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Copyright Warning & Restrictions
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reproductions of copyrighted material.
Under certain conditions specified in the law, libraries andarchives are authorized to furnish a photocopy or other
reproduction. One of these specified conditions is that thephotocopy or reproduction is not to be “used for any
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may be liable for copyright infringement,
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would involve violation of copyright law.
Please Note: The author retains the copyright while theNew Jersey Institute of Technology reserves the right to
distribute this thesis or dissertation
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HEAT TRANSFER BY NATURAL CONVECTION
IN HORIZONTAL FLUID LAYERS
A STATISTICAL CORRELATION OF EXPERIMENTAL DATA
BY JOHN L. O' TooLE
A THESIS SUBMITTED TO THE FACULTY OF THE DEPARTMENT OFCHEMICAL ENGINEERING OF NEWARK COLLEGE OF ENGINEERING
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THEDEGREE OF MASTER OF SCIENCE IN CHEMICAL ENGINEERING.
APPROVAL OF THESIS
FOR DEPARTMENT OF CHEMICAL ENGINEERINGNEWARK COLLEGE OF ENGINEERING
FACULTY COMMITTEE
APPROVED
NEWARK, N. J.JUNE, 1959
Lib raryCollege of Engineering
THE RATE OF HEAT TRANSFER DUE TO NATURAL CONVECTION UN CONFINED,
HORIZONTAL 0 FLUID LAYERS CAN BE EXPRESSED BY EQUATIONS INVOLVING THE
DIMENSIONLESS PARAMETERS NUSSELT NO,, RAYLEIGH NO., AND PRANDTL NO.,
WHERE RAYLEIGH NO. EQUALS THE PRODUCT OF GRASHOF AND PRANDTL NOS.
STATISTICAL CORRELATION BY STEPWIS REGRESSION OF
EXPERIMENTAL DATA OBTAINED FROM FOUR DIFFERENT INVESTIGATORS, ON BOTH
GASES AND LIQUIDS, YIELDS THREE SEPARATE EQUATIONS CORRESPONDING TO
THREE DISTINCT REGIMES OF CONVECTION. THE TOTAL RANGE SPANNED BY THE
DATA IS FROM THE ONSET OF CONVECTION TO RAYLEIGH = 10 9 .
THE EQUATIONS ARE
RAREGIME RANGE EQUATION
.816INITIAL 1600..3000 Nu = 0.00238(RA)
LAMINAR 3000..105 Nu = 0.221(RA).256
TURBULENT 105°4109 Nu = 0.0891 (RA)·316 (PR)·0853
THE STANDARD ERROR OF ESTIMATE, WHICH IS THE PROBABLE ERROR IN THE
PREDICTION OF NU FROM A REGRESSION EQUATION, IS OBTAINED FOR EACH
REGIME. A VERY HIGH DEGREE OF ACCURACY WAS OBTAINED FOR ALL. CORRELATIONS&
STD, ERROROF ESTI MATE
TABLE OF CONTENTS
SUBJECT PAGE
INTRODUCTION 1
THEORY 3
EXPERIMENTAL DATA 8
THE CORRELATION
TABLE 1 *I SOURCES OF THE DATA 8
TABLE if As FEATURES OF THE APPARATUS USED BY
SOURCES OF THE DATA 17
TABLE III · EXPERIMENTAL DATA I8.26
TABLE IV P.4 SUMMARY OF REGRESSION ANALYSIS 13
TABLE V · RESULTS OF COMPUTER ANALYSIS 27
TABLE VI COEFFICIENTS OF CORRELATION AND
DETERMINATION 28
FIG. 1 PRANDTL NO. OF AIR VS TEMPERATURE BETWEEN 28 & 29
FIG, 2 THERMAL CONDUCTIVITY OF AIR VS TEMPERATURE'
FIG. 3— NUSSELT No, VS RAYLEIGH No.: EXPERIMENTAL
DATA
FIG. 4 NUSSELT NO. VS RAYLEIGH No.: REGRESSION
EQUATIONS
Flo. 5 - NUSSELT NO./(PRANDTL NO.)• 0853 VS
RAYLEIGH No.: TURBULENT REGIME
APPENDIX 1 DIMENSIONAL ANALYSES 29
APPENDIX Il · DERIVATION OF THE COEFFICIENT OF
THERMAL EXPANSION OF AIR 31
TABLE OF CONTENTS (CO NT
SUBJECT PAGE
APPEND IX III - CORRECT ION OF THE DATA OF
MULL AND RE (HER 33
APPEND IX 1V - STEPWISE LINEAR MULTIPLE REGRESS ION 35
TABLE OF SYMBOLS 39
BIBLIOGRAPHY AND REFERENCES 40
HEAT TRANSFER BY NATURAL CONVECTI~N.
iN HORIZONTAL FLUID LAYERS
I NTRODUCT 1 ON
HEAT TRANSFER BY NATURAL OONVECTION IN CONFINED, HOR.ZONTAL~
FLUrO LAYERS WAS FURST INVESTIGATED ~ORMALLY BY BENARD (1)* IN
1901 AND ANALYZED BY LORD RAYLEIGH [N 1916 (2). SUNCE THAT TiME A
CONSIDERABLE AMOUNT OF WORK HAS APPEARED IN THE LITERATURE OONSISTING
LARGELY OF MATHEMATICAL ANALYSES DEFINING THE OONDITIONS UNCER WHIOH
A HEATED LAYER FIRST BEOOMES UNSTABLE AND BEGINS TO OIROULATE. PRIOR
TO 1950 ONLY A SMALL AMOUNT OF EXPERIMENTAL WORK HAD BEEN DONE. SfNOE
THEN, HOWEVER, THREE PAPERS HAVE BEEN PUBLISHED (12, 20$ 21) REPORTING
EXPERIMENTAL HEAT TRANSFER DATA OVER A WIDE RANGE OF THE PHENOMENON
AFTER THE ONSET OF CONVEOTION. WORKING INDEPENDENTLY, WITH DIFFERENT
FLUIDS, AND IN DIFFERENT RANGES, THE AUTHORS DIFFERED IN THEIR
OORRELATIONS AND NO COMPREHENSIVE RELATIONSHIp WAS DEVELOPED.
THIS THESIS PRESENTS AN ENGINEERING CORRELATION BY STEPWISE
MUll'lPLE REGRESSION OF THE AVAiLABLE DATA. THIS METHOD IS A POWERFUL
STATISTICAL TOOL THAT PROVIDES A CORRELATING EQUATION FOR A MUlTIPllOITY
OF PARAMETERS AND, AT THE SAME TIME, YIELDS AN ESTIMATE OF THE AOCURACY
OF THE CORRELATiON AND THE SIGNIFIOANOE OF THE INDIVIDUAL PARAMETERS.
To THE AUTHORtS KNOWLEDGE, THIS THESIS IS THE FIRST USE OF STEPwiSE
MULTIPLE REGRESSION iN THE ANALYSIS AND CORRELATION OF HEAT TRANSFER
DATA.
THREE OF THE FOUR SETS OF DATA (12, 20, 21) WERE OBTAINED
DIRECTLY FRCM THE AUTHORS ~N THE FORM OF THE D_MENSIONLESS
PARAMETERS NUBSELT. RAYLEIGH, GRASHOF, AND PRANGTL NUMBERS. (THESE
WILL BE ABBREVIATED Na, RA, GR, PR HEREAFTER.) THE OTHER SET (5)
WAS CALOULATED FROM RAW DATA TABULATED IN THE PAPER$
* NUMBERS IN PARENTHESES REFER TO REFERENOES LIBTED IN THE BIBLIOGRAPHY.
THEORY
THE SYSTEM
WHEN A TEMPERATURE DIFFERENOE Is APPLIED AOROSS A HORIZONTAL
LAYER OF FLUID THAT as INF(NfTE iN EXTENT AND OONFINED BETWEEN
OONDUOTING HORIZONTAL SURFAOES, A TEMPERATURE GRADIENT WILL BE ESTABLISHED
ACROSS THE LAYER AND HEAT wtLL fLOW BY OONDUCTION (AND, IN THE OASE
1, THE HOT SURFAOE is BENEATH THE LAYER, THE FLUID WiLL TEND TO (f/l4t,V,.l(
01 ROULATE ~ to A DENS ITY GRAD lENT AOROSS THE LAYER B NDUOED BY THE
TEMPERATURE GRADIENT. THIS OIROULATION M CONVEOTION RESUL T5 I N AN
- I -
INOREASE IN THE RATE OF HEAT TRANSFER OVER THAT DUE TO OONDUCTION ALONE.
TH8S THESIS OONSIDERS THE MAGNITUDE OF THE HEAT TRANSFER DUE TO
OONVEOTION.
ONSET OF CONVEOTiON
LORD RAYLEiGH (2) FOUND BY ANALYSIS THAT NO OONVEOTION WOULD TAKE
PLAOE UNTIL A ORaTIOAL VALUE OF THE RAYLEIGH NUMBER (RA = GR X PRJ WAS
EXOEEDEDo HIS ANALYSIS HAS BEEN OONrlRMED AND EXTENDED BY OTHERS
(3,4,9, II) AND THE ORITIOAL RA OF 1707.8 OALOULATED BY JEFFRIES (3)
HAS BEE N V E R I FIE D EX PER t MEN TAL L V ( 6 _ 7, to, 1 2, I 4 II 20) I>
HEAT TRANSFER - DrMENSIONAL ANALystS
THE RAYLEIGH - JEFFRlES ANALVSIS DOES NOT YIELD A FUNOTiONAL
RELATIONSHIP BETWEEN THE RATE OF HEAT TRANSFER AND THE SYSTEM VARIABLES
AND CONSTANTS AFTER CONVECTION HAS BEGUN. THERE HAVE aEEN VERY FEW
ATTEMPTS iN THE L~TERATURE TO DERIVE SUOH A RELAT(ONSH~P CHIEF~Y
BECAUSE THE DiFFERENTIAL EQUATIONS DERiVED FROM THE BASIC FLOW
EQUATION DO NOT YIELD TO liNEARIZATION (16~ 19). HOWEVER~ BATOHELER (16)
HAS DERIVED THE FOLLOWING EXPRESSIONS FOR THE OASE OF CONVEOTION IN
VERTioAL LAYERS:
lAMiNAR RE(UON
TURBULENT REGION
...
... Nu = F OF (RA)o25
Nu = F OF (RA)e33
THE RAYLEIGH - JEFFRIES ORITERION SUGGESTS THAT THE RATE OF HEAT
TRANSFER IS SOME FUNOTtON OF RA AND SUOH A FUNOTION OAN BE DERIVED BY
DIMENSIONAL ANALYSIS.
IF WE POSTULATE THAT AT STEADY"'STATE CONDITIONS THE OVERALL RATE
OF HEAT TRANSFER PER UNIT AREA H IS A FUNCTION OF
AREA BY CONDUOTION AND
~ = THERMAL CONO_CTTVITY OF THE FLUID
T = TEMPERATURE DIFFERENOE
L = THIOKNESS OF THE FLUID LAYER
e = DENSITY OF THE FLUID
~ = DYNAMIC VISCOSITY OF THE FLUID
o = SPECIFIC HEAT AT CONSTANT PRESSURE OF THE FLUID
, - COEFF,OlENT OF THERMAL EXPANSION OF THE FLU.'
Q = GRAV2TATIONAl CONSTANT
(ALL CONSTANTS ARE EVALUATED AT THE MEAN
TEMPERATURE OF THE LAYER)
THEN THE FOLLOWING ExPRESSION OAN BE DERIVED BY DIMENSIONAL ANALYSIS:(23)
(SEE ApPEND 1')( 1)
F OF (~ G Y L 3 e 2 ) N ( fl'2. )
H Is A FORM OF NUSSELT NUMBER AND fS THE RATfm OF OVERALL KY/L
RATE OF HEAT TRANSFER DUE TO CONVECTiON AND CONDUCTION TO THE RATE OF
HEAT TRANSFER DUE TO OONDUCTION ALONE. WHEN IT is EQUAL TO ONE, NO
HEAT Is BEfNG TRANSFERRED BY OONVECTIONo
~ G T 1.3 f 2 I S THE GRASHOF NUMBER.. t1 CAN BE REGlARDED AS THE 2
R A 1 BOO FeU 0 Y ANT FOR C E S " (~G T)>> TOT HE V I S 00 U s r 0 ROE 15 (r / (' ), 0 F
THE FLUID (22).
THE SISNIF"lCANCE OF THE CRn'()CAL RA DiSCOVERED BY LORD RAYLE1GH ,
THEN, IS THAT THE BUOYANT FORCE DUE TO THE TEMPERATURE GRADIENT IS
BALANCED BY THE vISCOUS FORCES IN THE LAYER UNTIL THE BUOYANT rORot
ATTAtNS A VALUE REPRESENTEP BY THE CRITICAL RA .. WHEN THIS HAPPENS, ('1 ~\A/
FLOW COMMENPES"
....!!2 IS THE PRANDTL NUMBER, WHICH IS THE RATIO OF MOMENTUM D·U-rUSI:vt1Y K
TO THERMAL D1FFUSrVITY IN THE FLUlo;t.E .. , THE RATIO OF THE FLutD
PROPERTY GOVERNING THE TRANSFER OF MOMENTUM BY VISCOUS FOROES TO THE
FLUID PROPERTY GOVERNING TRANSFER OF HEAT BY TEMPERATURE mIFFERENOE.
THE DIMENSIONAL ANALYSIS ASSUMES THAT HEAT TRANSFER THROUGH THE
LAYER is NOT AFFEOTED BY THE PARTIOULAR GEOMETRY OF THE FLOW PATTERNS
OR THE ABSOLUTE MAGNITUDE OF THE TEMPERATURE DIFFERENCE. 1M ADDITION.
IT ASSUMES THAT THE PHYS!OAL OONSTANTS CAN BE EVALUATED AT THE MEAN
TEMPERATURE OF THE LAYERo
MODES OF CONVEOTION . AFTER THE ONSET OF CONVECTION~ DIFFERENT MODES OF FLOW HAVE BEEN
OBSERVED (6, 7, 8, 12~ 20, 21). INITlALLY THE FLOW TAKES PLACE fN
D1s0REET OELLS THAT HAVE CROSS SECTIONS OF REGULAR POLYG~NS. As RA
INCREASES THE MODE CHANGES AS SHOWN BY CHANGES iN THE SHAPE AND WIDTH
OF THE CELLS. FINAllY, THE ORDERLY CELLS DISAPPEAR ALTOGETHER AS THE
OONVECTION CHANGES FROM LAMINAR TO TURBULENT FLOW.
A NUMBER OF ANALYSES HAVE BEEN PUBLISHED PREDIOTING 1HI RA AT
WHICH MODE TRANSITIONS OOOUR (3, 9, 14), AND SOME EXPERIMENTERS HAVE
TRIED TO CORRELATE THEIR HEAT TRANSFER DATA WITH THE CHANGES IN MODE
THAT THEY OBSERVED. HOWEVER, MALKUS (15, 17) REPORTED THAT THE HEAT
TRANSFER FUNOTION OHANGES ONLY WITH THE CHANGE FROM LAMINAR TO TURBULENT
FLOW AND DOES NOT OORRESPOND TO CHANGES !N MODE WITHIN THESE REGlUES.
FORM OF CORRELATION
THIS THESiS PRESENTS AN EUPER1CAL CORRELATION OF Nu VI THREE
D~UENSIONLESS PARAMETERS ~ RA, PRJ AND A/o 2 , WHERE A/02 IS THE RATte
OF AREA -TO THICKNESS SQUARED OF THE APPARATUS USED IN OBTAINING THE DATA.
THESE PARAMETERS WERE SELECTED FOR THE FOLLOWINB REASONS:
I. RA IS PREDIOTED BY DIMENSiONAL ANALYSIS.
2~ PR IS REPORTED (20, 21) TO BE AN ADDITIONAL PARAMETER
WITH RA IN THE TU~BULENT REBION.
3. A/o2 I S DES I GNEO TO TEST THE EXPER IMENTAL DATA F'OR
AGREEMENT WITH THE ASSUMPTION OF INFrNfTE EXTENT OF
THE lAYERa
E:XPEFUMENTAL DATA
SOURCES
THE DATA WERE OBTAINED FROM THE FOUR SOURCES LISTED iN TABLE I~
IN THREE CASES THE DATA WERE OBTAINED DIRECTLY FROM THE AUTHORS. IN
THE OTHER CASE {MULL AND REIHER) RA. DATA WERE TAKEN FROM A TABULATION
IN THE PAPER, AND THE PARAMETERS CAl.OULATED FROM IT.
TABLE
SOURCE AND RANGE OF EXPERIMENTAL DATA
DATE
SOURCE PUBLISHED
DEGRAAF" AND VAN DER HELD(12) 1952
GLOBE AND" DROPK 1 N (21 ) 1958 WATER, ILICONE o U.S 9 MERCURY
MULL ANI!! REI HErd5) 1930 AIR
SCHM I DT AND S H. VESTON (20) 1958 WATER,. GLYCOL~ HEPTANE, SILIOONE 01'1..8
ApPARATUS AND PROCEDURE
RA
RANGE
103 .... 105
r05 .109
10 3 ... 0106
103
... '°5
No. OF"
DATA
26
56
17
106
ALL OF" THE SOURCES OF DATA USED SIMILAR APPARATUS, DIFFERING
PRINCIPALLY iN DETAILS OF CONSTRUOTION AND METHODS OF MEASURING THE
NET RATE OF HEAT TRANSFER. A COMPARISON OF THE iMPORTANT FEATURES OF
* THE1R APPARATUS is MADE iN TABLE II. IN GENERAL g THE FLUID LAYER WAS
CONFINED BETWEEN TWO PARALLEL METAL PLATES AND AN ENCLOSING WALL O~
INSULATING MATERIAL. SPACiNG WAS FiXED EITHER BY SMALL SPACERS BETWEEN
PLATES OR BY THE ENCLOSURE. THE LOWER PLATE WAS EQUiPPED WITH ELEOTRICAL
HEATERS g WHilE THE UPPER PLATE WAS'DESIGNED TO AOOOMMODATE A FLOW OF
COOLING WATER OR AIR. TEMPERATURE DI~FERENCE WAS MEASURED BY
*' Ps. 17
THERMOCOUPLES EMBEDDED BN THE PLATES. HEAT TRANSFER RATE WAS
CALCULATED EiTHER BY MEASURING THE POWER OONSUMPTION OF THE HEATERS
OR THE HEAT ACQUIRED BY THE OOOLING MEDIUM .. THE PROPERTIES OF THE
FLaRe WERE TAKEN AT THE ARiTHMETIC MEAN TEMPERATURE OF THE HOT AND
COLD PLATES. IN THE EXPERIMENTS ON AIR, THE HEAT TRANSFER DUE TO
RADIATION WAS DETERMINED OVER THE TEMPERATURE RANGE OF THE
EXPERiMENTS BY INVERTING THE NOT AND COLO SURFAOES; I.E .. , BY HEATING
FROM THE TOP""AND OORREOTING FOR CONDUCTION BY CALCULATION. EAOH OF
THE EXPERIMENTAL RUNS WAS OORREOTED BY SUBTRACTING THE APPROPRIATE
RADIATION VALUE FROM THE MEASURED TOTAL HEAT TRANSFER RATE.
THE EXPERIMENTAL PROOEDURE WAS THE SAME FOR All THE SOURCES.
A TEMPERATURE DIFFERENCE WAS APPLIED TO THE FLUID LAYER, THE SYSTEM
WAS ALLOWED TO REACH EQUILIBRIUM AND MEASUREMENTS OF T., T2~ AND Q (OR
ELECTRICAL POWER CONSUMPTION) WERE MADEo
CERTAIN DISCREPANCIES UN EXPERIMENTAL TECHNIQUE ARE WORTHY OF
NOTE BECAUSE THEY BEAR ON THE ACCURACY OF THE DATA.
BOTH GLOBE AND DROPKIN AND SCHMIDT AND SILVESTON, WHO EXPERIMENTED
WITH LIQUIDS, DETERMINED THE HEAT TRANSFER BY MEASURING ELECTRICAL
POWER CONSUMPTION. THE APPARATUS OF THE LATTER WAS ELABORATELY
INSULATED TO GUARD AGAINST HEAT LOSS, AND OORRECTIONS, WHERE NECESSARY,
WER E APPL I ED.. How EVER J GLO BE AND ORO PK I N JI NE I TH ER I NSULA'~'t:O THE I'R
EQUIPMENT NOR CORRECTED FOR lOSS. AT lEAST THE AUTHORS ARE SILENT
ON THIS POINT AND THEiR DIAGRAMS AND DISCUSSIONS REVEAL NO SUCH
PRECAUTlONS. IF SUOH IS THE CASE, THEIR Nu SHOULD TEND TO BE HIGHER
AT PARTIOULAR RA. THE APPARATUS OF DE GRAAF AND VAN DER HELD WAS NOT
INSULATED EITHER, BUT THEY MEASURED THE HEAT TRANSFER RATE BY
MEASURING THE HEAT ACQUIRED BY THE COOLING WATER. IN ADDITION, THEY
CORRECTED FOR EXTRANEOUS LOSSES AT THE SAME T8ME THAT THEY OORRECTED
FOR RADIATION.
... 10 ~
DE GRAAF AND VAN OER HELD HAVE POINTED OUT (12, 13) THAT
MULL AND REIHER CAL~ULATED THE THERMAL COEFFICIENT OF EXPANSION
FOR AIR INCORRECTLY, USING 1/273 [NSTEAD OF 1fT CI\VG.) THEREF"ORE,
THE DATA OF AND REIHER WAS RECALCULATED BEFORE artNG USED IN
THiS CORRELATION" ApPENDIX I! CONTAINS THE DERiVATION OF A ~ I/TAv90
ApPEND1X I I I DESCRIBES THE CALCULATION OF MULL AND REIHER'S NUSSELT
AND GRASHOF NUMBERS USlNG THE CONSTANTS OF DE GRAAF AND VAN DER HELD
THE EXPERIMENTAL DATA O~N AIR (5,12) WAS PLOTTED BY THE ORIGINAL
AUTHORS iN THE fORM OF Nu VS GR, SINOE PR Is NEARLY A CONSTANT FOA
AIR OVER THE TEMPERATURE RANGE OF THEIR EXPERiMENTS. RA FOR THESE
DATA WAS CALCULATED FROM THE OR.lG:SNAL GR (RECALOULATED 1N THE CASE Of'
MUll" po.NO RElHER) USING THE PR THAT ARE PLOTTED VERSUS TEMPERATURE IN
FiG" I .. ALTHOU\3H TAE VARIATION WITH TEMPERATURE IS NOT LARG!::" THIS
WAS DONE TO REDUCE ERROR IN THE CORRELATION~
CORRELATION
THE DATA WAS ANALYZED STAT1ST~DALLY BY THE METHOD OF STEPWiSE,
LtNEAR, MULTIPLE, REGRESSiON (25). LINEAR MULTIPLE REGRESSION
CONSlSTS OF F[NDING BY THE METHOD OF LEAST SQUARES THE FUNCTION
OF THE FORM
THAT BEST FiTS THE DATA. THEREFORE, THE HYPOTHETIOAL FUNCTiON
Nu = A (RA)B 1 (PR)B 2 (A/0 2 )B 3 WAS EXPRESSED IN LOGARITHMIC FORM~
THE LEAST SQUARES BEST F1T lS THAT EQUATION THAT RESULTS IN tHE:
MlNtMUM STANDARD DEViATION OF THE DISTRIBUTION OF EXPERiMENTAL DATA
ABOUT THE REGRESSrON. K MORE DETAILED EXPLANATION OF THE METHOD
AND THE INTERPRETATiON OF THE STATISTiCS IS SEVEN [N ApPENDiX IV.
IN STEPWISE MULTfPLE REGRESSION, THE VAR'ABLES ARE ADDED TO THE
REGRESSION ANALYSiS ONE AT A TIME IN THE ORDER OF THEIR CONTRIBUTION
TO THE GOODNESS OF FIT. THIS PROCEDURE HAS SEVERAL ADVANTAGES. AT
EACH STEP IT PRovIDES THE FOLLOWING:
I. A REGRESSION EQUATiON FOR EACH OF THE VARIABLES
INCLUDED IN THE REGRESSION UP TO THAT STEP.
2. AN ESTIMATE OF THE PROBABLE ERROR IN THE PREDlOTION
OF Y FROM THE REGRESSION EQUATloN~
- 12 -
3. ESTIMATE OF THE PROBABLE ERROR IN THE REGRESSION ~
OOCFFiC~ENTS 8,. 8 2, 8 3 ••••••
4. AN ESTIMATE OF THE SIGNIFICANCE OF EACH VARiABLE
iN THE REGRESSION.
5. REJECT,oN OF INSIGNIFICANT VARiABLES.
THE CALCULATIONS FIRST WERE CARRIED OUT MANU~~LY TO DETERMINE
THE REGIMES INTO WHtCH THE DATA SHOULD SE DIViDED. THEN, FOR ACCURACY.
THE DATA WERE ANALYZED ON AN 1.8.M. 704 COMPUTER USING A STEPWISE.
LINEAR, MULTiPLE REGRESSION PROGRAM DEVELOPED BY THE COMPUTING CENTER
OF THE Esso RESEARCH AND ENGINEERING CO. THE RESULTS OF THE TWO
ANALYSES WERE IN COMPLETE AGREEMENT. TABLE I I 1* LISTS THE EXPERIMENTAL
DATA AND THEIR LOGARITHMS, WHICH WERE THE INPUT TO THE COMPUTER. ALL
THE DATA WERE ASSUMED TO HAVE EQUAL WEIGHT.
RESULTS
THE EXPERIMENTAL DATA CORRELATED IN THREE DISTINCT REGIMES
OF CONVECTION - INITIAL, LAMINAR, AND TURBULENT. THE RANGE COVERED
BY EACH REGIME, THE PARAMETRIC EQUATIONS FOR EACH REGIME, THE
STANDARD ERRCR OF ESTIMATE O~ Nu FROM EACH EQUATION, AND THE STANDA~D
ERROR OF THE COEFFICIENTS. ARE LISTED IN TABLE IV.
*COEFFICIENTS THROUGHOUT THIS PAPER REFER TO THE COEFFICIENTS OF
THE LOGARiTHMIC EQUATION. TH£SE BECOME EXPONENTS OF THE PAR'METERS
IN THE POWER •• W EQUATION.
* TABLE I I I - Ps. 18
TABLE 11 SUMMARY OF REGRESSION ANALYSI.
STD~ ERROR STD.ERROR OF COEF. OF ESTIMATE RA PA ...... %r..;.......,_
LAMINAR ~',IJ006
TURBUl.ENT
THE FOllO.'N8 CONCLUSIONS. CAN BE DRAWN FROM THE ANALYSIS:
I. CONVECTIVE HEAT TRANSFER IN HORIZONTAL LAYERS CAN BE
EXPRESSED ACCURATELY BY THREE DISTINCT EQUATIONS G
2. Nu CAN BE EXPRESSED AS A SIMPLE POWER LAW FUNCTiON
OF RA, AND, iN THE TURBULENT REGIME, OF RA AND PA.
3. A/0 2 is NOT A SIGNU''"lCANT CORRELATING PARAMETER AND
THE ASSUMPTION OF LAYERS OF iNFINITE HORIZONTAl. EXTENT
IS JUSTiFIED.
+7.19 ... 6.73
THE VALUE OF THE CRITtCAL RA CALCULATED FROM THE EQUATiON OF THE
iNITIAl. REGIME is 1640 .. THIS IS IN REASONABLE AGREEMENT WITH THE
ANALYtiCAL VALUE OF 1707 .. 8 (3) AND WiTH EXPERIMENTAL VALUES REPORTED
IN THE LITERATURE. (6, 7, 20, 21)
IN FIG. 3, Nu VB RA OF THE EXPERIMENTAL DATA Is PLOTTED TOGETHER
WITH THE EQUATIONS OF THE THRE[ REGIMES. THE TURBULENT EQUATION ts
PLOTTED WITH VARIOUS PH AS PARAMETERS. IN FIG$ 5 THE TURBULENT REGIME
/ 0853
IS OOMBINED iNTO ONE EQUATION BY PLOTTING Nu PR- VS RA. IT CAN BE
~ 14 ...
SEEN FROM THESE FIGURES THAT THREE REG!NES COMPLETELY DESCRIBE
THE DATA4 CHANGES IN MOOE OF CCNVECTtON WITHIN THESE REGiMES
APPARENTLY DO NOT AFFECT THE HEAT TRANSFER tUNCTION.
iN FIG. 4, THE EQUATIONS OF THE THREE ~EaIMES ARE PLOTTED AS
Nu VB RA, SiMILAR TO FIG. 3, EXCEPT THAT THE EXPECTED TRANSITlGNS
BETWEEN LAMiNAR AND TURBULENT REGIMES ARE SHOWN AS DASH EO LINES. IT
CAN BE SEEN THAT THE TRANSITION FROM INiTIAL TO LAMiNAR REGIME IS
SHARR, WHiLE THE TRANSITION FROM LAMINAR TO TURBULENT REGlME DEPENDS
ON THE PR OF THE FLU1@. PREVIOUS INVESTIGATORS HAYE NOT REALIZED THIS.
FLUIDS HAVING A HIGH PR UNDERGO A SHARP TRANSITION, WHILE THOSE WITH
SMALL PH GO THROUGH A MORE GRADUAL TRANSITION~ THE DATA, ALTHOUGH
SPARSE IN THtS REGION, SUPPORTS THIS OONOLUSION EXCEPT FOR MERCURY, WHICH
INEXPLICABLY DEVIATES FROM iTS EXPECTED PATTERN IN THE DIRECTION OF
LOWER Nu ..
I III FIG.. 5 ,t T CAN B ESE EN T HAT THE D A T A I NTH E T RAN SIT I 0 III Ii E G I {l III
OORRELATES WELL WITH THE REST OF THE REGIME$ EVEN THE LOWER MERCURY
POINTS FALL WiTHIN THREE SiGMA LIMITS AND~ THEREFORE, CANNOT BE
E~CLUD[O ON STATISTiOAL GROUNDS. THEREFORE, THE TURBULENT REGRESSION
EQUATION CAN BE USED TO PREDIOT Nu RN THE TRANSITION REGION.
DISOUSSION OF THE ANALYSIS OF DATA
TABLE V* SUMMARIZES THE COMPUTER PRlMT-OUT or THE STEPWISE
REGRESSION ANALYSIS .. AN EXPLANATION or THE STATISTIOS IS GIVEN IN
ApPENDIX IV.
* PG III 27
SllNIFrCANCE OF THE CORRELATING PARAMETERS
THE COMPUTER ANALYSIS SHOWED THAT All OF THE CORRELATING
PARAMETERS WERE STATISTICALLY SIGNIFICANT IN ALL OF THE REGIMES;
t.E., THE BEST CORRELATION INCLUDED ALL THREE PARAMETERS. HOWEVER.
THE IMPORTANT STATISTICAL MEASURE IS THE DEBREE OF SiGNIFICANCE OF
THE PARAMETERS, WHICH is GiVEN BY THE COEFFICIENTS OF DETERMINATaON.
THESE ARE TAB'I.H.I.TE:D IN TABLE vt~· AT EACH STEP OF THE REGIH:SSIONo
THE OOEFFICIENT OF OETERMINATION (EQUAL TO THE SQUARE OF THE MULTIPLE
CORRELATION COEFFICIENT) MEASURES THE PER CENT OF THE TOTAL VARIANCE
aN Nu ACCOUNTED FOR BY THE REGRESSION AT EACH STEP; i.E., AS EAOH
PARAMETER IS ADDED TO THE REGRESSION ANALYSIS. THE RESiDUAL REPRESENTS
THE VARIANOE IN Nu THAT IS NOT ACCOUNTED FOR BY THE REGRESSION
(SCATTER OF DATA ABOUT THE REGRESSION LlNE) AND IS ASSUMED TO BE
RANDOM ERROR. IT CAN BE SEEN FROM TABLE VI THAT RA IS VERY HIGHLY
SIGNiFICANT WHILE~ EXOEPT FOR ONE OASE, PR AND A/o 2 A~E LESS THAN THE
RES IDUAl6WE CONCLUDE, THEREfORE, THAT PR AND A/o 2 ARE NOT S IQNlf HCAtH
PHYSICALLY AND THAT THE CORRELATION FOUND BY THE REGRESSION IS
SPURIOUS; I.Ee, DUE TO CHANCE. THE VARIANCE REDUCTION ATTRIBUTED TO
PR AND A/D 2 ~s REGARD€D AS ERROR AND THE PROPER REGRESSION 18 THE
RESULT Of STEP ONE, INVOLVING ONLY RA. THE EXCEPTION is PR IN THE
TURBULENT REGIME. ITS SIGNIFiCANCE Is ALSO SHOWN BY THE SUBSTANTIAL
REDUCTION OF THE STANDARD ERROR OF ESTIMATE AFTER PR rs ADDED TO THE
REGRESsiON (TABLE V). FOR THIS REGIME WE CONCLUDE THAT THE PROPER
CORRELATING EQUATION IS THE RESULT OF STEP TWO~ iNVOlViNG RA AND PR.
THE MULTIPLE CORRELATION COEFFiCIENTS FOR ALL THREE REGIMES
REFLECT VERY HIGH DEGREES OF CORRELATiON. THEREFORE. WE CONCLUDE
THAT THE FUNCTIONAL RELATIONSHIP NW = F OF (RA) OR F OF (RA)(PR)
AOCURATELY DESORIBES THE PHYSiOAl PHENOMENON~ THERE is NO EWtDEHOE
iN THE GRAPHS OF THE DATA TO SUGGEST THAT THERE ARE MORE REGIMES THAN
THE THREE ALREADY DESCRIBED.
STANDARD ERROR ESTIMATE
THE STANDARD ERROR OF ESTIMATE as THE STANDARD DEVIATiON OF THE
DISTRIBUTION OF EXPERIMENTAL VALUES OF Nu ABOUT THE REGRESSIOH. fT
HAS THE USUAL SI~T~STtOAL S1GNIF10ANOE; I.E.# APPRoxiMATELY 2/3 OF
THE DATA (S EXPECTED TO FALL WITHIN t ONE STANDARD ERROR OF ESTIMATE
(S~GMA) OF THE VALUE PREDiCTED BY THE REGRESS~ONo IN THIS CASE, THE
STANDARD ERROR OF ESTIMATE IS GIVEN IN LOGARITHMIC FORM BY THE
REGRESSION ANALYSIS. IN ORDER TO EXPRESS (T IN TERMS OF Nu IT MUST
BE STATED AS A PERCENTAGE. THIS ACCOUNTS FOR THE FACT THAT BTS POSITIVE
AND NEGATlVE NUMERiCAL VALUES ARE DIFFERENT. IF IT is DESIRED TO
CALCULATE TWO OR THREE S$GMA LIMITS, THEN THE LOGARITHMIC VALUE MUST BE
USED TO CALCULATE THE APPROPRIATE PERCENTAGE~
ERROR OF THE REGRESSI0N
THE STANDARD ERROR OF A COEFFICIENT IS A MEASURE OF THE PROBABLE
ERROR (+ ONE SIGMA) OF THE COEFFICIENT ON THE AVERAGE. HOWEVER, IT IS A .... SPECIFIC PROPERTY OF REGRESSIONS THAT THIS ERROR IS NEGLIGIBLE AT THE
CENTER OF THE RANGE OOVERED BY THE EXPERIMENTAL DATA AND LARGE AT EITHER
ENDo (SEE ApPENDIX IV) THEREFORE, THE REaRESSION EQUATION SHOULD NOT BE
EXTRAPOLATED BEY0ND THE RANGE OF THE DATA WITHOUT FURTHER S~BSTANTIATDON.
TABLE 11
COMPARISON OF THE ARATUS USED By THE SOURCES OF THE DATA
DE GRAAF AND VAN DER HELD
MuJ..'l. AN D RE UIER
SHAPE OF FLUID CHAMBER SQUARE PRISM REOTANGULAR PRISM
AREA OF FLUaD CHAMBER, M2
,,0'. AM. OR LENGTH;\VIOTH,
MM
CONTAINER WALL MATERIAL
WAS THERE A GUARD HEATER?
METHOD OF HEATING
METHOD OF COOLING
MAXe TEMPo, DIFFERENOE, C
WERE LIQUIDS DEGASSEO?
METHOD OF PLATE TEMP.MEASUREMENT
METHOD OF HEAT TRANSF'ER MEASUREMENT
RADiATION CORRECT .'ON
430/430
6.9.22 .. 9
GLASS
No
No
ELEOTRIOAL
WATER
146
100
3 THERMOCOUPLESlN
10 10/612
12-196
WOOD
YES
YES
ELEOTRIOAL
WATER
146
29
15 THERMOCOUPLES IN
HOT PLATE; HOT PLATE; MEAN COOLING 6 THERMO-WATER TEMP. COU PL ES tiN
COLD PLATE
HEAT ACQuiRED ELECTR.OA~ BY 'OOOL.'I'NS POWER CON-WATER SUMPTION
YES
GLOBE AND DROPKIN
CYLINDER
0.0'3,0.014
127,134
35 ... 66
VI Pl EX! Ea. AS U ,
"PYFU:X II
No
No
ELECTRIOAL
Au~, WATER
92
27
YES
1 THERMOCOU Pl.£: IN EACH PLATE
ECTRIOAl POWER CONSUMPTION
SCHMIDT AI«l SU .. VEaTON
C'(ll NO ER
199
1.45 ... 13.0
&tPLEX'f'SLAsM
YES
YES
ELECTRIOAL
WATER
70 (EST.)
50
YES
3 THERMOCOUPLES IN EACH PLATE
EL E<1lTR • CAL POWER CONSUMPTiON
TA
BtE
J..
!!
DA
TA
~in)UROES:
MR
:::::
MU
LL
AN
D
RE
t H
ER
(5
) S
8
::::
SCH
MR
DT
A
ND
S
ILV
ES
TO
N
(20
) G
D
::::
GL
OB
E
AN
D
DR
OP
KIN
(2
1)
DV
::::
: D
E
GR
AA
F A
ND
V
AN
D
ER
H
EL
D
(12
)
TU
RB
UL
EN
T REGIME~
RA
105
TO
J0
9
SO
UR
OE
F
LU
,I!)
R
A
X
10
-5
PR
A
/g2
-~
LO
G_B
I<
j...o~
1a ~e
R~
1_oGAL~?
Lo
a
NUIllP
MR
A'iR
6
.32
.6
99
6
9.4
6
.67
5
.80
07
-.
15
55
i 4
18
4 1
4 .$
24
1
5 .. 0
3 .7
04
6
9.4
5
.75
5
.70
16
...
. 152
4 "
,,75
97
6.4
.6
93
6
9.4
6
.80
5
.80
62
-.
1593
It
418
325
6.0
9
.68
9
69
.4
6.5
5
5.7
84
6
.....
1618
Il
4181
62
52
.6
.70
0
16.,
2 1
4.
15
6.7
21
0
....
1549
i .
20
95
BC
II149
2 ro
2
5.8
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06
1
6.2
1
0.8
3
6.4
i 16
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1512
tt
i 1
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S8
WA
TE
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2.4
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6.0
7
30
6.
5.5
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5.3
92
7
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32
2
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40
4
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5.4
3
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6.
6.5
9
5.5
79
8
&73
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tt
.81
89
HE
PT
AN
E
1.2
5
6.2
6
32
90
. 4
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5
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69
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96
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3.5
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2
.62
63
Suo
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AK
3 1
.86
3
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4.
5111
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5.2
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5
1.5
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1
3.0
87
8
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GO
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C
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S U
. t
CO
NE
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n.
Lt5
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75
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4.9
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8.3
8
5.
1790
3
.94
20
.6
9' I
.9
23
2
2.0
1
85
00
. tt
9
.53
5
.30
10
3
.92
94
It
.9
79
J 6
.20
7
36
0.
It,
JO.9
0 5
.79
24
34
186
69
It
L,0
374
.
150
C.S
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H.
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2n
o
16
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48
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8.3
24
3
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204
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1
.68
84
67
60
t I.
n
66
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8.8
30
0
1.0
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4
n 1
.82
54
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49
20
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2.5
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8.6
92
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1.0
96
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1.7
71
6
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10
1541
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4
2.9
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tt
1
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26
TABL
E II
r
(Co
rnio
) --
TURB
ULE
NT
RE
GlU
E:
RA
10
5 TO
10
9
SOU
ROE
FL
UIO
R
A
x 10
... 5
PR
A
/o,2
N
u LO
G RA
LO
G PR
lO
G
ALD
2 lo
t N
u.
--
II'
i y--~
GO
IS
O
G .. S
. 4
.91
.. 6
9 I
I S
U ••
CON
E ou
. 11
40
i5~5
it
34.,
3 8
.05
69
1
.19
03
at
, .5
35
3
39
1
17
. It
3
2.7
7
.59
22
f .
23
04
"
1.5
14
6
-"
280
17
. f~
3
0.4
7
.44
72
t.
23
04
it
J .
48
29
"
49
5
17
. n:
34
.3
7.6
94
6
J .2
30
4
It;
1.<l
t535
3 76
0 1
5&
5
It;
41 ..
0
7.8
80
8
1.'
90
3
It
1.6
12
8
--.
954
15
. n
39
.8
7.9
79
6
I. f
761
n 1
.59
99
-.
80
5
17
. it
4
L.6
7
.90
58
t .
23
04
It
1
.61
91
-.
-.
15}2
15
.. It
· 4
6.5
8
" 17
96
'.1
76
' n
t.6
q7
4
.. 55
27
1241
It
6
5.0
8
.. 742
5 T
.07
92
it
1
.8 '2
9
" ..
-..
WAT
ER
2490
20
3 n
33
.2
8.3
96
2
.36
17
lic
L~
!.52
' 1
(t)
1934
2 .
• 5
a~
3041
9 8
.28
65
.3
97
9
It
1.4
90
0
-. -
J384
2l1
li5
at 2
9.2
8
. J4
1 I
.39
79
.,
f .4
65
4
-.
890
3.5
if
2
8.4
7
.94
94
.5
44
1
It
1.4
53
3
579
3.8
it
2
7.f
7
.76
27
41
5798
It
f .
. 43
30
-
388
4.3
It
2
4.3
7
.. 588
8 .6
33
5
n'
f.3
85
6
183
4.9
i.
19
.3
7.2
62
4
.69
02
it
1
.28
56
-.
"
80
cA
5.7
tt
1
7.8
6
.90
53
.. 7
559
It
r.25
04-
.-
42
.8
5.5
at
15
.4
6.6
31
4
0740
4 ••
f 0 H
375
f 1
6.8
5
.7
tt
19
.0
7.0
67
4
.75
59
It
1
.27
88
-.
50
G.8
. S
I L
I co
NE
0 u.
1
12
. 2
99
. It
2
4.1
7
.04
92
2
.47
57
It
1
.38
20
19
i 2
45
. n
26
.5
7.2
81
0
2.3
89
2
It
f .4
23
2
-.
40
.7
36
4.
It
16
.7
6.6
09
6
2.5
61
1
n r .
22
27
--
t L,8
4
02
. It
12
.8
6.0
7}
9
2.6
04
2
i~
1.1
07
2
TABL
E II
I (C
ON
TrD
) -
TURB
ULE
NT
RE
GIM
E:
RA
105
TO
109
SOU
RCE
FLU
ID
RA
X
10
.. 5
.E.!!
A/D
2 ~
LOG
RJI).
LO
G PR
LO
G~A/
D2
GO
M
EROU
RY
9.8
7
.02
32
3
.19
4
.91
5
.99
43
...
J .6
34
5
4115
038
.69
t 1
35
. .0
21
8
It
6~63
6.5
44
J ....
, .6
61
5
•• ~ 8
2 J.
5 3
076
.0
23
5
it
3.0
7
5.5
75
2
... 1
.62
89
it
.4
87
1
~
6.1
4
.02
33
at
3
.87
5
.78
82
-1
.63
26
It
.5
87
7
,
16
.5
.02
27
tt
5
.61
6
.21
75
....
1.6
44
0
it
.74
90
-
9.5
7
.02
29
Q
; 5
.22
5
.98
09
...
1.6
40
2
It
.. 7J7
7
~
7.9
2
.02
31
I~
: 4
.38
5
.89
87
...
f .6
36
4
UI
.64
15
-
4.6
7
.02
32
tt
30
:67
5.6
69
3
... f .
63
45
it
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64
7
13
.2
.02
28
n
6.2
0
6.
i 206
...
1.64
21
tt
.79
24
~
220
.02
22
tt
7
. r I
6
.34
24
...
1.6
53
6
lct
.85
19
~
34
.3
.. 0
21
4
tt
8.0
8
6.5
35
3
.. 1
.66
96
n
.90
74
1;
35
.8
.02
12
tt
7
.87
6
.55
39
...
1.6
75
7
ft
.89
60
I\
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36
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12
n
7.6
4
6.5
58
7
.... 1
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57
tt
.8
83
1
0 -
5t.
.0
20
3
if
8.3
2
6.7
07
6
... t
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25
u
.92
01
11
-9
.05
.0
23
1
Q
5.5
6
5.9
56
6
... 1
.63
83
tt
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45
' 5
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3
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4.5
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5.7
38
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... f
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53
2
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5
n 3
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5
.. 571
7 ....
1..6
289
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1
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6
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5.2
12
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... 1
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02
718
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n 2
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5
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54
....
1.6
27
1
lit
.40
65
2
4.7
.0
22
7
t 1
.28
7
.68
6
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27
...
1.6
44
0
1.0
52
3
.88
54
5
1.6
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20
0
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9.7
2
6.71
.26
... t
.69
90
u
.98
77
1
3.0
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23
8
U:
6~02
6.1
13
9
-1.6
23
4
If
1117
796
56
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35
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9;5
9
6.7
48
2
-1.6
28
9
it
.98
18
92
08
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32
u
10
.9 t
6".9
676
... 1
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34
5
tt
J .0
37
8
16
8.9
1110
222
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J3.8
6
7.2
27
6
.... 1
.65
36
It
1
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18
3
33
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tt
1
6.8
8
7.5
23
4
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6
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, .2
27
4
TAB
LE II
I (C
ON
TfD
) -
.
lAM
iNA
R
RE
GIM
E:
RA.
3000
TO
10
5
SO
UR
GE
F
LU
,~o
&
PR
p.J
02
Nu
LOB
RA
-L
oe
PR
LO
G
A/0
2
1..9
6 N
Q.4
> -
--O
V A
IR
3560
07
10
76
1.
1.5
2
3.5
51
4
.... 1
487
2.8
81
4
.18
18
6
49
0
0707
7
61
. 2
. 19
3
.81
22
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".15
06
2.8
81
4
.34
04
8
28
0
.70
5
76
1.
2.2
6
3.9
18
0
-.'
5'8
2
.88
14
.3
54
1
94
20
.7
03
7
61
. 2
.50
3
.97
40
-.
15
30
2
.88
14
03
979
40
00
.7
09
7
61
. '-'
,63
3 .. 6
021
M.1
494
2.8
81
4
.21
22
52
80
.70
8
76
1.
1.8
0
3 .. 7
226
.....
'50
0
2.8
8'4
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3
70
70
.7
07
76
f ..
2.0
2
3.8
49
4
-.1
50
6
2 .. 8
81
4
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54
8
72
0
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76
1.
\2 ..
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3 .. 9
40
5
-.1
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4
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tOO
11
1702
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t II
2
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4
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43
...
. 153
7 2
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14
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74
8
I fO
OO
• 6
99
76
L •
2.3
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4.0
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4
....
J552
2"
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4 .. 3
729
11
J i
100
• 698
76
1 •
2.3
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4.0
45
3
......
J561
2
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76
6
rv
2340
0 • 7
09
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1 •
3.3
3
4.3
69
2
.... 1
494
2.3
63
6
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--
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60
0
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231
Ell
3.7
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4.6
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231
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68
100
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6
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4.0
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2&34
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. 154
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2
1660
0 .7
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2
31
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4
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. 148
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- 28 -
TABtE !l COEF'F':I,Ct EMTS OF MtH. T.
CQRRELATIO.N AND DETt:RMINAT J<JM~
% DUE TO VARIABLE COEF. OF COEF& OF ENTER I,N Q
REG~fME STEP ENTERiNG MU,T.CORRE"'t - MULT., DETERM, % VAR i A Bh.!
INITiAL 1 RA .9624 92.6 92.6 2 PR .9690 93.9 1 .. 3 3 A/o2 .9803 9601 2.2
RES IOU AI.. 3.9 3.9
LAMINAR I RA .9691 93.9 93.9 2 A/o2 .9736 94.8 0.9 3 PR .9758 95 .. 2 0.4
RESIDUAL 4.8 4@8
TURBULENT I RA .. 9302 86.3 86.3 2 PR 09902 98.0 J 1.7 3 A/D2 .9908 98.2 0 .. 2
RESIDUAL fo8 1..8
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... 29 ...
APPENDIX l TRANSFER
DUAENSrONAl ANALYSIS OF' HEAT~BY CONDUCTION AND CONVECTH1N ACROSS
A HORIZONTAL FLUID LAYER, BOUNDED BY INFINiTE, RiIID, CONDUOTING
SURFACES AND HEATED FROU BELOW.
ASSUMPTiON:
IF THERE IS NO CONVECTION, HEAT WILL BE TRANSFERRED ONLY BY
OONDUCT'ON ANa THE HEAT TRANSFER RATE PER UNIT AREA CAN BE EXPRESSES
AT STEADY-STATE CONDITIONS WHERE HEAT 1$ BEING TRANSFERRED BY BOTH
CONDUCTION AND CONVECTioN THE NET HEAT TRANSFER RATE PER UNIT AREA~
HCC$ WILL BE LARGER THAN Hc SO THAT
Hcc = Hcc = A FUNCTION OF T$L» AND THE FLUID CONSTANTS.
H·c 'I$ST /L
THE FOLLOWING WiLL BE THE BASiC UNITS:
T LENGTH L HEAT QUANTITY - Q
MASS .... M TEUPERATURE ... T
THEREFORE THE PHYSIOAL CONSTANTS AND VARIABLES WILL HAVE THE FOLLOWING
MEASUR E: FOR MULAI:::
K = THERMAL CONDUCTIVITY
T TEMPERATURE DiFFERENCE
D = DEN$ lTY
M DYNAMIC VISCOSITY
c = SPECIFIC HEAT(AT CONSTANT PRESSURE)
OOEFFIOIENT or THERMAL EXPANSION
:: ACCELERATION DUE TO GRAVITY
::::: FORO~ or 8UOYANCY PER UNIT MASS
T
M/L3
M/LT
Q/TM
I/T
L/T2
L/T2
- 30 ...
CAGT) REPRESENTS TR£ BUOYANCY FOROE IN THE FLUID UNDER THE
INFLUENCE CF A TEMPERATURE-iNDUCED DENSITY GRADIENT AND IT is
CONVENIENT TO TREAT IT AS AN ENTITY HAVING THE UNITS l/T2.
IF WE POSTULATE THAT
TNEN THE EXPRESSION 'N PARENTHESES MUST BE DIMENSIONLESS, THAT fS
(M/l3) A (l) B (MilT) C (Q/Tl T) D {Q/rM)E (l/T2)F
THEREFORE~ EQUATING THE SUM OP THE EXPONENTS OF INDIVIDUAL
DIMENSIONS ON THE RIGHT TO ZERO, WE OBTAIN THE FOLLOWING tNOrCIAL
EQUATIONS:
o + E :: 0
"'D ... E :::: 0 ... C ... D ... 2F = 0
.,
RETAINiNG E AND~F WE CAN EXPRESS THE REMAINING aNDICES AS
FOLLOWS:
A ~ 2F C :::;: E'" 2F
B :::: 3F
THEREFORE
- 31 ...
APPEND IX
FOR BASES, THE THERMAL OOEFFIOIENT OF EXPANSION "IDS EQUAL TO
I /T08 WHERE TO C MEAN ABSOLUTE TEMPERATURE.
CONSIDER A OONF~NED HORIZONTAL LAYER OF GAS, AT ATMOSPHERIO
PRESSURE. AOROSS WHIOH A TEMPERATURE GRADIENT T2 ... TI HAS eEEN
APP1.IEI3.
T ::::::; ABSOLUTE TEMPEAATURE OF THE HOT SURFAOE '2
TI :: ~ ~ tt n OOL D SURFAOE
To ::::: MEAN ABSOLUTE T EM PER A Ttl A E
::::: 00EFFI01ENT OF THERMAL EXPANS ION,
T ::::::; T2 ... T ::::: 0 TO -T, = (T 2 ... T 1)/2
ASSUMING CONSTANT PRESSURE THROUGH THE LAYER AND V~~IDITY OF THE
PERFEOT GAS LAW> THEN
V2 .... VI = NR (T 2 ... T I) ::::: Va (T 2 '"" T I ) P -To
V "" VI C V (I + ~ T) ... Vo ( 1 .... f1'T) ::: Va(2~T) =: V ~ (T ... T ) 2 0 021
TN ER EFORE
- 32 ~
THE EXPRESSION V - Va (1 + T) IS BASED ON THE ASSUMPTION THAT
IS A OONSTANT$ THIS is VALID ONLY OYER SMAL~ INOREM£NTS OF
TEMPERATURE~ TAK[NG~ AT THE MEAN TEMPERATU~E INSTEAD OF TI OR T2
IS MOR£ ACCURATE BECAUSE aT REDUCES THE ~NTERVAL OVER WHICH IS
ASSUMED CONSTANT, AND BECAUSE ERRORS TEND TO CANOEL OUT.
"" 33 ...
APPENDIX ill
CORREOT10N OF T~E DATA OF MULL AND REIHER (5)
THE DATA OF BOTH MULL AND REIHER (5) ANG DE GRAAF AND
VAN DER HELO' (COMMUW·IOATION FROM THE AUTHORS) OONTAINS TABULATIONS
OF THE FOLLOWING QUANTITIES:
GRASHor No. AND NUISELT No.;
THERMAL CONDUOTiVITY OF AIR AT DIFFERENT TEMPERATURES.
TI AND T2• THE TEMPERATURE GRADIENT AOROSS THE LAYER;
L. THE LAYER THIOKNESS~
DE GRAAF AND VAN DER HELD HAVE SHOWN BY RECALCULATION OF MULL AND
RE1HERtS [lATA (13) THAT THESE AUTHORS USED 1/273 AS THE COEFFICIENT
OF THERMAL EXPANSloN FOR AIR AND THAT THEY USED DIFFERENT VALUES
FOR THE CONSTANTS IN GR AND Nu. A MORE ACCURATE VALUE FOR THE
I EXPANSiON COEFFICIENT IS /TAVa (SEE ApPENDiX I I) WHERE TAVG IS THE
MEAN ABSOLUTE TEMPERATURE iN THE LAYE~~
IN ORDER TO PUT ALL THE AIR DATA ON THE SAME BASIS, THE DATA
or MULL AND REIHER WAS RECALCULATED USING I/TAVG AND THE CONSTANTS
(THERMAL OONDUCTIVITY$ DENSITY, AND VISCOSITY) OF DE GRAAF AND
VAN DER HELO, SiNCE THEIR WORK WAS MUCH MORE RECENT THAN THAT OF
MULL AND RE I HER(4.
THE CORRECTIONS WERE MADE AS FOLLOWS:
NUSSEt T No I)
NU:::J H KT /1.
WHERE H - MEASURED RATE or HEAT TRANSFER
PER UNIT AREA DUE TO CONVEOTION
AND OONDUOTION
APPENDIX
K :::: THERMAL CONDUCTIVITY
T ~ TEMPERATURE DIFFERENCE
ACROSS THE LAYER
L = LAYER THIOKNESS
FJB~ 2 SHOWS THE THERMAL OONDUOTIVITIES VS TEMPERATURE USED
BY 80TH MULL AND REIHER AND DE GRAAF AND VAN DER HELD. THE CURVES
ARE PLOTTED FROM TABULATIONS IN THE RESPEOTIVE AUTHOR'S DATAe
CORREOTlON WAS MADE AS FOLLOWS:
CORRECTED Nu ::: ORIGiNAL Nu X K OF MULL AND REIHER K OF DE GRAAF & VAN DER RELD
GRASHOF' No,
GR ~@G T L3 e 2 WHERE .(1 :::: THERMAL CO E F F i C lEN T OF
2 r EXPANSiON
G :::: GRAVITATIONAL OONSTANT
T :::: TEMPERATURE DIFFERENCE
AORO 5S THE LAYEF!
L :::::: LAYER THICKNESS
P - DENSITY
~ ::::: VISCOSITY
TAVG :::: MEAN ABSOLUTE TEMPERATURE
CORREOTION WAS MADE AS FOLLOWS:
GR OF DE GRAAF AND VAN DER Hn.o a~G e2 :::: Z
.),t, 2
Z WAS PLOTTED AGAINST TAVG. THEN, FOR EACH OBSERVATION OF MULL AND
REIHER, Z WAS TAKEN rROM THE OURVE AT THE APPROPRIATE TAVG AND
T.L 3 0 f' Mu L L A NO REI HER X Z ::0 COR RE:O TED G R
- 35 -
APPENDIX IV ""'*"'*" - il:O_ """"'-"
STEPWISE MULTIPLE LiNEAR REGRESS ION
REFERENOES: 25. 26, 27, 28, 29
MULTIPLE LINEA~ REGRESSION
MULTIPLE LiNEAR REGRESSION BV THE METHOD OF LEAST SQUARES
CONsiSTS OF F[NDING THE CONSTANT AND COEFFICiENTS OF AN EQUATION OF
THE FORM
Y =.A + 81 XI + 92 X2 + B3 X3 ~IHI $. WHERE Y :::: PREDICTED VALUE 01'" THE DEPENDENT VARIABLE
Y :::::: MEASUR EO VALUE ,. u 1'1 U'
X :;::: ARBITRARY VALUES OF THE INDEPENDENT VARIABLES
X ::: ME ASUR ED VALUES 11 it It " A ::::: CONS TANT ::::: Y INTERCEPT
8 :.:::: COEFFIOIENTS OF THE INDEPENDENT VARIABLES
Y ::::: AVER AGE OF MEASURED DEPENDENT VARIABLE
X :::: AVER/H~E OF ME ASUR ED INDEPENDENT VARIABLE
• :::::: STANDARD DEVIATiON
V ::::: VARIANOE :::: S2
N ::::: NUMBER OF OBSERVATIONS OR PIEOES OF' DATA
§o THAT THE SUM OF SQUARES OF THE DEVIATlONS (V ~ Y) IS A MINIMUM~
FOR THE CASE OF SIMPLE REGRESSION, THE OALOULATION PROOEDURE CAN
BE EXPRESSED AS FOLLOWS:
(VI ~ YI) ~ VI • (A + 8 XI) - VI - A - B XI
i (y • y)2 =fJV'" 11.\ • BX)2
To SOLVE FOR II AND B WHEN
( V ... II ... BX)2
THESE EQUATIONS REDUCE TO THE FOLLOWING INVOLVING SIMPLE
2 SUMMATIONS CN V, X, X 3 AND XV:
... Z(y) + AN + B i.(x) =: 0
... z (x y) + A (X)\ + B (x 2) !II:: 0
THE SUMMATION EQUATtONS CAN BE SOLVED SIMULTANEOUSLY FOR A AND 8,
BUT IN PRACTICE THESE EQUATIONS ARE COMBINED, ELIMiNATING A, SOLVED FOO
B, AND A IS DETERMINED FROM
V = A + B X.
IN MULTIPLE REGRESSION, THERE ARE ADDITiONAL EQUATIONS FOR EAOH
VARIABLE ADDEDIII
THESE ARE SOLVED SIMULTANEOUSLY FOR 911 82' B3 ••••• AND A IS
FOUND FROM
ha STEPwiSE PROCEDURE THE StMULTfoJIl EOUS EQUATIONS ARE SOLVED rOR
ONE VARIABLE AT A TIME BY MATRIX ALGEBRA, WHICH SIMPLIFIES THE
PROCEDURE (25). AT EACH STEP STATiSTICAL iNFORMATION IS OBTAINED THAT
PERMITS EVALUATION or THE SiGNIFiCANOE or THE OORRELATING PARAMETERS.
.... 37 -
STANDARD ERROR OF ESTIMATE AND CORRELATION COEFFIOIENi
THE BAsro APPROACH Ot LINEAR REGRESSION MAY aE DESCRIBED AS
FOLLOWS:
IF THERE is NO CORRELATION BETWEEN THE DEPENDENT AND AN
tNOEPENDENT VARIABLE, THEN THE BEST ESTIMATE OF Y Is ~ AND THE
PROBABLE ERROR OF ESTIMATE Is Sy, THE STANDARD DEVIATION. IF THERE IS
CORRELATION BETWEEN Y AND X, THEN THE REGRESSiON CAN BE REGARDED AS
ACCOUNTING FOR SOME OF THE VARIANCE (EQUAL TO Sy2) [N y. THE RESIDUAL
VARIANCE WHICH THE REGRESS ION DOES NOT ACOOUNT FOR is REGARDED AS
ERR 0 R OR MAY BED U E TOO THE R PAR AM E T E R S J SUO HAS PR ! III THE T U R 8 U LEN T
THE MEASURE OF THE RESIDUAL ERROR VARiANOE IS
(y ~ y)2 = V = MEAN SQUARE OF RESIDUALS OF
WHERE OF = DEGREES O~ ~REEOOM
= N ~ NO. OF OONSTANTS IN TH[
REGRESSION EQUATION@
Sy, THE STANDARD DEVIATION O~ THE RESIDUALS OR
STANDARD ERROR OF ESTIMATE.
THEREFORE»
TOTAL VARIANCE 8N Y'" EPt:ROR VARIANOE:: VARIANOE OONTRHlUTlON
IF ALL THE EXPERIMENTAL DATA WERE TO FALL ON THE ~EGRESSION LtNE,
THEN ALL OF" THE VARIANC( IN Y WOULD BE ACOOUNTED FOR 13'1' THE 'REGIHSS&ON
OF Y UPON X AND THE RATIO
... 38 M
APPENDIX 11. (CONT'O)
WHERE Sx AND VX REFER TO THE Y VARIANCE OONTRIBUTIONS OF X.
fS THE OOEFFIOIENT OF MULTIPLE CORRELATiON.
VARY BETWEEN 0 AND ONE DEPENDING ON THE SiGNIFICANOE
OF THE CORRELATION.
OF MULTIPLE DETERMfNATION.
IT fS A MORE ACCURATE ESTiMATE OF SIGNIFICANCE, SINCE VARIANCES
ARE ADDITIVE.
ERROR OF THE REGRESSION
ASSUMiNG THAT PERFECT CORRELATiON HAS NOT BEEN ACHIEVED, THE
COEFFICIENTS OF THE REGRESSION ARE SUBJECT TO UNCERTAINTY AS MEASURED
BY THE STANDARd ERROR OF THE COEFFICIENTS. iT CAN BE REGARDED AS THE
STANDARD DEVIATION OF THE POPULATION OF REGRESSION LINES THAT CAN BE
DRAWN THROUGH INDiViDUAL EXPERIMENTAL DATA AND THE POINT X, Y SINCE
x, Y MUST BE A POtNT ON THE REGRESSION LINE. THEREFORE, THE UNCERTAINTY
IN THE COEFFICIENT WILL BE ZERO AT i, ~ (ASSUMED TO BE NEAR THE CENTER
OF THE RANGE OF DATA) AND LARGE AT EITHER END. THAT IS, THE REGRESSION
LINE CAN BE VrSUALIZEC A8 PIVOT1NG ABOUT X, Y AS FAR AS ITS P~OBABLE
ERROR IS CONOERNED. THE EFFECT OF THIS PROPERTY OF REGRESSldNS IS TO
INOREASE THE PROBABLE ERROR OF THE REGRESSION AT THE EXTREMES OF THE
RANGE OF DATA AND TO MAKE EXTRAPOLATION BEYOND THAT RANGE HAZARDOUS.
EQUATfONS FOR ESTIMATING PROBABLE ERROR AT ANY POINT ON A REGRESSION
ARE GIVEN IN THE REFERENCES.
T
To
TI ... T2 , T
K
fl
f E'
L., 0
A
C
Q
H
V
S
Sy
Sa
B
lOG A
Nu
RA
GR
PR
A/o2
:=
:=
:=
::::
==
:=
::
::::I
::::I
::::
::
::
:=
:::Ii!
::::
::::
::
-:::;:;
::::
:=
:::
#
SYMBOLS
ABSOLUTE TEMPERATURE
ABSOLUTE MEAN TEMPERATURE OF THE LAYER
TEMPERATURE DIFFERENCE
THERMAL CONDUOTIVITY
V J SCO S I TY
THERMAL COEFFiCIENT OF EXPANSION
DENS1TY
LAYER TH!CKNESS
AREA
SPECIFIC HEAT
HEAT QUANTITY
RATE OF HEAT TRANSFER PER UNIT AREA
VOLU ME
STANDARD DEVIATION
STANDARD ERROR OF ESTiMATE
STANDARD ERROR OF THE COEFFICIENT
OOEFFIOIENT OF LOGARtTHMIO EQUATION AND EXPONENT
OF POWER LAW EQUATION
CONSTANT Of LOGARITHMIC EQUAT!ON
Nuss EL T NO e
RAYLEIGH No. :: GR X PR
GRASHOF No o
PRANDTL No.
RATIO OF AREA TO THIOKNESS SQUARED OF EXPERIMENTAL
APPARATUS
... 40 "
B I Bli OGRAPHY
(I) BENARD, H. g ANN. CHEM. PHYS.a 23, 62 (190t) (REPORTED IN 12, 20)
(2) lORD RAYLEHlH, P'IlIL. MAG o , 32, 29 (1916)
(3) JEFFREYS, H.~ PHt<l.. MJUle, 2,833 (1926) AND ROYAL SOCe LONDON.II PROC. A., 118.11 195 (1928)
(4) low, A. R., ROYAL SOC. LONDON, PROC. A, 125, 180 (1929)
(5) MULt, I., AND REIHER, H .. , BErH .. Z. GESUNDHEITS * INGe, SERIES 1,28 (1930) ,
(6) SCHMIDT, R. J .. , AND MILVERTON, S. W., ROYAL SOC. LOND9N, PRoe. A, 152, 586 (1935)
(7) SCHMiDT, R. J., AND SAUNDERS, O. A., ROYAL SOC. LONDON, PROO. A, 165, 216 (1938)
(S) CHANDRA, K., ROYAL SOC. LONDON, PROC. A, 164, 231 (t938)
(9) PELLEW, A. , AND SOUTHWELL, R. V., ROYAL Soe. LONDON, PRoe. A, 176, 312 (1940)
( 10) JACOB. M .. , TRANS. ASME, 68, IS9 ( 1946)
(II) SUTTil2N , o. G., ROYAL Soo. LONDON, PROO. A, 204, 297 (l95f)
( '2) DE GRAAF, J.~.A., AND VAN DrR HELD, EoF .. M .. , ApPLt. SO'8 • R'ES .. A, THE HAGUE, 3, 393 ( 1952)
( 13) DE GR AAF , J .. G .. A .... AND VAN DrR HELD, E .. F.M .. , ApPLti~SO 'i. RES. A, THE HAGUE, 4, 460 ( 1954)
( 14) MAlKUS, W~V.R .. , ROYAL Sao. LONDON, PRoe. AO Il 225,185 (1954)
( 15) at it II It It II et 225,,196 (1954)
( 16) BATOHELOR" G. Ko,jll Qu AR T. App .. MATH .. , 12, 209 ( 1954)
( 17) MAlKUS, w. V. R .. , AND VERON IS, G., J .. FLUID MECH., 4, 225 ( 1958)
(rs) PEARSON, J. R. A •• J .. FLUID MEOH .. , 4, 489 (1968)
(19) POOTS, G., QUART. J. OF MECH. AND App. MATH., PT. 3, 11, 257 (1958)
... 41 ...
&IBLIOGRAPHY (CONTIO)
(20) SCNM1DT g R. J., AND SSLVESTON, p. l., 2ND NAT. HT. TRANS. CONF.jI AICHE - ASME, CHICAGO" PT. 24, (1958)
(21) GLOBE, S., AND DROPKJN, D., 2ND NAT. HT. TRANS. CONF • ., AtCHE .... ASNIE, CHIOAGO, 58-HT ... 21 (1958)
(22) nAN INTRODUCTiON TO HEAT TRANSFER" BY M. FJSHENDEN AND O. A. SAUNDERS OXFORD (1950)
(23) ·PHYSIOAL SIMILARITY AND DIMENSIONAL ANALYSIS" BY Vi.. J. Du N CAN EDWARD ARNOLD & CO., LONDON (1953)
(24) HILSENRATH, J., ET Al., NAT@L. BUR. OF SToS. CIRO. 564, 1955 ~EPORTED IN ~PROPERTIES OF GASES AND LIQUIDS· BY R. C. REID AND T. K. SHERWOOD, MOGRAW HILL, N. Y. (1958)
(25) "STEPWISE PROCEDURE FOR CALOULATiON OF MULTIPLE REGRESSION" BY M. A. EFROYMSON Esso RESEARCH AND ENGJNEERING CO., LiNDEN, N. J. DELIVERED AT GORDON RESEARCH CONFERENCE ON STATISTICS, AUGUST i-12, 1955
(26) "ELEMENTARY STATISTICAL ANALYSiS" BY S. S. WILKS PRINCETON UNIV. PRESS, PRINCETON, N. J. (1948)
(27) "METHODS OF STATISTICAL ANALYSIS· BY C. H. GOULDEN JOHN WilEY & SONS, N. Y. (1939)
(28) IISTATISTICAL METHODS IN RESEAROH AND PRODUCTION" BY O. L. DAVIES, OLIVER ANO BOYD, LONGON (1949)
(29) "METHOOS OF CORRELATION ANALYSIS" BY M. EZEKUEJ.., JOHN WILEY AND SONS, N. Y. (1941)
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