Steady-State Validation of Advective Bar Elements Implemented in the Aria Thermal Response Code

SANDIA REPORT
SAND2017-1341
Unlimited Release
Printed February 2017
Steady-State Validation of Advective Bar
Elements Implemented in the Aria
Thermal Response Code
Brantley H. Mills and Oscar W. Deng
Prepared by
Sandia National Laboratories
Albuquerque, New Mexico 87185 and Livermore, California 94550
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ii
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Validation of Advective Bar Elements
iii
SAND2017-1341
Unlimited Release
Printed February 2017
Steady-State Validation of Advective Bar Elements
Implemented in the Aria Thermal Response Code
Brantley H. Mills and Oscar W. Deng
Thermal Sciences and Engineering Department
Sandia National Laboratories
P.O. Box 5800
Albuquerque, NM 87185-MS0825
Abstract
A validation effort has been performed for the advective bar element model
implemented into the Aria thermal response code used to model convection
heat transfer. The goal of this effort is to provide credibility to the model for its
use in high-consequence thermal analyses at Sandia National Laboratories. In
this report, steady-state experiments reported in the literature with sufficient
documentation about the experimental setup and measurements were used to
validate the model. Models of the experiments using advective bar elements
were created for three relevant geometries including: pipe flow, annular flow,
and flow through a helical pipe. Overall, all three geometries agreed well with
the experimental measurements within conservative estimates for the
experimental and numerical uncertainty providing confidence in the use of the
advective bar element model in more complex and consequential thermal
models.

iv
Acknowledgements
We would like to thank Victor Brunini and Sam Subia for their
assistance with resolving the issues encountered running the advective bar
element model in Aria. In addition, we would like to extend our thanks to Nick
Francis and Dean Dobranich for their extremely valuable comments and
feedback on this work.

v
Contents
Introduction.............................................................................................. 1
Pipe Flow.................................................................................................. 5
Experimental Setup and Advective Bar Model............................................................ 5
Experimental Comparison .......................................................................................... 9
Annular Flow.......................................................................................... 15
Experimental Setup and Advective Bar Model.......................................................... 15
Experimental Comparison ........................................................................................ 19
Helical Pipe Flow ................................................................................... 23
Experimental Setup and Advective Bar Model.......................................................... 23
Experimental Comparison ........................................................................................ 28
Recommendations................................................................................. 37
Summary ................................................................................................ 39
Appendix A—Material Properties ......................................................... 40

vi
Figures
Figure 1. Schematic of the experimental setup used in Reference 9 ............... 6
Figure 2. Mesh used in the advective bar model for the experiments in
Reference 9................................................................................... 7
Figure 3. Outlet temperatures from the model compared to experimental
measurements for Cases 1-5....................................................... 10
Figure 4. Average wall temperatures from the model compared to
experimental measurements for Cases 1-5.................................. 10
Figure 5. Average Nu from the model compared to experimental
Figure 7. Average wall temperatures from the model compared to
experimental measurements for Cases 6-9.................................. 13
Figure 8. Average Nu from the model compared to experimental
Figure 9. Schematic of the experimental setup used in Miller et al. (a) and a
more detailed schematic of the annular test section (b) (Reference
16).............................................................................................. 16
Reference 16............................................................................... 18
measurements from Miller et al................................................... 20
Figure 12. Inner wall temperatures from the model compared to experimental
Figure 13. Nusselt numbers from the model compared to experimental
Figure 14. Schematic layout of experimental setup used in Reference 18...... 24
Figure 15. Picture of coils used in the experiment (Reference 18).................. 24
Reference 18............................................................................... 26
measurements for constant temperature conditions.................... 30
Figure 18. Average outer wall temperatures from the model compared to
experimental measurements for constant temperature conditions31
Figure 19. Average inner heat transfer coefficient from the model compared to
Figure 20. Average inner Nusselt number from the model compared to
Figure 21. Average overall heat transfer coefficient from the model compared
to experimental measurements for constant temperature
conditions................................................................................... 32
Figure 22. Nusselt numbers at various flow rates from the model compared to
experimental measurements for both constant temperature

vii
(isothermal) and constant heat flux (non-isothermal) cases with
water as the test fluid ................................................................. 33
experimental measurements for 20% glycerol in non-isothermal
conditions................................................................................... 34
Figure 24. Overall heat transfer coefficient at various flow rates from the
model compared to experimental measurements for water in
isothermal conditions ................................................................. 35
Figure 25. Overall heat transfer coefficient at various flow rates from the
model compared to experimental measurements for 10% glycerol
in isothermal conditions ............................................................. 35

viii
Tables
Table 1. Summary of Sampled Experiments from Kolář (Reference 9).............. 7
Table 2. Summary of Sampled Experiments from Miller et al. (Reference 16) 17
Table 3. Physical Dimensions of Helical Coil-I (Reference 18)........................ 25
Table 4. Summary of Experimental Conditions for Coil-I (Reference 18)........ 25
Table 5. Typical Boundary Conditions used for Simulations (Reference 19)... 27
Table 6. Material Properties for Air at 1 atm (Reference 11)........................... 40
Table 7. Material Properties for Subcooled Water at 1 atm............................ 40
Table 8. Material Properties for Brass (Reference 11) .................................... 41
Table 9. Material Properties for 10% Glycerol ............................................... 41
Table 10. Material Properties for 20% Glycerol ............................................. 42
Table 11. Material Properties for Mild Steel .................................................. 42

Introduction
1
Introduction
The Aria thermal response code developed at Sandia National Laboratories
is a finite element multi-mechanics module based on the Sierra Mechanics
framework for solving coupled PDEs. Among the many modeling capabilities
within Aria is a reduced-order model for the advection of energy associated
with fluid flow coupled convectively to a conductive solid body. In this
conjugate heat transfer model, fluids flowing internally through solid volumes
are represented using 1D bar elements that are thermally coupled to the
surrounding 3D solid using empirical heat transfer coefficient (HTC)
correlations. The primary advantage of this method is that relevant fluid flows
can be integrated in very large 3D thermal models without a significant
increase in setup or computation time.
This 1D fluid flow model within Aria, referred to as the ‘advective bar’
model here for brevity since it incorporates advection in the energy equation for
the bar elements, has already seen significant use at SNL.1,2,3 This emphasizes
the need to properly verify and validate the new model’s implementation into
Aria particularly with its use in high-consequence thermal models. The work
described in this report aims to support that effort through validation of the
advective bar model where previous efforts have focused primarily on code
verification.4
Conventional techniques for incorporating convection heat transfer in large
3D thermal models typically use an empirical HTC correlation for relevant
geometries with a specified constant reference temperature and velocity.
Alternatively, the 3D continuity, momentum, and energy conservation
equations for fluid flow may be coupled with the model if more accuracy is
required. In circumstances where convection does not significantly affect the
relevant temperature profiles in the solid volume, the former method may be
sufficient for analysts. Even if the convection is relevant, if the velocity and
temperature of the flow does not change significantly through the solid, then
an empirical HTC correlation with constant reference temperature may still be
acceptable. However, for cases where the solution is strongly influenced by
convection and the fluid temperature can change considerably as the flow
travels over the solid, coupling a model of the flow itself may be required.
Obviously, incorporating the full 3D fluid flow equations into an existing
1 Dobranich, D., “Ther2 Qualification Activity—Test-Design Simulation Support,” SAND2014-
17390, September 2014.
2 Dobranich, D., Hetzler, A., Francis, N. “Ther2-WR Qualification Activity—Model Validation
Evidence in Support of Environmental Specifications,” SAND2015-9654, November 2015.
3 Dobranich, D., Mills, B., “Ther2-JTA Qualification Activity – Model Validation Evidence,”
SAND2016-0967, January 2016.
4 Mills, B., “Verification of Advective Bar Elements Implemented in the Aria Thermal Response
Code,” SAND2016-0271, January 2016.

Introduction
2
thermal model may not be desired due to the large computational expense. For
cases where the flow can be described as unidirectional and gradients
perpendicular to the flow direction are small, modeling the flow as 1D (as in the
advective bar model) may be sufficient to accurately capture the effects from
the flow.
The mass, momentum, and energy conservation equations that govern a
1D flow system are as follows:
  0





z
w
t

(1)
zz
p
z
w
w
t
w zz










 
 (2)
  S
z
T
k
zz
wT
c
t
T
c pv 














 
 (3)
where 𝜌 is the fluid density, 𝑧 is the spatial coordinate in the flow direction, w
is the fluid velocity, 𝜏 𝑧𝑧 is the viscous stress, 𝑆 is a source term, 𝑇 is the fluid
temperature, 𝑐 𝑣 is the constant volume specific heat, 𝑐 𝑝 is the constant
pressure specific heat, 𝑘 is the fluid thermal conductivity, and 𝑝 is the
pressure. All fluid properties and velocities in the model are cross-sectionally
averaged values. For a limited class of problems in which changes of fluid
momentum are small and when there are negligible local changes of density
with time, the net effect of solving Eqs. 1 and 2 can be captured using simply
the steady term of Eq. 1. This is the current implementation of the advective
bar model in Aria at the time this document is written. Implicit in this current
model is the ability to evaluate the density over a limited range of pressure.
Future work will include modeling of all three conservations equations in their
form presented in Eqs. 1-3.
The conservation equations are discretized using the finite element method,
and 1D bar elements model the fluid volume. The 1D fluid bar elements are
coupled to the surrounding 3D volume through a convection boundary
condition on the surrounding solid. That is, the fluid temperature at each bar
node is used as the reference temperature in Newton’s Law of Cooling along
with an appropriate empirical heat transfer coefficient (HTC) correlation on the
surfaces nearest to that bar node. Note that the empirical HTC correlations in
the model are required to account for gradients perpendicular to the flow (that
are otherwise removed with the cross-sectional averaging) that strongly impact
the heat transfer to and from the surface. Additionally, the fluid velocity at a
bar node is also used to calculate a local Reynolds number Re at the surface
nearest to the node for the HTC correlation. The advantage of this method (as
opposed to specifying a constant reference temperature and velocity over the
entire surface) is that it allows for the HTC correlations to be more accurately

Introduction
3
evaluated locally as the fluid temperature and velocity changes through the
volume. Furthermore, with large changes in temperature, temperature-
dependent properties can be reevaluated locally. In Aria, empirical HTC
correlations may be chosen from an extensive library already integrated in the
code base for both laminar and turbulent flow. Correlations are also available
for free convection as well.
A suite of verification activities have been performed to support the
implementation of the advective bar model into Aria (Reference 4). These
activities included: a mesh resolution study to demonstrate convergent solution
behavior as the mesh is uniformly refined, a visual inspection of the mapping
of bar nodes to the coupled surfaces for various meshes, and a solution
comparison between the advective bar model in Aria and a CFD commercial
software program, ANSYS Fluent®, for simple geometries. Ultimately, the mesh
refinement study showed solution convergence for simple pipe flow in both
temperature and velocity. In addition, for pipe flow and annular flow with
different mesh discretizations and dimensions, the mapping was observed to be
performed correctly. Guidelines were provided in the report to create
appropriate meshes for the advective bar elements and the surrounding
volume. Simulations using the advective bar model in Aria also provided
comparable solutions in temperature and velocity to Fluent.
The work presented in this document is aimed at supplementing the work
performed in Reference 4 with a validation study of the advective bar model.
The validation strategy used for this analysis focuses on sufficiently
documented steady-state experiments found in the literature for relevant
geometries based on work already performed at Sandia. These geometries
primarily include pipe flow and annular flow. A literature search for relevant
experiments is performed first to identify candidate geometries. Then, models of
these experiments are developed in Aria using advective bar elements. Reported
temperatures in the fluid and walls from the experiments in the literature are
then compared directly with the simulation results to validate the model within
acceptable levels of experimental and numerical uncertainty. In addition to
pipe flow and annular flow, validation of flow through a helical pipe is also
performed to demonstrate the versatility of the advective bar model. The overall
goal of this effort is to document the capability of the advective bar element
model in order to provide confidence in its use for high-consequence thermal
models.
The advective bar element model is unique in that it uses empirical HTC
correlations in the convection boundary condition to capture the heat transfer
to and from the surrounding surface. As a result of this, the accuracy of the
solution is heavily dependent on the correlation that is used. In the literature,
the results of relevant experiments are often reported as non-dimensionalized
heat transfer coefficients, or Nusselt numbers Nu. In the validation of the
model, it is important to not solely compare reported Nu from the experiments
with the reported Nu from the simulation as that only evaluates the empirical

Introduction
4
correlation and not the model itself. Instead, it is important to directly compare
reported temperatures or Nu calculated from the solution temperatures in the
validation of the model. However, if the model is functioning correctly, Nu
calculated from solution temperatures should be equivalent to the correlation.
The remainder of this report is organized as follows. First, the validation of
the advective bar model for pipe flow is presented including a brief description
of the experimental setup described in the literature. Next, validation for
annular flow using the advective bar model is discussed also including a brief
description of the experiment described in the literature. Then, validation for a
helical pipe is performed. Finally, some general recommendations are
presented based on the results of this work and the results of this effort are
summarized.

Pipe Flow
5
Pipe Flow
Experiments documenting heat transfer in pipe flow are numerous in the
literature, and most of these experiments were performed in the early to mid-
20th century. The most relevant experiments described in the literature were
those that were used in the development of experimental Nu correlations5,6.
These experiments also include many famous works like Dittus and Boelter7 of
the Dittus-Boelter correlation and Sieder and Tate8 of the Sieder-Tate
correlation for heat transfer in turbulent pipe flow. However, often for brevity,
many of the works do not include all of the temperature measurements in the
works as needed for the validation herein. Therefore, many of these works
aren’t suitable for validation of the advective bar model.
An experimental work was obtained that did possess the necessary
experimental measurements to validate the advective bar model in Aria.
Although there also may be other suitable works in the literature, the
experiments performed by V. Kolář9 possessed sufficient detail specifically with
regards to temperature measurements of the fluid and the surrounding
surfaces. The next section describes the experimental setup in Kolář and the
thermal model that was developed to represent the experiment using advective
bar elements.
Experimental Setup and Advective Bar Model
The experimental setup used in Kolář is depicted in Figure 1. In this
experiment, either air or water was pumped through a tubular brass test
section 80 cm long, with an inner and outer diameter of 2.6 and 3.3 cm. Steam
from a boiler was pumped over the outer surface of the test section and allowed
to condense on the surface keeping a roughly constant surface temperature
over the length of the test section. Fluid temperature was measured on the
inlet and outlet of the test section using mercury thermometers and six
thermocouples were placed at different lengths along the inner wall surface of
the test section to measure wall temperature. The six thermocouples along the
length of the test section were averaged to calculate an average wall
temperature used in the calculation of the heat transfer coefficient.
5 Bialokoz, J., Saunders, O., “Heat Transfer in Pipe Flow at High Speeds”, Proceedings of the
Institution of Mechanical Engineers, 170, 389 (1956)
6 Ruppert, A., Schlunder, E., “Heat Transfer and Pressure Drop of Two-Phase-Two-Component-
Flow in Horizontal Smooth and Rough Tubes”, AIChE, 4-4 (1974)
7 Dittus, F., Boelter, L., “Heat Transfer in Automobile Radiators of Tubular Type”, Int. Comm.
Heat and Mass Transfer, 12, 3 (1930)
8 Sieder, E., Tate, G., “Heat Transfer and Pressure Drop of Liquids in Tubes”, Industrial and
Engineering Chemistry, 28, No. 12, 1429 (1936)
9 Kolář, V. “Heat Transfer in Turbulent Flow of Fluids Through Smooth and Rough Tubes”, Int.
J. Heat and Mass Transfer, 8, 639 (1965)

Pipe Flow
6
Figure 1. Schematic of the experimental setup used in Reference 9
Four different test sections were included in the experimental setup to
investigate the effect of wall roughness on the heat transfer coefficient. Of the
test sections used in experiments, data from the test section using a smooth
tube (labeled Tube 0 in Reference 9) and a tube with a relative roughness e/D =
0.044 (labeled Tube 1) was compared to the advective bar model. Air was used
as the working fluid in the selected smooth tube cases and water was used as
the working fluid in the selected rough tube cases. A sample of five experiments
using the smooth tube and four experiments using the rough tube spanning
the range of Re explored in the study were chosen for the validation. The
experimental data from those experiments is summarized in Table 1.

Pipe Flow
7
Table 1. Summary of Sampled Experiments from Kolář (Reference 9)
Case Tube Fluid 𝒎̇ (kg/s) Re 𝑻𝒊 (K) 𝑻 𝒐 (K) 𝑻̅ 𝒘 (K)
1 Smooth air 0.0375 94522 313.3 328.9 372.4
2 Smooth air 0.0257 64633 313.4 329.9 372.4
3 Smooth air 0.0146 36811 310.1 329.6 372.8
4 Smooth air 0.0043 10924 308.2 331.7 373.0
5 Smooth air 0.0023 5752 308.8 330.7 372.8
6 Rough water 1.722 116930 306.6 309.6 339.2
7 Rough water 0.520 34331 301.8 311.7 349.4
8 Rough water 0.255 16878 300.9 312.8 357.3
9 Rough water 0.132 8511 298.6 312.8 365.1
The mesh used in the advective bar model for the validation is depicted in
Figure 2. The mesh consisted of 3200 hexahedral elements in the test section
(with 40 elements along the length) and 20 advective bar elements for the fluid.
As is typical in an advective bar element model, the mesh was not contiguous
and the bar elements were centered along the axis of the pipe. Material
properties used in the model for the air, water, and brass are provided in
Appendix A.
Reference 9

Pipe Flow
8
Adiabatic boundary conditions were specified on the inlet and outlet of the
pipe with the assumption that losses were negligible along the length. The
Gnielinski correlation10 for turbulent heat transfer through a circular pipe was
applied in this model as a convection boundary condition on the interior
surface of the pipe to capture the heat transfer to the working fluid. This
convection model was available through a library of HTC correlations provided
in Aria. A very long entrance length leading up to the pipe entrance was
included in the experimental setup such that entrance effects were deemed
negligible and the thermal entrance effects correction factor available in Aria
was not applied for the correlation. However, the film gradient correction factor
was included in this model. The Gnielinski correlation used in this model was
also specifically developed for smooth tubes. As defined in heat transfer texts11,
as a first order approximation, the Gnielinski correlation can be applied to
rough tubes using the appropriate friction factor. For lack of a better
correlation compatible with the advective bar model, this was the correlation
applied to the rough test section using an appropriate friction factor for each
respective Re.
A convection boundary was applied to the outer surface of the test section
to model the condensing steam on the outer surface of the pipe. Values for the
HTC were calculated manually from an empirical correlation (Reference 11) and
input as a constant average value over the surface of the pipe. The correlation
is as follows:
 
 
 413
729.0











DTT
hkg
h
ssatl
f glvll


(4)
where 𝜌 is the density, 𝑔 is the gravitational constant, 𝑘 is the thermal
conductivity, 𝜇 is the dynamic viscosity, ℎ′ 𝑓𝑔 = ℎ𝑓𝑔 + 0.68𝑐 𝑝,𝑙(𝑇𝑠𝑎𝑡 − 𝑇𝑠) is the
modified latent heat of vaporization, 𝑐 𝑝 is the specific heat at constant
pressure, 𝑇𝑠𝑎𝑡 is the saturation temperature, 𝑇𝑠 is the surface temperature, the
subscript l refers to liquid properties, and the subscript v refers to vapor
properties. For experiments conducted with air, the HTC on the outer surface
of the pipe was sufficient (>10,000 W/m2K) to maintain the wall temperature
near the saturation temperature of the steam. That is, the solution was not
sensitive to its value and a constant value of 10,000 W/m2K was selected for all
flow rates. For the cases with water, the wall temperature was strongly
dependent on this correlation and values of 8,800, 9,600, 10,600, and 12,500
were calculated for cases 6-9 as defined in Table 1, respectively. Note that wall
10 Gnielinski, V., “New equations for heat and mass transfer in turbulent pipe and channel
flow,” International Chemical Engineering, Vol. 16, No. 2, 359-367 (1976)
11 Incropera, F., et al., Fundamentals of Heat and Mass Transfer, John Wiley & Sons, Inc., 6th
Edition, Hoboken, New Jersey, 2007.

Pipe Flow
9
temperature was not able to be held at the steam temperature in the
experiments when water was used as the fluid.
Experimental Comparison
Using the advective bar element model defined above, the outlet
temperature, average wall temperature, and calculated Nusselt numbers Nu
from the steady-state solution were compared to the values reported in Kolář
for the selected cases described in Table 1. Nu for each case was calculated
from the reported temperatures in the model by the following equation:
iD
kh
Nu  (5)
where 𝑘 is the thermal conductivity of the fluid evaluated at the mean fluid
temperature, 𝐷𝑖 is the inner pipe diameter, and ℎ̅ is the average heat transfer
coefficient. The average heat transfer coefficient is calculated using the
following relation:
 
lms
iop
lm TA
TTcm
T
q
h







(6)
where 𝑞′′̅̅̅ is the average surface heat flux, 𝑚̇ is the mass flow rate, 𝑐 𝑝 is the
specific heat at constant pressure, 𝑇𝑜 is the outlet temperature, 𝑇𝑖 is the inlet
temperature, 𝐴 𝑠 is the surface area of the inside of the pipe, and ∆𝑇𝑙𝑚 is the
log-mean temperature difference defined as:
 










ow
iw
io
lm
TT
TT
TT
T
ln
(7)
and 𝑇 𝑤
̅̅̅̅ is the average inner wall temperature of the pipe. Calculating Nu using
the reported temperatures as opposed to using the reported heat transfer
coefficient assured that the model as a whole was being validated as opposed to
only the correlation itself. The outlet temperature, average wall temperature,
and Nu from the model are provided in Figure 3, Figure 4, and Figure 5,
respectively, for Cases 1-5.

Pipe Flow
10
measurements for Cases 1-5
Figure 4. Average wall temperatures from the model compared to experimental

Pipe Flow
11
Figure 5. Average Nu from the model compared to experimental measurements
for Cases 1-5
Experimental uncertainty was not explicitly reported in Kolář; however, an
estimate of the experimental uncertainty was calculated from the description of
the instrumentation used to measure the temperatures and flow rates. Other
experimental error likely existed that was not being taken into account but was
ignored for conservatism. For inlet and outlet temperatures, mercury
thermometers were used with an uncertainty of ±0.05°C. Wall temperatures
were measured with custom copper-constantan thermocouples and assumed to
have an accuracy of ±1°C. Mass flow rate measurements were assumed to be
accurate within ±3% of the reported measurements. These errors were
propagated in the calculation of Nu and all reported in the figures with a ±2σ
uncertainty, where σ is the standard deviation.
Only the largest source of error in the model, the empirical heat transfer
coefficient on the interior surface, was considered for the numerical
uncertainty. As mentioned previously, the Gnielinski correlation was used as
the empirical heat transfer coefficient correlation for this model and has an
uncertainty of ±10% (two standard deviations) of the calculated value. The error
resulting from the use of the correlation provided in Eq. 4 used on the exterior
of the pipe was not included as it produced little effect on the resulting wall
temperatures.
As observed in Figure 3 through Figure 5, the solution from the model
agreed well with the experimental data for almost all of the data. The only
differences in solution that exceeded the experimental and numerical

Pipe Flow
12
uncertainty were observed in the outlet temperature and Nu of the lowest flow
rate tested in Case 5. However, in Case 4 with only slightly higher Reynolds
numbers, excellent agreement was observed for all reported measurements
indicating that the differences observed in Case 5 may simply be a result of the
correlation possessing higher uncertainty at flow rates that approach the
transition region. Error in the mass flow rate measurement may also be more
significant at lower flow rates than originally estimated (±3%). A more rigorous
uncertainty analysis may resolve this discrepancy but was not pursued here
due to the excellent agreement at all other Re. The results indicate that the
Aria advective bar model agrees well with the data over a very large range of
turbulent Re.
Next, Cases 6-9 were examined using this model. The outlet temperature,
average wall temperature, and Nu from the model are now provided in Figure 6,
Figure 7, and Figure 8, respectively.

Pipe Flow
13
Figure 7. Average wall temperatures from the model compared to experimental
Figure 8. Average Nu from the model compared to experimental measurements
for Cases 6-9

Pipe Flow
14
Recall that for cases with rough inner surfaces, a correlation was not
available through the library provided within Aria to appropriately handle this
scenario. The use of the Gnielinski correlation for circular pipe flow with an
appropriate friction factor was described as an option to provide a first order
approximation (the traditionally specified uncertainty of ±10% is no longer
valid), and this was the approach taken in this analysis for lack of a better
correlation. Nevertheless, despite the very high relative roughness (e/D =
0.044), for low to moderate Re approximately less than 4×104, the correlation
provided reasonable agreement within the experimental and numerical
uncertainty. This model also used water as opposed to air in Cases 1-5
indicating compatibility with other fluids. For Re approximately greater than
4×104, error in the solution far exceeded the estimated uncertainty indicating a
different empirical correlation was required.
In summary, an advective bar model has been created that simulates an
experimental setup of turbulent heat transfer in a circular pipe provided in
Reference 9. Note that entrance effect correction factor available in Aria for the
Gnielinski correlation used in this model was not applied as the experimental
setup provided enough entrance length such that entrance effects could be
ignored. For smooth tubes, the model agreed well with the provided
experimental measurements within experimental and numerical uncertainty
with the exception of the very lowest flow rates explored. For rough tubes, the
model also showed reasonable agreement with the experimental data for low to
moderate Re. Validation of the advective bar model is now explored in the
following section for annular flow.

Annular Flow
15
Annular Flow
The use of the advective bar model with annular flow is of particular
interest at Sandia as this fundamental geometry has already been applied to
several existing projects (References 1-3). Therefore, documenting the
validation of this model for simple annular geometries will provide additional
confidence in the analysis performed in those works. Experiments describing
turbulent heat transfer for annular flow are as numerous as pipe flow in the
literature where most of these experiments were again performed in the early to
mid-20th century. A literature search resulted in several candidate
experimental studies12,13,14,15. However, as discussed for pipe flow geometries, it
was necessary that the applicable experimental studies possessed sufficient
description of the experimental setup and measurements for a sufficient
validation of the model. The steady-state experiment reported in Miller et al.16
was found to have sufficient information about the setup and the temperature
measurements to pursue validation. The next section describes the
experimental setup in Miller et al. and the thermal model used to represent the
experiment developed using advective bar elements.
The experimental setup used in Miller et al. is shown below in Figure 9.
The flow loop is specifically identified in Figure 9a. In this experiment, de-
ionized water is released from an elevated storage tank at a specified flow rate
and allowed to flow through a vertically oriented annular test section into a 5
gallon storage tank used in the flow measurement. The water could be
preheated using a steam heater before the annular test section to reach the
desired inlet temperature. Temperature of the flow was measured using in-line
thermometers and pressure was also measured at the test section exit.
12 Kays, W., Leung, E., “Heat Transfer in Annular Passages – Hydrodynamically Developed
Turbulent Flow with Arbitrarily Prescribed Heat Flux”, Int. J. Heat and Mass Transfer, 6, 537
(1963).
13 Quarmby, A., “Some Measurements of Turbulent Heat Transfer in the Thermal Entrance
Region of Concentric Annuli”, Int. J. Heat and Mass Transfer, 10, 267 (1967).
14 Reynolds, W., Lundberg, R., McCuen, P., “Heat Transfer in Annular Passages. General
Formulatino of the Problem for Arbitrarily Prescribed Wall Temperatures or Heat Fluxes”, Int.
J. Heat and Mass Transfer, 6, 483 (1963).
15 Stein, R., Begell, W., “Heat Transfer to Water in turbulent Flow Internally Heated Annuli,”
A.I.Ch.E Journal, 4, No. 2, 127 (1958).
16 Miller, P., Byrnes, J., Benforado, D., “Heat Transfer to Water in an Annulus”, A.I.Ch.E
Journal, 1, No. 4, 501 (1955).

Annular Flow
16
Figure 9. Schematic of the experimental setup used in Miller et al. (a) and a
more detailed schematic of the annular test section (b) (Reference 16)
The annular test section is depicted in Figure 9b. Water entered the test
section from the bottom and was abruptly directed to flow vertically around a
heater rod that comprised the inner cylinder of the annulus. Rubber sleeves
held the ends of a 40.64” long glass tube that comprised the outer cylinder. A
glass tube was used to insulate the fluid from the environment. Flow then
exited from the top of the test section. The heater rod had an outer diameter of
(a)
(b)

Annular Flow
17
1.588 cm, and the glass tube had an inner diameter of 2.134 cm. Two spacers
(with a cross-sectional flow area of half of the annular region) centered the
heater rod inside the glass tube to create the annulus.
The heater rod itself (identified as Rod 15 in Miller et al.) was a custom
design for the experiments with a heated section of approximately 10.2 cm. The
heater rod consisted of a nichrome resistance element wound around a ceramic
core inside an aluminum jacket. A thin layer of insulation separated the
aluminum and the nichrome. Two grooves in the aluminum jacket along the
length of the rod housed two copper-nickel (type-T) thermocouples used to
measure the surface temperature on either side of the rod on the heated
section. Variation in the thickness of the insulation separating the jacket and
the nichrome often resulted in temperature measurements that differed up to
10°C between the two thermocouples. Two other heater rods were used in the
set of experiments to confirm that the depth of the embedded thermocouples in
the heater rod (Rod 19) and the length of the heated section (Rod 20) did not
affect the results. Neither change was reported to have had a significant effect;
therefore, only one heated length was explored here (10.2 cm) and the
differences between the remaining two rods (i.e. Rod 15 or Rod 19) were
ignored. Note that Rod 15 possessed two embedded thermocouples on either
side of the rod and Rod 19 only included one embedded thermocouple.
Six representative experiments from those reported in Miller et al.
(Reference 16) were simulated using the advective bar model in Aria. The
experimental data from those experiments is summarized in Table 2. These
cases were selected to span the range of fluid velocities, inlet temperatures,
and heater rod powers. Average wall temperatures 𝑇̅ 𝑤 on the heater rod surface
were reported from an average of the two embedded thermocouples in the case
of Rod 15 or the single measurement in the case of Rod 19.
Table 2. Summary of Sampled Experiments from Miller et al. (Reference 16)
Case Rod# Fluid 𝒎̇ (kg/s) Re 𝑻𝒊 (K) 𝑻 𝒐 (K) 𝑻̅ 𝒘 (K)
1 15 water 0.1517 5563 296.0 296.4 350.5
2 15 water 0.2942 10520 295.0 295.2 310.6
3 15 water 0.2942 10613 295.5 296.2 326.9
4 15 water 0.3310 21453 325.3 325.5 344.9
5 15 water 0.3310 22230 327.5 327.9 366.3
6 19 water 0.6070 22141 296.0 296.2 313.4
The mesh used in the advective bar model for the validation is depicted in
Figure 10. This mesh consisted of 10,240 hexahedral elements in the outer
pipe, 10,240 hexahedral elements in the inner hollow cylinder, and 64 bar
elements for the fluid consistent with the advective bar model. Only the glass

Annular Flow
18
tube test section between the rubber sleeves was modeled. As with the
advective bar element model for the pipe in the previous section, the mesh was
not contiguous and the bar elements were centered along the axis of the
annulus despite the fact that the fluid was defined to flow in the annular region
between the two cylinders. However, as long as the specified coupled surfaces
are properly defined on the inside surface of the outer pipe and the outside
surface of the inner cylinder, this does not affect the bar node mapping in this
geometry (see Reference 4). The inner hollow cylinder, defined as the aluminum
jacket of the heater rod in this problem, was the only meshed part of the heater
rod to reduce the total number of elements in the problem. Since this was a
steady-state problem, the surface temperature on the heater rod was
equivalent to the experimental measurements as long as the same power was
added to the heated length as measured from the heater rod in the experiment.
However, the transient profile of the surface temperature would not be in
agreement with the measurements. The heated section for the set of
experiments extended from approximately 19.0 cm to 29.2 cm along the length
of the test section. Conduction was allowed along the length of the aluminum
heater jacket, but its effect was found to be negligible. Material properties used
in the model for the water, and outer glass tubing, and aluminum jacket are
provided in Appendix A.
Reference 16
Heater rod modeled as
hollow in the advective bar
model

Annular Flow
19
Adiabatic boundary conditions were specified on the inlet and outlet of the
annulus with the assumption that losses were negligible at these locations
axially. The Gnielinski correlations17 for turbulent heat transfer through an
annular region were applied in this model as a convection boundary condition
on the heater rod surface and inner glass tube surface to capture the heat
transfer to the working fluid. These correlations were derived based on which
surface of the annular region was experiencing heat transfer where the other
surface was defined as adiabatic. Strictly speaking, heat transfer was occurring
on both surfaces of the annular region in this problem, but the vast majority of
heat transfer was occurring on the inner heater rod surface. Therefore, the
correlation where the heat transfer originates from the inner surface was
applied. The heat transfer occurring on the glass tube surface was very small
and could effectively be ignored in the problem; however, convection on this
surface was still included in the model. These correlations were also available
through a library of HTC correlations provided in Aria. In addition, this
problem did not feature a very long entrance length leading up to the heated
region of the rod, so the thermal entrance effects correction factor available in
Aria was applied for this correlation in addition to the film gradient correction
factor. Note that Case 1, defined in Table 2, was outside the bounds of
applicable Re for the correlation used in this model (Re > 1×104, Reference 17).
Using the annular advective bar element model defined above, the outlet
temperature, average wall temperature, and calculated Nusselt numbers Nu
from the steady-state solution were compared to the values reported in Miller et
al. and summarized in Table 2. The calculations for these values were similar
to those for pipe flow described in the previous section, but are repeated here
to illustrate subtle differences in the geometry. Nu for each case was calculated
from the reported temperatures in the model by the following equation:
hD
kh
Nu  (8)
temperature, 𝐷ℎ = 𝐷𝑜,𝑖 − 𝐷𝑖,𝑜 is the hydraulic diameter, 𝐷𝑜,𝑖 is the inner diameter
of the outer pipe, 𝐷𝑖,𝑜 is the outer diameter of the inner pipe, and ℎ̅ is the
average heat transfer coefficient. The average heat transfer coefficient is
calculated using the following relation:
T
q
h


 (9)
17 Gnielinski, V., “Heat Transfer Coefficients for Turbulent Flow in Concentric Annular Ducts,”
Heat Transfer Engineering, Vol. 30, No. 6, 431-436 (2009).

Annular Flow
20
where 𝑞′′̅̅̅ is the average surface heat flux output from the heater rod (assuming
negligible variation axially along the heated section), ∆𝑇 is the temperature
difference between the fluid and the wall at the thermocouple location.
As before, by calculating Nu from the solution temperatures as opposed to
using the reported heat transfer coefficient from the correlation assured that
the advective bar model as a whole was being validated as opposed to only the
correlation itself. The outlet temperature, average wall temperature, and Nu
from the model are provided in Figure 11, Figure 12, and Figure 13,
respectively, for Cases 1-6 listed in Table 2.
measurements from Miller et al.

Annular Flow
21
Figure 12. Inner wall temperatures from the model compared to experimental
Figure 13. Nusselt numbers from the model compared to experimental

Annular Flow
22
Experimental uncertainty in the model was briefly described in Miller et al.
with a claim that the heat transfer coefficients calculated from the experiments
were accurate within approximately ±8% of the reported values. This
uncertainty was applied to the experimental Nu values in the previous figures.
Fluid temperatures were reported to be measured with a glass thermometer
with an accuracy of ±0.2°C. Finally, the copper-nickel thermocouples
embedded in the heater rod were assumed to have an uncertainty of ±1°C for
lack of a better recommendation. Other experimental error likely exists that is
not being taken into account or is being underestimated but this was ignored
for conservatism.
As before with pipe flow, the largest source of uncertainty in the model was
the empirical heat transfer coefficient correlation used on the heater rod
surface. Overall recommendations for the uncertainty of the Gnielinski
correlations for annular flow (Reference 17) were not found, so they were
assumed to be accurate within ±15% (two standard deviations) of the
calculated value. Uncertainty from the correlation applied on the outer glass
tube surface was not applied as it was not significant to the solution. Other
sources of error in the model including material properties and errors in the
heat flux were assumed negligible to be more conservative.
As observed in previous figures, the solution from the model agreed well
with the experimental data for nearly the entire set of experimental
measurements. Only the case with the lowest mass flow rate was shown to
exceed the numerical and experimental uncertainty. As before, this may be a
result of the Re being below the minimum Re for which the HTC correlation
was applicable (Re > 1×104) or the fact the that mass flow rate measurements
may be less accurate at lower flow rates. This disagreement was not pursued
further as the other experimental values agreed well with the model within
conservative estimates for experimental and numerical uncertainties. Instead,
a recommendation is stated here to use caution when applying the empirical
correlations near the edges or outside of their applicable ranges.
In summary, a model of the experiment performed in Miller et al. (Reference
16) was created using advective bar elements in Aria. This model was
demonstrated to agree well with the experimental measurements provided in
Miller et al. within conservative estimates for the experimental and numerical
uncertainty with the exception of a case at the lowest mass flow rates. The
disagreement occurred as the flow rate fell below the bounds of the applicable
Re for which the empirical heat transfer coefficient correlation used in this
model could be applied. Validation of the advective bar model is now explored
in the following section for a more complex geometry of a helical pipe to
demonstrate the versatility of the advective bar model.

Helical Pipe Flow
23
Helical Pipe Flow
All the preceding validation has been performed for unidirectional flow in
one coordinate direction. To truly demonstrate the capabilities of the advective
bar model and provide more confidence in its ability to model more complex
flow, the advective bar model is validated for flow moving in a three-
dimensional path with a more complex flow field. Outside the fundamental
experiments for turbulent heat transfer in pipes and annuli discussed above
are many application specific studies too complicated for model validation.
However, the helical pipe flow heat transfer experiments performed by Pawar
and Sunnapwar18,19 provided all the necessary measurements and data to
recreate a comparable model for validation without excessive complexity. The
next section describes the experimental setup in Pawar and Sunnapwar as well
as the model developed using advective bar elements to replicate the
experiment.
To investigate the performance of the advective bar model in a more
complex geometry, a helical pipe geometry was chosen with spiraling defined
about the z-axis. Pawar and Sunnapwar performed an experiment consisting of
a mild steel helical pipe submerged in a hot water bath with cold water used as
the working fluid that was cooled in a separate helical pipe submerged in a cold
water bath. A schematic of the experimental setup is provided in Figure 14. A 1
horsepower centrifugal pump was used to pump the chilled water. The water
was pumped through the helical pipe upwards from the bottom port and exited
the helical pipe through the top port. The hot water tank was insulated with 50
mm thick Thermocol, or polystyrene foam, covered with asbestos material.
Three electrical heaters, two 3 kW heaters and one 5 kW heater, were mounted
at the bottom of the hot water tank. The cold water tank had a constant flow of
cold water at 30°C and 0.3292 kg/s. Six type K thermocouples were evenly
distributed and mounted to the outer wall of the pipe, and five thermocouples
were used to measure the temperature of the hot water bath at different
locations in the tank. Mercury thermometers were used at the inlet and outlet
of both the hot water coil and cold water coil. While the experiment included
three slightly differing helical pipes as depicted in Figure 15, only the first coil,
labeled as Coil-I, was used for the validation in this study. The dimensions of
Coil-I are provided in Table 3.
18 Pawar, S. S., and Vivek K. Sunnapwar. "Experimental Studies on Heat Transfer to
Newtonian and Non-Newtonian Fluids in Helical Coils with Laminar and Turbulent Flow."
Experimental Thermal and Fluid Science 44 (2012): 792-804. Web.
19 Pawar, S. S., and Vivek K. Sunnapwar. "Experimental and CFD Investigation of Convective
Heat Transfer in Helically Coiled Tube Heat Exchanger." Chemical Engineering Research and
Design 92.11 (2014): 2294-312. Web.

Helical Pipe Flow
24
Figure 14. Schematic layout of experimental setup used in Reference 18
Figure 15. Picture of coils used in the experiment (Reference 18)

Helical Pipe Flow
25
Table 3. Physical Dimensions of Helical Coil-I (Reference 18)
Δ (a/R) L (mm) Do (mm) Di (mm) do (mm) di (mm) N Pitch (mm)
0.0757 8195.47 300 274.6 25.4 20.8 9.5 29.15
The experiment was run under both isothermal as well as non-isothermal
conditions. Under isothermal conditions, one 3 kW heater was manually
controlled while the other two heaters were kept fully on to maintain a constant
hot water bath temperature of 62.5°C. Under non-isothermal conditions, all
three heaters were kept at full power for a total of 11 kW. Data for the non-
isothermal cases were collected in 15 minute increments once the hot water
bath reached a temperature of 54°C. Various fluids, flow rates, and
experimental conditions were tested in the experiment. These are summarized
in Table 4 below.
Table 4. Summary of Experimental Conditions for Coil-I (Reference 18)
Test fluids Flow rates, m3/s × 10-5 Experimental conditions
Water 3.3967-16.8350 Isothermal
Water 3.41-25.97 Non-isothermal
10% Glycerol 3.3967-16.8350 Isothermal
20% Glycerol 3.62-26.11 Isothermal
20% Glycerol 3.62-26.11 Non-isothermal
The geometry created for the simulation was comprised of a helical pipe of
the dimensions listed in Table 3 above. Missing from the advective bar pipe
model were the 90° entrance and exit elbows connecting the ends of the helical
pipe to the hot water tank inlet and outlet ports. This extra pipe was
determined to have a relatively small effect on the overall results and was
accounted for in the simulation uncertainty.
The mesh consisted of the pipe mesh and the working fluid represented as
advective bars defined along the axis of the helix. The mesh is pictured in
Figure 16. The pipe mesh had 2 elements through the thickness, 20 elements
along the circumference of the pipe, and 200 elements along the length of the
pipe for a total of 8000 elements. The advective bar network was made up of
400 bar elements, exactly double the number of pipe mesh elements along the
length of the helix. Mills (Reference 4) suggested that any value for the ratio of
bar elements to pipe mesh elements along the length of a pipe was acceptable
so long as the pipe and bar elements do not align with each other. Initially, a
mesh of 200 bar elements and 100 hexahedral elements along the pipe length
was used, but this resulted in solution convergence issues. Using twice as

Helical Pipe Flow
26
many total elements corrected these issues and allowed the simulation to
converge properly.
Reference 18
Pawar and Sunnapwar listed the simulation parameters used in each of the
CFD simulations, including the boundary conditions in Table 5 and mild steel
properties provided in Appendix A. While the grade of mild steel was not
specified explicitly, the properties used for the CFD study closely resemble that
of a 3% carbon steel. Properties for water and both glycerol solutions can also
be found in Appendix A. Properties for water were obtained from the NIST
Webbook of Thermophysical Properties of Fluid Systems20 and properties for
the glycerol solutions were obtained from Physical Properties of Glycerine and
Its Solutions21 and Righetti et al.22.
20 "Thermophysical Properties of Fluid Systems." Thermophysical Properties of Fluid Systems.
N.p., n.d. Web. 26 March 2016.
21 Physical Properties of Glycerine and Its Solutions. New York: Glycerine Producers'
Association, 1963. Print.
22 Righetti, M. C., G. Salvetti, and E. Tombari. "Heat Capacity of Glycerol from 298 to 383K."
Thermochimica Acta 316.2 (1998): 193-95. Web.

Helical Pipe Flow
27
Table 5. Typical Boundary Conditions used for Simulations (Reference 19)
# Inlet temp (°C) Mass flow rate (kg/s) Reynolds number
1 32 0.04148 4321
2 37.5 0.06811 7098
3 39.5 0.08417 8768
4 40 0.09383 9776
5 41 0.12894 13436
6 42 0.1665 17347
Heat transfer from the pipe to the internal fluid flow was modeled using the
Dittus-Boelter correlation (Reference 7 and 11). While the correlation library
provided within Aria did contain external free convection correlations, at the
time this document was written Aria was not configured to work with external
correlations. Therefore, to model the isothermal experiments, heat transfer
from the external hot water bath to the pipe was specified using a constant
heat transfer coefficient manually calculated using the Churchill-Chu23
correlation for external free convection from a cylinder. The value was input as
a constant average value over the surface of the helical pipe. The correlation is
as follows:
2
278169
61
Pr
559.0
1
387.0
6.0

































Ra
D
k
ho (10)
where 𝑘 is the thermal conductivity, 𝐷 is the outer pipe diameter, 𝑅𝑎 = 𝐺𝑟𝑃𝑟 is
the Rayleigh number, 𝐺𝑟 = (𝑔𝛽(𝑇𝑠 − 𝑇∞)𝐷3) 𝜐2⁄ is the Grashof number, 𝑔 is the
gravity constant, 𝛽 is the expansion coefficient of the fluid, 𝑇𝑠 − 𝑇∞ is the
difference between the wall surface temperature and the external fluid
temperature, 𝜐 is the kinematic viscosity, and 𝑃𝑟 is the Prandtl number. Test
fluid properties were obtained using the bulk temperature of the fluid for each
flow case which was taken as the average temperature between the prescribed
inlet temperature and the outlet temperature. A constant reference
temperature of 335.65 K was used corresponding to the target bath
temperature of the isothermal case. External water properties were obtained
23 Churchill, Stuart W., and Humbert H.s. Chu. "Correlating Equations for Laminar and
Turbulent Free Convection from a Horizontal Cylinder." International Journal of Heat and Mass
Transfer 18.9 (1975): 1049-053. Web.

Helical Pipe Flow
28
using this same temperature. For the non-isothermal case, a constant heat flux
of 16313 W/m2 radially inward was applied to the exterior surface of the helical
pipe. An initial pipe temperature of 327.15 K was used to match the initial
recorded temperature in the non-isothermal cases.
Using the advective bar element model defined above, the outlet
temperature, average wall temperature, inner heat transfer coefficient,
calculated Nusselt numbers Nu, and calculated overall heat transfer coefficient
from the steady-state constant temperature solution were compared to the
values reported in Pawar and Sunnapwar for all six flow cases. Nu for each
case was calculated from the inner heat transfer coefficients derived from the
simulation solutions:
i
i
D
kh
Nu  (11)
temperature, 𝐷𝑖 is the inner pipe diameter, and ℎ̅ is the average inner heat
transfer coefficient. The inner average heat transfer coefficient is calculated
using the following relation:
n
h
h
n
i
i
i

 1
(12)
where ℎ𝑖 is the inner heat transfer coefficient at thermocouple 𝑖 and 𝑛 is the
total number of thermocouples mounted to the helical pipe. Thermocouples
were evenly distributed along the length of the helix, so it was not necessary to
area-weight the heat transfer coefficient. The overall heat transfer coefficient 𝑈 𝑜
was calculated using the following relation:
oio hhU
111
 (13)
where ℎ𝑖
̅ is the average inner heat transfer coefficient and ℎ 𝑜
̅̅̅ is the calculated
average outer heat transfer coefficient.
For helical geometries in the literature, a new dimensionless parameter 𝑀
based on Re, was often defined to more accurately characterize the transition
between laminar and turbulent flow. To be consistent with the results reported
by Pawar and Sunnapwar, all results are plotted in terms of 𝑀 defined as:

Helical Pipe Flow
29
18.0
64.0
26.0 







R
a
Re
M (14)
where 𝑎 is the inner radius of the pipe and 𝑅 is the average radius of the helix.
Mujawar and Rao24 established a criteria for Newtonian fluids where flow is
classified as laminar if 𝑀 ≤ 2100. In the case of the coil investigated, this
equates to 𝑅𝑒 ≤ 9153. It is important to note that this is significantly different
from the flow criteria used in the advective bar model for the internal heat
transfer correlation where the flow was classified as laminar only if 𝑅𝑒 < 3000.
Pawar and Sunnapwar noted the various uncertainties in their
instrumentation as well as their measurements and results. The recorded hot
water bath temperature had a ±0.2°C error margin. The thermocouples used
along the pipe and throughout the hot water bath were calibrated to a ±0.1°C
error limit. A digital temperature indicator used to record all thermocouple
temperatures was also calibrated to a ±0.1°C error limit. The helix inlet
temperatures have an uncertainty of ±2°C. Inlet and outlet temperatures were
measured by mercury thermometers calibrated to a limit of error of ±0.1°C.
Repeatability of measured parameters obtained through multiple trials was
±1.57% for temperatures and ±5.6% for test fluid flow rates. All heat flux,
Nusselt number, and inner heat transfer coefficient results were reported to
have a ±5.67% uncertainty and all overall heat transfer coefficient results were
reported to have a ±1.67% uncertainty.
For the constant temperature case where water was the test fluid, ±2σ
uncertainty in the model was quantified using an incremental Latin Hypercube
Sampling (LHS) study excluding the overall heat transfer coefficient. A total of
40 samples were used to determine the 95% confidence interval for each
solution. The study revealed converging behavior in the standard deviation
between an initial sample size of 5 and the final sample size of 40. The
parameters varied in the LHS study included the three largest contributors to
the overall uncertainty: the inner heat transfer coefficient, outer heat transfer
coefficient, and mass flow rate. As previously mentioned, the Dittus-Boelter
correlation was used for the inner pipe heat transfer and has an uncertainty of
±15%. Also mentioned above, the Churchill-Chu external natural convection
correlation for a cylinder (Eq. 10) was used to calculate the average outer heat
transfer coefficient and was noted in Atayılmaz and Teke25 to have an
uncertainty of ±20%. The flow rate was varied by ±5.67% to match the
24 Mujawar, B. A., and M. Raja Roa. "Flow of Non-Newtonian Fluids through Helical Coils."
Industrial & Engineering Chemistry Process Design and Development Ind. Eng. Chem. Proc. Des.
Dev. 17.1 (1978): 22-27. Web.
25 Atayılmaz, Ş. Özgür, and Ismail Teke. "Experimental and Numerical Study of the Natural
Convection from a Heated Horizontal Cylinder." International Communications in Heat and
Mass Transfer 36.7 (2009): 731-38. Web.

Helical Pipe Flow
30
repeatability of the experiment. For outlet temperatures, an error estimate was
added to the upper margin to account for the extra pipe length not included in
the pipe geometry. The overall heat transfer coefficient uncertainty was
calculated by propagating the inner and outer heat transfer correlation
uncertainties.
Outlet temperature, outer wall temperature, inner heat transfer coefficient,
Nu, and overall heat transfer coefficient from the model are provided in Figure
17, Figure 18, Figure 19, Figure 20, and Figure 21, respectively, for the six flow
cases where water was the test fluid and constant temperature conditions.
Figure 17 through Figure 21 showed good agreement between the advective bar
model and the experimental results across almost all data points. The only
data point where uncertainties could not account for the difference is the outlet
temperature for the slowest flow rate in Figure 17. This error is not reflected in
the heat transfer coefficient or Nu results which suggests the cause to be the
heat transfer correlation used for internal flow in the helix. A turbulent heat
transfer coefficient was used in this study for all cases, including the slowest
flow rate where a laminar correlation may be more appropriate.
measurements for constant temperature conditions

Helical Pipe Flow
31
Figure 18. Average outer wall temperatures from the model compared to
experimental measurements for constant temperature conditions
Figure 19. Average inner heat transfer coefficient from the model compared to

Helical Pipe Flow
32
Figure 20. Average inner Nusselt number from the model compared to
Figure 21. Average overall heat transfer coefficient from the model compared to
Figure 22 shows a comparison of Nusselt numbers between the model and
experiment for both constant temperature and constant heat flux cases and

Helical Pipe Flow
33
water as the test fluid. One observation from the figure was that the non-
isothermal data from the experiment did not agree with the isothermal data
and may be an indicator of a difference in the experiment not explicitly defined
in the text. It should be noted, that the model agreed with the isothermal data
from the experiment for both boundary conditions. One possible explanation
for this discrepancy was the manner in how the data was collected for the non-
isothermal case. Recall that data was collected in 15 minute increments once
the hot water bath reached a temperature of 54°C. It was possible that the
experiment did not reach steady-state conditions, which could account for the
difference in Nu between the two data sets. Due to the unexplained differences
in the data, a more detailed validation with the non-isothermal cases was not
pursued.
experimental measurements for both constant temperature (isothermal) and
constant heat flux (non-isothermal) cases with water as the test fluid
Figure 23 shows a comparison of Nusselt numbers between the model
and experiment for the non-isothermal case where 20% glycerol is the working
fluid. Turbulent HTC correlations were used on the inner surface for all flow
rates despite the slowest flow case of 20% glycerol where the Re is below the
turbulent flow criteria. The model data has very good agreement with the
experimental data across the whole range of M.

Helical Pipe Flow
34
experimental measurements for 20% glycerol in non-isothermal conditions
Figure 24 and Figure 25 show a comparison of overall heat transfer
coefficients for water and 10% glycerol under isothermal conditions. Due to a
lack of reported data from the glycerol experiments, the outer heat transfer
coefficient calculated using wall temperatures from the water data was used for
the isothermal glycerol simulations. As previously mentioned, the overall heat
transfer coefficient uncertainty was quantified by propagating the known HTC
correlation uncertainties. The data shows good agreement within the numerical
uncertainty bounds.

Helical Pipe Flow
35
Figure 24. Overall heat transfer coefficient at various flow rates from the model
compared to experimental measurements for water in isothermal conditions
Figure 25. Overall heat transfer coefficient at various flow rates from the model
compared to experimental measurements for 10% glycerol in isothermal
conditions

Helical Pipe Flow
36
To conclude, an advective bar model was created that simulated the
experimental setup of turbulent heat transfer in a helical pipe provided in
Pawar and Sunnapwar. The model showed very good agreement with most of
the experimental data for all fluids and conditions within experimental and
numerical uncertainty.

Recommendations
37
Recommendations
The purpose of this section is to provide guidance for analysts who may
incorporate the advective bar element model in their work. It summarizes and
modifies recommendations provided in Reference 4, and makes further
suggestions on the use of the model. At the time of this document, the
advective bar model is still under development and may undergo additional
changes in the future. Some of these recommendations may no longer be valid
in future versions of the model, so analysts are advised to be cautious when
applying them.
A critical characteristic of the advective bar model is the mapping of the bar
element nodes to the surrounding surfaces. Reference 4 has provided evidence
for the proper mapping for pipe flow and annular flow with varying cross
sectional areas and spatial discretizations. However, for complex geometries
(such as those where the flow travels through three dimensional space), one
should be extremely cautious of the mapping and check the results visually.
Fortunately, Aria already provides analysts the ability to visualize the mapping
in post-processing visualization software. For example, a geometry that could
result in improper mapping at the time of this work includes a helical pipe with
a decreasing pipe diameter and decreasing spacing between coils.
As a general guideline, Reference 4 called for the use of ¼ the number of
bar elements as there are elements in the solid in the flow direction. Then, the
number of bar elements should be refined by a factor of 2 and the solution
repeated to assure that the solution is not dependent on the number of bar
elements. Note that while this is still the general recommendation, the number
of bar elements may need to exceed the number of elements in the solid in the
flow direction (as with the helical pipe model in this work) to achieve acceptable
solution convergence. However, it is still advisable to avoid having the nodes
between the bar elements and the surrounding solid aligned in the flow
direction. This may result in improper or irregular mapping as discussed in
Reference 4.
The correction factor for the Gnielinski HTC correlations (References 10
and 17) to correct for thermal entrance effects should be applied with caution
when using the advective bar element model. As demonstrated with the pipe
flow model in this report, should the geometry possess enough unheated
starting length, then the application of this correction factor will likely
overestimate the true heat transfer coefficient. However, if the geometry does
not have sufficient unheated starting length as with the annular flow case
explored in this report, then the application of this factor is necessary to
achieve sufficient heat transfer coefficients. This conclusion is also reported in
Reference 3 for similar annular flow without sufficient unheated starting
length. In the helical pipe flow case explored in this report, the total length of
flow geometry was sufficient that the effect of the correction factor was
negligible.

Recommendations
38
As reported throughout this document, the heat transfer coefficient
correlation is typically the largest source of uncertainty in the model. It is
important to attempt to use the most accurate correlations available to achieve
the best solution. Only use this model when the uncertainty provided by the
correlation is acceptable for the problem. In addition, as demonstrated for the
low flow rate case in the annular flow geometry, it is important to apply the
correlation only within the range of valid Re to assure that the uncertainty
provided by the correlation is valid. Furthermore, this report recommends that
empirical HTC correlations provided in the library within Aria undergo some
form of code verification to assure their proper implementation. An order
verification test using the Method of Manufactured Solutions would be one
approach to achieve this. Regression tests should be added to guarantee the
correlations and model itself remain valid through continued development of
the advective bar element model.
Finally, the validation performed in this report only investigates steady-
state experiments primarily because of the limitation of finding sufficient
documentation in the literature to perform an adequate transient validation. If
capable, an inexpensive table-top validation experiment, as described in
Appendix B of Reference 4, would be sufficient to provide the necessary
information to validate the advective bar model for transient solutions. In
addition, the experimental setup would be useful to provide more accurate
empirical HTC correlations for geometries of interest to the works described in
References 1-3. It is therefore recommended that such an experiment be
performed to provide more credibility for the use of this model in high-
consequence thermal models.

Summary
39
Summary
A 1D fluid flow model has been implemented into the Aria thermal response
code to model convection heat transfer from 3D solids. The advantage of this
model is that it provides analysts the ability to model energy advection
associated with convective flow in very large 3D models without a significant
increase in computational cost. To provide confidence and credibility in the
implementation of this model for high-consequence problems, a validation
effort was undertaken to supplement the verification activities previously
reported on this model. This approach used experiments described in the
literature for relevant geometries of interest already used at Sandia, and builds
advective bar models in Aria that can be validated using the reported
experimental measurements.
Three flow geometries including pipe flow, annular flow, and flow through a
helical pipe were chosen for the validation. In addition to investigating different
geometries, this model was validated for different fluids (air, water, and
glycerol), surface conditions (smooth pipe and rough pipe), flow rates, and
boundary conditions. For the pipe flow model, the model agreed well with the
experimental measurements within experimental and numerical uncertainty
with the exception of the lowest flow rate explored. It was suggested that the
discrepancy in this low flow case was a result of the larger error in the
correlation as the flow approached the transition region and larger
experimental uncertainties in the flow measurement for this case. Further
investigation into the flow for a rough walled pipe showed that as a first-order
approximation, the advective bar model with the Gnielinski HTC correlation can
be used to accurately model heat transfer for moderate Reynolds numbers (~1-
3×104).
For the annular flow geometry, the model agreed well with the experimental
measurements reported in the literature within experimental and numerical
uncertainty with the exception of the lowest flow rate. However, this particular
flow rate was outside the acceptable bounds of Reynolds numbers for which
this correlation was derived. This model also successfully applied the thermal
entrance effects correction factor in the validation. For the helical pipe flow
geometry, the model also agreed well with the experimental measurements
using water and the glycerol mixtures as the working fluid. The validation of
this geometry using the advective bar model illustrated the versatility of the
model outside of typical pipe or annular flow. For other reported cases not
using a constant temperature boundary condition on the outer pipe surface,
there was insufficient documentation for differences in test procedure and
resulting data to use this information as validation data.

Appendix A—Material Properties
40
The following temperature-dependent material properties were used for air,
water, and brass in the analysis presented in this document for both Aria and
ANSYS Fluent®. The data is provided in tabular form and is linearly
interpolated for temperatures between the defined values. Table 6 provides the
material properties for air, Table 7 provides the material properties for
subcooled water at 1 atm, and Table 8 provides the material properties used for
brass. Table 9 and Table 10 provide the material properties used for 10%
glycerol and 20% glycerol mixtures, respectively. Table 11 provides the material
properties for mild steel.
Table 6. Material Properties for Air at 1 atm (Reference 11)
𝑻 (K) 𝝆 (kg/m3) 𝒄 𝒑 (J/kg·K) 𝒌 (W/m·K) 𝝁 (μPa·s)
100 3.5562 1032 0.00934 7.11
150 2.3364 1012 0.0138 10.34
200 1.7458 1007 0.0181 13.25
250 1.3947 1006 0.0223 15.96
300 1.1614 1007 0.0263 18.46
350 0.995 1009 0.03 20.82
400 0.8711 1014 0.0338 23.01
450 0.774 1021 0.0373 25.07
500 0.6964 1030 0.0407 27.01
600 0.632 1051 0.0469 30.58
Table 7. Material Properties for Subcooled Water at 1 atm26
𝑻 (K) 𝝆 (kg/m3) 𝒄 𝒑 (J/kg·K) 𝒌 (W/m·K) 𝝁 (μPa·s)
275 999.94 4213.5 0.56459 1682
280 999.91 4200.9 0.57409 1434
285 999.52 4192.4 0.58352 1239
290 998.8 4186.6 0.59278 1084
295 997.81 4182.9 0.60174 958
26 NIST Chemistry WebBook, “Thermophysical Properties of Fluid Systems”, National Institute
of Standards and Technology, Accessed 2016, http://webbook.nist.gov/chemistry/fluid/

41
300 996.56 4180.6 0.61032 854
305 995.08 4179.5 0.61846 767
310 993.38 4179.2 0.62609 694
315 991.5 4179.6 0.63319 631
320 989.43 4180.5 0.63975 577
325 987.19 4181.9 0.64575 530
330 984.79 4183.7 0.65122 490
335 982.23 4185.8 0.65615 454
340 979.54 4188.3 0.66058 422
345 976.7 4191.2 0.66453 394
350 973.7 4194.5 0.66803 369
Table 8. Material Properties for Brass (Reference 11)
𝑻 (K) 𝒄 𝒑 (J/kg·K) 𝒌 (W/m·K)
100 - 75
200 360 95
300 380 110
400 395 137
600 425 149
Table 9. Material Properties for 10% Glycerol
𝑻 (K) 𝝆 (kg/m3) 𝒄 𝒑 (J/kg·K) 𝒌 (W/m·K) 𝝁 (mPa·s)
295 1030.3 3786.497 0.549157 1.32
300 1028.8 3784.643 0.555621 1.167
305 1027.1 3783.87 0.562085 1.04
310 1025.2 3783.817 0.56855 0.933
315 1023.1 3784.394 0.575014 0.844
320 1021 3785.42 0.581478 0.767
325 1018.6 3786.897 0.587943 0.701
330 1016.2 3788.734 0.594407 0.644

42
335 1013.6 3790.841 0.600871 0.594
340 1010.8 3793.307 0.607335 0.550
345 1007.9 3796.134 0.6138 0.511
350 1004.9 3799.321 0.620264 0.477
Table 10. Material Properties for 20% Glycerol
𝑻 (K) 𝝆 (kg/m3) 𝒄 𝒑 (J/kg·K) 𝒌 (W/m·K) 𝝁 (mPa·s)
295 1061.3 3390.093 0.505661 1.887
300 1059.5 3388.687 0.511267 1.649
305 1057.6 3388.24 0.516874 1.454
310 1055.5 3388.434 0.52248 1.293
315 1053.3 3389.187 0.528087 1.158
320 1051.0 3390.341 0.533693 1.044
325 1048.5 3391.894 0.5393 0.946
330 1045.9 3393.768 0.544907 0.863
335 1043.2 3395.881 0.550513 0.791
340 1040.4 3398.315 0.55612 0.728
345 1037.5 3401.068 0.561726 0.672
350 1034.5 3404.142 0.567333 0.624
Table 11. Material Properties for Mild Steel
𝝆 (kg/m3) 𝒄 𝒑 (J/kg·K) 𝒌 (W/m·K)
7850 620 52

Distribution
43
Distribution:
1 MS-0836 01514 B. Mills
1 MS-0825 01514 O. Deng
Electronic copies only:
1 MS-0836 01514 B. Mills
1 MS-0825 01514 O. Deng
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1 MS-0825 01514 R. Hogan
1 MS-0828 01514 J. Hartley
1 MS-0828 01512 A. Headley
1 MS-0836 01541 S. Subia
1 MS-0836 01514 D. Dobranich
1 MS-0836 01513 S. Roberts
1 MS-0836 01514 N. Francis
1 MS-0840 01514 J. Tencer
1 MS-0840 01513 T. Koehler
1 MS-0840 01512 T. O’Hern
1 MS-9957 08253 V. Brunini
1 MS-0899 Technical Library, 9536
(electronic copy)

Steady-State Validation of Advective Bar Elements Implemented in the Aria Thermal Response Code

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