How to Figure True Temperature Difference in Shell-and-Tube ...

_,,_.. 1'.)'­TechnologyHow to figure true temperaturedifference in shell ..and-tube exc;hangersThe charts shown here greatly simplify al'l often onerous choreBY DALE L. GULLEYEngineering Consultant, Tulsa, Okla.WANT TO CALCULATE thetrue temperature difference in shelland-tubeexchangers? Here is a simplifiedmethod fDr doing it for anyof the common flDW patterns.And here are new, previDusly unpublishedcDrrectiDn - factor chartsfDr divided-flow and split-flow patterns.'These charts enable the designerto liberalize his, ratings. PreviDusly,normal single-sheIl-pass charts, wereused, and the cDrrection factor, F,was kept above about 0.9. With thenew charts" designs will sDmetimescall for fewer' shells, and therefDrewill be mOore eCDnomical.The basic equatiDn for designingheat exchangers, is:A = Q/U ATWhere:A = external surface Df tubes,sq ftQ = heat lDad, Btu/hourU = Dverall heat-transfer coeffi­Cient' Btu/hDur-sq ft- of.AT =, mean temperature d iffe r­enee, OF.The problem is to find the trueAT.The simplest temperature differenceDCCurs when an exchanger hasconstant temperature Dn both sides:for instance, when steam cDndensesDn one side and a pure organk boilson the oth.er. With steam at 366 0F. and the Dther fluid bDiling at266 0 P., the temperature differencewould be366 - 266 = 100 0 F.This wDuld be the temperaturedifference used in EquatiDn 1. Thisholds true no. matter what the two.fluid flow:patterns are.108t 2 .....Pattern for countercurrent flowTIt1. Countercurrent flowFig. 1 is a simplified diagram Oofan exchanger in CDuntercurrent flDW.When there is sensible-heat transferinvDlved, this type Df flow givesthe mDst efficient temperature drivingforce and, the biggest temperaturecrDSS, because the Dutlet temperatureof the hot stream can becODler than the outlet temperatureof the cold stream. For example:Hot Cold1200 150 i100 80The AT for this flow pattern isdefined by Equation 2:GTD -LTDAT=-----­ (2)'GTDIn-­LTDWhere:--t 1Fig. 1GtD = greatest terminal temperaturedifferenceLTD = least terminal tern per a­ture differenceAT = lOog mean temperature difference(LMTD)Again referring tOo Fig. 1, the terminaltemperature differences areT 1 - t2 and T2 - t 1 . Assuming thatT 2 - t1 is the greater temperaturedifference:(T 2 - t 1 ) - (T 1 - t 2 )AT = -------- (2a)T2 - t1In(--­T1 -t22. Cocurrent flowFig. 2 is a simplified diagram ofan exchanger in cocurrent flow.Equation 2 is still useful, but theterminal temperature differences willPattern for cocurrent flow--t 2Fig. 2THE OIL AND GAS JOURNAL • SEPTEMBER 14, 1964

e different. For example, one dif­ference will be T 1 - t 1 , and the Pattern for one-shell pass; two-tube pass other will be T 2 - t2' t 2This type 'Of flDW pattern is sel­ f r- ...., i T1dDm used. It is not very effiCfentand therefore -will not cDol a givenfluid as. much as the CDuntercurrentflDW will. The hot 'Outlet temperaturecan 'Only approach the cold'Outlet; it cannot crDSS it.But use can be made of this inabilitytD cross temperatures. Forexample, in wax and. asphaltic cDolers,coeUITent flow is' used tD makesure that the sDlidificatiDn point willnDt be reached. If CDuntercurrentflow were used, there would be danger'Of cooling below design whenthe exchanger is clean..9.8.7.5[;1" I I I, ,,! I+T 1 -t 1 t 2 -t 1~~IC?.~ C!C! C!1C!:Jt c .. \LC~tt~~~~~oo...o LnRood', ~tl('l'\LIllI •.t 2-t T -T1 1 2X=--.· R=-­T1 = Inlet temperature shell sideT2::: Outlet temperature shell sidet1 = Inlet temperature tube sidet2::: Outlet temperature tube side;a.t 1and errDr, but a much faster chartmethod is cDmmDnly used. Thischart method is based on applyingaCDrrectiDn factDr tD the log-me antemperaturedifference. Then, thetrue temperature difference for thisflDW pattern will be:LlTc = LMTD (F) (3)Where LMTD is defined byEquatiDn 2F = cDrrectiDn factDrIf there is. CDnstant temperature'On either side; F will be 1.0.Fig. 4 is the LMTD cDrrectionfactDr fDr a 'One-shell pass, two-ormoretube-pass exchanger. SeveralpublicatiDns I 2 3 give cDrr~ctiDn-factDrcurves fDr one tD six shells inseries. and even number of tubepasses.TD use the correctiDn curves it is_i\~_ ..Q~c:i~~ --'••••\ • ......-.~ .~\..... ~~-~~~~~r...::..,T2 + Fig. 3necessary tD calculate two dimensiDnlessparameters. The parameter'On the curves is called R and isequal to:T 1 -T 2 WCR = Dr -- (4)t2 -t 1weWhere:wc = heat capacity of tube fluid,Btu/OF.we = heat capacity 'Of shell fluid,Btu/OF.The variable 'On the abscissa iscalled X and is defined by:t2 -t1x = -----"-'-''---'-- (5)T 2 -t 1As shown in Fig. 4, at high valuesof R it is difficult tD read F accurately.TD 'Overcome this prDblem,the parameters 'Of R and X can beredefined:b,~'5:.f).~'l,. '1,\,MTD correction factors 1 shell pass 2 or more tube passes '"" Fig. 4.1 .2 .3 .4 .5 .6xTHE OIL AND GAS JOURNAL • SEPTEMBER 14, 1964.7 .8 .9 1.0109

o-:,)smallest temperature rangeWhen R = 1, Equations 6 and 7 break down, and Equations 6a andR= ~~ 7 a should be used:greatest temperature rangeXgreatest temperature range X n - 2n = (6a)X =(Sa)Tl~hIt follows that R will always beX n - 2ny'2 (----'----­1 or less, and that the curves will 1 - X n - 2nalways have a relatively flat slope. F n - 2n = (7a)For example, when the shell fluid (2/Xn-2n~ - 1 - R + y'R2 + 1cools from 190 0 to 90 0 and the In [----------­tube-side fluid heats from 80 0 to (2/X n - 2n) ....,. 1 - R - y'R2 + 190 0 • Using Equations 4 and 5:Where n = number of shell passes.Whenever there is a temperature190 - 90 cross in one exchanger, the thermalR = = 10 Use Equation 6 to calculate Xdesign should be checked very care­90 - 80 for the desired number of shells infully. (This occurs at a correctionseries. Then use this value in Equafactorof approximately 0.8, or90 - 80 tion 7 to obtain F.when the correction factor is lessX == = 0.091 Suppose it is desired to find the190 - 80 than 0.8 with shell passes in series.)MTD correction factor for the fol­This should be done for two realowingtemperature conditions:A slight misreading of X cansons:make an appreciable difference inHot' Cold 1. It may be more economical toF. But by using Equations 4a anduse more shells in series, especially410Sa we find:404 i if the units are made of an expen­1400 204 Isive alloy.90 - 80 2. 'The exchanger may not work10 200R = = 0.1 as well as it was designed for. At190 - 90 R = 10/200 = 0.05 close temperature. approaches theMTD correction factors tend to190 - 90 X =200/206 = 0.971 break down at low values of F.X = = 0.91 Very seldom are three tube passes190 - 80Using Equation 6 and three shellsin series as an example, (or a larger odd number) used.Now, the correct value of 0.826 is easily read from Fig. 4. '1 - 0.05 X 0.9711 - ( )1/Equations 4a and Sa should only31 - 0.971be used with the normal one-shellX 3-6 = = 0.70pass and even number of tube passes. 1- 0.05 x 0.971They are inadequate for divided 0.05 - ( )1 /3flow or split flow. In this latter type 1 - 0.971of flow pattern the relationship R =we/We is violated.Then, from Equation 7,Sometimes R will be outside thenormal range of chart values, andy'(0.05)2 + 1 1 - 0.70more than six shell passes in seriesIn (-----­0.05 - 1 1 - 0.05 x 0.70will be needed. In these cases useF 3-6 = = 0.988the follOwing equations, based on(2/0.70) - 1 - 0.05 + y'(0.05)2 + 1work by Bowman: 4 In [1-RX(2/0.70) - 1 - 0.05 - y'(0.05)2 + 11- ( - - ) l/n "Manufacturing difficulties m a k eI-XTo calculate the corrected MTDevery type but a fixed-tube-sheetX n - 2n =(6) by computer, block diagraming inunitundesirable.corporating Equations 6 and 7 canFischer6 derived a trial-and-errorR-( be found. 5I-Xsolution for a one-sheIl-pass". threetUbe-passunit. One tube pass is iny'R2+1 1 - X n - 2n cocurrent flow and the other twoIn (-----are in countercurrent flow. SinceR-1 1 - R X n - 2n more than one-half of the tubeFn- 2n = (7) passes are in countercurrent flow,(2/X n - 2n) - 1 - R + y'IP + 1 the correction factor is higher than~[ ] in an even-tube-pass unit. A correc­(2/Xn- 2n) - 1 - R - y'R2 + 1 . tion-factor chart for one shell is110 THE OIL AND GAS JOURNAL • SEPTEMBER 14, 1964

2. A vapor belt or other type ofTwo-shell pass with long baffleenlarger.te 2If the first type is used, then Fig.6a is better. Since there are twotsmaller nozzles entering rather thanone big one, fewer tubes will be re­1-,----- moved for impingement protection.Thus, in this arrangement there are~more tubes in a given size shell.If the flow is reversed (flowingtFig; :5up) as in a thermosiphon reboiler,t 1where it is desired to remove tubesfor vapor escape area, the samegiven. For more than one shell the 5. Divided-flow, one-tube holds true.results are tabulated.passIf a vapor belt is used, then the4. Longitudinal baffle The divided - flow, single - tubebetter.The number of tubes will.nozzle arrangement in Fig. 6b isSometimes when a two-shell cor­ pass correction factor is seldom usedrection (or greater) is required, it by itself, but is useful to'develop remain the same in a given shell, 'is possible to use a longitudinal baf­ correction factors for divided flowis.The two outlet nozzles will beno matter what the inlet nozzle sizefle. This flow arrangement is illus­ multitube pass and split flow.trated in Fig. 5.In the past there has been some smaller than the two inlet nozzlesFor four or more tube passes, a confusion about what is divided shown in Fig. 6a, when condensing2-4 correction factor is used. This floW and what is split flow. This downward, and they can be placedassumes two things:was resolved in the 1959 issue of closer to the tube sheets. This leaves1. There is a perfect seal be­ TEMA.3 TEMA refers to divided less space between the nozzle andtween the shell and the long baffle. flow as being shell type J. Divided the tube sheet and a more effective2. There is no conduction through flow is illustrated in Fig. 6.flow pattern is obt~ned.the long baffle. Frequently the question arises, A procedure has been developedThe effect 9f the second assump­ which of the nozzle arrangements for constructing a divided-flow, onetioncan be ignored with small-shell shown in Fig. 6 should be used? The tube - pass correction - factor chart, 8fluid-cooling ranges. If the cooling answer lies mainly in what type of based on a method used by K. A.range is large and there is a large impingement protection is used. The Gardner.9temperature driving force across the two common types are: To develop a chart, an expressionfor X is used in terms. of Rbaffle, then baffle conduction will 1. Removing tubes under the nozand¢:have to be cgnsidered. Whister 7 pre­ zle.sents an equation for F, for two¢(2R+l)/2 -shell-two tube passes incorporating 1 (1 -X) (¢(2R-l)/2 - 1)baffle conduction. X= +------------------- (8)(2 R + 1) ¢(2R+l)/2 (2 R - 1) 4>(2R-l)/2If the two assumptions are valid,and there is a long baffle and twoSolving for X:tube passes, no MTD correction fac­tor is necessary because the shell (¢(2R-l)/2 - ¢-l) (2 R - 1) + (¢(2R-l)/2 - 1) (2 R + 1) side and the tube side will be in X = ------------~---- (9)counterflow. (2R + 1) (2R 4>(2R-l)/2 - 1) --------------------------------------------Divided flowTo calculate F, a value of ¢ isassumed. Then X is calculated from~--------~-~----~ Equation 9. For a given value ofR, Equation 10 is used to find F:(1 - X)/(1 - RX) tl~ ---.. t 2 F = In (10) (R - 1) In 4>t 1 -: ! I I 'I' I I i:Xni , , ,(0) If it is desired to develop 'chartsfor niore than one shell pass, a formof Equation 6 can be used. Let nbe the number of shell passes in(b)series. Then:RX -1)n - 1~t2X -1= (11)Fig.6X -1)n - R.THE OIL AND GAS JOURN,AL • SEPTEMBER 14, 1964 111

....'.:,.0.1 0.2 0.4 0.5 0.6 On,e divided shell jOOne-tube pass 1$1rrrITTTTTITT Tj T lti9 t2T20.90.8O.0.70.1 0.2 0.30.4 0.5 0.6X=(t'2.- t,~(rl- 9o. o.B o.~1.0Xn and R are plotted to give themultishell-pass charts.There are two values of R thatrequire a different calculation procedure.These values are R = 0.5,or R = 1.0.When R = 0.5, Equation 9 breaksdown.Taking the limit of Equation9 as (R - 0.5) approaches 0, wehave:(cp - l)/cp + In cpX= (12) 2+Incp Equation 12 is used with Equation10 to give F.When R = 1.0, Equation 9 isstill used to calculate X, but Equation10 will break down when usedfor F. Taking the limit of Equation10:XF=(1 - X) In cp(13)Equation 9 is used with Equation13 to give F.\Vhen developing charts for more,than one shell, Equation 11 willalso break down. In that case, useEquation 6a to find the correct valueof X for plotting F.Fig. 7 is an F chart for singletubepass and one divided shell.Charts for the two, three, and fourdivided shells may be obtained fromthe author.Developing these charts by handis quite tedious, but they lend themselvesto computer application verywell.A Fortran program is availablefor calculating F values, upon requestfrom the author.6. Divided-flow, two-tubepassA trial-and-error solution for Fhas been presented 8 to use dividedflow,one-tube pass correction:1. Assume a value for the intermediatetemperature between tubepasses. A good starting point is:tb = t1 + 0.6 (t2 - t1) (14)2. Calculate the LMTD in thelower tube pass by using Fig. 7.3. Calculate the LMTD in theupper tu be pas s using Fig. -7.4. C a I cui ate the intermediatetemperature (tb)' Since temperaturerise in each pass is proportional toits LMTD:tb = t1 +(t2 - t1) (LMTD)L(LMTD)L + (LMTDh(15)5. If tb doesn't check the assumedvalue, start again using thelast value of tb'6. Calculate the corrected MTDby weighting the MTD's of theupper and lower tube pass, usingEquation 16:t2 - t1 Atm =(16) tb - t1 t2 - tb ._---+--­(LMTD)L (LMTDh 7. The correction factor is backedout by using Equation 3. Example:The shell fluid cools from 200 0to120 0 F., and the tube fluid heatsup from 80 0 to 120 0 F.112 THE OIL AND GAS JOURNAL • SEPTEMBER 14, 1964

, '. I0.1 0.3 0.4 0.5 0.6 0.7 0.8 1.0I.80.90.9 110­eUco ....... c:e'"5Q)10­10­e0.8" ~I­~IIu.:..TITI.. . ita~====~j:Jtl ,0.7 T2One divided··shellTwo-tube pass.0.1 0.2 0.31. From Equation 14, tb = 80 +0.6 (40) = 104.2. LMTD in the lower tube pass:Hot fluid Cold fluid Difference 200 - 104 96 120 - 80 = 40 LMTD = 64R =3.33 X = 0.2Lltm = LMTD X F = 64 X 0.92= 58.93. LMTD in the upper tube pass:Hot fluid Cold fluid Difference200 - 120 80120 - 104 16LMTD = 39.8R=5.0X=0.167Lltm = LMTD X F =0.882 = 35.14. From Equation 15,(40) (58.9)tb = 80 +58.9 + 35.139.8 )< = 105 0.5 0.6X=(t~- tJ(r,- t 1)5. Since the calculated value oftb did not match the value used, usethe calculated value and go back tothe start.The final value of tb is 105.5.6. With the final value of tb theMID's are:Lower Lltm = 57 and upper Lltm= 32.From Equation 16,~".' 40Lltm= = 44.525.5/57 + 14.5/327. F = 44.5/57.6 = 0.773A computer is almost a must tocalculate the many different pointsrequired for an F chart.A Fortran program can be writtenfor this type of flow. The biggestproblem in putting it on thecomputer is to calculate the correctionfactor for divided-flow, onetubepass. When doing it by hand,as in the above example, use an Fchart. Since the F chart' was developedby trial and error, we donot have an equation for it. Thereinlies the trouble.0.7 0.8 0.9 1.0Solving Equation .9 for cp0.80.7isn'tpossible by the usual means. Andwithout cp it is impossible to calculateF.In the computer program developedby the author, cp is found bytrial and' error. To converge on thevalue of cp, use. Equation 17 as afirst approximation:cp=2[ F/(R+O.5) (17)2 - 1.05 X (2 R + 1)After this value is calculated; usethe first derivative method of converg~nce.Fig. 8 is the F chart for one dividedshell and two tube passes.The dashed line extended across thechart shows when the outlet temperatureof the hot side is ~qual tothe outlet temperature of the coldside. In ,_ comparing this with thenormal one,-shell pass, the low pointis approximately 0.8 while the dividedshell is approximately 0.775.Three F charts for divided-flow,two-tube pass, from two to fourTHE OIL AND GAS JOURNAL •SEPTEMBER . 14, 1964113

.\'........ -.....,(" tSplit flow·t. 2 t TI r -,~--I tFig. 9- f2 T2shells in series, respectively, maybe obtained from' the author.Split flow. At times, when a highcorrection factor is desired, a long. baffle will give too much pressuredrop. In this case split flow can beused. This type of flow is shown inFig. 9.Some of the correction - factorcharts already developed can beused. By analyzing two dividedflow,one-tube passes in series, itcan be seen that it is equivalent toa split-flow-shell, two-tube passes.In like fashion, the four dividedshell,one-tube pass can be used fortwo split-flow shells in series withtwo tube passes each. ­If it is desired to use more splitflowshells, in series, use Equation11. Using the divided-flow correctionfactors in this manner assumeswe have the same perfect conditionslisted under longitudinal baffle.When you have split flow andfour or more tube passes, then thepreviously mentioned charts obtainablefrom' the author can be used,one for a single shell, and one fortwo shells.Author Dale L. Gulley lives at2714 S. 75th E. Ave., Tulsa, Okla.References1. Kern, D. Q., "Process Heat Transfer":McGraw-Hill, 1950.2. McAdams, "Heat Transmission": Thirdedition, McGraw-Hill, 1951.3. Standards of TEMA: Fourth edition,1959.4. Bowman, R. A., "Mean-TemperatureDifference Correction in Multipass Exchangers":Ind. and Engr. Chern., 28, 1936, pp.541-544.5. Gulley, D. L., "Use Computers to SelectExchangers": Petroleum Refiner, 39,1960, pp. 149-156.6. Fischer, F. K., "Mean - Temperature­Difference Correction in Multipass Exchangers":Ind. and Engr. Chern., 30, 1938, pp.377-383.7. Whister, A. M., "Correction for HeatConduction Through Longitudinal Baffle ofHeat Exchanger": Trans. ASME, 69, 1947,pp. 683-685.8. Gulley, D. L., "Make This CorrectionFactor Chart to· Find Divided Flow ExchangerMTD":Petro/Chem Engineer, July1962, pp. 143-145.. 9. Gardner, K. A., "Mean TemperatureDifference in Multipass Exchangers": Ind.and Engr. Chern., 33, 1941, pp. 1495-1500.AcknowledgmentTo Western Supply Co. for use of itselectronic computer and to Paul Buthodfor mathematical assistance.BOOKSGUIDE DU PETROLE ET DELA PETROCHIMIE (Petroleum andPetrochemical Guide). 1964 annualedition. Published by Editions O. Lesourd,252 Faubourg Saint-Honore,Paris 8, France. Price 100 francs.(Franco 104.20 francs.)The 34th annual edition of this outstandingdirectory came off the pressin Paris in May. It contains more than1,000 pages of names and addressesof oil, petrochemical, and service companies,plus phone numbers, personnel,company relationships, and otherdata.Here, briefly, is an outline of thisuseful book:Part 1. Oil. This listing gives detailson administrative and professionaloil groups, including memberships.It lists French oil companiesaccording to their functions, i.e., explorationand production; transportationand storage; engineering, refining,importing, and distribution; lube oils;and natural gas and products.Part 2. Petrochemicals and plasticmaterials. Here you will find informationon various French petrochemicalcompanies and organizations, includingnames and titles of companyofficials.Part 3. Oil and petrochemical market.This inCludes names of companiesand other information on firmsdoing business in the Common Marketand the OECD countries, as wellas North Africa :;tnd the Middle East.Part 4. . This includes a generalindex (Petrole-Chimie-Telephone) ofoil, chemical, and supply and equipmentcompanies. The listings includenames of administrators, directors,heads of departments, and technicalpersonnel. This section involves morethan 3,000 namees of oil personnel.In addition to the basic guide, youmay subscribe to the monthly supplementswhich keep the book up to date.Price of the monthly supplements is20 francs per. year.FEDERAL TAX TREATMENTOF INCOME FROM OIL ANDGAS. By Stephen L. McDonald.Brookings Institution, 1775 MassachusettsAve. N.W., Washington, D.C.163 pages. Paper $2, cloth $3.50.This is a summary of a symposiumon oil-depletion tax provisions heldby a group of professional economistsat Brooking Institution. Bulk of thevolume is a background paper preparedin advance by Stephen L. Mc­Donald of University of Texas. Theremainder is a summary of conferencediscussion.The volume is perhaps the best recentstatement of views of economistson percentage depletion and expensingof intangible drilling costs. It outlinesthe history of these provisionsand describes their application in variousproducing situations and their effectson the petroleum industry.It discusses the theories, justifications'and criticisms of these tax provisionsfrom the standpoint of differentialrisk, the wasting asset ofpetroleum resources, conservation, nationalsecurity, and capital-gains taxation.Conclusions are that reduction ofpercentage rates would tend to raiseprices in the long run, but this mightbe offset by competition, greater- efficiency,and reduction in rental bonuses.Increased revenue to the U.S.Treasury could vary from a very li~t1eto $2 billion annually' depending onhow the industry operated.The book notes that it considerseconomic aspects only, not other factorsin public policy.NEW GUIDE TO MORE EF­FECTIVE WRITING IN BUSINESSAND INDUSTRY. By Robert Gunning.Published by Industrial EducationInstitute, 221 Columbus Ave.,Boston 02116. 358 pp. $12.50.This book gives you ideas to improveeverything you write.These are techniques and formulasyou can app~y at once to every writingproblem.Step-by-step guidance is given fororganizing thoughts and putting themdown on paper, getting ideas across,writing to get the action you want,how to do a better job of writing,and how to write faster.Note: The Oil and Gas Journal maintainsa book department. Write to the Book Department,PO Box 1260, Tulsa, Okla. 74101,for copies of the book list. Often books reviewedhere may be purchased from thissource.114 THE OIL AND GAS JOURNAL • SEPTEMBER 14, 1964