1

Experimental Evaluation of Shape Factor of Axis Symmetric Sunken

Structures

Dhananjay R. Mishra

Department of Mechanical Engineering, Jaypee University of Engineering & Technology,

A.B. Road, Guna-473226, Madhya Pradesh (India)

Abstract:

This paper presents the dependence of a shape factor for the fully sunken axis

symmetrical structures (viz. cubical, square prismatic, pyramidal, and cylindrical)

corresponding to the depth and their orientation. Experimental evaluations of the shape factor

on reduce scale models are carried out in laboratory using thermal simulation method for

different sets of conditions. The method has been used to determine shape factor, which can

be used to determine heat loss from ground to structure or structure to groud fully sunken

with the different orientation. Maximum and minimum value of shape factor for set-I and II

condition are recoded as 90.18 and 9.93 respectively. In set –III it will varies from 16.49 to

35.28. At 2D L shape factor of set-VI leads by 17.26 % as compared to set VII. Where as

set- IX leads by 33.47% as compaired to set VIII. It would help for designing building

structure of fully buried nature for creating thermal comfort.

Keywords: Thermal simulation; heat transfer; buried structure.

Symbols

D Depth of model below earth surface, m

F Shape factor of cubical structure placed about its lateral surface on horizontal plane,

set I condition (dimensionless)

'F Shape factor of cubical structure placed about its edge and lateral surfaces are equally

inclined to horizontal surface, set II condition (dimensionless)

''F Shape factor of cubical placed on horizontal surface about its corner in such way so

that all edges are equally inclined to it, set III condition (dimensionless)

IVF Shape factor of square prismatic structure placed about its lateral surface in Set IV

condition (dimensionless)

VF Shape factor of square prismatic structure placed on horizontal plane about its one of

the edge of the lateral surface in Set V condition (dimensionless)

VIF Shape factor triangular pyramid structure placed about its base on horizontal plane, set

VI condition (dimensionless)

Corresponding Author, mobile number: +91 9893808251,

E-mail address: dhananjay.mishra@juet.ac.in; dm30680@gmail.com (Dhananjay R. Mishra).

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VIIF Shape factor square pyramid structure placed about its base on horizontal plane, set VI

condition (dimensionless)

VIIIF Shape factor of cylindrical structure placed about its end surface on horizontal plane,

set VIII condition (dimensionless)

IXF Shape factor of cylindrical structure placed about its lateral surface on horizontal plane,

set IX condition (dimensionless)

I Current on heating bulb or element, A

K Thermal conductivity of ground or sand, W m C

L Characteristic length of object, m

Q

Rate of heat loss from building to ground, W

r Radius of the sphere, m

sT Temperature of surface of building or model,

C

gT Temperature of the earth surface, C

V Voltage across the heating element or bulb, V

1cosh D R

1. Introduction

An evaluation of heat transfer between the ground and an earth coupled structure is

essential for a rational thermal design of the latter for creating human comfort. The solution of

the corresponding three dimensional Fourier equation of thermal conduction with relevant

boundary conditions is in general very difficult if at all possible; the analytical solutions of

highly symmetrical structure can be obtained only for structures with high symmetry like a

sphere or an infinite cylinder with horizontal axis [1,2]. Evaluation of the heat transfer through

the building structure to surrounding ground is needed three-dimensional heat conduction

relation with the appropriate boundary conditions for the computation of heat transfer between

structure and earth surface. Numerical methods for few axis symmetrical (like cylinder and

sphere) sunken structures are available but it needs large computational time [3,4]. Effect of

different earth surface treatment of surrounding surface for heating and cooling of earth-

integrated building structures are reported by Sodha and his associates [5,6]. The theoretical

basis for dynamic simulation (arbiter time variation) heat transfer between ground and

structure using electrical simulation for fully buried structures of periodic nature has been

reported by Sodha. These suggested methods are needed to reduce scale models for simulation

purpose which can further scale up for bigger realistic situation [7]. Mishra et al.[8] have

been validated the basis for the experimental simulation of heat transfer between ground and

structure proposed by Sodha [9,10]. Geometical optimization of heat storage unit using shape

factor of structure has been reported by Solé et al. [11]. Experimental evaluation of heat

transfer between full scale fully or partially buried structures and ground is not feasible on

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account of the corresponding high cost; moreover, the large variations in size and shape of the

structure make a specific experiment to have little relevance with the real problem. wall-to-

excavation shape factor concept use for the preliminary design of deep cement mixing walls

for identification of cause of excavation failure has been reported by Waichita et al. [12].

Influence of dam geometry and satellment of rock fill dams based on shape fator unding finite

element analysis has been reported by Sukkarak et al. [13]. Dependence of berming;

corresponding dependence on the slope of berming has been reported by Sodha and Mishra

[14]. CFD simulation and analysis of the temperature distribution within the green houe, solar

heat gain and heat loss using sing shape factor has been reported by Tong et al. [15]. Shape

fators of three different desgine and its optimization using RSM and validation with P test for

the steady state heat transfer between swimming pool water and surrounding ground has been

reported by Somwanshi et al. [16]. Utilistion of shape factor of device/ equipement for the

analysis of heat transfer from surface to ambient or ambient to surface have been reported by

various researchers [17–22]. Evaluation of heat transfer from fully sunken structure to the

surrounding ground or surrounding ground to sunken structure is a comples phenomena.

Although fully sunken building structure is a popular method for degine of bunkers and

partially sunken for heigh rise buildings, one has to adopt approximation of doubtful merits for

the evaluation of heat transfer from fully sunken structure to the surrounding ground or

surrounding ground to sunken structure. From the brief account neumerical simulation is

difficult and highly time consuming for realistic cases. This brigs us possibility of thermal

simulation experiment, which would enables one to evaluate heat transfer from fully sunken

structures to the ground on resuce scale structure in small time duration. The shape factor of

fully buried axis symmetrical structures using thermal simulation method is presented in this

paper. Dependence of shape factor corresponding to depth and orientation are reported in this

paper.

2. Experimental setup

A reduced scale model i.e. dry fine silica sand in a wooden box of a dimensions 1×1×1 m

(Fig.1a) treated as a semi-infinite medium. Hollow sphere made of the copper material having

98% purity (chosen on account of large thermal conductivity) is energized with the

incandescent lamp used for evaluation of the thermal conductivity of silica sand. A constant

DC power source of 24V ,

The symmetrical standard structures are modeled (made of a copper sheet of thickness 1.5

mm) viz. Hollow copper sphere for determination of the thermal conductivity of the silica

sand.

Fig.1a

An incandescent lamp is placed inside the box in all the cases and the steady state

(normally after 8–10 h) difference of temperature between the surface of the structures and the

simulated earth surface was noted (by a temperature indicator DTI-039T) corresponding to

eight values of the power consumption (product of the current and potential difference) by the

lamp.

Fig.1b

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Actual photograph and schematic arrangement of the experimental setup are illustrated in

Figs. 1a and 1b respectively. Shape factor is a parameter used for the prediction of the thermal

behavior of sunken structures of symmetrical in nature with different depth and orientation.

Figs.2 and 3 represents a photograph of the cubical structure of 10 side and square prism of

base side 5 and axis 10 long. Figs. 4a and 4b represent the photograph of the triangular and

square pyramid of base side 5cm and axis 10 cm long. A photograph of the cylindrical

structure of 5 base diameter and 10 axis length is shown in Fig. 5. The rate of heat transfer

from a sphere of a radius at a depth of earth surface and shape factor of the hollow sphere of

0.015 radii has been used as a simulation structure[13,14]. As heat transfer through the

structure can be evaluated with the Eq.1.

' s gQ F rK D r T T

(1)

where, 3 51 sinh sinh

' 4 1 cosh 1 cosh3 1 cosh5 ...2 sinh3 sinh5

F e e e

(2)

Rearranging Eq.1 we can get

' s gK Q rF D r T T

(3)

A different set of experimentations are carried out for the determination of the shape factor of

structures for their different orientations.

Fig.2

Fig.3

Fig.4a

Fig.4b

Fig.5

3. Observation, result and discussion

The thermal conductivity K of sand is in general dependent on the temperature; however,

within the range of parameters of interest, the variation is less than the accuracy of

determination of K . In any case the experimental simulation does not allow for the

temperature-dependent thermal conductivity. The thermal conductivity of simulating media

(sand) has been experimentally evaluated using thermal simulation method and tabulated in

Table 1.

Table 1

On the basis of six continuous observations and using Eq.2, it has been recorded as 0.07

0.060.62 W mK Variation of the shape factor of a hollow copper cube of 10cm long side

with response to different depth and orientations of the structure are reported with the help of a

different set of the condition in

5

Fig.6. Three different sets of conditions for cubical structure is used to determination of

the shape factor, maximum value of shape factor is recorded 90.18 at D L ratio 2 for set I of

a cubical structure place on horizontal surface and minimum value of 'F is 9.93 is recorded

for set II cubical structure placed in HP about its one of the corner and its edges are equally

inclined to HP at D L ratio 10. ''F is varied from 35.28 to 16.49 while the experimentation

for set III when the cubical structure is placed on edge and its lateral surfaces are equally

inclined to HP and end surfaces are perpendicular to the HP.

Fig.6

Shape factor cubical structure for Set I, II and III conditions one can get from Eq. 4,6 and 8

respectively :

2

0.1971 5.0624 48.927F D L D L (4)

2 0.99R (5)

2

' 0.1866 3.4513 26.168F D L D L (6)

2 0.9959R (7)

2

'' 0.3552 6.469 46.11F D L D L (8)

2 0.981R

The shape factor of the square prism of base side 5 cm and axis 10 cm long set IV and V

square prismatic structure placed about its lateral surface and edge of lateral surface and axis

kept parallel to the horizontal surface respectively. Experimental result for the evaluation of

shape factor is shown in Fig. 7.

Fig.7

Experimental results for the fully sunken structure at different D L ratio and orientation

are shown in Fig.8. Maximum 30.32 value of shape factor for IV condition is recorded while

experimentation at a D L ratio of 2 and a minimum value of shape factor 8.73 at a D L ratio

10 in set V condition. Shape factor at D L ratio 10 for set IV is 15.32% higher as compared to

set V. It indicates maximum comfort within the underground structure of square prismatic

structure in the fully buried case with this relation and orientation. Eq. 9 and 10 can be utlised

for the theoretical evaluation of shape factor for the IV and V set of conditions respectively.

2

0.058 3.2779 37.086IVF D L D L (09)

2 0.9924R (10)

2

0.1204 4.073 37.24VF D L D L (11)

6

2 0.9867R (12)

Thermal simulation result of shape factor for the triangular and square pyramid is shown in

Fig.8 as a set VI and VII, when its base placed on the horizontal surface and an axis

perpendicular to it.

Fig.8

24.35 is a maximum value of shape factor recorded in set VI. Best comfort in comparison of

triangular and square pyramid shape structure can be achieved in case of a square pyramid (set

VII) as it has 3.49 at 10D L . At a 2D L shape factor for set VI leading by 17.26% and

it maintains it throughout the thermal simulation experimentation. Maximum deviation is

observed at 2D L in set VI 52.68% as compared to the set VII. Eq. 13 and 15 are the

empirical relations obtained from the experimental result can be utilized for the neumerial

evaluation of shape factor of the structure for setVI and VII condition respectively.

2

0.1204 3.5433 26.618VIF D L D L (13)

2 0.9839R (14)

2

0.0896 3.1957 29.954VIIF D L D L (15)

2 0.9912R (16)

Experimental result for the cylindrical structure is shown in Fig. 9 for two different set of

conditions named as set VIII and IX.

Fig.9

Set VIII indicated that the cylindrical structure kept about its end surface on horizontal plan

and set IX resembles the condition when it is placed on its lateral surface on the horizontal

plan at different D L ratio. At a ratio of D/L= 2 shape factor of set IX is 33.47% higher as

compared to set VIII condition. Set IX condition has given us higher shape factor throughout

the experimentation as compare to the set VIII. It indicates that set VIII condition is optimal

for creating thermal comfort in the fully buried structure of cylindrical shape. The minimum

value of 3.52 is recorded for it at a ratio of 10. For set VIII and IX condition Eq.17 and 19

can be utlised for the evaluation of shape factor of the structure respectively.

2

0.0633 1.4454 11.644VIIIF D L D L (17)

2 0.9953R (18)

2

0.116 2.5123 18.099IXF D L D L (19)

7

2 0.9926R (20)

4. Conclusion

The shape factor of fully buried axis symmetric structure depends on D/L ratio. It is an

essential input parameter for the evaluation of thermal performance of fully sunken structure.

By knowing shape factor of the fully buried structure heat transfer from ground to structure

or structure to ground easily evaluated. It would help for designing building structure of fully

buried nature for creating thermal comfort.

REFERENCES

[1] Shelton, J.A.Y. "Underground storage of heat in solar heating systems", Solar Energy,

17, pp. 137–143(1975).

[2] Mishra, D.R. and Tiwari, A.K. "Sunken effect on building structures", I-Manager's

Journal on Mechanical Engineering, 2, pp. 41–45 (2008).

[3] Givoni, B. "Underground longterm storage of solar energy—An overview", Solar

Energy, 19, pp. 617–623 (1977).

[4] Patil, K., Srivastava, V. and Baqersad, J. "A multi-view optical technique to obtain

mode shapes of structures", Measurement: Journal of the International Measurement

Confederation, 122, pp. 358–367 (2018).

[5] Deshmukh, M.K., Sodha, M.S. and Sawhney R.L. "Effect of depth of sinking on

thermal performance of partially underground building", International Journal of

Energy Research, 15, pp. 391–403 (1991).

[6] Martinopoulos, G. "Solar Energy in Buildings: Reference Module in Earth Systems

and Environmental Sciences", 2016.

[7] Sodha, M.S., Sawhney, R.L. and Jayashankar, B.C. "Estimation of steady state ground

losses from earth coupled structures by simulation", International Journal of Energy

Research, 14, pp. 563–571 (1990).

[8] Mishra, D.R., Sodha, M.S. and Tiwari, A.K. "Validation of the basis of experimental

simulation of heat transfer between a building and surrounding earth", SESI Journal ,

21, pp. 36–48 (2013).

[9] Sodha, M.S. "Simulation of periodic heat transfer between ground and underground

structures", International Journal of Energy Research, 25, pp. 689–693 (2001).

[10] Sodha, M.S. "Simulation of dynamic heat transfer between ground and underground

structures", International Journal of Energy Research, 25, pp. 1391–1394 (2001).

8

[11] Solé, A., Falcoz, Q., Cabeza, L.F. and Neveu, P. "Geometry optimization of a heat

storage system for concentrated solar power plants (CSP)", Renewable Energy, 123 ,

pp. e95 (2018).

[12] Waichita, S., Jongpradist, P. and Jamsawang, P. "Characterization of deep cement

mixing wall behavior using wall-to-excavation shape factor", Tunnelling and

Underground Space Technology, 83 pp. 243–253 (2019).

[13] Sukkarak, R., Jongpradist, P. and Pramthawee, P. "A modified valley shape factor for

the estimation of rockfill dam settlement”, Computers and Geotechnics, 108, pp. 244–

256 (2019).

[14] Sodha, M.S. and Mishra, D.R. "Shape factor for bermed wall", Heat Mass Transf. Und

Stoffuebertragung, 47, pp. 1143–1146 (2011).

[15] Tong, G., Christopher, D.M. and Zhang G. "New insights on span selection for

Chinese solar greenhouses using CFD analyses", Computers and Electronics in

Agriculture, 149, pp. 3-15 (2017).

[16] Somwanshi, A., Dixit, A. and Tiwari, A.K. "Shape factor for steady state heat transfer

between swimming pool water and wsurrounding ground", Fundamentals of

Renewable Energy and Applications, 4, pp. 2–5 (2013).

[17] El-samadony, Y.A.F.A.F., El-maghlany, W.M. and Kabeel, A.E.E. Influence of glass

cover inclination angle on radiation heat transfer rate within stepped solar still, DES.

384, pp. 68–77 (2016).

[18] Ruivo, C.R. and Vaz, D.C. "Prediction of the heat gain of external walls: An

innovative approach for full-featured excitations based on the simplified method of

Mackey-and-Wright", Applied Energy, 155, pp. 378–392 (2015).

[19] Amarasinghe Vithanage D., Devižis, A., Abramavičius, V., Infahsaeng, Y.,

Abramavičius, D., MacKenzie, R.C.I., Keivanidis, P.E., Yartsev, A., Hertel, D. ,

Nelson, J., Sundström, V. and Gulbinas, V. "Visualizing charge separation in bulk

heterojunction organic solar cells", Nature communications, 4, pp. 23-34 (2013).

[20] Boulton, C., Dedekorkut-Howes, A. and Byrne, J. "Factors shaping urban greenspace

provision: A systematic review of the literature", Landscape and Urban Planning, 178,

pp. 82–101 (2018).

[21] Kiwan, S. and Khammash, A.L. "Investigations into the spiral distribution of the

heliostat field in solar central tower system", Solar Energy, 164, pp. 25–37 (2018).

[22] Chel, A., Tiwari, G.N. and Singh, H.N. "A modified model for estimation of daylight

factor for skylight integrated with dome roof structure of mud-house in New Delhi

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(India)", Applied Energy, 87, pp. 3037–3050 (2010).

Figure captions

Fig. 1a: Actual photograph of the experimental setup

Fig. 1b: Schematic diagram for determining thermal conductivity of sand

Fig. 2: Photograph of cubical structure

Fig. 3: Photograph of square prismatic structure

Fig. 4a: Photograph of triangular pyramid structure model

Fig. 4b: Photograph of square pyramid structure model

Fig. 5: Photograph of cylindrical model

Fig. 6: Variation of shape factor with respect to different depth for Set I, II and III of a cubical

structure

Fig. 7: Variation of shape factor with the depth of square prism.

Fig. 8: Variation of shape factor with the depth of triangular (set VI) and square (set VII)

pyramids.

Fig. 9: Variation of shape factor with the depth of cylinder placed on its end surface (set VIII)

and placed on its lateral surface (set IX).

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Table Caption

Table 1: Determination of thermal conductivity of sand

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Figures

Fig.1a: Actual photograph of the experimental setup

Fig.1b: Schematic diagram for determining thermal conductivity of sand

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Fig.2: Photograph of cubical structure

Fig.3: Photograph of square prismatic structure

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Fig.4a: Photograph of triangular pyramid structure model

Fig.4b: Photograph of square pyramid structure model

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Fig.5: Photograph of cylindrical model

Fig.6: Variation of shape factor with respect to different depth for Set I, II and III of a cubical

structure

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Fig.7: Variation of shape factor with the depth of square prism.

Fig.8: Variation of shape factor with the depth of triangular (set VI) and square (set VII)

pyramids.

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Fig.9: Variation of shape factor with the depth of cylinder placed on its end surface (set

VIII) and placed on its lateral surface (set IX).

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Table

Table 1: Determination of thermal conductivity of sand

Sr. No. D/r F V (volts) I(amp.) sT C

gT C K W mK

01 1 16.86 12.01 0.32 62 40 0.69

02 2 15.31 12.01 0.34 64 36 0.64

03 3 14.46 12.03 0.35 65 32 0.59

04 4 13.94 12.06 0.37 68 30 0.56

05 5 13.69 12.02 0.37 66 29 0.59

06 6 13.60 12.61 0.42 69 28 0.63

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Dr. Dhananjay R. Mishra is an Assistant Professor Senior (SG) in Department of

Mechanical Engineering at Jaypee University of Engineering & Technology, Guna. He

received his Ph.D. in Mechanical Engineering from National Institute of Technology Raipur,

India in 2016. He has published more than 50 research papers in reputed peer reviewed

national and international journal. He is supervising four students, leading to the Ph.D. degree

from Jaypee University of Engineering & Technology and National Institute of Technology

Raipur. He is associated as an editor to ‘International Journal of Thermodynamics &

Catalysis’ , associate editor of International ‘Journal of Applied Research’ and editorial board

member of nine other peer reviewed international Journals. He has an association with the

Suprabha Industries Ltd., Lucknow, as an Assistant Production Manager, During 2002 -2004,

as a Lecturer in Mechanical Engineering Department, Rungta College of Engineering and

Technology, Bhilai, C.G. (During Sept. 2005 to March 2006), as a Lecturer in Mechanical

Engineering Department, Shri Shankaracharya College of Engineering and Technology,

Bhilai, C.G.(During March 2006 to July 2007) as an Assistant Professor in Department of

Mechanical Engineering at Disha Institute of Management & Technology and also carried out

responsibility of Academic Administrator (July,2007 to July,2012). Since July 2012, he is

associated as an Assistant Professor (SG) in Mechanical Engineering Department, JUET,

Guna (M.P.). He has established State Level Energy Park at DIMAT funded by MNRE

through Nodal agency CREDA. He has also completed a research project titled as

“Determination of the parameters affecting heat transfer between ground and fully or partially

underground structures” sponsored by Chhattisgarh Council of Science and Technology at

Disha Institute of Management & Technology, Chhattisgarh.