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Energy and Buildings, 18

(1992) 35-43 35

A n a l y s is o f t h e a c c u r a c y a n d s e n s i t i v i ty o f e ig h t m o d e l s t o p r e d i c t t h e

p e r f o r m a n c e o f e a r th - to - a ir h e a t e x c h a n g e r s

A. Tzaferis and D. Liparakis

T.E.I. Pireus , Thiv on 250 P. Ralli, GR-122 24 Aega leo (Greece)

M. Santamouris

Labora tqry of Meteorology, Univer sity of Athens, Ippokr atous 33, GR-105 80 Athens (Greece)

A. Argiriou*

Protechna Ltd, Themistokleous 87, GR-105 83 Athens (Greece)

(Received October 24, 1991)

A b s t r a c t

Eight different algorithms predicting the performance of earth-to-air heat exchangers are evaluated. The

sensitivity of the methods to the inlet air temperature, air velocity, pipe length, pipe radius and the pipe

depth is examined. Two different types of experiments have been designed and performed, and exper imental

results have been compared with the set of the predicted values. The relative accuracy of each algorithm

is estimated and reported for both cases.

1 . I n t r o d u c t i o n

Passive and hybrid cooling systems and techniq ues

can contr ibute decisively to the cooling of buildings

reducing theref ore the peak electricity load of util-

ities and prese rving th e e nvi ronm ent [1 [.

The use of the gr ound as a sink for dissipation

of the excess heat o f buildings is a well-researched

area [2-4]. Hybrid ground cooling systems and

especially earth-to-air heat exchang ers have gained

an increasing acceptance during the last years

[5-15]. Here air is circulated through buried pipes

using fans. Due to the te mpera ture difference be-

tween the ground and the air, the air temperature

decreases. The main advantages of the system are

its simplicity, high cooling potential, low capital,

operational and maintenance cost and environmental

protection.

The use o f such a system requ ires a quite difficult

dimensioning process, involving optimization of its

length, diameter, dep th and airflow. For this purpo se

various algorithms have been proposed. Each al-

gorithm is deduced from and is validated for a

specific set of parameters, i.e., a certain depth, a

flow rate, a length, a diameter, etc.

*Author to whom correspondence should be addressed.

However there is an important dependence and

variability of the system's per forma nce as a functi on

of its thermal and geometr ical characteristics. Con-

sequently such algorithms are not automatically

accurate for every set of parameters and therefore

a more detailed validation is required.

The aim of the pr esent pa per is to contribute to

the validation of the algorithms by examining their

accurac y for various data sets, and also to investigate

their sensitivity upon the more important param-

eters. For this purpose experiments have been un-

dertaken and two sets of experimenta l data have

been obtained and used for validation. The overall

work offers a compl ete analysis of the subject and

contributes towards a more accurate design of earth-

to-air heat exchangers.

2 . T h e a l g o r i t h m s

The eight examined algorithms can be classified

in two groups:

Thos e algorithms which first calculate the con-

vective heat transfer from the circulating air to the

pipe and then the conductive heat transfer from

the pipe to the ground and inside the ground mass

[16]. The necessary input data are:

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36

-- the geometrical characteristics of the system;

-- the therma l characteristi cs of the ground;

--

the thermal characteristics of the pipe;

-- the undisturbed ground temperature during the

operation of the system.

Those algorithms which calculate only the con-

vective heat transfer from the circulating air to the

pipe [17-21]. In this case the necessary input data

are:

-- the geometr ical characterist ics of the system;

-- the therm al characte ristics of the pipe;

-- the te mperature of the pipe surface.

A . T h e S c h i l l e r a l g o r i t h m [ 16 ]

In this algorithm, the two hea t transfer processes

determining the behaviour of the buried pipe as a

heat exchanger are treated separately. The first

process is the convective heat transfer between the

air circulating in the pipe and the internal surface

of the pipe. The second process is the heat transfer

by conduction from the internal surface of the pipe

to the ground. The ground is thus divided theo-

retically into coaxial cylindrical elements.

In order to better simulate the heat transfer pro-

cess in the ground, the thermal resistance of the

ground was considered to be time-dependent, in

order to take into account the variation of its thermal

properties with temperature.

The air temperature at the outlet of an element

of the pipe can be obtaine d by the following relation:

x

T~, x+1 =Tt, x - - - (1)

~hcr

with x, x+ 1 the axial positions of the inlet and

the outlet of the element. The air temperature at

the outlet of the pipe is obtained by applying this

relation for all the elements of the pipe in order

to cover its entire length.

B . S a n t a m o u r i s a l g o r i t h m [ 17 ]

According to this algorithm, the air temperature

at the outlet of a buried pipe is given by the following

expression:

Tf, o = (Tf, ~ - Tp,~o) exp( -Sa )[1 + ( B i S a ) 5

Fo

[ o

_j exp( - B i F o ) I i ( 2 [ B i S a Fo] '5)

0

F o S d F o ] + Tpwo (2)

The dimensionless parameters B i , S a a n d F o

depend on the the rmophysical properties of the pipe

and of the air and on the characteristics of the

airflow.

C . T h e R o d ~ i g u e z e t at . a l g o r i t h m [ 18 ]

This model allows the prediction of the air tem-

perature circulating in the pipe at any position x

from the inlet. The relation is:

T~, x = Tpwo+ (T~,, - Tpwo) exp pfc---~--2R~ (3)

This relation is based on the assumption that the

temperature of the external wall of the pipe is

maintained constant during the process. The air

temperature at the outlet of the pipe is calculated

by eqn. (3) with x - - L , L being the length of the

pipe.

D . T h e L e v i t e t a l. a l g o r i t h m [ lP ]

The air temperature in the pipe at a distance

x 1 from the inlet is calculated by solving the

steady-state heat balance equation between the air

and the ground at a distance D from the pipe,

through a finite differences scheme. The resulting

relation is:

T ~ + 1 _ T ~ r h c ~ - U A d x U A d x

- + - - T s x , D )

4 )

rhcf ~hc~

with A = 2 ~ r D , U=the overall heat transfer coeffi-

cient,

Tg ( x ,

D) = the ground temperature at length

x at a distance D from the pipe and dx the length

of the elem entary part of the pipe.

The air temperature at the outlet of the pipe is

calculat ed by applying eqn. (4) consecutivel y for

all the elements of the pipe.

E . T h e S e r o a e t a l. a l g o r i t h m [ 20 ]

The buried pipe is divided in elementary parts.

The air temperature at the outlet of each element

is given by the relation:

[(1 - U/2 )Tf , l + UTpwo]

Tf, o = (5)

(1 U / 2 )

where

U = (2 rrR dXUw)/~Cf

Uw is the overall heat t ransf er coefficient betwee n

the air and the external wall of the pipe. The air

temperature at the outlet of the pipe is calculated

by applying eqn. (5) consecutively for all the ele-

ments of the pipe.

F . T h e E l m e r a n d S c h i l l e r a l g o r i t h m [ 21 ]

Again the pipe is divided in elementary cylinders

of length dx. The air temperature at the outlet of

each element is given by the relation:

T f , x + 1 = V f ,

x - - Qx /T~c f

6 )

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w h e r e t h e h e a t f l o w r a t e p e r u n i t l e n g t h o f t h e

e l e m e n t a t a d i s t a n c e x f r o m t h e i n le t o f t h e p i p e

e q u a l s :

Q= = U w T f , x -

Tpwo)2~rR~dx 7)

T h e a i r t e m p e r a t u r e a t th e o u t l e t o f t h e p i p e i s

c a l c u l a t e d b y a p p l y i n g e q n . 6 ) c o n s e c u t i v e l y f o r

a ll e l e m e n t s o f t h e p i p e .

G . T h e S o d h a e t al . a l g o r i t h m [ 22 ]

I n t h i s c a s e t h e a i r t e m p e r a t u r e a t t h e o u t l e t o f

t h e p i p e i s g iv e n b y t h e r e l a ti o n :

T , , o = TpwoCa n ) 8 )

w h e r e

a n ) = C a - 1 ) e x p -

f i n ) + 1 + F ( n , f l )

9 )

~j = Tz. , /Tpwo 10 )

2 ~rRiL U

[~= ( 11)

~ c f

n = ( x / L )

1 2 )

W h e n t h e a i r t e m p e r a t u r e a t th e o u t l e t o f t h e

p i p e h a s t o b e c a l c u l a t e d ,

x = L

c o n s e q u e n t l y n - - 1.

K n o w i n g / 3 , t h e p a r a m e t e r

F ( n , f l ) ,

e q n . 8 ) t h e a i r

t e m p e r a t u r e a t th e o u t l e t o f t h e p i p e i s o b t a i n e d .

H . T h e C h e n e t a l . a l g o r i t h m [ 2 3]

T h i s a l g o r it h m s o l v e s t h e t h r e e - d i m e n s i o n a l t ra n -

s i en t h e a t b a l a n c e e q u a t i o n b e t w e e n t h e a i r a n d t h e

e x t e r na l w a l l o f t h e p i p e. T h e N e u m a n n - t y p e b o u n d -

a r y c o n d i t i o n a t th e e x t e r n a l s u r f a c e o f th e p i p e i s

a l s o t a k e n i n t o a c c o u n t . B y a p p l y i n g a f in i t e d i f-

f e r e n c e s s c h e m e , t h e f o l l o w i n g e q u a t i o n g i v i n g t h e

a ir t e m p e r a t u r e a t t h e o u t l e t o f an e l e m e n t o f t h e

p i p e i s o b t a i n e d :

T ~ + I _ 2 U ~

2 U w )

pfCfRo---~

Tp w o d ~+ T ~f 1 P fCf Ro

V d x

(13)

T h e a i r t e m p e r a t u r e a t t h e o u t l e t o f t h e p i p e i s

c a l c u la t e d b y a p p l y i n g eq n . 1 3 ) c o n s e c u t i v e l y f o r

a l l t h e e l e m e n t s o f t h e

p i p e .

3 . S e n s i t i v i t y a n a l y s i s

T h e a l g o r it h m s p r e s e n t e d a b o v e h a v e b e e n in -

t r o d u c e d i n to c o m p u t e r p r o g r a m s i n o r d e r t o si m -

u l a t e t h e b e h a v i o u r o f t h e e a r t h - t o - a i r h e a t e x -

c h a n g e r s a n d t o i n v e s t ig a t e t h e i m p a c t o f e a c h i n p u t

p a r a m e t e r , b y m e a n s o f a s e n s i ti v i t y a n a l y s i s. T h e

s e n s i t i v i t y o f a l l t h e m o d e l s , r e g a r d i n g t h e i n l e t a i r

37

t e m p e r a t u r e , t h e g r o u n d t e m p e r a t u r e , t h e a ir v e -

l o c it y , th e l e n g t h o f t h e p i p e , a n d t h e i n te r n a l p i p e

r a d i u s a n d th e d e p t h , h a s b e e n i n v e s ti g a t e d . S o m e

o f t h e a l g o r i th m s r e q u i r e t h e p i p e w a l l t e m p e r a t u r e

a s a n i n p u t. I n t h a t c a s e , t h e p i p e w a l l t e m p e r a t u r e

h a s b e e n c a l c u l a t e d f i r st b y t h e S c h i l le r a l g o r i t h m

a n d t h e n u s e d a s a n i n p u t f o r t h e s t u d i e d a l g o r it h m .

T h e m a i n p a r a m e t e r d e t e r m i n i n g t h e o u t l e t a i r

t e m p e r a t u r e i s t h e i n le t a i r t e m p e r a t u r e . T h e o u t l e t

a i r t e m p e r a t u r e h a s b e e n c a l c u l a t e d u s i n g a ll t h e

a l g o r it h m s , w i t h t h e i n l e t a i r t e m p e r a t u r e r a n g i n g

f r o m 2 5 C t o 7 5 C . F o r t h e o t h e r i n p u t d a t a ,

t y p i c a l v a l u e s u s e d d u r i n g t h e o p e r a t i o n o f t h e

e a r t h - t o - a i r h e a t e x c h a n g e r s h a v e b e e n s e l e c t e d .

T h e s e v a l u e s a r e T g = 2 5 C , V = 5 m s - I , R i =

1 2 5 x 1 0 - 3 m , L = 3 0 m , z = l . 5 m . T h e r e s u l t s

o b t a i n e d a r e i l l u s t r a t e d i n F i g . 1 . F r o m t h e r e s u l t s

i t

c a n b e s e e n t h a t a l l m o d e l s p r e s e n t a s i m i l a r

l in e a r b e h a v i o u r r e g a r d i n g t h e i n l e t t e m p e r a t u r e

v a r i a ti o n s . Al l m o d e l s e x c e p t t h e S c h i l le r a n d S o d h a

e t a l .

a l g o ri th m s p r e d i c t a l m o s t t h e s a m e v a l u e o f

t h e o u t l e t a i r t e m p e r a t u r e . T h e S c h i ll e r a n d S o d h a

p r e d i c t i o n s a r e a l s o l i n e a r f u n c t i o n s o f t h e i n l e t a ir

t e m p e r a t u r e b u t t h e S c h i ll e r a l g o r i th m A ) g i v e s

s i gn i fi c a n tl y h i g h e r t e m p e r a t u r e s w h i l e t h e S o d h a

e t a l . a l g o r i th m G ) g iv e s l o w e r v a l u e s .

F i g u r e 2 i l l u st r a te s t h e o u t l e t a i r t e m p e r a t u r e v s .

g r o u n d t e m p e r a t u r e a t t h e d e p t h o f th e e x c h a n g e r ,

u s i n g t h e s a m e i n p u t d a t a a s b e f o r e , b u t a s s u m i n g

a c o n s t a n t i n l e t a i r t e m p e r a t u r e o f 3 5 C. A s it c a n

b e s e e n , t h e o u t l e t t e m p e r a t u r e i s p r e d i c t e d a s a

l i ne a r f u n c t i o n o f t h e g r o u n d t e m p e r a t u r e . A l l m o d e l s

p r e d i c t a lm o s t th e s a m e o u t l et t e m p e r a t u r e e x c e p t

t h e S c h i l l e r m o d e l t h a t p r e d i c t s v a l u e s h i g h e r o f

a b o u t 1 .5 C f o r l o w g r o u n d t e m p e r a t u r e s a n d t h e

S o d h a e t a l . m o d e l w h i c h p r e d i c t s l o w e r v a lu e s o f

a b o u t 0 . 5 C . T h e d i f f e r e n c e s b e t w e e n t h e m e a n

p r e d i c t i o n s d e c r e a s e f o r t h e S c h i ll e r m o d e l a n d

i n c r e a s e f o r th e S o d h a

e t a l .

m o d e l a s t e m p e r a t u r e

i n c r e a s e s .

T p o [ C )

6 0

5 0

4 0

3 0

J

i i i I

2 5 3 0 3 5 4 0

2 0 ~ ~ ~ ~ ~ ~ ~

2 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0

Tpi O)

~ A - - + - 8 - -X '~ -- - 8 - O - -X - -E - e - - F ~ G ~ H

Fig. 1. Ou tlet air tempera ture va riation vs. inlet air

temperature.

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3 9

Tpo (O)

32

30

28

2 6 ~ t ~ ~ t ~ ~ ' ~ t ~ t ~ ~

~2 14 16 18 2 22 24 26 2.8 3 3.2 3.4 3.6 3.8 4

z r n )

~A -4--.B +C -8- D --x-E ~F - -G --~--H

~ 8 . 6 . O u ~ e~ ~ t e ~ e ~ a ~ e ~ ~ i o ~ ~ s. d e ~ t h .

depths; the outlet temperature drop is about 5 C

for the first 3 m, but bet ween 3 and 4 m the decrease

is only about 0.5 C. It must be noted here that

the difference between the Schiller model mean

predictions i ncreases with depth, while the difference

between the Sodha model and the mean predictions

remains practically constant for all depths.

4 . C o m p a r i s o n w i t h e x p e r i m e n t a l d a t a

In order to evaluate and validate the predicted

performance of the algorithms discussed above, the

performance of a PVC horizontal pipe, buried at a

depth of 1.1 m, was measured. The pipe was 14.8

m long, with a diameter of 0.15 m. The thickness

of the pipe wall was 0.01 m. Ambient air, heated

by a 2 kW electric heater, circulated in the pipe

by means of a fan. Two experiments have been

performed.

During the first experiment the air circulated

cont inuo usly for 9 days, in June 1982, with a velocity

of 4.5 m s- 1. The inlet and ou tlet air temperature s,

ground surface temperature and ground temperature

at various depths were measured [24].

The aim of the second experiment was to test

the behaviour of the system by applying an inter-

mittent air circulation. The air circulated from 08 :00

until 20: 00, with a velocit y of 10.5 m s -I. Mea-

surement s were perfo rmed for 15 days, during June

and July, 1983. The temperature was measured at

the same experimental points as for the first ex-

perime nt [25]. T he thermal diffusivity of the ground

was determin ed experimenta lly and used as input

data for the models [26].

Figure 7 illustrates the measured inlet air tem-

perature and the ground temperature at the depth

of the pipe, 1.1 m, for the first experiment. Figure

8 illustrates the outlet air temperature obtained by

this expe rimen t (solid line) and the simulation results

obtained by the various algorithms as a function

of time. The curve representing the experimental

data presents oscillations on a daily period. This

is due to the fact that the inlet air temperature is

not thermostatically controlled, but ambient air was

heated by applying a constant heating power. In

this graph it can be observed also that the mean

daily outlet temperature increases with time. This

is due to progressive heating of the ground layers

close to the pipe (see Fig. 7), which reduces the

heat transfer rate between the air circulating in the

pipe and the ground.

As can be seen from Fig. 8, while the models

can follow the long-term dynamic behaviour of the

outlet air temperatu re, their predict ions of the max-

imum and minimum values on a daily basis is much

less accurate. In order to quantify the accuracy of

the various models regarding the experimental re-

stilts, the root-mean-square error between experi-

mental data and calculated values was calculated

(Fig. 9) f or all days of the ex perim ent. It is observed

that except for the Schiller and the Sodha algorithms

which present the highest r.m.s, error, all the other

algorithms present an r.m.s, erro r of the same order,

i.e., of about 3.4%.

The comparison between measured data of a

typical day during the second experiment and the

corresponding simulation results is illustrated in

Fig. 10 and the corresp onding inlet air and ground

temperature are given in Fig. 11. From this Figure

it is observed that the results from all models are

very close to the curve, except the Schiller and the

Sodha models. The Schiller algorithm gives outlet

air temperature values which are about 3-4 C

higher than the error band, while the Sodha algorithm

gives values which are located in the middle of the

error band. Comparisons between measurements

and calculated values were made for all the days

of the exper iment and results obtained were similar

to the ones already discussed.

The r.m.s, error between measurements and cal-

culated values is illustrated in Fig. 12. The error

was calculated for all the measurements obtained

during the 15 days of the exp eriment, for both the

upper and lower limits of the error bands. These

results are illustrated in Fig. 12. The Schiller al-

gorithm presents the highest r.m.s, error. The error

of all other model s regarding u pper and lower limit

of measurement s is respectively of about 2.98% and

4.7%, except the Sodha algorithm which gives an

error of the same order in both cases, i.e., 3.53%

and 3.25% respectively.

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4

T C)

6 5 -

6 ~

5 5

5 8 -

4 5

4 0

3 5

2 5

~ - e I I I I I I I I ] I ] I I I I I I I I I I I I I I

~ ~ ~ ~ ~ ~ ~ ~ ~_ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

T i ~ e ( 6 )

~ . 7. Ir~t ni~ and ~ o un d t~mp~r~t~r~s ~t 1.1 m d~pth~ d~r i~ th~ cor~tant ~ circulation xp ~d m~ nL

i -- In t e r Ai~ I

e m p e r a t u r e

--

C ~ o ~ d

T em p era t u re

4 6 -

4

I

',/,'i

I

. i ~ - ~ ~ t 1

i i ~ ~ i ~ ; ' ~ ~

~ r , ~ ~ ~ : ~ ~ , ~ ; ~ E

/ ~ . ~ - : ~ , ~ ~ : ~

~

:

',

~.~,,.~ ~ ~, , . ~ :~; ~ ~

~ ; ,~ ~ . :~~ .~ ~ ~ ~ : ~ , ~ ~ : , ~ : ~ ~

] ~ , ~ ~ , : : : . , : ~ ~

~ _ ~ ~ ~ - - ~. , ~. ~ . ~ ~ : i ~

; ~ : : [ , ~ ~ ,, ~ , ~ . f ~ , . ~ , ~

~ ~x ~ ,' ' ~ / ~ : ~ Z : , , :

~ ~ , ~ h

' ,~ ~ , , ~ , '

~ ~ t ~ , " , ~ ~ .' , ~ ; " )

. ( r , ~ ] ; ;

~ ; , ' . ' ~ , " . ; '

~ / ~ ~ , , , . ~ : , . ~ , ,

: . : , M . ; , , . ~ , , .,

i~,..,..,,,., ~ ~: '~

3 4 [ ' ..~,,~,;

~ 2 I I I I I i I I I

9 1 2 2 e 3 3 3 6

4 2 5 5

6 6

7 9

8 2 9 e ~ e le 1 1 1 2

3 6 ~ ?

T i ~ e

( ~ )

~ g . 8 . M e ~ e d ~ d p r e ~ c te d o u de t t e m p e r a t e d ~ ~ e c o ~ t ~ c ~ c ul a t io n e x p e ~ e n t .

~ ; t . .

' .~ ~ I ~ : , - ~ ' , : ~

' , . ~ ~ .

, : ~

~ ? , . ~

, ~ D~ ~ i ~ X

..... , L < ~

~ j ~ ~ , ~+ ~

,. ~7 ~ .7, .. .. ~, ,.

.~, . ; .

~ ~

13 13 15 15 16 17 17 IB 19 20 21

8 I 4 2 5 8 6 9 2

H e a s ~ e ~ e n t s

- - S c ~ i | | e ~

..... Santa~ou~is

R o d ~ i ~ u ez

-- Levit

S e ~ o a

E l m e ~

S o d Ha

C He~

8/11/2019 0378-7788-2892-2990049-m

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(~)

6

0

Schiller Santarnour. Rodr iguez Levit Soroa Ermer Sodha Chon

lgorithm

Fig. 9. The r.m.s, errors bet ween measured and predict ed values

for the constant air circulation experiment.

41

air temperature. After that limit, the changes do

not influence the performance of the system. The

models have been used to simulate the performance

of a heat exchanger, operating under two different

experimental conditions. The r.m.s, error in each

case has been calculated. The mean value of this

error was found to be of about 3.5% for all cases

except for the models, which gave higher r.m.s.

errors.

N o m e n c l a t u r e

Cf

5 . C o n c l u s i o n s h

In this paper, eight algorith ms for the predic tion 11

of the performance of cylindrical earth-to-air heat

exchan gers have been examine d. A sensitivity anal- L

ysis has shown that the main parameter s determining rh

the air temperature at the outlet of the exchanger

Qx

are the inlet air temperature and the ground tem-

perature which is a function of the depth at which Ro

the exch ange r is placed. The sensitivity analysis t

has shown also that only change s up to a certain T~

limit of length and of the di amete r of the pipe and T~, x

of the air velocity in the pipe can modify the outlet

specific heat at constant pressure of the

air (J kg -~ C -~)

convective heat transfer coefficient in-

side the pipe (W m -~ C -~)

modified Bessel funct ion of the first kind

of order 0

length of the pipe (m)

air ma ss flow rate (kg m -~)

heat flux per unit length of the pipe at

lengt h x (W m-1)

outer radius of the pipe (m)

operating time (s)

ground temperature (C)

air temperatur e in the pipe at a distance

x from the inlet of the pipe (C)

53-

52-

51

58

49-

48-

4 7-

46

T

(C) 45

44

43

42

41

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i

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T i m e < h

Fig. 10. Measured and predicted outlet temperatures during the intermittent air circulation experiment.

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F i g. 1 1 . I n l e t a n d g r o u n d t e m p e r a t u r e s a t 1 .1 m d e p t h ) d u r i n g t h e i n t e r m i t t e n t a ir c i r c u l a t i o n e x p e r i m e n t .

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~)

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Schi llerSantamour Rodr iguez Lev i l Seroa Elmer S odha Chen

A l g o r i t ~ q m

m U p p e r C a s e m L o w e r C a s e

F ig . 1 2 . T h e r .m . s , e r r o r s b e t w e e n m e a s u r e d a n d p r e d i c t e d

v a l u e s f o r t h e i n t e r m i t t e n t a i r c i r c u l a t i o n e x p e r i m e n t .

Tpwo

Tp~,o

V

z

t e m p e r a t u r e o f t h e e x t e r n a l w a l l o f t h e

p i p e a t l e n g t h x C )

m e a n t e m p e r a t u r e o f th e e x t e r n a l w a l l

o f t h e p ip e C )

a ir v e l o c i t y i n t h e p i p e m s - 1 )

d e p t h o f t h e p i p e m )

Greek symbol

p~

d e n s i t y o f t h e a i r k g

m - ~ )

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