Experimental study of single-phase flow and heat transfer in rectangular channels under uniform and non-uniform heating

Journal Pre-proofs Experimental Study of Single-Phase Flow and Heat Transfer in Rectangular Channels under Uniform and Non-uniform Heating Rulei Sun, Gongle Song, Dalin Zhang, Jian Deng, G.H. Su, F.A. Kulacki, Wenxi Tian, Suizheng Qiu PII: DOI: Reference:

S0894-1777(19)31713-3 https://doi.org/10.1016/j.expthermflusci.2020.110055 ETF 110055

To appear in:

Experimental Thermal and Fluid Science

Received Date: Revised Date: Accepted Date:

13 October 2019 4 January 2020 20 January 2020

Please cite this article as: R. Sun, G. Song, D. Zhang, J. Deng, G.H. Su, F.A. Kulacki, W. Tian, S. Qiu, Experimental Study of Single-Phase Flow and Heat Transfer in Rectangular Channels under Uniform and Nonuniform Heating, Experimental Thermal and Fluid Science (2020), doi: https://doi.org/10.1016/j.expthermflusci. 2020.110055

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2020 Published by Elsevier Inc.

Experimental Study of Single-Phase Flow and Heat Transfer in Rectangular Channels under Uniform and Non-uniform Heating Rulei Sun a, Gongle Song a, Dalin Zhang a*, Jian Deng b*, G. H. Su a, F. A. Kulacki c, Wenxi Tian a, Suizheng Qiu a a School

of Nuclear Science and Technology, State Key Laboratory of Multiphase Flow in Power Engineering, Shaanxi Key Lab. of Advanced Nuclear Energy and Technology, Xi'an Jiaotong University, No. 28, Xianning West Road, Xi'an, China b Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China, Chengdu, China c Department of Mechanical Engineering, University of Minnesota, 111 Church St. S. E., Minneapolis, MN 55455, United States

ABSTRACT Characteristics of single-phase convective heat transfer in a rectangular channel are reported for heated and non-heated conditions. Results indicated that the friction factors were basically the same in the laminar region. However, in the transition and quasi-turbulent region, the heated friction factor was slightly lower than the non-heated values, and the value of nonuniform heating was slightly higher than that of uniform heating. The transition started critical Reynolds number increased from 2700 without heating to about 3000 with heating only due to the viscosity decreased with increasing temperatures. Due to the transition started and ended Reynolds number increased simultaneously under heating, the width of flow transition was almost the same as that without heating. The transverse power distribution influenced the average Nusselt number of the channel, although measured differences were small for uniform and non-uniform heating conditions. For the heat transfer data under uniform heating conditions obtained in this study, 15 correlations were evaluated. It is found that the existing correlations underestimated the experimental data for Re < 3000 and overestimated the data for 3000 < Re < 6000. A large database of heat transfer in rectangular channels including the experimental data obtained from this study and other researchers' work was established, and new heat transfer correlations for laminar (Re < 3000) and transition (3000 < Re < 10000) flows were developed. The Dittus-Boelter correlation was confirmed to provide a good prediction for the heat transfer for Re > 10000, with almost all data being

*Corresponding author: E-mail address: [email protected] (Dalin Zhang), [email protected] (Jian Deng)

within ± 20% errors. KEYWORDS: Rectangular channel, Single-phase water, Friction factor, Heat transfer correlation, Power distribution

NOMENCLATURE

cp

Specific heat, J/(kg∙K)

Greek symbols

De

Hydraulic diameter, m

α

Aspect ratio (width-to-depth)

f

friction factor, Eq. (9)

β

Thermal expansion coefficient,

G

mass flux, kg/m2s

δ

Uncertainty

Gr

Grashof number, Gr  g  T f  T w  De 3 v 2

λ

Thermal conductivity, W/(m∙K)

Gz

Graetz number, Gz  RePr De L

μ

Dynamic viscosity, Pa∙s

h

Heat transfer coefficient, W/(m2∙K)

ν

Kinematic viscosity, m2/s

k

Thermocouple index in copper block

ρ

Density, kg/m3

L

Length, m

υ

Specific volume, m3/kg

m

Mass flow rate, kg/s

η

Thermal efficiency

N

Heating element number

θ

Inclination angle, o

Nu

Nusselt number, Nu  hDe 

P*

Dimensionless power factor, Fig. 4

Subscripts

∆P

Pressure drop, Pa

a

Acceleration

Po

Poiseuille number, Po  f Re

c

Local

Pr

Prandtl number, Pr  c p  

exp

Measured value

q

Heat flux, W/m2

Cu

Copper

Qa

Heat absorption of water

f

Fluid

Qe

Electric heating power

fric

Friction

Re

Reynolds number, Re  uDe 

g

Gravitational

1/K

2

Recr

Critical Reynolds number

h

Hydrodynamic

Reqt

Start of quasi-turbulent flow regime

in

Inlet

Ret

Start of turbulent flow regime

out

Outlet

T

Temperature, K

pre

Predicted value

u

Velocity, m/s

S

Steel

y

Measurement parameter

t

Thermal

z

Distance between thermo-couples, m

w

Wall

1. INTRODUCTION The narrow rectangular channel exhibits excellent thermal-hydraulic and mechanical performance. For example, it has small heat transfer temperature difference, simple manufacturing and surface finishing technology, and it is not susceptible to cause impurity precipitation due to fluid scouring [1]. In recent years, it has been widely used in the field of efficient heat transfer, such as nuclear reactor, plasma, electronic cooling, and spacecraft thermal control, etc. Especially, as nuclear energy has become more and more important in energy utilization, various advanced reactors, integrated reactors and mobile nuclear power platforms have been proposed and designed. It is worth noting that these advanced reactor types and systems have increased requirements for compactness while balancing performance and safety. Therefore, compared with rod fuel elements, plate fuel elements are favored because of their compact structure, high volume power density, low center temperature, high fuel consumption and safe operation [2, 3]. Figure 1 shows a cross-sectional view of a typical standard plate fuel element of the MTR (Materials Testing Reactor) research reactor. The gap between adjacent fuel plates is generally 1-3mm, so a rectangular channel with larger aspect ratio is formed. Therefore, the coolant flow and heat transfer characteristics may be significantly different from those of conventional channels and tubes. In addition, due to the influence of the irradiation target, fuel arrangement and other factors, the fuel plate usually has a significant unevenness in the power distribution along its width. This non-uniformity also varies with axial position and the operating conditions of the reactor [4]. Therefore, the study of single-phase forced convection flow and heat transfer in a rectangular channel under lateral uniform and non-uniform heating will be the focus of this investigation. 3

Fig. 1. Cross section of standard fuel elements of a MTR reactor core.( fuel meat indicates the fuel part of each plate) [5] Many experimental and theoretical studies of single-phase flow and heat transfer in narrow channels under uniform heating conditions have been reported. Ma et al. [6] experimentally investigated single-phase flow and heat transfer in deionized water flowing vertically upward in a rectangular channel. They concluded that the non-heated friction factor and average Nusselt number can be predicted via traditional relations. The transition from laminar to turbulent flow was 2500 < Re < 4000, and the critical Reynolds number was slightly larger than that for circular tubes. Caney et al. [7] measured vertical upward singlephase pressure drop and heat transfer coefficients in a 1 mm square channel and found that the Shah-London [8] and Blasius [9] relations can predict friction factor data quite well. In addition, the heat transfer measurements were in good agreement with those of classical literature. Ghione et al. [10] studied flow characteristics in the rectangular channel based on the data from the SULTAN-JHR experiment [11, 12]. The gap sizes of the test section were 1.51 and 2.16 mm. The medium was water flowing vertically upwards, and the test sections were uniformly heated on both sides. It was confirmed that the traditional Blasius relation underestimated the experimental value in the turbulent range of 1.0×104 < Re < 3.1×105. Similarly, the Dittus–Boelter [13] relation underestimated the turbulent heat transfer coefficient, especially at high Reynolds number. The authors also found that reducing the channel gap size may enhances heat transfer. Wang et al [14] carried out an experimental 4

study of single-phase flow and heat transfer in a 2.0 mm × 40 mm rectangular channel. Their results showed that the transition zone was 2700 < Re < 3800, and the critical Reynolds number for the laminar-to-turbulent transition was larger than that for conventional channels. Friction factors can be well correlated by the Troniewski-Ulbrich [15] correlation for developed laminar and turbulent flow and the Bhatti-Shah [16] correlation in the laminar-toturbulent transition region under non-heated conditions. The authors also found that the Prandtl number had a significant effect on the heat transfer characteristics in turbulent flow. Sudo et al. [17] experimentally studied single-phase heat transfer characteristics of vertical upward and downward flow in a 2.25 mm × 50 mm channel. It was found that the Dittus-Boelter relation accurately predicted experimental data for Re > 4000, and the influence of buoyancy could be neglected. The transition zone was 2000 < Re < 4000, and the effect of the buoyancy would become apparent when Re < 700. Jo et al. [18] studied the heat transfer characteristics of single-phase flow in a rectangular channel of 2.35 mm × 54 mm. It is found that there was no significant difference between vertical upward and downward flow. The existing correlations underestimated the experimental data and cannot predict the higher heat transfer coefficient near the entrance. Liang et al. [19, 20] carried out steady and transient experiments on flow and heat transfer characteristics of single-phase water flow in narrow rectangular channels with 1.0 mm, 1.8 mm and 2.5 mm gaps. The critical Reynolds number for laminar to turbulent transition decreased significantly and the friction factor was larger than that in conventional channel. Wang et al. [21] carried out an experimental study on the flow and heat transfer characteristics of the transition zone and found that the upper and lower limits of the critical Reynolds in the transition zone increased with decreasing the Prandtl number. The friction factor increased with Prandtl number at certain Reynolds numbers in the transition zone. Li et al. [22] studied the effect of blisters formation of plate fuel on flow and heat transfer performance. The large thermal resistance of gaseous fission products resulted in a relatively high maximum temperature in the fuel plate and additional structure rupture. There are still relatively few studies on the thermal-hydraulic characteristics of rectangular channels under transverse non-uniform heating condition. Jo et al. [23] used the 5

Monte Carlo N-Particle (MCNP) code to calculate an equilibrium reactor core with 22 fuel assemblies. The power distribution of the fuel element in the lateral direction was obtained, and its effect on the fuel temperature was investigated. Al-Yahia et al. [24] analyzed the influence of transverse power distribution on flow instability in rectangular channels. The results showed that the thermal power and mass flow rate were almost independent of the heat flux distribution, but the pressure drop and temperature distribution characteristics were different. This could be attributed to the disturbance of transverse velocity profile caused by the distribution of bubbles on the cross section. Kim et al. [25] studied the characteristics of the ONB in a rectangular channel under laterally uniform and non-uniform heating. It is found that the lateral wall temperature was related to the power distribution profiles and thus affected the position of the ONB point. At the same mass flow rate and inlet subcooling, the thermal power at the ONB point under non-uniform heating was less than that under uniform heating. However, there was almost no difference in the local heat flux and wall temperature at the ONB point. The heat transfer coefficient of non-uniform heat flux in subcooled boiling region was lower than that under uniform heat flux. Kim et al. [26] also investigated the effect of transverse power distribution on the minimum pressure drop in rectangular channels. The pressure drop-flow rate curves under uniform and non-uniform heating power were obviously different. The inlet mass flow rate at the minimum pressure drop point under nonuniform heating was much smaller than that under uniform heating. This phenomenon was analyzed in depth using numerical calculations, considering the effect of power profile on void fraction and lateral flow redistribution. While great advancements have been made in the study of flow resistance and heat transfer in narrow rectangular channels, there is still no unified understanding of the flow and heat transfer characteristics in rectangular channels under uniform heating, and some researchers’ conclusions conflict with each other. Standard correlations like those of traditional circular tubes are still lacking. In addition, there is no relevant report on the effect of non-uniform heat flux on single-phase friction factor and heat transfer coefficient. Therefore the purpose of this study was to further carry out the single-phase flow and heat transfer experiments under uniform heating in rectangular channels. Based on the 6

experimental data, the applicability of the traditional circle correlations and the existing rectangular correlations were evaluated, and then more universal heat transfer correlations were established. The effect of transverse non-uniform heating on flow and heat transfer was also compared to a base case with uniform heating. 2. EXPERIMENTAL APPARATUS 2.1 Flow Loop The experiments were carried out on the water circuit of the Thermal Hydraulics Laboratory of the School of Nuclear Energy Science and Technology, Xi'an Jiaotong University [27] (Fig. 2). The working fluid is deionized water, which is regularly monitored and replaced during the experiment. The flow loop is divided into the main loop and the cooling water loop. The main circuit system comprises a high-pressure plunger pump (3S752/32, Ningbo Heli Mechanical Pump Co. Ltd), mass flowmeter (N5D101B1GB4N1B, Xi'an Dongfeng Machinery & Electronic Co., Ltd), regenerator, preheater, experimental section, condenser, water tank, and other equipment and pipe network accessories. The cooling water loop includes a condenser, circulating water pump, cooling water tank and other pipe network accessories. Its function is to cool the working fluid of the main circuit and ensure the safe and stable operation of the loop. Among them, the high-pressure plunger pump provides the power needed for coolant flow up to 2 t/h. The stainless-steel serpentine coil preheater uses Joule heating to heat the experimental working medium, and the heating power source adopts multiple sets of low voltage and large current DC current sources. The regenerator is a shelland-tube heat exchanger which fully utilizes the waste heat of the working medium leaving the test section. The test circuit pipes and accessories are made of 316 stainless steel. Deionized water passes through valves and filters by water tank and is pumped into two channels by high-pressure plunger pump. One is a bypass system designed to regulate flow and pressure, and the other is the main circuit system of the experiment. In the main loop, the experimental medium passes through the regulating valve and the flow meter in sequence, and then enters the shell-and-tube regenerator. The feed water absorbs the heat of the hightemperature medium coming out of the test section and converts it into its own sensible heat. It then flows into the preheating section where it is heated up to the temperature required for 7

the inlet of the test section and flows vertically upward into the test section. Finally, the outlet flow is returned to the water tank through the regenerator and the condenser. Cooling Loop Tower

L3

Test section

Tank Filter

∆P

T Tw6 Tw5 Tw4

Valve Pump P

Filling tank

Tw2 Tw1

T

L1

Filter Supply pump

Thermocouples

Tw3

L2

Drain valve

Power System

Filter Condenser

Cooler Flowmeter

Drain valve

Regenerator

P

P

Valve Tank

Preheater

Valve

Piston pump

Fig. 2. Schematic of the experimental loop. 2.2 Test Section The test section was designed for double-sided indirect heating mode, and the overall arrangement was symmetrical along the axial and width directions. It included the rectangular flow channel, heating elements, thermal copper blocks, inlet and outlet reducers (inclination angles of 15° with reference to numerical simulation) and adapters, insulation plates, support structures and parameter measurement modules (Fig. 3(a)). The cross section of the test section was 2.0 mm × 60.0 mm with an aspect ratio of α = 30 and the length was 1000 mm. The effective heating area was 700 mm long and 56 mm wide. A 2.0 mm wide non-heating zone was left on both sides in the width direction, considering the influence of the cladding of the actual plate-type fuel element. A 145 mm non-heated section was left between the inlet and the heating section to allow the flow passing through the variable diameter joint to develop before entering the heating zone. The same distance was left between the heating section and the outlet to reduce the influence of the outlet wake on the heating zone (Figs. 3(b), (d)). The test section contained ten heating elements arranged symmetrically on both sides of the rectangular channel. The heating element was made of Ni20Cr80 high-strength electro8

thermal alloy and finely processed into flat strip structure. Different heating elements on the same side were insulated by adjacent gaps. A high-purity oxygen-free copper conductive block was placed between each heating element and the wall surface of the rectangular channel on which there existed several air gaps along the axial direction to block the heat conduction. Six rows of thermocouple holes were arranged in the axial direction, and two 1.5 mm diameter K-type thermocouples were placed in each row with a spacing of 8.0 mm. Based on the temperature data measured by thermocouples at different heights, the temperature of the inner wall surface of the channel could be calculated according to the Fourier heat conduction law (Fig. 4). A 0.5 mm thick high thermal conductivity aluminum nitride (AlN) ceramic sheet was placed between the copper block and the heating element for electrical insulation. In addition, a thin layer of high-temperature inorganic heat conductive filler was coated between the copper block, the aluminum nitride (AlN) ceramic sheet and the heating element, and between the copper block and the outer wall of the channel to minimize the heat transfer resistance at the interface. A set of 1.5 mm diameter thermocouples were arranged at a distance of 15 mm from both ends of the effective heating section to measure the temperature of the inlet and outlet fluid. Pressure and differential pressure transmitters were used to measure inlet pressure and pressure drop of fluid flow in the test section. The detailed description of position for pressure taps and support plates were shown in Figs. 3a,b. The data acquisition system and interface panel based on LabVIEW were used to collect and store test data. According to the literature [28, 29], the length of hydrodynamic entrance and thermal entrance can be calculated using the following formula, Lh ,laminar De  0.05Re

(2)

Lh ,turbulent De  4.4 Re1 6

(3)

Lt ,laminar De  0.1RePr

(4)

Lt ,turbulent De  10

(5)

Within the parameters of flow and heat transfer we have investigated, the hydrodynamic 9

entrance length was 0.19 m < Lh,laminar < 0.51 m in the laminar region and 0.054 m < Lh,turbulent < 0.064 m in the turbulent region. In addition, the thermal entrance length was 1.7 m < Lt,laminar < 3.6 m in the laminar flow zone and Lt,turbulent ≈ 0.038 m in the turbulent zone. The prediction of the hydrodynamic entrance length mentioned above was based on the assumption that the inlet velocity profile is evenly distributed. In this study, the inlet and outlet of the test section were connected to the circular pipe of the circuit through the reducer. According to Rohsenow et al. [30] and Lee et al. [31], the length of hydrodynamic entrance would be greatly reduced due to the wake effect of the reducer. In the design stage of the test section, the numerical simulation was carried out for the channel with a reducer angle of 15 degrees, and the Reynolds number was 3000. The numerical results also verified the above discussion that the flow development length was greatly reduced to 0.24 m, the flow field changed more smoothly, and the vortex strength at the inlet and outlet reducer was weakened. As can be seen from Fig. 3(b) that the distance between the high-pressure terminal of the differential pressure transmitter and the inlet was 0.5 m. The effective heating length was 0.71 m, and the first row of thermocouples was 0.32 m away from the inlet. Therefore, the flow was considered to be hydrodynamically fully developed in flow resistance and heat transfer experiments, while the thermal entrance effect only needed to be considered for laminar heat transfer.

10

(a) Assembly drawing. (side view)

(c) Cross section of assembly. (top view)

(b) Rectangular channel.

(d) Cross section of the channel. 11

Fig. 3. Details of test section. (not drawn to scale)

Fig. 4. Thermocouple position diagram. (a) Side view (1/2 symmetry). (b) Front view. All dimensions are given in mm. (not drawn to scale) 3. PROCEDURE AND DATA REDUCTION 3.1 Procedure Deionized water with a conductance < 0.8 μs/cm was used as working fluid. The flow rate and inlet pressure were adjusted to the required value by controlling the pump outlet valve and the back-pressure valve of the loop system, and then the preheater was heated gradually until the inlet water temperature of the test section reached the set values. Under the monitoring of the data acquisition system, when the change of temperature was within ± 0.1 °C and the pressure difference was within ± 0.1 kPa over 5 min, the system was considered to be at steady state. The flow rate, inlet pressure, differential pressure and temperature were then recorded at a frequency of 1 kHz for 5 min. To investigate the effects of transverse power distribution on single-phase flow and heat transfer, three wall power profiles were chosen: uniform, polynomial and sinusoidal (Fig. 5). The continuous lines in the figure represented the ideal condition, and the stepped lines were the actual heating power distribution. As described in section 2.2, the heating component consisted of ten heating elements, each of which had a separate power module. The power of 12

each set of heating elements was adjusted according to the designed power distribution profile. During the experiment, the total thermal power was kept constant under the same operating conditions for the three different power distributions. The range of experimental parameters was shown in Table 1. Table1. Range of the parameters. Parameter

Range

Flow direction Pressure (MPa) Mass flux (kg/m2s) Inlet temperature (°C) Heat flux (kW/m2) Heat flux distribution Total power (kW)

Upward 0.2-0.22 134-975 30-90 0-22 Uniform/non-uniform 0.4-1.8

Non-uniform Series1 Non-uniform Series2 uniform

1.6

P*Dimensionless power factor

1.4

1.2

1.0

0.8

0.6

0.4

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

x(m)

Fig. 5. Power distributions profile. 3.2 Data Reduction The total pressure drop due to the fluid flowing through the test section is,  p   p g   p a   p fric   pc

(6)

where p ,  p fric ,  p g ,  p a ,  p c indicated the total pressure drop of the test section, frictional pressure drop, gravitational pressure drop, acceleration pressure drop and local pressure drop, respectively. The local pressure drop is caused by various singularities along 13

the flow path such as abrupt changes in free-flow areas, bends, and valves[30, 32]. Because the flow area remained constant along the effective test section, so pc  0 and, pa 



u2

u1

 udu  G  u 2  u1 

(7)

where G is the mass flux, and u1, u2 respectively represented the average velocity at inlet and outlet. Equation (2) is expressed as a function of ρ or υ,

 1 1  pa  G2     G2  υout  υin   ρout ρin 

(8)

In single-phase flow, the fluid was super-cooled water and the temperature change was less than 15 °C. The density changes between the inlet and outlet were thus very small, so

pa  0 . Thus, the impact of accelerated pressure drop was ignored,

pg  gL

(9)

   out  in  / 2

(10)

Therefore, the single-phase frictional pressure drop is,  p fric   p   p g

(11)

Consider the water temperature in the pressure tube of the differential pressure transmitter is different from that in the test section, ptest   gL  p fric   gL

(12)

where ptest is the total pressure drop measured, ρ is the density of water in the pressure tube of the differential pressure transmitter,

 is the average density of water in the test section, and

L is the effective length of the measurement section. Therefore, the formula for calculating single-phase friction pressure drop is,  p fric   ptest   gL   gL

(13)

The friction factor under the non-heated condition is usually calculated by the Darcy formula,

f 

2p fric  De LG 2

(14) 14

The overall thermal efficiency of the test section is,



Qa mcp Tf ,out  Tf ,in   Qe Ui Ii

(15)

where m is the mass flow rate, cp is the constant pressure specific heat capacity, T f , in is the inlet temperature, and T f ,out is the outlet temperature, Ui refers to the heating voltage, and

I i is the heating current. Since the temperature difference between the two thermocouples on the copper block was smaller than the measurement error, the heat flux cannot be obtained by measurement. In this paper, the heat flux of a single copper block is estimated using the overall thermal efficiency.

U i I i

qi 

(16)

Aw

The average heat flux on the whole heated wall is, q 

1 N

N

q i 1

(17)

i

where Aw is the area of heating wall corresponding to a single copper block, and N is the number of heating elements. The conductive filler between the copper block and the wall surface of the channel is very thin, and the influence of the interfacial thermal contact resistance was neglected. The temperature distribution on the inner wall of the channel corresponding to each copper block is,

Tw,in  i, j   TCu  i, j , 2  

2 103 qi

Cu



2 103 qi

S

(18)

6

Tw,in  i    Tw,in  i, j  / 6

(19)

j 1

where S was the thermal conductivity of the channel wall, and Tw ,in  i  is the average temperature of the inner wall surface corresponding to each copper block. The average temperature of the inner wall surface of the entire rectangular channel is calculated by the 15

weighted average, T w ,in 

 T  i   q    q  w ,in

i

(20)

i

The average heat transfer coefficient is,

h  q Tw,in  Tf 

(21)

Tf  Tf ,in  Tf ,out  / 2

(22)

and the average Nusselt number is, Nu  hDe  f

(23)

where De is the hydraulic diameter and  f is the thermal conductivity of the liquid. 3.3 Experimental Uncertainty The total experimental uncertainty of the results mainly came from the first-order uncertainties of the instruments and the zero-order uncertainty of the data acquisition system. Assuming that the instrument measurement errors were subject to uniform distribution, the standard uncertainty for any measurement parameter could be estimated [33],   x    ins  x    acq  x  2

2

(24)

where x represents a measurement parameter; ins  x is the expected error of the measuring instrument, and acq  x is the systematic error of the data acquisition system. The maximum error of the data acquisition system is 0.02%. The measurement parameters and uncertainties of the instruments were shown in Table 2. For the derived parameters with several independent linear variables, e.g. y  f  x1 , x2 , xn  , the relative uncertainty is [34], 2

2

x  x  x    1    2   ...   n  y  x1   x2   xn 

y

2

(25)

The results were shown in Table 3.

16

Table 2. Measurement parameters and instrument errors. Parameters

Instruments

Mass flow rate Pressure

Coriolis mass flow meter Pressure transmitter (3051T, Rosemount) Differential pressure transmitter (3051CD, Rosemount) T-type sheathed thermocouple K-type sheathed thermocouple Hall Effect Current Transducer Voltage transmitter Millimeter scale Vernier caliper

Pressure drop Water Temperature Wall Temperature Current Voltage Length Width/Height

Calibration range 0-350 kg/h 0-1.0 MPa

Instrument errors ±0.2% ±0.075%

0-6.22 kPa

±0.1%

298-385 K 298-405 K 5-350 A 0-10 V -

±0.5 K ±1.5 K ±0.5% ±0.5% ±0.5 mm ±0.01 mm

Table 3. Uncertainty of derived parameters. Parameter Heat flux Friction factor Nu Uncertainty 3.27-7.43% 2.08-4.04% 5.41-13.44% (δy/y)

4. Verification of Experimental System The friction factor for fully developed fluid flow in conventional channels has been extensively studied, and accurate relations have been developed. For fully developed laminar flow in a circular tube, the theoretical equation for calculating friction factor was,

Po  f Re  64

(26)

For the steady turbulent flow in smooth tubes, the Blasius relation was most widely used, and many researchers [7, 35-37] pointed out that it can also be used to calculate with good accuracy the friction factor in mini- and micro-channels, f 

0.3164 Re0.25

2300  Re  105

(27)

The calculation of the friction factor in non-circular channels has also been extensively studied. However, due to the particularity and diversity of channel geometries of the various investigations, there was no consensus for prediction. For fully developed laminar flow in simple geometric channels, Kakac et al. [38] have given Po numbers of several different channel geometries. The Po number for smooth rectangular channel varied with the aspect 17

ratio of the channel (Table 4). Shah and London [8] and Spiga [39] also have given analytical solutions of friction factor of fully developed laminar flow in rectangular channels. Table 4. Po number in smooth rectangular channels [22]. Aspect ratio

Po

1.0

57

1.43

59

2.0

62

3.0

69

4.0

73

8.0

82

∞

96

Turbulent flow is much more complex than laminar flow. Although many researchers have made extensive and in-depth studies on it using various techniques and methods, the conclusions of the various investigations vary considerably (Table 5). Based on a large number of literature research, several correlations with high credibility have been are selected for comparison to our results. Table 5. Friction factor correlations. Reference

Correlation

Shah-London [8]

Laminar:

f 

2 3 96  1  1.3553  1.9467  1.7012    Re  0.9546 4  0.2537 5 

Laminar:

f 

Troniewski-Ulbrich [15]

64 0.16 Re*  Re  2  * Re 0.3164 0.16 Turbulent: f  *0.25 Re*  Re  2  Re

Spiga-Morino [39]

Laminar:

f 

Blasius [9]

Turbulent: f 

2 3 96  1  1.20244  0.88119  0.88819    Re  1.69812 4  0.72366 5 

0.3164 Re0.25

18

13  0.0154Cv   f  f c   0.012   0.85 64   

Sadatomi et al. [40]

0.25 Turbulent: f c  0.3164 Re

 1  1.3553  1.9467 2  1.7012 3  Cv  96    0.9546 4  0.2537 5   Kakac-Yener [41]

Turbulent:

f  f c (1.0875  0.1125 ) f c  0.3164 Re0.25 f  f c (1.0875  0.1125 )

8 1.5 Turbulent: f c  0.0216  9.2  10 Re

Bhatti-Shah [16]

3

f c  5.12  10  0.4572 Re

2300
0.311



Kandlikar et al. [9]

4000
 1.01612  *  0.268  L De  f   0.0929   Re L De   Turbulent:   2 11  (2   )  Re *  Re    3 24 

The experiments on single-phase non-heated flow resistance were carried out first. Pressure drop data were processed and the friction factor was obtained (section 3.2, Fig. 6). It readily apparent that the flow could be divided into laminar zone (Re < 2700), transition zone and turbulent zone (Re > 4000). The critical Reynolds number for the laminar-to-turbulent transition was ~2700, which had a significant hysteresis compared to the conventional channel for which the critical Reynolds number was ~1800 < Re < 2200 [42, 43]. The transition zone boundary was 2700 < Re < 4000, which was close to the conclusion obtained by Ma. et al [6]. In addition, the friction factors in the laminar flow zone and the turbulent zone were predicted using the equations in Table 5 and compared with the experimental values. The predicted performance was measured using the weighted mean absolute percent error (WMAPE) [44] and the normalized root mean square error (NRMSE) [45, 46] (Table 6), which are defined as follows. error 

Predicted value  Measured value  100% Measured value

(28)

19

N

WMAPE 

 Predicted value  Measured value i 1

N

 (Measured value)

 100%

(29)

i 1

NRMSE 

1 N 2  Predicted value  Measured value  N i 1 100% 1 N (Measured value) N i 1

(30)

The predicted WMAE and NRMSE errors for all equations were within 10%, indicating that they could be used for engineering approximate predictions without significant inaccuracy. Among them, the Troniewski-Ulbrich equation had the smallest WMAE and NRMSE values in the laminar flow region, and most of the experimental data were within ± 10% relative error. The Kakac-Yener equation had the best predictive performance in the turbulent region, and most of the experimental data were within ±5% relative error. Therefore, it also showed that the experimental system was suitable for conducting relevant experimental research. 0.18

Test_data Troniewski-Ulbrich [15] Kakac-Yener [37]

0.16 0.14

+10%

0.12 0.1

-10%

f

0.08

0.06

Re=4000 Re=2700 +5%

0.04

-5%

1000

Re

10000

Fig. 6. Non-heated single-phase flow resistance. Table 6. Comparison of experimental data with correlations for single-phase flow resistance. Authors Shah-London [8] Troniewski-Ulbrich [15] (laminar) Spiga-Morino [39]

WMAPE NRMSE (%) (%) 7.1 8.5 3.0 4.1 6.8 8.2 20

Blasius [9] Troniewski-Ulbrich [15] (turbulent) Sadatomi et al. [40] Kakac-Yener [41] Bhatti-Shah [16] Kandlikar et al. [9]

8.6 2.0 3.9 1.5 9.4 2.2

8.8 2.5 4.3 1.9 9.5 2.5

For the verification of heat transfer data, the comparison between the experimental data and the data of Sudo et al. [17], Jo et al [18], Wang et al. [14], etc. shown in Fig.11 can play this role to a certain extent. Because the development of heat transfer correlations in rectangular channels is still immature, and there are no reports of heat transfer coefficients in rectangular channels with the same geometric dimensions as in this study. Therefore, we selected some results obtained within rectangular channels with geometric dimensions close to this study. By comparison, it is found that the experimental data and literature data are in good agreement.

5. RESULTS 5.1 Heated friction factor

Figure 7 shows the single-phase flow friction factors under heating conditions, together with the non-heated values for better comparative study. It follows that the distribution law of friction factors under heated and non-heated conditions was basically the same in the laminar flow region. However, in the transition and quasi-turbulent zone, the heated friction factor was slightly lower than the non-heated values, and the value of non-uniform heating was slightly higher than that of uniform heating. This could be attributed to the effect of lateral secondary flow due to the lateral temperature gradient caused by non-uniform heating. In addition, the transition started critical Reynolds number increased from 2700 without heating to about 3000 with heating. As Everts and Meyer [47] found that the start of transition occurred at the same moment in time along the entire horizontal test section, the critical Reynolds numbers increased with the increasing heat flux only due to the viscosity decreased with increasing temperatures. Meanwhile, the free convection effects in mixed convection caused the transition to occur earlier, and accelerated the transition from laminar to turbulent (the width of the transitional flow regime decreased). However, the free convection effects 21

caused by the buoyancy were smaller in the vertical upward flow of double-sided heating rectangular channels in this study, so forced convection dominates. Sudo et al. [17] also found that the effect of buoyancy on vertical upward flow in a rectangular channel heated on both sides is not significant. Therefore, the mass flow rate corresponding to the transition started critical Reynolds number in this study kept almost constant for both heated and nonheated conditions. In addition, since the transition started (Recr) and ended (Reqt) Reynolds number increased simultaneously under heating, the width of flow transition was almost the same as that without heating. Bashir and Everts et al. [48] also draw a similar conclusion for the forced flow in a vertically uniform heated circular tube. 0.16 f_Isothermal f_Non-uniform Series1 f_Non-uniform Series2 f_Uniform

0.14 0.12 0.048

0.1

0.046 0.044 0.042 0.04

0.08

0.038

0.036

f

0.034

0.032

0.06

0.03

2500

3000

3500

4000

4500 5000 5500

Upper limit line 0.04

Lower limit line 1000

Re

10000

Fig. 7. Single-phase flow resistance. 5.2 Forced convection heat transfer

Heat transfer characteristics of single-phase forced convection in rectangular channels were studied under transversely uniform and non-uniform heating conditions according to the procedure described in section 3.1. The overall average Nusselt number of the test section at different transverse heating power profile was obtained and shown in Fig. 8. The error bars represented the maximum standard uncertainty of the data. It can be qualitatively found that the overall average Nusselt number of the non-uniform heating series 1 was larger, and the value of the non-uniform heating series 2 was smaller, and the value of the uniform condition was centered. But the differences between the three different series of data was very small, 22

and most of them were still within the range of error bars. Therefore, due to the relatively large uncertainty caused by experimental measuring equipment and different thermal efficiencies, it was difficult to accurately describe the results by quantitative analysis methods. Research on the convective heat transfer in circular tubes is mature. Correlations for Nusselt number for different flow regimes are commonly found in various manuals and textbooks. However the study of heat transfer characteristics in narrow rectangular channels is relatively immature. Therefore combined with the experimental data, the traditional circular and the existing rectangular relations have been evaluated. It is worth noting that, as mentioned in section 2.2, developing laminar heat transfer needs to be considered, while the entrance effect can be ignored for turbulent heat transfer. Table 7 gives some selected relations for comparison with experiments. 40

Power_uniform Power_non-uniform1 Power_non-uniform2

35

30

Nu

25

20

15

10

5 1000

2000

3000

4000

5000

6000

7000

Re

Fig. 8. Mean Nusselt number under three heating power distribution. Table 7. Heat transfer correlations for comparison. Reference

Correlation

HartnettKostic [49]

 1  2.0421  3.0853 2   Nu  8.235   3 4 5   2.4765  1.0578  0.1861 

Description Rectangular, fully developed laminar, Re < 2200

23

Sieder-Tate [50]

13   Nu  1.86  Re Pr De L   f   w 

Circle, thermally developing (constant flux), Re < 2200 Circle, simultaneously developing (constant flux), 0.7 < Pr < 7 or RePrD/L < 33 for Pr > 7 Rectangular, 2.35×54 mm, upward flow, Re < 3000

0.14

0.086  Re Pr De L  StephanNu  4.364  0.3 Preußer [51] 1  0.11Pr  Re De L  1.33

Jo et al [18]

Nu  2.0129Gz0.3756 Nu  4.36  Nu1  Nu2





1 0.54 0.46 0.84 Pr 0.2 LtMCD  0.72  Re De  Pr 0.34 LtMCD L 1 0.08 Meyer-Everts Nu2   0.207Gr 0.305  1.19  Pr 0.42  Re De   L  LtMCD  L [52] 2.4 Re Pr 0.6 De LtMCD  for L>LtMCD Gr 0.57 LtMCD  L for L
Hausen [53]

  De  2 3    f  Nu  0.116  Re 2 3  125  Pr 1 3 1        L     w 

Meyer et al._1 [54]

Nu   0.017 Re  30.3 Pr 0.33 Gr 0.08

DittusBoelter [13]

Nu  0.023Re0.8 Pr0.4

Gnielinski [55]

Nu 

Circle, transitional and fully developed turbulent, 3000 < Re < 5×106

2

Meyer et al._2 [54]

Nu  0.018 Re 0.25  Re  500 

Liang et al. [19]

Nu  0.00666Re

Pr

Ma et al. [6]

Nu  0.00354RePr

0.4

Ghione et al._1 [56]

  Nu  0.0044 Re 0.96 Pr 0.568  f   w

1.07

0.933

Circle, transitional, 2200 < Re <104 Circle, transitional flow, 2115 104

 f 8 Re 1000  Pr 12 1  12.7  f 8   Pr 2 3  1

f  1.82log10  Re   1.64 

0.14

Circle, developing and fully developed laminar flow, 48
 Pr  Pr 0.42    Prw 

0.11

Circle, quasi-turbulent and turbulent, 0.5
0.4

Rectangular, 2.0×40 mm, upward flow, 4000 < Re < 1.3×104, 3.9 < Pr < 4.1    

0.14

Rectangular, 2.161×50 mm, upward flow, 104 < Re < 2.69×105, 1.2 < Pr < 5.94 24

Ghione et al._2 [56]

  Nu  0.00184 Re1.056 Pr 0.618  f   w 

Jo et al [18]

Nu  0.0058Re0.9383 Pr0.4

Rectangular, 1.509×50 mm, upward flow, 104 < Re < 1.77×105, 1.18 < Pr < 5.7 Rectangular, 2.35×54 mm, upward flow, Re > 5000

0.14

Table 8 shows the comparison of the predicted values of the relation and the experimental values under uniform heating conditions. Statistical parameters including the WMAPE, NRMSE, and maximum and minimum values of relative error are also shown. For Re < 3000, the latest Meyer and Everts [52] correlation showed the best prediction performance, which fully considered the developing and developed flow in both forced and mixed convection in a circular tube. The Hartnett-Kostic [49] and Jo et al. [18] correlations established from experimental data of rectangular channels also performed well. Other traditional circular tube equations, such as Sieder-Tate [50] and Stephan-Preußer [51], had large prediction errors. In addition, the predictions of all the correlations underestimated the experimental data to a large extent (Fig. 9). For Re > 3000, predicted values of all the correlations overestimated the experimental values except Meyer et al.[54]. The Ma et al. [6] and Jo et al. [18] correlations showed relatively good performance for the experimental data in this study (Fig. 10). Within the experimental data range, the applicability of the traditional tube correlations for rectangular channels was relatively poor. Even for some rectangular channel correlations, their generality was limited because most of them were developed based on limited experimental data within a specific channel. Table 8. Comparison of experimental data with correlations for single-phase heat transfer Correlation

Data range

Hartnett and Kostic [49] Sieder and Tate [50] Stephan and Preußer [51] Jo et al. [18] Meyer and Everts [52] Hausen [53] Meyer et al._1 [54] Dittus and Boelter [13] Gnielinski [55]

Re < 3000 Re < 3000 Re < 3000 Re < 3000 Re < 3000 3000 < Re < 6300 3000< Re < 4000 4000 < Re < 6300 4000 < Re < 6300

WMAPE (%) 16.03 33.72 21.96 18.33 8.56 16.46 21.10 31.53 18.91

NRMSE (%) 16.92 33.81 22.07 18.69 10.16 17.48 21.92 32.39 19.57

Maximum error (%) -5.59 -28.94 -17.60 -10.94 1.19 27.86 -4.15 47.4 28.11

Minimum error (%) -25.81 -40.25 -26.87 -26.86 -20.46 6.51 -35.55 13.75 5.84 25

Meyer et al._2 [54] Liang et al. [19] Ma et al. [6] Ghione at al._1 [56] Ghione at al._2 [56] Jo et al. [18]

4000 < Re < 6300 4000 < Re < 6300 4000 < Re < 6300 4000 < Re < 6300 4000 < Re < 6300 4000 < Re < 6300

11.43 17.44 10.47 14.85 13.43 8.57

12.95 18.52 11.66 17.02 15.53 9.68

22.07 28.51 19.55 29.74 28.17 17.3

-2.85 3.08 -2.66 -1.81 -2.87 -6.05

15

Test data_uniform Hartnett and Kostic [49] Sieder and Tate [50] Stephan and PreuBer [51] Jo et al [18] Meyer and Evert et al [52]

14 13 12 11 10

Nu

9 8 7 6 5 4 3 2 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000

Re

Fig. 9. Comparison of experimental data with correlations for Re < 3000. 50

45

40

Nu

35

Test data_uniform Hausen [53] Dittus and Boelter [13] Gnielinski [55] Liang et al. [19] Ma et al. [6] Ghione et al._2.161×50 [56] Ghione et al._1.509×50 [56] Jo et al [18] Meyer et al._2 [54] Meyer et al._1 [54

30

25

20

15

10 2500

3000

3500

4000

4500

5000

5500

6000

6500

Re

Fig. 10. Comparison of experimental data with correlations for Re > 3000.

26

5.3 New correlations for rectangular channels

We have established a large heat transfer database, which not only contains the experimental data of this study, but also existing data extracted from the literature about the vertical upward and double-sided heating of the rectangular channel. The hydraulic diameter of the data ranges from 3.48 mm to 4.76 mm, and the aspect ratio from 20 to 30. The detailed description of the experimental database is shown in Table 9. The distribution of all data points is shown in Fig. 11. According to the intersection of the data trend lines and with reference to the research of Everts and Meyer[47], there were three feature points, namely Recr (~3000), Reqt (~4000) and Ret (~10000): Recr denoted the point of departure from the laminar heat transfer, Reqt represented the point of transition to quasi-turbulent flow, and Ret represented the point of transition to turbulent flow. Therefore we divided heat transfer into four zones: the laminar zone (I), transition, quasi-turbulent and turbulent zone (III). The transition and quasi-turbulent regions were unified as the transition zone (II) in this study. Table 9. Database of single-phase forced convection heat transfer in rectangle channels. Reference Sudo et al. [17] Jo et al [18] Wang et al. [14] Liang et al. [19] This study

Description 2.15 mm × 50 mm, upward flow, double-sided heating, 300
Data Points 265 204 120 286 80, 82

27

Nu/Pr0.4

100

Sudo et al. [17] Jo et al. [18] This study_uniform This study_Non_uniform series 1 This study_Non_uniform series 2 Wang et al. [14] Liang et al. [19]

Ret

Reqt

10

Ⅰ

Ⅲ

Ⅱ

Recr

1000

Re

10000

100000

Fig. 11. Heat transfer database. Forced convection heat transfer in laminar flow, zone I, is susceptible to the entrance effect. The entrance effect region is further divided into thermal entry and combined entry (simultaneously developing of thermal and hydraulic). Generally, the effect of the thermal entrance length on heat transfer is represented via Graetz number, but the effect of hydraulic entrance length on heat transfer is a function of Reynolds number and tube diameter. However, only the influence of thermal development was considered here for the following reasons. Jo et al [18] indicated that the fluid has fully developed before entering the effective heating section. Sudo et al. [17] and Wang et al. [14] did not mention whether their flow was fully developed in their research. But in Sudo et al. [17], discussion mainly centered on the Nusseltversus-Graetz number relation. Wang et al. [14] used buffer chambers at the entrance and exit of their test section and designed the hydraulic development length before the effective heating test section. Therefore, the optimal fitting of laminar flow data has used to develop the laminar convection heat transfer equation of rectangular channel considering the effect of thermal entry. The new correlation based on the form of Sieder-Tate [50] was assumed,

Nu  C1Gza

1

(31)

Through linear regression, the best fitting correlation was,

Nu  4.57Gz0.194

(32) 28

where the Re < 3000, 11.7 < Gz < 1045.0, 3.8 < De<4.5, 20 < α < 30. The predictive performance of the correlation was shown in Fig. 12. Most of the experimental data were within the ± 25% error lines. 25

Test data Nu=4.57Gz0.194 ±25%

20

+25%

15

Nu

-25% 10

MAE=9.68% RMSE=12.6%

5

10

100

1000

Gz

Fig. 12. Comparison of laminar data with best-fitting correlation. The Dittus-Boelter [13] equation has been widely used in the prediction of turbulent heat transfer in circular tubes because of its simple structure, convenience and good prediction accuracy. Our data from the vigorous turbulent zone (III) was used to verify its applicability in rectangular channels. The results in Fig. 13 showed that almost all data were within ± 20% of the error range. Therefore, the Dittus-Boelter correlation was still applicable to the prediction of convective heat transfer in the turbulent region of rectangular channels.

29

200

Test data Dittus and Boelter [44] ±20%

180 160 140 120

Nu/Pr0.4

100 80

60

WMAE=10.39% NRMSE=13.35%

40

10000

20000

30000

40000

50000

Re

Fig. 13. Comparison of turbulent data with best-fitting correlation. In the transition zone (II), the new empirical relationship based on the Hausen [53] form was, Nu  C2 (Rea2  B）Pr 0.4 (1  De / L)2/3

(33)

Based on the linear regression analysis, of the data, the best fitting equation was, Nu  0.212(Re0.6  83.4）Pr 0.4 (1  De / L) 2/3

(34)

where 3000 < Re < 10000, 3.5 < De < 4.5, and 20 < α < 30 Approximately 84.4% of the experimental data were within ± 20% error lines, and about 98.7% of the data points were within ± 30% error lines (Fig. 14).

30

70

Nu/Pr0.4

40

Test data Nu=0.212(Re0.6-83.4)Pr0.4(1-D/L)2/3 ±20% ±30%

MAE=11.33% RMSE=14.49%

10

7

2000

4000

6000

8000

10000

Re

Fig. 14. Comparison of transitional data with best-fitting correlation. 6.

CONCLUSION Measurements of heat transfer coefficients and friction factors in single-phase flow in a

rectangular channel was reported, and the present data supplemented the existing experimental data under non-heated and uniform heat flux conditions. The influence of transverse power distribution on flow and heat transfer characteristics have been investigated and analyzed. Our main conclusions were, (1) The distribution law of friction factors under heated and non-heated conditions was basically the same in the laminar flow region. However, in the transition and quasiturbulent zone, the heated friction factor was slightly lower than the non-heated values, and the value of non-uniform heating was slightly higher than that of uniform heating. (2) The transition started critical Reynolds number increased from 2700 without heating to about 3000 with heating only due to the viscosity decreased with increasing temperatures (mass flow rate was almost constant). Due to the transition started (Recr) and ended (Reqt) Reynolds number increased simultaneously under heating, the width of flow transition was almost the same as that without heating. (3) The transverse wall power distribution influenced the average Nusselt number of the 31

channel, and the ordering of values depended on the form of the power distribution, although measured differences were small for uniform and non-uniform power distributions. Due to the relatively large uncertainty of the experimental parameters, there was no quantitative analysis. (4) For the heat transfer data under uniform heat flux, 15 correlations were evaluated. It was found that the existing correlations underestimated the experimental data for Re < 3000 and overestimated the experimental data for 3000 < Re < 6000. (5) A large database of heat transfer experiments in rectangular channels has been established, and new heat transfer correlations for laminar (Re < 3000) and transition region (3000 < Re < 10000) were developed. The Dittus-Boelter [13] correlation still had a good prediction for the heat transfer characteristics of rectangular channels in the vigorous turbulent (Re > 10000). Due to the influence of parameter range and measurement uncertainty, there is no quantitative analysis of the effect of lateral power distribution on friction factor and heat transfer. Therefore, we will conduct in-depth research through the improvement of measurements and numerical experiments.

ACKNOWLEDGEMENTS The authors would like to thank the support from Natural Science Foundation of China (Grant No. 11675127). CONFLICT OF INTEREST The authors declare that there is no conflict of interest.

REFERENCES [1] K. Thulukkanam, Heat exchanger design handbook, CRC press, 2013. [2] E. Ishibashi, K. Nishikawa, Saturated boiling heat transfer in narrow spaces, International Journal of Heat & Mass Transfer, 12(8) (1969) 863-866. [3] Y. Fujita, H. Ohta, S. Uchida, K. Nishikawa, Nucleate boiling heat transfer and critical heat flux in narrow space between rectangular surfaces, International Journal of Heat & Mass Transfer, 31(2) (1988) 229-239. [4] S.J. Miller, H. Ozaltun, Evaluation of U10Mo fuel plate irradiation behavior via numerical and experimental 32

benchmarking, Idaho National Laboratory (INL), 2012. [5] M. Arkani, M. Hassanzadeh, S. Khakshournia, Calculation of six-group importance weighted delayed neutron fractions and prompt neutron lifetime of MTR research reactors based on Monte Carlo method, Progress in Nuclear Energy, 88 (2016) 352-363. [6] J. Ma, L. Li, Y. Huang, X. Liu, Experimental studies on single-phase flow and heat transfer in a narrow rectangular channel, Nuclear Engineering and Design, 241(8) (2011) 2865-2873. [7] N. Caney, P. Marty, J. Bigot, Friction losses and heat transfer of single-phase flow in a mini-channel, Applied Thermal Engineering, 27(10) (2007) 1715-1721. [8] R.K. Shah, A.L. London, Laminar Flow Forced Convection in Ducts, Advances in Heat Transfer, in, Academic Press, New York, 1978. [9] S. Kandlikar, S. Garimella, D. Li, S. Colin, M.R. King, Heat transfer and fluid flow in minichannels and microchannels, elsevier, 2005. [10] A. Ghione, B. Noel, P. Vinai, C. Demazi¨¨re, Assessment of thermal¨Chydraulic correlations for narrow rectangular channels with high heat flux and coolant velocity, International Journal of Heat and Mass Transfer, 99 (2016) 344-356. [11] G. Willermoz, A. Aggery, D. Blanchet, S. Cathalau, C. Chichoux, J. Di-Salvo, C. Döderlein, D. Gallo, F. Gaudier, N. Huot, HORUS3D code package development and validation for the JHR modeling, in:

Proc. Int.

Conf. PHYSOR, 2004. [12] C. Chichoux, J.-M. Delhaye, P. Clement, T. Chataing, Experimental study on heat transfer and pressure drop in rectangular narrow channel at low pressure, in:

International Heat Transfer Conference 13, Begel House Inc.,

2006. [13] F. Dittus, L. Boelter, Heat transfer in automobile radiators of the tubular type, International Communications in Heat and Mass Transfer, 12(1) (1985) 3-22. [14] C. Wang, P. Gao, S. Tan, Z. Wang, C. Xu, Experimental study of friction and heat transfer characteristics in narrow rectangular channel, Nuclear Engineering and Design, 250 (2012) 646-655. [15] L. Troniewski, R. Ulbrich, Two-phase gas-liquid flow in rectangular channels, Chemical Engineering Science, 39(4) (1984) 751-765. [16] M. Bhatti, Turbulent and transition flow convective heat transfer in ducts, Handbook of single-phase convective heat transfer,

(1987).

[17] Y. Sudo, K. Miyata, H. Ikawa, M. Ohkawara, M. Kaminaga, Experimental study of differences in singlephase forced-convection heat transfer characteristics between upflow and downflow for narrow rectangular channel, Journal of nuclear science and technology, 22(3) (1985) 202-212. [18] D. Jo, O.S. Al-Yahia, A. RAGA'I M, J. Park, H. Chae, Experimental investigation of convective heat transfer in a narrow rectangular channel for upward and downward flows, Nuclear Engineering and Technology, 46(2) (2014) 195-206. [19] Z.H. Liang, Y. Wen, C. Gao, W.X. Tian, Y.W. Wu, G.H. Su, S.Z. Qiu, Experimental investigation on flow and heat transfer characteristics of single-phase flow with simulated neutronic feedback in narrow rectangular channel, Nuclear Engineering and Design, 248 (2012) 82-92. [20] Z.-H. Liang, H. Zhang, S.-Z. Qiu, G.-H. Su, Experimental investigation on thermal-hydraulic characteristics of narrow rectangular channels with simulated neutronic feedback, Nuclear Engineering and Design, 273 (2014) 668-679. [21] C. Wang, P. Gao, S. Tan, Z. Wang, Forced convection heat transfer and flow characteristics in laminar to turbulent transition region in rectangular channel, Experimental Thermal and Fluid Science, 44 (2013) 490-497. [22] L. Li, D. Fang, D. Zhang, M. Wang, W. Tian, G. Su, S. Qiu, Flow and heat transfer characteristics in plate33

type fuel channels after formation of blisters on fuel elements, Annals of Nuclear Energy, 134 (2019) 284-298. [23] D. Jo, C.G. Seo, Effects of transverse power distribution on thermal hydraulic analysis, Progress in Nuclear Energy, 81 (2015) 16-21. [24] O.S. Al-Yahia, T. Kim, D. Jo, Flow Instability (FI) for subcooled flow boiling through a narrow rectangular channel under transversely uniform and non-uniform heat flux, International Journal of Heat and Mass Transfer, 125 (2018) 116-128. [25] T. Kim, O.S. Al-Yahia, D. Jo, Experimental study on the Onset of Nucleate Boiling in a narrow rectangular channel under transversely non-uniform and uniform heating, Experimental Thermal and Fluid Science, 99 (2018) 158-168. [26] T. Kim, Y.J. Song, O.S. Al-Yahia, D. Jo, Prediction of the minimum point of the pressure drop in a narrow rectangular channel under a transversely non-uniform heat flux, Annals of Nuclear Energy, 122 (2018) 163-174. [27] J. Zhang, H. Yu, M. Wang, Y. Wu, W. Tian, S. Qiu, G. Su, Experimental study on the flow and thermal characteristics of two-phase leakage through micro crack, Applied Thermal Engineering, 156 (2019) 145-155. [28] T.L. Bergman, F.P. Incropera, D.P. DeWitt, A.S. Lavine, Fundamentals of heat and mass transfer, John Wiley & Sons, 2011. [29] Y. Cencel, J. Cimbala, Fluid mechanics–fundamentals and applications,

(2006).

[30] W.M. Rohsenow, J.P. Hartnett, Y.I. Cho, Handbook of heat transfer, McGraw-Hill New York, 1998. [31] P.-S. Lee, S.V. Garimella, D. Liu, Investigation of heat transfer in rectangular microchannels, International Journal of Heat and Mass Transfer, 48(9) (2005) 1688-1704. [32] Y. Bai, Q. Bai, 13 - Hydraulics, in: Y. Bai, Q. Bai (Eds.) Subsea Engineering Handbook (Second Edition), Gulf Professional Publishing, Boston, 2019, pp. 315-361. [33] K. Zhang, Y. Hou, W. Tian, Y. Zhang, G. Su, S. Qiu, Experimental investigations on single-phase convection and two-phase flow boiling heat transfer in an inclined rod bundle, Applied Thermal Engineering, 148 (2019) 340-351. [34] Y. Wang, K. Deng, B. Liu, J. Wu, G. Su, A correlation of nanofluid flow boiling heat transfer based on the experimental results of AlN/H2O and Al2O3/H2O nanofluid, Experimental Thermal and Fluid Science, 80 (2017) 376-383. [35] B. Dai, M. Li, Y. Ma, Effect of surface roughness on liquid friction and transition characteristics in microand mini-channels, Applied Thermal Engineering, 67(1-2) (2014) 283-293. [36] M. Zhang, R.L. Webb, Correlation of two-phase friction for refrigerants in small-diameter tubes, Experimental Thermal and Fluid Science, 25(3-4) (2001) 131-139. [37] D.F. Sempértegui-Tapia, G. Ribatski, Flow boiling heat transfer of R134a and low GWP refrigerants in a horizontal micro-scale channel, International Journal of Heat and Mass Transfer, 108 (2017) 2417-2432. [38] S.k. Kaka, R.K. Shah, W. Aung, Handbook of single-phase convective heat transfer,

(1987).

[39] M. Spiga, G.L. Morino, A symmetric solution for velocity profile in laminar flow through rectangular ducts, International communications in heat and mass transfer, 21(4) (1994) 469-475. [40] M. Sadatomi, Y. Sato, S. Saruwatari, Two-phase flow in vertical noncircular channels, International Journal of Multiphase Flow, 8(6) (1982) 641-655. [41] S. Kakac, Y. Yener, A. Pramuanjaroenkij, Convective heat transfer, CRC press, 2013. [42] K.V. Sharp, R.J. Adrian, Transition from laminar to turbulent flow in liquid filled microtubes, Experiments in fluids, 36(5) (2004) 741-747. [43] A. Darbyshire, T. Mullin, Transition to turbulence in constant-mass-flux pipe flow, Journal of Fluid Mechanics, 289 (1995) 83-114. [44] E.D. Fylladitakis, A.X. Moronis, M. Theodoridis, Analytical Estimation of the Electrostatic Field in Cylinder34

Plane and Cylinder-Cylinder Electrode Configurations, International Journal of Electrical & Computer Engineering (2088-8708), 6(6) (2016). [45] A. Zendehboudi, X. Li, A robust predictive technique for the pressure drop during condensation in inclined smooth tubes, International Communications in Heat and Mass Transfer, 86 (2017) 166-173. [46] M. Aghbashlo, J. Müller, H. Mobli, A. Madadlou, S. Rafiee, Modeling and simulation of deep-bed solar greenhouse drying of chamomile flowers, Drying Technology, 33(6) (2015) 684-695. [47] M. Everts, J.P. Meyer, Heat transfer of developing and fully developed flow in smooth horizontal tubes in the transitional flow regime, International Journal of Heat and Mass Transfer, 117 (2018) 1331-1351. [48] A. Bashir, M. Everts, R. Bennacer, J. Meyer, Single-phase forced convection heat transfer and pressure drop in circular tubes in the laminar and transitional flow regimes, Experimental Thermal and Fluid Science, 109 (2019) 109891. [49] J.P. Hartnett, M. Kostic, Heat transfer to Newtonian and non-Newtonian fluids in rectangular ducts, in: Advances in heat transfer, Elsevier, 1989, pp. 247-356. [50] E.N. Sieder, G.E. Tate, Heat transfer and pressure drop of liquids in tubes, Industrial & Engineering Chemistry, 28(12) (1936) 1429-1435. [51] K. Stephan, P. Preußer, Wärmeübergang und maximale Wärmestromdichte beim Behältersieden binärer und ternärer Flüssigkeitsgemische, Chemie Ingenieur Technik, 51(1) (1979) 37-37. [52] J.P. Meyer, M. Everts, Single-phase mixed convection of developing and fully developed flow in smooth horizontal circular tubes in the laminar and transitional flow regimes, International Journal of Heat and Mass Transfer, 117 (2018) 1251-1273. [53] H. Hausen, Neue Gleichungen fur die Warmeubertragung bei freier oder erzwungener Stromung, Allg. Wärmetech, 9 (1959) 75-79. [54] J.P. Meyer, M. Everts, N. Coetzee, K. Grote, M. Steyn, Heat transfer coefficients of laminar, transitional, quasi-turbulent and turbulent flow in circular tubes, International Communications in Heat and Mass Transfer, 105 (2019) 84-106. [55] V. Gnielinski, New equations for heat and mass transfer in the turbulent flow in pipes and channels, NASA STI/recon technical report A, 75 (1975) 8-16. [56] A. Ghione, B. Noel, P. Vinai, C. Demazière, Assessment of criteria for Onset of Flow Instability in vertical narrow rectangular channels with downward flow, in, 2018.

35

Highlights 

The critical Reynolds number increases only due to the viscosity decreases with increasing temperature.



The influence of transverse power distribution on friction factor is small.



The average Nusselt number is affected by the form of the transverse power distribution.



15 correlations are evaluated for the heat transfer data.



New heat transfer correlations for laminar and transition region are developed.

36

Conflict of Interest Statement

No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication. Neither the entire paper nor any part of its content has been published or has been accepted elsewhere. It is not being submitted to any other journal.

37