MIET1081 - Advanced Thermofluids: Conduction Theory & Shape Factors

This report delves into the concept of shape factors within advanced thermofluids mechanics, providing a theoretical foundation and exploring a real-life application through analysis. It begins by defining the shape factor and its relationship to heat transfer, thermal resistance, and dimensionless quantities. The report presents fundamental equations for heat exchange and discusses their derivation. A practical application is then examined: heat loss from buried steam pipes, including a detailed analysis using given parameters such as pipe length, diameter, burial depth, and soil thermal conductivity. The analysis includes assumptions made, such as steady operating conditions and two-dimensional heat transfer, to calculate the heat loss rate from the pipe. The report concludes by referencing relevant studies on thermal performance, heat conduction, and numerical analysis of heat transfer problems. Desklib provides students access to past papers and solved assignments.

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ADVANCED THERMOFLUIDS MECHANICS –SHAPE FACTOR
Assignment A: Conduction Theory and Real Life Application and Analysis
Name
Class (Course)
Professor (Tutor)
School (University)
The City and State
The Date

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Theory
Shape factor alludes to a value that is influenced by an article's shape however is free
of its measurements. Yovanovich postulated the characteristics of S, the conduction Shape
Factor for a confined three dimensional shaped body dependent on a boundary’s basic
gradient of temperature slope along an outward-facing perpendicular, n projected from the
surface of the body:
For the specific issue of the 3-D fenced in area framed between a warmed internal and cooled
external limit, the boundary is assessed at the inward surface, Ai, where the dimensionless
temperature increment is characterized as:
The general scale length is utilized in making the shape factor dimensionless as shown:
The the total rate of heat transfer, Q and thermal resistance, R relates to shape factor as
follows:
Where the area of the inside surface is choosen as the feature length and is raised by the
power of half, and therefore:

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The figure below shows the various notations used for indicating dimensions of the diffent
shapes of inside bodies
Comparative documentation is utilized for the elements of the external limit shapes.
Hollands and Hassani came up with the conduction shape factor framework for the
dimensionless quantity, S (shape factor) for three dimensional walled in area hole with a
regular hole dividing:
All shapes of bodies are approximated using the sphere’ S∞:
And using the foundations and arguments on 1D heat transfer, the approximation of S0 can
be made:
where δ denotes the least spacing of the gap

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Equations
An essential class of heat exchange issues for which straightforward arrangements are gotten
incorporates those which include two surfaces kept up at steady temperature T1 and T2. The
heat exchange from one surface at T1 to the next surface at T2 can be denoted as q=sk (T1-
T2) where s represents shape factor and k represents heat conductivity of the material
 Shape factor can be identified with resistance to heat obstruction:
q=sk ( T 1−T 2 )=T 1−T 2
sk = T 1−T 2
Rt where Rt=1 /sk
 One dimensional heat exchange can utilize shape factor moreover. For example, heat
exchange inside a plane mass of thickness L is q=kA ( ∆ T
L ,∧s= A /L). This considers
heat movement in straightforward geometries, for example, vast plane dividers since
heat moves in such geometry can be approximated as one dimensional, and basic
explanatory arrangements can be gotten effectively.
Real-life applications
Shape factor is used in insulation to reduce heat transfer and save energy which directly
translates to money. The choices on the appropriate measure of protection depend on warmth
exchange investigation which is trailed by financial examination to decide the money related
estimation of the misfortune. This is widely applied in calculating the heat loss through walls
in winter. The shape factor is also widely applied in the calculation of heat loss from buried
steam pipes. The steam pipes usually convey steam for industrial heating or power

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generation. Another application is the calculation of heat exchange between heated and cool
fluid pipes.
Analysis
For our analysis, we are going to focus on the heat toss from buried steam pipes.
Consider a pipe which is 30m long having a diameter of 0.1m conveying heated water for a
county supply steam chemical plant. The pipe is sunk 0.5m into the soil. The outer ground
level temperature of the earth is 10-degree celsius and the pipe outer-wall surface temperature
is 80-degree celsius, we are tasked to determine the heat loss rate from the pipe taking the
soil’s heat conductivity to be 0.9W/Mk
The assumption in this case is:
 The soil’s heat conductivity is invariant
 There are steady operating conditions existing
 The heat transfer is 2-D, and there exists no changes along the axis change in
the axial directions,
The shape factor is given by
s= 2 πl
ln ( 4 z
D ) since z is greater than one and half times the diameter of the pipe D,
where z denotes the depth to which the pipe is sunk. Substituting:
s= 2 π × 30
ln (4 × 50
10 )
=62.9 m
The constant heat transfer hence, q=sk ( T 1−T 2 ) = 62.9× 0.9× (80-10)

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= 3963W
This quantity of thermal energy is conducted to the ground level of the soil from
the outer walls of the pipe, and is lost to the atmosphere either through radiation or
convection.

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References
Duquette, J., Rowe, A. and Wild, P., 2016. Thermal performance of a steady state physical
pipe model for simulating district heating grids with variable flow. Applied energy, 178,
pp.383-393.
Ge, Y., Zhang, Y., Weaver, J.M. and Dobson, P.S., 2017. Dimension-and shape-dependent
thermal transport in nano-patterned thin films investigated by scanning thermal microscopy.
Nanotechnology, 28(48), p.485706.
Kakaç, S., Yener, Y. and Naveira-Cotta, C.P., 2018. Heat conduction. CRC Press.
Rana, S.K., 2016. A study of two dimensional heat conduction in exchanger tubes of non-
circular cross-section (Doctoral dissertation).
Zamolo, R. and Nobile, E., 2017, November. Numerical analysis of heat conduction
problems on 3D general-shaped domains by means of a RBF Collocation Meshless Method.
In Journal of Physics: Conference Series (Vol. 923, No. 1, p. 012034). IOP Publishing.