhead sprinkler irrigation. Agric. For. Meteorol., 37 :159--174