1.3. The Mean Temperature Difference 1.3.1. The Lgarithmic Mean Temperature Difference 1. Basic Assumptins. In the previus sectin, we bserved that the design equatin culd be slved much easier if we culd define a "Mean Temperature Difference" (MTD such that: A * Q t = (1.25 * U ( MTD In rder t d s, we need t make sme assumptins cncerning the heat transfer prcess. One set f assumptins that is reasnably valid fr a wide range f cases and leads t a very useful result is the fllwing: 1. All elements f a given stream have the same thermal histry. 2. The heat exchanger is at a steady state. 3. Each stream has a cnstant specific heat. 4. The verall heat transfer cefficient is cnstant. 5. There are n heat lsses frm the exchanger. 6. There is n lngitudinal heat transfer within a given stream. 7. The flw is either ccurrent r cunter-current. The first assumptin is wrthy f sme nte because it is ften mitted r stated in a less definitive way. It simply means that all elements f a given stream that enter an exchanger fllw paths thrugh the exchanger that have the same heat transfer characteristics and have the same expsure t heat transfer surface. In fact, in mst heat exchangers, there are sme flw paths that have less flw resistance than thers and als present less heat transfer surface t the fluid. Then the fluid preferentially fllws these paths and underges less heat transfer. Usually the differences are small and d nt cause serius errr, but ccasinally the imbalance is s great that the exchanger is very seriusly deficient. Detailed analysis f the prblem is t cmplex t treat there, but the designer learns t recgnize ptentially trublesme cnfiguratins and avid them. The secnd, third, furth, fifth, and sixth assumptins are all straight- frward and are cmmnly satisfied in practice. It shuld be nted that an isthermal phase transitin (biling r cndensing a pure cmpnent at cnstant pressure crrespnds t an infinite specific heat, which in turn satisfies the third assumptin very well. The seventh assumptin requires sme illustratin in terms f a cmmn and simple heat exchanger cnfiguratin, the duble pipe exchanger. 2. The Duble Pipe Heat Exchanger. A duble pipe heat exchanger essentially cnsists f ne pipe cncentrically lcated inside a secnd, larger ne, as shwn in Fig. 1.20. One fluid flws in the annulus between the inner and uter pipes and the ther in the inner pipe. In Fig. 1.20, the tw fluids are shwn as entering at the same end, flwing in the same directin, and leaving at the ther end; this cnfiguratin is called ccurrent. In Fig. 1.21, pssible temperature prfiles are drawn fr the temperatures f the 25
fluids in this exchanger. (We have shwn the ht fluid in the annulus and the cld fluid in the inner pipe, but the reverse situatin is equally pssible. Ntice that the utlet temperatures can nly apprach equilibrium with ne anther, sharply limiting the pssible temperature change. If we had pltted the lcal temperatures vs. quantity f heat transferred, we wuld get straight lines, a cnsequence f the assumptin that the specific heats are cnstant. A cuntercurrent heat exchanger is diagrammed in Fig.1.22 and a pssible set f temperature prfiles as a functin f length is shwn in Fig. 1.23. Als bserve that the maximum temperature change is limited by ne f the utlet temperatures equilibrating with the inlet temperature f the ther stream, giving a basically mre efficient heat exchanger fr therwise identical inlet cnditins cmpared t the ccurrent arrangement. Fr this reasn, the designer will almst always chse a cuntercurrent flw arrangement where pssible. If ne stream is isthermal, the tw cases are equivalent and the chice f ccurrent r cuntercurrent flw is immaterial, at least n grunds f temperature prfiles. 3. The Lgarithmic Mean Temperature Difference. The analytical evaluatin f the design integral Eq. 1.23 can be carried ut fr bth ccurrent and cuntercurrent flw if the basic assumptins are valid. The details f the derivatin are nt relevant here and can be fund in a number f standard textbks (e.g. Ref. 6. Fr the ccurrent exchanger, the result is: MTD = ti ( T t ti ln ( T t (1.26 and fr the cuntercurrent case, MTD = t ( T ti t ln ( T t i (1.27 Fr the special case that (T i t = (T t i, eqn. (1.27 reduces t: MTD = ( T t = ( T t i i (1.28 The definitins f MTD's given in Eqns. (1.26 and (1.27 are the lgarithmic means f the terminal temperature differences in each case. Because f its widespread imprtance in heat exchanger design, Eq. (1.27 is cmmnly referred t as "the lgarithmic mean temperature difference," abbreviated as LMTD. 26
1.3.2. Cnfiguratin Crrectin Factrs n the LMTD 1. Multiple Tube Side Passes. One f the assumptins f the LMTD derivatin was that the flw was either cmpletely ccurrent r cmpletely cuntercurrent. Fr a variety f reasns, mixed, reversed r crssflw exchanger cnfiguratins may be preferred. A cmmn case is shwn in Fig. 1.24 - a neshell-pass, tw-tube-pass design (a 1-2 exchanger, fr shrt: Nte that n the first tube side pass, the tube fluid is in cuntercurrent flw t the shell-side fluid, whereas n the secnd tube pass, the tube fluid is in ccurrent flw with the shell-side fluid. A pssible set f temperature prfiles fr this exchanger is given in Fig. 1.25. Nte that it is pssible fr the utlet tube side temperature t be smewhat greater than the utlet shell-side temperature. The resulting temperature prfiles then might lk like Fig. 1.26. The maximum pssible tube utlet temperature that can be achieved in this case, assuming cnstant verall heat transfer cefficient, is t = 2T t, max i (1.29 Since this requires infinite area and all f the ther assumptins being rigrusly true, ne wuld rdinarily stay well belw this limit r lk fr anther cnfiguratin. An alternative arrangement f a 1-2 exchanger is shwn in Fig. 1.27, and a pssible set f temperature prfiles is given in Fig. 1.28. In this case t * cannt exceed T. In spite f the very different appearance f these tw cases, it turns ut that they give identical values f the effective temperature difference fr identical temperatures. 27
The prblem f cmputing an effective mean temperature difference fr this cnfiguratin can be carried ut alng lines very similar t thse used t btain the LMTD. The basic assumptins are the same (except fr the pure ccurrent r cuntercurrent limitatin, thugh in additin it is assumed that each pass has the same amunt f heat transfer area. Rather than calculate the MTD directly hwever, it is preferable t cmpute a crrectin factr F n the LMTD calculated assuming pure cuntercurrent flw, i.e. MTD F = (1.30 LMTD where F = 1 indicates the flw situatin is equivalent t cuntercurrent flw, and lwer values very clearly and directly shw what penalty (ultimately expressed in area required is being paid fr the 1-2 cnfiguratin. It is imprtant t remember that the LMTD used in Eq. (1.30 is t be calculated fr the cuntercurrent flw case, Eq. (1.27. The crrectin factr F is shwn in Fig. 1.29 fr a 1-2 exchanger as a functin f tw parameters R and P defined as (in terms f the nmenclature given n the chart: T1 T2 R = t t 2 t2 t1 P = T t 1 1 1 Range f shell fluid = Range f tube fluid Range f tube fluid = Maximum temperature difference (1.31a (1.31b The chart given here is adapted frm the Standards f the Tubular Exchanger Manufacturers Assciatin (9 and is almst identical t the ne in Kern (7. The crrespnding chart in McAdams (8 uses entirely different symbls, but is in fact identical t the ne given here. Hwever, there are ther different (but finally equivalent frmulatins and each ne shuld be used carefully with its wn definitins. Examinatin f the chart reveals that fr each value f R, the curve becmes suddenly and extremely steep at sme value f P. This is due t the tube-side temperature appraching ne f the thermdynamic limits discussed abve. It is extremely dangerus t design an exchanger n r near this steep regin, because even a small failure f ne f the basic assumptins can easily render the exchanger thermdynamically incapable f rendering the specified perfrmance n matter hw much excess surface is prvided; the first assumptin is especially critical in this case. Therefre, there is a generally accepted rule-f-thumb that n exchanger will be designed t F < 0.75. Besides, lwer values f F result in large additinal surface requirements and there is almst always sme way t d it better. The discussin t this pint has centered n the 1-2 exchanger. Larger numbers f tube-side passes are pssible and frequently used. Kern discusses the prblem briefly and pints ut that crrectin factrs fr any even number f tube-side passes are within abut 2 percent f thse fr tw passes, s it is cmmn practice t use Fig. 1.29 fr all 1-n exchangers where n is any even number. Other cnfiguratins will be discussed later. Kern, McAdams, and Perry's Handbk (10 give fairly extensive cllectins f F charts. 28
29
2. Multiple Shell-Side Passes. In an attempt t ffset the disadvantage f values f F less than 1.0 resulting frm the multiple tube side passes, sme manufacturers regularly design shell and tube exchangers with lngitudinal shell-side baffles as shwn in Fig. 1.30. If ne traces thrugh the flw paths, ne sees that the tw streams are always cuntercurrent t ne anther, therefre superficially giving F = 1.0. The principle culd be extended t multiple shell side passes t match multiple tube side passes but this is seldm r never dne in practice. Even the prvisin f a single shell-side lngitudinal baffle pses a number f fabricatin, peratin and maintenance prblems. Withut discussing all f the pssibilities, we may bserve that there may be, unless very special precautins are taken, will be, thermal leakage frm the ht shell-side pass thrugh the baffle t the ther (cld pass, which vilates the 6th assumptin. Further, there may even be physical leakage f fluid frm the first shell-side pass t the secnd because f the pressure difference, and this vilates the 1st assumptin. A recent analysis has been made f the prblem (Rzenmann and Tabrek, Ref. 11, which warns ne when the penalty may becme severe. 3. Multiple Shells in Series. If we need t use multiple tube side passes (as we ften d, and if the single shell pass cnfiguratin results in t lw a value f F (r in fact is thermdynamically inperable, what can we d? The usual slutin is t use multiple shells in series, as diagrammed in Fig. 1.31 fr a very simple case. Mre than tw tube passes per shell may be used. The use f up t six shells in series is quite cmmn, especially in heat recvery trains, but sner r later pressure drp limits n ne stream r the ther limit the number f shells. Qualitatively, we may bserve that the verall flw arrangement f the tw streams is cuntercurrent, even thugh the flw within each shell is still mixed. Since, hwever, the temperature change f each stream in ne shell is nly a fractin f the ttal change, the departure frm true cuntercurrent flw is less. A little reflectin will shw that as the number f shells in series becmes infinite, the heat transfer prcess appraches true cunter- current flw and F 1.0. It is pssible t analyze the thermal perfrmance f a series f shells each having ne shell pass and an even number f tube passes, by using heat balances and Fig. 1.29 applied t each shell. Such calculatins quickly becme very tedius and it is much mre cnvenient t use charts derived specifically fr varius numbers f shells in series. Such charts are included in Chapter 2 f this Manual. 30
4. The Mean Temperature Difference in Crssflw Exchangers. Many heat exchangers - especially air-cled heat exchangers (Fig. 1.32 - are arranged s that ne fluid flws crsswise t the ther fluid. The mean temperature difference in crssflw exchangers is calculated in much the same way and using the same assumptins as fr shell and tube exchangers. That is, MTD = F (LMTD (1.32 where F is taken frm Figure 1.33 fr the cnfiguratin shwn in Fig. 1.32. Recall that the LMTD is calculated n the basis that the tw streams are in cuntercurrent flw. 31