Circular flat plate heat exchanger for solar concentrator

Circular Flat Plate Heat Exchanger for Solar Concentrator M. H. Cobble Associate Professor of Mechanical Engineering, University of Delaware, Newark, Delaware, USA

N analysis is given of a heat exchanger of the flatplate type having circular boundaries t h a t could be used just out of the focus of a concentrator of the paraboloid of revolution type to heat liquids. The heat exchanger is shown in Fig. l, and it consists of two parallel plates, through which an incompressible ideal fluid is flowing radially while being heated. Two types of heating are investigated. In both cases the upper plate is insulated. In the first case, the lower plate temperature is assumed constant, and in the second case the heat rate per unit area to the lower plate is assumed constant. The temperature field for both cases is found, and an expression for Nusselt N u m b e r as a function of Graetz N u m b e r is developed. The energy equation for an inviscid fluid in cylindrical coordinates can be written as

B 07'

0~7'

r Or

OZ2

CASE

Equation (3) is subject to the following boundary conditions: 1. T(r, 0) = 7',~

0T 2. OZ (r, Zo) = 0 3. T(rl, Z) = 7'+ The solution to Eq. (3) using the given boundary conditions is 7'(r, Z) = T., +

4(7'~- 7'w) 7r

£e-L~d • ,=0 I O'T

lOT

++T7 (1)

= k L °~ + ; o,-: + ~ 0%-,+ o z , ] Assumptions:

(3)

1

F(~,++1)-72F,2-~,23

aT 07' Ve 07' aT' 1 pc~ ~ [ + V. -Or + - r ~ + Vz o z j -

r+

Eq. (2) becomes

L~J (2n + 1)

(4) . (2n+ 1)~-Z sin 2Z0

Equating the heat rate picked up b y the fluid, based on an average exit, temperature To,,,: at radius r2, to iFlow in at %

'\ !

I. c~ = c, (true for incompressible liquids) 2. T = T ( r , Z ) 3. V~ 4. V o

=

O~T

A

~h

r

27rpZo r

Vz = 0

O~T

5. - ~ << - ~ k 6,--

~

o~

pep

~-'r - T.

Using the assumptions in Eq. (1) gives

1) OT

I(A

r

Or

lr- 0r(r,o): - ~

02T OZ2

Allowing A ----I=B

(2)

Dz FIG. 1.--Radial flow between plates•

k

t h a t convected through the lower surface using a film coefficient h, one obtains

OL

Manuscript received February 27, 1962. 164

#wv[Te -- Ta~,,] = h~r[r2~ -- rl~l

2

Tw

(5)

Solar Energy

,oo¢ [ Illlll

K=

I

!/I

/

I

Ill-

I

i~]!i;

~

~ i!!

I

I

I I l~t

I,IIII

v

111111 I :I~[ ~ I

IO

IO0

I ]illllll

I00

~z:, [,.c, _,]

I000

~= rhCp 2~Z.k

Fro. 2.

FIG. 3.

The average temperature at radius r2 is defined as 1

f Z o T(ro , Z) dZ

(6)

The solution to Eq. (3) using the new boundary conditions is ~z0 . ~z [ z T(r,Z) = Te +2-k~oB (r2 - rfl) + - ~ - - F ~ - k ~ ° -- 1

Using Eq. (4) in Eq. (6) yields F<2~+l)--]2F~2~-~2- ] Ta,,% = Tw -~- 8(Te -

- L ~ /

T.,) ~

7r2

L T /

2hZo -

8z+

F

1_]]

(9)

8Zo~ ~(M^

(13)

~Zo [Ta~,~- %]k

(14)

a Graetz No. as (2n+l)

¢:

2\

27rZ0k

(11)

{

8 1 ++o=0

i5

(15)

and a dimensionless modulus ,I, as 2z02 xII = (r2 2 -

A plot of N u vs q~ using Eq. (11) for various values of •I, is shown in Fig. 2. CASE

Nu

(lO)

r~2)

Eq. (5) can be rearranged to yield l

2 - rl 2)

Tar,,= % + ~ ( r 2 By defining a Nusselt No. as

and a dimensionless modulus as

rl ~)

OT 1. " ~ (r, O) = --~

(16)

Eq. (14) can be rearranged to yield

Nu = ~(~ - 1)

II

Equation (3) is subject to the following boundary conditions

(17)

Equation (17) is plotted in Fig. 3 for xI, = 1. For values of ,I, other than 1, the result is evident. EXAMPLES

OT 2. ~-~ (r, Zo) = o

Vol, 6 , No. 4; 196~

(6)

Using Eq. (12) in Eq. (6) yields

+ - ,~2(M - M) L2~kZo -

3. T(rl, Z) ffi %

7'.... 2 = Z0 1 f0 z° T(r2, Z) dZ

(8)

7r2]C

a Graetz No. as

Nu =

(12)

n~rZ cos Z~

The average temperature at radius r2 is defined as

By defining a Nusselt No. as N~t

2~Zo~-~ e-(~o-o ) [ ~ ] -~k ~1 n~

(7)

(2n + 1)2

n =o

/

n~" 2 r2_r12

..

.

. •

I a . Determine the average exit temperature of a fluid which enters a heat exchanger of type Case I. The following properties are assumed known:

165

FIG. 4.

°1

I0

I

8z.

I000

I00

rm ,

~=~'(¢-¢) LZ~kz. j By repeating the previously outlined steps for problem l a one would arrive at Eq. (20). Tw in Eq. (20) must now be determined. The local heat rate per unit area (q,) is defined as

8Z°2 I rhc~ -- 1 ] = 100 1. • -- ~2(r~2 -- rl 2) 2~rkZ0 2. '~

8Z0~ ~2(r22 -

rx2)

0.1

3. T~ = 100 deg F

qt = --k OT (r, 0) = 0z(r, 0) OZ

4. Tw = 200 deg F

From Fig. (2) the Nusselt Number when • = 100, 0 . 1 , is

Using Eq. (4) in Eq. (21) one obtains

~I, =

Nu

To)

~ 2~ ~T ~

~(r,O)--

N u ~ 38

Zu

Equation (5) can be rearranged to give

R - (p + ,],

1

q"~

(18)

~(r22

hO

ftl

16kZo(T~ - T,)B ~

(19)

qav =

~2(r2~.

r12)

J L

~B 2

(22)

~(r, O)2~rrdr

(23)

(1 - R)T~ + 2RTw T~,.~ = 1+ R

•[1-e-F(~"+"7'F~-"~77L 2Zo J L z -I/_ By defining a special Nusselt Number as

T~,~, ~ 155 deg F

Ib. Determine the average exit temperature of a fluid which enters a heat exchanger of type Case I, where the average heat rate/unit area is known, but the wall temperature is not known. The following properties are assumed known: n2 ) L 2 ~ r k Z ° - 1

8Zo2 ~*(r~2 -

r~2)

(24)

(20)

In this case R --~ 0.38, and

~(r2 ~ -

1

2.,n=0 (2n + 1) 2

Equation (18) can be rearranged to yield

~ 100

0.1

3. T, = 100 deg F = 2000 B t u p e r h r p e r sq f t

5. Zo = ~ ft 6. k = 0.36 B t u p e r h r p e r ft p e r d e g F 166

2Zo

n=O

Using Eq. (22) in Eq. (23) yields Nil

4. ~

e L

The average heat rate per unit area (~av) is defined as

If one lets

2. ~I,

_I-(2.+1)~721-,~-,,~7

~

Te -- "1',~~

,I, + ,I~ = T~ + St'av r 2 - 27',,~

1.'}

(21)

~.~ Zo

Nu'

(25)

T~)k

2(T~-

and a Graetz Number as 8Zo~

[mcp

= 7r2(r22- rl 2)

1]

(9)

2rZok

Equation (24) can be rearranged to N u ' = (I,

n=o (2n + 1) 2

1 - e

(26)

A graph of Eq. (26) is shown in Fig. 4. Using Fig. 4 the special Nusselt Number, when (I, -- 100, is N u ' ~-- 8.8

Equation (25) can be rearranged to give Solar Energy

qav "1'~ = T~ + 2 k N u ' ~-- 126 deg F

Using this value of T~ in Eq. (20) yields T~

"~ 114 deg F

I f . Determine the average exit t e m p e r a t u r e of ~ fluid which enters a heat exchanger of t y p e Case I I . T h e following properties are a s s u m e d : 1. ~

2. 'V

?hCp 27rZ0 k

100

2Zo~ (r~~ -- rl 2)

0.1

3. ~/ = 2500 B t u p e r h r p e r sq ft 4. T~ = 100 deg F 5. k = 0.36 B t u per h r p e r ft p e r deg F

6. Zo = j ~ f t

Using Eq. (17), when cI, = 100, ~I, = 0.1, the Nusselt N u m b e r is N u = 9.9

E q u a t i o n (14) can be rearranged to give ~Zo " =

~+

Nuk

Thus 7',~T2 --~ 158 deg F

NOMENCLATURE A = constant, sq ft per hr B = constant, dimensionless cp = specific heat at c o n s t a n t lb,n per deg F

Vol. 6, No. ,~, 1962

pressure,

Btu

per

c~ = specific heat at constant volume, B t u per lbm per deg F h = film coefficient, B t u per hr per sq ft per deg F k = conductivity, B t u per hr per ft per deg F ~h ---- mass flow rate, lbm per hr n -- integer, dimensionless Nu - - Nusselt No., dimensionless Nu' = special Nusselt No., dimensionless q = heat rate per unit area, B t u per hr per sq ft q~, = average heat rate per unit area, B t u per hr per sq ft qt = local heat rate per unit area, B t u per hr per sq ft r = coordinate, ft rl = fluid entrance radius, ft r~ = fluid exit radius, ft R = modulus, dimensionless T = temperature, deg F 7'~v, = average t e m p e r a t u r e at radius r, deg F Te = fluid entrance temperature, deg F Tw = wall temperature, deg F t = time, hr Vr = radial velocity, ft per hr Vz = velocity in Z-direction, ft per hr Vo = velocity in O-direction, ft per hr Z = coordinate, ft Z0 = distance between plates, ft -- diffusivity, sq ft per hr 0 = coordinate, radians p = density, lbm per cu ft = Graetz No., dimensionless = modulus, dimensionless

167