Natural convection heat transfer from horizontal concentric and eccentric cylinder systems cooling in the ambient air and determination of inner cylinder location

Abstract

Heat transfer characteristics of horizontal copper concentric cylinders in the case of natural convection was investigated numerically and experimentally. While the inner cylinder had an electric heater to keep it at a constant temperature, annulus was filled with water. There were two different test sections as bare and concentric cylinder systems located in different ambient temperatures in a conditioned room for the comparison of the results. Comparison of average Nusselt numbers for the air side of the concentric cylinder system and the effective thermal conductivity of the annulus were calculated with both experimental data, numerical results and a well-known correlation. Annulus and the air side isotherms and streamlines are shown for Ra_L = 9 × 10⁵–5 × 10⁶ and Ra = 2 × 10⁵–7 × 10⁵ respectively. Additionally, a numerical study was conducted by forming eccentric cylinder systems to determine the optimum location of inner cylinder to maximize the heat transfer rate. Comparison of heat transfer rates from bare and concentric horizontal cylinders were done under steady state conditions. Heat transfer enhancement, the effect of the decrease in condensing temperature of the inner cylinder surface on COP of an ideal Carnot refrigeration cycle and rise in COP were determined in the study. Also the optimum location of inner cylinder to maximize the heat transfer rate was determined as at the bottom quadrant of outer cylinder.

1 Introduction

Nowadays, the use of energy efficiently is a common subject including majority of the industrial, infrastructure and household machines. Especially for the refrigeration applications, reducing energy consumption, in other words, decreasing compressor power cause an increase in COP by lowering the condensing temperature which depends on the increase in the total heat transfer coefficient or condenser’s heat transfer surface area. Besides, household refrigerators have usually such condenser tubes whose radiuses are quite smaller than the critical one. Therefore, the sizes should be larger up to the critical one by covering the bare condenser pipe in order to enhance the heat transfer.

Cylinder coverage can be done as one of the techniques of insulation or surface enhancement. According to Incropera and DeWitt [1], curved surfaces such as circular cylinder having a radius smaller than a certain critical size, adding insulation to the surface increases the heat transfer. This occurrence is about the critical radius which should be taken into account for the case of the fluids’ stationary situation in the annulus. In general the fluid in the annulus does not remain stationary because of the different surface temperatures of concentric cylinders. According to Cengel [2], natural convection effect in annulus increases the heat transfer, for that reason, numerical and experimental studies are performed to determine the natural convection heat transfer from horizontal concentric cylinders.

In open sources, there are a lot of works and their correlations regarding natural convection heat transfer from horizontal cylinder. Morgan [3] proposed empirical correlations to determine average Nusselt number. Fand and Brucker [4] made a comparison of some correlations obtained from their work regarding the natural convection from an infinitely long horizontal isothermal cylinder and proposed a new correlation based on own experimental data in the range between 10⁻⁸ < Ra < 10⁸ and 0.7 < Pr < 10⁴. Misumi et al. [5] and Kitumura et al. [6] performed an experimental study on the flow of natural convection from horizontal cylinders and they showed the inconsistency about the average Nusselt numbers among their results and the others in the literature empirical correlations as well. Misumi et al. [5] emphasized the disagreement on the predictability of McAdams [7] and Kutateladze [8]’ correlations for the Nusselt numbers due to their insufficient techniques both on the experimental apparatus and measurement system.

Churchill and Chu [9] correlated a practical equation to determine average Nusselt numbers over horizontal cylinder including the range of Rayleigh and Prandtl numbers benefitting from Churchill and Usagi's [10] model. Raithby and Hollands [11] investigated heat transfer over elliptic cylinders of arbitrary eccentricity at a persistent surface temperature, and offered equations for laminar and turbulent situations of natural convection. Fand et al. [12] used Raithby and Hollands’ [11] one and included fluid specifications at average film reference temperature. Fujii et al. [13] did experiments of the free convection from horizontal cylinder in water and in various oils and stated a reference temperature. Pera and Gebhart [14] investigated the flow over the cylinder’s surface affected by thermal effects and observed the wake formation over cylindrical surfaces practically.

Kuehn and Goldstein [15] performed a numerical study and used the equations of energy and Navier–Stokes benefiting from the finite-difference over-relaxation method for an isothermal horizontal circular cylinder validated for 10⁰ ≤ Ra ≤ 10⁷. As a result of their analysis, the boundary layer equations for θ > 130° have missing points to characterize heat transfer. From one of newer studies, Reymond et al. [16] calculated natural convection heat transfer from a horizontal cylinder confined with water, they informed that the average Nusselt number distribution show a maximum at the bottom of the cylinder (θ = 0°) and as the boundary layer developing it decreases towards the top (θ = 180°) around the circumference of a cylinder. Merkin [17] developed a model regarding the solution of the full partial differential equations for the natural convection boundary layer over cylinders of general rounded cross-sectional shape. Muntasser and Mulligan [18] benefitted from the local non-similarity modelling and proposed local results for various Prandtl numbers. The outcomes were derived from lower stagnation point (θ = 0) to the 150°. Farouk and Guceri [19] performed heat transfer analyses nearby a single cylinder numerically. According to their analyses, heat transfer increase reaches its peak point at the bottom and it decreases just before the top of the cylinder.

Natural convection occurrences caused by body forces between two concentric horizontal cylinders has been studied in recent years commonly. Itoh et al. [20] proposed an empirical equation using the average numbers of Nusselt and Grashof. Raithby and Hollands [21, 22] investigated laminar and turbulent flows during natural convection from various types of cylinders and spheres. Their study includes proposition of a formula regarding with the effective thermal conductivity for their specified test sections. Their model has the heat transfer per unit length of annulus for inner and outer surface and combination of these two formulas and conduction equation. Their correlation has a relation with k_eff/k having the geometric factor (F_cyl). The error bands for the proposed equation and experimental results from open sources were plotted 10 and 35% for D_o/D_i ≅ 3 and D_o/D_i > 3 respectively. As a result of their study, when the inner diameter of cylinder is small in comparison to the annulus spacing, curvature influences has significance and the proposed equation may lose its accuracy. When the curvature influences are effective, authors proposed two new correlations for heat transfer through annulus. Kuehn and Goldstein [23] performed an experimental study on the natural convection from horizontal circular cylinders and offered an equation related with heat transfer by conduction for the flows of laminar and turbulent. Waheed [24] prepared a numerical study regarding natural convection from a concentric horizontal cylinder for 10² ≤ Ra ≤ 10⁵, 10⁻³≤ Pr ≤ 10³ using finite difference method. Finally, the heat function profiles were influenced by the numbers of Rayleigh and Prandtl, and the temperature and the diameter ratios. The numerical results were confirmed with Kuehn and Goldstein [25]’ experimental results. A theoretical work has been performed by Leong and Lia [26] on natural convection in concentric cylinders with a porous sleeve by perturbation technique and Fourier transformation. Shu et al. [27] did a numerical study on the determination of the flow and thermal fields in arbitrary eccentric annulus benefitting from differential quadrature technique. They regimented the natural convection its work on the arbitrary eccentric cylinders for Ra = 10⁴ and radius ratio of 2.6 considering the influences of inner cylinder position on total Nusselt number, flow field and thermal field investigation. Kumar [28] researched natural convection using some gasses flowing in a horizontal annulus under constant heat flux condition. Nguyen et al. [29] tried to solve the natural convection problem in a horizontal annulus benefitting from regular perturbation technique for the flow of cold water. They used not only circular concentric geometries but also various cylinders such as elliptical ones [30], triangular ones [31, 32] and square ones [33]. A method regarding heat transfer development in the horizontal annulus is includes the use of ultrafine solid particles in the base fluids. Abu-Nada et al. [34] and Abu-Nada [35] studied the natural convection from horizontal annulus aiming to determine the influences of the nano-particles.

Constant surface temperature situations are usually used for heat transfer from horizontal concentric cylinders. Investigation of heat transfer from concentric cylinders placed in the ambient air and constant surface temperature applied merely to inner cylinder is the goal of this work. Hence, the natural convection is valid in the annulus and the ambient air as well. Meanwhile there isn’t any comparable conditions investigated previously, the former works focused on two sections: outer side of the horizontal cylinder and inner side of the annulus.

Isothermal concentric horizontal cylinders has been investigated by many researchers so far. On the contrary, constant temperature situation was valid for inner horizontal cylinder surface and concentric cylinder system was positioned in the ambient air having constant temperature in this work. Therefore, the outer cylinder surface had a temperature gradient and finally, experimental and numerical investigation were performed on the angular surface temperature distribution. Besides, geometric factor (F_cyl) is found to be a significant parameter and there is a turning point on the diameter ratio D_o/D_i = 3. According to Raithby and Hollands [21], the deviation between the correlation including the geometric factor and experimental results has high values for diameter ratios larger than 3 for water. Results of experiments of this work are evaluated for validation of the correlations for highly curved geometries. Novel equations can be proposed for the phenomenon stated above. For this target, experimental and numerical investigations will be prolonged in the similar way.

One more success of existing work is reducing the condensing temperature particularly in home fridges. Hence, two horizontal concentric cylinders were practiced with the purpose of decreasing the condensing temperature. First of all, experiments are done with bare and concentric horizontal cylinders under steady-state conditions. Comparison of heat transfer amounts from bare and concentric horizontal cylinders was performed and heat transfer improvement was obtained noticeably. The 2D numerical heat transfer studies was accomplished by a commercial CFD software (FLUENT).

2 Experimental study

A view of the experimental facility is depicted in Fig. 1. The tests were done in a conditioned room, for that reason the desired ambient temperature has been provided. One m length bare copper cylinders having diameters of 9.45 and 4.8 mm were used. To have a constant temperature in the axial direction, copper material and thick wall thickness (1 mm) was preferred for test cylinders. Five notches were putted at equal intervals on the axial surface. Three more notches were putted in angular way with 90° for the determination of circumferential temperature distribution. The mean surface temperature (T_c) is measured with the thermo elements connected to the notches in the axial way. Two pieces of isolating items made of XPS is putted on the cylinder’s endpoints in order to reduce the end losses, heat transfer rate by conduction to the supports as well.

Fig. 1

Schematic view of the experimental setup; 1 conditioned room, 2 test cabin, 3 test cylinder, 4 resistant wire, 5 insulating material, 6 thermocouples on cylinder surface, 7 thermocouples in the environment, 8 humidity sensor, 9 datalogger, 10 computer, 11 DC power supply, 12 power meter (a), comprehensive illustration of the test cylinder and position of the thermocouples (b) [From Atayil [41], with permission from Elsevier]

Concentric/eccentric cylinder system have 2 copper cylinders whose annulus was filled with water. One m length outer cylinders have 66 mm outer diameter and 2 mm wall thickness. Bare test cylinders were located inside the outer cylinders concentrically and two different concentric cylinder systems were prepared. A special centering structure formed and positioned in the annulus to enable accurate concentric positioning of inner cylinder. Five thermocouples placed on the outer surface of the test cylinder exactly in the same angular direction of the brazed thermocouples on bare cylinder. The average outer surface temperature (T_w) is determined with five thermocouples in the axial direction and also surface temperature observed circumferentially with additional seven thermocouples located every 45° in the angular direction. The steady state condition is considered when variation of all temperatures and especially the average temperatures stay in the range of ±0.1–0.2 °C for 15 min as shown in Fig. 2. Locations of thermocouples on test cylinder are shown in Fig. 1b.

Fig. 2

Distribution of the variation of inner cylinder surface temperature of the concentric cylinder for 30 °C (a), 40 °C (b), 50 °C (c), 60 °C (d)

Constant temperature was applied to bare cylinders and inner cylinders of concentric cylinder systems. The inner cylinders were connected to a DC power supply with 10 A–60 V value to give the required electrical power for desired constant surface temperature. Although the adjustable voltage and current values given by the power supply can be read from the indicators on the device, a power meter with a resolution of ±0.1 W is used to measure the input power accurately.

The experiments were performed in the conditioned room in order to provide the desired ambient temperature and the natural convection conditions. In the room with the dimensions of 4000 mm × 4900 mm × 2550 mm the environment temperature between 5 and 50 °C and the relative humidity between 20 and 95% can be adjusted to the required value. Ambient environment should be quiescent since the experiments are made under natural convection conditions. As the result of the velocity measurements made with a hot-wire calibrated with LDA from the different points in the room it is determined that air velocity varies between 0 and 0.25 m s⁻¹. A test cabin was constructed from Plexiglas which has four lateral surfaces and floor with the dimensions of 800 mm × 1250 mm × 1300 mm. A gap on the closed surfaces in the protection cabin with only its top open can cause a stack effect and thus an unwanted bulk fluid movement will affect the temperature and velocity fields. It is determined that the required quiescent environment was provided as the result of velocity measurements made inside the test cabin. It is stated that the differences between the experimental data obtained from the pre-studies on heat transfer in horizontal cylinder under conditions of natural convection are caused by choosing the insufficient heat measuring system or designed test cabin. To minimize the errors mentioned in the pre-studies highly sensitive measuring tools were used, the test cabin volume was well determined and the experiments were made in the conditioned room to keep the ambient temperature stable at the required values.

The work station connected to the data acquisition unit of the room is used for data collection. By means of HP VEE based software program, the all instant experimental data and required average values can be observed graphically. The program was developed on demand and brought into use. The mentioned features help to determine the steady state condition making analysis after the experiments. The measured values for bare and concentric cylinders obtained from experimental study are summarized in Table 1.

Table 1 Test cylinders of bare and concentric’ thermal boundary conditions

3 Data reduction

The dimensional analysis generally shows that natural convection heat transfer from horizontal cylinders depends on Rayleigh number. Natural convection heat transfer was studied not only at the air side but also in the annulus experimentally. Accordingly the calculations are summarized in two parts: the air side and the annulus.

In the experiment facility, sensitively measured input electrical power, Q_e, gives the total natural convection heat transfer from the outer surface of horizontal bare and concentric cylinders. Under steady state conditions the energy equation for the horizontal test cylinders based on the first law of thermodynamics is expressed as below:

$${\text{Q}}_{\text{e}} = {\text{Q}}_{\text{conv}} + {\text{Q}}_{\text{rad}} + {\text{Q}}_{\text{cond}}$$

(1)

Since two pieces of insulating material is placed on the endpoints of the cylinders the conduction heat loss is neglected:

$${\text{Q}}_{\text{conv}} = {\text{Q}}_{\text{e}} - {\text{Q}}_{\text{rad}}$$

(2)

Heat transfer from the horizontal cylinders surfaces by radiation can be represented by:

$$\left. {\begin{array}{*{20}l} {{\text{Q}}_{\text{rad}} = \frac{{{\text{E}}_{\text{bc}} - {\text{E}}_{{{\text{b}}{\infty }}} }}{{\frac{{1 -\upvarepsilon_{\text{c}} }}{{{\text{A}}_{\text{c}}\upvarepsilon_{\text{c}} }} + \frac{1}{{{\text{A}}_{\text{c}} {\text{F}}_{{{\text{c}}{\infty }}} }} + \frac{{1 -\upvarepsilon_{{\infty }} }}{{{\text{A}}_{{\infty }}\upvarepsilon_{{\infty }} }}}}} \hfill & {\text{for bare cylinder}} \hfill \\ {{\text{Q}}_{\text{rad}} = \frac{{{\text{E}}_{\text{bw}} - {\text{E}}_{{{\text{b}}\infty }} }}{{\frac{{1 -\upvarepsilon_{\text{w}} }}{{{\text{A}}_{\text{w}}\upvarepsilon_{\text{w}} }} + \frac{1}{{{\text{A}}_{\text{w}} {\text{F}}_{{{\text{w}}\infty }} }} + \frac{{1 -\upvarepsilon_{\infty } }}{{{\text{A}}_{\infty }\upvarepsilon_{\infty } }}}}} \hfill & {\text{for concentric cylinder}} \hfill \\ \end{array} } \right\}$$

(3)

$${\text{E}}_{\text{bw}} =\upsigma \cdot {\text{T}}_{\text{w}}^{4} ,\quad {\text{E}}_{\text{bc}} =\upsigma \cdot {\text{T}}_{\text{c}}^{4}$$

(4)

$${\text{E}}_{{{\text{b}}\infty }} =\upsigma \cdot {\text{T}}_{\infty }^{4}$$

(5)

The emissivity of horizontal bare and concentric cylinder surface (ɛ_c, ɛ_w) should be determined accurately to calculate the heat transfer by radiation from Eq. 3. Therefore the outer surface temperature of the heated horizontal bare and concentric cylinder is measured with T type thermocouple that calibrated with reference thermometer and thermal camera simultaneously. Thermal camera’s emissivity value is adjusted till the measured two temperatures become equal. As a result, emissivity is found as 0.5 in this study. Also the obtained value is in the suggested range of Thermal camera’s emissivity tables for oxidized copper. The convection heat transfer which is the driving force of the plume can be represented by:

$$\left. {\begin{array}{*{20}l} {{\text{Q}}_{\text{conv}} = {\text{hA}}({\text{T}}_{\text{c}} - {\text{T}}_{\infty } )} \hfill & {\text{for bare cylinder}} \hfill \\ {{\text{Q}}_{\text{conv}} = {\text{hA}}({\text{T}}_{\text{w}} - {\text{T}}_{\infty } )} \hfill & {\text{for concentric cylinder}} \hfill \\ \end{array} } \right\}$$

(6)

Nusselt number is defined as:

$${\text{Nu}}_{\text{o}} = {\text{hD/k}}$$

(7)

The air thermo physical properties (k, ν) in non-dimensional parameters were evaluated at the mean film temperature

$$\left. {\begin{array}{*{20}l} {{\text{T}}_{\text{f}} = ({\text{T}}_{\text{c}} + {\text{T}}_{\infty } )/2} \hfill & {\text{for bare cylinder}} \hfill \\ {{\text{T}}_{\text{f}} = ({\text{T}}_{\text{w}} + {\text{T}}_{\infty } )/2} \hfill & {\text{for concentric cylinder}} \hfill \\ \end{array} } \right\}$$

(8)

$$\left. {\begin{array}{*{20}l} {{\text{Gr}} = \frac{{{\text{g}}\upbeta({\text{T}}_{\text{c}} - {\text{T}}_{\infty } ){\text{D}}^{3} }}{{\upnu^{2} }}} \hfill & {\text{for bare cylinder}} \hfill \\ {{\text{Gr}} = \frac{{{\text{g}}\upbeta({\text{T}}_{\text{w}} - {\text{T}}_{\infty } ){\text{D}}^{3} }}{{\upnu^{2} }}} \hfill & {\text{for concentric cylinder}} \hfill \\ \end{array} } \right\}$$

(9)

where

$$\upbeta = 1/(273 + {\text{T}}_{\text{f}} )$$

(10)

$${\text{Ra}} = {\text{Gr}}\,{ \Pr }$$

(11)

Heat transfer rate through the annular space between the cylinders by natural convection per unit length is expressed as:

$${\dot{{\text{Q}}}} = \frac{{2\uppi{\text{k}}_{\text{eff}} }}{{\ln (\text{D}_{\text{o}} /{\text{D}}_{\text{i}} )}}({\text{T}}_{\text{c}} - {\text{T}}_{\text{w}} )$$

(12)

Effective thermal conductivity (k_eff) of the annulus which includes both convection currents and conduction is determined experimentally from Eq. 12.

On the other hand Raithby and Hollands [21] gave two equations for heat transfer per unit length of annulus considering the heat transfer from inner surface and to the outer surface. They suggested a correlation by combining these equations and Eq. 12:

$$\frac{{{\text{k}}_{\text{eff}} }}{\text{k}} = 0.386\left( {\frac{\Pr }{0.861 + \Pr }} \right)^{1/4} \frac{{\ln ({\text{D}}_{\text{o}} - {\text{D}}_{\text{i}} )}}{{{\text{L}}_{\text{c}}^{3/4} \left( {{\text{D}}_{\text{i}}^{ - 3/5} + {\text{D}}_{\text{o}}^{ - 3/5} } \right)^{5/4} }}{\text{Ra}}_{\text{L}}^{1/4}$$

(13)

where the Rayleigh number for the annulus is determined from

$${\text{Ra}}_{\text{L}} = \frac{{{\text{g}}\upbeta({\text{T}}_{\text{c}} - {\text{T}}_{\text{w}} ){\text{L}}_{\text{c}}^{3} }}{{\upnu^{2} }}{ \Pr }$$

(14)

The water thermo physical properties (k, ν, β) in non-dimensional parameters were evaluated at (T_c + T_w)/2 and the characteristic length is the spacing between the cylinders:

$${\text{L}}_{\text{c}} = ({\text{D}}_{\text{o}} - {\text{D}}_{\text{i}} )/2$$

(15)

However it is stated that if the inside cylinder diameter becomes small compared with the annulus spacing, curvature effects could become important and Eq. 13 may fail. If a curvature effect is important in a problem, they suggested two new Eqs. (16, 17) for heat transfer per unit length of annulus;

$${\text{Q}} = \frac{{2\uppi\,{\text{k}}}}{{\ln \left[ {1 + \left( {2/\left( {0.386\left( {\frac{\Pr }{0.861 + \Pr }} \right)^{1/4} {\text{Ra}}_{\text{Di}}^{1/4} } \right)} \right)} \right]}}\left( {{\text{T}}_{\text{c}} - {\text{T}}_{\text{M}} } \right)$$

(16)

$${\text{Q}} = \frac{{2\,\uppi\,{\text{k}}}}{{\ln \left[ {1 - \left( {2/\left( {0.386\left( {\frac{\Pr }{0.861 + \Pr }} \right)^{1/4} {\text{Ra}}_{\text{Do}}^{1/4} } \right)} \right)} \right]}}\left( {{\text{T}}_{\text{M}} - {\text{T}}_{\text{w}} } \right)$$

(17)

Rayleigh numbers are determined from;

$${\text{Ra}}_{\text{Di}} = \frac{{{\text{g}}\upbeta({\text{T}}_{\text{c}} - {\text{T}}_{\text{M}} ){\text{D}}_{\text{i}}^{3} }}{{\upnu^{2} }}\Pr$$

(18)

$${\text{Ra}}_{\text{Do}} = \frac{{{\text{g}}\upbeta({\text{T}}_{\text{M}} - {\text{T}}_{\text{w}} ){\text{D}}_{\text{o}}^{3} }}{{\upnu^{2} }}\Pr$$

(19)

Combining the Eqs. 12 and 16 effective thermal conductivity (k_eff) is determined from;

$$\frac{{{\text{k}}_{\text{eff}} }}{\text{k}} = \frac{{\ln \left( {{\text{D}}_{\text{o}} /{\text{D}}_{\text{i}} } \right)}}{{\ln \left[ {1 + \left( {2/\left( {0.386\left( {\frac{\Pr }{0.861 + \Pr }} \right)^{1/4} {\text{Ra}}_{\text{Di}}^{1/4} } \right)} \right)} \right]}}\quad \frac{{\left( {{\text{T}}_{\text{c}} - {\text{T}}_{\text{M}} } \right)}}{{\left( {{\text{T}}_{\text{c}} - {\text{T}}_{\text{w}} } \right)}}$$

(20)

Raithby and Hollands [21] stated that there is no way of explicitly eliminating T_M from Eqs. 16 and 17 so it is obtained by trial and error process using these equations.

One of the objectives of this study was to improve the performance of the domestic refrigerators and deep freezers by decreasing the condensation temperature. Thus the effect of the condensation temperature on coefficient of performance (COP) of a Carnot refrigerator was investigated. The COP of a Carnot refrigerator can be expressed in terms of temperatures as;

$${\text{COP}}_{\text{R}} = \frac{1}{{({\text{T}}_{\text{c}} /{\text{T}}_{\text{e}} ) - 1}}$$

(21)

4 Uncertainty analysis

Amongst many error analysis methods, uncertainty analysis method which is firstly proposed by Kline and McClintock [36] is most widely used one for experimental studies. In this experimental study, the uncertainty analysis method which is more sensitive compared to others is used. If the independent variables that cause errors in experiments are chosen as, input electrical power, local cylinder surface temperatures and the environment temperatures, the uncertainty of Nusselt number (Nu_o) can be defined as follows. Uncertainty analysis is not necessary for radiation heat transfer because it is calculated theoretically and the errors neglected on length measurements;

$${\text{w}}_{\text{h}} = \pm \left[ {\left( {\frac{1}{{{\text{A}} \cdot \left( {{\text{T}}_{\text{w}} - {\text{T}}_{\infty } } \right)}}{\text{w}}_{{{\text{Q}}_{\text{e}} }} } \right)^{2} + \left( {\frac{{ - {\text{Q}}_{\text{conv}} }}{{{\text{A}} \cdot \left( {{\text{T}}_{\text{w}} - {\text{T}}_{\infty } } \right)^{2} }}{\text{w}}_{{{\text{T}}_{\text{w}} }} } \right)^{2} + \left( {\frac{{{\text{Q}}_{\text{conv}} }}{{{\text{A}} \cdot \left( {{\text{T}}_{\text{w}} - {\text{T}}_{\infty } } \right)^{2} }}{\text{w}}_{{{\text{T}}_{\infty } }} } \right)^{2} } \right]^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}$$

(22)

$${\text{w}}_{{{\text{Nu}}_{\text{o}} }} = \pm \left[ {\left( {\frac{\text{D}}{\text{k}}{\text{w}}_{\text{h}} } \right)^{2} } \right]^{1/2}$$

(23)

As the result of the calculations made the average uncertainty is found as 1.03% for heat transfer coefficient (Nusselt number).

If the independent variables that cause errors in experiments are chosen as input electrical power, local inner cylinder surface temperatures and outer cylinder surface temperatures the uncertainty of effective thermal conductivity (k_eff) can be defined as follows.

$${\text{w}}_{{{\text{k}}_{\text{eff}} }} = \pm \left[ {\left( {\frac{{\ln ({\text{D}}_{\text{o}} /{\text{D}}_{\text{i}} )}}{{2\uppi\left( {{\text{T}}_{\text{c}} - {\text{T}}_{\text{w}} } \right)}}{\text{w}}_{{{\text{Q}}_{\text{e}} }} } \right)^{2} + \left( {\frac{{ - {\dot{{\text{Q}}}}\ln (\text{D}_{\text{o}} /{\text{D}}_{\text{i}} )}}{{2\uppi\left( {{\text{T}}_{\text{c}} - {\text{T}}_{\text{w}} } \right)^{2} }}{\text{w}}_{{{\text{T}}_{\text{c}} }} } \right)^{2} + \left( {\frac{{{\dot{{\text{Q}}}}\ln ({\text{D}}_{\text{o}} /{\text{D}}_{\text{i}} )}}{{2\uppi\left( {{\text{T}}_{\text{c}} - {\text{T}}_{\text{w}} } \right)^{2} }}{\text{w}}_{{{\text{T}}_{\text{w}} }} } \right)^{2} } \right]^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}$$

(24)

As the result of the calculations made the average uncertainty is found as 2.38% for k_eff. Table 2 shows the whole results of the uncertainty analysis.

5 Numerical analysis

ANSYS FLUENT software was employed in the numerical part of this study. The solution domain as taken two dimensional since the temperature of the cylinder does not change significant in the axial direction and therefor grid generated in two dimensional geometry is shown in Fig. 3a. Detailed view of the solution grid with finer mesh inside the annulus and outside the concentric cylinder system is shown in Fig. 3b.

Fig. 3

Illustration of 2D model’s solution grid for the concentric cylinder regarding its complete view in the test cabin (a), detailed view (b)

Boundary conditions are determined as constant temperature at the lateral and bottom surface of test cabin and the inner copper cylinder surface, and pressure-inlet for the upper surface of the test cabin. The lateral and bottom surface temperatures of the test cabin and inner copper cylinder surface temperature are set to the ambient temperature and average inner cylinder temperature obtained from experimental study. The measured experimental values for bare and concentric cylinder systems are given in Table 1.

Segregated solution method is used for the solution. The governing equations are solved consecutively in this method. In order to obtain a converged solution, several iterations of the solution loop must be performed because the governing equations are non-linear (and coupled). Each iteration consists of the steps illustrated in Fig. 4.

Fig. 4

Overview of the segregated method [37]

The problem has been solved using standard laminar flow model and surface-to-surface (S2S) radiation model at steady state conditions. The surface-to-surface radiation model can be used for the calculation of radiation heat transfer in an enclosure of gray-diffuse surfaces. The energy exchange between two surfaces depends in part on their size, separation distance, and orientation which are accounted for by view factors. For pressure–velocity coupling discretization the semi-implicit method for pressure linked equations (SIMPLE) algorithm has been used. For continuity and momentum the residual values were taken 10⁻³ and for energy 10⁻⁶.

It should be noted that Ansys Fluent uses finite volume for discretization of the governing equations by dividing the solution domain into small control volumes and resulting algebraic equations are solved iteratively. Moreover, a grid independence test had been performed before the calculations done. The calculations were performed for the number of cells approximately 10,000, 20,000, 40,000, 60,000, 100,000. The difference between the heat fluxes and temperatures for the cell numbers of 60,000 and 100,000 were less than 1%. So considering the computer capacity and solution time, the cell number was chosen as 57,000.

Table 2 Uncertainty of measured variables and calculated values

6 Results and discussion

Natural convection heat transfer from a horizontal cylinder has been studied numerically and experimentally for over 50 years but it is reported by the researchers [1, 2] that the obtained results show high levels of deviation among each other due to various reasons. Morgan [3], after a wide literature research, proposed empirical correlation equations for average Nusselt number and pointed out that the results of the experimental study show deviations ranging between 3 and 36% according to the Rayleigh number. The discrepancy in the average Nusselt number between the well-known previous experimental studies in the literature was defined by one or some of these factors together; wrong designed test cabin, undersized measuring systems, heat conduction to the supports and the temperature measurement locations, distortion of the temperature and velocity fields by fluid movements. Fand and Brucker [4] reported that there is approximately 50% difference between average Nusselt numbers calculated from Churchill and Chu [9] and Raithby and Hollands [21] ’s equations for air at Ra = 1. The difference can be calculated as 43% for Ra = 10⁵ (our experimental study range) and Pr = 0.7. Recently, Misumi et al. [5] investigated natural convection flows around large horizontal cylinders experimentally and discussed the discrepancy on the average Nusselt numbers between their results and the previous empirical correlations. They stated that previous empirical equations such as McAdams [7] and Kutateladze [8] may not predict the Nusselt numbers correctly because of poor descriptions on the experimental apparatus and measurement techniques. Besides Kitumura et al. [6] pointed out that some of the previous researchers have derived the equations by averaging the existing experimental data obtained by others. However, the data show considerable scatters between the researchers, so that a simple average of the data may cause the deviations against the real Nusselt numbers.

As it is stated above generally the wide dispersion in the published experimental results was attributed to distortion of the temperature and velocity fields by bulk fluid movements, the use of under sized test cabin or existence of the temperature measurement system and supports.

In the present study, the test cabin was constructed at proper dimensions and all the surfaces except the top were made impermeable to minimize these factors. Different from the other experimental studies made in the air, all the experiments were performed in the conditioned room (approximate cost including very sensitive state-of-art measurement and control devices is $1,000,000) to keep the environmental temperature constant during the experiments and to have experimental runs at different environmental temperatures. Besides two pieces of insulating material made of XPS is placed on the cylinder’s endpoints to minimize the end losses (heat conduction to the supports). As mentioned above, according to the results of the studies on natural convection heat transfer from horizontal cylinders, 25% deviation between the experimental and the numerical mean Nusselt numbers is fairly good and acceptable.

Heat transfer from outside the concentric cylinder system and inside the annulus were investigated for Ra_L = 7.7 × 10⁵–6.9 × 10⁶ and Ra = 2 × 10⁵–6.4 × 10⁵ respectively. Heat transfer coefficient outside the concentric cylinder system and the corresponding Nusselt number were calculated using experimental and numerical data under steady state conditions. Convection heat transfer rate was calculated using Eqs. (2–5) in the experimental part of this study, while it was calculated numerically from the difference between total and radiation heat transfer rates. Figure 5a represents the calculated Nusselt and Rayleigh numbers from the experimental data and results of numerical study for concentric cylinder system. The deviation between Nusselt numbers obtained from experimental study and results of numerical analysis stay in the range of ±25% as seen in Fig. 5b. Average Nusselt number increases with increasing Rayleigh number for inner cylinder diameters of 4.8 and 9.45 mm.

Fig. 5

Change in experimental and numerical mean Nusselt numbers versus Ra numbers for outer cylinder boundary-layer conditions (a), comparison of data (b)

Effective thermal conductivities (k_eff) of the annulus were also determined by using experimental data and results of numerical solutions. Ra_Di and k_eff/k values calculated using the experimental data and numerical solution results are given in the Fig. 6a. For comparison with the results of this study, it is also calculated according to Raithby and Hollands [21] using Eq. 20. The deviation of k_eff/k values for the annulus stay in the range ±10% as seen in Fig. 6b.

Fig. 6

Experimental and numerical k_eff values’ alteration with Ra number for annulus (a), comparison of data (b)

Isotherms and streamlines are presented in Fig. 7 in the annulus and the air side for ambient temperature of 20 °C for inner cylinder diameters of 4.8 and 9.45 mm. The bottom region is stagnant and the temperature distribution is divided hot and cold regions by an imaginary line below the inner cylinder for both inner cylinder diameters while the fluid is re-circulating in top half of the annulus and making the outer layer warmer. Also, the temperature difference between the points θ = 0° and θ = 180° increases by increasing the temperature difference between inner cylinder surface and ambient temperatures. Also, it is very interesting to note how the plume region is affected by increasing the temperature difference between inner cylinder surface and ambient temperatures as shown in Fig. 7.

Fig. 7

Illustrations of isotherms (left) and streamlines (right) for 9.45/66 mm at T_∞ = 20.4 and T_c = 30.2 °C (a), T_∞ = 20.8 and T_c = 40.1 °C (b), T_∞ = 21.2 and T_c = 50.7 °C (c), and for 4.8/66 mm at T_∞ = 20.8 and T_c = 31.6 °C (d), T_∞ = 21.0 and T_c = 41.5 °C (e), T_∞ = 21.3 and T_c = 52.8 °C (f)

Figure 8 represents the comparison of experimental and numerical average outer surface temperatures of the horizontal concentric cylinder systems (T_w) for inner cylinder diameters of 4.8 and 9.45 mm in a ±5% range. As seen from Fig. 8, experimental data and results of numerical solution are very close and the deviation between experimental and numerical values less than 5%.

Fig. 8

Evaluation of the experimental and the numerical average outer surface temperature of the horizontal concentric cylinder

Figure 9 shows the experimental and numerical angular surface temperature distribution outside the concentric cylinder systems for ambient average temperature T_∞ = 20.4 and T_∞ = 20.8 °C for 9.45/66 and 4.8/66 mm, respectively. It is seen that there is a good agreement between numerical and experimental values.

Fig. 9

Evaluation of experimental and numerical angular outer surface temperature distribution of the concentric cylinders during ambient temperature of 20 °C for 9.45/66 and 4.8/66 mm

It is determined that the location of the inner cylinder is important for maximizing the heat transfer. Changing the location of inner cylinder downwards by the ordinate, stagnant regions can be dominated by convection. For this aim a numerical study has been conducted to determine the most proper location that maximizes the heat transfer for 4.8/66 mm eccentric cylinder system. Natural convection occurred in the annulus side and heat transfer difference for four different position of inner tube have been analyzed. Heat transfer rate is calculated as 10.25 W m⁻¹ for the inner tube’s center position. The maximum heat transfer rate has been obtained for inner tube’s down vertical axis position as shown in Table 3. Heat transfer enhancement was determined as 4.5% increase in comparison to its center position. Heat transfer rates and outer tube surface temperatures for other positions of inner tube were given in Table 3. Moreover, temperature contours occurred for different positions can be seen from Fig. 10.

Table 3 Numerical results for the different location of inner cylinder (T_∞ = 20, T_c = 30 °C)

Fig. 10

Temperature contours for center position (a), down position at vertical axis (b), 20° with vertical axis position (c), 45º with vertical axis position (d)

Heat transfer rates from bare and concentric cylinders were compared with each other under steady-state conditions. The heat transfer augmentation can be seen in Fig. 11 for a constant temperature of inner cylinder. Also, this can be used for decreasing the condensation temperature for a given heat transfer rate by critical radius effect. Besides this, filling the annulus with water also increases the heat transfer due to convection currents.

Fig. 11

Heat transfer rates from tested cylinders under steady-state conditions during the ambient temperatures of 20 and 30 °C

Effect of the condensation temperature on COP_R of a Carnot refrigeration cycle is shown in Fig. 12. As an example if the heat flux is 15 W m⁻¹, COP_R increases from 3.73 for bare cylinder to 5.32 for 9.45/66 mm concentric cylinder system under steady-state condition at evaporation and ambient temperatures −15 and 20 °C respectively. Under the same conditions COP_R increases from 3.07 for bare cylinder to 5.17 for 4.8/66 mm concentric cylinder system. It is found that the enhancement in the coefficient of performance of a Carnot refrigeration cycle is 42.6 and 68.4% for 9.45/66 and 4.8/66 mm concentric cylinder systems respectively. Also if the inner cylinder is located on vertical axis and 5 mm away from the bottom of the outer cylinder, coefficient of performance of a Carnot refrigeration cycle increases by 1.01% according to concentric case.

Fig. 12

Alteration of COP_R with condensing temperatures (T_c) for various evaporating temperatures (T_e) from −15 to 0 °C having constant Q = 15 W m⁻¹ under steady-state conditions during the ambient temperatures of 20 and 30 °C

It should be noted that this study should be considered continuations of authors’ previous publications’ [38–42] in the literature. Due to page limitation, only critical results have been presented. Detailed information on the issue can be obtained from authors’ publications [38–42].

7 Conclusion

The main purpose of this study is improving the performance of the domestic refrigerators by decreasing the condensation temperature. For this aim, critical radius effect and natural convection currents in annular space between the concentric cylinders are employed. Natural convection heat transfer from heated horizontal concentric cylinder systems with the diameters of 9.45/66 and 4.8/66 mm (most used condenser tube diameter for domestic refrigerators: 4.8 mm) were investigated experimentally and numerically under steady-state conditions. In order to determine heat transfer enhancement, heat transfer rates from bare horizontal cylinder and concentric horizontal cylinder system were compared each other.

The effective thermal conductivity (k_eff) of the annulus was determined experimentally, numerically and compared with the Raithby and Hollands [21]. The deviation between the calculated k_eff/k values for annulus stay in the range of ±10%.

Non-uniform temperature distribution occurred on the outer cylinder surface and resulting angular surface temperature distribution was investigated both experimentally and numerically. Increasing the temperature difference between inner cylinder and ambient temperature results increase in the non-homogeneity of the angular surface temperature distribution.

It is also observed that Raithby and Hollands [21] correlation (Eq. 13) fails for experimental data of the present study because of strong curvature effects (D_o/D_i = 6.98 and D_o/D_i = 13.75 for the diameters of 9.45/66 and 4.8/66 mm respectively) and the experimental data were validated using suggested equations for highly curved geometries (Eqs. 16, 17).

Natural convection heat transfer from concentric horizontal cylinders can be analyzed using CFD with proper boundary conditions and mesh size.