Thermal Design of Condensers ; the Limiting Case of Zero Mass Transfer Resistance

W. ROETZEL, Pretoria, Südafrika

Abstract. The validity of a condensing curve is accepted. As Zusammenfassung. Es wird die Gültigkeit einer Kondensations- an alternative to the incremental calculation of the heat trans- kurve angenommen. Als Alternative zu einer schrittweisen fer area and pressure drop a rapid calculation method is pre- Berechnung von Austauschfläche und Druckverlust wird eine sented, in which the local overall heat transfer coefficient schnelle Berechnungsmethode beschrieben, bei welcher der and differential pressure drop is calculated at only three points örtliche Wärmedurchgangskoeffizient und differenzielle Druck- (Simpson's rule). In the case of desuperheating, total condens- verlust nur an drei Stellen berechnet werden muß (Simpson- ing and subcooling there is one such point in each of the three Regel). Bei überhitztem Dampf mit vollständiger Kondensa- sections. tion und Kondensatunterkühlung liegt je ein Berechnungs- punkt im Überhitzungs-, Naßdampf- und Flüssigkeitsgebiet.

Nomenclature

Symbols B boiling point Cp specific heat capacity at constant pressure (of the b just on the verge of condensing (Tw = Tq) hot stream) C convection (forced) F heat transfer area D dew point h specific enthalpy of the hot stream E effective k local overall heat transfer coefficient g just containing condensate throughout entire cross- m mass flow rate (of hot stream) section (Tmax = TB) T temperature, if not indexed with s, w or max i counter for reference points 1 and 2 L adiabatic mixing temperature (of hot stream) liquid or condensate film m X, Y, Z dummy variables middle reference point R radiation a local heat transfer coefficient (of the hot stream) s film surface or at interface A finite difference determined at the temperature T S thickness Ts s V vapour X thermal conductivity w at the wall surface (on the hot side)

Subscripts Superscripts 0 uncorrected ' cold stream 1, 2 reference points * special reference value

1. Introduction

Recently [1] it was shown how non-isothermal con- a finite mass transfer resistance is taken into account. densers can be calculated in the limiting case of in- Therefore this paper shows also how the heat transfer finite mass transfer resistance, the condensate precip- area and the pressure drop can be determined with a itating as fog. rapid calculation method. The current paper investigates the other limiting case of zero mass transfer resistance in both phases, 2. The Change of State of the Hot Stream the entire condensate precipitating at the cooled wall or the condensate film surface. We consider a countercurrent heat exchanger in This limiting case occurs only when a single vapour which inside the tubes a superheated vapour mixture is condensed; but also with mixed vapours without is first desuperheated then completely condensed and inerts present the mass transfer resistance may be very finally the condensate is subcooled. The coolant is a low so that this limiting case then represents a good single phase in turbulent flow. We assume that the approximation for the actual problem. At least it can enthalpy and the composition of vapour and con- be regarded together with the first case [1] as a good densate are functions only of temperature, and that starting point for a rapid calculation method in which these functions are given in functional or tabular form W. Roetzel: Thermal Design of Condensers; the Limiting Case of Zero Mass Transfer Resistance 61

mv/m mum temperature for turbulent single phase flow can be calculated according to [1] Eqs. (14) and (18). In the following chapter we look at the condensa- tion which takes place between the points "b" and

hB hn h hB 3. Incremental Calculation of the Heat Transfer Area Fig. 1. Condensing curves for enthalpy and vapour fraction 3.1 The Local Heat Balance Fig. 1 illustrates an example of the dependence of the enthalpy on temperature and mass fraction of The total heat flux occurs between the film surface vapour. When the vapour and the condensate have a at temperature Ts and cold stream at temperature T'. uniform temperature throughout the cross-section the This heat flux causes the enthalpy change: condensation begins at the point where the vapour temperature T(TV) is equal to the dewpoint Tp. The - (Ts - T') • d F condensation is completed when the boiling point m • d h (2) la' + w w l temperature Tg of the mixture is reached. In the 1 S /A + 6 Al special case of a single vapour T0 = Tg and the con- which is equal to that of the cold stream (no heat densing curve is a horizontal line between and hB. losses to the vicinity) In the case of a real condenser the enthalpy of the hot stream is given by m • dh = m' • Cp' • dT' . (3)

mv mi From the vapour core to the film surface sensible h = hT + -T • Cpv • + —• Cp • s m (Tv - Ts) L (Tl - Ts) heat is transferred by convection and (usually negli- gible) radiation. This heat flux cools the superheated superheating sub cooling (1) vapour in the flow direction and desuperheats the condensing vapour from the vapour core temperature where hxs is the enthalpy according to the condensing Tv to the film surface temperature curve evaluated at the condensate film surface temper- ature Ts. - a (Tv - Ts) • dF = mv • Cpv • d Tv + The condensation begins at that point of the heat exchanger where the wall temperature (on the vapour + Cpv-(Tv - Ts)-dmv . (4) side) is equal to the dewpoint temperature. This is The heat transfer coefficient a is composed of one for point "b" (TWjb = Td) as already defined in [1] and shown in Fig. 2. radiation and one for convection (turbulent forced The point "g" is that point of the heat exchanger convection). where the total vapour is just condensed. With film condensation and the film thickness growing gradually a = aR + ac . (5) in the flow direction the maximum temperature in the centre of the cross-section at point "g", from whereon The convective coefficient ac has to be determined the tube contains condensate throughout, is equal to from the usual heat transfer coefficient ac 0 by the correction for the mass flux to the wall [2, 3, 4]: the boiling point, Tmax>g = TL?maX;g = TB. The maxi-

<*C = <*C,0 ' Y (6)

\ where

Tb \ \l/^X Tmax Cp •d m \ i —U v v Z = (7) Y, ac,o • d F Tg ^^ T0 For condensation, this dimensionless group is positive because dm /dF is negative. g b v Heat transferred Combining Eqs. (4) to (7) leads to: Fig. 2. Longitudinal temperature profiles for countercurrent m • Cp • dT flow v v v <*E ' (Tv - Ts) • dF (8)

Wärme- und Stoffübertragung 7 (1974) No. 1 62 W. Roetzel: Thermal Design of Condensers; the Limiting Case of Zero Mass Transfer Resistance 62

with the effective heat transfer coefficient 4.1 Finding of Tb and Tg

The same considerations are valid as for the finding aE = aR + aC)0 • gZ _ j • (9) of the temperatures Ta and Tj, in [ 1 ] and need not be repeated here. However, an iterative determination of

For small values of Z the correction factor becomes Tb and Tg is inconvenient and should be avoided in a one. rapid calculation method. Assuming in the desuperheating and subcooling zone constant ratios of the heat capacities and transfer 3.2 The Local Heat Transfer Coefficient for coefficients of the hot and the cold fluid leads to the Condensation following iteration-free approach; First in each of the two sections at one reference At points "b" and "g" the heat transfer coefficients point "i", represented by a pair of Tj and Tj which are those for single phase convective heat transfer. fulfil the heat balance, the wall temperature Twl Immediately following the condensation point "b" we (i = 1 for desuperheating zone) and the maximum may assume a very thin laminar film; the local film temperature TLmax = Tmax2 (i = 2 for subcooling zone), resistance and adiabatic mixing temperature Tl can be using Eqs. (14) and (18) of [1] have to be determined. calculated according to [5] and [6] with high accuracy. Then the reference values Tj must be calculated This accuracy decreases with the distance from "b", according to the film tends to become rippled and later turbulent and the film thickness is no longer uniform along the m • Cp • T; — rh' • Cp' • Tj circumference. A calculation of the film resistance (10) rh-Cp - m'-Cp' according to [5] and [6] also in the turbulent region yields a conservative value. Alternatively, the heat where i = 1 for the desuperheating zone and i = 2 for transfer coefficient can be calculated according to the subcooling zone. other correlations developed for turbulent flow regime, The temperatures Tj, and Tb can now be calculated for instance that of Narayana Murthy and Sarma [7], from The local pressure drop can be calculated conserva- tively as for laminar film condensation with the differ- Tb - Tf T{, - Tf Tr TT (11) ence velocity between vapour and film surface (see Ti - Tf 1\ - Tf Twi " Ti [6]), neglecting the deceleration forces.

The heat transfer area (and pressure drop) can be and the temperatures Tg and Tg from calculated incrementally in the usual manner by intro- ducing in Eqs. (2), (3), (7) and (8) the finite differ- Tg - T2 Tg - T2 Tb - TI ences AF, Ah, AT', Amv, and ATV. For the iterative T? - Ti Tmax 2 — Tl*2 (12) procedure at each step one should give the value of

Amv. Then the condensing curve yields immediately For the derivation of Eq. (12) it was assumed that the

the new value of h/j-s and Ts. The actual new enthalpy dimensionless maximum temperature according to according to Eq. (1) or the change Ah can be found Eqs. (14) and (18) of [1] is constant in the range only iteratively usings Eqs. (2), (3) and (7) to (9) considered, which is a good approximation. because the new values of Tv and Tl are not yet known. Once the final value of Ah has been found the small 4.2 Calculation of the Heat Transfer Area and area AF is given by Eq. (2). The sum of all AF — s Pressure Drop is the required total area. Once the points "b" and "g" have been found the areas necessary for desuperheating and subcooling can 4. Rapid Calculation Methods be calculated in the conventional manner. For this calculation the heat transfer coefficients at the refer- For cost-optimized design which already requires ence points "1" and "2", needed for the previous repetitive procedures an incremental calculation is determination of the points "b" and "g", should be prohibitively time consuming and fast but reliable used. calculation methods for the heat transfer area and the This procedure is well known and need not be dis- pressure drop are required. We consider the case of cussed here. The following considerations deal with desuperheating, condensing and subcooling. As is done the condensing section between the points "b" and "g". for the incremental calculation the heat exchanger is Considering the method of Colburn and Hougen [8] broken down in three sections. and the rapid calculation methods developed for single

Wärme- und Stoffübertragung 7 (1974) No. 1 W. Roetzel: Thermal Design of Condensers; the Limiting Case of Zero Mass Transfer Resistance 63 phase heat exchangers [9] to [12], leads with Eqs. (2) The pressure drops with subscripts b, m and g are and (3) to the conclusion that for the integration of calculated with the local properties and flow rates. the area as in [8, 9] and [10] the following variables Replacing p by p'-Eq. (18) is also valid for the cold should be used: stream. The effect of changing pressure on the gas density can be taken into account, as shown in [12],

X = T' (13) The pressure drops Apb and Apg should be deter-

mined, as Yb and Yg, with the friction factors deter- and mined at the reference points "1" and "2" and these friction factors could also be used to calculate the m'-Cp' m'-Cp' J_ Sw 5l Y (14) pressure drops of the desuperheating and subcooling k-(T-T') Ts — T' a' Xw zones. Thus the heat transfer area and frictional pressure where drop can be calculated. The local pressure drop and heat transfer coefficients have to be determined only 7b,g=- / Y-dX. (15) at three points: one in the desuperheating, one in the Xb condensing and one in the subcooling zone. The method is also valid for the case of infinite mass trans- Because in the case of total condensation the heat fer resistance discussed in [1], The difference in both transfer coefficients at points "b" and "g" are known, cases is only the local heat transfer coefficient and a three-point integration (using Simpson's rule) as used pressure drop of the vapour in the condensing zone at in [9] is preferred: point "m". This demonstrates that the mass transfer resistance in a mixed vapour condenser can be taken into account by a modified heat transfer coefficient Fb,g = 1 • (Y + 4 • Y + Yg) (X - Xg) (16) b m b lying between the two limiting cases of infinite and zero mass transfer resistance. where Y is determined at that point of the heat m If the vapour is not condensed totally with or with- exchanger where out inerts present the same method can be applied. In that case point "g" is replaced by the vapour out- (17) let and must be treated as point "m"; we have one xm - 2 ' <-Xb + ' single phase point and two condensing points.

This point can be found by the heat balance using

Eq. (3). With T'm and the known constant or temper- 4.3 The Special Case of a Single Vapour ature dependent Cp' first the enthalpy hm at that point can be determined. Incremental calculations have In the special case of a single vapour a higher accu- shown that around that point the superheating and racy can be attained with the same expenditure if the subcooling terms in Eq. (1) can be neglected because following method based on [11] and [12] is applied. in that region both terms are small and partially com- In the procedure described in 4.2 instead of Eq. (13) pensate each other. Then the wanted temperature and (14) the following equations must be used: Tsm follows directly from the condensing curve and Y can be determined according to the righthand term m X = In (Ts - T') , (19) of Eq. (14).

The values of Yb and Yg should be determined (Tg- T'). m'-Cp' 1 •I n I I ^w A) according to the first expression in Eq. (14) using the Y = - - m-Cp • -7 + T— k • (T — T') \ a A overall heat transfer coefficient k. This overall heat w transfer coefficient should be calculated with the heat (20) transfer coefficients again at the reference points "1" and "2" which are available from the previous deter- At points "b" and "g" the first expression of Eq. (20) mination of the points "b" and "g". should be used and at point "m" the second expression. From the derivation of the pressure drop method This method is more accurate for two reasons. Firstly, [12] it follows that this method can be applied also the integration is more accurate with strong changes with the variable Y according to Eq. (14) and with of temperature difference, because Y according to changing vapour flow rate (neglecting deceleration Eq. (20) is then less variable than Y according to forces): Eq. (14). (See also example in [11]). Secondly, the accuracy of the condensation heat transfer coefficient Yb-Apb + 4-Ym-Apn + Yg-Apg Ap = (18) may be higher because now the middle point "m" is Yu + 4 • Y™ + Y„ closer to point "b" than in the general method

Wärme- und Stoffübertragung 7 (1974) No. 1 64 W. Roetzel: Thermal Design of Condensers; the Limiting Case of Zero Mass Transfer Resistance

(Eqs. (13) and (14)) and the condensate film may still 6. Roetzel, W.: Laminar film condensation in tubes; calcula- be laminar and uniform along the circumference (as tion of local film resistance and local adiabatic mixing assumed in [5] and [6]). Thus for a single vapour this temperature. Intern. J. Heat Mass Transfer 16 (1973) method should be preferred to the general one des- 2297/2304. cribed before. 7. Narayana Murthy, V., Sarma, P. K.: Condensation heat transfer inside horizontal tubes. The Canadian Jorunal of Both methods can be applied also for other flow Chemical Engineering 50 (1972) 547/549. arrangements as occur in air-cooled cross-flow heat 8. Colburn, A. P., Hougen, O. A.: Design of cooler-conden- exchangers. Neglecting secondary effects one merely sers for mixtures of vapours with non-condensing gases. divides the area according to Eq. (16) by the ratio of Ind. Engng Chem. 26 (1934) 1178/1182. the true mean temperature difference to that valid for 9. Hausen, H.: Wärmeübertragung im Gegenstrom, Gleich- countercurrent flow (mean temperature difference strom und Kreuzstrom. Technische Physik in Einzeldar- correction factor). stellung. Berlin/Göttingen/Heidelberg: Springer 1950. 10. Hausen, H., Binder,-!.: Vereinfachte Berechnung der Wärmeübertragung durch Strahlung von einem Gas an eine References Wand. Intern. J. Heat Mass Transfer 5 (1962) 317/327. 11. Roetzel, W.: Berücksichtigung veränderlicher Wärmeüber- 1. Roetzel, W.: Thermal design of non-isothermal condensers; gangskoeffizienten und Wärmekapazitäten bei der Bemes- the limiting case of infinite mass transfer resistance. sung von Wärmeaustauschern. Wärme- u. Stoffübertragung Wärme- u. Stoffübertragung 6 (1973) 228/234. 2 (1969) 163/170. 2. Eckert, E. R. G.: Einführung in den Wärme- und Stoffaus- 12. Roetzel, W.: Calculation of single phase pressure drop in tausch. Dritte neubearb. Auflage. Berlin/Heidelberg/New heat exchangers considering the change of fluid properties York: Springer 1966. along the flow path. Wärme- u. Stoffubertragung 6 (1973) 3. Colburn, A. P., Drew, T. B.: The condensation of mixed 3/13. vapours. Trans. Am. Inst. Chem. Engrs. 33 (1937) 197. 4. Marschall, E.: Wärmeübergang bei der Kondensation von Dämpfen aus Gemischen mit Gasen. Abhandlung des Dr. W. Roetzel, Deutschen Kältetechnischen Vereins Nr. 19, Karlsruhe: Chemical Engineering Research Group, C. F. Müller 1960. Council for Scientific and Industrial Research, 5. Roetzel, W.: Heat transfer in laminar film condensation; P.O.Box 395, local film resistance and local enthalpy allowing for vari- Pretoria, South Africa able viscosity and subcooling. Wärme- u. Stoffübertragung 6 (1973) 127/132. Received August 17, 1973