NASA

,,.

,

_

E

....

NASA

Technical_Memo

rand_urn

105751

....

_......

5_

i...... Solar Radiation

on a C!tenarY

Collector

M. Crutchik Tel-Aviv

University

Tel-Aviv,

Israel

and J. Appelbaum Lewis Research Cleveland,

Center

Ohio

(NASA-TM-1057_I) CATENARY 32

\ \ t

SOLAR COLLECTOR

N92-32243

RADIATION (NASA)

p Unclas

NASA G3/33

0110079

SOLAR

M.

RADIATION

ON A CATENARY

J.

Crutchik

Faculty

of

University

Tel-Aviv

69978,

AppelbaumJf

National

Engineering

Tel-Aviv

COLLECTOR

Israel

Aeronautics

Space

Administration

Lewis

Research

Cleveland,

and

Center,

OH 44135

Abstract

A

tent-shaped

collector

is

structure

presented.

The

with shadow

producing

a

self

shading

effect

radiation

on

the

collector

are

collector

operating

"]'Current

address:

This

work

Associate

was at

on

Mars

done

NASA

flexible

cast

by

determined.

the

Research

photovoltaic one

side

The

direct

analysed.

for

University,

while Lewis

is

surface

Tel-Aviv

a

An

the

location

Faculty

author Center.

example

was

of a

Work

of

blanket

of

the

Viking

Engineering, National funded

collector

beam, is

the

given

for

Lander

as

on

a

the

diffuse

and

the

insolation

catenary other the

side albedo on

the

1 (VLI).

Tel-Aviv

Research under

acting

69978,

Council

NASA

Grant

-

Israel. NASA

Research

NAGW-2022.

Mission little by

to

Martian

installation

time,

a

photovoltaic

power

generation

pressurized

combination

of

cables,

PV

with

the

is due

blanket

increase:

is in

load

plane,

2.

Fig.

to

the

According

will

to

A

and

by

and

the

with

1.

to

The

structural

cable

which

shape

of

some

and

The

main

Of

c.atenary

be

accomplished

blanket

for

with

design

PV

contribution PV

a

cable

fc(Y)

is given

the

array

blanket. to

catenary

iteratively. distributed

constant However, along

the

k

can

when

the

y-axis

and

be

The

a

array in

the

The

shape

of

the

cable

tension a

respect

and

uniformly tO

the

Y±Z

by:

(1)

determined

blanket the

uses

stress

kEcosh( ) ,]The

solar

the

carrying with

requires

self-deploying

the

blanket. in

fc(Y),

a

for

the

curve,

manipulations,

can

designed

reduction

gravity

that

PV

the

between

supply

flexible

deploy

supports

force a

is

power volume

a

array

rolled.

the

a small

The

or

A

with

support

optimization

under

in

structure

Fig.

folded

power.

stowed

expansion.

an

the

electric

and

columns

in

area.

[2]

[1], gas

tension

'take

require

tent-shaped

either

determined

will weight

in

beams

blanket

blanket

distributed

array.

proposed

a

INTRODUCTION

light

is

using

structure

being

(PV)

mechanism

is stowed

surface

1.

is

catenary

using fairly

the

taut,

curve

i.e.,:

2

may

condition the

load

fc(0) may

be approximated

be

=

H

assumed by

and

solving

uniformly

a parabola

[2],

H

f_(y)

which

-- _

simplifies

the

Because of on

its

these

diffuse for

of

sides

the

the

of

Fig. beam

albedo

location

Appendix

shape

B in

results,

the

calculation.

the

(side

and

(2)

(D - y)2

a catenary-tent-co!lector,

2).

In

insolation

insolation

of

Viking

article on

on

the

Lander

we

the

analyze

collector

collector.

1.

Results

self-shading shadow

example

the

We

for

parabolic

occurs

shape

is calculated.

An for

the

effect

the

and

also

on

area.

one Based

determine

planet

Mars

approximation

is

the

is given given

in

B.

2.

The is

this

a

shaded

alternately

SHADOW

catenary-tent-collector depend

on

shaded

is self

the

sun

in a given

_sr

< Wc -

"_c

-

the

collector

"Vsr -

the

sun

shading.

position.

day

if

at

CALCULATION

In

The

size

general,

of

both

the sides

shadow of

and

the

the

side

collector

which will

be

sunrise

90 °

(3)

where

The

azimuth eq.

(3)

paper

we

analyze

the

the

are

is not

south-north

replacing

azimuth

angles

when

azimuth

the

sunrise.

measured

from

satisfied, shape

direction. sun

at

azimuth

only and The angle

size results

true

south

positively

one

side

of

of

the

shadow

can

"Vs by

the

be

the

collector cast

generalized difference

3

in a

clockwise

is shaded

on

a

for between

all

direction. the

times.

catenary-tent-collector an

arbitrary the

solar

In

days this

facing

oriented and

In

the

tent collector

by

azimuthangles,i.e., :

With

_s

-

_c

reference

(4)

to

Fig.

3,

the

shadow

cast

by

line

ON

can

be

divided

into

three

distinct

cas6s:

It

will

of

case

(i)

Pz

(

L

,

P_ (

(ii)

Pz

(

L

,

P_

(iii)

Pz

_> L

,

Pu (

be

shown

later

(ii) The

or

case

D

(s)

_> D

that

D

for

the

case

Px

_> L,

Py

_> D

the

shadow

takes

the

shape

(iii).

components

Px and

Px

= H cos6

sinw/simx

Pu

= H(sino

cos6

P_ of

the

line

OP was

derived

in

[3,4]

and

are:

(6)

and cosw

-

cos¢

sinS)/sinet

(7)

where sinct

= sino

sin8

+ coso

cos8

and 6

7

-

solar

declination

O -

local

latitude

tx -

sun

to -

solar

elevation hour

angle

angle angle

7

:

cosw

(8)

either

G2se _i):

In collector

Pz < L _ Ps < D

this

case

takes

the

penetrates

the

order

calculate

to

determined.

solar

shape

collector,

Since

YE is obtained

the

NTEExM and point

the the

shaded

catenary

numerically

H cosh

_

The component

rays

x_ is then

the

as shown

In Fig.

T is a point area,

the

function

from

+

penetrate

4.

surface Point

and

the

shadow

on

the

E is a point

where

the

ray

to the collector.

In

where the ray is tangent

components

fc(Y)

the.solution

collector

of

is hyperbolic,

of the following

the

points

the solution equation

E

and for

=

_

the

must

be

component

(see Appendix

A):

= I + g

given by (see Fig.

4)

Pz XE

T

(I0) YE

Point T is in y-z plane, i.e.,xr = 0 and YT is obtainedfrom the solutionof (seeAppendix A): Yr = D - k sinh-l(H/Ps)

It is easier

to obtain

the shaded

Ash = L N Es - 0.5

The unshaded

A_sh

xET

(11)

area

Ash

from

the "streched

Ey

out" collector,

Fig.

4(b),

i.e.,

(12)

area Aush is given by:

= A - Ash

= L(N Nn - N Es)

where A is the area of one

+ 0.5 x_T

side of the collector,

i.e.,

Es

(13)

A

= L N

Defining

a relative

t. = (2

The

Nn

length

(14)

shaded

L N E_

of

an

arc

Yl Y2 :

-

area

xL-T

+

t.,

we obtain

E_)/2L

between

x/1

by

two

[fc'(y)]

N Nn

points

(15)

Yl and

Y2 is given

by:

2 dy

(16)

1

where

fc'(Y)

is the

derivative

of

the

function

fc(Y).

dy

= k sinh(D/k)

The

length

of

the

arcs

in eq.

(15)

thus

become:

:" E

N

Nn

=

_/1

+ [fc'(Y)]

N

Ev

=

x/1

+

_/1

+ fc'(y)

T Eu

:

[fc'(y)]

2 dy

2 dy

= k sinh(D/k)

[(

= k sinh

T

Case

(ii):

P_ < L

This at

point

U.

case

Pu

is shown

The

coordinates

derived

in

+ sinh

)

+ sinh

ye

D

(18)

(

D k- YT

)]

(19)

D

in Fig. of

YE = D

As

>

YE l_- D

(17)

5. point

The

array

touches

the

collector

at

the

foot,

point

E,

E are:

(20)

Appendix

A,

and

Pz

(21)

xE = _

where

(D

- Yv)

Yu is determined

from

the

solution

k[ .h(O Y°)] " - 1

The

coordinates

eq.

(11).

The

Ash

:

Aus_

The

where

_. :

1

the

arc

case

N Nn

and

are

area

OH

(22)

the

same areas,

as for Fig.

case

5(b),

(i), are,

i.e.,

xr

= 0

and

Yr is given

(23)

(24)

is obtained

+

as:

(25)

is given

[fc'(y)]

by:

2 dy

: k sinh

T

is given

The

solar

in eq.

>_ L

ray

intersection collector,

by

respectively:

Nn

T Nn

_/1

:

the

A):

Yv = _

unshaded

xE'T

Appendix

xr T Nn

length

Pz

intersects

0.5

T

(see

xr T Nn 2 L Nt'_Nn

(iii):

of

-

shaded

--

Case

A

relative

point

shaded

: 0.5

T Nn

and

of

+ _

of

(

D - YT

_

)

(26)

(17).

, Pu < D

may

either

was

already

eqs.

(9)

intersect analysed and

(10)

the in

collector case

are solved.

or (i).

may To

fall

determine

outside

the

whether

collector.

The

the

ray

solar

If Fig.

6.

xE

< L

The

than

there

coordinates

is an

of

point

intersection.

If

xE

> L than

there

is

no

intersection,

E are:

xE = L

and

(27)

YE is determined

from

the

solution

- y_ + L _ k

k cosh

The

see

coordinates

of

of

- cosh D

(see

point T

are xr

Appendix

=

0

Yr

=

- YE k

and

A):

L H Px

(28)

is given by

eq.

(11).

The

shaded

and

unshaded areas, Fig. 6, are respectively:

Ash

The

= L N

Aush

= L(N

relative

shaded

_. = (2 N

where

N

N,,

The on

E_

the

Nn

-

N Ev

area

+ 0.5

Ey - T E_)/2 N

Pz

T E_ are

>_ L

Px/Pu

of

(29)

(30)

T Es)

is:

N E_, and

case

ratio

- 0.5 L T E_

the

,

Pu ray

N.

(31)

given

in eqs.

-> D corresponds OP

either

(see

Fig.

7).

For

Px/Pu

< L/D

the

solution

is as for

case

(ii)

for

Pz/Pu

>_ L/D

the

solution

is as for

case

(iii)

For

a

with

the

solar

O'M

(see

Fig.

south-north noon. 3) ....

oriented For

the ::

collector, after

noon,

(17)-(19).

to

case

(ii)

a

function

or

to

case

(iii)

depending

of

time

is symmetrical

and

the

shadow

as

the

shadow

calculations

are

with

respect

to

line

3.

With and

the

the

beam

determined.

insolation

in

Whr/day

and

albedo

diffuse insolation.

to

any

A

in

the

previous or

components

will

facing

oriented

the

Whr/m2-day

north-south

arbitrary

section,

tent

on be

beam

the

irradiance

added

be obtained

or

catenary-tent-collector to

the

beam

eatenary-tent-collector

may

in W

to

will

using

eq.

W/m 2 can

get

the

beam

(3).

In

irradiance

P,

dPb

where

:

Gb

between

is

the

.cosO :

global

be considered.

A

(4).

irradiance

calculating in

on the

watts

(W),

Gb COS0 dA

the

beam

solar

rays

cos_(y)

both

sides

irradiance

of on

the an

on an unshaded

tent

depends

unshaded

side

side

is given

on

the

self-shading

we

resort

to

condition

Fig.

8.

dA

:

L dS

dS is the

unit

the

in

normal

sincx + sinl3(y)

length

sin/3(y)

= Ifc'(y)l/,,/1

cost3(y)

:

1/-,/1

+

beam

(32)

irradiance

= L _/1

of

by:

\

and

The

W/m 2

normal

to

to the

surface,

cos0

the and

solar

rays,

dA are

and

given

0

is

the

by

(33)

coscx cosns

and

where

be

lrradiance

The eq.

developed

and

generalization

Beam

results

The

irradiance

SOLAR RADIATION CALCULATION

+ [fc'(y)]

of

the

+

[fc'(y)]

collector

[fc'(Y)]

2

(34)

2 dy

z

(Fig.

8)

and

(3S)

(36)

angle

Therefore,

eq.

dPb

= Gb

Integrating unshaded

Using

i.e.,

(32)

(37)

of

Pb

= Gb

the

angle

Pb

= Gb

the

beam

collector

L(sinc_

eq. side

reduces

+ [fc'(Y)l

in

the

L(D

sine

the

irradiance

i.e.,

the

angle

0

Oh(y)

varies

W/m

2,

= v/I +

and

with

eq.

(1)

on

8,

the

beam

irradiance,

in

W,

a

on

with

along

on

the

38)

the

a

we obtain:

simx

+ sine

catenary

conclusion

on

beam

in

in Fig,

beam

the

obtain

+ H cosc_ cos'_s)

incoming

since

we

i.e.,

irradiance

irradiance

(37)

cos-_,)dy

[0,D],

L(H 2 + D2) 1/2 (cose

beam

irradiance,

cosa

interval

e as defined

This

multiplying

the

collector,

MNMsNs.

The

to:

collector

is also

Valid

for

collector

that

is

irradiance

with

the

collector

y.

Using

eqs.

collector

is:

Gb [simx [fc'(y)]2

39)

cos_x cos'_s)

is any

non-flat

partially

the

is not

equivalent

factor

(33),(35)

shaded

along and

that

shape

(1-_,)

uniform

to

(36),

for

the the

a

flat

plate

collector.

can

as

of

not

be

a flat

y axis

of

variation

obtained

plate

by

collector

the

collector,

of

the

beam

+ COSC_C0S'_slfc'(Y)l]

we obtain:

(40) Gb(y)-

With

reference

coshi____)

to

Fig.

[sinc_÷ cosa cos_s'sinh(-_)']

9(a),

the

beam

irradiance

10

on

the

partially

shaded

side

of

the

collector

for

p_h

=

case

(i)

is given

/?

Oh(y)

dAl

by:

+

T

where

Go(Y)

Oh(y)

dA2

1_"

is given

= x dS

dAl

/,°

by

eq.

(40),

and

(41)

= x cosh(-_.Y_ dy k 1¢ !

_-

(42) s

where

x and

y ale

related

X = ( YE x_ YT )(Y

Substituting

these

results,

by

-

(Fig.

9(a))

(43)

Yr)

we

obtain

the

beam

irradiance

on

the

partially

shaded

side

collector:

(45)

Similarly,

the

P_'h

and

the

beam

= Gb

beam

irradiance

for

fD[ c°sc'uT sin_' +

irradiance

for

case

cos'is

case

(iii)

(ii)

(Fig.

9b)

sm -" h/D-y _---k---)i]

(Fig.

9c)

is:

( D XEyT)

is:

11

(Y-

YT)dy

(46)

of

the

I In winter, on

this

side

irradiance

side

A

is given P_,_

therefore,

is

"is

1

by eq.

given is

by

= P_A + P_

summer,

early

and

the

the beam

thus

the

afternoon

Pb

For

other

hours

:

the

P_

hours

or eq.

eqs.

(,45), by

L[D

unshaded,

(39).

Side

(46)

or

B may

(47)

('_s- 180°).

therefore

beam

be partially

where

The

the

the

shaded

collector

total

irradiance

beam

and

azimuth

irradiance

PbA

the

beam

is

180 ° ,

on

sincx

and

late

PbA is given

PbB is given

irradiance

in by

by eq.

the

eqs.

(38)

afternoon, (45),

with

the

side

(46)

or

azimuth

A

is

(47).

partially

Side

"ts-180

°,

B is unshaded,

i.e.,

(49)

+ H cosc_ cos(ws-180°)]

on

both

sides

of

shaded,

the

tent

collector

for

early

morning

and

is:

+ PbB

of

the

is:

morning

irradiance

irradiance

beam

(38)

always

(48)

in

beam

PbB = Gb

remains

in winter

Pb

therefore,

2)

replaced

catenary-tent-collector

In

(Fig.

(47)

the

(50)

day,

side

B will

be partially

12

shaded

and

eq.

(48)

applies.

late

Diffuse

lrradiance

The

diffuse

irradiance

dPa

= Gab

G_h

-

dA

on

an

incremental

area

dA

is given

by

(51)

FaA-s

where

F,_a-s

For

diffuse

irradiance

-

factor

of

dA

dS,

the

view. factor

view

sufficiently

Fda-s

Using

small

= 0.5[1

(34),

(36)

and

dPa

= 0.5

Gab

L(_/1

for

P,_ = 0.5

Since skies);

using

the eq.

in the

Gab

diffuse

view

P,t

= Gab

eqs.

(17)

with

respect

to

for

the

dA

L(N

factor

for

Nn

is

(51)

2

[0,D]

+ D)

also of

eq.

+ [fc'(y)]

interval

applies

is given

by:

+ 1)dy

we get

= 0.5

Gab

independent

to the

other

a catenary

we obtain:

the

of side

collectors

and

(53)

E

1 + k

diffuse

L[k

of

orientation

the

can

irradiance

sinhiD/k)

the

on

one

side

of

+ D]

of

the

tent:

(53)

the

collector

(for

isotropic

tent.

now

be calculated

by defining:

(54)

A Fa-s

= 0.5

sky

(52)

(52)

irradiance

(53)

The

Fa-s

y

a horizontal:Surface

+ cos/3(y)]

eqs.

Integrating

on

one

obtains:

° ]

(55)

sinh(D/k)

13

Albedo The

albedo

Pat

= al Oh A FA-c

where

al

-

the

G^

-

global

FA-a

The

view

FA-g,

are

obtaining

be determined

albedo

view

factor

of

related

F,4-G

eq.

on

of

area

expression:

A

a horizontal

area

A with

with

respect

surface

respect

to

to ground

the

sky,

FA-s,

and

with

respect

to

the

ground,

[5]:

+ FA-a

with

the

factor

factor

by

by using

(56)

irradiance

-

F,i-s

Using

can

= 1

(57)

(55):

[

= 0.5

o ]

1

eq.

(14),

(17),

Pat

= 0.5

al

the

albedo

(58)

k sinh(D/k)

(56)

and

(58)

we

obtain

the

albedo

irradiance

on

one

side

of

the

tent

as:

Since also

to

other

G^ L[k

sinh(D/k)

irradiance

side

of

the

-

is independent

on

Mars

surface

example [6]

refers at

the

the

orientation

of

the

collector

eq.

(59)

applies

tent.

4.

The

(59)

D]

to

a

locations

Example

south-north of

Viking

facing Lander

14

catenary-tent-collector VLI

(Latitude

deployed -

22.3°N,

on

the

Longitude

-

47.9°W) and in an autumnday Ls = 200° apply the

and,

therefore,

catenary

collector

in k = 2.53

and

fc(Y)

Figure

side

are:

D

catenary

shows

different

the

hours the

shadow

self-shading

effect

is

O=22.3ON,

the

be even

The

the

the

beam

that

the

typical

Table

more

= 2m

for

side

0=32 ° for

is quite

=

during

the the

1.5m.

year day.

Since

eq. The

fe(0)

(3)

does

dimensions

= H,

this

not of

results

(60)

B of

shadow

pronounced

L

of

1]

on

The

time

by:

-

shapes

this

shaded

and

is given

day.

effect

partially

0.395y)

shapes

the

1

based

are

catenary

shown

higher

small

the

calculations

during

in

the

collector

are

latitudes.

Mars,

-

on insolation a place

Insolation

on

side

q (kWhr/mZ-day)

on

A

radiation

on side

BEAM

Q(kWhr-day)

catenary-tent-collector

insolations

insolation diffuse

1.

the

of

Table

albedo

for

H

For

based

Fig. It

for on

10(b).

hours.

In

= 22.3°N

section As

is interesting

noon

O

to

2.

For

expected,

the

note

for

summer,

that

the

shadow

less.

in

and

-

shadow

shadow

insolation

is shown

-- 3m,

be

equation

of

comparison,

will

B will

= 2.53[cosh(1.186

10(a)

for

the

only

[6].

data

sides

A

is

lower

B

comprises where

the

and by

atmosphere

O=22.3°N,

VLI

[6].

59.6% of

than the

collector,

Total

on

global

consists

mainly

O=22.3°N,

COMPONENT

Ls=200

The

kWhr-day

DIFFUSE

COMPONENT B

at B in

46.6%

a catenary-tent

side

for

table

° and

shows

and

kWhr/m2-day.

side

A.

It

is

insolation. of

the

a1=0.22

beam, As

interesting

This

dust

albedo

diffuse expected, to

characteristic

Ls=200

particles.

°,

ai=0.22

ALBEDO

GLOBAL

COMPONENT

INSOLATION

12.65

5.11

17.76

16.27

0.89

34.92

2.246

0.907

1.577

1.445

0.079

3.101

15

note is

5.

The blanket

article

that

analyses

falls

freely

characteristics

(portability

space

Because

planets.

account

in

the

the

collector

and

albedo

The

numerical

solar

is

the on

of

shape,

radiation

radiation

and

were

example

sides

of

there

a

a

central

that

are

is

a

catenary-tent-collector support). desirable

Therefore,

used" in

determination

the

calculated

to

on solar

6.

data

Collector

Ash

shaded

Aush

unshaded

D

length

of

FA-c

view

factor

of

area

A with

respect

to ground

F,4-s

view

factor

of

area

A with

respect

to sky

FdA-s

view

factor

of

incremental

Gb

direct

Gdh

diffuse

Oh

global

irradiance

H

height

of

k

catenary

a collector,

area

beam

on

global Mars.

[m 2]

of

of

the

the

the

NOMENCLATURE

A

area

of

determine

radiation

shape

a collector,

collector,

collector,

[m]

[m]

irradiance,

irradiance

[m 2]

area

with

respect

to sky

[W/rn 2]

on a-horizontal on

dA

a horizontal

surface, surface,

[m]

constant

16

[W/m2] [W/m _]

:

kind

solar

effect

the

[1]

This for

self-shading

calculation.

also

is based

area,

of

simplicity)

its

calculated

performance

both

and

CONCLUSIONS

beam

area

plants

must

be

of

the

radiation.

radiation

flexible

collectors

power

that and

of

(a

on

has

on

outer

taken

into

shadow The

the

on

diffuse

collector.

L

collector

Ls

areocentric

Pat

albedo

irradiance,

Pb

direct

beam

irradiance

on

an

direct

beam

irradiance

on

a partially

direct

beam

irradiance

on

an

on

a partially

PbA,

PbB

width,

longitude

respectively, psh

lash. bA' --bB

direct

[m] of

sun

(for

Mars)

[W] unshaded

side

of

a collector,

shaded

side

side

A and

shaded

side

unshaded

of

beam

irradiance

a collector. B of

A and

[W ]

Pz,Ps

x and

0

insolation,

[kWhr-day]

q

insolation,

[kWhr/m2-day]

IX

sun

y components

of a shadow

q'c

collector

"/s

sun

azimuth

"s/st

sun

azimuth

6

solar

£

characteristic

9

angle

length.

[m]

altitude

relative

[W] [W]

a collector,

[W]

respectively,

_J

the

azimuth

at

sunrise

declination

angle angle

between

solar

shaded

local

latitude

solar

hour

of ray

the

collector

and

the

normal

to the

area

angle

17

collector

B of

a collector,

7.

[1]

A.J.

Colozza,

Array',

[2]

[3]

[4]

F.B.

NASA Beer,

New

J.

J.

Appelbaum,

Systems"

Solar

J.

J.

H.C.

Bany,

A.F.

187119,

1967;

pp.

"Shadow

Appelbaum Cell,

and

23,

"The

Effect

Sarofim,

20,

for

pp.

Radiative

a Self-Deploying

PV Tent

1991. Engineers:

of

Static

Adjacent

497-507, of

pp.

of

and

Dynamics

258-273.

Effect

Vol.

Vol.

Analysis June

Mechanlcal

YOrk,

Energy,

Solar

Hottel,

Report

Johnston,

McGraw-Hill,

Bany,

Optimization,

Contract

E.R.

Collectors"

[s]

"Design,

REFERENCES

Solar

Collectors

in Large

Scale

1979.

Shadowing

201-228,

on

the

Design

of

Field

of

Solar

1987.

Transfer,

McGraw

Hill,

New

York,

1967,

25-39.

[6]

J.

Appelbaum

353-363,

D.J.

Flood,

"Solar

Radiation

1990.

18

on

Mars',

Solar

Energy,

Vol.

45,

pp.

pp.

APPENDIX Calculation Calculation Point

of

Point

E and

may

be

T for

case

h

of points

E and

T

(i)

E

Point Fig.

A1

z

The

E

(a).

The

= H(1

of

YE, i.e.,

line

the

equation

projection

of

the

solar

ray

NP

on

the

z-y

plane,

is:

(A.1)

the

projection

the

solution

yE_

line,

eq.

(A1)

with

the

catenary

equation

fc(Y)

xE is on

line

yields

the

of:

YE = k + _ I +

]_

coordinate

from

y/Ps)

k[Dcosh_

The

projection

-

intersection

coordinate

calculated

OP,

(A.2)

i.e.,

P.r

xE : V_ yE

and

finally

Point

the of

the

(A.3)

coordinate

zE is calculated

from

eq.

(A1).

T

arc line

Point

T

N

Nn,

AB.

fc'(y)

is determined Fig.

From

=-sinh(

A1 eq.

by (b),

i.e.,

a sun at

ray this

that point,

its the

projection derivative

on

the of

z-y

fc(Y)

plane

is tangent

is equal

to the

to slope

(1)

(A.4)

D k ----_y)

19

Since

A--B and

Yr

The

N P_ are

= D -

coordinate

Calculation

A blanket,

sun

(D

the

slope

of

line

AB is -H/Py,

thus

obtaining:

(A.S)

= 0 and

ytl and

ray

Figs.

lines,

k sinh-l(H/P_)

xr

o[

parallel

xg

that

5 and

- Yv)/xE

the

[or

coordinate

case

touches

is calculated

from

the

fc(Y)

equation.

(ii)

the

foot

It

follows

A.l.(a).

zr

of

the

tent

at

point

E passes

through

point

U on

that

(A.6)

= P_/Px

and

zv D - Yv

from

which

_

H P_

by

(A.7)

solving:

H DH + Fvv Yv =

fc(Y)

is given

Calculation

o/

Referring

(y_

fc(Yv) D - Yv

Yu is determined

fc(Ye)

where

_

-

y¢

in eq.

for

to

yv)/L

case

Figs.

(A.8)

(1).

The

coordinate

x_: is then

calculated

from

eq.

(A.6).

(iii)

6 and

A.I

(b),

we may

write

(A.9)

= Ps/P-

and

zv - z.____g _ fc(Yv) Y_ - Yv YE -

fc(YE) Yv

_ -

H

(A.10)

2O

the

From eqs. (A.9) and (A.10), yt is determtnedby: fc(YE- L P--_)- fc(YE)

where

fc(Y)

is given

in eq.

(A.11)

L H

(1).

APPENDIX

Using

the

calculation. procedure fc(Y)

The for

parabolic error

approximation

is

small

is the

calculating parabolic

the

and

points

the

fc(Y)

results

for

approximation

=.(1

for

the

(eq.

B

are

shadows

(2)),

greatly very

are

simplifies

similar the

to

same

as

the in

the

mathematical

catenary

case.

The

but

now

Appendix

A

i.e.,

rc y :y(, -2H

(B.1)

Case (i)

The

coordinates

xE: = D • _--_

of

2

points

-

XT = 0

With arc

these is given

g(y)

results,

the

by eq.

=

(16)

arcs and

N

E and

T are:

, y_

= D 2

-

, yr

= D(1

-

Nn, for

N the

, zt- = H

_)

E_ and

_/I + [fc'(y)]2 dy =

(B.2)

1

, Z r = H(_py)2

T

parabolic

_--_y-

Es

can

now

approximation

I +

21

_

(O - y)

(B.3)

be determined. we have:

dy

The

length

of

an

the

solution

of

g(y)

The

Case

arc

= _-_

lengths

integral

sinh -1

_

(y

-

D)

(B.4)

+

therefore:

N N_ = g(D)

- g(0)

(B.5)

N

Ev

-- g(YE)

-

g(0)

(B.6)

T Ey

= g(y_:)

-

g(Yr)

(B.7)

(ii)

using

eq.

eq.

(A.8)

(A.6)

with

XT

=

expression

T Nn

0

.

eq.

we obtain

,

The

is:

are

Solving

and

the

(2)

the

YE = D

= g(Yr)

T

Nn

-

is given

Yv (Fig.

coordinates

,

YT -- D (2--P-_) 1 D

for

for

of

5)

zr

in:

x_:. The coordinates

of

points

E and

T are:

(B.9)

zE = 0

,

results

= H (2__y) 2

(B.10)

by:

g(D)

(B.11)

22

whereg(y) is given in eq. (4).

Case

(iii)

Solving

eq.

(A.11)

with

eq.

(2)

for

YE (Fig.

6)

results

in:

(B.12)

The

coordinates

XE = L

XT=

of

,

points

YE = D

E and

1 -

T are:

_

+ L 2Pz

0 ZE

=

z,:

0

23

.(D

Table1. -

Insolationon a catenary-tentcollector,_=22.3ON,Ls=200 °,

BEAM side Q(kWhr-day)

A

12.65

COMPONENT side

B

Total

5.11

a1=0.22

DIFFUSE

ALBEDO

GLOBAL

COMPONENT

COMPONENT

INSOLATION

17.76

16.27

0.89

34.92 |

q (kWhr/m2-day)

2.246

O. 907

Figure

1. 577

1 .--Artist's

1.445

view of a catenary-tent-collector.

24

0.079

3.101

H M a n

Figure 2.QCatenary-tent-collector:

definitions and orientation.

N

S .,,_J_

N

Py / =Y /

/ X

D Figure 3.--Catenary

...........

=7"

collector:

25

Mn

basic shadow parameters.

N

M

w

N i

/

5

_-N

I t

t Ex

_1"

'[Ey

d

_N_9y

p

\E

]

I,

Mn

Nn

D (o) Figure 4.--Catena_ shadow shape.

(b) collector-case(i)

(a) shadow parameters,(b)

N

stretched-out

w

!i

N

s-,jJ_ ,U

,T

l,

I

_/

\ _ \

/,"

/. ,,.

11 Nn

/p x

E

D

(o) Figure 5.--Catena_ shadow shape.

(b) collector-case(ii)(a)shadow

26

parameters,(b)

stretched-out

L

U

W

.

/I

S._

/

_'U

'S

_N / /

_ .y×E

p,z_/

M.

N.

x (a) Figure 6.--Catenary shadow shape.

'_

(bl collector - case (iii) (a) shadow parameters,

(b) stretched-out

Z

Z

W °

.-

W

N

E

I

NnT_

H

.....

/

Y

ex

D

"t" X

(a) Figure 7,--Catenary

collector-

(b) shadow shape for Px > L, Py > D (a) Px/Py

27

>_ L/D,

(b) Px/Py

< LID.

'

Z

E

N

No//_s W RMAL

Y

Figure 8.--Catenary

collector - solar radiation calculation on an unshaded side.

28

_

Mn

x

(o.) j IZ

L

N i

T

(b)

_

H /

; ,,__i'-_ 2

--d_ ° (C)

Figure 9.mCatenary collector - beam radiation on a partially shaded side (a)(D, (b) case (iD,(c)case (iii).

29

7h

lOh

8h

12h

14h

16h

17h

if

0.023

0.20

_":069

0.024

O. 023

O. 20

14h

16h

0.69

(a) 8h

7h

(_=

0

IOh

0.57

I70

0.33

12h

0.33

0.34

17h

0.57

0.70

(b) Figure lO.iCatenary collector - shadow shapes on side day for (a) (l' = 22.3._N, (b) 4J = 32°N.

B at different

hours of the

Y/ HI ;

,,_-___ky/,

I Px

/

H

/

,,

I

X

/

_

'_\

,/,/

x

(a)

(b) Figure A.l--Catenary

collector

- calculation

3O

of points (a) E, (b) T.

REPORT

DOCUMENTATION

PAGE

Form

Approved

OMB

No, 0704-0188

Pul_iicreport|ngburden for this collectionof informatiOn is estimatedto average 1 hour per response, includingthe time for reviewinginstructions,searching existingdata sources. gathering and maintainingthe data needed, and completing and reviewingthe collectionof information. Send comments regardingthis burden estimate or any other aspect of this collectionof information,includingsuggestiOnsfor reducingthis burden, to Washington HeadquartersServices. Directoratefor information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington,VA 22202-4302, and to the Office of Management and Budget, PaperworkReduction Project (0704-0188), Washington, DC 20503. 1. AGENCY

USE ONLY

(Leave blank)

2. REPORT

DATE

3. REPORT

TYPE

July 1992 4. TITLE

AND

Technical

COVERED

Memorandum

5. FUNDING

SUBTITLE

Solar Radiation

AND DATES

on a Catenary

NUMBERS

Collector

WU-506-41-I 6.

M.

Crutchik

and

7. PERFORMING

National Lewis

J. Appelbaum

ORGANIZATION

Aeronautics Research

Cleveland,

NAME(S)

and

Space

8." 'PERFORMING ORGANIZATION REPORT NUMBER

AND ADDRESS(ES)

Administration

E-7156

Center

Ohio

44135-3191

9. SPONSORING/MONITORING

AGENCY

NAMES(S)

AND ADDRESS(ES)

10.

National Aeronautics and Space Administration Washington, D.C. 20546-0001

11.

SUPPLEMENTARY M.

12a.

Tei-Aviv Center,

University,

Cleveland,

DISTRIBUTION/AVAILABILITY

Unclassified

NASA TM-I05751

Ohio

Faculty

of

Engineering,

TeI-Aviv

69978,

Israel.

J. Appelbaum,

NASA

Lewis

44135.

12b.

STATEMENT

DISTRIBUTION

CODE

- Unlimited

Subject Category

ABSTRACT

SPONSORING/MON_ORING AGENCY REPORTNUMBER

NOTES

Crutchik,

Research

13,

1

AUTHOR(S)

(Maximum

33

200 words)

A tent-shaped structure with a flexible photovoltaic blanket acting as a catenary collector is presentcd. The shadow cast by one side of the collector on the other side producing a self shading effect is analyzed. The direct beam, the diffuse and the albedo radiation on the collector are determined. An example is given for the isolation on the collector operating on Mars surface for the location of Viking Lander 1 (VLI).

14.' SUBJECT

Catenary

TERMS

collector;

Flexible

photovoltaic

blanket;

Solar radiation;

Self shading;

Shadow

15.

NUMBER

OF PAGES

16.

PRICE'CODE

20.

LIMITATION

32

shape and size 17. SECURITY CLASSIRCATION OF REPORT

Unclassified NSN 7540-01-280-5500

A03 18. SECURITY CLASSIFICATION OF THIS PAGE

Unclassified

19. SECURITY CLASSIFICATION OF ABSTRACT

OF ABSTRACT

Unclassified Standard Form 298 (Rev, 2-89) Prescribed by ANSI Std, Z39-1 El 298-102