aiche journal volume 23 issue 6 1977 [doi 10.1002%2faic.690230602] karl gardner; jerry taborek --...

7/30/2019 AIChE Journal Volume 23 Issue 6 1977 [Doi 10.1002%2Faic.690230602] Karl Gardner; Jerry Taborek -- Mean Tempe

1/10

Mean Temperature Difference:A ReappraisalThe derivation of the mean temperature difference in heat exchangers

is based on a number of assumptions or idealizations, the most importantones being that the heat transfer coefficient is constant throughout the ex-changer; that the temperature of either fluid is constant over any cross sec-tion of its nominal path, that is, complete mixing, no stratification or by-passing; and that an additional assumption for shell and tube exchangers,which has not always been fully recognized, is that within one baffle cross-ing the shell fluid temperature change is small with respect to its overallchange, (that is, number of baffles is large). In actual exchangers, any of theabove assumptions are frequently subject to various degrees of invalidation.This paper examines the effects of deviating from the first two assumptionsand presents a new solution to the third.

KARL GARDNERLiqui d Met al Engineer ing Center

Canaga Park. Californiaand

JERRY TABOREKHeot Tran sfer Research, Ins.Alharnbra, Cal i forn ia

SCOPEAs indicated by the title, this paper examines the mag-nitude of potential error in evaluation of the mean tem-

perature difference (MTD) in heat exchangers caused bydeparture from a number of simplifying assumptions nec-essary for a straightforward solution of the MTD equa-tions. Attention in this paper is confined to the three sig-nificant ones.

1. The assumption that the overall coefficient is con-stant throughout the exchanger is most frequently violated,especially for fluids with steep viscosity characteristics andlarge temperature changes. Existing literature on this sub-ject is scattered and confined to a few specific cases.

2. Bypassing in heat exchangers occurs if the flow splitsinto several branches owing to unequal flow resistances.

Each stream is exposed to different conditions of heattransfer, and the mixed exit temperature can be severelydistorted against idealized assumptions. While this prob-lem has been qualitatively realized for some ideal condi-tions, solutions for (or even awareness of) the potentiallydisastrous results in industrial applications have not beenproperly recognized.3. The consequences of the third assumption, that thenumber of baffles in a shell and tube exchanger is large,have never been analyzed before. The potential error isparticularly apparent in a one shell pass-one tube pass ex-changer, customarily considered as operating in counter-flow. However, when only a few baffles are used, the flowin each baffle space is in cross flow, resulting in a possiblesevere MTD penalty.

CONCLUSIONS A N D SIGNIFICANCEWhile simplifying assumptions for solutions of the mean

temperature difference (MTD) in heat exchangers are nec-essary for explicit solutions, most of the assumptions areviolated in practical applications. This paper focuses onthree assumptions considered the most frequent and mostserious ones. For the first two assumptions, namely, changeof the overall coefficient between hot and cold terminaltemperatures and effects of fluid bypassing and unequalheat transfer, a critical review of the conditions leading topotential errors is presented. Remedial actions are recom-mended on a systematic basis as good as the present stateof the art permits. A comprehensive review of the meagerand scattered literature sources on the subject is given.

The third assumption, that the number of cross flowbaffles in a shell and tube exchanger is Earge, is appraised

by comparison of the conventional MTD to that of an ex-changer synthesized from pure unmixed cross flow ele-ments, all other conventional assumptions remaining valid.A lower bound to the MTD is thus established (the upperbound being the conventional MTD) over the range fromno baffles at all to any arbitrary number. Interpolationformulas between the two are proposed. Although atten-tion is confined to single-pass, TEMA type E shells withone- or two-pass tubes, the general procedure requiredfor other configurations is apparent.As a final comment and advice for caution, the authorswould like to emphasize that use of complex, sophisticatedcomputer programs will resolve some of the problemstreated in this paper but will never become a substitutefor a sound engineering judgment and individual analysisof critical cases.

Underlying all heat exchanger calculations is the basicdQ =U(T- )dA (1)

This equation is strictly valid for pointwise values of thecoefficient U with respect to heat transfer surface dA andAlChE Journal (Vol. 23, No . 6) November , 1977 Page 777

any arbitrary temperature-heat release relationship, ex-pressed as d Q vs. temperature change. In this sense,actually all the terms of Equation (1) can be interrelatedbut must be then solved only on a pointwise (or step-wise) basis. Rearranging Equation (1) and integrating,we get

equation


2/10

dQ1U ( T - )If we calculate stepwise U and corresponding AT and plotl / [ U ( T - t ) ] vs. Q, the area under such curve willrepresent the required surface A. A similar procedurewas demonstrated in a classical paper by Colburn ( 1933).The pointwise calculations can get quite involved andare, for all but simple problems, practically solvable onlyby sophisticated computer programs, and even in suchcase the techniques of doing so are just emerging.For many flow arrangements and heat transfer proc-esses, especially in no phase change, the integration ofEquation (1) can be performed under a number of re-strictive assumptions. These are, in approximately de-creasing order of importance:1. The overall heat transfer coefficient U is constantthroughout the exchanger.2. For pure counterflow or cocurrent flow, the tem-perature of either fluid is uniform over any cross sectionof its path. For multishell pass and multitube pass ar-rangements, this applies within any shell or tube pass.In other words, the thermal history of any particle ineither stream is identical to any other particle in thatstream, that is, complete mixing, no stratification or by-passing. For cross flow arrangements, each fluid is con-sidered either completely mixed or unmixed normal tothe flow, and mixed or unmixed between passes.3 . For baffled shell and tube exchangers, the heat trans-ferred in each baffle compartment is small compared tothe overall; that is, the number of baffles is large.4. The flow rate and specific heat of each fluid is con-stant.5. There is no change of phase of either fluid in onlya part of the exchanger. If condensation or boiling occurs,it must do so uniformly over the entire surface and insuch a way that equal quantities of heat are exchangedfor equal changes of the fluid temperature. This producesa linear plot of heat exchanged vs. temperature.6. There is equal heat transfer surface in each tube orshell pass.7. Heat losses to surroundings are negligible.

If all the above restrictive assumptions are consideredvalid, integration of Equation ( 2 ) for counterflow (orcocurrent flow) yields the elementary solution for themean temperature difference At, as Atlog,which becomesa function of the hot and cold terminal temperaturedifferences Dh and D,:

For various combinations of counterflow and cocurrentflow such as occur in multitube pass exchangers, A t , isobtained by means of a correction factor F :

At, =F A t l o g (4)The F factor formulation will be used throughout this

paper as being the most illustrative one, representingdirectly the penalty against the best possible case ofcounterflow. HecalcuIations to the sometimes preferredpresentation as number of transfer units NTU are easilypossible through the following relationships:

1(NTU).8F =where

However, in actual exchangers several of the aboveassumptions are frequently violated to some extent, re-sulting in various degrees of potential inaccuracy. Thefirst tiiree asumptions are considered not only the mostimportant ones but also the most misunderstood ones,being either not at all or only fragmentarily documented.Consequently, the remaining content of this paper isdevoted to review of the problem regarding deviationsfrom assumptions 1 and 2 and presents a solution toassumption 3, which has, to the best knowledge of theauthors, never been considered before.EFFECTS OF VARIATION OF HEAT TRANSFERCOEFr lC lENT U

Only in rare cases will the assumption of constant Ube strictly valid. A commonly used simple equation formfor heat transfer coefficient suggests that for turbulentfiow approximately the following relationship holds:

h k0.6c0.4,,-0.4 ( 7 )For liquids, the k and c variations with temperatureroughly oflset their eflects (except in extreme cases ofapproaching critical temperatuie) so that h remainsmainly dependent on p. Thus, for liquids with steep pvs. 2 characteristic, 12 can change significantly betweeninlet and outlet, For gases, all properties involved gen-erally increase with temperature, but only mildly so,and thus gas coefficients are relatively stable.

Analytical solutions to several flow arrangements existin the literature, in all cases under assumption of linearvariation of U between inlet and outlet. These are re-viewed in the following.Countercurrent Flow Exchangers

For countcrcurrent flow, an analytical soIution wasderived by Coiburn (1933) as the logarithmic mean ofthe products U J , D ,and U c D J l , nder the assumption that Uvaries linearly between the hot and cold end of the ex-changer:

A In (U{ ,D, )/ (U,D,,) U h D c + L c D h 2UhDc -k UcDhA- ( u h D c- UCDh)- _

(8)This compares to the constant Uavg coefficient formulationcombined with Atlog as follows:

Uavgcan be interpreted as :1. Arithmetic average between Uh and uc (two cal-culations for U ) .2 . U calculated from shell stream and tube stream hvalues, each referred to the arithmetic average tempera-ture of that stream (single calculation for U ) .3. Colburn (1933) also derived a n alternate methodto Equation (8) by which a caloric temperature is de-fined so that if used for evaluation of UavPand used inEquation ( 9 ) , a coriect result will be obtained.The question obviously arises how much error is com-mitted by using Equation (9 ) against Col1,urns Equation( 8 ) . Expressing both equations in terms of D J J D , = D= (1 - P ) / ( 1 - P R ) and ( U h / U c ) = U nd takingthe ratio of Equation (8 ) to Equation i s ) , we get anexpression for the error ratio or correction factor F:

ZDU( 1/D - /U) In D(1+U)D - 1) In (U/D)= (10)

Page 778 November, 1977 AlChE J ourna l (Vol. 23, No. 61


3/10

This is shown graphically in Figure 1.Values of F


4/10

F

0 . 6 0 t -10.50t j i0 1 2 3 4 5

"h/'cFig. I . Error ratio or correct ion factor F evaluated from Equation

(10).sential for manual calculations only. The almost universaluse of computer programs for design of all standardtypes of exchangers on high-speed computers makes i tnow possible to employ stepwise calculations or numericalanalysis methods, even though the computational com-plexity is considerable.EFFECTS OF BYPASSING

Another important but frequently violated assumptionfor At,n validity is the condition that the thermal historyof all fluid particles is identical (assumption 2 ) . In prin-ciple, we are concerned here with a situation where,for reasons of unequal flow resistances, two or morestreams develop within a heat exchanger and becauseof their flow path are not heated (or cooled) to thesame degree and, in the extreme case, exchange heatonly through mixing, which can be continuous or sudden.

For baffled shell and tube exchangers, it was shownby Tinker (1958) and later by Palen and Taborek (1969)that the shell side flow is not uniform but rather com-

I Hot Stream (Isothermal) 1

- lowPathFig. 2. Temperature change within a f low element.

posed of the main cross flow stream and usually threeidentifiable bypass streams: between baffle and shell(E stream), between baffle holes and tubes (A stream),and between tube bundle and shell wall ( C stream).The magnitude of these streams can vary widely de-pending on the geometries involved, the driving pressuredifferential between baffles, and, as the resistances ofthe individual streams are function of Re, the flow regime.In other exchanger types, a somewhat different caseof bypassing was experimentally demonstrated by Weier-man et al. (1975) in flow across in-line finned tube banks,producing a dramatic error in At,. Still another type ofbypassing or maldistribution can occur in tube side flowfor laminar cooling where part of the tubes can freeze up,as shown by Mueller (1974) and further confirmed byKao (1975).The bypassing problem is schematically illustrated inFigure 2. Stream 1 is fully heat transfer effective, suchas the cross flow stream in shell and tube exchangers.Stream 2 is partially effective, such as the C stream,while stream 3 is completely ineffective. A mixed outlettemperature is then obtained, which shows an apparenttemperature difference driving force which is much largerthan the one effectively present. Thus, uncorrected by-passing will $ways result in underdesign of exchangersor, if applied for interpretation of experimental data,will result in underestimation of the true film coefficientfor the fully effective stream. A secondary, but not lessimportant, effect is, of course, the decrease of velocityof the effective stream, resulting in a reduced film co-efficient.For countercurrent shell and tube exchangers, theeifects of bypassing on At, were first recognized by\Vhistler (1947) and a simplified analysis presented ona baffle-by-baffle basis. The difficulty at that time wasthe estimation of the bypass streams magnitude, whichbecame more realistically possible thi ough the work ofTinker (1958), later refined by Palen and Taborek (1969).Short (1960) also recognized the effects of bypassingAt f l t but did not develop a practically applicable method.Palen and Taborek (1969) developed a qualitative em-pirical method for shell and tube exchangers in the formof A t f l Lpenalty factor which is function of the E streamrelative magnitude and the ratio of the terminal temper-ature approach to the shell side temperature rise. Theeffect of Reynolds number is also observed to be of greatimportance, as it will affect the degree of mixing.Fisoher and Parker (1969) developed an analyticalmethad for the effect of bypassing on At, in a 1-2 ex-changer. Here, the bypass stream is considered as thesum of the A, C, and E streams and as being thermallyfully ineffective. Complete mixing is assumed to occurat each baffle crossing. The analysis is not presented inthe usual dimensionless terms, but a number of graphsfor the correction factor F as function of P for discretecombinations of bypass fractions and number of bafflescrossed is shown. While the simplified assumptions wereprobably necessary in order to permit a manageable solu-tion, their validity can be somewhat questioned. Forexample, the A stream (tube-to-baffle hole) is relativelyeffective and mixes immediately with the cross flow stream;the C stream is at least partially effective, or can bemade so by use of sealing strips, and also mixes continu-ously with the cross flow stream. The most ineffectiveE stream (baffle to shell) is considered independent of Re,which is contrary to what was observed from experi-mental data, such as shown by Bell (1963) and Palenand Taborek (1969).A generalized analytical solution for an elementary

Page 780 November , 1977 AlChE Journal (Vol. 23, No . 6)


5/10

A 1-1 counterf lowI I

t,

C1 1

C 1-2, countercurrent return T I a t stat ionary head end

LI - - - - I

E 1-2, cocurrent return T I a t stat ionary head endFig. 3. Synthesis o f simulated 1-1 an d 1-2 shell and tube heat exchangers from identical basic cross flow elements.system of two streams with arbitrary mixing was pre-sented by Bell and Kegler (1975) and compared to databy Weierman et al. (1975) . It is suggested that this typeof analysis will be necessary for realistic solution of thebypass problem in tube banks and shell and tube ex-changers as well. However, a much more systematicset of data than presently available would have to begenerated, including variation of all the basic parameters.

In summary, there is no generalized method availablein published literature which would dependably accountfor the effects of bypassing in tube banks or shell andtube exchangers. In the latter, the effective cross flowstream for well-designed exchangers varies from about70% in turbulent flow to as low as 20% in deep laminarflow. Thus, the real problem is the combination of effectson both temperature difference as well as the true effec-tive film coefficient. Similar to the conclusion of the pre-vious chapter, it appears that a stepwise calculationprocedure on fast digital computers may be the onlyright answer, assuming the availability of methods, suchas shown by Palen and Taborek (1969), to predict theAlChE Journal (Vol. 23, No. 6)

magnitude of the bypass streams and their respectiveheat transfer effectiveness.EFFECTS OF SMALL NUMBERS OF BAFFLES

The following discussion and derivation is basically aparametric study in the sense that all other conventionalassumptions are considered valid while only one, in thiscase assumption 3, is changed and its effects observed bycomparison to those of the original. An assumption cus-tomarily made in deriving a mean temperature difference(MTD) correction factor F for shell and tube heat ex-changers is that the temperature of the shell fluid maybe considered uniform over any cross section normal tothe shell axis. This assumption is a good approximationto the physical situation only to the extent that the tem-perature change in a single compartment between adja-cent baffles is a small fraction of the total temperaturechange of the shell fluid through the exchanger, Withsegment-cut baffles so disposed as to encourage cross flowof the shell stream over all tube passes in alternating

November, 1977 Page 781


6/10

directims in successive baffle compartments, validationof the assumption requires a large number of baffles pershell in either single shell pass, single tube pass (1-l) ,or single shell pass, two tube pass (1-2) exchangers. Lim-itations of time and space confine attention here to ex-changers of the types described above.At least four questions are raised by the precedingparagraphs:1.How large is a large number of baffles in that context?

2 . What MI'D correction is required for a lesser num-ber?3 . Does such correction remain independent of shellfluid flow direction in 1-2 exchangers as it does whenthe number of baffles is large?4. On what basis can the foregoing questions be an-swered quantitatively and conservatively?Basis for A n o l y s i s

Taking the last question first, we propose to considera 1-1 exchanger with (X-1) cross flow baffles as anarray of X geometrically identical pure crossflow ex-changers arranged in series counterflow, as shown inFigure 3a. (The cocurrent series array is of little interestin practice.) A 1-2 exchanger, by similar reasoning, isconsidered as a much more complex array of 2X identicalpure cross flow elements (for X-1 baffles), as shown inFigures 3b, c, d, and e . The four alternative arrays coverthe cases where the shell stream at the tube return endof the shell crosses the tube bundle counter to the tubeflow direction in the return head (or U tubes), Figures3b and c, or in the same direction, Figures 3d and e . Ineither situation, a distinction must be maintained be-tween even and odd numbers of shell stream cross flowpasses, at least until X increases to such a value that theneed for such distinction clearly no longer exists, Atsuch a point (or number of cross flow passes), largenumber will have been defined in response to Question 1above.In all cases, the entire shell stream is first assumed topass in uniformly distributed cross flow through eachcomponent element of the array; that is, there are noaxial components of the shell stream past effective surfaceand no bypassing to complicate what is intended to bea lower bound solution for the MTD correction factor.It is necessary also to specify the type of cross flowinvolved within each element and the mixing effectsof passage from one element to the next. It is assumedthat both streams are unmixed within an element (as

appropriate to widely spaced baffles) and that the num-ber of tube layers crossed per element is sufficiently high(say >6) to justify use of pure unmixed cross flow ther-mal relationships for each element. It is further assumedthat both streams are completely mixed between ele-ments. (Actually, the tube stream is unmixed betweenelements except in the return head of 1-2 exchangers,and the shell stream between elements alternates frommixed, as it turns and passes through a baffle opening, tounmixed, as it crosses from one tube pass element to theother in 1-2 exchangers),Stevens et al. (1957) obtained results for two andthree cross flow exchangers in series-counterfIow, unmixedbetween elements on one stream and mixed on the other.Comparison of these results with those for both streamsmixed between elements and consideration of a discus-sion by Landis support the authors' conclusion that al-though some slight sacrifice of conservatism is involved,the foregoing paragraphs constitute a suitable quantita-tive and conservative basis for analysis in response toQuestion 4 above.1 - 1 Exch ongers with Cross Flow Boffles

For an array of X identical heat exchangers connectedin series-counterflow on both streams, Figure 3a , Bowman(1936) has shown that the MTD correction factor Ffor the array as a whole is identical to that F' of anysingle element of the array and that

P- PRp' = I-,1/x R-1 [ X - ( X - ) ]l - -R( l - p )1 - RP- PR.y '= I-,1/x R-1 [ X - ( X - ) ]l - -R( l - p )1 - R

where P and P' apply, respectively, to the array as awhole and to any single element.An explicit form F' equation for unmixed-unmixed crossflow of considerable accuracy is available in an ingeniousgeneral approximation by Roetzel and Nicole (1975).Figure 4, which has a much denser population of curvesof constant R than any previously published for unmixedcross flow, is based on computer solution OF their generalequation for MTD correction factors, using the appropriateconstants provided in their paper. The accuracy is verygood for F >0.7 and R >0.4. Figure 5 shows, for R =1, the variation of F , with P and X as calculated fromEquation (1 2 ) and Figure 4.

P' = ThermalEfficiencyFig. 4. MTD correct ion factors for cross f l o w hea t exchangers w i th

bo th s t reams unmixed ; ca l cu la t ions .

Page 782 November , 1977 AlChE Journal (Vol. 23, No . 6)


7/10

Equation (12) could, of course, be solved for P in-stead of P. If this is done, it is then possible, with Figure4 as a basis, to construct separate charts of F , vs. P and Rfor any desired values of X . This is not an attractive al-ternative to the use of one chart and one simple equation.Conclusions on 1-1 Cross Flow B af f led Cou nter f low Exchangers

The unattractive alternative referred to above was pur-sued to a point where it became obvious that there is nocut08 point defined solely i n terms of X beyond whichthe customary value, F =1, for counterflow may be usedwith impunity. As in all MTD correction factor charts,there are areas where F , is satisfactorily close to unity,but they remain strong functions of P and R, as well as X.The following very simple procedure, however, definesthe minimum number of bundle crossings Xmin requiredto provide an MTD correction factor F, greater than0.98. If Figure 4, or an equivalent table or equation,is entered with P instead of P and the resulting F des-ignated as F(P, R ), henand F , may be taken as 1.0 without further ado if theactual X equals or exceeds this number. This relation isempirical rather than derived and, for the moment atleast, is not thoroughly understood. I t does, nevertheless,provide the answer to Question 1 for 1-1 counteiflowexchangers.If X is less than the value obtained from Equation(13), the appropriate procedure to answer Question 2has already been furnished in the paragraphs immediatelypreceding these conclusions. With the use of Equation(12) and Figure 4, the procedure is not onerous forhand calculations and not at all so for computerized so-lutions.

Some thought must also be given to the fact thataxial flow of the shell stream has been assumed neg-ligible in the derivation for F,. Without going as far asto recommend it, the authors suggest that the followingequation provides a reasonable approximation to thetrue situation:

Xmin=1+20[1 - (P,R ) ] (13)

where N is the total number of tubes in the tube bundle,N o is the number of tubes in one baffle opening and Fis the MTD correction factor appropriate to these quan-tities in combination with P, R, and X . This amounts toa primitive interpolation between the upper bound solu-tion (F=1 ) and the lower bound solution ( F =F , ) ,1-2 Exchangers with Cross Flow Baff lesN o such simple relation as Equation (12) is to befound for 1-2 exchangers synthesized from cross flowelements as shown in Figures 3b, c, d, and e, even whenboth streams are considered mixed between adjacent ele-ments. Nevertheless, MTD correction factors for the ex-changer as a whole can be determined by use of therelations utilized previously by Gardner (1941, 1942,1945): . f 1 - p 1

I n \ 1 - P R ) PF , = .-.-, (15)( R - l ) B R-*l ( 1 - P ) B=2BX= - U A

UA BB=--- -wc 2xw c

P = (P , R , X )AlChE Journal (Vol. 23, No. 6)

PFig. 5. MTD correction factors for simulated 1-1 exchangers withR = 1, showing effects of number of shell crossflow passes (circlednumbers).

P = (B, R ) (19)The last of these is simply a graph, table, or equationproviding values of P for an unmixed cross flow exchangerwhen B and R are given. Table 1 shown in the paperby Stevens et al. (1957) is used for this purpose here,supplemented on occasion by the Roetzel and Nicole(1915) reference.

Once an explicit equation for the relationship indicatedas Equation (18) is provided, the procedure is simple,working the equations in reverse order. For numericalevaluation of F,, the details of the procedure are givenby Gardner (1942).The deveiopment of a suitable form of the relationshipindicated by Equation (18), while straightforward, isnot at all simple. With reference to any one of Figures3b , c, d, or e, it involves building up the exchangerfrom left to right block by block, each block consistingof two vertically stacked elements constituting one crossflow pass of the shell stream. Regardless of the stageof completion of the built-up exchanger under considera-tion, it may be viewed at that stage as the combinationof one accumulation of blocks to the left characterizedby P,-l and a single b!ock to the right consisting oftwo elements, each characterized by P. The eight tem-peratures, T I ,Ta, b, Tz, and tl , ta, tb, and tz , when com-bined in appropriate temperature difference ratios todefine P, PXpI, , and R, are sufficient to define P also interms of R and X , but only by iteration from x =2 tox =x.The net results of the foregoing procedures, which aretoo tedious and space consuming to detail here, are twosets of recurrence formulas, one for counterflow in theleft-hand (re turn) block, the other for concurrent flow.Each set requires the alternate use of equations for P,for even and odd x until the desired value X is reached.Fortunately, these equations are the same whether theshell fluid enters at the stationary head end or the returnend of the shell. Thus, no distinction need be made be-tween the MTD correction factors for Figures 3b and c ,nor between those of Figures 3d and e . The answer toQuestion 3, posed earlier, is that other things (includingreturn flow orientation and X ) being equal, the correc-tion factor is independent of shell flow direction in 1-2exchangers.Counter f low Return

November, 1977 Page 783


8/10

PFig. 6. MTD correct ion factors for s imulated 1-2 exchangers withR 1 , showing effects of number of shell cross flow passes ondorientat ion of tube return shell f low.

where

terest, that is, Fx >0.75, and for R = 1 only. This isdone in Figure 6 over the range 0.7 >P >0.3.The heavy central line in the family of curves shownis the customary line for 1-2 exchangers with R =1 anda large number of bafRes. The uppermost (so lid) line isthat for a single crossing of a two-pass bundle with coun-terflow return, a normal arrangement, but not often usedin cylindrical shells. The lowermost (dashed) curve isalso for one shell crossing but with concurrent returnwhich, by itself, is never used. Reading inward from eitherof these extremes, it is seen that increasing the numberof shell stream crossflow passes by unit increments suc-cessively collapses the two curves with the same valuesof X closer to the heavy line for large X and that, atconstant P , they are very nearly equally spaced aboveand below it for X >1. This variance, as a percentageof the conventional F , for 1-2 exchangers, is summarizedin Table 1 for a minimum acceptable F of 0.75 for R =1 and for both 0.75 and 0.85 for R =0.2 and 5. (F, =

(21)for even x

1-pz-1 ,i1- 'PZ-lRY, =P' (for odd x >1ii (24)where (1- 'R) (1- x-1) +P'Pz-1R1- 'R[P' +P,-1[1- P'(R +1)11, =P'

0.75 at R =1 and F , =0.85 at 0.2 or 5 are compatiblewith an acceptability criterion of Popt =0.9Pma,.)C o n c l u s i o n s o n Cross Flow B a f f l e d 1-2 Ex c hangers

The equations given above generate both an upperbound and a lower bound solution for F,, as shown in

C o c u r r e n t R e t u r n(25)I =P'[2 - P'(R +l ) ]

(26)for even x

1 YxR P'RPz =- - 1- [P' +Pz-i[l- P' (R +1111 )

where P'( 1- 'R)Y, = (1- 'R[P'+ P,-1[1 - ' ( R +111 I >orP , =P'

+[l - ' ( R +l)]{P' +P,-1[1 - ' (R +1)1>

(1- 'P,-lR) for odd x >1 (28)N u m e r i c a l R es u l ts f o r 1-2 Ex c hongers

Either of the foregoing two sets of equations is solvediteratively for P by starting with Equation (20 ), forexample, to determine P1, sing this result in Equation(21) with x = 2 to find P 2 , using this result in Equation( 2 3 ) with x = 3 to find P3, and so on, alternating be-tween Equations (21) and (2 3) until the required valueof X is reached. This, then, gives the value of P requiredin the procedure, that is, the numerical solution of Equa-tion (18) for counterflow return. For cocurrent return,substitute Equations (25), (26), and (28) for ( 20) , (21),and (23).Values of F, have been determined as a function of Pat R = 0.2, 1, and 5, X = 1-5 inclusive and for bothcounterflow and cocurrent return blocks. For clarity ofpresentation, these are displayed on magnified scale forthat region of a correction factor chart normally of in-Page 784 November, 1977

Figure 6 and Table 1. The higher value occurs only fortwo combinations of X and return flow orientation-counter-flow return and odd X or cocurrent return and even X.The lower value is obtained with counterflow return andeven X or with concurrent return and odd X . This some-what complicates the reply to the questions initially raised.

The authors' answer to Question 1 is that the largenumber sought, at and above which the conventional 1-2TABLE.PERCENTA RIA N CEROM CONVENTIONAL,

R 1 o 0.2 and 5.0?; 0.75 0.75 0.853 a4.9 e6.9 k2 .64 ~ 2 . 9 74.5 ~ 1 . 65 e1.9 k3.1 kO.9

Upper sign for counterflow return.Lower sign for cocurrent return.

AlChE J o u r n a l (Vol. 23, No. 6)


9/10

correction factors may be used with impunity, is X = 5(4 baffles). The basis, as before, is that a discrepancy of-2 % from the customary F , is , for practical purposes,negligible.Question 2, what to do when X


10/10

a, bcfhmaxopt12

Ccc=intermediate points along stream=cold end of counterflow exchanger=cocurrent tube pass=counterHow tube pass=ho t end of counterflow exchanger=maximum obtainable=economic optimum=inlet of a stream=outlet of a stream

LITERATURE CITEDBell, K. J., Bull. No. 5, Univ. Del. Eng. Exp. Station (1963)., and W. H. Kegler, Analysis of Bypass Flow Effects inTube Banks and Heat Exchangers, AIChE No 7, AIChE-ASME 15th Natl. Heat Tiaiisfer Conf., San Francisco, Calif.(1975).Bowman, R. A,, Mean Temperature Difference Correction inMultipass Exchangers, Ind. Eng. Chem., 28, No. 5, 541( 1936).,A. C. Mueller, and W. M . Nagle, Mean TemperatureDifference in Design, Trans. ASME, 283 (1940).Cagla an, A. N., and P. Buthod, Factors Correct Air-Cooleran lS& T Exchanger LMTD, Th e Oil and Gas Journal, 91

( 1976).Colburn, A . P., Mean Temperature Difference and Heat Trans-fe r Coefficient in Liquid Heat Exchangers, Ind. Eng. Chem. ,25 , No. 8, 873 ( 1933).Fisher, J. , and R. 0. Paiker, New Ideas on Heat ExchangerDesign,Hydrocarbon Processing, 147 ( 1969).Cardner, K. A., Variable Heat Transfer Rate Correction inMultipass Exchangers, Shell-Side Film Controlling, Trans.ASME, 31 (1945)., Mean Temperature Difference in Unbalanced-PassExchangers,Ind. Eng. Chem ., 33, No . 10, 1215 (1941)., Mean Temperature DXference in an Array of Identi-cal Exchangers, ibid., 34, No . 9, 1083 ( 1942).

Kao, S., Analysis of Multipass Heat Exchangers with VariablePropeities and Transfer Hate, 1.Heat Transfer, 509 (1975).Mueller, A. C. , Criteria for Maldistribution in Viscous FlowCoolers, Trans. of the 5th Intl. Heat Transfer Conf., PaperHE1-4, V, 170, Tokyo, Japan (1974).Palen, J. W . and J. Taborek, Solution of Shell Side Flow Pres-sure Drop and Heat Transfer by Stream Analysis Method,Che m. Eng. Piogr. Symposium Ser., No. 29, 63, 53 (1969).Ramalho, R. S., and F. M. Tiller, Improved Design Methodfor Multipass Exchangers, Chem. Eng. , 87 (1965).Roetzel, Wilfried, Heat Exchanger Design with Variable Trans-fer Coefficients for Crossflow and Mixed Flow Arrangements,Intern. J. Heat Mass Transfer, 17, 1037 ( 1974)., and F. J. L. Nicole, Mean Temperature Difference forHeat Exchanger Design-A General Approximate ExplicitEquation, J. Heat Transfer, 5 ( 1975).Short, B. E., Shell-and-Tube Heat Exchangers: Effect of By-Pass and Clearance Streams on the Main Stream Tempera-ture, ASME Paper No. 60-HT-16, ASME-AIChE HeatTransfer Conf., Buffalo, N.Y. (1960).Sieder, E. N., and G. E. Tate, Heat Transfer and PressureDrop of Liquids in Tubes, Ind. Eng. C hem., 28, No. 12,1426 (1936j. .,Stevens, R. A., J. Fernandez and J. R. Woolf, Mean Tem-uerature Difference in One. Two and Three-Pass Crossflowheat Exchangers, Trans. ASME, 79, 287 ( 1937).Tinker, T., Shell-Side Characteristics of Shell-and-Tube HeatExchangers-A Simplified Rating System for CommercialHeat Exchangers, ibid. , 80, 36 ( 1958).Weierman, C., J. Taborek, and W. J. Marner, Comparison ofthe Performance of Inline and Staggered Banks of Tubeswith Segmented Fins, paper presented at the AIChE-ASME15th Natl. Heat Transfer Conf., San Francisco, Calif. (1975).Whistler, A. M., Effect of Leakage Around Cross-Baffles in aHeat Exchanger, Petrol. Refiner, 26, No. 10 , 114 (1947).Manuscript rece ived September 14 , 1976; revision received May 31,and accepted June 6, 1977.

Dispersion in Layered Porous MediaExperimental data on directional dispersion coefficients in transversely

isotropic media, a special case of anisotropic media, were determined inmodels consisting of al ternat ing layers of sintered glass and unconsolidatedglass beads. Values of appropriate parameters to relate the velocity to thedirectional dispersion coefficient were determined for a variety of perme-abilities and angles. Sensitivity of these parameters with respect to anglewas determined, and results are presented which will permit estimate ofwhen this effect is significant.

SCOPE

M. B. MORANVILLED. P. KESSLER

andR. A. GREENKORN

School of Chemical EngineeringPurdue UniversityWest Lafoyette, Indiana 47907

Porous media occur in many areas of chemical engi-neering, petroleum engineering, and hydrology. The usualdifficulties with charac terization of porous media occurbecause of their nonuniformity, heterogeneity, an d anisot-ropy. Nonuniformity and heterogeneity are concernedwith spatial variations of the media; anisotropy is con-cerned with variation with angle. To treat a general anis-otropic medium is difficult; therefore, some special case ofanisotropy is usually treated.In this paper, we trea t dispersion in transversely iso-tropic porous media, which are a special subset of aniso-tropic porous media. Transversely isotropic media charac-Texas.M. B. Moranville is with the Shell Development Company, Houston,

terize layered systems. Most petroleum and water reser-voirs are layered. Many packed beds are packed in layers.The objective was to characterize the dispersion (mixing)of a fluid as it flows through such a medium as it relatesto the angle the velocity makes with axis of isotropy andth e velocity of flow. To take a single example, such knowl-edge is extremely impor tant in the production of petroleumfrom underground deposits when this production is beingaccomnlished by miscible displacement, that is, displace-ment of petroleum from the rock formation using a misci-ble fluid. The significant theory leading up to the treat-ment in this article is reviewed, and here we give resultswhich permit the directional dispersion to be related bothto velocity and to angle.

Page 786 November, 1977 AlChE Journal (Vol. 23, No. 6 )

aiche journal volume 23 issue 6 1977 [doi 10.1002%2faic.690230602] karl gardner; jerry taborek --...

Documents