# Local Heat Transfer Dynamics in the In-Line Tube Bundle under Asymmetrical Pulsating Flow

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Computational Domain and Boundary Conditions

_{1,2}/D = 1.3. The 2D in-line tube bundle was employed to simplify the mathematical model. Satisfactory agreement with the experimental data with a 2D formulation was obtained in [28,40,41] for systems similar to the flow field studied in this work. A constant temperature t

_{w}= 42 and a no slip condition (U

_{x}= 0, U

_{y}= 0) were set on all walls of the tubes. A constant flow temperature t

_{f}= 27 was set to the computational domain at the inlet., and a symmetry condition (∂U

_{x}/∂n = 0, ∂U

_{y}/∂n = 0, ∂p/∂n = 0) on the top and bottom of the domain. At the outlet, constant pressure P = 101,325. Water was chosen as the working fluid. The thermophysical properties of water were calculated depending on the flow temperature.

_{p}〉τ,

_{p}〉 is the pulsation flow velocity averaged over time τ for the corresponding negative values of instantaneous pulsation velocity u

_{p}:

_{1}is the start time of the pulsation period; τ

_{2}is the end time of the pulsation period.

_{1}+ T

_{2},

_{1}is the first pulsation half-period; T

_{2}is the second half-period. The steady flow velocity u

_{st}is equal to the pulsation flow velocity 〈u

_{p}〉 averaged over the pulsation period:

_{inlet}, Q

_{1}, Q

_{2}, Q

_{4}, Q

_{5}, Q

_{6}are the volumetric flow rate of hydraulic model elements according to Figure 3. The flow rate in hydraulic accumulator 1 is defined by the relation:

_{1}and H

_{h}are pressure in node 1.1 and on the surface of the liquid in the hydraulic accumulator 1, respectively. S

_{h}

_{0}is the connecting hole area with node 1.1 at the bottom of the hydraulic accumulator. D

_{j}

_{1}, k

_{1}, and S

_{1}are the diameter, area, and hydraulic resistance coefficient in the hydraulic accumulator 1, respectively. g is the gravity acceleration. z

_{1}is the liquid level in the hydraulic accumulator.

_{2}, l

_{3}, l

_{4}are the length of the hydraulic model elements according to Figure 3. k

_{2}, k

_{3}, k

_{4}are the hydraulic resistance coefficient of the hydraulic model elements according to Figure 3. S

_{2}, S

_{3}, S

_{4}are the area of the hydraulic model elements according to Figure 3. D

_{j}

_{2}, D

_{j}

_{3}, D

_{j}

_{4}are the diameter of the hydraulic model elements according to Figure 3. H

_{4}is the pressure in node 1.4.

_{p.c.}

_{0}is the connecting hole area with node 1.5 at the bottom of the pulsation chamber. H

_{4}is the pressure on the surface of the liquid in pulsation chamber 5. z

_{2}is the liquid level in the pulsation chamber.

_{6}is the length of pipe 6. k

_{6}, k

_{7}are the hydraulic resistance coefficient of pipe 6 and valve 7. S

_{6}is the area of pipe 6. D

_{j}

_{6}is the diameter of the hydraulic model elements according to Figure 3.

_{p.c.}(t) was set to generate a pulsating flow in the pulsating chamber. The shape of the H

_{p.c.}(t) dependence was obtained on an experimental setup in [42]. The system of Equations (1)–(8) was solved by Newton’s iterative method [43].

#### 2.2. Modeling Approach

_{max}/D = 0.2. The minimum cell size in the near-wall region is r

_{min}/D = 3.16 × 10

^{−2}. The grid size in the near-wall region expanded in the radial direction with a factor of 1.2. The number of layers in the near-wall region was 10. The convergence of a grid solver with similar systems to this work was carried out by Kim et al. and Mulcahey et al. [29,46]. In the present study, the maximum mesh size was 0.2 mm. In works [29,46], a sufficient element size was 0.2 mm and higher. The mesh of the computational domain is shown in Figure 5. Mathematical modeling was carried out in Ansys Fluent, with a coupled solution algorithm based on the pressure-based solver. For all calculations, the SIMPLE algorithm was used. Except for the energy equation, the residual was less than 10

^{−4}for all of the governing equations. A convergence criterion of 10

^{−6}was used for the energy equation for the residuals. The time step was 0.01 s. For time stepping, the first order implicit transient formulation was used.

#### 2.3. Methodology for Evaluating Results of Simulation

_{o}was found by Equation (9) over the surface, and over time 〈q

_{o}〉 using Equation (10). The local heat flux q

_{φ}averaged depending on the azimuth angle—φ (Figure 6) by Equation (11), and q

_{φ}averaged over time by Equation (12).

_{s}was averaged over the area around the cylinder for x = [x

_{1}; x

_{2}] = [0; 1.3D] y = [y

_{1}; y

_{2}] = [0; 1.3D] (Figure 6) and was found by Equation (13). The temperature averaged over space and time 〈t

_{s}〉 was found by Equation (14).

_{a}averaged over the sector area around the cylinder with the thickness L = 0.15D (Figure 6) was found according to Equation (16), and over space and time 〈U

_{a}〉 according to Equation (17). The local fluid velocity U

_{φ}is averaged depending on the azimuth angle φ according to Equation (18), and U

_{φ}is time-averaged 〈U

_{φ}〉 according to Equation (19).

_{x}

_{,y}, is the velocity component.

_{eff}averaged over the annular region around the cylinder λ

_{eff}

_{,a}, a was determined by Equation (20), and over space and time 〈λ

_{eff}

_{,a}〉 by Equation (21). The local effective thermal conductivity λ

_{eff}

_{,φ}and local effective thermal conductivity averaged over time 〈λ

_{eff}

_{,φ}〉 were found according to Equations (22) and (23), respectively.

_{turb}are the thermal and turbulent thermal conductivity of the water. λ, λ

_{turb}in Equations (20)–(23) was defined similarly to Equations (16)–(19).

_{o}was found according to Equation (26), and over space and time 〈Nu

_{o}〉 according to Equations (26) and (27). The local Nusselt number averaged over one-eighth of the region around the cylinder Nu

_{φ}and over time 〈Nu

_{φ}〉 were determined by Equations (28) and (29).

#### 2.4. Model Verification

## 3. Results and Discussion

#### 3.1. Effect of Amplitude and Frequency of Pulsations on the Flow and Heat Transfer Characteristics

_{a}〉 = 〈U

_{a}

_{,p}〉/U

_{a}

_{,st}, heat flux 〈δq

_{o}〉 = 〈q

_{o}

_{,p}〉/q

_{o}

_{,st}, Nusselt number 〈δNu

_{o}〉 = 〈Nu

_{o}

_{,p}〉/Nu

_{o}

_{,st}, temperature difference 〈δΔt〉 = 〈Δt

_{p}〉/Δt

_{st}, and effective thermal conductivity 〈δλ

_{eff}

_{,a}〉 = 〈λ

_{eff}

_{,a,p}〉/λ

_{eff}

_{,a,st}. The averaged over space and time velocity, heat flux, Nusselt number, and effective thermal conductivity were higher with higher amplitude. The temperature difference was almost the same (change less than 0.2%). The augmentation of heat transfer with an increase in the amplitude of pulsations correlated with the increase in the flow velocity and the effective thermal conductivity. The temperature difference stayed the same (change less than 0.3%) with increased pulsation frequency, while an increase in the heat flux and velocity was observed. The increase in the effective thermal conductivity was no more than 0.5%. The maximum augmentation of the averaged over space and time velocity, heat flux, and effective thermal conductivity was 3.18, 2.32, 1.4 times, respectively, at A/D = 4.5, f = 0.5 Hz. The maximum increase in the Nusselt number averaged over space and time across the entire studied range was 106% at f = 0.5 Hz (St = 0.114), A/D = 4.5, the minimum was 25% at f = 0.166 Hz (St = 0.038), and A/D = 1.25, which is consistent with the data from other authors. In [24], with symmetric pulsations in the in-line tube bundle, an augmentation of 42% was obtained at St = 0.3. The Strouhal number was higher than in our study. However, the amplitude of the velocity pulsations was 0.42 without reciprocating flow. In [27], at higher amplitudes of A/D = 18.5, a heat transfer augmentation of 200% was achieved. In [26], by a pulsating flow at St = 0.45, the enhancement of heat transfer occurred only at the front of the second cylinder in the tube bundle. Other cylinders in the tube bundle were not much affected by the external pulsation. The lack of heat transfer enhancement was associated with a low pulsation amplitude without reciprocating flow. The velocity pulsation amplitude was 0.1. In [29], the heat transfer enhancement ratio in the tube bundles increased with the pulsation frequency, which also agreed with the data presented in this study.

_{o}〉 was observed at the maximum frequency and amplitude of pulsations. The minimum augmentation of the Nusselt number 〈δNu

_{o}〉 corresponded to the minimum frequency and amplitude of pulsations. In the studied range, an increase in frequency had a greater effect on the heat transfer enhancement in comparison with the amplitude of pulsations. For example, at a fixed frequency f = 0.25 Hz with an increase in the amplitude A/D from 1.25 to 4.5, the augmentation of the Nusselt number averaged over space and time at A/D = 1.25 was 1.35 and 1.57 times at A/D = 4.5. With an increase in the frequency f from 0.166 Hz to 0.5 Hz at a fixed amplitude A/D = 3, the Nusselt number averaged over space and time at f = 0.166 Hz was 1.35 and 1.94 times at f = 0.5 Hz.

_{a}, δq

_{o}, δNu

_{o}, δΔt, and δλ

_{eff}

_{,a}for one period of pulsations at a frequency f = 0.5 Hz. The increase in instantaneous values of the flow velocity δU

_{a}, heat flux δq

_{o}, and the effective thermal conductivity δλ

_{eff,a}averaged over the area around the cylinder was higher, so the higher the pulsation amplitude. The instantaneous values of the heat flux and effective thermal conductivity at A/D = 4.5 during the period of pulsations increased up to 3.57 and 2.23 times compared to the steady flow, respectively. The instantaneous values of the flow velocity increased up to 9.5 times for a separate pulsation phase, while by the end of the pulsation period at τ/T > 0.8, the instantaneous flow velocity was less than the steady flow. The maximum instantaneous augmentation temperature difference δΔt = 1.31 was observed at the minimum pulsation amplitude, which is the opposite to δU

_{a}, δq

_{o}, δNu

_{o}, and δλ

_{eff,a}. The averaged instantaneous values of the Nusselt number δNu

_{o}increased up to 3.37 times at amplitude A/D = 4.5 and a pulsation phase τ/T = 0.3. A slight decrease in the Nusselt number δNu

_{o}up to 0.9 times, in a pulsating flow compared to a steady flow, was observed at an amplitude A/D = 1.25 and a pulsation phase τ/T = 0.3. When considering the dynamics of instantaneous values of increases in the effective thermal conductivity and the Nusselt number over time, two peaks were observed (Figure 9b,e). The first peak was observed up to τ/T = 0.25, which corresponded to the first half-period of pulsations T

_{1}; the second peak was observed after τ/T = 0.25, which corresponded to the second half-period of pulsations T

_{2}. The fluid flow in the tube bundle was characterized by a reciprocating flow. Therefore, the first maximum of the values δNu

_{o}and δλ

_{eff}

_{,a}, was associated with the acceleration of the fluid flow during its flow in the opposite direction (Figure 4). The second maximum was related to the acceleration of the fluid flow during its flow in the forward direction (second half-period of pulsations T

_{2}). For instantaneous values of heat flux and velocity (Figure 9a,c) there was one significant peak during T

_{1}and a second less noticeable one during T

_{2}. Both peaks were also associated with the accelerations of the fluid flow from pulsation. The more significant value of the first velocity peak (Figure 9c) was associated with the asymmetric nature of the flow pulsations at the inlet to the computational domain. The maximum velocity amplitude values at the tube bundle inlet were higher at the first half-period of pulsation than the second half-period of pulsation. The velocity amplitude values were the difference between the steady flow velocity and instantaneous pulsating velocity. Differences between the velocity amplitude values were higher with higher pulsation amplitude A/D. At the minimum value of the pulsation amplitudes A/D = 1.25, the velocity values of the peaks were close (Figure 9c). Obviously, with decreasing pulsation amplitude, the peak of the first half-period of pulsation will become smaller than the peak of the second half-period of pulsation.

#### 3.2. Effect of Amplitude and Frequency of Pulsations on Local Flow Characteristics and Heat Transfer

_{φ}〉, the effective thermal conductivity 〈δλ

_{eff}

_{,φ}〉, the velocity 〈δU

_{φ}〉, and the Nusselt number 〈δNu

_{φ}〉 with the amplitude A/D and the frequency f of pulsations. Increasing the pulsation frequency had a greater effect on the Nusselt number averaged over one-eighth of the cylinder area 〈δNu

_{φ}〉 compared to increasing the pulsation amplitude. The local Nusselt number 〈δNu

_{φ}〉, in a pulsating flow compared to a steady flow at φ = 0° and 180°, increased from 4 to 9.5 times with increasing frequency (Figure 12b). As the amplitude increased, the Nusselt number 〈δNu

_{φ}〉 at φ = 0° and 180° increased from 4 to 5.5 times (Figure 12a). The minimum enhancement of the local Nusselt number 〈δNu

_{φ}〉 was observed at φ = 90°, at A/D = 3, f = 0.166 Hz 〈δNu

_{φ}〉 = 1.04. When f = 0.25 Hz and A/D = 3 for φ = 90°, there was a decrease in the local value of the Nusselt number by 1%. The maximum increase in velocity 〈δU

_{φ}〉 in a pulsating flow with an increase in the amplitude and frequency of pulsations (Figure 10c and Figure 11c) was observed at φ = 0° and 180°, which is consistent with the increase in the local heat flux (Figure 10a and Figure 11a) and the Nusselt number (Figure 12). In the A/D range from 3 to 4.5, a slight decline in the growth 〈δU

_{φ}〉 was observed for φ = 0° and 180° (Figure 10c), while the values of 〈δq

_{φ}〉, 〈δNu

_{φ}〉 also declined at φ = 180° and decreased at φ = 0° (Figure 10a and Figure 12a). The maximum growth of the effective thermal conductivity 〈δλ

_{eff}

_{,φ}〉, with increasing A/D and f was observed at φ = 45° (Figure 10b and Figure 11b), and the minimum at φ = 0° and 180°, which was the opposite to 〈δq

_{φ}〉, 〈δNu

_{φ}〉, 〈δU

_{φ}〉. The maximum absolute values of 〈δq

_{φ}〉, 〈δNu

_{φ}〉, 〈δU

_{φ}〉 were observed at φ = 0° and 180°, and the minimum at φ = 45° and 90°.

_{φ}, δq

_{φ}, δNu

_{φ}, δλ

_{eff}

_{,φ}are shown in Figure 13 and Figure 14. The values of local heat flux δq

_{φ}and local Nusselt number δNu

_{φ}in the front (φ = 0°) and back part (φ = 135°, 180°) of the cylinder increased up to 14–29 times, which is consistent with an increase in the values of δU

_{φ}. The instantaneous value of the local Nusselt number at the front of the cylinder (φ = 0°) increased up to 29 times (Figure 14d) during the first pulsation period T

_{1}, which corresponded to the reciprocating flow. The augmentation of the instantaneous values of the local Nusselt number in the front part of the cylinder was associated with the restructuring of the flow structure, as the result of the reciprocating flow. The maximum augmentation of δNu

_{φ}in the back of the cylinder (φ = 180°) reached 16 times. The maximum augmentation at (φ = 180°) was observed during the second period of pulsations T

_{2}, which corresponded to the flow acceleration, which led to heat transfer intensification.

_{eff}

_{,φ}a rise to 1.4 was observed at φ = 45°, A/D = 1.25 (Figure 14a) during the second half-period T

_{2}, while the instantaneous local Nusselt number δNu

_{φ}at φ = 45° did not increase (Figure 14c). Consequently, an increase in δλ

_{eff}

_{,φ}to 1.4 is insufficient for heat transfer augmentation. It can be assumed that the heat transfer enhancement at A/D = 1.25 was mainly associated with the increase in the flow velocity. When A/D = 4.5, the instantaneous effective thermal conductivity δλ

_{eff}

_{,φ}increased up to 3.5 times at φ = 45°, 90°, and 135° (Figure 14b), which is consistent with the growth of δq

_{φ}and δNu

_{φ}for this region of the cylinder.

#### 3.3. Contour Plots of Temperature, Effective Thermal Conductivity, and Plots of the Velocity Vector

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

A | Dimensional amplitude of pulsation, m |

A/D | Dimensionless relative amplitude of pulsation |

D | Diameter of the tubes in tube bundle, m |

D_{j} | Diameter of the hydraulic model elements, m |

f | Frequency of pulsation, Hz |

F(φ,r) | Area of the sector around the cylinder in the tube bundle, m^{2} |

g | Gravity acceleration, m s^{−2} |

H | Pressure in the hydraulic model elements, mH2O |

H_{h} | Pressure on the surface of the liquid in the hydraulic accumulator, mH2O |

H_{p.c.} | Pressure on the surface of the liquid in the pulsation chamber, mH2O |

k | Hydraulic resistance coefficient of the hydraulic model elements |

l | Length of the hydraulic model elements, m |

Nu | Nusselt number |

P | Pressure, Pa |

q | Heat flux, W m^{−2} |

Q | Volumetric flow rate in the hydraulic model elements, m^{3} s^{−1} |

Re | Reynolds numbers |

r_{min}/D | Minimum cell size in the near-wall region |

S | Area of the hydraulic model elements, m^{2} |

s/D | Relative transverse and longitudinal pitch of tube bundle |

S_{h}_{0} | Hole area at the bottom of the hydraulic accumulator, m^{2} |

S_{p.c.}_{0} | Hole area at the bottom of the pulsation chamber, m^{2} |

St | Strouhal number |

t_{f} | Flow temperature at the inlet of the tube bundle, °C |

t_{s} | Temperature around central cylinder in the fifth row of the tube bundle, °C |

t_{w} | Tube bundle wall temperature, °C |

u | Flow velocity at inlet of the tube bundle, m s^{−1} |

U | Velocity around central cylinder in the fifth row of the tube bundle, m s^{−1} |

U_{x,y} | Velocity components, m s^{−1} |

y_{max}/D | Maximum mesh size related to the tube diameter |

z_{1} | Liquid level in the hydraulic accumulator, m |

z_{2} | Liquid level in the pulsation chamber, m |

T | Period of the pulsation, s |

T_{1} | First half-period of the pulsation, s |

T_{2} | Second half-period of the pulsation, s |

x | x-coordinate |

y | y-coordinate |

Greek symbols | |

η | Kinematic viscosity, m^{2} s^{−1} |

λ | Thermal conductivity, W m^{−1} K^{−1} |

λ_{eff} | Effective thermal conductivity, W m^{−1} K^{−1} |

λ_{turb} | Turbulent thermal conductivity, W m^{−1} K^{−1} |

τ | Time, s |

τ_{1} | Start time of the pulsation period, s |

τ_{2} | End time of the pulsation period, s |

Δt | Temperature difference, °C |

Subscripts | |

st | Steady flow |

p | Pulsating flow |

o | Averaged value over the surface of the cylinder wall in the tube bundle |

a | Averaged value over the annular area around the cylinder in the tube bundle |

φ | Averaged value depending on the azimuth angle |

δ | Enhancement factor |

Notations | |

〈 〉 | Averaged value over one period of the pulsation |

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**Figure 3.**A scheme of the hydraulic model to generate a pulsating flow: 1—hydraulic accumulator, 2, 4, 6—pipes, 3—tube bundle, 5—pulsation chamber, 7—valve.

**Figure 4.**The pulsating velocity at f = 0.5 Hz, A/D = 3, Re = 500: 1—experimental data [42]; 2—numerical simulation.

**Figure 6.**The location of the azimuth angle and the annular region around the cylinder of the fifth row of the tube bundle to determine the flow characteristics and heat transfer.

**Figure 8.**The effect of the pulsations on the flow and heat transfer characteristics averaged over space and time. (

**a**) Variation of the velocity ratio 〈δU

_{a}〉 = 〈U

_{a}

_{,p}〉/U

_{a}

_{,st}, heat flux ratio 〈δq

_{o}〉 = 〈q

_{o}

_{,p}〉/q

_{o}

_{,st}, Nusselt number ratio 〈δNu

_{o}〉 = 〈Nu

_{o}

_{,p}〉/Nu

_{o}

_{,st}, temperature difference ratio 〈δΔt〉 = 〈Δt

_{p}〉/Δt

_{st}, and effective thermal conductivity ratio 〈δλ

_{eff}

_{,a}〉 = 〈λ

_{eff}

_{,a,p}〉/λ

_{eff}

_{,a,st}with A/D at f = 0.25 Hz. (

**b**) Variation of 〈δU

_{a}〉, 〈δq

_{o}〉, 〈δNu

_{o}〉, 〈δΔt〉, 〈δλ

_{eff}

_{,a}〉 with f at A/D = 3.

**Figure 9.**The effect of pulsations on instantaneous flow and heat transfer characteristics averaged over space at f = 0.5 Hz for one period of pulsation. (

**a**) Variation of heat flux ratio δq

_{o}= q

_{o}

_{,p}/q

_{o}

_{,st}with τ/T. (

**b**) Variation of effective thermal conductivity ratio δλ

_{eff}

_{,a}= λ

_{eff}

_{,a,p}/λ

_{eff}

_{,a,st}with τ/T;.(

**c**) Variation of velocity ratio δU

_{a}= U

_{a}

_{,p}/U

_{a}

_{,st}with τ/T. (

**d**) Variation of temperature difference ratio δΔt = Δt

_{p}/Δt

_{st}with τ/T. (

**e**) Variation of Nusselt number ratio δNu

_{o}= Nu

_{o}

_{,p}/Nu

_{o}

_{,st}with τ/T.

**Figure 10.**The effect of pulsations on the local flow and heat transfer characteristics at f = 0.25 Hz. (

**a**) Variation of heat flux ratio 〈δq

_{φ}〉 = 〈q

_{φ}

_{,p}〉/q

_{φ}

_{,st}with A/D. (

**b**) Variation of effective thermal conductivity ratio 〈δλ

_{eff}

_{,φ}〉 = 〈λ

_{eff}

_{,φ,p}〉/λ

_{eff}

_{,φ,st}with A/D. (

**c**) Variation of velocity ratio 〈δU

_{φ}〉 = 〈U

_{φ}

_{,,p}〉/U

_{φ}

_{,,st}with A/D.

**Figure 11.**The effect of pulsations on the local flow and heat transfer characteristics at A/D = 3. (

**a**) Variation of 〈δq

_{φ}〉 with f. (

**b**) Variation of 〈δλ

_{eff}

_{,φ}〉 with f. (

**c**) Variation of 〈δU

_{φ}〉 with f. The effect of pulsations on the local flow and heat transfer characteristics at A/D = 3. (

**a**) The variation in the heat flux ratio 〈δq

_{φ}〉 = 〈q

_{φ}

_{,p}〉/q

_{φ}

_{,st}with f. (

**b**) Variation in the effective thermal conductivity ratio 〈δλ

_{eff}

_{,φ}〉 = 〈λ

_{eff}

_{,φ,p}〉/λ

_{eff}

_{,φ,st}with f. (

**c**) Variation in the velocity ratio 〈δU

_{φ}〉 = 〈U

_{φ}

_{,,p}〉/U

_{φ}

_{,,st}with f.

**Figure 12.**The effect of pulsations on the local Nusselt number. (

**a**) Variation in the Nusselt number ratio 〈δNu

_{φ}〉 = 〈Nu

_{φ}

_{,p}〉/Nu

_{φ}

_{,st}with A/D at f = 0.25 Hz. (

**b**) Variation of 〈δNu

_{φ}〉 = 〈Nu

_{φ}

_{,p}〉/Nu

_{φ}

_{,st}with f at A/D = 3.

**Figure 13.**The effect of pulsations on the instantaneous local heat flux and velocity at f = 0.5 Hz for one period of pulsation. (

**a**) Variation of heat flux ratio δq

_{φ}= q

_{φ}

_{,p}/q

_{φ}

_{,st}with τ/T at A/D = 1.25. (

**b**) Variation of δq

_{φ}with τ/T at A/D = 4.5. (

**c**) Variation of velocity ratio δU

_{φ}= 〈U

_{φ}

_{,,p}〉/U

_{φ}

_{,,st}with τ/T at A/D = 1.25. (

**d**) Variation of δU

_{φ}with τ/T at A/D = 4.5.

**Figure 14.**The effect of pulsations on the instantaneous local heat flux and velocity at f = 0.5 Hz for on period of pulsation. (

**a**) Variation of effective thermal conductivity ratio δλ

_{eff}

_{,φ}= λ

_{eff}

_{,φ,p}/λ

_{eff}

_{,φ,st}with τ/T at A/D = 1.25. (

**b**) Variation of δλ

_{eff}

_{,φ}with τ/T at A/D = 4.5. (

**c**) Variation of the Nusselt number ratio δNu

_{φ}= Nu

_{φ}

_{,p}/Nu

_{φ}

_{,st}with τ/T at A/D = 1.25. (

**d**) Variation of δNu

_{φ}with τ/T at A/D = 4.5.

**Figure 15.**The temperature contour plots for the three central rows of the tube bundle at a steady flow.

**Figure 16.**The effective thermal conductivity contour plots for the three central rows of the tube bundle at a steady flow.

**Figure 17.**The velocity vector plots for the three central rows of the tube bundle at a steady flow.

**Figure 18.**The instantaneous vector plots at different phases of pulsations for the three central rows of the tube bundle at A/D = 3, f = 0.5 Hz: (

**a**) τ/T = 0.005 (

**b**) τ/T = 0.025; (

**c**) τ/T = 0.07; (

**d**) τ/T = 0.125; € τ/T = 0.185; (

**f**) τ/T = 0.32; (

**g**) τ/T = 0.65. (

**h**) The velocity at tube bundle inlet for one period of pulsations and seven phase definition.

**Figure 19.**The instantaneous effective thermal conductivity contour plots at different phases of pulsations for the three central rows of the tube bundle at A/D = 3, f = 0.5 Hz: (

**a**) τ/T = 0.005 (

**b**) τ/T = 0.025; (

**c**) τ/T = 0.07; (

**d**) τ/T = 0.125; (

**e**) τ/T = 0.185; (

**f**) τ/T = 0.32; (

**g**) τ/T = 0.65.

**Figure 20.**The instantaneous temperature contour plots at different phases of pulsations for the three central rows of the tube bundle at A/D = 3, f = 0.5 Hz: (

**a**) τ/T = 0.005 (

**b**) τ/T = 0.025; (

**c**) τ/T = 0.07; (

**d**) τ/T = 0.125; (

**e**) τ/T = 0.185; (

**f**) τ/T = 0.32; (

**g**) τ/T = 0.65.

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**MDPI and ACS Style**

Haibullina, A.; Khairullin, A.; Balzamov, D.; Ilyin, V.; Bronskaya, V.; Khairullina, L.
Local Heat Transfer Dynamics in the In-Line Tube Bundle under Asymmetrical Pulsating Flow. *Energies* **2022**, *15*, 5571.
https://doi.org/10.3390/en15155571

**AMA Style**

Haibullina A, Khairullin A, Balzamov D, Ilyin V, Bronskaya V, Khairullina L.
Local Heat Transfer Dynamics in the In-Line Tube Bundle under Asymmetrical Pulsating Flow. *Energies*. 2022; 15(15):5571.
https://doi.org/10.3390/en15155571

**Chicago/Turabian Style**

Haibullina, Aigul, Aidar Khairullin, Denis Balzamov, Vladimir Ilyin, Veronika Bronskaya, and Liliya Khairullina.
2022. "Local Heat Transfer Dynamics in the In-Line Tube Bundle under Asymmetrical Pulsating Flow" *Energies* 15, no. 15: 5571.
https://doi.org/10.3390/en15155571