## Correlations for heat exchange and pressure drop coefficients

### Introduction

Given the number and importance of heat exchanger applications in modern societies, whether for energy production, chemistry, the petroleum sector, refrigeration, air conditioning, a growing need has arisen in recent decades to know precisely the values of heat exchange and pressure drop coefficients, especially for boiling or condensing fluids.

Research has been undertaken in many countries on this subject, which has given rise and continues to give rise to a large number of publications on the correlations used to determine them.

One of the great difficulties today is to be able to choose the correlations adapted to a given problem among all those that are published. Some review articles are available, but they are not always reliable. Errors frequently exist in the formulas that are given, as the definitions of the variables taken into account may differ from one publication to another...

In this page, we review the correlations that were added in 2021 to Thermoptim's external classes, referring the interested reader to the publications from which they are derived, as well as to the external classes in which they are implemented.

The versions of Thermoptim that allow you to use these new features are 2.72, 2.82 and Demo 2.82. Previous versions are not compatible.

These correlations relate to the following cases:

simple flows

saturated boiling

nucleate boiling

condensation

exterior of finned exchangers

It should also be noted that these correlations are generally implemented in external classes of type FlowConfig, but they are sometimes implemented in TechnoHX.

This** ****page provides you with some details on how these classes are structured**, in addition to the detailed explanations provided in volumes 3 and 4 of the Thermoptim reference manual.

If the implementations of these correlations do not meet your needs, you can supplement them with other classes built in a similar way.

It is important to bear in mind that a large number of correlations have been published in recent years.

They depend on many parameters, such as the fluid involved, the hydraulic diameter, the orientation and geometry of the pipes...

We have coded a number of them, but it is impossible to implement them all and you may have to introduce others that better suit your needs.

You will be able to do this by adapting existing classes, which is usually not a very difficult task.

Note that the current classes are not optimized in order to provide the results of the different correlations in the result file output.txt.

If you wish, you can easily customize them by removing calculations that you do not need.

Explanations on the development of external classes are given in **this page** of the portal.

### Simple flows

The correlations giving the **Nusselt number** for simple flows are classic: MacAdams inside the tubes, in the IntTubeConfig java class, and Colburn in the ExtTubeConfig java class.

### Saturated boiling

The TechnoEvaporator class used the Gungor-Winterton correlation calculated on an average basis, which was criticizable. It is now implemented by discretizing the evaporation zone into 100 elements, which is much more accurate.

Five new correlations for estimating exchange coefficients during boiling were also implemented, those of (Shah, 1982), (Borishanskiy, 1971), (Kim and Mudawar, 2013), (Kandlikar, 2017) and (Saitoh, 2007).

For **pressure drops**, the correlation of (Lockhart-Martinelli, 1949) is one of the most used, even if it is not among the most precise according to some authors: many publications state that it significantly overestimates them. It would seem that those of (Müller-Steinhagen & Heck, 1986) and (Sun & Mishima, 2009) or that of (Friedel, 1979) are much more precise. All three have been implemented in the TechnoSteamGenerator class.

### Nucleate boiling

The calculation of nucleate boiling is much more delicate than that of saturated boiling. This is a relatively new area of investigation and many publications exist, without any really relevant summaries being available.

As the wall temperature Tw increases, three thresholds appear:

when Tw reaches Tsat, bubbles may begin to appear,

but they only do so with some delay, when Tw = Tw,onb (Onset of Nucleate Boiling). A complementary flow of the purely convective flow then appears q_fc. It is equal to q_nb for nucleate boiling.

boiling becomes complete when Tw > Tw, fdb (Fully Developped Boiling), which corresponds to a quality x greater than or equal to 0.

In the TechnoSteamGenerator class, the ONB is detected by a relation such as (Thom, 1965), and the FDB by a relation such as (Bowring, 1962).

It is considered that the exchange coefficient h_fnb is equal to that calculated for saturated boiling, depending on the correlation chosen.

For the transition between the ONB and the FDB, a ramp is made between the local hfl and hlvf reached during the FDB. More sophisticated methods are proposed in the literature, but they are complex to implement and their robustness is not guaranteed.

The exchanger is divided into 100 parts and the different thermal equilibria are calculated. The wall temperature is determined for each interval, assuming that the exchange coefficient on the hot side remains constant.

Different ONB detection correlations have been implemented, and the one that leads to the highest value is retained. Just replace it with another one if you wish.

In the same way, different correlations of OFB detection (or FDB or NVG) have been implemented, and the one that leads to the highest value is retained. h_fdb is estimated to be equal to that of saturated boiling.

### Condensation

The Nusselt number for laminar condensation **outside a horizontal tube** can be estimated by Nusselt's theory with the gravity modification of Dhir and Lienhard (1971). This correlation is implemented in the ExtHorCondConfigDhir.java class.

A **variant for vertical tubes** is implemented in the ExtVertCondConfig.java class.

The correlation presented by (Lévy, 1990) for **condensation outside horizontal tubes used in steam plant condensers** is implemented in the ExtHorCondConfig.java class.

For the interior of horizontal tubes, the correlation of (Shah, 1979) continues to be often used, although the author later proposed a more complex one. We implemented a variant modified by (Bivens, 1994) in the CondConfig.java class.

### External flow of tube and fin exchangers

Unlike the previous correlations that generally provide the Nüsselt number Nu, those relating to the air side of finned exchangers are generally related to the Chilton–Colburn factor j.

Two correlations can be used if experimental data are available, those of (Morisot, 2002) and of (Martin and Ginielski, 2000).

Three predictive correlations assume a more or less detailed knowledge of the geometry of the finned exchanger. These are those of (Kim and Jacobi, 2000) of (Manglik and Bergles, 1995), and of (Wang, Chi and Chang, 2000). They are implemented in the FinnedAirCoilConfigKJ.java, FinnedAirCoilConfigMB.java, and FinnedAirCoilConfigWCC.java classes.

They use the correlation parameterization screen to define the geometric quantities they need, relating to:

the spacing of the fins Fp,

the outer diameter of the tubes,

the number N of aquifers,

the transverse distance Pt between them,

their spacing Pl.

An example of an explained parameterization is given, in addition to the **simple exchanger** used to illustrate the technological sizing and the off-design calculation mode.

### References

W.W. Akers, H. Deans, O. Crosser, Condensing heat transfer within horizontal tubes, Chem. Eng. Prog. Symp. 54 (1955) 89–90.

Bivens, D. B. and Yokozeki, A., "Heat Transfer Coefficients and Transport Properties for Alternative Refrigerants" (1994). InternationalRefrigeration and Air Conditioning Conference. Paper 263.

Borishanskiy, B.M., Andreevskij, A.A., Fromzel, V.N., Fokin, B.S., Cistgakov, V.A., Danilowa, G.N. and Bikov, G.S., (1971). Heat transfer during two-phase flows (in Russian). Teploenergetika 11, pp. 68-69.

Bowring R.W. , Physical model based on bubble detachment and calculation of steam voidage in subcooled region of a heated channel, Institute for Atomenergi, Halden, Norway, Report No. HPR-10, 1962.

Dhir V. K., Lienhard J. H., Laminar Film Condensation on Plane and Axisymmetric Bodies in Nonuniform Gravity, Journal of Heat Transfer-transactions of The Asme, 39, 97, 1971

Friedel, L., (1979). Improved friction pressure drop correlation for horizontal and vertical twophase pipe flow. European Two-Phase Flow Group Meeting, paper E2, Ispra, Italy.

Ghione A., Assessment and improvements of thermal-hydraulic correlations and methods for the analysis of the Jules Horowitz Reactor, Thesis for the Degree Of Doctor Of Philosophy, Chalmers University of Technology, Goteborg, Sweden, 2017

Gungor, K. E.; Winterton, R. H. S.: A general correlation for flow boiling in tubes and in annuli. Int. J. Heat Mass Transfer 29 (1986) 351–358

Kandlikar, Ch. 8. Boiling, Multiphase Flow Handbook, Second Edition by Crowe, Clayton T. Michaelides, Efstathios Schwarzkopf, John D., 2017

Kim G. J., Jacobi A. M. Condensate Accumulation Effects on the Air-Side Thermal Performance of Slit-Fin Surfaces, ACRC CR-26, University of Illinois at Urbana-Champaign I, 2000.

Kim, S.M. and Mudawar, I., 2013, “Universal Approach to Predicting Saturated Flow Boiling Heat Transfer in Mini/Micro-Channels Part II. Two-Phase Heat Transfer Coefficient,”International Journal of Heat and Mass Transfer, Vol. 64, pp. 1239-1256

Lévy W., Condenseurs par surface dans les centrales thermiques, Article B 1 540, Techniques de l'Ingénieur, 1990

Lockhart, R.W. and Martinelli, R.C.,Proposed correlation of data for isothermal two-phase, two-component flow in pipes. Chem. Eng. Progr. 45 (1949), pp. 39-48.

Manglik, R. M., Gergles, A.E. Heat transfer and pressure drop correlations for the rectangular offset strip fin compac heat exchanger, Experimental Thermal and Fluid Science 1995; 10:171-180, Elsevier

Martin H., Gnielinski V., Calculation of heat transfer from pressure drop in tube bundles and, 3rd European Thermal Sciences Conference 2000, E.W.P. Hahne, W. Heidemann and K. Spindler (Editors),

Morisot, O., Marchio, D. (2002). Simplified Model for the Operation of Chilled Water Cooling Coils Under Nonnominal Conditions. HVAC&R Research 8(2): 135-158.

Müller-Steinhagen, H. and Heck, K., (1986). A simple friction pressure drop correlation for twophase flow in pipes. Chem. Eng. Process. 20, pp. 297-308.

Ohrby, Fredrik, Numerical modeling of subcooled nucleate engine cooling systems, Master's thesis in the Applied Mechanics programme Chalmers University of Technology, Goteborg, Sweden 2014

R. Rabiee, M. Désilets, M. Proulx, M. Ariana, M. Julien, Determination of condensation heat transfer inside a horizontal smooth tube, International Journal of Heat and Mass Transfer 124 (2018) (816-828)

Saitoh, S., Daiguji, H. and Hihara, E., (2007). Correlation for boiling heat transfer of R134a in horizontal tubes including effect of tube diameter. Int. J. Heat Mass Transfer 50, pp. 5215-5225.

Shah, M.M. 1982. Chart correlation for saturated boiling heat transfer: equations and further study. ASHRAE Trans. 88(1):185–196.

Shah, M. A General Correlation for Heat Transfer During Flow Condensation Inside Pipes, Journal of Heat and Mass Transfer 22, 1979, 547-56

Sun, L. and Mishima, K., (2009). Evaluation analysis of prediction methods for two-phase flow pressure drop in mini-channels. Int. J. Multiphase Flow 35, pp. 47-54

Thom J. R. S. , W. M. Walker, T. A. Fallon, and G. F. S. Reising, Boiling in subcooled water during annuli. In Proceedings of Symposium on Boiling Heat Transfer in Steam Generation Units and Heat Exchangers, 1965

Thome J. R and Cioncolini, A. Encyclopedia of Two-Phase Heat Transfer and Flow I, Fundamentals and Methods, Volume 3: Flow Boiling in Macro and Microchannels, 2015

Wang, Chi-Chuan, Chi Kuan-Yu, Chang Chun-Jung Heat transfer and friction characteristics of plain fin-and-tube heat exchangers, part II: Correlation August 2000, International Journal of Heat and Mass Transfer 43(15):2693-2700 DOI: 10.1016/S0017-9310(99)00333-6

Wang, J, Y. Huang, Y. Wang, Visualized study on specific points on demand curves and flow patterns in a single-side heated narrow rectangular channel, Int. J. Heat Fluid Flow 32 (2011) 982–992.