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Comparative CFD case study on forced convection: Analysis of constant vs variable air properties in channel flow

  • Ahmed Hikmet Jassim , Salah M. Salih EMAIL logo and Kadhum Hassan Ali
Published/Copyright: March 20, 2025
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Open Engineering
From the journal Open Engineering Volume 15 Issue 1

Abstract

In this study, a steady-state forced convection heat transfer (HT) of air flow in a two-dimensional channel with a circular cross-section is numerically investigated. The analysis considers two heat sources at uniform temperatures along the lower surface of the mini-channel, with the upper surface remaining adiabatic to facilitate energy exchange. The heat sources are placed at distances L1 = 3.5 m and L2 = 1.5 m on the bottom surface. The finite element method is used to solve momentum-energy equations using Computational fluid dynamics (CFD) software, under constant and variable air properties. HT rates are computed for Reynolds numbers (Re ≤ 2,000) and Prandtl number (Pr = 0.713). The study evaluates the effects of Reynolds number, air thermo-physical properties, and thermal boundary conditions on hydrodynamic and thermal behavior. Results show that changes in the Nusselt number are significantly influenced by Re number, heat source configuration, and air properties. HT rate increases with Reynolds number, highlighting notable differences in centerline temperature, velocity, and conductive heat flux along the lower wall with variable air properties, with a maximum HT rate difference of 14% at T in = 20°C. Pressure also decreases with increasing Re number that shows good agreement between CFD results and empirical Shah equation.

Keywords: forced convection; flow inside mini-channel; air properties; channel flow; CFD

Nomenclature

C p

specific heat capacity (J/kg K)

D

diameter of the pipe (m)

h

heat transfer coefficient (W/m² K)

k

thermal conductivity (W/m K)

L

length of the pipe (m)

P

pressure (Pa)

Pr

Prandtl number

q=

heat flux by conduction (W/m2)

Re

Reynolds number

r

radial coordinate (m)

T

temperature (K)

uc

centerline velocity (m/s)

V

axial velocity (m/s)

V → =

velocity vector

X

axial coordinate (m)

Greek symbols

μ

viscosity (Pa s)

ρ

density (kg/m3)

Subscripts

avg

average

b

bulk

f

fluid

in

inlet

w

wall

x

local

1 Introduction

In the extensive field of fluid dynamics and thermal management, mini-pipes play a crucial role in both the transportation and thermal regulation of working fluids [1,2]. Heat transfer (HT) dynamics within mini-circular ducts are crucial to design efficient compact heat exchangers particularly in the entrance region. Typically, these systems feature short passageways with diverse cross-sectional shapes including triangular, rectangular, and non-circular configurations. These geometries accommodate a broad spectrum of fluids from gases to viscous liquids like automotive oils, which are used in various industrial and automotive cooling or heating processes. Historical models for predicting HT in these areas include the works of both circular ducts and parallel plate channels for the purpose of convective HT [3,4,5,6,7]. Moreover, the HT parameters were heavily studied to understand the performance of many different thermal engineering applications of interest [8,9,10,11,12,13,14].

Generally speaking, the main contributions of this study involve the numerical modeling advancements of thermal engineering applications using the finite element method (FEM) in a comprehensive Computational fluid dynamics (CFD) based analysis of forced convection HT concerning the effect of variable air properties in a mini-channel. This research gives sensible ideas that might further improve the design of a heat exchanger and optimization of a thermal system by studying the influence of different placements of the heat source as well as that of the Reynolds number on the HT rates and fluid dynamics. Variable vs constant air properties show that there are critical differences in the distribution of temperature, velocity, and conductive heat flux, all critical for operational accuracy and performance improvement engineering applications. Besides, the study compared its findings with the empirical Shah equation since it did validate the approach for modeling and brought about a refinement in understanding, hence giving value to the existing literature in thermal behaviors in practical engineering context. This paper is crucial for developing more effective thermal management strategies and advancing the overall knowledge of mechanical-thermal engineering.

2 Literature review

The recent surge in CFD research has significantly advanced the understanding and optimization of forced convection processes in various engineering applications. The meticulous analysis conducted by Hossain et al. [15] demonstrated the intricate behavior of turbulent forced convection over a spinning cylinder within an enclosed channel, emphasizing the complex interaction between airflow patterns and HT efficiencies. This study aligns with the broader efforts to refine simulation accuracy, as seen in the study by Nasution et al. [16], who integrated a genetic algorithm with fuzzy inference systems to enhance the predictive capabilities of CFD models in water convection within copper metal foam tubes.

In parallel, a recent investigation into steam condensation under natural convection conditions inside tubular bundles offered insights into the dual-phase HT mechanisms, which are critical to optimize industrial heat exchangers and enhance energy efficiency [17]. Kim and Cho [18] further contributed to this field by assessing various turbulence models to understand the HT phenomena in rectangular channels under forced and mixed convection scenarios. Their findings concluded the necessity to select appropriate turbulence models to capture the nuances of convective HT accurately. Moreover, the experimental and simulation work by Xu et al. [19] on convective HT in the crossflow around circular cylinders provided valuable data on condensation effects, relevant to designing more efficient thermal management systems in both industrial and environmental applications. Moreover, Ahmed et al. [20] explored the forced convection of non-Newtonian nanofluids in a sinusoidal wavy channel by the employment of a methodology that follows the perspective of response surface to dissect the interdependencies of flow parameters on transferred heat rates for applications that require enhanced HT with geometric modifications.

CFD-based investigation for space-optimization of cooling systems around lithium-ion battery packs has been performed to simulate airflow [21]. The method reflected the increased demand due to thermal control requirements put forth by the electronic and automotive industries. The study was able to show that airflow management has potential great power in the consideration of cooling efficiency, hence the battery longevity and performance effect. Similarly, Salim et al. [22] presented a study on the thermal dynamics of aluminum foam heat sinks. They used a transient three-dimensional CFD model to describe the transient behavior of aluminum foam heat sinks in a horizontal channel. The work provided a framework for enhancing heat dissipation in the compact spaces typical of modern electronic devices. Finally, energy analysis of freezer cabinets using phase change materials for convective HT was an example where CFD tools integrate in the food storage industry as a way of improving energy consumption patterns during several operational phases [23]. Taken together, these studies give a solid base for new and on-going research in the field of forced convective systems optimizers across the most diverse applications, as well as future research, by using advanced CFD techniques to answer the theoretical and practical challenges in HT technology.

In engineering analysis, the FEM has gained significant traction due to its robustness in tackling complex structural and fluid dynamics challenges. For instance, Dąbrowska et al. explored the validation of simulation models for 316L steel which is fabricated using additive techniques [24]. Similarly, Al-Haddad and Mahdi applied FEM in conjunction with data-driven approaches to model aircraft undercarriage landing gear [25]. Further, the integration of FEM with neural networks has been showcased by Al-Haddad et al. for predicting thermal heat flux distribution in electric vehicle battery cells [26]. Al-Haddad et al. utilized FEM in demonstrating UAV fault diagnosis approaches, progressively [27,28]. Additional literature that broadens the context of FEM’s application includes works by Fattah et al. on the analysis of interfering strip footings [29], and by Al-Haddad et al., who investigated sustainable building designs through aerodynamic shading devices using FEM [30]. Notable works by Al-Haddad et al. on the application of FEM in structural analyses of steel beam-column connections [31], and Decker et al.’s investigation of vehicle chassis axle fractures also enrich the discussion [32]. The extensive application of FEM across various domains is further exemplified by research on greywater flow characteristics [33], interdisciplinary studies using Euler methodology alongside CFD technologies [34], and development of hydropneumatic benches for pipeline valve testing [35].

In the present study, a two-dimensional numerical investigation on laminar forced convection of air inside mini-channel with constant and variable properties of air is presented. In this study, the mini-channel is subjected to two different heat sources with a uniform temperature for two cases, while the remaining upper surface of mini-channel is kept adiabatic. The main aim of this study is to determine how the heat sources configuration and varying properties (VP) of air affect the HT rate and pressure drop of flow in a mini-channel.

3 Model formulation

Figure 1 illustrates the geometry of problem under consideration, which contains copper pipe with length of 5.0 m and circular cross-section with diameter of (0.06)m. The flow is a steady state incompressible flow with laminar forced convection of air flowing inside a mini-pipe. The mini-pipe has two isothermal heat sources of 100 and 200°C. The two heat sources are placed on different lengths, taken as L 1 = 3.5 m, and L 2 = 1.5 m on the bottom surface of the mini-pipe. The remaining upper surface is isolated. The flow enters the mini-pipe with a uniform cold temperature and a uniform axial velocity.

Figure 1 Schematic of geometry problem and physical model.
Figure 1

Schematic of geometry problem and physical model.

The governing equations for steady-state incompressible laminar flow are represented as follows:

Continuity equation:

(1) ∇ ⋅ ρ f V → = 0 ,

Momentum equation:

(2) ∇ ⋅ ( ρ f V → V → ) = − ∇ P + ∇ ⋅ ( μ f ∇ 2 V → ) .

Energy equation:

(3) ∇ ⋅ ( ρ f V → C p f T ) = ∇ ⋅ ( K f ∇ T ) .

4 Boundary conditions

The governing equations for fluid flow are nonlinear and coupled partial differential equations as they necessitate resolution through the application of suitable boundary conditions. The flow boundary conditions at the mini-pipe inlet, are a uniform axial velocity V in, and temperature T in. On the walls, no slip boundary conditions are applied for momentum equations. The remaining upper surface is adiabatic condition, for the two cases:

Case 1: In this case, the mini-pipe has two isothermal heat sources placed on the different lengths, taken as L1 = 3.5 m and L2 = 1.5 m on the bottom surface of the mini-pipe and maintained at Ts1 = 100°C, and Ts2 = 200°C, respectively.

Case 2: In this case, the mini-pipe has two isothermal heat sources placed on the different length taken as (L1 = 3.5 m and L2 = 1.5 m) on the bottom surface of the mini-pipe and maintained at Ts1 = 200°C and Ts2 = 100°C, respectively.

The temperature fields in addition to the flow at the outlet section of the air-pipe were both assumed to be a fully-developed flow. The fields within the pipe itself were also assumed to be fully-developed with a value of L/D ≈ 100 based on the study by Jahanbin and Javaherdeh [36]. The local Nusselt number along the hot lower wall is defined as

(4) Nu x = h x D k f ,

where (D) is the diameter of pipe and (h x) is defined as:

(5) h x = q x T w − T b ,

The average of HT coefficient is expressed as

(6) h avg = 1 L ∫ 0 L h x d x ,

and the average Nusselt number becomes

(7) Nu avg = h avg D k f .

The local HT coefficient was exported from the COMSOL software and imported in MATLAB program to determine the average HT coefficient by using Simpson’s rule of integration to evaluate Equation (6) and then calculate average Nusselt number.

5 Air thermo-physical properties

In this section, the material properties of air were defined (C p, ρ, μ, k, and Pr). Some of the properties strongly depend on temperature while others do not. Table 1 shows the percent difference in those properties based on these extremes, with a rather large temperature range of 20–250°C. This is done by examining Appendix (C) – Properties of dry air at atmospheric pressure, Coulson et al. [37].

Table 1

Different percentage of the material properties of dry air at atmospheric pressure properties, Ref. [37]

C p P μ × 10−6 K Pr
T ∞ (20°C) 1006.1 1.2042 18.17 0.02564 0.713
T ∞ (250°C) 1034.4 0.6748 27.64 0.04095 0.698
Difference% based on T ∞ (20°C) 2.813 44 52.12 59.7114 2.104

Based on these calculations, it is clear that for air in the given temperature range, ρ, μ, and k strongly depend on temperature while the others depend weakly on the temperature. Therefore, C p and Pr will be set as constants while ρ, μ, and k will be modeled as VP.

The following equations are utilized for the purpose of calculating the properties of air as they vary with the surrounding temperature [37]:

(8) ρ = P o M w R T , [ kg / m 3 ] ,

(9) k = 10 − 3.723 + 0.865 log 10 ( T ) , [ W / m K ] ,

(10) μ = 6 × 10 − 6 + 4 × 10 − 8 T , [ Pa ⋅ s ] ,

where P o represents the pressure in the atmosphere with a value of 101.3 kPa, M w is the air’s molecular weight with a value of 0.0288 kg/mol, and the universal constant of gas is said to be R = 8.314 J/mol K.

6 Numerical solution

A CFD software suite is utilized to address this particular case study. This software offers a robust and dynamic platform designed for the purposes of modeling and resolving a broad range of engineering and scientific challenges that are predicated on partial differential equations. The CFD software leverages the FEM to rigorously solve the model. It executes the finite element analysis by means of incorporating adaptive mesh refinement and error control through the use of various advanced numerical solvers to enhance the precision and efficiency of simulations [38,39].

Three different amount of mesh elements 1.616 × 103, 6.464 × 103, and 2.5856 × 104 of triangular-type are used to ensure the grid independency of numerical solution. Table 2 shows the mean Nusselt number (Num) and maximum centerline velocity (uc) for different numbers of element at Re = 1,000 based on which it is found that the mesh size of around 6.464 × 103 gives approximately 4% deviation of Num compared to the mesh size of 1.616 × 103; whereas, the results from 6.464 × 103 mesh elements deviate up to 6% as compared to those from the finest one 2.5856 × 104. Therefore, a mesh of around 6.464 × 103 elements was sufficient for the numerical investigation purposes.

Table 2

Grid independency of mesh elements in the present analysis at Re = 1,000

Mesh element Num u c(max)
1.616 × 103 3.17352 0.541013
6.464 × 103 3.2996 0.551906
2.5856 × 104 3.5171 0.557589

7 Results and discussion

7.1 Validation of the results

The COMSOL 3.5a software is validated by means of the comparison between the progressed results with the data already available from previously published literature. The laminar flow classical problematic scheme is discussed and compared for the air-pipe conjunction.

For the purpose of acquiring this result, first a laminar flow and HT of air through a pipe with 0.06 m in diameter and 1 m in length at Re = 1,000 are developed as a case study. It is assumed that air flow enters the pipe with uniform velocity and temperature. The thermal-related boundary condition for the wall is kept on a constant temperature value to conduct the analysis required of this study.

Additionally, the Nusselt number which is localized, was obtained from the adopted program on the computer and compared with those acquired from Equations (8) and (9) of the localized Nusselt number directly calculated for circular pipes under the same boundary conditions of constant wall temperature, as illustrated in Figure 2.

(11) Nu x = 1.077 X ⁎ − 1 / 3 − 0.70 , X ⁎ ≤ 0.01 3.657 + 6.874 ( 10 3 X ⁎ ) − 0.488 e − 57.2 X ⁎ , X ⁎ ≥ 0.01 ,

where Nu x = h x D / k f , X ⁎ = [ ( x / D ) / ( Re Pr ) ] .

Figure 2 Comparison of the numerical local Nusselt number with the empirical Shah equation for air under constant wall temperature at Re = 1,000.
Figure 2

Comparison of the numerical local Nusselt number with the empirical Shah equation for air under constant wall temperature at Re = 1,000.

Again, a good agreement is apparently displayed between the results obtained by COMSOL 3.5a solution in present work and that of Shah equation [40,41]. Thus, these results confirm the validity of the computational scheme used in the present investigation.

The conductive heat fluxes along the lower wall for each case are plotted together in Figure 3, at different values of Re (100, 500, and 1,000) for constant properties of air. As illustrated in the figure, the conductive heat flux is augmented with an increase in Reynolds number. Consequently, the HT intensifies and allows the temperatures within the fluid domain to more rapidly approach the wall temperature. A crossover point at 3.5 m results from the diminished discrepancy between the bulk temperature and wall temperature.

Figure 3 Variation in conductive heat flux variation along the pipe length for CP of air: Case 1 and Case 2.
Figure 3

Variation in conductive heat flux variation along the pipe length for CP of air: Case 1 and Case 2.

Figure 4 shows the conductive heat fluxes normalized to the baseline case 1 for CP and VP of air at different values of Re. The conductive heat flux increase is maximum at VP of air at high Re as compared to the CP of air.

Figure 4 Comparison of conductive heat flux variation along the pipe length for CP and VP of air, Case 1.
Figure 4

Comparison of conductive heat flux variation along the pipe length for CP and VP of air, Case 1.

The velocity profile at the centerline of the mini-pipe along pipe lower wall are illustrated in Figure 5, for various values of Re. One can clearly see that the value of centerline velocity is increasing, as Re increases due to the enhancement of hydrodynamic flow inside mini-pipe.

Figure 5 Centerline velocity variation along the pipe length for CPs of air, Case 1.
Figure 5

Centerline velocity variation along the pipe length for CPs of air, Case 1.

Figure 6 shows how varying air properties affect the centerline velocity along lower wall of pipe as compared to the constant properties of air.

Figure 6 Comparison of centerline velocity variation along the pipe length, at (Re = 500) for Case 1.
Figure 6

Comparison of centerline velocity variation along the pipe length, at (Re = 500) for Case 1.

The bulk temperature through the pipe for each case are plotted together in Figure 7, at different Re (500, 1,000, and 1,500) for constant properties of air. As shown in the figure, the bulk temperature is decreased with increased Re number, because the exchange energy is reduced between air layer and heated wall. The temperatures in the fluid domain more quickly reach the wall temperature. Beyond point3.5 m, the bulk temperature is raised due to heat sources configuration on the lower wall of pipe.

Figure 7 Variation in bulk temperature along the pipe length for CP of air: Case 1 and Case 2.
Figure 7

Variation in bulk temperature along the pipe length for CP of air: Case 1 and Case 2.

In Figure 8, the effects of thermo-physical properties of air on the local HT of air at Re = 1,500 are presented. It is seen that Nux decreases and then due to heat sources configuration on the lower wall of pipe, at point 3.5 m, the (Nux) is raised due to higher isothermal temperature.

Figure 8 Comparison of local Nusselt number along lower wall of pipe at Re = 1,500, Case 1.
Figure 8

Comparison of local Nusselt number along lower wall of pipe at Re = 1,500, Case 1.

The effects of thermo-physical properties of air and Reynolds number on the average HT of air are presented in Figure 9. The influence of the thermo-physical properties of air for VP on HT is increasing when compared to the CP of air. The overall HT throughout the domain is greater, because of the increase in local HT coefficient. Also, the decrease in thermal conductivity of air for VP is temperature dependent. Also, the max. difference of HT rate for VPs of air based on (T in = 20°C) is 14%.

Figure 9 Variation in average Nusselt number within thermally developing region at different Reynolds number.
Figure 9

Variation in average Nusselt number within thermally developing region at different Reynolds number.

The HT enhancement is achieved at the cost of pressure loss. Figure 10 shows the pressure variation along the mini-pipe length. The plot shows that maximum pressure loss is achieved in high Re number and minimum in low Re number.

Figure 10 Pressure variation along the pipe length, for constant properties of air, Case 1.
Figure 10

Pressure variation along the pipe length, for constant properties of air, Case 1.

8 Conclusion

The results demonstrated that the maximum conductive heat flux occurs on lower wall of pipe for VP of air as compared to the CP of air, for each case. Also, the maximum difference of HT rate for variable properties of air based on T in = 20°C is 14%. Pressure also decreases with increasing Re number. It is concluded that the local Nusselt number of wall decreases gradually through the pipe length.

However, the incremental change in the average Nusselt number is strongly dependent on Re number, heat sources configuration, and thermo-physical properties of air. The results of the variable properties of air and Case 1 of the heat sources configuration on the lower wall of mini-pipe show a significant increment in the HT rate when compared to Case 2. Results clearly show that the type of heat sources configuration and thermo-physical properties of air considered is very important on the forced convection HT heating application.

Acknowledgements

The authors are grateful for the reviewer’s valuable comments that improved the manuscript.

  1. Funding information: Authors state no funding involved.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. A.H.J. and S.M.S. conducted the experimental simulations. A.H.J. and S.M.S. developed the model and performed the writing of the original manuscript and K.H.A. edited and reviewed the initial writing of the manuscript.

  3. Conflict of interest: Authors state no conflict of interest.

  4. Data availability statement: All the data, models, and codes generated or used during the study appear in the submitted article.

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Received: 2024-05-24
Revised: 2024-07-20
Accepted: 2024-07-27
Published Online: 2025-03-20

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/eng-2024-0081
Keywords for this article
forced convection; flow inside mini-channel; air properties; channel flow; CFD
Creative Commons
BY 4.0

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