Journal of the Brazilian Society of Mechanical Sciences
Print version ISSN 0100-7386
J. Braz. Soc. Mech. Sci. vol.22 n.1 Rio de Janeiro 2000
http://dx.doi.org/10.1590/S0100-73862000000100010
Turbulent Heat Transfer and Pressure Drop in Pinned Annular Regions
Angela Ourivio Nieckele
Pontifícia Universidade Católica do Rio de Janeiro - Departamento de Engenharia Mecânica - 22453-900 Rio de Janeiro, RJ Brazil
nieckele@mec.puc-rio.br
Francisco Eduardo Mourão Saboya
Universidade Federal Fluminense - Departamento de Engenharia Mecânica- 24210-240 Niterói, RJ Brazil
fsaboya@mec.uff.br
Experiments were performed to determine average heat transfer coefficients and friction factors for turbulent flow through annular ducts with pin fins. The measurements were carried out by means of a double-pipe heat exchanger. The total number of pins attached to the inner wall of the annular region was 560. The working fluids were air, flowing in the annular channel, and water through the inner circular tube. The average heat transfer coefficients of the pinned air-side were obtained from the experimental determination of the overall heat transfer coefficients of the heat exchanger and from the knowledge of the average heat transfer coefficients of the circular pipe (water-side), which could be found in the pertinent literature. To attain fully developed conditions, the heat exchanger was built with additional lengths before and after the test section. The inner circular duct of the heat exchanger and the pin fins were made of brass. Due to the high thermal conductivity of the brass, the small tube thickness and water temperature variation, the surface of the internal tube was practically isothermal. The external tube was made of an industrial plastic which was insulated from the environment by means of a glass wool batt. In this manner, the outer surface of the annular channel can be considered adiabatic. The results are presented in dimensionless forms, in terms of average Nusselt numbers and friction factors as functions of the flow Reynolds number, ranging from 13,000 to 80,000. The pin fin efficiency, which depends on the heat transfer coefficient, is also determined as a function of dimensionless parameters. A comparison of the present results with those for smooth sections (without pins) is also presented. The purpose of such a comparison is to study the influence of the presence of the pins on the pressure drop and heat transfer rate.
Keywords: annular regions, heat transfer coefficients, pressure drop, pin fins, turbulent flow.
Introduction
Finned walls are commonly employed in heat exchange devices to enhance the heat transfer rates by forced convection. An interesting example is that of annular regions with pin fins attached to inner wall of the annulus. The performance of the pinned surface is determined mainly by the knowledge of the pin efficiency, the average heat transfer coefficient of the pinned region and the friction factor. The first and second parameters permit the calculation of the heat transfer rate, while the third allows the determination of the pressure drop. The geometry of the flow passage is also an important factor.
A survey of the available pertinent literature reveals a great lack of information on turbulent heat transfer and pressure drop for annular channels with pinned internal wall. Carnavos (1979) reported experimental results for turbulent heat transfer and pressure drop in internally finned tubes with longitudinal fins. In this work, tubes with 21 different types of longitudinal fins were considered.
For finned annuli and turbulent flow, Patankar, Ivanovic and Sparrow (1979) have presented heat transfer and pressure drop coefficients. In this research, the authors have analyzed the performance of continuous longitudinal fins attached to the annulus inner wall. The thermal boundary conditions were constant heat flux at the inner surface and the outer boundary adiabatic. The fin efficiency was not considered, since it was taken equal to 100 percent.
Results that can be applied to annular ducts with rectangular segmented fins were reported by Braga and Saboya (1986). Plain sections were also studied. The results consisted of heat transfer coefficients, friction factors and fin efficiency.
With regard to turbulent heat transfer in finned annuli, one of the most extensive sets of results is that reported by Braga and Saboya (1988). The results consisted of average heat transfer coefficients and fin efficiency for annular regions with pins and segmented fins as well as for pinned triangular sections. Results for pressure drop are not reported.
Experimental data related to internally finned equilateral triangular ducts are available in Braga and Saboya (1996). Turbulent flow was considered in this research and the results are concerned with heat transfer and pressure drop. The influence of the fin efficiency was taken into account.
Taborek (1997) presented interesting calculation procedures for design of double-pipe heat exchangers with longitudinal finned tubes. Although some approximations are made, his method can be useful in engineering applications and is directly related to the present work.
The subject investigated in this research deals with the experimental determination of the average heat transfer coefficients and friction factors for turbulent flow through annular channels with pin fins attached to the inner wall of the annulus. The measurements were carried out by means of a double-pipe heat exchanger. The number of pins in a same cross-section (transversal row) was 8, while the number of transversal rows was 70, totaling 560 pins. The pin fin efficiency, which depends on the average heat transfer coefficient, is also determined as a function of dimensionless parameters.
The working fluids were air, flowing in the annular duct, and water through the internal circular tube. A schematic diagram of the double-pipe heat exchanger employed in the experiments is presented in Fig. 1.
The average heat transfer coefficients of the pinned air-side were obtained from the experimental determination of the overall heat transfer coefficients of the heat exchanger and from the knowledge of the average heat transfer coefficients of the circular pipe (water-side), which could be found in the pertinent literature.
Nomenclature
A = total frontal area of the air flow, area, m^{2} | Nu = average Nusselt number of air, average Nusselt number, dimensionless Greek Symbols DP = pressure drop, Pa | Subscripts a relative to air |
To attain fully developed conditions, the heat exchanger was built with additional lengths before and after the test section. The inner circular duct of the heat exchanger and the pins were made of brass. The inner surface of the annular channel was practically isothermal due to the material (brass) high thermal conductivity and small thickness of the tube (3.18 mm). Besides, the temperature variation of the water flowing in the inner tube was not high (less than 1.0 deg. C). The external tube was made of an industrial plastic which was insulated (see Fig. 1) from the environment by means of a glass wool batt. In this manner, the thermal boundary conditions of the present work are constant temperature over the surface of the inner tube, the external tube being adiabatic.
The results are presented in dimensionless forms, in terms of average Nusselt numbers and friction factors as functions of the flow Reynolds number. The range of the Reynolds number, for the air-side flow, was from 13,000 to 80,000. A comparison of the present results with those for smooth sections (without pins) is also presented. The purpose of such a comparison is to study the influence of the presence of the pins on the pressure drop and heat transfer rate.
Figure 1 contains dimensional nomenclature for the heat exchanger cross section designated by D, H, b, t, D_{1}, g, D_{2}, D_{3}. The actual values of the apparatus dimensions are: D = 5.556 mm; H = 19.05 mm; b = 11.113 mm; t = 3.175 mm; D_{1} = 31.75 mm; g = 45^{o}; D_{2} = 38.10 mm; D_{3} = 76.20 mm. The length of the pinned annular test section is L = 1,025 mm. The pin rows are displayed in staggered fashion forming isosceles triangles, as shown in Fig. 2, which is a plan view of the array of pins. The pin longitudinal and transversal pitches, designated by S_{L }and S_{T}, respectively, are defined in Fig. 2. The values of these geometric parameters are: S_{L} = 14.44 mm and S_{T} = 14.96 mm. The dimensionless geometrical parameters which govern the problem can be taken as: D_{2} / b = 3.43; D / b = 0.5; D_{3} / b = 6.857; L / b = 92.23 and S_{L} / S_{T} = 0.97.
From geometric considerations, the number of pins, N_{T}, in a same cross-section (transversal row) and the number of transversal rows , N_{L}, can be written, respectively, as:
As mentioned before, N_{T} = 8 and N_{L} = 70. For the experiments, 102 data runs were performed by varying the water and air mass flow rates.
Test Apparatus and Experimental Method
The description of the experimental apparatus is facilitated by reference to Fig. 3, which is a schematic side view of the test section arrangement. The experiments were performed in an open-loop air flow circuit. Upon traversing an upstream blower, the air passes successively through a plenum chamber, a calibrated flow meter (venturi), a valve to control the air flow, the heat exchanger and is finally discharged to the atmosphere. Two pressure taps, to measure the pressure drop across the venturi tube, are used (see Fig. 3).
The air mean temperatures at inlet and exit of the annulus were given by calibrated thermocouples, a precision voltmeter and a thermocouple switch. Three thermocouples were used at the inlet. The same procedure was used at the exit.
As shown in Fig. 3, hot water is pumped from a reservoir where the water temperature was kept constant by means of a thermostatic heater. The water mass flow was measured by a calibrated rotameter. Two pressure taps were installed, respectively, before and after the rotameter to make possible pressure measurements. The water circuit is designed in such way that the heat exchanger works in counter flow mode. The hot fluid (water) operates in a closed-loop circuit and its mass flow rate is controlled by the valve shown in Fig. 3 (before the heat exchanger).
The water average temperatures at inlet and exit of the internal circular pipe section were measured by three thermocouples at the inlet and three more at the exit. Since the flow is turbulent (flat velocity profiles in the cross sections of the ducts), the average temperatures determined at the entrance and exit of the test section, in both fluids (air and water), are practically equal to the bulk temperatures.
The uncertainty of the thermocouples used in the apparatus was 0.1 deg. C. Typically, the experimental uncertainty of the measurements of the air and water mass flow rates was 2%.
For the measurements of the air pressure drop through the pinned annular section, two pressure taps were installed, respectively, at the inlet and exit of the section. All the pressure drop determinations (across the venturi, the rotameter and annulus) were made with two precision manometers working in conjunction with a pressure switch. The experimental uncertainty associated to the pressure measurements was, typically, 2%. The ambient air temperature and relative humidity were also monitored by a precision thermometer and hygrometer, respectively. The atmospheric pressure was monitored by a barometer of column of mercury with a smallest scale division of 0.1 mm.
The experiments consisted of 10 sequences of runs. Each sequence was characterized by a pre-selected value of the air mass flow rate through the pinned annular duct. During each sequence, the water mass flow rate through the smooth inner tube assumed several values, totaling 102 runs distributed among the 10 sequences. During each run, all the relevant parameters (temperatures, pressures, mass flow rates and environmental conditions) were registered. A new sequence of data runs was initiated by fixing a new air mass flow rate. Once the measurements were performed, it is possible to determine the friction factors in the pinned annulus as well as the overall heat transfer coefficient of the double-pipe heat exchanger. With the knowledge of the overall heat transfer coefficient, the average heat transfer coefficient of the pinned annular region can be calculated. The method also requires the knowledge of the pin efficiency and of the average heat transfer coefficient of the water in the inner circular tube.
Although the average heat transfer coefficients of the air-side region does not depend on the water mass flow rate, the experimental procedure used afforded great confidence to the results, since it was possible to check the repeatability of the test apparatus. All measurements were made under steady-state conditions and the fluid properties were calculated at the mean value between the inlet and outlet bulk temperatures. From the determination of the mass flow rates in the venturi tube and in the rotameter, the air and water Reynolds numbers can be, respectively, obtained.
Data Reduction Procedure
The main purpose of this section is to determine the average heat transfer coefficient of the pinned annular region (air-side) as function of the air Reynolds number. This is accomplished by the computation of the overall heat transfer coefficient of the double-pipe heat exchanger used in the experiments. In addition, the friction factor of the pinned annulus, also a function of the air Reynolds number, was calculated from the pressure measurements.
The heat transfer rate of the exchanger, Q, is determined from:
where is the mass flow rate, c_{p} is the specific heat at constant pressure and T is the absolute bulk temperature of the working fluids. The subscripts e and i refer, respectively, to the exit and inlet of the heat exchanger test section, while the subscripts a and w refer to the fluids air and water, respectively. Preliminary runs, using low water flows to increase the the temperature changes, showed that the energy balance of the heat exchanger was satisfied to within 3%, attesting to the good quality of the thermal insulation. Nevertheless, the error in the heat transfer rate was minimized by the arithmetical mean expressed by Eq. (3).
The overall heat transfer coefficient of the heat exchanger, U, is given by:
where A_{t} is the total area of the heated region of the air duct, including the area of the pins and DT_{log} is the logarithmic mean temperature difference (LMTD) that can be written as:
In Eqs. (3) and (5), the mass flow rates and all temperatures were directly measured. Then, the heat transfer rate, Q, and the logarithmic mean temperature difference, DT_{log}, can be calculated. Next, by means of Eq. (4), the overall heat transfer coefficient of the heat exchanger, U, can be evaluated.
On the other hand, according to Kreith (1973), the overall heat transfer resistance of a heat exchanger is equal to the sum of the internal, wall and external resistances. Thus, if the total area of the pinned region, A_{t}, is selected as the reference area it is possible to write:
where h and h_{w} are the average heat transfer coefficients at the test section of the air and water sides, respectively, A_{1} is the internal area of the water circular tube (equal to p D_{1} L), R_{k} is the wall heat transfer resistance, h is the efficiency of the pinned region and represents the ratio between the actual and the ideal (if the pins were isothermal at the base temperature, T_{p}) heat transfer rates.
According to Mills (1992), the region efficiency, h, and the wall heat transfer resistance, R_{k}, are given by the following equations:
In Eqs. (7) and (8), h_{p} is the pin efficiency, A_{p} is the heat transfer area of the primary surface and k is the thermal conductivity of the internal tube material (brass, same pin material). Denoting the total pin heat transfer area by A_{f}, it is clear that A_{t} = A_{p} + A_{f}. The pin efficiency was obtained from Kern and Kraus (1972) and can be written as:
where
It is seen that the efficiency of the pinned region, h, is a function of the pin geometry, pin material (k), and average heat transfer coefficient of the air-side, h.
The average heat transfer coefficient of the water-side, h_{w} in Eq. (6), was determined from the notorious equation of Dittus and Boelter (1985), for turbulent flows inside pipes. For the present situation (water flowing as hot fluid through the inner pipe), this equation is written as:
where Nu_{w} is the average Nusselt number of the water, D_{1} is the internal diameter of the inner circular tube of the heat exchanger (see Fig. 1), k_{w} is the water thermal conductivity, Re_{w} and Pr_{w} are the Reynolds number of the water flow and the water Prandtl number, respectively. These dimensionless parameters are given by:
In Eqs. (12) and (13), A_{w}= (p )/4 is the frontal area of the water flow (cross section of the internal pipe), m_{w} is the water dynamic viscosity, is the water mass flow rate and c_{pw} is the specific heat at constant pressure of the water.
The last two terms on the right-hand side of Eq. (6) are very small when compared with the first one. This fact shows that the air-side heat transfer resistance (air-side thermal resistance) is the most relevant factor in the composition of the overall heat transfer coefficient, U, of the heat exchanger. In this manner, the experimental uncertainties associated with the average heat transfer coefficient of the water, h_{w}, and with the thermal conductivity, k, of the internal tube material will have a very small influence on the computation of the average heat transfer coefficient of the air, h. This is one of the main features of the present experimental method.
As mentioned before, the experiments consisted of 10 sequences of runs. Each sequence possessed a fixed air mass flow rate. During a sequence, the water mass flow rate was varied. For any particular sequence, Eq. (6) shows that (1 / U) is a linear function of (1 / h_{w}) with a known constant slope equal to (A_{t} / A_{1}). As the water mass flow rate, , varies, (1 / h_{w}) also changes. It should be noted that, when the air mass flow rate, , is fixed, (h h) is also constant. Naturally, the geometric dimensions and R_{k} are known constants. Since U and h_{w} are given by Eqs.(4) and (11), for all data runs, and the slope (A_{t} / A_{1}) is known, one can determine easily, by the method of least squares, the curve-fitting equations of the parallel straight lines that represent the dependence of (1 / U ) with (1 / h_{w}). An important property of any straight line fit, by the method of least squares, is that the curve passes through the point characterized by the mean values of the ordinates and abscissas. Then, Eq. (6) gives:
where n is the number of data runs of the sequence of runs under consideration. One should recognize from Eq. (6) (making (1 / h_{w}) = 0) that [ 1 / (h h) ] + R_{k} is the ordinate intercept of the straight line representative of the sequence of runs.
Equations (7), (9) and (14) constitute a transcendental system of equations that was solved by an iterative method to yield the average heat transfer coefficient of the air-side, h, which is one of the objectives of the present research. It should be observed that h_{p}, given by Eq. (9), is a function of h through c defined by Eq. (10).
In Eq. (14), the average heat transfer coefficient of the air, h, refers to a pinned annular duct if the pin fins were isothermal (h = 1) at the base temperature, T_{p}. The fact that the pins are not isothermal is taken into account by the region efficiency, h. In this way, h does not depend on the pin material, whereas the region efficiency, h, does. Then, it is clear that the results should be presented separately for h and h, and not for the product (h h) as one might think.
The dimensionless representation of h is the average Nusselt number, Nu, of the air pinned region, expressed by the following equation:
where k_{a} is the air thermal conductivity and D_{h} is the hydraulic diameter based on the smooth annular duct (see Fig. 1) given by:
It is clear that Nu is a function of the Reynolds number of the air flow expressed by:
where m is the air dynamic viscosity and A is the total frontal area of the air flow (cross section of the annulus) that can be written as:
To determine the friction factor, f, for the pinned annular test section, as function of the annulus Reynolds number given by Eq. (17), the following expression was employed:
where DP_{t} is the total pressure drop (in the pinned annulus) and r is the air density, which is calculated as a mean value between the values of the inlet and exit of the test section. Denoting by DP_{p} the pressure drop due only to the presence of the pins, it is possible to define a loss coefficient per transversal row of pins, K, as:
The friction factor, f_{s}, for the smooth annulus (without pins) can be written as:
where DP_{s} is the pressure drop. For the smooth annulus, with the same dimensions of those of the present work, Braga and Saboya (1986) reported the following correlation for f_{s}:
where Re_{s} is the Reynolds number of the flow in the smooth annular section with the same definition of Eq. (17).
Since DP_{t} = DP_{p} + DP_{s} it is easy to show that:
Once f is determined by Eq. (19), Eq. (23) yields the loss coefficient, K.
Results and Discussion
In order to perform the measurements, the desired air and water mass flow rates were set and the thermostatic heater was turned on. The equipment was allowed to operate until the steady-state condition was reached, which was verified by monitoring the fluid temperatures by means of the previously mentioned thermocouples working in conjunction with a thermocouple switch and a precision voltmeter. The total numbers of experimental data runs was 102.
Figure 4 presents the overall thermal resistance, (1 / U), as function of (1 / h_{w}). Each sequence of runs is represented by a straight line fit having the Reynolds number of the air flow, Re, as parameter. Figure 4 also shows the experimental points for the pinned annulus. It can be observed that there are nine sequences of runs (9 straight lines) in Fig. 4. The experiments consisted of ten sequences. One sequence, for Re = 77,072, was not included in Fig. 4 because it is almost coincident with the sequence of runs for Re = 78,285.
The straight line fits were obtained by the method of least squares with the constraint that the slope (A_{t} /A_{1}) is known and equal to 2.261, for all sequences. Thus, all the straight lines are parallel, as shown in Fig. 4. The mean deviation of the curves in Fig. 4 in relation to the experimental points was only 1.6%, which attests to the good quality of the data. As mentioned before, [ 1 / (h h) ] + R_{k} is the ordinate intercept of the straight lines in Fig. 4 and this fact could be used to obtain 1 / (h h). However, for convenience, Eq. (14) was employed.
The results for the average Nusselt number of the pinned annulus, as function of the air Reynolds number, can be expressed as:
Equation (24) represents a curve-fitting of the experimental data, by the method of least squares, with a mean deviation of 2.2% in relation to the data points. Although air was the fluid used in the experiments with Prandtl number (Pr = m c_{pa} / k_{a}) equal to 0.7, it is possible, as suggested by Patankar, Ivanovic and Sparrow (1979), to generalize the results for other fluids by introducing into Eq. (24) the factor Pr^{0.4}. It is obtained:
Equation (25) is not restricted to air. In Eq. (25), Pr_{g}, Re_{g}, and Nu_{g} are, respectively, the Prandtl number, the Reynolds number and the average Nusselt number of any fluid flowing through the pinned annular duct. The definitions of these dimensionless parameters are the same given by Eqs. (13), (17) and (15), respectively.
Information related to average Nusselt number, Nu_{s}, for air flow in smooth annuli with the same geometric dimensions of those of present research, can be found in Braga (1987). The following correlation was reported:
In Eq. (26), Nu_{s} has the same definition of Eq. (15).
Figure 5 presents a comparison of the average Nusselt numbers of air, Nu and Nu_{s}, given by Eqs. (24) and (26), respectively. The comparison is made by taking Re = Re_{s}. Figure 5 also contains experimental data for the pinned annular duct. It should be observed that the Reynolds numbers of the pinned and smooth sections have the same definition given by Eq. (17). It is seen, from observation of Fig. 5, that, for the same Reynolds numbers, the Nusselt numbers for the pinned annulus are approximately 200% higher than those for the smooth annular duct. However, it should be remarked that for the same length L, the installation of pins considerably increases the pumping power.
The friction factor, f, of the pinned annular section, as defined by Eq. (19), can be expressed by the following expression:
Equation (27) represents a curve-fitting of the experimental points, by the method of least squares, with an average dispersion of less than one percent in relation to the data points.
Figure 6 compares Eqs. (22) and (27), for the same Reynolds numbers, Re_{s} and Re. It should be observed the low degree of dispersion presented by the experimental data and Eq. (27). As expected, the friction factors for the plain annulus are much smaller than the corresponding values for the case of the annulus with pins.
The loss coefficient, K = K (Re), is now calculated by means of Eqs. (22) and (23), and the experimental data for the friction factor, f, of the pinned annulus. The following curve-fitting equation was obtained by the method of least squares:
In Fig. 7, the solid line passing through the data points represents Eq. (28) with a mean deviation from the experimental points of less than 2%. As expected, the loss coefficient, K, reduces as the Reynolds number increases.
A fair comparison between pinned and smooth annular ducts can be made by imposing the same heat transfer areas and pumping powers for both geometries. The condition of same heat transfer areas yields:
In Eq. (29) L_{s} and L are the lengths of the smooth and pinned annulus, respectively. By using Eqs. (22) and (27), the condition of same pumping powers yields the following relation for the Reynolds numbers, Re and Re_{s}, of the two situations under consideration:
Finally, the ratio (Nu / Nu_{s}) can be determined from Eqs. (24), (26) and (30) yielding:
Equations (30) and (31) are valid only when the pinned and smooth annular ducts have the same heat transfer areas and pumping powers.
In Fig. 8, the ratio Nu / Nu_{s} is plotted as function of Re. As shown therein, the ratio almost reaches 2.25 for Re = 10,000 and then decreases to approximately 1.25 for Re = 80,000. As far as compactness and heat transfer enhancement are concerned, this is an important outcome since Eq. (29) shows that L_{s} - L is about 90% of L (L is shorter than L_{s}).
Equation (30) shows that the imposed conditions for comparison require Re_{s} be higher than Re. Nevertheless, Fig. 8 indicates that Nu is considerably higher than Nu_{s}. The increase in the heat transfer coefficient with the presence of pins is most probably due to the generation of horse-shoe vortices around the pins. Saboya and Sparrow (1974) have shown the presence of these vortices around the tubes in the flow of air in plate fin and tube heat exchangers.
Uncertainty Analysis
An uncertainty analysis for the experiments was performed using the well-known Kline and McClintock (1953) methodology. This methodology of describing uncertainties in experimental works is also reported by Moffat (1988). Such a method consists of the evaluation of the responses of a given parameter to changes in each of the variables used in the data reduction procedure for the computation of the parameter under consideration.
Typically, the experimental uncertainty associated to the air Reynolds number was 2.5% with maximum value of less than 4%. For the air Reynolds number the most relevant variable is the air mass flow rate as shown by Eq. (17).
For the determination of the experimental uncertainty of the average Nusselt number, Nu, of the air pinned region, Eqs. (14), (15) and (7) were employed. In this case the most important variable is (1 / U)_{m}, which depends on Q and DT_{log}. Typically the uncertainty on Nu was 7% with a maximum value of less than 9%. As previously mentioned, the variables (1 / h_{w})_{m} and R_{k} have a minor influence.
For the friction factor, f, and loss coefficient, K, the uncertainty analysis gave, typically, a value of 3% with a maximum of less than 4%. In this case the most relevant variables are the pressure drop and air the mass flow rate, as shown by Eqs. (19) and (23).
It has been demonstrated that the experimental technique used in the present investigation is a powerful tool for obtaining average transport coefficients. All the correlations presented in this work have a very low degree of uncertainty. Also, it should be mentioned the small scattering of the experimental data in Figs. 4 – 7.
Concluding Remarks
In practical applications, one is interested in the determination of the actual rate of heat transfer, Q, exchanged by the pinned annular region. This is given by the expression:
Inspection of Eq. (32) shows that the knowledge of h and h is fundamental to obtain Q. The present work gives h and Nu (consequently h) for the proposed pinned annulus geometry. If T_{p}, T_{ia} and the mass flow rate of air, , are known, and the exit bulk temperature, T_{ea}, is specified, Q can be calculated from Q = c_{pa} (T_{ea} - T_{ia}). Then, by Eq. (32), the total heat transfer area, A_{t}, (or the length, L, of the annular section) can be calculated.
Another important practical application is the determination of the total pressure drop, DP_{t}, through the annular pinned region. It can be obtained from Eq. (19) as:
In Eq. (33), the friction factor, f, is given as one of the results of the present research.
Equations (9) and (10) show that the pin efficiency, h_{p}, depends only on the flow Reynolds number (through h) when the geometry, fluid properties and pin material are kept constant. It can be shown that the pin efficiency decreases as the flow Reynolds number increases and the thermal conductivity of the pin material decreases. The Nusselt number expressed by Eq. (24), for the geometry under study, is an exclusive function of the Reynolds number of the air flow. It does not depend on the thermal conductivity of the pin material since it is associated to an ideal isothermal pin (h_{p} = h = 1). In many applications, the pin efficiency is less than one and Eqs. (7) and (9) must be used in order to determine the actual heat transfer rate Q.
It should be observed that the region efficiency, h, is a known function of the Reynolds number of the air flow and thermal conductivity of pin material. In this way, the results of the present experimental research are not restricted to brass, which was the material used in the construction of the test section. In addition, if the working fluid is not air, one should use Eq. (25) to calculate the average heat transfer coefficient.
Since local heat transfer coefficients over the pins are not available, the pin efficiency was determined by employing the average heat transfer coefficient that is, by definition, constant. Such an approximation is a common practice in heat exchanger design and the error introduced in the pin efficiency is small as reported by Rocha, Saboya and Vargas (1997). Such an error decreases as the pin efficiency approaches unity. In the present experiments, the pin material was brass yielding high efficiency values (typically 96%).
It is important to note that the results presented in this work are applicable to configurations having dimensionless geometrical parameters equal to those of the experiments. These parameters are (D_{2} / b), (D / b), (D_{3} / b), (L / b) and (S_{L} / S_{T}). Since one can consider the flow as being fully developed, the requirement of same (L / b) may be ignored. In this case, the present average Nusselt number results are to be used for sufficiently long sections of the finned annulus. For short sections the results should not be used. Although the results of the present investigation are not applicable to non-similar configurations, (different dimensionless geometrical parameters) they may be used to check the validity of analytical models. Finally, the results of the present experimental research (friction factors, Nusselt numbers and fin efficiency) are directly applicable to heat exchanger design and should be of great value for the engineer and researcher.
Acknowledgments
The authors gratefully acknowledge the Brazilian agency, National Council of Scientific and Technological Development-CNPq, for the financial support provided during the course of the present research through grant 302130 / 84-5(RN).
References
Braga, C. V. M., 1987, "Theoretical and Experimental Analysis of the Thermal and Hydraulic Performances of Smooth and Finned Annular Regions" (in Portuguese), Ph.D. Thesis, Department of Mech. Engineering, PUC-Rio, Rio de Janeiro, Brazil. [ Links ]
Braga, C. V. M. and Saboya, F. E. M., 1986, "Turbulent Heat Transfer and Pressure Drop in Smooth and Finned Annular Ducts", Eighth International Heat Transfer Conference, San Francisco, California, Vol. 6, pp. 2831-2836. [ Links ]
Braga, C. V. M. and Saboya, F. E. M., 1988, "Analysis of Thermal Performance of Pinned and Finned Walls Subjected to Convection" (in Portuguese), Second Brazilian Meeting on Thermal Sciences, Aguas de Lindoia, São Paulo, Brazil, pp. 229-232. [ Links ]
Braga, S. L. and Saboya, F. E. M., 1996, "Turbulent Heat Transfer and Pressure Drop in na Internally Finned Equilateral Triangular Duct", Experimental Thermal and Fluid Science, Vol. 12, N^{o} 1, pp. 57-64. [ Links ]
Carnavos, T. C., 1979, "Coding Air in Tubular Flow with Internally Finned Tubes", Heat Transfer Engineering, Vol. 1, N^{o} 2, pp. 41-46. [ Links ]
Dittus, F. W. and Boelter, L. M. K., 1985, "Heat Transfer in Automobile Radiators of the Tubular Type", Int. Comm. Heat Transfer, Vol. 12, pp. 3-22. [ Links ]
Kern, D. Q. and Kraus, A. D., 1972, "Extended Surface Heat Transfer", McGraw-Hill, New York. [ Links ]
Kreith, F., 1973, "Principles of Heat Transfer", Intext Educational Publishers, New York. [ Links ]
Kline, S. J. and McClintock, F. A., 1953, "Describing Uncertainties in Single Sample Experiments", Mech. Eng., pp. 3-8. [ Links ]
Mills, A. F., 1992, "Heat Transfer", Richard D. Irwin, Inc., Homewood, IL. [ Links ]
Moffat, R. J., 1988, "Describing the Uncertainties in Experimental Results", Experimental Thermal and Fluid Science, Vol. 1, pp. 3-17. [ Links ]
Patankar, S. V., Ivanovic, M. and Sparrow, E. M., 1979, "Analysis of Turbulent Flow and Heat Transfer in Internally Finned Tubes and Annuli", J. Heat Transfer, Vol. 101, pp. 29-37. [ Links ]
Rocha, L. A. O., Saboya, F. E. M. and Vargas, J. V. C., 1997, "A Comparative Study of Elliptical and Circular Sections in One- and Two-Row Tubes and Plate Fin Heat Exchangers", Int. J. Heat and Fluid Flow, Vol. 18, N^{o} 2, pp. 247-252. [ Links ]
Saboya, F. E. M. and Sparrow, E. M., 1974, "Local and Average Transfer Coefficients for One-Row Plate Fin and Tube Heat Exchanger Configurations", Journal of Heat Transfer, Vol. 96, pp. 265-272. [ Links ]
Taborek, J., 1997, "Double-Pipe and Multitube Heat Exchangers with Plain and Longitudinal Finned Tubes", Heat Transfer Engineering, Vol. 18, N^{o} 2, pp., 34 – 45. [ Links ]
Manuscript received: October 1999. Editor: Atila P. Silva Freire.