NATURAL CONVECTION IN HEAT EXCHANGERS - A NEW INCENTIVE FOR MORE COMPACT HEAT EXCHANGERS?

Proceedings of Fifth International Conference on Enhanced, Compact and Ultra-Compact Heat Exchangers: Science, Engineering and Technology, Eds. R.K. Shah, M. Ishizuka, T.M. Rudy, and V.V. Wadekar, Engineering Conferences International, Hooken, NJ, USA, Septemer 25. CHE25 7 NATURAL CONVECTION IN HEAT EXCHANGERS - A NEW INCENTIVE FOR MORE COMPACT HEAT EXCHANGERS? Thomas Aicher and Holger Martin 2 Fraunhofer-Institut für Solare Energiesysteme ISE, Freiurg, Germany, thomas.aicher@ise.fraunhofer.de 2 Universität Karlsruhe (TH), Thermische Verfahrenstechnik, Karlsruhe, Germany, martin@tvt.uka.de ABSTRACT Natural convection in the la is very often studied at single surface elements or in single channels. The flow and heat transfer characteristics in these cases are more or less well understood and prediction of flow rates and heat transfer coefficients can usually e otained from standard textook formulae. While the ehaviour of a single channel can safely e used to predict what happens in a undle of parallel tues in forced flow, the same is not true for natural convection. Starting from relatively simple cases of natural convection in single channels as well as in undles of parallel ones, it is shown, that natural convection in undles does ehave completely differently. In the case of mixed convection in a vertical tue undle, the effect of natural convection may lead to severe reductions in overall performance, ut also depending on the operation parameters to an enhancement of heat transfer! Till now, the textooks and handooks on heat transfer do not even mention these possiilities, that may lead to a numer of prolems in heat exchanger operation practice. T T 2 one may call it a single phase heat i...andit isa direct contact alanced counter current h L single horizontal T d T 2 channel Fig. Natural Convection in a Single Horizontal Channel. WHAT IS THE PROBLEM? The natural convective flow situation in a single horizontal channel is shown in Fig.. This arrangement may e called a single phase heat pipe as it does create a circulating flow etween the cold side (T ) and the hot side (T 2 ) with the hotter fluid on top of the colder one in two horizontal layers. d T T Ra c =87 L=Z Ra c =5.6 Ra=gd 4 β T/(Lνκ) T *G. I. Taylor (954), * * J. Unger (98) * ** single vertical Fig. 2 Natural Convection in a Single Vertical Channel. While the circulating flow in case of the horizontal channel starts as soon as there is a temperature difference greater zero, this is not true for the case of a vertical channel as shown in Fig. 2. Only if a critical value Ra c of the (dimensionless) temperature difference Ra is surpassed, the circulating flow will start. For Ra<Ra c the fluid remains at rest. The critical Rayleigh numers (temperature differences) have een calculated from a staility analysis for a cylindrical vertical tue y Taylor (954) and for a vertical parallel plates duct y Unger (98). The definition of Ra, as well as the values of Ra c T 48

from the literature are given in Fig. 2. Diameter (or gap width) d and length L enter the Rayleigh numer as: d 4 /L. Nu 2 8 6 4 2 Ra c [(Ra - Ra c) L/ d] Nu = + Ra 92 Ra c 2 4 6 8 Ra Gz = Ra/96 7/3 Ra = c 96 7/3= 56.9 Fig. 3 Approximate solution to the natural convection prolem in a single vertical parallel plates channel. The single channel natural convection cases (Fig. and 2) may e solved in a much simpler way (approximately) y a -D-model, taking into account their nature as direct contact alanced counter current heat exchangers. This is shown for the parallel plates duct in Fig. 3. The -D model in this case leads to simple explicit formulae for the flow rate (or Graetz numer Gz) as a (linear) function of temperature difference (Rayleigh numer) and predicts the critical Rayleigh numer aout 3% higher than the more rigorous 2-D approach from the literature. Looking at a tue undle (or a numer of parallel plate ducts), the situation is different: No circulation will take place within each one of the single parallel tues, ut in pairs of tues in that case. 2 Ra c =5.6 counter current heat exchanger -D approximate model gives Ra c aout 3% larger than 2-D solution y Unger d L=Z Rayleigh-Bénard Convection Ra c =384* *for adiaatic tue walls g Determined Natural Convection Ra c = Fig. 4 Natural convection in tue undles. Z L L Rayleigh numer: Ra=gd 2 Zβ T/(νκ) d to e multiplied y (L/d) 2 to arrive at the correct form of Ra for the undle. From this we can find, that Ra c,single tue (L/d) 2 had to e less than Ra c,tue undle for circulation within a single tue to occur in a undle. Using the values from Figs. 2 and 4, namely Ra c,single tue =87 and Ra c,tue undle =384, we find (L/d) 2 < 384/87 or L/d <.59. So, in case of tues with lengths greater than.6 diameters, the single tue internal convection will never happen in a undle. Using again a - D approximate solution, which is certainly sufficient for an engineering approach, the temperature profiles in the tues of the undle as a function of the individual flow rates are shown in Fig.5. Θ.8.6.4.2 3 m = (ρu)c p L/λ dimensionless mass velocity m=3.3.2.4.6.8 x.3 Fig. 5 Temperature profiles in the channels of tue undles (see Fig. 4) in natural convection as a function of flow rate m The heat transferred from the hot to the cold side y the comined positive and negative flows in every second channel has een calculated from the same -D model y Martin (992) and the results are shown oth for the horizontal, and for the vertical tue undle as Nusselt numers versus Rayleigh numer in Fig. 6. For the sake of simplicity, the horizontal undle in this case consists of only two tues (or two tue rows) in a vertical distance Z as shown in Fig. 6. For sufficiently high Rayleigh numers, the heat transfer coefficient is directly proportional to the temperature difference. The heat flux therefore is proportional to the square of T. This of course is the same dependency as found for the single channels (see Fig. 3) in this limit. The difference in ehaviour of single tue versus tue undle can e seen in the fact, that the flow pattern occurring in a undle differs from that in a single tue, ecause the greater degree of freedom for the flow, as soon as more than one tue is availale for a circulation. m=3 temperature profiles in the tues with m as a parameter 3 Diameter (or gap width) d and length L enter the Rayleigh numer as: d 2 L, (for L=Z) i. e. the Rayleigh numers of the single channel cases, containing d 4 /L have 49

5 Z Q proportional T 2 + Nu Nu = Ra/28 - unexpected tueside ackflow in a vertical shell-and-tue heat exchanger 5 Z 384 5 Ra 5 Ra=gd 2 Zβ T/(νκ) flow rate + Fig. 6. Nusselt vs. Rayleigh numers for natural convection in horizontal (upper curve) and vertical (lower curve) tue undles with adiaatic walls. MIXED CONVECTION IN BUNDLES Typically in a classical heat exchanger, we do have forced convection. Natural convection alone is rather unusual, ut mixed convection, i. e. fluid flow driven y pressure and density gradients, is exactly what happens in every heat exchanger. Usually, however, the effects due to density gradients are neglected in heat exchanger design. The question, whether natural convection may e safely neglected, is mainly answered on the asis of the well known single channel ehaviour. Judging from single channel ehaviour, an additional natural convection will usually increase (or only marginally decrease) the heat transfer coefficients compared to forced convection alone (see Aicher and Martin, 997). That s why neglecting natural convection in general is thought of eing justified y staying on the safe side in heat exchanger design. An example from industrial practice taught us, more than ten years ago, that this assumption may e completely wrong (Martin and Pajak, 994). A single tue pass vertical shell and tue heat exchanger with many segmental affles on the shell side had een designed and uilt for heat recovery to operate in counter flow at an efficiency of 9 to 97%. However, the efficiencies measured in operation were only 6 to 77%. The overall heat transfer coefficient, calculated from the measured temperatures, assuming ideal counter current operation reached only aout one third of the design value. After checking a numer of possile reasons for the unexpectedly low performance of that apparatus, we came to the conclusion, that superimposed natural convection might (see Fig. 7) e the reason of such a drastic deviation from the predicted design. Fig. 7. Unexpected tueside ackflow in a vertical shelland-tue heat exchanger This hypothesis has later een verified a) y tracer measurements of residence time distriutions in the undle, that actually showed, that ackflow existed in some of the tues. ) y -D model calculations on the flow and temperature distriutions in that shell-and-tue apparatus. c) y changing the flow directions of oth tue-side and shell-side fluid in the shell-and-tue apparatus, which indeed resulted in a stale situation avoiding ackflow and resulting in the high efficiencies of around 95% as expected from design. See Martin and Pajak (994) and Aicher et al. (999) for details. + z + NTU=5 = Fig. 8. Temperature distriution without ackflow in a vertical shell-and-tue heat exchanger ideal counter flow. To demonstrate the effect of unwanted ackflow in a part a of the tues in a vertical undle, the model equations from (Aicher et al., 999) have een used again here in a simpler form. We assume a alanced counter flow heat exchanger with NTU=5. If there is no ackflow (as shown in Fig. 8), the temperatures vary linearly with the tue 5

length, and the efficiency would e ε=5/6. The case with ackflow can e calculated analytically as the solution of three coupled ordinary differential equations for the temperatures in the tueside upflow (suscript, flow rate=+ in the fraction -a of the tues), the tue-side ackflow (suscript 2, flowrate in a fraction a of the tues), and the laterally mixed shell-side flow (suscript 3, assuming no ackflow or axial mixing). The (normalized) temperatures from this analytical solution can e written as: T j (z)=a j +B j exp(r 2 z)+ C j exp(r 3 z) j=,2,3 () Normalized axial coordinate: < z <. The roots of the characteristic equation (r =), r 2, r 3, and the coefficients A j, B j, and C j are listed in the appendix. These coefficients have een otained from the differential equations and from the oundary conditions: T ()=T 2 (), (+)T ()=+ T 2 (), T 3 ()= (2, 3, 4) The variation of the heat transfer coefficients with individual flowrates has not een taken into account in this calculation for the sake of simplicity. NTU, therefore, has een assumed to e inversely proportional to the flowrate. Figure 9 shows the results of such a calculation from eqns. (-4) and the coefficients as given in the appendix. In the paper y Aicher et al. (999) the calculations were carried out including the effects of flowrates on the heat transfer coefficients (htc), ut for a demonstration of the principle, the assumption of constant htc may e acceptale here. channels cannot e used directly in undles. The flow patterns occurring in undles differ from those in single channels in as far as flow in two opposite directions in these cases is not oserved within one channel, ut only in separate tues. From Figs. 8 and 9 we have seen, that ackflow in part of the tues, due to superimposed natural convection, may consideraly decrease the performance of a vertical counter current heat exchanger, especially so, if operated in the high NTU, high efficiency range. In order to avoid ackflow in these cases, it is to e recommended to have the hot end of the heat exchanger at the top, not the ottom. Internal circulation, however, does increase the local flowrates and, therefore, may increase the overall performance in other cases. In any case it is clear, that the influence of natural convection should e taken into account in heat exchanger design. Yet, so far nearly nothing on that topic is found in the textooks as well as in the handooks of heat transfer and heat exchanger design. To check, whether ackflow in some tues is possile, we need critical Rayleigh numers, which depend on the Graetz numer in laminar flow, and additionally on the Prandtl numer and d/l in turulent flow. Only recently, Nickolay (2) has developed a Ra I, Ra II vs. Graetz-chart from analytical calculations as well as experimental data with tue undles. Ra no stale steady state solution Ra II. no ackflow + NTU=5.8 = a=.7 =.7 Gz.6 Ra I + z T.4.2...2.4.6.8. z Fig. 9. Temperature distriution with ackflow in a vertical shell-and-tue heat exchanger MAY BACKFLOW IN A TUBE BUNDLE ALSO BE USEFUL? It has een shown y a numer of relatively simple examples, that the results for natural convection in single Fig. Nickolay s staility chart for laminar mixed convection in vertical tue undles. The Rayleigh numer here is defined with d 4 /L as the characteristic volume. For Ra < Ra I no ackflow is oserved. For Ra > Ra II there exists no stale solution. In the range etween the two limits stale solutions with a certain numer of tues in ackflow do exist. The limiting value for low Graetz numers is Ra min =768. Extensions into the turulent flow regime have also een calculated y Nickolay (2). The curves in Fig. have een otained from the analytic formulae: 5

Ra I =32/(Θ/Gz) (5) Ra II =32/(dΘ/dGz) (6) and an empirical approximation for the volumetric mean temperature Θ = Θ vm, otained from a volume averaged enthalpy content which is different from the caloric mean temperature (corresponding to an average enthalpy flux): Θ=[+(24/Gz) 5 +(8/Gz) 5/3 +(3/Gz) 5/2 ] -/5 The derivative dθ/dgz in Ra II can e found analytically from this formula. This approximation has een shown to e in very good agreement to the more rigorous series calculations y Nickolay (2). Furthermore the experimental oservation of ackflow in tue undles verified the existence and importance of these two limiting Rayleigh numers. When increasing the temperature difference in Ra= gd 4 β T/(Lνκ), ackflow in one or more tues is oserved, when Ra II is reached. When reducing Ra, ackflow remains stale and only vanishes, when Ra I is reached! In the range etween the two critical Rayleigh numers, the actual flow pattern, and the performance of the apparatus depends on the history of its operation efore! A NEW INCENTIVE FOR MORE COMPACT HEAT EXCHANGERS? CONCLUSIONS Stale operation without ackflow can always e reached, if the Rayleigh numer stays elow Ra min =768. Scaling down an existing (shell-and-tue) heat exchanger to a smaller volume, when keeping its flowrates (nud 2 ), efficiency and pumping power constant, requires in the laminar flow range that (7) a) NTU and, therefore, Gz is kept constant Gz=(nud 2 /κ)/(nl), or nl=c t (8) ) p=32ηl(nud 2 )/(nd 4 ), or nd 4 /L=c h (9) From these scaling laws, one can see, that the length L decreases inversely with the numer n of parallel tues, while, eliminating L from eqns. (8) and (9) shows, that d 2, too, is inversely proportional to n. So douling the numer of parallel tues in a undle, will decrease L, d 2, and the Rayleigh-numer Ra y a factor of one half. Compact heat exchangers are definitely less affected y unwanted ackflow prolems, than the classical shelland-tue heat exchangers with tue diameters in the order of centimeters. NOMENCLATURE Latin symols a part of tues in ackflow, A area, m 2 ackflow ratio, c constants d diameter (gap width), m Gz Graetz numer, Gz=RePr d/l L length, m m dimensionless flowrate (Fig. 5), M shell-side NTU, n numer of parallel tues, N tue-side NTU, Nu Nusselt numer, αd/λ Pr Prandtl numer, Pr=ηc p /λ Ra Rayleigh numer, Ra=gL 3 β T/(νκ) (with L 3 replaced y d 4 /L or d 2 L if appropriate) Re Reynolds numer, Re=ud/ν u flow velocity, m/s z coordinate in flow direction, Greek symols α heat transfer coefficient, W/(m 2 K) λ thermal conductivity, W/(m K) κ thermal diffusivity, m 2 /s η viscosity, Pas ν kinematic viscosity, m²/s ρ density, kg/m 3 Suscripts tue-side upflow 2 tue-side ackflow 3 shell-side flow h hydrodynamic task t thermal task REFERENCES Aicher, T., Zur üerlagerten freien und erzwungenen Konvektion in lotrechten Rohründelwärmeüertragern, PhD Dissertation, Fortschr. Ber. VDI Reihe 9, Nr., Düsseldorf, 997. Aicher, T., Martin, H., New correlation for mixed turulent natural and forced convection heat transfer in vertical tues, Int. J. Heat Mass Transfer, 4 (997) 5, 367-3626. Aicher, T., Martin, H., Polt, A., Rayleigh-Bénard convection in vertical shell-and-tue heat exchangers, Chem. Eng. Process. 38 (999) 579-584. 52

Martin, H., Rayleigh-Bénard-Konvektion in Rohründeln, Archive of Applied Mechanics 62 (992) Martin, H., Pajak, D., Rayleigh-Bénard Convection in Tue Bundles, Heat Transfer 994, Proc. th Int. Heat Transf. Conf. Brighton, UK, Vol. 7, 6-NC-2A, pp. 3-7. Nickolay, M., Naturumlaufströmungen in durchströmten Rohr- und Spaltündeln, PhD Dissertation, Aachen, Shaker, 2. Taylor, G.I., cited from: Wooding, R. A., The staility of a viscous liquid in a vertical tue containing porous material. Proc. Roy. Soc. London, Ser. A 252 (959) 2-34. Unger, J., Ausgeildete Kanalströmung zwischen Behältern unterschiedlicher Temperatur, ZAMM 6 (98) T23-T233. APPENDIX Coefficients, to e used in eqns. (-4) for the temperatures in a shell-and-tue hx with ackflow N_ = N a/(+) N_2 = N (-a)/ N_3 = M a N_4 = M (-a) NA = - (N_ N_2 N_3 N_4)/2 NB = ((N_ + N_2)^2 + (N_3 + N_4)^2-2 (N_ + N_2) (N_3 N_4))^(/2) 565-57. r_2 = NA + NB/2 r_3 = NA NB/2 e_2 = exp(r_2) e_3 = exp(r_3) K_B = N_/(r_2 + N_) K_B2 = N_2/(r_2 N_2) K_C = N_/(r_3 + N_) K_C2 = N_2/(r_3 N_2) K = e_2 e_3 ((K_B + K_B2) (K_C + K_C2)) - e_2 (((+) K_C + K_C2) (K_B + K_B2)) + e_3 (((+) K_B + K_B2) (K_C + K_C2)) A_ = A_3 A_2 = A_3 A_2 = e_2 e_3 ((K_B + K_B2) (K_C + K_C2))/K B_ = K_B B_3 B_2 = - K_B2 B_3 B_3 = e_3 (K_C + K_C2)/K C_ = K_C C_3 C_2 = - K_C2 C_3 C_3 = - e_2 (K_B + K_B2)/K 53