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Comparison work about different empirical formulas for the boiling heat transfer coefficient in separated heat pipe system

Abstract

Thermosiphon is a kind of heat transfer equipment with high thermal conductivity. It can transfer heat for long distance by boiling and condensing. Thus, it is important to study the heat transfer character of the boiling process in the evaporator. In this paper, we have compared mainly seven different heat transfer formulas by other researches and our experiment results. Such formulas include Rohsenow formula (1951), Cooper formula (1984), Labuntsov formula (1973), Kutateladeze formula (1990), Kruzhilin formula (1947), Imura formula (1979) and Hasna formula (2017). Rohsenow formula is derived in theoretical for nucleate boiling process in pool boiling. Cooper formula is mainly for refrigerating coolants. Imura formula is an empirical formula based on the experiment result of the two phase closed thermosiphon. Hasna formula has the same pattern as the Cooper formula but it is used for separated heat pipe system. Labuntsov formula, Kutateladeze formula and Kruzhilin formula are all for the nucleate boiling. Four coolants are used for comparison including R744, R134a, R22 and R410A. The results show that, Imura formula’s calculating results are lower than the experiment results. Cooper formula accords very well with the experiment results for Freon coolant, but for R744, the relative error is as large as 62%. Rosenow formula’s average error is ~29%, and the variation trend is close to the experiment results.

1 INTRODUCTION

The separated heat pipe system which can also be called loop thermosiphon is a kind of heat transfer equipment. There are mainly two kinds of thermosiphon, as can be seen in Figure 1. The first kind is the separated heat pipe system and the second is the two phase closed thermosiphon. Thermosiphon mainly contains four parts: evaporator, condenser, vapor pipe and liquid pipe. Inside the evaporator, coolant absorbs heat and changes its phase from liquid to vapor, then vapor comes to the condenser through vapor pipe, in the condenser, vapor releases heat to heat sink and becomes liquid. Finally, liquid coolant flows back to the evaporator through liquid pipe. The condenser is placed higher than the evaporator, such system can be driven by gravity instead of pump.

The scheme of the separated heat pipe system and two phase closed thermosiphon, adapted from reference [1].

Figure 1.

The scheme of the separated heat pipe system and two phase closed thermosiphon, adapted from reference [1].

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Although these two kinds of thermosiphon are all driven by gravity, there is a little difference in the flow patterns. For two phase closed thermosiphon, the vapor and liquid flows in the opposite direction while for the separated heat pipe system, the flowing direction is the same. In separated heat pipe system, which is also called loop heat pipe system, the coolant flows as a circle. As a result, from evaporator to condenser and from condenser to the evaporator is the same direction, also it can be regarded as the counterclockwise circulation.

Thus, the boiling pattern in the two phase closed thermosiphon is more likely to be the pool boiling, while the separated heat pipe system is the combination of pool boiling and convection boiling. As a result, it is important to study the boiling heat transfer coefficient for the separated heat pipe system.

There are four kinds of coolants used as the working fluid for the heat pipe system in our paper, including: R744, R134a, R410A and R22. Among these four kinds of coolants, R744 (carbon dioxide) is an environmental friendly coolant, however as the working pressure is much higher than Freon, it is not used generally, R22 is a kind of traditional coolant which will be obsoleted in the future because it will threaten the ozone sphere. R134a and R410A are all environmental friendly coolant. Detailed information of different coolants can be seen in Table 1.

Table 1.

Detail information of different coolants.

Coolant	ODP	P(MPa)@20°C	Critical pressure	Relative molecular weight
R744	0	5.73	7.38	44.0
R22	0.045	0.91	4.974	86.5
R134a	0	0.57	4.067	102.0
R410A	0	1.45	4.95	86.0

Coolant	ODP	P(MPa)@20°C	Critical pressure	Relative molecular weight
R744	0	5.73	7.38	44.0
R22	0.045	0.91	4.974	86.5
R134a	0	0.57	4.067	102.0
R410A	0	1.45	4.95	86.0

Note 1: R410A is a kind of non-azeotropic refrigerants, thus the critical pressure is estimated in the T-S diagram.

Note 2: GWP means global warming potential and ODP means Ozone Depression Potential. ODP larger than 0 means such coolant can destroy the ozone layer. The lower the GWP, the more the environmental friendly.

Note 3: Since the GWP of each coolant in different references have different values, we have not listed the GWP in the table. However, the GWP of R744 is much lower than other Freon coolants.

Table 1.

Detail information of different coolants.

Coolant	ODP	P(MPa)@20°C	Critical pressure	Relative molecular weight
R744	0	5.73	7.38	44.0
R22	0.045	0.91	4.974	86.5
R134a	0	0.57	4.067	102.0
R410A	0	1.45	4.95	86.0

Coolant	ODP	P(MPa)@20°C	Critical pressure	Relative molecular weight
R744	0	5.73	7.38	44.0
R22	0.045	0.91	4.974	86.5
R134a	0	0.57	4.067	102.0
R410A	0	1.45	4.95	86.0

Note 1: R410A is a kind of non-azeotropic refrigerants, thus the critical pressure is estimated in the T-S diagram.

Note 2: GWP means global warming potential and ODP means Ozone Depression Potential. ODP larger than 0 means such coolant can destroy the ozone layer. The lower the GWP, the more the environmental friendly.

Note 3: Since the GWP of each coolant in different references have different values, we have not listed the GWP in the table. However, the GWP of R744 is much lower than other Freon coolants.

In order to compare different empirical formulas, we have done an experimental work. In the experiment, the boiling heat transfer coefficient in the separated heat pipe system has been studied. The heating power is ~100–700 W, the tube inside diameter is 8 mm, the length of the evaporator is 1.515 m. For these four coolants, the heat transfer coefficient of R134a has the lowest uncertainty (3.48–8.55%), while R744 has the highest (10.04–25.18%). The experiment details can be seen in reference [1].

2 INTRODUCTION TO DIFFERENT EMPIRICAL FORMULAS AND CALCULATION ERROR ANALYSIS

2.1 Introduction to different empirical formulas

The paper has compared mainly seven different heat transfer formulas and our experiment results. Such formulas include Rohsenow [2] formula (1951), Cooper [3] formula (1984), Labuntsov [4, 5] formula (1973), Kutateladeze [5, 6] formula (1990), Kruzhilin [5, 7] formula (1947), Imura [8] formula (1979) and Hasna [9] formula (2017). Rohsenow formula is derived in theoretical for nucleate boiling process in pool boiling.

Rohsenow’s formula is derived in theoretical [2, 10]. In his theory, the boiling process can be regarded as a special kind of liquid heat transfer process. As the heat is mainly transferred directly from heating surface to liquid and bubbles may enhanced the heat transfer process. Thus, the Nusselt number (Nu) can be written as equation (1).

N u_{l} = C {Re}_{l}^{m} \Pr^{n}

(1)

Here, the diameter and velocity of the fluid needs to know before calculating the Renold number (Re). According to Rohsenow’s [2] theory, the bubble will leave the heating surface when the surface tension (F_t) is equal to buoyancy force (F_f).

\begin{matrix} F_{f} = ρ g V \propto g (ρ_{l} - ρ_{v}) D_{b}^{3} \\ F_{t} = σ L \propto σ D_{b} \end{matrix}

(2)

Thus, the leaving diameter is D_b in equation (3), and V is the characteristic velocity in Re number.

D_{b} \propto \sqrt{\frac{σ}{g (ρ_{l} - ρ_{v})}}, V \propto \frac{q}{ρ_{l} h_{f g}}

(3)

Now, substitute D_b and V in equation (1), we can get equation (4), which is the Rohsenow’s formula:

h = \frac{c_{p l} q^{0.67}}{h_{s f} C_{w l}} {(\frac{1}{μ_{l} h_{s f}} \sqrt{\frac{σ}{g (ρ_{l} - ρ_{g})}})}^{- 0.33} {(\frac{c_{p l} μ_{l}}{k_{l}})}^{- n}

(4)

Cooper [3] has studied the pool boiling heat transfer coefficient for refrigerating coolants and has given his formula, see equation (5). p_r means working pressure/critical pressure. M is the Relative molecular weight. Detailed information can be seen in Table 1. ε is the surface roughness, for common pipe, ε is ~0.3–0.4, in our calculation, as the tube surface roughness is unknown, thus the lgε is omitted. Cooper’s formula is written as equation (6). Such ε can be omitted because the heat transfer coefficient results calculated by equations (5 and 6) are similar. For example, for R744, when heating power is 109 W, the relative difference is ~5%, when heating power is 696 W, the relative difference is ~2%. But for R134a, the relative difference is higher which is ~16–19%.

h_{e} = 55 p_{r}^{0.12 - 0.2 \lg ε} {(- \lg p_{r})}^{- 0.55} M^{- 0.5} q^{0.67}

(5)

h_{e} = 55 p_{r}^{0.12} {(- \lg p_{r})}^{- 0.55} M^{- 0.5} q^{0.67}

(6)

Hasna [9] has obtained his formula based on the cooper’s formula. Thus, Hasna’s formula has the same pattern as the Cooper formula but it is used for separated heat pipe system. In Hasna’s experiment, the working fluid is water and the heat flux changes from ~2 W/cm² to 20 W/cm².

h_{e} = 7704 M^{- 0.5} q^{0.157} {(p_{r})}^{0.12} / {[- \log (p_{r})]}^{0.55}

(7)

Imura [8] formula is an empirical formula based on the experiment results of the two phased closed thermosiphon. For the two phase closed thermosiphon, as the vapor flows upward and liquid flows downward, the boiling pattern inside the evaporator is more likely to be the pool boiling. In his formula, the relationship between the h and thermophysical property is considered. In equation (8), subscript l means liquid, g means vapor, c_p is the specific heat at constant pressure, q is the heat flux, r is the latent heat, μ is viscosity, g is the gravity accelerate, ρ is density, k is the thermal conductivity.

h_{e} = 0.32 \frac{ρ_{l}^{0.65} k_{l}^{0.3} c_{p l}^{0.7} g^{0.2} q_{h}^{0.4}}{ρ_{g}^{0.25} r^{0.4} μ_{l}^{0.1}} {(P_{i n} / P_{a})}^{0.3}

(8)

Labuntsov formula [4, 5] (equation (9)), Kutateladeze formula [5, 6] (equation 10a–c) and Kruzhilin formula [5, 7] (equation (11)) are all for the nucleate boiling.

h_{e} = 0.075 [1 + 10 {(\frac{ρ_{v}}{ρ_{l} - ρ_{v}})}^{0.67}] {(\frac{λ_{l}^{2}}{υ_{l} σ (273.15 + T_{e})})}^{0.33} q^{0.67}

(9)

\begin{matrix} N u_{⁎} = 3.37 \times 10^{- 9} K^{- 2} M_{⁎}^{- 4} (a) \\ N u_{⁎} = \frac{h_{b} l_{⁎}}{k_{l}}, K = \frac{h_{f g} h_{b}}{c_{p} q}, M_{⁎}^{4} = \frac{\frac{σ g}{ρ - ρ_{g}}}{{(\frac{p}{ρ_{g}})}^{2}} (b) \\ \begin{matrix} h_{b} = {[3.37 \times 10^{- 9} \frac{k_{l}}{L_{b}} {(\frac{h_{f g}}{c_{p} q})}^{- 2} M_{⁎}^{- 4}]}^{1 / 3}, & L_{b} = \sqrt{\frac{σ}{g (ρ_{l} - ρ_{g})}} (c) \end{matrix} \end{matrix}

(10)

\begin{matrix} \frac{h L_{b}}{k_{l}} = 0.082 {[\frac{h_{f g} q}{g (T_{s} + 273.15) k_{l}} \frac{ρ_{v}}{ρ_{l} - ρ_{v}}]}^{0.7} {(\frac{(T_{s} + 273.15) c_{p} σ ρ_{l}}{h_{f g}^{2} ρ_{g}^{2} L_{b}})}^{0.33} \Pr^{- 0.45} \\ L_{b} = \sqrt{\frac{σ}{g (ρ_{l} - ρ_{g})}} \end{matrix}

(11)

2.2 Error between experiment and formula analysis

In order to compare the error between our experiment results and the other researcher’s formulas, we have defined the following errors:

The first error is the calculation error, as can be seen in equation (12). Here, h_cal means the calculation results by other researcher’s formulas and h_exp is our experiment results.

e_{1} = \frac{h_{c a l} - h_{\exp}}{h_{\exp}} \times 100 %

(12)

The second error is the root-mean-square error. As for a given coolant, there are seven heating fluxes. For each condition, it has a calculation error, in order to know the average error of the seven conditions, the root-mean-square error can be used. In equation (13), N is the total calculation number.

e_{2} = \sqrt{\frac{\sum_{i = 1}^{N} e_{i}^{2}}{N}}

(13)

In fact, the above mentioned two errors cannot represent the stability of the empirical formula. For example, for one formula, the calculation error is higher for some condition while it is lower at the other condition. But for another formula, at all condition, the error may roughly be the same. The root-mean-square error may be similar for these two formulas but in fact, the stability of the second empirical formula may be better than the first one. As a result, we use the standard deviation to evaluate the stability, as can be seen in equation (14). When calculation, we can use the function STDVP in excel to calculate the standard deviation. There are four coolants and each coolant has seven conditions, thus the standard deviation of one formula is for 28 conditions.

σ = \sqrt{\frac{\sum_{i = 1}^{N} {(e_{i} - μ)}^{2}}{N}}

(14)

3 RESULTS AND ANALYSIS

According to the calculation results:

For R134a coolant, Imura’s formula is ~34% lower than the experiment results. Besides, with the increasing of the heating power, the error also comes higher. Rohsenow’s equation has the best results, the average error is just 9%. For R22, Cooper’s results are more close to the experiment, Kruzhilin’s variation trends are more similar but the results are lower than the experiment results.

For R410A, Imura’s formula is ~45% lower than the experiment results. The variation trend is just the similar with R134a. The variation trends of Kutateladze formula (average error: 31%), Labuntsov formula (average error: 37%), Kruzhilin formula (average error: 60%), and Cooper formula (average error: 25%) are similar, for higher heating power, the error comes lower. For Hasna’s formula, his experiment’s heating flux range is from ∼2 to 20 W/cm², and our experiment’s heating flux range is from 0.26 to 1.84 W/cm², thus, when the heating power is ~700 W, the Hasna’s formula may be more accurate to calculate our experiment condition, the results also shows that, the higher heating power, the lower the error between Hasna’s formula and our experiment results (Table 2).

Table 2.

The average relative error between formula and experiment results.

Coolant	Imura	Kutateladze	Labuntsov	Rohsenow	Kruzhilin	Hasna	Cooper
R134a	0.34	0.28	0.48	0.09	0.59	0.24	0.28
R22	0.36	0.31	0.40	0.43	0.55	0.28	0.17
R410A	0.45	0.31	0.37	0.32	0.60	0.25	0.25
R744	0.39	0.62	0.56	0.22	0.58	0.99	0.62
Average	0.38	0.41	0.46	0.29	0.58	0.54	0.37

Coolant	Imura	Kutateladze	Labuntsov	Rohsenow	Kruzhilin	Hasna	Cooper
R134a	0.34	0.28	0.48	0.09	0.59	0.24	0.28
R22	0.36	0.31	0.40	0.43	0.55	0.28	0.17
R410A	0.45	0.31	0.37	0.32	0.60	0.25	0.25
R744	0.39	0.62	0.56	0.22	0.58	0.99	0.62
Average	0.38	0.41	0.46	0.29	0.58	0.54	0.37

Table 2.

The average relative error between formula and experiment results.

Coolant	Imura	Kutateladze	Labuntsov	Rohsenow	Kruzhilin	Hasna	Cooper
R134a	0.34	0.28	0.48	0.09	0.59	0.24	0.28
R22	0.36	0.31	0.40	0.43	0.55	0.28	0.17
R410A	0.45	0.31	0.37	0.32	0.60	0.25	0.25
R744	0.39	0.62	0.56	0.22	0.58	0.99	0.62
Average	0.38	0.41	0.46	0.29	0.58	0.54	0.37

Coolant	Imura	Kutateladze	Labuntsov	Rohsenow	Kruzhilin	Hasna	Cooper
R134a	0.34	0.28	0.48	0.09	0.59	0.24	0.28
R22	0.36	0.31	0.40	0.43	0.55	0.28	0.17
R410A	0.45	0.31	0.37	0.32	0.60	0.25	0.25
R744	0.39	0.62	0.56	0.22	0.58	0.99	0.62
Average	0.38	0.41	0.46	0.29	0.58	0.54	0.37

For R744, Imura’s formula is also lower than our experiment results. And the errors are within 32–42%. Which means the errors of the seven condition are very stable. For Kutateladze formula, Labuntsov formula and Cooper formula, when the heating power is lower, the error is also not very large. However, when the heating power comes larger, the error also comes really large. Take the cooper’s formula for example, if the heating power is 109 W, the error between our experiment result and Cooper’s formula is ~9%, however, when the heating power is 696 W, the error is 114%. Here is the reason why at higher heating power, the formula’s error comes larger. As for these formula, they are mainly for pool boiling. However, for the boiling in the heat pipe system, it is more likely to be the combination of pool boiling and convection boiling. For heat pipe system, higher heating power also means higher flowing velocity of the liquid inside the evaporator, higher flow rate may enhance the heat transfer coefficient at first, however when the flow rate comes much higher, it may bother the bubble’s formation, thus, the heat transfer coefficient in the evaporator comes lower than pool boiling (Figure 2).

The results of calculation for different formulas and experiment.

Figure 2.

The results of calculation for different formulas and experiment.

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Here, we can make a comparison between there seven formulas:

Imura’s formula:
The calculation results for every coolants are all lower than experiment results. The average error is 38%, besides, the standard deviation is just as low as 0.08. According to the Imura’s equation, it considered the relation between heat transfer coefficient and many thermophysical properties of the coolants, which is very comprehensive. As for the reason why all the calculation results are all lower than the experiment result, it maybe the flow pattern. As for the combination of the flow and pool boiling, the heat transfer maybe enhanced by the coolant flow, however, when the flow rate comes much higher, the heat transfer maybe weaken. In fact, maybe the Imura’s equation does not consider the enhancing effect by the convection boiling. All in all, it is for two phase closed thermosiphon, the convection or flow factor is not considered.
Kutateladze’s formula, Labuntsov’s formula, Kruzhilin’s formula:
All of these three formulas are for the pool boiling. The total average error of Kruzhilin is 58%. For Freon coolants, the errors of Kutateladze’s formula and Labuntsov’s formula are not very large, but for R744, for higher heating power, the error can even as high as 128% (Kutateladze, 696 W). In a word, these formulars are for pool boiling, which is not very suitable for our experiment thus, the error is also very big.
The standard deviations of Kutateladze’s formula, Labuntsov’s formula, Kruzhilin’s formula are 0.39, 0.41 and 0.05 which means Kruzhilin’s formula has the best stability.
Cooper’s formula and Hasna’s formula:
Cooper’s formula is a classical and widely used equation. Hasna’s formula is obtained based on the form of Cooper. Cooper’s formula is more suitable for Freon coolants. The standard deviation of Cooper for Freon is 0.1, but for all coolants it is 0.37. For Hasna’s equation, the higher the heating power, the lower the error. Besides, it is mainly for water in sub-atmospheric. Thus it is not suitable for R744.
Rohsenow’s formula:
Rohsenow’s formula is derived in theoretical. For different coolant-heating surface, it has different Cwl and n. In our calculation, we let Cwl = 0.0049 and n = 1.7 (n-pentane-Lapped cooper). In fact, if there is the experiment results for cooper and each different coolant, the results can be more accurate. All in all, this formula is suitable for all the four coolants and the average error of all coolants is 29%. The standard deviations is 0.18.

4 CONCLUSION

At different working condition, different formulas show different results. These formulas include Rohsenow formula (1951), Cooper formula (1984), Labuntsov formula (1973), Kutateladeze formula (1990), Kruzhilin formula (1947), Imura formula (1979) and Hasna formula (2017). All in all, Imura, Rohsenow, Cooper and Hasna’s formula shows the better results.

Imura formula’s calculating results are all less than the experiment results. For example, it is 34% less than the experiment results for R134a while, for R744, it is 39% less than the experiment results, but the variation trend is very close to the experiment results. Cooper formula accords very well with the experiment results for Freon coolant, the relative average error is ~17–28%, but for R744, the relative error is as large as 62%. Rohsenow’s formula average error is ~29%, and the variation trend is close to the experiment results. For Hasna’s equation, when calculating the Freon coolants, the higher the heating power, the lower the error. However, it is not suitable for R744.

FUNDING

This paper is supported by National Key R&D Program of China (2016YFB0601600).

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APPENDIX

Nomenclature
Nu	Nusselt number
Re	Renold number
Pr	Prandtl number
$ρ$	Density
$D_{b}$	The bubble’s diameter
$σ$	Surface tension
$h_{f g}, r$	Latent heat
h	Heat transfer coefficient
c_p	specific heat at constant pressure
q,q_h	Heat flux
$μ$	Viscosity
$k, λ$	Heat conductivity
$p_{r}$	Working pressure/critical pressure
M	Relative molecular weight
P_a	Atmospheric pressure
l	Liquid
v	Vapor

Nomenclature
Nu	Nusselt number
Re	Renold number
Pr	Prandtl number
$ρ$	Density
$D_{b}$	The bubble’s diameter
$σ$	Surface tension
$h_{f g}, r$	Latent heat
h	Heat transfer coefficient
c_p	specific heat at constant pressure
q,q_h	Heat flux
$μ$	Viscosity
$k, λ$	Heat conductivity
$p_{r}$	Working pressure/critical pressure
M	Relative molecular weight
P_a	Atmospheric pressure
l	Liquid
v	Vapor

Nomenclature
Nu	Nusselt number
Re	Renold number
Pr	Prandtl number
$ρ$	Density
$D_{b}$	The bubble’s diameter
$σ$	Surface tension
$h_{f g}, r$	Latent heat
h	Heat transfer coefficient
c_p	specific heat at constant pressure
q,q_h	Heat flux
$μ$	Viscosity
$k, λ$	Heat conductivity
$p_{r}$	Working pressure/critical pressure
M	Relative molecular weight
P_a	Atmospheric pressure
l	Liquid
v	Vapor

Nomenclature
Nu	Nusselt number
Re	Renold number
Pr	Prandtl number
$ρ$	Density
$D_{b}$	The bubble’s diameter
$σ$	Surface tension
$h_{f g}, r$	Latent heat
h	Heat transfer coefficient
c_p	specific heat at constant pressure
q,q_h	Heat flux
$μ$	Viscosity
$k, λ$	Heat conductivity
$p_{r}$	Working pressure/critical pressure
M	Relative molecular weight
P_a	Atmospheric pressure
l	Liquid
v	Vapor

© The Author(s) 2019. Published by Oxford University Press.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact [email protected]

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December 2023	11
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February 2024	23
March 2024	21
April 2024	20
May 2024	17
June 2024	9
July 2024	5
August 2024	8
September 2024	14
October 2024	16
November 2024	14
December 2024	11
January 2025	16
February 2025	8
March 2025	16
April 2025	20
May 2025	2