HEAT EXCHANGERS HEAT EXCHANGERS: TYPES

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1 HEAT EXCHANGERS Heat exhangers: types... 1 Plate heat exhangers... 4 Shell-and-tube heat exhangers... 5 Boilers... 6 Charateristis... 8 Heat exhanger analysis Overall heat-transfer oeffiient Fouling Heat exhanger modelling Non-dimensional method of heat-exhanger modelling: the NTU approah Preliminary sizing HEAT EXCHANGERS: TYPES Heat exhangers are off-the-shelf equipment targeted to the effiient transfer of heat from a hot fluid flow to a old fluid flow, in most ases through an intermediate metalli wall and without moving parts. We here fous on the thermal analysis of heat exhangers, but proper design and use requires additional fluiddynami analysis (for eah fluid flow), mehanial analysis (for losure and resistane), materials ompatibility, and so on. Heat losses or gains of a whole heat exhanger with the environment an be negleted in omparison with the heat flow between both fluid flows; i.e. a heat exhanger an be assumed globally adiabati. Thermal inertia of a heat exhanger is often negligible too (exept in speial ases when a massive porous solid is used as intermediate medium), and steady state an be assumed, reduing the generi energy balane to: openings t,ed e (1) time Ε = W + Q+ h m = m h + m h where the total enthalpy ht has been approximated by enthalpy (i.e. negligible mehanial energy against thermal energy), and means output minus input. Although heat flows from hot fluid to old fluid by thermal ondution through the separating wall (exept in diret-ontat types), heat exhangers are basially heat onvetion equipment, sine it is the onvetive transfer what governs its performane. Convetion within a heat exhanger is always fored, and may be with or without phase hange of one or both fluids. When one just relies in natural onvetion to the environment, like in the spae-heating hot-water home radiator, or the domesti fridge bak-radiator, they are termed 'radiators' (in spite of onvetion being dominant), and not heat exhangers. When a fan is used to fore the flow of ambient air (or when natural Heat exhangers page 1

2 or artifiial wind applies, like for ar radiator) the name heat exhanger is often reserved for the ase where the ambient fluid is duted. Other names are used for speial ases, like ondenser for the ase when one fluid flow hanges from vapour to liquid, vaporiser (or evaporator, or boiler) when a fluid hanges from liquid to vapour, or the ooling tower dealt with below. Devies with just one fluid flow (like a solar olletor, a spaeraft radiator, a submerged eletrial heater, or a simple pipe with heat exhange with the environment) are never named heat exhangers. The basi designs for heat exhangers are the shell-and-tube heat exhanger and the plate heat exhanger, although many other onfigurations have been developed. Aording to flow layout, heat exhangers are grouped in: Shell-and-tube heat exhanger (STHE), where one flow goes along a bunh of tubes and the other within an outer shell, parallel to the tubes, or in ross-flow (Fig. 1a shows a typial example of STHE; details presented below). Plate heat exhanger (PHE), where orrugated plates are held in ontat and the two fluids flow separately along adjaent hannels in the orrugation (Fig. 1b shows details of the interior of a PHE; more details are presented below). Open-flow heat exhanger, where one of the flows is not onfined within the equipment (or at least, like in Fig. 1, not speifially piped). They originate from air-ooled tube-banks, and are mainly used for final heat release from a liquid to ambient air, as in the ar radiator, but also used in vaporisers and ondensers in air-onditioning and refrigeration appliations, and in diretlyfired home water heaters. When gases flow along both sides, the overall heat-transfer oeffiient is very poor, and the best solution is to make use of heat-pipes as intermediate heat-transfer devies between the gas streams; otherwise, finned separating surfaes, or, better, diret ontat through a solid reuperator, are used. Contat heat exhanger, where the two fluids enter into diret ontat (simultaneous heat and mass transfer takes plae). Furthermore, the ontat an be ontinuous, i.e. when the two fluids mix together and then separate by gravity fores, as in a ooling tower, or the ontat an be alternatively with a third medium, usually solid, as in regenerative heat exhangers (RHE), like the rotating wheel shown in Fig. 1d (the hot gas heats the wheel whereas the old gas retrieves that energy). When the heat-exhange proess between the hot and the old fluids is delayed signifiantly, the term 'thermal energy storage' is used instead of RGE. There is always some ontamination by entrainment of one fluid by the other, although many times it is irrelevant (as in air-onditioning heat-reuperators), or even intended (as in ooling towers). Notie also that, if the mixed-up fluids do not separate, as in open feed-water heaters or in evaporative oolers, the devie is not named heat exhanger but just heater or ooler. Heat exhangers page 2

3 Fig. 1. Types of heat exhanges: a) shell-and-tube, b) plates, ) open-flow, d) rotating-wheel. Additionally, heat exhangers may be lassified aording to the type of fluid used (liquid-to-liquid, liquid-to-gas, gas-to-liquid, gas-to-gas), aording to phase hanges (vaporisers, ondensers), aording to relative flow diretion (ounter-flow, o-flow, ross-flow), aording to area density (transfer area per unit volume) or hannel size, et. In terms of the smallest hydrauli diameter of the two flows, Dh, or the area density, β (typial orrelation is β 3/Dh), heat exhangers may be grouped as: Conventional or non-ompat heat exhangers, if Dh>5 mm, or β<400 m 2 /m 3. Compat heat exhangers, if 1<Dh/mm<5, or 400 m 2 /m 3 <β<3000 m 2 /m 3. Many times, the terms ompat-heat exhanger (CHE) and plate-heat-exhanger (PHE) are used indistinguishably. Meso heat exhangers if 0.2<Dh/mm<1, or 3000 m 2 /m 3 <β< m 2 /m 3. Miro heat exhangers if Dh<0.2 mm (or β> m 2 /m 3 ). Human lung alveoli are typially 0.2 mm in size and have some m 2 /m 3. Heat exhangers are used to promote thermal energy flows at intermediate stages in proess engineering, or as a final heat release to the environment, ambient air in most ases, whih renders non-ontat devies as STHE and PHE) rather ineffiient and reourse is to be made of ontat heat exhangers as the wet ooling towers treated aside. A speial ase is that of marine engineering, where seawater is plenty available in the environment, greatly alleviating the thermal problem for heat-exhangers, but at a ost in materials ompatibility (upro-nikel or titanium must be used instead of opper or aluminium), sine seawater is very orrosive and plenty of miroorganisms. In order to mitigate the effets of seawater on heat exhangers, and to minimise hull-pass-throughs, only one entral heat exhanger is ooled by seawater (a PHE usually), and all other required heat exhangers use lean fresh-water as an intermediate fluid loop to finally disharge the energy at the seawater exhanger (entralised ooling system); different fluid loop layouts an be used, normally grouping several thermal loads by proximity of loation and by temperature level. For the latter, two levels are onsidered: high-temperature level (HT-iruit), say at >50 ºC like for engine ooling iruits (main engine and auxiliaries), and low-temperature level (LTiruit), say at <50 ºC like for engine-oil-lubriation ooling, air-onditioners and refrigerators, eletroni equipment, and so on; instead onneting the HT-iruit to the LT-iruit by means of a heat exhanger, it is better to use a partial mixing of the streams (regulated by a thermostati valve). The standard design value for final heat release in ships is a seawater temperature of 32 ºC, to allow for operation in all seas, in spite of fat that the largest share of operating time for most ships takes plae in seas at 15 ºC to 20 ºC (initial ost annot be avoided, but operation osts an be minimised by adjusting the seawater flow). Heat exhangers page 3

4 Heat exhangers are widely used in proess ontrol to promote or quenh hemial reations (by heating or ooling, respetively). The food industry makes use of heating to kill pathogen miroorganisms (sterilisation), either after anning, or before pakaging; the latter is most onveniently made for liquid stuff in heat exhangers. Sterilisation, i.e. the inativation of all miroorganisms, requires hightemperature proessing, typially at 120 ºC or more (i.e. under pressure, for aqueous stuff); to kill even the most resistant spores. In the pasteurisation proess, however, a quik heating to 60 ºC or 70 ºC is applied to kill most bateria without protein denaturising, but other miroorganisms remain, what implies that quik ooling after pasteurisation is required (what makes heat pumps so onvenient), and that vauum or refrigeration is needed afterwards. The time-for-pasteurisation (or for sterilisation) depends on the miroorganisms and the holding temperature; tabulated values are usually given for a proessing temperature of T0=120 ºC, and an be extrapolated to other temperatures with the logarithmi law ln(t/t0)= m(t T0)/T0, with m 100. Plate heat exhangers A plate heat exhanger, PHE, is a ompat heat exhanger where thin orrugated plates (some 0.5 mm thik, bended 1 or 2 mm) are staked in ontat with eah other, and the two fluids made to flow separately along adjaent hannels in the orrugation (Fig. 1b). The losure of the staked plates may be by lamped gaskets, brazing (usually opper-brazed stainless steel), or welding (stainless steel, opper, titanium), the most ommon type being the first, for ease of inspetion and leaning. Additionally, a frame (end-plates and fixing rods) seures together the plate stak and onnetors (sometimes PFHE, standing for plate-and-frame heat exhanger, is used instead of PHE). Plate assembly is skethed in Fig. 2. Suitable hannels, sometimes helped by the gaskets, ontrol the flow of the two fluids, and allow parallel flow or ross flow, in any desired number of passes, one pass being most used. They have large ondutane oeffiients (up to K=6000 W/(m 2 K) for liquid-to-liquid use), are ideally suited for low-visosity fluids, the number of plates an be adjusted to the needs, and the transfer surfae aessible to leaning (the latter two advantages only for gasket assemblies; in any ase, the gaskets should be hanged if dismounted). The projeted area of plates is usually taken as nominal heat transfer area, in spite of the real urved surfaes and lost spae in gaskets and ports. Heat exhangers page 4

5 Fig. 2. Plate heat exhanger (single plates, exploded assembly, assembled). Mayor limitations in PHEs are: maximum allowed pressure (usually below 1 MPa, although there are designs with 4 MPa), temperature range (usually limited to 150 ºC by the gasket material, although there are designs allowing 400 ºC), and prize (but brazed PHEs are about half prize of servieable PHEs). Although typial PHE appliation is in liquid-to-liquid heat-transfer, speial plate designs have been developed for phase-hange appliations. Higher working pressures and still goof thermal performane an be ahieved with hybrid plate-shell heat exhangers, where a plate stak is welded inside a shell (i.e. a kind of STHE with plates instead of tubes). The PHE was developed in the 1920s in the food industry (for the pasteurization of milk), but they are taking over all markets now beause of its ompatness and effiieny (3 to 10 times more than STHE). They are used for proess heating, ooling, in all ryogeni appliations, and as an intermediate step in domesti water heaters, where onsumable hot water (hot tap water) is produed in an intermediate heat exhanger from losed-loop fuel-fired hot water, to minimise solid depositions. PHE are often named CHE (ompat heat exhangers), although the word ompat an be added to any type of heat or mass transfer unit with speifi area >10 3 m 2 /m 3. Shell-and-tube heat exhangers The shell-and-tube heat exhanger (STHE) is the most ommon type of heat exhanger. It originated from the jaketed-oil distiller, and is used in heavy industries (steam ondensers, boilers), and residential hotwater and heating systems (fire-tube water heaters). Several details of STHEs are presented in Fig. 3. a ) b ) Heat exhangers page 5

6 ) d) e) Fig. 3. Shell-and-tube heat exhangers (STHE): a) Small mounted STHE (in opper); b) Cut-out of a similar STHE; ) Shemati diagram of a STHX with one shell pass and one tube pass; d) Constrution details with one shell pass and one tube pass (legend: 1, Tube-fluid exit; 2, Gasket; 3, Shell-fluid entrane; 4, Tubes; 5, Shell; 6, Buffles; 7, Purge; 8, Expansion diaphragm; 9, Tube holding plate; 10, Gasket; 11, Tube-fluid entrane; 12, Head over; 13, Shell-fluid exit; 14, Drain; 15, Fastener; 16, Tube holding plate; 17, Head over); e) Constrution details of a STHX with two tube-passes and one shell-pass. As an be seen in Fig. 3, eah fluid an flow along several passes. In the tube-side, every set of tubes that the fluid travels through, before it makes a turn, is onsidered a pass. In the shell-side, a pass aounts for eah main flow diretion hange. A standard ode of pratie in heat exhanger design and operation is that of TEMA (Tubular Exhangers Manufaturers Assoiation). Other important soure of standards are ASME (Amerian Soiety of Mehanial Engineers; Boiler and Pressure Vessel Code), and, for heating and ooling loads, ASHRAE (Amerian Soiety of Heating, Refrigerating and Air Conditioning Engineers). Boilers Basially, a boiler is a losed vessel or arrangement of enlosed tubes in whih water is heated to supply steam (to drive an engine or turbine, or to provide heat); when other liquid than water is used, the boiler is more often named vaporiser (or evaporator). A seond meaning of boiler is a domesti devie burning Heat exhangers page 6

7 solid fuel, gas, or oil, to provide hot water, espeially for entral heating (better alled a heater). Closely related to boilers are pressure ookers, i.e. strong hermetially sealed pots in whih food may be ooked quikly under pressure at a temperature above the normal boiling point of water (in this ase the intention is not to supply steam but to generate it for pressurising; the higher the pressure, the higher the boiling temperature). Most boilers are fuel-fired, thus, they an be viewed as shell-and-tube heat exhangers (Fig. 4), where the hot fluid is the burnt gases, and the old fluid the water stream. Heat transfer by radiation is important in boilers (and in furnaes) beause of the high temperatures (some 2000 K). In most boilers, the air for ombustion is previously heated by the exhaust gases in the stak. Typial effiienies, measured as water enthalpy hange divided by the ombustion enthalpy (most often based on the standard low heating value of the fuel), are around 100% in modern ondensation boilers (where part of the water vapour dissolved in the flue gases is ondensed), around 90% for large non-ondensing boilers, and around 80% for modern small non-ondensing boilers. A boiler is often the largest energy onsumer both at domesti and at industrial level, thus, great savings may be obtained by their proper seletion, operation and maintenane. Fig. 4. A boiler is basially a burner and a heat exhanger. Boiler development is linked to the steam-engine development, the driver of the Industrial Revolution. Hero of Alexandria, around 150 b.c., was the first to use a boiler to produe motion (a freely-spinning hollow-sphere boiler, exhausting through two hollow tubes oppositely bended), but it was Danis Papin, in 1679, the first to develop a pressure vessel and the first safety valve. Improvements were added by T. Savery in 1698, T. Newomen in 1705, and J. Watt in Boilers were basially large kettles with a fire underneath, until Rihard Trevithik, in 1804, implemented the first fire-tube boiler (a single fire-tube within the water tank), and made the first steam-loomotive. The Stephenson (father and son) greatly improved steam loomotives (with boilers working up to 0.5 MPa), and S. Wilox in 1856 (G.H. Babok joined Wilox in 1866, reating one of the biggest boiler firms), developed the water-tube boiler, inherently safer beause the high steam pressure only ats in the tubes (small), and not in the muh larger shells. Rudimentary water-tube boilers were built before Wilox's (a patent by Blakey dates 1766) but were ineffiient and not muh used. There were many boiler explosions in the XIX. and early XX. until the first ASME Code for Pressure Vessels of 1915 was widely adopted. Large steam-flow-rates are needed in many proess industries. On board a ship, steam may be required for: Propulsion of steam-turbine vessels: all nulear vessels, and nearly all LNG arriers, are powered by steam turbines requiring high-pressure (e.g. 10 MPa), high-temperature (e.g. 800 K) steam. For other steam appliations, any positive-pressure saturated-steam may work, with 0.7 Heat exhangers page 7

8 MPa being ommon (up to 2 MPa in large heavy-fuel arriers). Boiler energy effiieny is in the range , inreasing with size. Heating the load. For semi-solid substanes: asphalt (>100 ºC), heavy-fuel-oil (>50 ºC), molasses (>50 ºC), et. Heating its own fuel. For day-servie store of marine fuel, and for main and day-servie stores of heavy fuels. Fuel heating used in all diesel ships with more than 10 MW engines. Heating the oil for the engine lubriation ('lub-oil'). Also for heating lub-oil/fuel mixtures for stowage or separation (bilge mud). Spae heating for rew and passenger. Cleaning steam for inert spaes (hold ompartments, engine room), and for habitable spaes (kithens, laundry, lavatories). Charateristis Boiler speifiation is made in terms of several variables, among whih the most important are: Capaity (mass flow rate prodution). Most ommon boilers are in the range from 2 kg/s to 10 kg/s of steam, supplied at typially 0.5 MPa to 5 MPa. Pressure level: o High pressure boiler (HPB, p>0.1 MPa), that may be Power-HPB (if D>0.4 m) or Miniature-HPB (if D<0.4 m). o Low pressure boiler (LPB, p<0.1 MPa). Sometimes, high pressure water-heaters (with p>1 MPa) are onsidered as steam-lpb. Fuel used (gas, oil, oal, nulear, eletrial). Water flow arrangement (water must always be pressurised to get high-temperature steam): o Pressure-vessel (Watt-1785) with water externally heated by the flue gases. o Pyro-tubular (Soth boiler). The flue gases penetrate or are reated inside the pressure vessel with water. Evans pv<2 MPa. Tubes work at ompression, and shell at expansion. o Aqua-tubular (drum boiler, Parsons-1894-TV-'Turbinia', 1.2 MPa, 1.5 MW). A nonpressurised vessel lodges the flame and flue gases, inside whih pressurised pipes arry the water and/or its vapour. Tubes work at expansion, and shell does not work. Cost (first ost, running ost). Et. In a fire-tube boiler (Fig. 5), hot flue gases from the burner are hannelled through tubes that are surrounded by the fluid to be heated. The body of the boiler is the pressure vessel and ontains the fluid. In most ases this fluid is water that will be irulated for heating purposes or onverted to steam for proess use. Fire-tube boilers are relatively inexpensive, easy to lean, and more ompat than water-tube boilers (although of smaller steam apaities, and not suitable for high pressure appliations (up to 2 MPa only). Heat exhangers page 8

9 Fig. 5. A fire-tube boiler as those used in steam loomotives (Steamboat.om). In a water-tube boiler, water flows through the tubes within a furnae in whih the burner fires into. The tubes are onneted to a steam drum on top and a mud drum at the bottom. Water-tube boilers typially produe steam or hot water for large industrial appliations (less frequently for heating appliations). They support higher pressure (up to 35 MPa) and temperature (900 K) than fire-tube boilers, but are more omplex, larger (up to 50 m high, up to 60 kg/s of steam), and more expensive than fire-tube boilers. In superritial boilers, water is heated at more than 22 MPa and onverted to superritial steam without any phase hange. Boiler design must onsider not only the boiler itself (the heat exhanger where water beomes steam), but the steam system (heaters, piping, ontrols), the feed-water system (water onditioning, venting, purging), the fuel system (fuel/air ratio, piping), and the exhaust system (draught). Boiler material may be opper, steel, or ast iron. Contrary to what might be believed, boiler water is an expensive item beause it must be de-mineralised (from typial 500 ppm total dissolved solids from the mains, down to 10 ppm), and de-gasified (CO2, O2 and N2; e.g. air-saturated water may have 10 mg/kg of dissolved oxygen, and it must be redued to some 10 µg/kg). The latter is ahieved in a devie alled deaerator, where feed-water to the boiler is heated and agitated by steam bubbling up to its vapour-equilibrium temperature at the working pressure of the deaerator (usually slightly above atmospheri pressure); non-ondensable gases (and some entrained steam) are released through a vent valve. Boiler safety is an important issue, sine steam pressurised vessels have a lot of exergy and their explosion is very violent (many asualties have ourred). As a minimum, two independent, stepped safety devies are implemented; they open fully when needed. Two types of safety valves an be distinguished: Reversible safety valves (will loses after low-pressure reovery; e.g. spring-loaded type). May be tested. Irreversible safety valves (for abnormal operation; permanent damage; must be replaed after use). Statistial test. Heat exhangers page 9

10 Besides safety devies, some pressure regulation devie may be required for use or added for safety. Contrary to safety devies, these relief valves open proportionally to exess pressure. Heat exhanger analysis The thermal analysis of a heat exhanger is based on the simple oaxial onfiguration (Fig. 6), where one fluid goes along a pipe, and the other fluid goes along the annular setion within a larger ylindrial sheath with openings at the ends, and in partiular, to the ounter-urrent onfiguration shown in Fig. 6a, more thermally-effiient than the o-flow set-up of Fig. 6b. The oaxial ross-setion is skethed in Fig. 6. Longitudinal temperature-profiles for the ounter-flow and the o-flow onfigurations are also shown, and a detail of the transversal temperature profile aross the wall separating the two fluids (Fig. 6d). The minimum temperature jump from one fluid to the other is alled the (temperature) approah of the heat exhanger. Fig. 6. Simple annular heat exhangers: a) sketh and temperature profile in a ounter-flow onfiguration, b) sketh and temperature profile in o-flow, ) ross-setion sketh, d) detail aross the separating surfae. Heat transfer from one fluid to the other is a ombined onvetion-ondution-onvetion proess aross the separating surfae, as detailed in Fig. 2d, where the hotter fluid is subsripted '1' and the older one '2'. The fat that the older flow surrounds the hotter one in Fig. 6 is irrelevant to the results here developed, although it may be advantageous in pratie to minimise heat-losses to the ambient (if its temperature differene to the ambient is lower than that of the hotter one), here negleted. Overall heat-transfer oeffiient At steady state, the loal heat flux through the wall separating the two fluids, from hotter to older (Fig. 6), is: T T dq = hda T T = k da = h da T T KdA T T (2) 1 2 ( ) ( ) ( ) s m Ls Heat exhangers page 10

11 where the latter identity serves to introdue the loal overall heat-transfer oeffiient, K (traditionally named also U), assoiated to the atual hoie of transfer area da (either the internal one, da=da1, the external one, da=da2, or an appropriate average da=dam, although in most irumstanes the separating solid is so thin that all areas are pratially the same, da=da1=da2=dam). If one introdues additional transfer fators to aount for possible extended surfaes (e.g. fins or grooves) in terms of fin effiieny assoiated to root area, ηfin (defined aside), and possible degradations in the onvetive oeffiients due to fouling, measured by an additional fouling resistane, Rf (often named 'fouling fator', although in that ase should be treated as another effiieny term like the ηfin above), to be explained below, one gets for the overall heat-transfer oeffiient, K: 1 K = 1 L 1 + R R η s f1 f 2 fin1h1 ks ηfin2h2 (3) To a first approximation, the ondutive thermal resistane, L/k in (3), an always be negleted (exept in low effiieny ases like distillers made of glass), and the onvetive resistane of liquids an also be negleted against that of gases. Table 1 below presents some typial K-values. In pratie, however, what matters is the average overall heat-transfer oeffiient for the whole heat exhanger, K, defined in (3), instead of its loal value defined by (2); we are using the same symbol for both the loal and the global value, and some ambiguity also in temperature naming, sine from now on T1 and T2 are going to be the bulk temperatures of the flows at a ertain ross-setion along the heat exhanger, i.e. what was named T1 and T2 in Fig. 6, instead of the temperatures at the wall. Notie, by the way, that the overall heat-transfer oeffiient, K, is also very onvenient in the study of heat transfer between two fluids by natural onvetion, like for windows and radiators, where typial values are K=1 W/(m 2 K) for a double-glass window, K=3 W/(m 2 K) for a single-glass window, K=6 W/(m 2 K) for a planar solar olletor, or K=10 W/(m 2 K) for a heating 'radiator'. In the ase of plate heat exhangers, the overall heat-transfer oeffiient, K, is poorly defined beause only the produt KA enters into the equations, and the ontat area, A, between the two fluids is diffiult to define and measure due to the orrugations. Fouling One generi problem of heat transfer is the effet of surfae ontamination on heat-transfer rates, that may be due for instane to hemial attak at the interfae between solids in heat ondution, rust build-up at walls in heat onvetion flows, or dust deposition in heat radiation surfaes. Fouling (i.e. dirt and depositions) is detrimental within heat exhangers beause it adds a thermal resistane to heat, it adds a fluid-dynami resistane to flow, and it is diffiult (sometimes impossible) to lean. Typial fouling fators used in heat transfer analysis are Rf=0.002 m 2 K/W for fuel oil, Rf=0.001 m 2 K/W for flue gases, Rf= m 2 K/W for air and refrigerant vapours, Rf= m 2 K/W for normal liquids, Heat exhangers page 11

12 and Rf= m 2 K/W for steam and for water (or sea water) below 50 ºC. If one fluid is a gas, fouling has a negligible effet on transfer rates. Fouling is typially due to algae growth on old surfaes, to salt deposition on hot surfaes, and to unfiltered dirt logging. All industrial iruits ooled with natural fresh or sea water, are subjeted to biologial fouling (a bio-film settlements of living organisms). Although physial sreening, physial leaning and hemial dosing an manage to get a long trouble-free operation, nowadays environmental restritions may impinge on that; e.g. hemial dosing with hlorine onentrations up to 10 mg/l were ommon in the past, but now hlorine disharge to the water environment have been severely restrited. Heat exhanger modelling Engineering pratie makes use of off-the-shelf heat exhangers (PHE or STHE), but thermal performane and design are best explained in terms of the simple parallel-flow onfigurations shown in Fig. 6 above, and in partiular for the ounter-flow ase in Fig. 6a. For a differential slie of length dx along the ounter-urrent heat-exhanger in Fig. 2a, the longitudinal energy balane for eah fluid, and the transversal energy balane aross the separating wall, assuming the hot fluid is subsripted '1', are: ( ) dq = m dt = m dt = KdA T T (4) where dq is, as in (2), the transversal heat flow from hot (T1) to old (T2) flows aross a differential annulus of length dx, m 1 1 and m 2 2 the thermal-apaity rates of eah stream (the perfet-substane model is assumed to ease writing, but phase hanges would be equally treated in terms of enthalpies instead of temperatures), and da the wetted annular are of length dx (either internal, external, or some average area) The negative signs assoiated to dt in (4) ome from the negative slopes of the temperature profiles in Fig. 6a. Integrating (4) from one end (x=0) to the other end (x=l) along the heat exhanger (Fig. 6a) yields: ( ) ( ) Q = m T T = m T T = KA T (5) 1 1 1,in 1,out 2 2 2,out 2,in 12,mean where T12,mean is an appropriate average, defined by (5) in terms of the overall heat-transfer oeffiient K and the hoie of wet area A. This average temperature approah an be obtained by integration of a ombination of the three equations in (4), and ulterior omparison with (5); i.e., from (4): d dt dt dt dt m m m m ( ) Q= = = = KAT d 1 T (6) and integrating the latter equality from x=0 to x=l: ( T T ) 1 1 T T T T T T T T = KA = KA = ( T1 T2 ) m 1 1 m 2 2 Q Q T12,mean ( ) ( ) 1 2 x= L 1,out 1,in 2,in 2,out 1 2 x= L 1 2 x= 0 ln x= 0 (7) Heat exhangers page 12

13 whih finally yields: 12,mean ( T T ) ( T T ) x L ( T1 T2) x= L ln ( T T ) 1 2 = 1 2 x= 0 T = T 1 2 x= 0 LMTD (8) what shows that, for the ounter-flow onfiguration in Fig. 6a, with onstant-thermal-apaity fluids, the appropriate mean-value of the temperature jump aross the separating wall, is the logarithmi mean temperature differene (LMTD) defined by (8), whih nearly oinides with the arithmeti mean when the approah at both ends is of the same order (the singularity for this ase in (8) an easily be irumvented). In an analogous manner, it an be demonstrated that the LMTD also applies to the oaxial heat exhanger in o-flow shown in Fig. 6b, but not in a more general ase of multi-pass or ross-flow heat exhangers, neither to the ase of highly-varying thermal apaities (as in two-phase mixtures); in those ases, an empirial orretion fator, F, an be defined to get an appropriate average approah, T12,mean=F T12,LMTD, in terms of the LMTD defined in (8). Thus, the thermal modelling of a heat exhanger redues to the three equations (5) with the eight variables: Q, m 1 1, T1,out, T1,in, m 2 2, T1,out, T1,out, and KA. Notie that K and A do not appear separately (as for m and ). Although a brute aounting would yield 8 7 6/(3 2)=56 possible ombinations for the trio of solvable unknowns, in pratie there are only a few interesting ases, namely: Testing problem. This is the ase where an atual heat exhanger is benh-tested by measuring all input and output flow variables: m 1 1, T1,out, T1,in, m 2 2, T1,out, and T1,out. We have the three equations (5) but only two unknowns Q, and KA, meaning that there is one redundany in the measurement (what might be used to hek that the overall heat losses were really negligible). As in general, only the produt KA an be omputed, but it is not diffiult to get an estimate of A from the external dimensions and the knowledge of the number of plates in a PHE or the number of tubes in a STHE. Conversely, K might be estimated by heat-onvetion orrelations in terms of the type of fluids, their veloities, and the geometry of flow, or even based on typial values for the overall oeffiient (e.g. K 2000 W/(m 2 K) for water-to-water heat exhangers). Performane: problem. Given the entry onditions for the two flows ( m 1 1,T1,in, m 2 2, and T2,in), and the size (measured by the transfer area, A), and assuming the overall heat-transfer oeffiient known, K, determine the heat rate exhanged, Q, and the output temperatures (T1,out and T2,out), i.e. three unknowns for the three equations in (4), if the LMTD approah is applied (otherwise, see the NTU approah below). Design problem. Given the entry onditions for the two flows ( m 1 1,T1,in, m 2 2, and T2,in), and the heat rate exhanged, Q,.and assuming the overall heat-transfer oeffiient known (K), determine the size (measured by the transfer area, A) and the outlet onditions: i.e. three unknowns for the three equations in (5). The step-by-step proedure to solve the design problem is: Heat exhangers page 13

14 1. Establish the governing equations (5) and perform the analysis of data and unknowns. 2. Compute the outlet temperatures from the flow-rate apaities and heat load. 3. Estimate the required transfer area using typial values of overall heat-transfer oeffiients (e.g. Table 1) and the TLMTD from (8), or other averaged approah. This order of magnitude may be good enough for non-ritial designs, and will help to hek more refined analysis, as following. 4. Selet a type of heat exhanger based on type of fluids and expeted loading onditions. 5. In the ase of STHE, selet whih fluid goes on the tube-side: either the dirtiest one (beause shells annot be leaned), or the high-pressure one (beause small tubes resist pressure better than large shells). In the ase of other types of heat exhangers, do it in a similar way. 6. Selet tube diameter, d, aording to available tube sizes. 7. Estimate the required flow ross-setion, A= m /( ρv), assuming a typial speed of v=1 m/s for liquids, or v=10 m/s for gases. 8. Estimate the number of tubes, N, from A = m /( ρv) = NπdL assuming a typial Reynolds number, Re=vd/ν, of Find if several tube-passes are needed by estimating the shell diameter, D, by D= 2d N, and heking that the resulting overall slenderness is around L/D Estimate pressure losses in the tube side (drag on pipe flow an be found in Fored and Natural Convetion, but other losses in bends and fittings may be relevant too). 11. Consider that the unertainty in the above proedure may be ±50%, and guess if a more refined design is needed at that stage (in that ase, one must resort to more speifi literature, or to experimentation, as a last reourse). Table 1. Order of magnitude values of global or overall heat-transfer oeffiients, K, for heat exhangers. Configuration Typial value K [W/(m 2 K)] Typial range K [W/(m 2 K)] Gas-to-gas heat exhanger at normal pressure Gas-to-gas heat exhanger at high pressure Liquid-to-gas or gas-to-liquid heat exhanger Liquid-to-liquid tubular heat exhanger Liquid-to-liquid plate heat exhanger Condenser, to a gas Condenser, to a liquid Vaporiser, to a gas Vaporiser, to a liquid Vaporiser, to a ondensing gas Exerise 1. A hot water stream of 0.2 kg/s at 80 ºC is to be ooled in an annular ounter-flow heatexhanger with 0.5 kg/s of ambient water at 15 ºC. The inner tube is made of stainless steel and has 20 mm internal diameter and 1 mm thikness, whereas the internal diameter of the outer tube has 30 mm in diameter. Find the onvetive oeffiients, the overall heat-transfer oeffiient, the heat flow-rate, and the exit temperatures. Heat exhangers page 14

15 Solution. A rough typial-value analysis yields h=5000 W/(m 2 K) from Table 6 in Heat Convetion, with a orresponding overall heat-transfer oeffiient of K=h/2=2500 W/(m 2 K) from (3), having negleted ondutive resistane aross the separating tube wall, and all kind of fouling; notie, however, that the typial K-value suggested in Table 1 above for liquid-to-liquid tubular heat-exhanger is just K=1000 W/(m 2 K). With the latter, and taking the temperature differene at inlet points instead of the orret LMTD-value, the heat flow rate would be Q = KA T = (80 15) = 7.8 kw, where the transfer 2 area has been approximated by the internal one, A = π DL = = 0.12 m. From the energy balane of eah stream, Q = m T, we finally get the outlet temperatures: Thot,out = Thot,in Q ( m ) = ( ) = 71 ºC, T = T + Q ( m ) = ( ) = 19 ºC. old,out old,in But the expeted unertainty in the h-values and K-values assumed above, some 50% or more, are only aeptable for an order-of-magnitude analysis. A more refined analysis requires a more preise determination of the onvetive oeffiients h, what is done through the empirial orrelations presented under Fored and natural onvetion, aside. Assuming the hot stream runs along the inner tube (the typial ase), its bulk speed is u=0.64 m/s, Re=12 700, Pr=2.7 (evaluated at 80 ºC), and using Dittus- Boelter orrelation for turbulent flows (12700>2300), Nu=0.023Re 0.8 Pr 0.3 =60, and h=knu/d=1800 W/(m 2 K). For the old stream, the hydrauli diameter must be used, Dh 4A/p=D2 D1=30 21=9 mm, what yields a bulk speed of u=1.53 m/s, Re=12 200, Pr=7.7 (evaluated at 15 ºC), Nu=0.023Re 0.8 Pr 0.3 =97, and h=knu/d=7260 W/(m 2 K). The proper K-value is now K=1/(1/h1+δ/k+1/h1)=1300 W/(m 2 K), and the heat flow-rate taking the temperature differene at inlet points instead of the orret LMTD-value, Q =10.7 kw, what yields Thot,out=67 ºC and Told,out=20 ºC. With these values, we an ompute a better approximation to the logarithmi mean temperature differene using (8), TLMTD=56 ºC (instead of the 80 15=65 ºC former approximation), and new, more aurate values for Q =9.2 kw, Thot,out=69 ºC and Told,out=19 ºC. We ould go on with further iterations, orreting the omputed material properties at an average bulk temperature between inlet and outlet (instead of the assumed inlet values), and so on, but there is usually no need to better fit final figures when the initial data unertainty is not well defined. Non-dimensional method of heat-exhanger modelling: the NTU approah The LMTD method above-explained offers a full model of a heat exhanger, with 8 variables linked by the three equations in (5) with the definition in (8) (supplemented, if needed, with the mentioned orretion fator F to aount for multi-pass or ross-flow onfigurations). However, solving the performane problem stated above, i.e. finding Q, T1,out and T2,out, given m 1 1, T1,in, m 2 2, T2,in, and KA, gives a non-linear system beause of the LMTD-expression (8), involving tedious iterations, what an be irumvented by another approah, based on a non-dimensional parameter introdued by Kays-London in 1955, namely the number of transfer units (NTU), N, defined in terms of the minimum of the two thermal mass-flow-apaities, ( m, as: ) min N KA ( m ) min (9) Two other non-dimensional parameters omplete the new model: the heat-exhanger effetiveness, η (sometimes denoted ε), and the heat apaity ratio,, defined by: Heat exhangers page 15

16 ( ) ( ) ( ) ( m ) ( ) min m ( ) ( ) ( ) ( ) ( ) Q m 1 1 T1,in T1,out m 2 2 T2,out T2,in KA T12,mean η Q = = = m T T m T T m T T max min 1,in 2,in min 1,in 2,in min 1,in 2,in max (10) (11) The heat apaity ratio,, ranges from =0 in the ase where one fluid hanges of phase (vaporisers or ondensers, beause then the energy hange in that fluid is without temperature hange and thus its thermal apaity tends to infinity), to =1 in the ase where both flows have the same m -value, as in most regenerative heat exhangers. When one fluid is liquid and the other gas, the m -value for the gas stream is usually muh lower than its ounterpart for the liquid, and 0. The effiieny of a heat exhanger is the ratio of heat-flow atually exhanged, Q, to the maximum heatflow exhangeable for given input onditions if the length were infinity, Q max. With this notation, the three equations (5) that govern heat-exhanger modelling, an now be stated as (assuming that the flow subsripted '1' has the lowest thermal apaity): T T T T T,, N, T T T T T 1,in 1,out 2,out 2,in 12,mean η = η = η = = η ( 1,in T2,in ) 1,in 2,in 1,in 2,in ( N) (12) where the latter equation has been set in the generi funtional form η=η(n,), instead of in terms of a suitable averaged approah (whih was demonstrated to be the LMTD in the ase of simple oaxial-tube heat exhangers, to be modified with the F fator for other onfigurations). We an now find the expliit form of η=η(n,) in the ase of the ounter-flow onfiguration in Fig. 6a, as done for the LMTD, namely, by ( ) ( ) T1 T2 1 1 T1,out T x= L 2,in ln = KA ln = N ( 1 ) T T m m T T 1 2 x= ,in 2,out ( T1,out T1,in ) + ( T1,in T2,in ) ( T T ) ( T T ) T1,out T2,in η + 1 ln = ln = ln = N( 1 ) T T 1 η 1,in 2,out 1,in 2,in 2,out 2,in 1 e η ( N, ) = 1 e N N ( 1 ) ( 1 ) (13) Thus, instead of the three equations (5) helped with (8) and an F-fator orretion, now we have the three equations (12) helped with (13) and another orretion as before for multi-pass or ross-flow onfigurations, whih in this NTU-method an be established analytially in some ases as a partiular η(n,)-funtions other than (13), as here presented analytially in Table 2, or presented in graphial form in most Heat Transfer texts. Of ourse, instead of using speial η(n,)-funtions, the old orretion fator, F, to be multiplied by the ounter-flow η(n,)-funtions (13) an be used, similarly as for the LMTDapproah. Heat exhangers page 16

17 Table 2. Effetiveness relations of some heat exhanger onfigurations. Configuration η=η (N,) N=N(η,) Coaxial pipes in ounter-flow N( 1 ) 1 e 1 1 η η = N = ln N( 1 ) 1 e 1 1 η Coaxial pipes in o-flow N( 1+ ) 1 e ln η ( 1+ ) 1 η = N = STHE with 1 shell-pass and any even η = 1 η 2 N = ln N 1+ tube-passes 2 1+ e N 1+ 1 e η Cross-flow, single pass, exp( N ) 1 ln ln ( 1 η ) + 1 ( m ) mixed, ( m ) unmixed η = 1 e N = min max Cross-flow, single pass, 1 ( ) ( m ) unmixed, ( m ) mixed ( ) 1 1 exp N ln ( 1 η ) η = e N = ln 1 + min max Any heat exhanger with =0 (phase hange) N η = 1 e N = ln ( 1 η ) ( ) ( ) The variety of flow onfigurations in a heat exhanger may be very wide, as depited in Fig. 7 for a nonstandard STHE, where the temperature profiles have been skethed. And not only heat-flow orrelations are needed, but frition orrelations too, to know the pressure losses on eah fluid. In atual pratie, after a preliminary analysis using the models presented above, reourse is made to the partiular data supplied by the manufaturer (one seleted). Fig. 7. Temperature profiles along a ross-flow shell-and tube heat exhanger. Preliminary sizing It is important to have simple dimensioning proedures to allow for preliminary analysis of thermal systems with heat exhangers. For instane, when studying a new heat-pump appliation, one has to hoose appropriately the operating temperatures, based on the heat soure and heat sink temperatures Heat exhangers page 17

18 available; from these ondenser and vaporiser temperatures seleted, and the seleted working fluid, one an solve the heat-pump yle and ompute the mass-flow and power requirements for the equipment, what serves for dimensioning the ompressor and make a sound seletion among the available offer. But if the fluid working temperatures are hosen too lose to the heat soure and heat sink values, very effiient and expensive heat exhangers will be required to ahieve that goal, and it ample margin is let, there will be no problem to find a heap heat exhanger to do the job, but at the expense of over-sizing the heat-pump ompressor. In onlusion, there is the omnipresent trade-off between saving on first ost at the expense of operation osts (by hoosing heap omponents with high onsumption), and saving on operation osts at the expense of initial ost (by hoosing more expensive omponents with lower onsumption). But one annot optimise everything at one, and engineers should not, beause many times their results are not to be implemented in the final produt but just iterative inputs to the design proess, to be disarded after an overall view of the proess ditates new hanges for the next iteration. That is why the modelling effort (be it of a heat exhanger or of another devie or proess), should be a proportional share of the whole effort at a time. In our ase, the reommendations, from simplest to more omplex modelling of heat exhangers, are: Just aount for a reasonable minimum temperature approah between the two fluids, e.g. T12,min=5 K (1 K approah might be enough with good plate heat exhangers for liquids, but 50 K may be required for gas-to-gas heat exhangers). At a later stage, use the LMTD-method or the NTU-method to find the required ontat area of the heat exhanger, A, for a typial overall heattransfer oeffiient, K, guessed from Table 1. A better model may be to aount for a reasonable heat-exhanger effiieny, e.g. η=0.8, and use (12) to find the outlet temperatures. At a later stage, use the LMTD-method or the NTU-method to find the required ontat area of the heat exhanger. A more aurate model onsiders the system of equations ((5) or (12), plus speifi relations for the F-fator or for η(n,)-funtion as in Table 2), and the unknowns as a separate proeduralmodule to be omputer-solved in onjuntion with the rest of the thermal problem. As the last resort of theoretial analysis, the overall heat-transfer oeffiient, K, an be omputed using (3) with the onvetive oeffiients, h, at eah side omputed with appropriate foredonvetion orrelations found aside, plus additional fouling data (see above) if neessary. Beyond this analysis, one may ondut detailed numerial simulation using a CFD pakage, and experimental testing. A final remark is that we have just foused on the thermal analysis and design of heat exhangers, i.e. on their heat-transfer duty, trying to inorporate general eonomi riteria always present in any kind of engineering projet, but, as said in the beginning, there are many other important aspets in heat exhanger analysis, as pressure-loss alulations for pumping needs, materials ompatibility with the fluids, strutural design, maintainability, ost... Bak to Heat and mass transfer Bak to Thermodynamis Heat exhangers page 18