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Light transmission losses due to structural members in multispan greenhouses under diffuse skylight conditions
J. ugric. Engng Rex (1987) 38, 193-207
Light Transmission Losses Due to Structural Members in Multispan Greenhouses under Diffuse Skylight Conditions D. L. CRITTEN* An accurate method of predicting light losses due to periodic structural members in the body of multispan greenhouses is presented. Losses due to stanchions are shown to be negligible, and members, such as roof glazing bars, which are not horizontal are successfully replaced by horizontal cylindrical members, of slightly increased diameter. Experimental measurements for three distinct greenhouse types are compared with predictions, and an accuracy of + 1 percentage point is obtained. The comparison shows that structural members behave as if they were nearly black, and that quite different designs lead to comparable losses, within + 13 percentage points.
1. Introduction In a previous paper’ the loss of light due to multispan gutters, treated as infinitely long black circular cylinders was predicted analytically, and reasonable agreement with experimental measurements was obtained. Some difficulty exists, however, in choosing the diameter of the cylinder to represent the gutter because of its complex cross section. If the process is to be extended to other structural members, such as the ridge, a similar problem will arise, the ridge usually being far from cylindrical. In the present paper, a more accurate approach is presented in which all horizontal long periodic structural members can still be assessed in a simple manner. As before, errors due to the finite size of the greenhouse are ignored. The analysis considers standard diffuse skylight conditions. In addition, we consider the results of an analysis’ in which it was shown that skylight can generally be separated into harmonic components. The principal harmonic radiance component is of the form (sin a), where a is the angle of elevation, which gives almost identical greenhouse light transmission to the dull overcast sky radiance, namely (1 + 2 sin a)/3. This analysis compares structural losses for the two radiances. Experimental measurements have previously been carried out only in single span greenhouses,3*4 and no record of the cross section of structural members has been made. Accordingly, comparison with experiment for the present theory relies on new measurements for which cross sectional data is known.
2. Theory 2.1. Light loss due to a regular array of obstructions In assessing light loss the detailed shape of any member is not important. Bat’ showed, for example, that a gutter could be accurately ,represented by an isosceles trapezium. The area projected normal to the light beam is what matters. In this work we approximate each
* Horticultural Engineering Division, IER, Silsoe, Bedford, UK Received 10 November 1986; accepted in revised form 26 March 1987 193 0021-8634/87/l
10193 + I5 $03.00/O
0
1987 The British Society for Research in Agricultural
Engineering
194
TRANSMISSION
LOSSES
IN
MULTISPAN
GREENHOUSES
1
Notation
angle of elevation : effective light blockage width, m D diameter of a cylinder, m h cylinder height, m I sky radiance, W m-2 str-’ 10 radiance at the zenith of the sky vault, W me2 str-’ L irradiance loss, W m-2 n number of cylinders required to represent a structural member N numbering system or number of cylinders in an array R distance from a specified point on
R, t
X X, P r * 4
the ground to a point along a cylinder’s axis, m shortest value of R, m transmission loss coordinate normal to the cylinder axis, m cylinder spacing, m D/X, h/X, XIX, angle between R and R, (Fig. 2)
overbar (-)
average value
member by a variable number of cylinders of equal diameter (D). Illustrative examples are shown in Fig. 2. A glazing bar is represented by a single cylinder (Fig. I, left), a gutter by two horizontally adjacent cylinders (Fig. I, middle) and a ridge by two vertically adjacent cylinders (Fig. 1, right). Inspection of Fig. 1 using different light beam elevations shows that a much better approximation of light obstruction is obtained in Fig. 1 (middle and right), than would be obtained by using a single cylinder approximation. By choosing different numbers of cylinders in a single line, any generally rectangular obstruction can be simulated. To obtain the angular width of a series of n equidiameter cylinders, we view the cylinder array in the plane containing R, and R, where R is the distance to the cylinder from any point X, and R, the shortest distance (Fig. 2). The result for any angle C$is shown in Fig. 3 (left) for horizontally, and Fig. 3 (right) for vertically adjacent members using two cylinders. For n cylinders: d = (l+(n-l)sina,) (Fig. 3, left, horizontally aligned cylinders) d = D( 1 + (n - 1) cos a,) (Fig. 3, right, vertically aligned cylinders) -D(n-(n-1) sin ao).
(1)
The approximation cos a- (1 -sin a) is gross, but for n = 2 experience shows a satisfactory accuracy is obtained. For high values of n, deterioration in accuracy might be expected.
Fig. 1. Representation
of various structural members by equidiameter contacting Glazing bars; (middle) gutter; (right) ridge
cylinders.
(Left)
D.
L.
195
CRITTEN
Fig. 2. View of a single cylinder de$ning linear and angular distance for horizontal cylinders
In general, therefore, d =
D(e+fsin
a,),
(2)
where e and f are defined in Eqn (1) for the described combinations of cylinders. A single cylinder is represented with e = 1 and f= 0. Since d is not a function of 4, the total irradiance loss (L) for an infinitely long obstruction array can be immediately written down using the equation derived by Critten’ for an array of single cylinders: Under a standard sky
where p = D/X,, q = h/X0, II/=X/X, and N is the cylinder reference number, such that (NX,+X) is the horizontal distance of the cylinder from the point of interest. Replacing D
Fig. 3. View of (a) horizontally and (b) vertically touching cylinders of diameter D in the plane of R and R,. (Left) d= D(l+sina,); (right) d= D(I+cosa,)
196
TRANSMISSION
by d from Eqn (1):
71
L may be conveniently
L=
IN
MULTISPAN
GREENHOUSES
4+3./m
N=m 2 L=310PgN=zm
2
LOSSES
4
‘I
. t3) >3
rewritten
N=m
jzop9N=c_m
A (N+$)2+q2
BV + [(N+t+b)2+rj2]3’2
Cf12 + [(N+~I)~+Y+I~ >’
(4)
B=le+:f.
C=if
2.2. The average loss In Eqn (3) the average value of L across a span is given by the integral of L with respect to @, normalized with respect to the integral of $, both from 0 to 1. Since the latter integral is unity, the resulting sum is over consecutive integrals, (L) is simply given by
(5) The form of Eqn (5) is discussed in the Appendix. Noting 00 1 _m Wd$ s
= nnlrl,
Then
(6) The external irradiance, obtained by integrating the irradiance due to dull overcast sky elements over the sky vault may be shown to equal 24431,. Then substituting for A, B, and
Fig. 4. Representation
of roof glazing bars by an equivalent horizontal member qf increased thickness
D.
L.
197
CRITTEN
Fig, 5. Obstruction produced by a vertical pole in a light beam of elevation a
C, regrouping, and normalizing transmission loss:
with respect
to external
irradiance
f = p( 1.40e + 0.99f). Using the first harmonic yielding:
gives the light (7)
(sin a) only (Section l), a similar expression can be calculated, f = p(1.27e+f).
(8)
Typically, p < 0.1 and e = 1. Hence the difference between the two forms is approximately one percentage point. 2.3. Representation
of roof glazing bars
Roof glazing bars do not form parallel horizontal obstructions, but are of sawtooth form. However, for low angle roofs (e.g. 26”) the undulating form can be replaced by a horizontal obstruction, of thickness increased by l/cos (roof angle) to allow for the extra length introduced by the undulations (Fig. 4). 3. Light loss due to a regular array of vertical poles
Referring to Fig. 5, consider first the light loss due to a single pole, height h, subject to a beam of elevation a. Then, for radiance symmetric in azimuth (A), dL = 2xlDh cos’ a da for an overcast sky
xl.7
loss L = $ I, Dh
(1 + 2 sin a) cos’ a da,
(9)
I0
where L is the total light removed by the pole over the whole ground area. Consider now a square array of poles, shown in plan view in Fig. 6. A pole is assumed at the centre of each square. Consider the loss introduced by a pole at position 0 into the square occupied by pole 1, say. This must equal the loss in square 0 due to pole 1. The argument is true for any of poles 1 to 8, and so the total loss in square 0 due to poles 0 to 8 equals the loss in the complete square due to pole 0. The process may be extended indefinitely, and so the total loss in square 0 is given by Eqn (9). The average loss L over square 0 is then simply L = L/X$ (10) and average transmission loss (Q is given by t = L/(2.4431,)
(11)
198
TRANSMISSION
LOSSES
IN
MULTISPAN
GREENHOUSES
Fig. 6. Matrix dejining areas associated with a square array of vertical poles for calculkng
total average
light loss
or .
(12)
Typically, p = 0.01, q = O-5, giving f = 0.6 percentage points. Thus iFfor stanchions can generally be ignored. 4. Comparison with experimental measurements Experimental measurements are normally carried out along a line of symmetry across a central span of a multispan greenhouse. For structural members parallel to this line, if these results are to compare with predictions from Eqn (7), the difference between the average loss and loss at the line of measurements must be small (< 1 percentage point). It has been shown’ that if v > 0.4, at a point midway between single cylinders, the loss is about 20% less than the average value. For a member causing a loss of five percentage points, this is equivalent to an error of one percentage point. Thus, u should be of this order for accurate prediction of losses. For structural members perpendicular to the line of measurement such as gutters, the average value of measurements is taken anyway and so the condition is not required. 4.1. Light measurements
carried out during the construction of a Simpson twinspan greenhouse at IER, Silsoe, Bedfordshire
Though not a multispan, average measurements over the central half of the twinspan should be substantially in agreement with multispan predictions. Measurements were carried out in three stages: (1) after erection of all weight-bearing elements; (2) insertion of glazing bars; and (3) completion of the house. Fig. 7 shows the schematic cross sections of structural members and Table 1 shows values of D, h, and X, leading to the irradiance loss for each member, the angled support being circular in cross section. A plan view of the cross bracing is shown in Fig. 8. Complex struts between the cross braces and purlins were ignored. At stage (3), the effect of the vent glazing bars and sills, which overlay the normal bars has not been included. Vent glazing bars occupy about one-sixth of the total glazing bar length, but are obscured for near zenith radiation. They are also on the outer roofs, remote from the central region.
(Section
3.3)
0.035
Roof glazing bar
Equivalent cylinder dia. D, m
0.06 0.10 0.035 0.03 0.06
* After correction
Table 1
1
2 2 1 1 1
No. of cylinders n
3.9
4.9 2.9 3.9 2.8 3.1
Height above sensor h, m
0.6
6.8 6.8 3.4 4.5 5.6
Periodic separation X0, m = p
0.058
0~009 0.015 0.010 0~007 0.011
(D/X,)
5.00
060 0.30 0.45 060 0.55
(hlX,J
= II
Stage 2 total
1
Stage 1 total
2 1 1 1 1
e
0
-1
f 1 0 0 0
1
leading to fractional light losses caused by structural members within the IER Simpson twinspan greenhouse (Figs 7 and 8)
Ridge Gutter Purlin Cross beam Angled support
Structural member
Measurements
0.182
0.091*
0.091
0.016 0.036 0.014 0.010 0.015
lossf
Fractional
TRANSMISSION
200
LOSSES
IN
MULTISPAN
GREENHOUSES
0.06 m
E
0.25 m
*------_---_~
u
j5m +------__F 0.145 m
0.03
0.03 m 0.04 m rl C
cl
0.03 m
I
m 0.04 m
F
Fig. 7. Simplified cross sections of structural members in the IER twinspan greenhouse used to calculate light loss. A, ridge; B, gutter; C, purlin; D, cross bracket (Fig. 8); E, angled bracket, taken as - - -; F, glazing bar
4.2. Light measurements carried out during the construction four-span Venlo at IER
of a
In this case, only two stages were measured: (1) after erection of stanchions, gutters and girders, and (3) after erection of glazing bars, glass and ridge, the Venlo design requiring the latter to be erected at the same time. The form of the girders, which run across the house, is shown in Fig. 9. Because girder members are co-planar with the vertical, directly beneath the
Fig. 8. Plan view of cross bracing in the IER twinspan glasshouse. g = gutter
201
D. L. CRITTEN
Fig. 9. Form of transverse girder in the IER Venlo greenhouse. Cross bracing represented by - - - ,f;rr light loss calculations
girder losses are reduced. However, measurements are taken between girders, and so overlap is ignored. The effect of the overlap on the average loss is very slight (-0.1%). Cross sections are shown in Fig. 10, and the cylinder losses are shown in Table 2. As with the Simpson twin span, the effect of vent glazing bars, which are smaller than the roof glazing bars, is ignored. The combined losses due to vent sill and slam rail, upon which the sill rests, are assumed to equal those due to a continuous cylinder, the vents occupying only about 40% of the total axial length. 4.3. Light measurements Robinsons
carried out during construction of a Jive-span multispan at A. R. Wills Ltd, Ramsey
These measurements were carried out during the summer period, and unfortunately conditions were only suitable for measurement at one stage, after all structure except ridges, gable-end glazing bars and roof vents were erected. At each stanchion position along the house, five vertically co-planar beams occur across the house, viz. rafters (2), trusses (2) and truss tie (Fig. 11). However, the average gain is again assumed unaffected by the overlap which affects a small length directly beneath the beams. The truss ties form a complex shape (Fig. II) and light losses are assumed equal to those due to a continuous straight cylinder of constant dimension across the house. Also, every sixth glazing bar is laid directly above a rafter, and totally obscured by it. Losses due to glazing bars are therefore reduced by 5/6 to account for this. Cross sections are shown in Fig. 12, and cylinder losses in Table 3. 0.24m ~----_-__--~ 0.06m
0.019
1-~0~025m
*____-__*
8
0.14 m
C
A
0.021 m A
10.036m v
---t 0,029m
0.030m 0,038m 0 F
D Fig. IO. Simplified cross sections of structural members in the IER four-span Venlo greenhouse used to calculate light loss. A, gutter; B, top and bottom girder channels; C, girder cross bracing; D, ridge; E, roof glazing bar; F, slam rail/sill
* After correction
(Section
3.3)
2.5 3.0 2.5 0.75 3.2 1.6
3.2 3.0 3.0 3.0
0.020 0.030 0.034
Periodic separation X0, m
Height above sensor h, m
Glazing bar Ridge Slam rail/vent
No. of cylinders n 2 2 1.75 1.85
Equivalent cylinder dia. D, m
0.1 0.03 0.03 0.013
sill
Table 2
0.027 0.009 0.021
0.03 1 0.010 0.010 0.005
(DlKJ = P
3.3 I.0 1.6
0.60 0.65 060 0.60
(W-W = ‘I
i Stage 2 total
Stage 1 total
leading to fractional light losses caused by structural members within the IER four-span Venlo greenhouse (Figs 9 and 10)
Gutter Girder top Girder bottom Cross piece
Structural member
Measurements
0.226
0,128
zm CA
0
D.
203
L. CRITTEN
1 Truss Fig. II. Rafters, trusses, and truss tie in Robinson multispan greenhouse. Truss tie represented by - _ for light loss calculations
4.4.
Comparison
of calculated
and measured
losses
In Table 4, calculated transmission losses at the various stages defined in Tables 1 to 3 are compared with measured losses where they are available. Predicted losses for glass cladding are obtained by using a measured reflective loss under diffuse overcast conditions for a 26” symmetric roof of 13x, and an absorption loss of 4%, giving a total of 17% (Refs 7, 8). These were then combined with structural losses as products of transmission, rather than addition of losses f TOT = 1 - c1 -%Fs)(l -&c>, (17) where tror is the total loss, Ts the loss due to structure and Tc the loss due to cladding. This reduces second order losses, which become significant (N 4%) as constituent losses approach 20%. The agreement between measured and predicted results is clearly good. 5. Discussion The comparison of experiment and predicted losses shown in Table 4 demonstrates the general validity of both Eqn (7) (Section 2), applied to periodic horizontal structures and the negligible contribution to light loss of the vertical stanchions (Section 3). The representation of non-horizontal structural elements, such as roof glazing bars by horizontal obstructions of increased diameter also appears satisfactory. The three greenhouses are quite different in the construction, and yet a maximum difference of less than two percentage points between measurements and prediction has been obtained. The general tendency is to over-predict 0.23 m t------_)
0.034m
0,,44m
I‘(
7
0.05m
0
1__~
0,025m
EOO4m
u
C
0
D
A
0,035m 0.03 m iI:r 0 0.03 m E
1
0.08m
F
+---_) 0.063m
_1{ 0,029m *-* 0.052m l-l
G
Fig. 12. Simplified cross sections of structural members in the A. R. Wills Robinson multispan greenhouse used to calculate right loss. A, gutter; B, truss; C, rafter; D, truss tie; E, roof glazing bar; F, purlin; G. ridge and vent hinges: H, vent sill
Table 3
4.0 3.6
2 2
0.06 0.025
Ridge and vent hinge Vent sill
* After correction
0.009 OXKkl
6.1 6.7
by rafter
0.35 040 040 0.70 0.70 060 4.30 1.0
0.018 OX)05 0.005 0.008 0.008 0.007 0.036 0.012
6.7 5.0 5.0 5.0 5.0 5.0 0.83 3.3
0.60 0.50
(hl&) = rl
(Dlxoi = P
Periodic separation X0, 112
obscured
2.4 2.0 2.2 3.4 3.6 3.0 3.6 3.3
2 2 2 1 1 1 1 2
0.12 0.025 0.025 O@IO OQ40 0.034 0.030 OJMO
Gutter Truss upper Truss lower Rafter upper Rafter lower Truss tie Roof glazing bars Purlins
(a) as Section 3.3, (b) for every sixth bar totally
Height above sensor h, m
of
cylinders n
Equivalent cylinder dia. D, m
Structural member
No.
Stage 2 total
2 1
Stage 1 total
-1
-1
1
1 1 1 0 0 0 0
.f
1 1 1 1 1 1 1 1 2
e
Measurements leading to fractional light losses caused by structural members within the A. R. Wills Robinson’s multispan (Figs 11 and 12)
0.193
0.016 0.009
0.168
0.043 0.012 0.012 0.011 0.011 0.010 0@47* 0.022
Fractional loss 7
D.
L.
205
CRITTEN Table 4
Calculated and predicted percentage light losses due to greenhouse structural members at various stages of erection
IER Simpson twinspan Stage (i) Stage 1 (ii) Stage 2 (iii) Complete house with glass Measurements
A. R. Wills Robinson multispan
IER Venlo
calculated
measured
calculated
measured
calculated
measured
9.1 18.2
10.0 20.0
12.8 22.6
12.3
16.8 19.3
152
32.2
31.0
35.8
34.0
32.9
taken over the central
region of each house
losses, by assuming that structures are completely “black”. Allowing, say, an additional transmission loss of 1% or 2% for stanchions, and other minor structural elements, a reflection coefficient of between 5% and 10% will bring experimental and predicted values into agreement for the Venlo and Robinsons houses. Predicted losses for the Simpson house are actually lower than measured values, but the house is not a multispan, and some structural members are excluded (vent sill, complex strut and vent drive pole). The losses due to these members can be estimated by the methods used in Table 3, and shown to produce an additional 2% loss over the whole house, as opposed to the central region, equivalent to a multispan losses. A comparison between predictions for completed houses can be made if the additional 27: referred to above is added to the Simpson house central area loss. Comparison between measured values is not appropriate, because different absorptive losses in the glass can occur. Losses due to completed houses are therefore approximately 34% (Simpson “multispan”), 36% (Venlo multispan) and 33% (Robinsons multispan). Thus the Venlo house is slightly worse than the other two, a 3% difference being significant, though a l’jO difference is probably not. The additional Venlo loss is undoubtedly due to the closeness of the gutters (Table 2). However, all the differences are small, and it is noteworthy that three quite different structural designs result in losses within f l_t% percentage points. In applying Eqn (7), equivalent circular diameters were judged by eye, and in some cases the fit was approximate. The method is therefore particularly useful because approximate methods of predicting losses produce reasonably accurate results. This is achieved because losses as a whole constitute a small (although important) fraction of the total incident light, and also because errors will partly average out. Examples showing how to fit the cylinders have already been shown in Fig. I. Referring to Eqn (7) taking II = 2, (e+f) equals 2.4 for horizontal and 1.8 for vertical cylinders. In fitting cylinders and choosing diameters therefore it is more important to fit the horizontal cylinders more accurately than the vertical ones. It is also obvious that a horizontal dimension is more significant than a vertical, because more light comes from higher than lower elevations, under a dull overcast sky. Hence, if difficulty is experienced in fitting cylinders to a member, a good horizontal fit should be made, and errors loaded into the vertical direction. From design drawings, tables such as 1 to 3 can readily be completed, and an accuracy of f 1 percentage point can be expected. 6. Conclusions The light losses due to periodic roof structural members that are generally horizontal in a multispan greenhouse under diffuse overcast conditions can be predicted by representing
206
TRANSMISSION
LOSSES
IN
MULTISPAN
GREENHOUSES
each member by touching multiple cylinders, followed by application of a simple algebraic relationship between light loss and cylindrical diameter. The light loss due to stanchions can be ignored. Roof structural members behave as nearly total absorbers with an effective reflectivity of less than 10% resulting in a slight over-prediction of losses. Houses with quite different structural design produce comparable light losses, and a “rule of thumb” value of 20% light loss due to structure can be applied. Additional losses due to cladding will depend on the cladding, but for 3 to 4 mm glass, a combined loss of 35% is appropriate. Small losses due to individual structural members can be added, but totalized structural losses should be combined with cladding losses by using the product of transmissivities, to reduce second-order errors. Under diffuse overcast conditions, the width of a horizontal structural member has a greater effect on light loss than the depth. Consequently, horizontal dimensions should be simulated more precisely than vertical when fitting cylinders to obtain light losses. Acknowledgements The author is indebted to Dr B. J. Bailey and Mr R. F. Cotton, who supplied light measurement data on the IER greenhouse structures, to Messrs. Robinsons of Winchester who carried out measurements on the Robinsons greenhouse during erection, and supplied drawings for light loss analysis, and to Dr B. J. Legg who suggested the basis for obtaining losses due to vertical poles (Section 3). References ’ C&en, D. L. Greenhouse light transmission: the shading effect of infinite parallel cylinders under diffuse light. Journal of Agricultural Engineering Research 1983, 28: 569-572 z Critten, D. L. A general analysis of light transmission in greenhouses. Journal of Agricultural Engineering Research 1986, 33: 289-302 3 Edwards, R. I.; Lake, J. V. Transmission of solar radiation in a large span east-west glasshouse. Journal of Agricultural Engineering Research 1964, 9(3): 245-249 4 Edwards, R. I.; Lake, J. V. The transmission of solar radiation in a small east-west glasshouse glazed with diffusing glass. Journal of Agricultural Engineering Research 1965, lO(3): 197-201 5 Bat, G. P. A. Greenhouse climate: from physical processes to a dynamic model. Thesis, Wageningen, 14 December 1983 6 Critten, D. L.; Legg, B. J. A general theory of the light transmittance of complex structures. Journal of Agricultural Engineering Research 1987, 36: 125-140 ’ Bowman, G. E. The transmission of diffuse light by a sloping roof. Journal of Agricultural Engineering Research (1970) H(2): loo-105 a Breuer, J. J. G. Energy measurements on various greenhouse coverings. Research Report No. 81-l. IMAG, Mansholtlaan lo-12 Wageningen
Appendix
The general form of Eqn (5) Eqn (5) states that, if cylinder overlap can be ignored, the average irradiance within an infinite set of long cylinders equals the total loss due to one cylinder. This result echoes an earlier conclusion that, to a first approximation, the loss within an infinite regular set of slightly attenuating structures equals the total loss due to one structure.’ Eqn (5) extends this result to regularly spaced discrete cylinders, without the need for spatial averaging.
D.
L.
207
CRITTEN
For any regular set of obstacles under diffuse or direct light:
EC1 x
0
s
L,(X) dx.
;
m
Here, L,is the irradiance loss due to any single obstacle within the set, and 1+9 Eqn (5) is replaced by (X/X,), with X0 constant. In this form, an experimentally based value of L can be obtained. This might be useful if structural members are covered in reflecting material, with an uncertain effective reflection coefficient.
Light Transmission Losses Due to Structural Members in Multispan Greenhouses under Diffuse Skylight Conditions D. L. CRITTEN* An accurate method of predicting light losses due to periodic structural members in the body of multispan greenhouses is presented. Losses due to stanchions are shown to be negligible, and members, such as roof glazing bars, which are not horizontal are successfully replaced by horizontal cylindrical members, of slightly increased diameter. Experimental measurements for three distinct greenhouse types are compared with predictions, and an accuracy of + 1 percentage point is obtained. The comparison shows that structural members behave as if they were nearly black, and that quite different designs lead to comparable losses, within + 13 percentage points.
1. Introduction In a previous paper’ the loss of light due to multispan gutters, treated as infinitely long black circular cylinders was predicted analytically, and reasonable agreement with experimental measurements was obtained. Some difficulty exists, however, in choosing the diameter of the cylinder to represent the gutter because of its complex cross section. If the process is to be extended to other structural members, such as the ridge, a similar problem will arise, the ridge usually being far from cylindrical. In the present paper, a more accurate approach is presented in which all horizontal long periodic structural members can still be assessed in a simple manner. As before, errors due to the finite size of the greenhouse are ignored. The analysis considers standard diffuse skylight conditions. In addition, we consider the results of an analysis’ in which it was shown that skylight can generally be separated into harmonic components. The principal harmonic radiance component is of the form (sin a), where a is the angle of elevation, which gives almost identical greenhouse light transmission to the dull overcast sky radiance, namely (1 + 2 sin a)/3. This analysis compares structural losses for the two radiances. Experimental measurements have previously been carried out only in single span greenhouses,3*4 and no record of the cross section of structural members has been made. Accordingly, comparison with experiment for the present theory relies on new measurements for which cross sectional data is known.
2. Theory 2.1. Light loss due to a regular array of obstructions In assessing light loss the detailed shape of any member is not important. Bat’ showed, for example, that a gutter could be accurately ,represented by an isosceles trapezium. The area projected normal to the light beam is what matters. In this work we approximate each
* Horticultural Engineering Division, IER, Silsoe, Bedford, UK Received 10 November 1986; accepted in revised form 26 March 1987 193 0021-8634/87/l
10193 + I5 $03.00/O
0
1987 The British Society for Research in Agricultural
Engineering
194
TRANSMISSION
LOSSES
IN
MULTISPAN
GREENHOUSES
1
Notation
angle of elevation : effective light blockage width, m D diameter of a cylinder, m h cylinder height, m I sky radiance, W m-2 str-’ 10 radiance at the zenith of the sky vault, W me2 str-’ L irradiance loss, W m-2 n number of cylinders required to represent a structural member N numbering system or number of cylinders in an array R distance from a specified point on
R, t
X X, P r * 4
the ground to a point along a cylinder’s axis, m shortest value of R, m transmission loss coordinate normal to the cylinder axis, m cylinder spacing, m D/X, h/X, XIX, angle between R and R, (Fig. 2)
overbar (-)
average value
member by a variable number of cylinders of equal diameter (D). Illustrative examples are shown in Fig. 2. A glazing bar is represented by a single cylinder (Fig. I, left), a gutter by two horizontally adjacent cylinders (Fig. I, middle) and a ridge by two vertically adjacent cylinders (Fig. 1, right). Inspection of Fig. 1 using different light beam elevations shows that a much better approximation of light obstruction is obtained in Fig. 1 (middle and right), than would be obtained by using a single cylinder approximation. By choosing different numbers of cylinders in a single line, any generally rectangular obstruction can be simulated. To obtain the angular width of a series of n equidiameter cylinders, we view the cylinder array in the plane containing R, and R, where R is the distance to the cylinder from any point X, and R, the shortest distance (Fig. 2). The result for any angle C$is shown in Fig. 3 (left) for horizontally, and Fig. 3 (right) for vertically adjacent members using two cylinders. For n cylinders: d = (l+(n-l)sina,) (Fig. 3, left, horizontally aligned cylinders) d = D( 1 + (n - 1) cos a,) (Fig. 3, right, vertically aligned cylinders) -D(n-(n-1) sin ao).
(1)
The approximation cos a- (1 -sin a) is gross, but for n = 2 experience shows a satisfactory accuracy is obtained. For high values of n, deterioration in accuracy might be expected.
Fig. 1. Representation
of various structural members by equidiameter contacting Glazing bars; (middle) gutter; (right) ridge
cylinders.
(Left)
D.
L.
195
CRITTEN
Fig. 2. View of a single cylinder de$ning linear and angular distance for horizontal cylinders
In general, therefore, d =
D(e+fsin
a,),
(2)
where e and f are defined in Eqn (1) for the described combinations of cylinders. A single cylinder is represented with e = 1 and f= 0. Since d is not a function of 4, the total irradiance loss (L) for an infinitely long obstruction array can be immediately written down using the equation derived by Critten’ for an array of single cylinders: Under a standard sky
where p = D/X,, q = h/X0, II/=X/X, and N is the cylinder reference number, such that (NX,+X) is the horizontal distance of the cylinder from the point of interest. Replacing D
Fig. 3. View of (a) horizontally and (b) vertically touching cylinders of diameter D in the plane of R and R,. (Left) d= D(l+sina,); (right) d= D(I+cosa,)
196
TRANSMISSION
by d from Eqn (1):
71
L may be conveniently
L=
IN
MULTISPAN
GREENHOUSES
4+3./m
N=m 2 L=310PgN=zm
2
LOSSES
4
‘I
. t3) >3
rewritten
N=m
jzop9N=c_m
A (N+$)2+q2
BV + [(N+t+b)2+rj2]3’2
Cf12 + [(N+~I)~+Y+I~ >’
(4)
B=le+:f.
C=if
2.2. The average loss In Eqn (3) the average value of L across a span is given by the integral of L with respect to @, normalized with respect to the integral of $, both from 0 to 1. Since the latter integral is unity, the resulting sum is over consecutive integrals, (L) is simply given by
(5) The form of Eqn (5) is discussed in the Appendix. Noting 00 1 _m Wd$ s
= nnlrl,
Then
(6) The external irradiance, obtained by integrating the irradiance due to dull overcast sky elements over the sky vault may be shown to equal 24431,. Then substituting for A, B, and
Fig. 4. Representation
of roof glazing bars by an equivalent horizontal member qf increased thickness
D.
L.
197
CRITTEN
Fig, 5. Obstruction produced by a vertical pole in a light beam of elevation a
C, regrouping, and normalizing transmission loss:
with respect
to external
irradiance
f = p( 1.40e + 0.99f). Using the first harmonic yielding:
gives the light (7)
(sin a) only (Section l), a similar expression can be calculated, f = p(1.27e+f).
(8)
Typically, p < 0.1 and e = 1. Hence the difference between the two forms is approximately one percentage point. 2.3. Representation
of roof glazing bars
Roof glazing bars do not form parallel horizontal obstructions, but are of sawtooth form. However, for low angle roofs (e.g. 26”) the undulating form can be replaced by a horizontal obstruction, of thickness increased by l/cos (roof angle) to allow for the extra length introduced by the undulations (Fig. 4). 3. Light loss due to a regular array of vertical poles
Referring to Fig. 5, consider first the light loss due to a single pole, height h, subject to a beam of elevation a. Then, for radiance symmetric in azimuth (A), dL = 2xlDh cos’ a da for an overcast sky
xl.7
loss L = $ I, Dh
(1 + 2 sin a) cos’ a da,
(9)
I0
where L is the total light removed by the pole over the whole ground area. Consider now a square array of poles, shown in plan view in Fig. 6. A pole is assumed at the centre of each square. Consider the loss introduced by a pole at position 0 into the square occupied by pole 1, say. This must equal the loss in square 0 due to pole 1. The argument is true for any of poles 1 to 8, and so the total loss in square 0 due to poles 0 to 8 equals the loss in the complete square due to pole 0. The process may be extended indefinitely, and so the total loss in square 0 is given by Eqn (9). The average loss L over square 0 is then simply L = L/X$ (10) and average transmission loss (Q is given by t = L/(2.4431,)
(11)
198
TRANSMISSION
LOSSES
IN
MULTISPAN
GREENHOUSES
Fig. 6. Matrix dejining areas associated with a square array of vertical poles for calculkng
total average
light loss
or .
(12)
Typically, p = 0.01, q = O-5, giving f = 0.6 percentage points. Thus iFfor stanchions can generally be ignored. 4. Comparison with experimental measurements Experimental measurements are normally carried out along a line of symmetry across a central span of a multispan greenhouse. For structural members parallel to this line, if these results are to compare with predictions from Eqn (7), the difference between the average loss and loss at the line of measurements must be small (< 1 percentage point). It has been shown’ that if v > 0.4, at a point midway between single cylinders, the loss is about 20% less than the average value. For a member causing a loss of five percentage points, this is equivalent to an error of one percentage point. Thus, u should be of this order for accurate prediction of losses. For structural members perpendicular to the line of measurement such as gutters, the average value of measurements is taken anyway and so the condition is not required. 4.1. Light measurements
carried out during the construction of a Simpson twinspan greenhouse at IER, Silsoe, Bedfordshire
Though not a multispan, average measurements over the central half of the twinspan should be substantially in agreement with multispan predictions. Measurements were carried out in three stages: (1) after erection of all weight-bearing elements; (2) insertion of glazing bars; and (3) completion of the house. Fig. 7 shows the schematic cross sections of structural members and Table 1 shows values of D, h, and X, leading to the irradiance loss for each member, the angled support being circular in cross section. A plan view of the cross bracing is shown in Fig. 8. Complex struts between the cross braces and purlins were ignored. At stage (3), the effect of the vent glazing bars and sills, which overlay the normal bars has not been included. Vent glazing bars occupy about one-sixth of the total glazing bar length, but are obscured for near zenith radiation. They are also on the outer roofs, remote from the central region.
(Section
3.3)
0.035
Roof glazing bar
Equivalent cylinder dia. D, m
0.06 0.10 0.035 0.03 0.06
* After correction
Table 1
1
2 2 1 1 1
No. of cylinders n
3.9
4.9 2.9 3.9 2.8 3.1
Height above sensor h, m
0.6
6.8 6.8 3.4 4.5 5.6
Periodic separation X0, m = p
0.058
0~009 0.015 0.010 0~007 0.011
(D/X,)
5.00
060 0.30 0.45 060 0.55
(hlX,J
= II
Stage 2 total
1
Stage 1 total
2 1 1 1 1
e
0
-1
f 1 0 0 0
1
leading to fractional light losses caused by structural members within the IER Simpson twinspan greenhouse (Figs 7 and 8)
Ridge Gutter Purlin Cross beam Angled support
Structural member
Measurements
0.182
0.091*
0.091
0.016 0.036 0.014 0.010 0.015
lossf
Fractional
TRANSMISSION
200
LOSSES
IN
MULTISPAN
GREENHOUSES
0.06 m
E
0.25 m
*------_---_~
u
j5m +------__F 0.145 m
0.03
0.03 m 0.04 m rl C
cl
0.03 m
I
m 0.04 m
F
Fig. 7. Simplified cross sections of structural members in the IER twinspan greenhouse used to calculate light loss. A, ridge; B, gutter; C, purlin; D, cross bracket (Fig. 8); E, angled bracket, taken as - - -; F, glazing bar
4.2. Light measurements carried out during the construction four-span Venlo at IER
of a
In this case, only two stages were measured: (1) after erection of stanchions, gutters and girders, and (3) after erection of glazing bars, glass and ridge, the Venlo design requiring the latter to be erected at the same time. The form of the girders, which run across the house, is shown in Fig. 9. Because girder members are co-planar with the vertical, directly beneath the
Fig. 8. Plan view of cross bracing in the IER twinspan glasshouse. g = gutter
201
D. L. CRITTEN
Fig. 9. Form of transverse girder in the IER Venlo greenhouse. Cross bracing represented by - - - ,f;rr light loss calculations
girder losses are reduced. However, measurements are taken between girders, and so overlap is ignored. The effect of the overlap on the average loss is very slight (-0.1%). Cross sections are shown in Fig. 10, and the cylinder losses are shown in Table 2. As with the Simpson twin span, the effect of vent glazing bars, which are smaller than the roof glazing bars, is ignored. The combined losses due to vent sill and slam rail, upon which the sill rests, are assumed to equal those due to a continuous cylinder, the vents occupying only about 40% of the total axial length. 4.3. Light measurements Robinsons
carried out during construction of a Jive-span multispan at A. R. Wills Ltd, Ramsey
These measurements were carried out during the summer period, and unfortunately conditions were only suitable for measurement at one stage, after all structure except ridges, gable-end glazing bars and roof vents were erected. At each stanchion position along the house, five vertically co-planar beams occur across the house, viz. rafters (2), trusses (2) and truss tie (Fig. 11). However, the average gain is again assumed unaffected by the overlap which affects a small length directly beneath the beams. The truss ties form a complex shape (Fig. II) and light losses are assumed equal to those due to a continuous straight cylinder of constant dimension across the house. Also, every sixth glazing bar is laid directly above a rafter, and totally obscured by it. Losses due to glazing bars are therefore reduced by 5/6 to account for this. Cross sections are shown in Fig. 12, and cylinder losses in Table 3. 0.24m ~----_-__--~ 0.06m
0.019
1-~0~025m
*____-__*
8
0.14 m
C
A
0.021 m A
10.036m v
---t 0,029m
0.030m 0,038m 0 F
D Fig. IO. Simplified cross sections of structural members in the IER four-span Venlo greenhouse used to calculate light loss. A, gutter; B, top and bottom girder channels; C, girder cross bracing; D, ridge; E, roof glazing bar; F, slam rail/sill
* After correction
(Section
3.3)
2.5 3.0 2.5 0.75 3.2 1.6
3.2 3.0 3.0 3.0
0.020 0.030 0.034
Periodic separation X0, m
Height above sensor h, m
Glazing bar Ridge Slam rail/vent
No. of cylinders n 2 2 1.75 1.85
Equivalent cylinder dia. D, m
0.1 0.03 0.03 0.013
sill
Table 2
0.027 0.009 0.021
0.03 1 0.010 0.010 0.005
(DlKJ = P
3.3 I.0 1.6
0.60 0.65 060 0.60
(W-W = ‘I
i Stage 2 total
Stage 1 total
leading to fractional light losses caused by structural members within the IER four-span Venlo greenhouse (Figs 9 and 10)
Gutter Girder top Girder bottom Cross piece
Structural member
Measurements
0.226
0,128
zm CA
0
D.
203
L. CRITTEN
1 Truss Fig. II. Rafters, trusses, and truss tie in Robinson multispan greenhouse. Truss tie represented by - _ for light loss calculations
4.4.
Comparison
of calculated
and measured
losses
In Table 4, calculated transmission losses at the various stages defined in Tables 1 to 3 are compared with measured losses where they are available. Predicted losses for glass cladding are obtained by using a measured reflective loss under diffuse overcast conditions for a 26” symmetric roof of 13x, and an absorption loss of 4%, giving a total of 17% (Refs 7, 8). These were then combined with structural losses as products of transmission, rather than addition of losses f TOT = 1 - c1 -%Fs)(l -&c>, (17) where tror is the total loss, Ts the loss due to structure and Tc the loss due to cladding. This reduces second order losses, which become significant (N 4%) as constituent losses approach 20%. The agreement between measured and predicted results is clearly good. 5. Discussion The comparison of experiment and predicted losses shown in Table 4 demonstrates the general validity of both Eqn (7) (Section 2), applied to periodic horizontal structures and the negligible contribution to light loss of the vertical stanchions (Section 3). The representation of non-horizontal structural elements, such as roof glazing bars by horizontal obstructions of increased diameter also appears satisfactory. The three greenhouses are quite different in the construction, and yet a maximum difference of less than two percentage points between measurements and prediction has been obtained. The general tendency is to over-predict 0.23 m t------_)
0.034m
0,,44m
I‘(
7
0.05m
0
1__~
0,025m
EOO4m
u
C
0
D
A
0,035m 0.03 m iI:r 0 0.03 m E
1
0.08m
F
+---_) 0.063m
_1{ 0,029m *-* 0.052m l-l
G
Fig. 12. Simplified cross sections of structural members in the A. R. Wills Robinson multispan greenhouse used to calculate right loss. A, gutter; B, truss; C, rafter; D, truss tie; E, roof glazing bar; F, purlin; G. ridge and vent hinges: H, vent sill
Table 3
4.0 3.6
2 2
0.06 0.025
Ridge and vent hinge Vent sill
* After correction
0.009 OXKkl
6.1 6.7
by rafter
0.35 040 040 0.70 0.70 060 4.30 1.0
0.018 OX)05 0.005 0.008 0.008 0.007 0.036 0.012
6.7 5.0 5.0 5.0 5.0 5.0 0.83 3.3
0.60 0.50
(hl&) = rl
(Dlxoi = P
Periodic separation X0, 112
obscured
2.4 2.0 2.2 3.4 3.6 3.0 3.6 3.3
2 2 2 1 1 1 1 2
0.12 0.025 0.025 O@IO OQ40 0.034 0.030 OJMO
Gutter Truss upper Truss lower Rafter upper Rafter lower Truss tie Roof glazing bars Purlins
(a) as Section 3.3, (b) for every sixth bar totally
Height above sensor h, m
of
cylinders n
Equivalent cylinder dia. D, m
Structural member
No.
Stage 2 total
2 1
Stage 1 total
-1
-1
1
1 1 1 0 0 0 0
.f
1 1 1 1 1 1 1 1 2
e
Measurements leading to fractional light losses caused by structural members within the A. R. Wills Robinson’s multispan (Figs 11 and 12)
0.193
0.016 0.009
0.168
0.043 0.012 0.012 0.011 0.011 0.010 0@47* 0.022
Fractional loss 7
D.
L.
205
CRITTEN Table 4
Calculated and predicted percentage light losses due to greenhouse structural members at various stages of erection
IER Simpson twinspan Stage (i) Stage 1 (ii) Stage 2 (iii) Complete house with glass Measurements
A. R. Wills Robinson multispan
IER Venlo
calculated
measured
calculated
measured
calculated
measured
9.1 18.2
10.0 20.0
12.8 22.6
12.3
16.8 19.3
152
32.2
31.0
35.8
34.0
32.9
taken over the central
region of each house
losses, by assuming that structures are completely “black”. Allowing, say, an additional transmission loss of 1% or 2% for stanchions, and other minor structural elements, a reflection coefficient of between 5% and 10% will bring experimental and predicted values into agreement for the Venlo and Robinsons houses. Predicted losses for the Simpson house are actually lower than measured values, but the house is not a multispan, and some structural members are excluded (vent sill, complex strut and vent drive pole). The losses due to these members can be estimated by the methods used in Table 3, and shown to produce an additional 2% loss over the whole house, as opposed to the central region, equivalent to a multispan losses. A comparison between predictions for completed houses can be made if the additional 27: referred to above is added to the Simpson house central area loss. Comparison between measured values is not appropriate, because different absorptive losses in the glass can occur. Losses due to completed houses are therefore approximately 34% (Simpson “multispan”), 36% (Venlo multispan) and 33% (Robinsons multispan). Thus the Venlo house is slightly worse than the other two, a 3% difference being significant, though a l’jO difference is probably not. The additional Venlo loss is undoubtedly due to the closeness of the gutters (Table 2). However, all the differences are small, and it is noteworthy that three quite different structural designs result in losses within f l_t% percentage points. In applying Eqn (7), equivalent circular diameters were judged by eye, and in some cases the fit was approximate. The method is therefore particularly useful because approximate methods of predicting losses produce reasonably accurate results. This is achieved because losses as a whole constitute a small (although important) fraction of the total incident light, and also because errors will partly average out. Examples showing how to fit the cylinders have already been shown in Fig. I. Referring to Eqn (7) taking II = 2, (e+f) equals 2.4 for horizontal and 1.8 for vertical cylinders. In fitting cylinders and choosing diameters therefore it is more important to fit the horizontal cylinders more accurately than the vertical ones. It is also obvious that a horizontal dimension is more significant than a vertical, because more light comes from higher than lower elevations, under a dull overcast sky. Hence, if difficulty is experienced in fitting cylinders to a member, a good horizontal fit should be made, and errors loaded into the vertical direction. From design drawings, tables such as 1 to 3 can readily be completed, and an accuracy of f 1 percentage point can be expected. 6. Conclusions The light losses due to periodic roof structural members that are generally horizontal in a multispan greenhouse under diffuse overcast conditions can be predicted by representing
206
TRANSMISSION
LOSSES
IN
MULTISPAN
GREENHOUSES
each member by touching multiple cylinders, followed by application of a simple algebraic relationship between light loss and cylindrical diameter. The light loss due to stanchions can be ignored. Roof structural members behave as nearly total absorbers with an effective reflectivity of less than 10% resulting in a slight over-prediction of losses. Houses with quite different structural design produce comparable light losses, and a “rule of thumb” value of 20% light loss due to structure can be applied. Additional losses due to cladding will depend on the cladding, but for 3 to 4 mm glass, a combined loss of 35% is appropriate. Small losses due to individual structural members can be added, but totalized structural losses should be combined with cladding losses by using the product of transmissivities, to reduce second-order errors. Under diffuse overcast conditions, the width of a horizontal structural member has a greater effect on light loss than the depth. Consequently, horizontal dimensions should be simulated more precisely than vertical when fitting cylinders to obtain light losses. Acknowledgements The author is indebted to Dr B. J. Bailey and Mr R. F. Cotton, who supplied light measurement data on the IER greenhouse structures, to Messrs. Robinsons of Winchester who carried out measurements on the Robinsons greenhouse during erection, and supplied drawings for light loss analysis, and to Dr B. J. Legg who suggested the basis for obtaining losses due to vertical poles (Section 3). References ’ C&en, D. L. Greenhouse light transmission: the shading effect of infinite parallel cylinders under diffuse light. Journal of Agricultural Engineering Research 1983, 28: 569-572 z Critten, D. L. A general analysis of light transmission in greenhouses. Journal of Agricultural Engineering Research 1986, 33: 289-302 3 Edwards, R. I.; Lake, J. V. Transmission of solar radiation in a large span east-west glasshouse. Journal of Agricultural Engineering Research 1964, 9(3): 245-249 4 Edwards, R. I.; Lake, J. V. The transmission of solar radiation in a small east-west glasshouse glazed with diffusing glass. Journal of Agricultural Engineering Research 1965, lO(3): 197-201 5 Bat, G. P. A. Greenhouse climate: from physical processes to a dynamic model. Thesis, Wageningen, 14 December 1983 6 Critten, D. L.; Legg, B. J. A general theory of the light transmittance of complex structures. Journal of Agricultural Engineering Research 1987, 36: 125-140 ’ Bowman, G. E. The transmission of diffuse light by a sloping roof. Journal of Agricultural Engineering Research (1970) H(2): loo-105 a Breuer, J. J. G. Energy measurements on various greenhouse coverings. Research Report No. 81-l. IMAG, Mansholtlaan lo-12 Wageningen
Appendix
The general form of Eqn (5) Eqn (5) states that, if cylinder overlap can be ignored, the average irradiance within an infinite set of long cylinders equals the total loss due to one cylinder. This result echoes an earlier conclusion that, to a first approximation, the loss within an infinite regular set of slightly attenuating structures equals the total loss due to one structure.’ Eqn (5) extends this result to regularly spaced discrete cylinders, without the need for spatial averaging.
D.
L.
207
CRITTEN
For any regular set of obstacles under diffuse or direct light:
EC1 x
0
s
L,(X) dx.
;
m
Here, L,is the irradiance loss due to any single obstacle within the set, and 1+9 Eqn (5) is replaced by (X/X,), with X0 constant. In this form, an experimentally based value of L can be obtained. This might be useful if structural members are covered in reflecting material, with an uncertain effective reflection coefficient.