# Thermally Driven Flow of Water in Partially Heated Tall Vertical Concentric Annulus

^{1}

^{2}

^{*}

## Abstract

**:**

^{3}to 6.6 × 10

^{4}, while the Prandtl number is 6.43, the numerical solution was obtained. The modelling result showing the measurement and transient behavior of different parameters is presented. The numerical results would be both qualitatively and quantitatively validated. The presentation of unstable state profiles and heat variables along the annulus are also discussed.

## 1. Introduction

^{5}, 0.5 < Pr < 5, radius ratio from 1 to 4 and aspect ratio varies from 1 to 20. The motion was found to be produces by the gradient of radial density induced by thermal boundary conditions, maintaining the internal cylinder heated and the outer cylinder cold. The motion also consists of a single cell with low Rayleigh numbers, while a multicellular motion on large Rayleigh numbers can be observed. Mochimaru [2] developed a way to improve the solution of the transient natural convective transfer of heat in enclosures. The equation of motion, energy and continuity can also be differentiated in the Fourier series using the additional trigonometric function formulas, reducing the variables and lowering the calculation time. Ho and Lin [3] carried out numerical experiments with finite difference methods on the natural convective transfer of heat for cold water. Numerical results were reached for the radius ratio of 2.6 with a Rayleigh number ranging from 10

^{3}to 10

^{5}. The inversion parameters were 0 to 1, the eccentricity was 0 to 0.8 and the cylinder orientation angle was 0 to π. The results indicates that the characteristics of transfer of heat and flow pattern are mostly affected by a combined effect caused by the water density inversion and the inner annulus cylinder position. El-Shaarawi and Al-Attas [4] investigate numerically, the behavior of laminar free convective transfer of heat in a vertical annulus with time for 4 < modified Gr < 50,000 and Pr = 0.7. At the small-time values, the temperature overshoot phenomena, that is, because of the superiority of conduction over the mode of convective transfer of heat, has been found to be more pronounced as the dimensionless annulus height decreases, that is, modified Gr when increases. El-Shaarawi et al. [5] investigated annulus geometric parameters and their effect on transfer of heat and rate of induced flow. The numerical results were provided with Pr = 0.7, showing that the geometry parameters like radius ratio and eccentricity had a significant impact on the results. Sankar and Younghae [6] studied the impact of a separate heating on convective transfer of heat for cylindrical annuli. During their study, the inner cylinder of the annulus had two separate source of heat like flush-mounted and the outer cylinder was maintaining at constant lesser temperature. The horizontal walls at bottom and top were kept adiabatic. The empirical tests reveal that at the bottom heater, the rate of heat transfer has always been higher, increasing as increase in radii ratio while decreasing as increase in the aspect ratio. Desrayaud et al. [7] conducted a comparative experiment to test the sensitivity of natural convection for four open boundary conditions in an asymmetric vertical heated channel and to define a benchmark solution for each of the boundary conditions. Their findings indicate that the return flow takes place at the outlet of channel and has also demonstrated that the change in flow patterns will not substantially change the flow rate of the outlet of the channel. The experiment and numerical analysis on a tall vertical open-ended concentric cylindrical annulus was also performed by Mustafa et al. [8,9,10]. Mohamad et al. [11] performed numerical investigation for the unsteady state, natural convection in the annular cylinders. Time needed for fully charging the storage tank and rate of heat transfer was calculated. It was found that a convection-operated storage tank reduces the thermal charging process time drastically compared with the thermally diffusion charging process. Lee et al. [12] conducted numerical analysis, in his research, unsteady three-dimensional incompressible Navier-Stokes equations are solved to simulate experiments. Unsteady time marching is proposed for a time sweeping analysis of various Rayleigh numbers. The accuracy of the natural convection data of a single horizontal circular tube can be guaranteed when the Rayleigh number based on the tube diameter exceeds 400. The aim of this work is to carry out a numerical study for detailed thermally induced behavior of water flow and also on the effect of partial heating in the open-ended Tall vertical annulus. Dynamic behavior of flow fluid is also analyzed for various fluxes of heats. In this study, the annuli had a radius ratio (outer radius to inner radius) equals to 1.184 and an aspect ratio (length to annular gap) of 352. Taken as a whole, we emphasize quantifying the impact of partial heating on design parameters like pressure distribution and coefficient of heat transfer for a very high aspect ratio and we further aim to identify flow physics as a flow pattern within the annuli.

## 2. Problem Formulation and Method of Solutions

_{1}= 21 (dimensionless height). From Z = Z

_{1}= 21 to Z = Z

_{2}= 300 is the heated region for inner cylinder while Z = Z

_{2}= 300 to Z = A = 352 is non heating zone near the exit for inner cylinder while the outer cylinder assumed adiabatic. In radial directions R

_{i}= r

_{i}/b and R

_{o}= r

_{o}/b.

- (a)
- Inner cylinder ($\mathrm{R}={\mathrm{R}}_{\mathrm{i}}$):
- Full Heating$\mathrm{U}=\mathrm{W}=0\mathrm{and}\frac{\partial \mathsf{\theta}}{\partial \mathrm{R}}=-1\mathrm{for}0\le \mathrm{z}\le \mathrm{A}$ (heating zone)
- Partial Heating$\mathrm{U}=\mathrm{W}=0\mathrm{and}\frac{\partial \mathsf{\theta}}{\partial \mathrm{R}}=-1{\mathrm{for}\mathrm{Z}}_{1}\le \mathrm{Z}\le {\mathrm{Z}}_{2}$ (heating zone)
- $\mathrm{U}=\mathrm{W}=0\mathrm{and}\frac{\partial \mathsf{\theta}}{\partial \mathrm{R}}=0\mathrm{for}0\le \mathrm{Z}{\mathrm{Z}}_{1}{\mathrm{and}\mathrm{Z}}_{2}\mathrm{Z}\le \mathrm{A}$ (non-heating zone)

- (b)
- Outer cylinder ($\mathrm{R}={\mathrm{R}}_{\mathrm{o}}$): $\mathrm{U}=\mathrm{W}=0\mathrm{and}\frac{\partial \mathsf{\theta}}{\partial \mathrm{R}}=0\mathrm{for}0\le \mathrm{Z}\le \mathrm{A}$
- (c)
- Inlet ($\mathrm{Z}=0)$: $\frac{\partial \mathrm{U}}{\partial \mathrm{Z}}=\frac{\partial \mathrm{W}}{\partial \mathrm{Z}}=0\mathrm{and}\mathsf{\theta}=0{\mathrm{for}\mathrm{R}}_{\mathrm{i}}\mathrm{R}{\mathrm{R}}_{\mathrm{o}}$
- (d)
- Outlet ($\mathrm{Z}=\mathrm{A})$: $\frac{\partial \mathrm{U}}{\partial \mathrm{Z}}=\frac{\partial \mathrm{W}}{\partial \mathrm{Z}}=0\mathrm{and}\frac{{\partial}^{2}\mathsf{\theta}}{\partial {\mathrm{Z}}^{2}}=0,{\mathrm{R}}_{\mathrm{i}}\mathrm{R}{\mathrm{R}}_{\mathrm{o}}$.

- (i)
- at inflow and solid walls of the annulus: $\frac{\partial {\mathrm{P}}^{\prime}}{\partial \mathrm{n}}=0$
- (ii)
- at out flow of the annulus: ${\mathrm{P}}^{\prime}=0$,

## 3. Validation of Scheme

## 4. Results and Discussion

#### Effect of Partial Heating on Different Parameter

_{h}which will be at the same point in the absence of any motion that would result in the departure of the pressure field from the hydrostatic variation imposed due to gravity by Gebhart et al. [19]. With buoyancy force and motion, the difference between these two (p − p

_{h)}, is the pressure change that arises through fluid motion. It is due to acceleration, viscous force and buoyancy force. The difference (p − p

_{h)}is called the ‘motion’ pressure field or ‘pressure defect’ by Mohanty and Dubey [20]

_{.}That is the actual static pressure p is decomposed into p

_{h}and p

_{m,}as

_{h}+ p

_{m}(Differential pressure = Hydrostatic pressure + Pressure defect).

## 5. Conclusions

^{3}to 6.61 × 10

^{4}). The study results show the rapid increase in the temperature for any radial location due to conduction having a maxima. As the convection becomes significant, the axial flow is developed and a steady state is observed below the maxima. The flow becomes parallel and fully developed with this eventual steady state which is known as temperature overshoot. The above phenomenon was observed for all Raleigh number. In addition, the convective radial velocity is getting higher by increasing Rayleigh number resulting in a negative effect on the fully developed region and positive effect on the thermal entrance length. The time taken to achieve equilibrium decreased with Rayleigh number due to the positive effect of this number on the buoyancy force. The numerically determined Nusselt number at steady state for fully and partially heated cases comes out to be 3.09 to 3.58 and 3.03 to 3.57, respectively, for Rayleigh number, Ra is 4.4 × 10

^{3}to 4.4 × 10

^{4}. At the beginning of heated wall, local heat transfer coefficient and Nusselt number decrease from very large values and then for the remaining length of annulus it gradually decreases and become constant. The variations in the Nusselt number along the annulus height represent the developing boundary layer at the entrance and fully developed flow in the remaining length. The average Nusselt number increases gradually with Rayleigh number and can be represented in terms of Rayleigh number by the following correlations.

_{a}= 3.049 + 0.00001 Ra − 8 × 10

^{−11}Ra

^{2}Mean deviation = ±3.15% (Fully Heated)

_{a}= 2.986 + 0.00001 Ra − 9 × 10

^{−11}Ra

^{2}Mean deviation = ±3.28% (Partially Heated)

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- De Vahl Davis, G.; Thomas, R.W. Natural convection between concentric vertical cylinder. Phys. Fluids
**1969**, 12, 198–207. [Google Scholar] [CrossRef] - Mochimaru, Y. New way for simulation of transient natural convection heat transfer. J. Fluid Mech.
**1987**, 8, 235–239. [Google Scholar] [CrossRef] - Ho, C.J.; Lin, Y.H. Natural convection heat transfer of cold within an eccentric horizontal cylinder annulus. J. Heat Transf.
**1988**, 110, 894–901. [Google Scholar] [CrossRef] - El-Shaarawi, M.A.I.; Al-Attas, M. Transient Induced Flow through a Vertical Annulus. JSME Int. J. Ser. B
**1993**, 36, 156–165. [Google Scholar] [CrossRef] [Green Version] - El-Shaarawi, M.A.I.; Mokheimer, E.M.A.; Jamal, A. Geometry effects on conjugate natural convection heat transfer in vertical eccentric annuli. Int. J. Numer. Method. Heat Fluid Flow
**2007**, 17, 461–493. [Google Scholar] [CrossRef] - Sankar, M.; Do, Y. Numerical simulation of free convection heat transfer in a vertical annular cavity with discrete heating. Int. Commun. Heat Mass Transf.
**2010**, 37, 600–606. [Google Scholar] [CrossRef] - Desrayaud, G.; Chénier, E.; Joulin, A.; Bastide, A.; Brangeon, B.; Cherif, Y.; Eymard, R.; Garnier, C.; Giroux-Julien, S.; Harnane, Y.; et al. Benchmark solutions for natural convection flows in vertical channels submitted to different open boundary conditions. Int. J. Therm. Sci.
**2013**, 72, 18–33. [Google Scholar] [CrossRef] [Green Version] - Mustafa, J. Experimental and Numerical Analysis of Heat Transfer in a Vertical Annular Thermo-Siphon. Ph.D. Thesis, Aligarh Muslim University, Aligarh, India, 2014. [Google Scholar] [CrossRef]
- Mustafa, J.; Siddiqui, M.A.; Fahad, S. Anwer Experimental and Numerical Analysis of Heat Transfer in a Tall Vertical Concentric Annular Thermo-siphon at Constant heat Flux Condition. Heat Transf. Eng.
**2018**, 40, 896–913. [Google Scholar] [CrossRef] - Mustafa, J.; Husain, S.; Siddiqui, M.A. Experimental Studies on Natural Convection of Water in a Closed Loop Vertical Annulus. Exp. Heat Transf.
**2017**, 30, 25–45. [Google Scholar] [CrossRef] - Mohamad, A.; Taler, J.; Oclon, P. Transient Natural Convection in a Thermally Insulated Annular Cylinder Exposed to a High Temperature from the Inner Radius Energies. Energies
**2020**, 13, 1291. [Google Scholar] [CrossRef] [Green Version] - Lee, J.H.; Shin, J.H.; Chang, S.M.; Min, T. Numerical Analysis on Natural Convection Heat Transfer in a Single Circular Fin-Tube Heat Exchanger (Part 1): Numerical Method. Entropy
**2020**, 22, 363. [Google Scholar] [CrossRef] [Green Version] - Saad, Y. Numerical Methods for Large Eigenvalue Problems; Society for Industrial and Applied Mathematics: Philadephia, PA, USA, 2011. [Google Scholar] [CrossRef]
- Cheng, L.; Armfield, S. A simplified marker and cell method for unsteady flows on non-staggered grids. Int. J. Numer. Methods Fluids
**1995**, 21, 15–34. [Google Scholar] [CrossRef] - Hasan, N.; Anwer, S.F.; Sanghi, S. On the Outflow Boundary Condition for External Incompressible Flows: A New Approach. J. Comput. Phys.
**2005**, 206, 661–683. [Google Scholar] [CrossRef] - Gresho, P.M. Incompressible fluid dynamics: Some fundamental formulation issues. Annu. Rev. Fluid Mech.
**1991**, 23, 413–454. [Google Scholar] [CrossRef] - Kawamura, T.; Takami, H.; Kuwahara, K. Computation of high Reynolds number flow around a circular cylinder with surface roughness. Fluid Dyn. Res.
**1986**, 1, 145–162. [Google Scholar] [CrossRef] - Amine, Z.; Daverat, C.; Xin, S.; Giroux-Julien, S.; Pabiou, H.; Ménézo, C. Natural Convection in a Vertical Open-Ended Channel: Comparison between Experimental and Numerical Results. J. Energy Power Eng.
**2013**, 7, 1265–1276. [Google Scholar] - Gebhart, B.; Jaluria, Y.; Mahajan, R.L.; Sammakia, B. Buoyancy Induced Flows and Transport; Hemisphere Publishing Corporation: Abingdon-on-Thames, UK, 1988. [Google Scholar]
- Mohanty, K.; Dubey, M.R. Buoyancy induced flow and heat transfer through a vertical annulus. Int. J. Heat Mass Transf.
**1996**, 39, 2087–2093. [Google Scholar] [CrossRef]

**Figure 1.**Schematic diagram (

**a**) Plan of the thermo-siphon (

**b**) computational geometry (not drawn to scale).

**Figure 6.**Temperature variation along the radial direction (

**a**) at mid-height with time (

**b**) at different axial length (Ra = 4.4 × 10

^{4}).

**Figure 7.**Variation of Temperature along the axial length with time for partial heated inner wall (

**a**) wall temperature (

**b**) liquid bulk temperature (Ra = 4.4 × 10

^{4}).

**Figure 8.**Contours of temperature profile at (

**a**) the beginning and (

**b**) the end of heating of the annulus.

**Figure 9.**Comparison of (

**a**) thermal boundary layer profile and (

**b**) temperature profile along the axial length for different Rayleigh numbers.

**Figure 10.**Variation of local and average radial velocity (

**a**) at a different radial location (

**b**) along the axial direction with time at mid-height (Ra = 4.4 × 10

^{4}).

**Figure 11.**Variation of radial velocities at different axial lengths along the radial direction for (

**a**) Heating zone (

**b**) Non-heating upper zone. (Ra = 4.4 × 10

^{4}).

**Figure 12.**Contours of radial velocity component for Ra = 4.4 × 10

^{3}at (

**a**) the beginning and (

**b**) heating end.

**Figure 13.**Comparison of radial velocity at (

**a**) the beginning and (

**b**) end of the inner heating wall of the annulus at different Rayleigh Number.

**Figure 14.**Variation of axial velocity at different axial positions along the radial length (

**a**) heating zone (

**b**) non-heating upper. (Ra = 4.4 × 10

^{4}).

**Figure 15.**Axial velocity contours at (

**a**) the beginning and (

**b**) the end of the annulus inner heated wall.

**Figure 16.**Pressure defect variation along the axial length over time (

**a**) fully heated annulus (

**b**) partial heated annulus (Ra = 4.4 × 10

^{4}).

**Figure 17.**Comparison of (

**a**) temperature (

**b**) Radial velocity variations at mid-height along the radial direction for partial and full heated annulus for different Ra.

**Figure 18.**Comparison of (

**a**) axial velocity variation at mid-height along the radial length (

**b**) pressure defect variation along the axial length for partial and full heated annulus for different Ra.

**Figure 19.**Variation of Local Nusselt number for different Ra over the axial length (

**a**) full (

**b**) partially heated.

**Figure 20.**Variation of average Nusselt number at different Ra for the fully and partially heated annulus.

**Figure 21.**Variation of mass flow rate with time (

**a**) at different axial locations (

**b**) for fully and partial hated annulus for different Rayleigh numbers.

b | Annular space (m) | L | Length of annulus (m) |

T_{a} | Ambient temperature (°C) | g | Gravitational force (m · s^{−2}) |

k | Thermal conductivity (W · m^{−1} · °C^{−1}) | q | Heat flux (W/m^{2}) |

r, R | Dimensional (m) and Non-dimensional Radial distance | α | Thermal diffusion coefficient (m^{2}/s) |

β | expansion coefficient (K^{−1}) | μ | Dynamic viscosity (N · s/m^{2}) |

ν | Kinematic Viscosity (m^{2} · s^{−1}) | C_{p} | =Specific heat (J · kg^{−1} · °C^{−1}) |

u, U | Dimensional and Non-dimensional radial velocity | w, W | Dimensional and Non-dimensional axial velocity |

z, Z | Dimensional (m) and Non-dimensional axial distance | T, θ | Dimensional and Non-dimensional temperature |

p, P | Dimensional and Non-dimensional pressure | Nu | Nusselt Number (hb/k) |

A | Aspect ratio (l⁄b) | t, τ | Dimensional and Non-dimensional time |

∂ | discrete | δ | change |

ε | computational space | n | Time level |

* | Predicted value | w | Wall |

i | Inner | o | Outer |

q,q_{w} | Heat flux (W/m^{2}) | $\in $ | iteration-error at current time-level |

ρ | Density (kg · m^{−3}) | i, j | i-th and j-th coordinate direction |

RR | Radius ratio | l | liquid |

Nu_{a} | Mean Deviation | |
---|---|---|

Fully heated | 3.049 + 0.00001Ra − 8 × 10^{−11} Ra^{2} | ±3.15% |

Partially heated | 2.986 + 0.00001Ra − 9 × 10^{−11} Ra^{2} | ±3.28% |

**Table 3.**Comparison of Thermal entrance length, average Nusselt number and axial bulk velocity for fully and partially heated case.

Ra | Thermal Entrance Length (Dimensionless) | Average Nusselt Number | ||
---|---|---|---|---|

Fully Heated | Partially Heated | Fully Heated | Partially Heated | |

4.4 × 10^{3} | 16.75 | 14 | 3.09 | 3.03 |

1.1 × 10^{4} | 27 | 23 | 3.19 | 3.14 |

2.2 × 10^{4} | 37.25 | 33.5 | 3.31 | 3.27 |

3.3 × 10^{4} | 45.85 | 41.5 | 3.39 | 3.37 |

4.4 × 10^{4} | 53.65 | 48.5 | 3.46 | 3.44 |

5.5 × 10^{4} | 60.75 | 54.5 | 3.53 | 3.51 |

6.61 × 10^{4} | 66.65 | 60 | 3.58 | 3.57 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mustafa, J.; Alqaed, S.; Siddiqui, M.A.
Thermally Driven Flow of Water in Partially Heated Tall Vertical Concentric Annulus. *Entropy* **2020**, *22*, 1189.
https://doi.org/10.3390/e22101189

**AMA Style**

Mustafa J, Alqaed S, Siddiqui MA.
Thermally Driven Flow of Water in Partially Heated Tall Vertical Concentric Annulus. *Entropy*. 2020; 22(10):1189.
https://doi.org/10.3390/e22101189

**Chicago/Turabian Style**

Mustafa, Jawed, Saeed Alqaed, and Mohammad Altamush Siddiqui.
2020. "Thermally Driven Flow of Water in Partially Heated Tall Vertical Concentric Annulus" *Entropy* 22, no. 10: 1189.
https://doi.org/10.3390/e22101189