# Laminar Pipe Flow with Mixed Convection under the Influence of Magnetic Field

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{3}O

_{4}) seems to be the most suitable raw material. A review on the subject counts 18 experimental studies on the enhancement of heat transfer in laminar and turbulent pipe flow by magnetic forces conducted between 2010 and 2020 [2]. All but one of these studies, as well as a more recent investigation [3], used water-based suspensions containing magnetite nanoparticles. Let us consider such suspensions as ferronanofluids.

- investigation of the governing equations to identify the relevant parameter space,
- numerical schemes incorporating the specific mechanisms relevant to magnetically affected ferronanofluid flow, and
- experiments intended to examine the switch ability of heat transfer.

## 2. Material and Methods

#### 2.1. Test Rig

^{®}) ensured a fully developed parabolic velocity profile up to the critical Reynolds number of 2300. The test section consisted of a pultruded copper pipe (Cu > 99.9%) with a roughness of R

_{a}= 0.66 µm. This extraordinarily small roughness gave at least the hope that no nanoparticles would stick to the surface due to roughness effects. This seems to be especially important in the vicinity of the magnets. The cross section of the outlet section (PLEXIGLAS

^{®}) is quadratic. It contains a static mixer with helical elements to create the mixing cup temperature measured at the outlet. All sections are insulated by an inner glass-fibre mat (two layers of 13 ± 2 mm, k = 0.04 W m

^{−1}K

^{−1}) and an outer ArmaFlex

^{®}shell (20 mm, k = 0.04 W m

^{−1}K

^{−1}, armacell, Germany). A constant heat flux is generated by an electric heater coiled around the pipe. The constancy of the heat flux over the pipe mantle is ensured by the tight coiling with a very uniform pitch of about 10 mm. Moreover, the high thermal conductivity of the copper pipe (k = 401 W m

^{−}

^{1}K

^{−}

^{1}) conducts the thermal energy provided by the heater nearly evenly in axial and circumferential direction. The test rig needs a filling of about 1100 mL of working fluid.

_{i}= 5.0, 18.8, 53.3, 73.3, 93.3, 115.0, 133.3, 163.3, 203.3, and 253.3 (x = 30 mm, 113 mm, 320 mm, 440 mm, 560 mm, 680 mm, 800 mm, 980 mm, 1220 mm, 1520 mm) on the outer mantle of the test section. The effective length of the test section employed to analyse laminar ferronanofluid flow is, at 1520 mm, longer than any other test section except one discussed in [2] and slightly longer than the one used by Tekir et al. [3].

#### 2.2. Magnets

^{®}Engineering e. K.) with a nominal holding force of 10.59 kg generate the magnetic field (Figure 1). Their main dimensions are 120 × 12 × 5 mm

^{3}(L × B × H). The maximal operation temperature of the magnets (t

_{max}= 80 °C) limits the temperature maximum during the experiments. The magnetic flux density on the magnet surfaces amounted to 0.1215 T. The magnetic poles are axially aligned along the height (5 mm) of the magnets.

_{i}≤ 93.3 (440 mm ≤ x ≤ 560 mm) downstream of the inlet of the test section. The poles are oriented in parallel to the test section. The distance between the centrelines of the magnets and of the test pipe r

_{cl}is 34 mm. Four magnet configurations (single magnet either above or below the pipe and two magnets above and below the pipe) for either attracting or repulsing are investigated. For the attracting and the repulsing configuration, two different poles and two equal poles, respectively, are in opposition. The main axes of the pipe and the magnets are oriented in parallel. A second distance between the centrelines of the magnets and the test pipe r

_{cl}= 55 mm is investigated for some selected cases to understand the influence of the magnetic field intensity. In all cases, the magnets are fixed in additive printed holders (Figure 1).

#### 2.3. Ferronanofluid

_{3}O

_{4}) suspension is stabilised by an organic chemical, anionic polymer. The appearance is an opaque, deep dark-brown. Optical inspection over a period of 6 months does not indicate sedimentation.

^{−3}, the particle concentration of the ferronanofluid ranges between 2.9 and 3.1 vol.%.

^{−3}and that of the ferronanofluid to 1190 kg m

^{−3}(Densito 30P, METTLER TOLEDO, Germany). The pH values are 8.9 and 9.1, respectively (SM Titrino 702, Metrohm AG, Switzerland). The ferronanofluid data are close to the reference values of 1.19 g/mL (22.8 °C) and a pH value of 9.0 given by the manufacturer [7].

^{−1}. The accuracy of the measurements is assumed to be less than 3% of the measurement point. The obtained data reveal that the base fluid has a mean of a 1.4-fold higher viscosity than water, while the ferronanofluid shows a 2.1-fold increase. The data found with the rheometer are confirmed by independent measurements employing an Ubbelohde capillary (metering range 1.0–10 mm

^{2}s

^{−1}) at 20 and 40 °C. The measured data are only partially used in Figure 2. Actually, measuring points are recorded at every 0.2 K temperature difference: between 20 and 40 °C, one measuring point every minute; between 40 and 60 °C, approximately once every 20 s. Dynamic viscosity is employed in combination with the density for the determination of the Reynolds number.

^{−1}K

^{−1}at room temperature [9]. The measured thermal conductivity is used to predict the local and the averaged Nusselt number.

^{−1}). At each temperature level, a waiting time of 30 min ensured thermal equilibrium. Regardless, the expected error is less than 3% of the measurement value (reproducibility 1%). Adding solid particles to a liquid mostly lowers its specific heat capacity. This is also the case for the ferronanofluid investigated here. The base fluid had a specific heat capacity of about 4.7% less than that of water and about 17.9% less than that of the ferronanofluid. However, due to the higher density of the base fluid and ferronanofluid, the products (ρ c

_{p})

_{BF}and (ρ c

_{p})

_{FNF}are nearly equal, having 14 to 15% less than the equivalent values of water.

#### 2.4. Experimental Procedure and Data Analysis

_{in}= 14.2. All tests are carried out at a constant total heat input of 150 W, which is equivalent to a specific heat of 1768 W m

^{−2}. The ferronanofluid flow is fully laminar at the entrance of the test section in all experiments. The inlet Reynolds numbers Re

_{in}varied between 127, 237, 750, and 1234.

_{x}, which is defined as

_{x}is set to 7.0% throughout the entire analysis.

## 3. Experimental Results

#### 3.1. Magnet Configurations

_{cl}= 34 mm, Re

_{in}= 236.6 ± 3.5). The baseline without magnetic field is identical in all plots (Figure 4, red). Up to x/d

_{i}≅ 50, the Nusselt number baseline values are falling, approaching the limiting value $\underset{\text{}x/{d}_{i}\text{}\to \text{}\infty}{\mathrm{lim}}N{u}_{x}\text{}=\text{}4.36$ of the exact solution for laminar pipe flow with constant heat flux [13]. The flow is entirely laminar without any secondary motion. Further downstream, the Nusselt number increased, reaching a maximum at about x/d

_{i}= 120, and decreased again. This departure from the expected monotonically falling curve throughout signal a secondary motion. The heating of the pipe wall generated a radial temperature profile that induced a pair of counter-rotating longitudinal vortices (Figure 3). This vortex pair consisted of two vertical upwardly oriented currents at the inner wall of the pipe and a stronger central down flow [13]. The laminar pipe flowed with pure thermal conduction as the heat transfer mechanism changed to a mixed convection. The thermal conduction is supported by a free convection. This caused an enhancement of the local heat transfer and therewith of Nu

_{x}. The more the temperature profile inside the flow is equalised, the less this augmentation is. Eventually, Nu

_{x}approached the value 4.36 again. Axial position and extent as well as the intensity of the vortex pair depended on the Reynolds number and heat input.

_{x}is increased right from the upstream edge of the magnet at x/d

_{i}= 73.3. The maximal enhancement ΔNu

_{x}= +0.65 (+ 11.5%) is achieved at x/d

_{i}= 93.3. The maximum follows a relaxation region where the Nusselt number is reduced and finally collapsed to the baseline Nu

_{x}distribution at x/d

_{i}= 113.3. Assuming that the relaxation length started at the rear end of the magnet, this length, 240 mm, is twice as long as the magnet itself.

_{i}= 93.3, the rear end of the magnet. The deterioration reached, with ΔNu

_{x}= −0.85 (−16.3%), a minimum at x/d

_{i}= 133.3 and ended at x/d

_{i}= 163.3. The relaxation length is now 420 mm, 3.5 times the magnet length, which indicated that it took quite a long time to re-establish the purely gravity-driven vortex pair.

_{i}= 53.3. For the two attracting magnets the heat transfer deteriorated up to x/d

_{i}= 163.3. The minimum is reached at x/d

_{i}= 113.3 with ΔNu

_{x}= −0.71 (−13.3%). For the repulsion case (Figure 4, last diagram), the situation is nearly identical with the configuration where the magnet is located above the pipe. The interpretation of these two cases with respect to the secondary flow is difficult. The opacity of the ferronanofluid did not allow optical access. However, one can definitively state that the gravity-driven vortex pair is significantly stalled in its motion and therewith heat transfer is lowered. Moreover, following the hypothesis from the beginning of this section—magnetite nanoparticles turn under the influence of magnetic fields such that they are always attracted by the nearest magnet—weakening of the gravity-generated secondary motion seems plausible.

#### 3.2. Intensity of Magnetic Field

_{cl}= 55 mm. The magnets are now approximately 20 mm further away from the flow, and their influence intuitively should already be significantly weaker. The first configuration—single magnet below the pipe—indicates actually no effect on the flow (Figure 5, orange). All other configurations cause a degeneration of the local Nusselt number, which is within experimental error and therefore identical. However, compared to the findings for r

_{cl}= 34 mm, the deterioration is weaker. Moreover, the extension of the magnetic influence ranges only from the upstream edge(s) of the magnet(s) at x/d

_{i}= 73.3 up to about x/d

_{i}= 133.3. The maximal reduction occurs in all cases at x/d

_{i}= 93.3 and ranged around ΔNu

_{x}= −0.43 (−7.5%). Further downstream, all data of the configurations with magnets collapse to the baseline results.

_{i}= 5.0) up to the last one (x/d

_{i}= 253.3). For that purpose the Nu

_{x}distribution is fitted with fourth-order polynoms and integrated. The averaged Nusselt numbers for Re

_{in}= 236.6 ± 3.5 are compiled in Figure 6. Clearly the weak and nearly equal influence of the configurations with the magnet(s) distantly placed (r

_{cl}= 55 mm) is visible. No change of the averaged Nusselt number is found for the case where the magnet is positioned below the pipe with r

_{cl}= 34 mm. For all other cases, more significant effects are found than for the distant positioning. However, even for the strongest influence found—two repulsing magnets with r

_{cl}= 34 mm—the net effect with $\mathsf{\Delta}\overline{Nu}\text{}=\text{}0.32$ (6.0%) is rather weak.

#### 3.3. Reynolds Number

_{in}= 236.6 ± 3.5, the Nusselt number distributions for the different magnet(s) configurations collapse to the baseline results within the experimental error. The only exception might be the configuration with two attracting magnets (Figure 7, third diagram), where at the front of the magnets (x/d

_{i}= 73.3) heat transfer is slightly lower compared to that of the baseline. As already pointed out, the achievable changes of the heat transfer depend on the interplay of the different forces acting on the flow. Clearly, the inertial force is significantly larger than the magnetic force at Re

_{in}= 1224.3 ± 3.5, so the influence of the magnet(s) is more or less negligible.

_{in}= 236.6 ± 3.5 (Figure 4) and 126.4 ± 2.4 (Figure 7) supports the above findings. The baseline of the latter flow showed a more pronounced peak, which indicated a stronger secondary motion. Similar changes as for Re

_{in}= 236.6 ± 3.5 are found for a single magnet below the increase of Nu

_{x}in the vicinity of the magnet-and a single magnet above-deterioration downstream of the magnet. The situation is different for the configurations with two opposite magnets. In both cases, the magnets created an increase of heat transfer in a region reaching far downstream. It is open to interpretation if this effect followed from a disturbance of the gravity-driven secondary motion or if additional secondary motions are generated by the magnetic field. Samsam-Khayani et al. (2020) showed numerically the existence of such magnetically caused secondary motion/vortices in an annular horizontal tube. While these flow structures are obviously axially oriented [5], this is not necessarily the case in other flow geometries, such as rectangular channels. Again, numerically, Goharkhah & Ashjaee (2014) [14] confirmed the evolution and decay of magnetically initiated vortex systems. The flow structures these authors found for an instantaneous magnetic field are oriented normally to the main flow of a channel [14].

_{in}= 749.9 ± 9.8). The obtained increase, with $\mathsf{\Delta}\overline{Nu}$ = 0.28 (5.2%), is rather weak. More pronounced is the increase at the lowest inlet Reynolds number (Re

_{in}= 126.4 ± 2.4) for the attracting ($\mathsf{\Delta}\overline{Nu}$ = 0.48, 7.7%) and for the repulsing configuration ($\mathsf{\Delta}\overline{Nu}$ = 0.56, 8.9%). Similar minor decreases appeared only at Re

_{in}= 236.6 ± 3.5, but for the configuration where a single magnet is placed above the pipe. In summary, the higher the Reynolds number at the inlet, the lower the influence of the magnet(s) on the overall heat transfer. At low Reynolds numbers, there is little effect either positive or negative.

## 4. Summary and Conclusions

- The pipe flow of the working fluid, a suspension of magnetite nanoparticles, exhibits a significant gravity-driven secondary motion already in the absence of magnetic influence. This motion is either hindered or assisted by the magnetic field. Which of these two cases occur and with what intensity depends on the spatial orientation of the magnetic force with respect to gravity and on the Reynolds number.
- The alteration may either be positive (enhancement) or negative (deterioration). The effect of the overall heat transfer of the pipe flow is rather weak and rarely exceeds 5% (positive or negative). The further the radial distance of the magnets from the flow or the higher the Reynolds number, the weaker are the effects on heat transfer.
- Based on data analysis, it is argued that of the three possible heat transfer enhancing mechanisms in laminar pipe flow under a radial magnetic field—viscosity-controlled percolation effect, inertia-controlled formation of secondary motion, and magnetic force-controlled dune formation—the second seemed the most likely to occur in our experiments. The first option is ruled out because of the comparably high Reynolds numbers, and the third because of the comparably weak magnetic forces.

_{in}= 236.6 ± 3.5 once again. The drawings on the top of the diagrams illustrate the basic state related to the region upstream of the magnets that is disturbed by the magnetic field. Such a perturbation can lead either to an enhancement or a reduction of heat transfer. Therefore, it can be imposed with the intension of controlling the flow and, therewith, the heat transfer. Downstream of the magnets the disturbance decays due to viscous effects and the flow approaches the basic state again. The possibility of performing such a control with permanent magnets seems rather limited. Further research should therefore focus on the use of instantaneous magnetic fields generated by electromagnets.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

c_{p} | specific heat capacity | (J kg^{−1} K^{−1}) |

d_{i} | inner diameter of test pipe | (mm) |

h | heat transfer coefficient | (W m^{−2} K^{−1}) |

K | thermal conductivity | (W m^{−1} K^{−1}) |

L | overall length of test section | (mm) |

Nu | Nusselt number | (-) |

Pr | Prandtl number | (-) |

Q | specific heat | (W m^{−2}) |

r_{cl} | distance between centrelines of magnet(s) and pipe | (mm) |

Re | Reynolds number | (-) |

t, T | temperature | (°C, K) |

x | streamwise coordinate | (mm) |

η | dynamic viscosity | (kg m^{−1} s^{−1}) |

ρ | density | (kg m^{−3}) |

Subscripts | ||

cl | centreline | |

in | inlet of test section | |

ou | outlet of test section | |

x | local coordinate in axial direction of pipe | |

w | wall | |

Abbreviations | ||

BF | base fluid | |

FNF | ferronanofluid |

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**Figure 1.**Test rig. Clockwise starting from the upper left photo: general view before closing the insulation (white: glass fibre mat, black ArmaFlex

^{®}shell, wires from Pt100), NdFeB magnet, and magnet holder (single magnet above the pipe, r

_{cl}= 55 mm). Orange arrow indicates direction of flow. The blue-framed area of the magnet (lower right photo) indicates one pole area. The other pole area is opposite (the area on which the magnet lies).

**Figure 2.**Thermophysical properties of water, base fluid and ferronanofluid. The plots show (

**a**) dynamic viscosity, (

**b**) thermal conductivity, (

**c**) specific heat capacity, and (

**d**) Prandtl number. The colours stand for DI-water (light blue, [8]), base fluid (green), and ferronanofluid (dark blue). Symbols indicate experimental data and full curves polynomial fits. Orange triangles indicate base fluid and purple triangles ferronanofluid viscosity measurements with Ubbelohde capillary.

**Figure 3.**Flow situation for mixed convection (water) without magnetic influence. The main flow points into the plane of drawing. The secondary motion (counter rotating vortex pair) is shown in blue and the centrally positioned down flow by the light-blue arrow. The magnets are shown only to illustrate their position with respect to the pipe and the secondary motion. Sketch not to scale.

**Figure 4.**Local Nusselt number along non-dimensional axial coordinate for the magnet configurations single magnet below and above and two attracting magnets (r

_{cl}= 34 mm). The plots show (

**a**) magnet below pipe, (

**b**) magnet above pipe, (

**c**) magnets below and above pipe attracting each other, and (

**d**) magnets below and above the pipe repulsing each other. Symbols indicate experimental data. Data points connected only to improve visibility. The colours stand for ferronanofluid without magnet(s) (red) and with magnet(s) (dark blue). Orange arrow indicates direction of flow. Black rectangles show length and position of magnet(s) with respect to pipe. Grey bar indicates region of magnet(s) with respect to x/d

_{i}.

**Figure 5.**Local Nusselt number along non-dimensional axial coordinate for the magnet configurations single magnet below and above and two attracting magnets (r

_{cl}= 55 mm). Symbols indicate experimental data. Data points connected only to improve visibility. The colours stand for ferronanofluid without single magnet (red), with single magnet below (orange), and with magnet(s) (green: ■ above, ♦ above and below attracting, and ● above and below repulsing). All other symbols and marks are as in Figure 3.

**Figure 6.**Comparison of averaged Nusselt number for different magnet configurations with Re

_{in}= 236.6 ± 3.5. Symbols indicate ferronanofluid without magnet (dark blue ■) and with magnet(s), for r

_{cl}= 34 mm (red ▲), and for r

_{cl}= 55 mm (orange ▼).

**Figure 7.**Comparison Local Nusselt number along non-dimensional axial coordinate for the magnet configurations single magnet below and above and two attracting magnets (r

_{cl}= 34 mm). The plots show (

**a**) magnet below pipe, (

**b**) magnet above pipe, (

**c**) magnets below and above pipe attracting each other, and (

**d**) magnets below and above the pipe repulsing each other. Squares indicate Re

_{in}= 126.4 ± 2.4 and lozenges Re

_{in}= 1224.4 ± 3.4.

**Figure 8.**Comparison of averaged Nusselt number for different inlet Reynolds numbers with r

_{cl}= 34 mm. Symbols indicate ferronanofluid without magnet (dark blue ■, (

**a**,

**b**)), magnet below (orange ▲, (

**a**)), magnet above (dark orange ▼, (

**a**)), two magnets attracting (light green ◆, (

**b**)), and two magnets repulsing (dark green ★, (

**b**)). Increases are marked with full and decreases with broken ellipses.

**Figure 9.**Characteristic of flow control. Magnet below pipe (

**a**) and magnet above pipe (

**b**). Symbols as in Figure 4. Green arrow acting on the Fe

_{3}O

_{4}nanoparticle represents gravity and purple arrow, magnetic force. The red areas above the plots indicate the zones of the base state. Grey areas show regions where the magnets affect the flow and therewith the local heat transfer. The grey marked relaxation regions stretch well downstream from the rear ends of the magnets.

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**MDPI and ACS Style**

Rudl, J.; Hanzelmann, C.; Feja, S.; Meyer, A.; Potthoff, A.; Buschmann, M.H.
Laminar Pipe Flow with Mixed Convection under the Influence of Magnetic Field. *Nanomaterials* **2021**, *11*, 824.
https://doi.org/10.3390/nano11030824

**AMA Style**

Rudl J, Hanzelmann C, Feja S, Meyer A, Potthoff A, Buschmann MH.
Laminar Pipe Flow with Mixed Convection under the Influence of Magnetic Field. *Nanomaterials*. 2021; 11(3):824.
https://doi.org/10.3390/nano11030824

**Chicago/Turabian Style**

Rudl, Johannes, Christian Hanzelmann, Steffen Feja, Anja Meyer, Annegret Potthoff, and Matthias H. Buschmann.
2021. "Laminar Pipe Flow with Mixed Convection under the Influence of Magnetic Field" *Nanomaterials* 11, no. 3: 824.
https://doi.org/10.3390/nano11030824