MODELLING OF HEAT TRANSFER DURING

5th European Thermal-Sciences Conference, The Netherlands, 2008

MODELLING OF HEAT TRANSFER DURING REFLUX CONDENSATION INSIDE RECTANGULAR CHANNELS AND EXPERIMENTAL VERIFICATION 1

T. Klahm1 , H. Auracher1 , F. Ziegler1 Berlin University of Technology, Institute for Energy Engineering, Germany

Abstract Reflux condensation refers to the condensation process in which the vapor enters an inclined or vertical condenser at the bottom and flows upward while the condensate flows downward due to gravity. In this study the heat transfer during reflux condensation in a rectangular channel of a hydraulic diameter of 7mm is modelled using the assumptions of the classical Nusselt theory for laminar film condensation on a vertical plate. The model is validated with experimental results for the refrigerant R134a. The film thickness distribution on the side walls of the channel is calculated numerically. The calculated results show that low inclination angles enhance heat transfer because of the smaller vertical flow length and hence smaller mean film thickness of condensate. Contrary to the tube geometry with the same hydraulic diameter heat transfer inside the rectangular channel with an inclination angle of 90◦ is underpredicted by the classical Nusselt theory.

Nomenclature Only symbols are defined which are not included in the common list of symbols in Journal of Heat Transfer (1999). Symbols B width of the channel (m) P Point in coordinate system U Circumference of the channel (m) Φ Flow length(m) ∆ Difference Superscript + dimensionless

1

Subscript m S W plate side top

Mean value Saturation state At the wall Plane of the plate Side wall Top plate

Introduction

Reflux condensation is a gravity controlled system. Vapor enters a vertical or inclined oriented condenser at the bottom and flows upward. The condensate flows downward counter-currently to the vapor due to gravity. Compact plate heat exchangers are increasingly used for reflux condensation. The hydraulic diameter of flow channels formed between the plates is between 5 and 10 mm and they can be inclined. To study reflux condensation in small channels and especially the effect of inclination Fiedler & Auracher (2004) carried out experiments with R134a in a test tube of 7 mm inner diameter and 500 mm length as an idealised single subchannel of a compact condenser. Other studies on reflux condensation with vertical tubes (Chen & Tien (1987); McNaught & Moore (1989)) and inclined tubes (McNaught & Moore (1989); Gross (1987); Gross (1992)), respectively, are focused on tubes with larger diameters. The reflux condensation in inclined closed two-phase thermosyphons is also a topic of research work. Those studies are, however, mainly focused on the influence of the inclination angle on


the maximum heat transport capability, i.e. on the maximum performance which is limited by flooding, see e.g. Huanzhuo et al. (1997), Groll & Roesler (1992) and Chen & Tien (1987). Fiedler & Auracher (2004) found that the inclination angle has a significant effect on heat transfer. The optimum inclination angle from the horizontal for heat transfer was found to be close to 45◦ . Here the heat transfer for R134a is increased by a factor of nearly 2 compared to a vertical arrangement. The reason is the nonuniform condensate film distribution around the tube perimeter in the inclination case. The film is very thin and consequently the heat transfer very good in the upper parts of the tube. Only at the bottom a thicker film exists where the main part of the condensate is drained to the lower tube end. A model for the condensation in an inclined tube supports this phenomenological explanation (Fiedler & Auracher (2004)). The model was used to calculate the film thickness at the tube wall based on the classical Nusselt theory (Nusselt (1916)) applied to the cylindrical geometry. The former results for a circular tube let us assume that the positive effect of inclination on heat transfer should be even better if a channel with a cross section is used where the ratio of thin film area with respect to the area occupied by the film at the bottom is larger. In this report the simple case of an inclined rectangular channel is studied where the wider walls are vertically arranged. The same hydraulic diameter is used as by Fiedler & Auracher (2004).

2

Model approach

The rectangular channel is divided into three parts: The side walls (height H), the bottom plate (width B) and the top plate (width B). Design-details of the test section are given in chapter 4.2. The channel is inclined (see figure 1) and the condensate that forms on the side walls flows vertically down due to gravity. It is collected at the bottom plate of the channel. The film thickness and the corresponding heat transfer resistance on the bottom plate is much higher than on the side walls and the amount of heat transferred at the bottom plate is therefore small. The top plate of the channel is idealized as an independent heat transfer surface. It is assumed that the liquid that condenses on that plate does not mix with the liquid at the side walls. Local heat transfer coefficients inside the channel at the side walls and at the top plate are calculated. The heat transfer at the bottom plate is assumed to be neglegible. Fiedler & Auracher (2004) showed that in reflux condensation the heat transfer inside an inclined tube can be calculated utilizing the classical Nusselt theory. Consequently the same is assumed for the rectangular channel. According to Nusselt the local heat transfer coefficient of the film can be calculated with k . (1) h= δ The thickness of the film (δ) is very small and there is no convective mixing inside it. The interaction between vapor and the liquid film is negligible. Another assumption is that all fluid properties are constant. In some cases where the change of a property with temperature is large, temperature dependent fluid properties have to be used. According to Nusselt the film thickness is a function of flow length Φ "

#1/4

4 · kl µl (TS − TW ) δ= Φ ρl (ρl − ρg )g · ∆hlg

.

(2)

Nusselt developed this equation using a vertical plate. The edge where the condensation process started was horizontal and the liquid flow was parallel. Consequently the film thickness along the width of the plate is constant. In the present case there are two vertical walls inside the rectangular cross section each with height H (see figure 1). This length coincides with the flow length Φmax if the inclination angle is 0◦ . In the case of 90◦ inclination angle Φmax


Figure 1: Definition of flow length and coordinate system

coincides with the length L of the channel and in both cases the film thickness is constant along the corresponding width of the plate L and H, respectively. With an inclination angle above zero two different areas of the plate exist. Even on an inclined plate the fluid flows down vertically because it is driven by gravity. The development of the condensate film starts at the topmost point on the plate. Figure 1 shows a coordinate system, which defines this edge as P(z,x=0). Along this line the film thickness is zero. The flow length to the bottom on most of the x0-area reaches the maximum flow length Φmax . The other part of the plate (z0-area) is characterised by a condensate film starting to develop at the edge (z=0,x) but the flow length to the bottom is smaller than the maximum length. Using the coordinate system in figure 1, Φ is calculated in every point P (x, z). For the x0-area: x+ · H Φ(x , z ) = cosβ +

+

for

z+ · L x+ · H ≥ sinβ cosβ

!

(3)

and in the z0-area by z+ · L Φ(x , z ) = sinβ +

+

for

x+ · H z+ · L < sinβ cosβ

!

.

(4)

The expressions in parenthesis denote the mathematical expression that was used to distinguish between the x0 and the z0-areas. To generalise the results, x and z have been turned into dimensionless parameters x+ = Hx and z + = Lz . Knowing Φ, the liquid film thickness across the whole plate is calculated from equation (2) and the local heat transfer coefficient from equation (1). Calculating the integral mean of the local values gives a mean heat transfer coefficient of the side wall hm,side . The mass flow on the side wall in x direction M˙ x and z direction M˙ z is calculated by numerical integration of dM˙ x = and dM˙ z =

ρl (ρl − ρg )g · cosβ 3 δ (x = H, z) · dz 3µl

ρl (ρl − ρg )g · sinβ 3 δ (x, z = L) · dx 3µl

(5)

,

(6)

respectively. They are summarized to the overall mass flow of the side wall M˙ side . The plate on the top of the channel is modelled using equation (2) with Φtop = z. To accommodate to the case of an inclined plate the gravity constant was replaced by the component that is in the plane of the plate: gplate = g · sinβ (7)


Using equation (7) and Φtop = z with equation (2) and (1) a local heat transfer coefficient was calculated. The integral mean of the local values is given as 4 ρl (ρl − ρg )g · kl3 ∆hlg sinβ = 3 4 · µl (TS − TW ) L "

hm,top

#1/4

.

(8)

The mass flow on the top plate M˙ top is given by ρl (ρl − ρg )g · sinβ 3 δ (z = L) · B M˙ top = 3µl

(9)

The comparison with experimental values in section 5 was done using the integral mean of hm,side and hm,top : Atop Aside hm = 2 · hm,side · + hm,top (10) A A with Aside = H · L, Atop = B · L and A = 2 · Aside + Atop . hm was used to calculate Nu (equation 15). Re (equation 16) was calculated with the overall mass flow M˙ l inside the channel which is the sum of 2 · M˙ side + M˙ top .

−5

x 10 4 2 0 1

1

0.5

+

x

+

0 0 z

Film thickness (m)

Model results Film thickness (m)

3

−5

x 10 4 2 0 1

1 0.5 0 1

1 0.5

Film thickness (m)

Film thickness (m)

−4

(c) β = 88◦

0 0

+

z

0.5

(b) β = 60◦

x 10

x+ 0 0 z+

+

x

(a) β = 10◦

0.5

1

0.5

0.5

−4

x 10 1 0.5 0 1

0.5

1 x+ 0 0 z+

0.5

(d) β = 90◦

Figure 2: Film thickness distribution on one side wall of the rectangular channel, R134a at 0.7 MPa, TS − TW = 3K

An example of the film thickness distribution on one side wall of the channel is depicted in figure 2. There are four diagrams that show the development of the film with changing inclination angle, constant fluid properties and temperature difference. In figure 2a the inclination angle is 10◦ to the horizontal. The film thickness distribution is almost the same as with a non inclined surface with the upper edge at z + = 0. With increasing inclination angle this picture changes not much. At 60◦ it is apparent that the z0-area marked in figure 1 becomes visible. There is


a steep increase of the film thickness in this area which flattens until the value of the x0-area is reached. With inceasing inclination angles the z0-area grows. At an inclination angle of 88◦ both areas are almost equally sized. The z0-area increases rapidly with increasing angel and finally it covers the whole plate at an inclination angle of 90◦ . It should be noted that the strong growth of the z0-area appears at different inclination angles if the ratio of length and height of the side walls is different. Consider for example a channel with square side walls. In this case the x0-area and z0-area are equally sized at an inclination angle of 45◦ . Inclination angle and aspect ratio of the sidewalls are related by const =

H · tanβ L

.

(11)

Thus, any aspect ratios are included in the presented film thickness distribution. Only the inclination angle has to be adjusted accordingly. Beside the contribution of different flow areas figure 2 reveals an important information about the maximum film thickness. Comparing figures 2a-d an increase of the maximum film thickness can be recognized from a to d. The film thickness is related to the flow length which increases with inclination angle. In section 5 the comparison with the experimental results will reveal that the decreased flow length at small inclination angles is the main reason for the improvement of heat transfer in inclined passages.

4 4.1

Experiments Setup

Details of the test loop are presented in Fiedler & Auracher (2004) and Klahm et al. (2007). It consists of an evaporator where overheated vapor (temperature is kept approximately 1K above saturation temperature) is produced, a plate heat exchanger that condenses residual vapor which was not condensed inside the test section, a vessel to collect the liquid and control the system pressure and a gear pump to charge the evaporator with the amount of liquid mass flow that is needed inside the test section. The vapor mass flow rate was measured with a Coriolis type flow meter(accuracy ±0.15%). The mass flow rate of the condensate is determined by collecting the liquid for a known period of time (accuracy ±1%). Inlet and outlet temperatures of the test section (see section 4.2) are determined with platine resistance thermometers of 1 mm diameter (accuracy ±0.1◦ C). The pressure difference in the test section is determined by a differential pressure transducer with a range of 0-50 mbar and an accuracy of ±0.25%. The pressure transducer that measures the system pressure is located at the inlet of the test section. 4.2

Test section

In figure 3a) a schematic of the rectangular test channel with the surrounding circular channel for the cooling water is depicted. The length (L) is 500 mm and the cross sectional area 15mm x 4.6 mm (H x B) corresponding to a hydraulic diameter of 7 mm. This enables a comparison with former results obtained with a circular test section (Fiedler & Auracher (2004)) of 7 mm inner diameter shown in figure 3b). 6 thermocouples of type K (Ni, Cr-Ni) with 0.25 mm diameter are installed in each of 5 cross sectional areas along the rectangular test channel (figure 3a) which consists of two parts, sealed in the contact area. The distance of the thermocouple tips to the inner wall is 0.25 mm. The leads are silver soldered into axial grooves along a distance of 5 mm and then taken out through the water flow channel and the outer wall where they are sealed to the outside. The cooling water flows countercurrently to the vapor. It enters and leaves the outer channel via 4 tubes on each side. Its flow rate is determined with a calibrated


C o o lin g W a te r IN

C o o lin g W a te r IN 5 0 m m

5 0 m m

1 5 0 m m

1 5 0 m m

4 5 0 m m

4 5 0 m m

T h e r m o c o u p le

C o o lin g W a te r O U T

C o o lin g W a te r O U T

a )

b )

Figure 3: Test section, a) Rectangular channel, b) Circular channel

impeller flow meter (accuracy ±2, 5%). To enable a variation of the inclination angle, the test section is connected by flexible tubes. 4.3

Experimental procedure and data reduction

Refrigerant R134a is used as test fluid. The pressure at the inlet of the test section is 0.7 MPa corresponding to a saturation temperature of 26.7◦ C. Each test run is carried out at constant system pressure, vapor mass flow rate (0.0005-0.0016 kg/s) and inclination angle β (30◦ - 90◦ to the horizontal). By varying the temperature of the cooling water inside the cooling jacket of the test section the condensate flow was increased. The maximum condensate flow rate is reached when flooding occurs (Liquid is dragged upward by vapor flow, see Fiedler & Auracher (2004)). The onset of flooding can be recognized by a sharp increase in pressure drop. All measurements are done without reaching the onset of flooding. Vapor mass flow rate was found to have no influence on the condensation process. It was adjusted mainly to avoid flooding or total condensation. Thus only the inclination angle was altered between the runs. The mean wall temperature TW is calculated as an arithmetic mean of the 30 thermocouple temperatures inserted into the channel wall. The difference to the inner wall surface temperature is not more than 0.1 K. This difference is within the uncertainty of the measurement and therefore not taken into account. The heat transfer coefficient is determined according to hm =

Q˙ UL(TS − TW )

.

(12)

The circumference of the rectangular channel is given by U = 2 · (B + H)

.

(13)

For L, B and H see figure 1. The total heat flow rate Q˙ is taken from the measured condensate flow rate M˙ l : Q˙ = M˙ l ∆hlg . (14) The results are presented in dimensionless Nu vs. Re-plots. As usual in studies on condensation, the Nusselt number is defined as Num =

hm (νl2 /g)1/3 kl

(15)


and the Reynolds number of the condensate is given by M˙ l Re = . (16) U · µl All fluid properties were determined from NIST´s Refprop software (Lemmon et al. (2002)) at test section inlet conditions.

5

Comparison of theory and experiment 0.8

Num

0.6

°

Model 10° 30° 45° 60° 90°

45 tube °

30

°

45

°

60

0.4

°

90

0.2 Num=0.925⋅(Re)−1/3

0 0

50

100 150 Re Figure 4: Nu vs. Re, experimental (symbols) and theoretical (lines) results, R134a at 0.7MPa

Figure 4 shows the experimental and theoretical results for refrigerant R134a. The experimental results (symbols without lines) show that the Nusselt number decreases with increasing Reynolds number as to be expected because of the increasing film thickness with increasing Re. The filled triangles show the maximum Nusselt-numbers of reflux condensation inside a tube with the same hydraulic diameter (Fiedler & Auracher (2004)). The heat transfer is significantly better in the rectangular channel. The inclination angle has a significant effect on heat transfer. At low inclination angles Nu is larger than at high inclination angles. The worst heat transfer results are encountered at vertical orientation. The solid lines represent calculated Nusselt numbers using equations (10) and (15) as outlined in section 2. The model predicts the heat transfer at small inclination angles of 45◦ and 30◦ and small Reynolds-numbers quite well. In this range the film thickness distribution changes not much as shown in figure 2a and b. The small change of film thickness inside the z0-area has little effect on heat transfer. Nevertheless the mean film thickness increases with increasing inclination angle because the liquid passes the increasing vertical √ distance Φmax (see figure 1). Due to Nusselt’s theory the mean film thickness grows with 4 Φmax . Thus the longer the distance Φmax the bigger the mean film thickness and the smaller the heat transfer coefficient. At Reynolds-numbers above 80 the model underpredicts the experimental Nusselt-numbers slightly. A possible reason may be an increased interaction of vapor and liquid with increasing waviness of the film surface and hence convective transport inside the film. At higher inclination angles e.g. 60◦ the deviation between theory and experiment reaches 10% and the values at 90◦ inclination are underpredicted by almost 50%. In figure 4 it is shown that the numerical calculation coincides with the well known Nusselt-correlation for a vertical wall as expected. However, the assumption that the vertical channel could be regarded as four separate vertical walls where an undisturbed film-flow is established is not supported by experimental results. Possible reasons are a contribution of convective heat transfer inside the falling film due to vapor shear at the film surface because of thicker films at 90◦ or a kind of Gregorig suction effect due to the radius of the film surface in the corners of the rectangular channel. This is under further investigation.


6

Conclusions

The dependence of heat transfer from inclination angle during reflux-condensation of R134a inside a rectangular channel of 7mm hydraulic diameter was modeled and experimentally investigated. Heat transfer inside inclined channels is significantly better than in vertical orientation. The increase of heat transfer coefficients results from a lower mean film thickness at the side walls of the channel which is attributed to the smaller vertical flow path of condensate. The numerical model and the experimental results agree in the range of 30◦ to 45◦ . The heat transfer in the range of 60◦ to 90◦ is underpredicted by the model. Overall, heat transfer in rectangular channels is better than in tubes with equal hydraulic diameters. Contrary to results in tubes the Nusselt theory does not hold for vertical channels. Reasons are under investigation.

Acknowledgements We like to thank the DFG for financial support, Clariant Inc. and Solvay Inc. for donations of technical fluids and refrigerants, respectively.

References Chen, S. L. & Tien, C. L. (1987), ‘General film Condensation Correlations’, Exp. Heat Transfer 1, 93–107. Fiedler, S. & Auracher, H. (2004), ‘Experimental and Theoretical Investigation of Reflux Condensation in an Inclined Small Diameter Tube’, International Journal of Heat and Mass Transfer 47(19-20), 4031–4043. Groll, M. & Roesler, S. (1992), ‘Operation principles and performance of heat pipes and closed two-phase thermosyphons’, Non-Equilibrium Thermodynamics 17, 91–151. Gross, U. (1987), ‘Experimentelle Untersuchung des W¨arme¨ uberganges bei der R¨ uckstromKondensation in einem geneigten Rohr’, Chem.-Ing.-Tech. 59(2), 168–169. Gross, U. (1992), ‘Reflux condensation heat transfer inside a closed thermosyphon’, Int. J. Heat Mass Transfer 35(2), 279–294. Huanzhuo, C., Tongze, M. & Groll, M. (1997), Performance limitation of micro closed twophase. thermosyphons, in ‘Proc. 10th International Heat Pipe Conference’, Stuttgart. Journal of Heat Transfer (1999), ‘Common list of symbols’, Journal of Heat Transfer 121(4), 770–773. Klahm, T., Auracher, H. & Ziegler, F. (2007), Heat Transfer to R134a and R123 during Reflux Condensation in Rectangular Channels, in ‘Proc. 22nd IIR International Congress of Refrigeration’. Lemmon, E. W., McLinden, M. O. & Huber, M. L. (2002), ‘NIST Reference Fluid Thermodynamic and Transport Properties (REFPROP), Version 7.0’, Software on CD-Rom. McNaught, J. M. & Moore, M. J. C. (1989), Reflux condensation in Vertical Tubes, Technical report, ESDU Data Item No. 89038, Heat Transfer Sub-series. Nusselt, W. (1916), ‘Oberfl¨achenkondensation des Wasserdampfes’, Zeitschrift des VDI 60(27).