Appendix A: THEORY FOR FINITE ELEMENT MODELING
There are two general approaches to thermal calculations when modeling a structure
containing both solids and fluids. Conjugate heat-fluid transfer calculations estimate both energy
and mass transfer in the solid and the fluid. This form of calculation requires that the user have
substantial computer resources, since a large number of finite elements and calculation iterations
are required for the calculations to converge. The alternative method separates fluid flow from
heat transfer. The fluid flow simulation calculates the flow rates and flow distributions in the
cooling channels, which is the most important information needed to estimate the convection
heat transfer at the solid/fluid boundary as a function of wall temperature. The pre-calculated
estimates of the convection heat transfer are used as inputs into the thermal model. This approach
has the advantage that we can account for the wide range of the convection heat transfer due to
the phase change of the cooling fluid, such as nucleate boiling, fully developed boiling and
critical heat flux. In addition, the approach substantially reduces the calculations times and
computer resources needed for the calculations. The only disadvantage is that this type of
modeling simplifies the geometry of the object. In reality heat transfer depends upon flow rate
and cooling tube geometry, which can vary in complex ways from location to location in the
object. Assigning the heat transfer characteristics of the solid/fluid boundary tends to simplify
the variations in convection heat transfer over the surfaces of the cooling tubes. Because of
limitations in our computer resources and in the number of nodes that could be used in our
academic version of the finite element modeling software, it was necessary that we use the
second calculation approach for our calculations.
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A. Heat Transfer Mechanisms and Thermal Modeling
There are three heat transfer mechanisms: conduction, convection and radiation. In our
thermal models, conduction was modeled, convection was accounted for using the techniques
described below and radiative heat transfer was ignored. Our preliminary calculations suggested
that the maximum temperature reached in the model would not exceed 1600 K (750W for
megavoltage target and 1200W for kilovoltage target). The radiative heat flux is less than 7.7
W/cm2 at this temperature{Yih & Wang 2001 YIH2001 /id} and can thus be safely ignored.
The convective heat transfer is a very complex phenomenon. As the wall temperature
increases, the processes that govern heat transfer pass through a number of distinct stages: forced
convection heat transfer (qfc), nucleate boiling, fully developed boiling (qb), and critical heat flux
(qchf). When the wall temperature of the cooling tube is below the boiling point of the fluid, the
heat transfer is dominated by forced convection heat transfer, which is governed by the thickness
of a stationary layer of fluid close to the wall, which in turn is due to the viscosity of the fluid.
This thin layer of fluid reaches an elevated temperature and acts as an insulating layer, reducing
the heat transfer across the solid/fluid boundary. Forced convection heat transfer is a function of
thermal properties and velocity of the coolant, the shape of cooling path and the temperature
difference between cooling wall and bulk of the fluid. We have used the equation for forced
convection suggested by Bjorge et. al.,{Bjorge, Hall, et al. 1982 BJORGE1982 /id} which
applies to fully developed turbulent flow in sub-cooled and regions of low thermal quality [The
thermal quality is defined by the vapor content of a liquid. When all of the liquid is changed into
vapor at the boiling temperature, the quality is 1.0. When the liquid and the vapor are mixed, the
thermal quality is between 0 to 1. When the temperature of liquid is lower than its boiling
temperature, the thermal quality has a negative value.]:
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qfc = 0.023(GD/μ)0.8 (μCp /k)0.3 k/D (T-Tb)
2
(1)
where G, D, μ, Cp, k, T, and Tb are mass flux, tube diameter, dynamic viscosity, specific heat,
thermal conductivity, temperature of the wall and temperature of the bulk of the coolant,
respectively. The dynamic viscosity was evaluated at the film temperature, which is the average
temperature of T and Tb .
As nucleate boiling starts, the thin layer of fluid is converted into a gas (boiling bubble)
that is much more efficient at transferring heat into the bulk of the fluid - due to the agitation
created by bubble motion. Fully developed boiling depends mainly on the thermal properties and
the applied pressure of the coolant. We have used the equation derived by Rohsenow to predict
the heat flux as a function of wall temperature.{Bjorge, Hall, et al. 1982 BJORGE1982 /id}
qb = A ·ΔTsat3
(2)
where ΔTsat is the temperature difference with respect to saturation temperature. The “A” is a
complex coefficient dependent upon the properties of water.{Bjorge, Hall, et al. 1982
BJORGE1982 /id} These properties (density, thermal conductivity, specific heat, enthalpy of
vaporization, surface tension, viscosity, and saturation temperature) have been obtained from the
literature.{Gray 1972 GRAY1972 /id}
When the bubble generation becomes too great, it may generate an insulating vapor
blanket and prevent the cooling liquid from reaching the hot surface. The heat flux where the
bubbles start to aggregate and unstable cooling conditions starts to occur is called the critical
heat flux. It depends on many parameters such as the geometry of cooling path, surface finish,
coolant velocity, pressure, and the size of the heat source. We have used an equation derived by
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Inasaka et.al. to predict the critical heat flux.{Inasaka & Nariai 1997 INASAKA1997 /id}
qchf = C Hfg G 0.4 µ 0.6 D -0.6
(3)
where: Hfg is latent heat of evaporation (J/kg), G is mass velocity (kg/m2s), µ is viscosity of
liquid, D is the tube inside diameter, and C is the correction factor proposed by Inasaka as a
function of pressure, mass velocity, and exit equilibrium quality of the fluid. Equation (3) applies
to low-pressure (0.1-7.0MPa), medium flow-rate (5.5-30Mg/m2s) and long heating length (10cm)
conditions, which closely match our physical situation (0.3MPa, ~8Mg/m2s).{Nariai, Inasaka, et
al. 1987 NARIAI1987 /id}
The total heat flux curve is formed by combining the three curves from each heat transfer
stage: forced convection heat transfer, fully developed boiling heat transfer, and critical heat
flux. We have used the mathematical relationship suggested by Bjorge et.al.{Bjorge, Hall, et al.
1982 BJORGE1982 /id} to combine the heat transfer curves for forced convection and fully
developed boiling.
qr = (q2fc + q2b[1-w3])1/2
(4)
w=(T-Tib)/(T-Tsat)
where T, Tib and Tsat are the temperature of solid surface, incipient boiling temperature, and
saturation temperature of liquid, respectively. To extend the temperature range to the region of
critical heat flux, the following equation has been used in this paper:
qtot = (q-2r + q-2chf [1-w3])–1/2
(5)
This generates a smoothly changing curve of heat flux as a function of temperature of the
solid/fluid boundary. Equation (5) can be used to generate a table of values for the total heat flux
versus wall temperature, which can then be used in the finite element calculations to represent
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the convection heat transfer coefficients.
B. Thermal-Mechanical Modelling and Thermal Damage
Once the thermal calculations have been completed, a temperature distribution
throughout the model is known. From this temperature distribution, the stress and strain in the
model can be calculated if the mechanical properties of the materials forming the model are
known. These quantities (coefficient of thermal expansion, Young’s modulus, Poisson’s ratio,
strength coefficient, and hardening exponent) have been obtained from the literature.{White &
Minges 1997 WHITE1997 /id}{Yih & Wang 2001 YIH2001 /id}{Upthegrove & Burghoff 1953
UPTHEGROVE1953 /id} Typical finite element calculations yield the material behavior for
monotonic loading conditions (e.g., linear stress/strain quantities). Unfortunately, the material
behavior under cyclic loading conditions (e.g., repeated thermal loading and unloading) is the
quantity needed to estimate the life-time of our target assemblies. Thus, we need to convert from
monotomic to cyclic material properties. One method of converting between linear stress/strain
(monotonic material properties) and nonlinear cyclic stress/strain is Neuber’s rule.{Neuber 1961
NEUBER1961 /id} We have used Neuber’s rule to convert linear monotonic stress/strain into
cyclic stress/strain using the equations below:
ε = σ/E +(σ/K’)1/n’
(6)
ε•σ = (Kσ S) • (Kε e)
(7)
where: K’ is the cyclic strength coefficient and n’ is the cyclic hardening exponent. Those can be
calculated from the material properties such as Young’s modulus (E), ductility (εf or true fracture
strain), and ultimate strength(σu).{Muralidharan & Manson 1988 MURALIDHARAN1988 /id}
Kσ is elastic stress concentration factor, Kε is the elastic strain concentration factor, S is the
engineering or nominal stress and e is the engineering or nominal strain. The (Kσ S) and (Kε e)
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represent the true stress and strain of interest, respectively and these quantities are output by the
linear finite element analysis. Equations (6) and (7) can be solved for cyclic stress (σ), or cyclic
strain (ε) by iteration or by various numerical techniques.
There are a number of approaches to convert linear into cyclic stress/strain which are not
as conservative as Neuber’s rule, such as the use of a fatigue notch factor instead of elastic
concentration factor,{Topper, Wetzel, et al. 1969 TOPPER1969 /id} use of a modified
relationship between linear and cyclic stress-strain, or use of a strain energy density rule.
However, we have used Neuber’s rule since it represents the most conservative technique of
converting between linear and cyclic stress/strain and thus likely to lead to safe estimates of the
maximum thermal load.
When a stress load is removed, the material behavior differs from that under stress
loading. This is known as the Bauschinger effect. Materials for which the hysteresis loop
(between loading and unloading stress) can be described by magnifying the cyclic stress/strain
curve by a factor of 2 are said to exhibit “Masing-type” behavior. This type of behavior is typical
of many common metals.{Stephens 2001 STEPHENS2001 /id} All the materials in our model
are assumed to exhibit Masing-type behavior. The following equations are used to model the
material deformation when heat loading is removed.
ε = σ/E +2• (σ/2K’)1/n’
(8)
ε•σ = (Kσ S) • (Kε e)
(9)
where the symbols are the same as in equations (6) and (7). Knowing the strain range ε, we can
obtain the strain amplitude, εa = ε/2, for the fatigue analysis.
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Once the strain amplitude is known, the modified universal slope method{Muralidharan
& Manson 1988 MURALIDHARAN1988 /id} can be used to estimate the number of loadings to
fatigue failure:
εa = 0.623(σu/E)0.832(2Nf)-0.09 + 0.0196(εf)0.155(σu/E)-0.53(2Nf)-0.56
(10)
where σu is ultimate strength, εf is true fracture strain (or ductility), and Nf is the number of
loading before fatigue failure. This equation can be solved for the number of loading to fatigue
failure, Nf, by an iterative or numerical method, given the strain amplitude, εa.
In most situations variable amplitude loading occurs, where the object is exposed to
different stress loads (i.e., different heat loading) for different operating modes. When variable
amplitude loading occurs, determining the fatigue life of a component requires a method that
determines how each load condition contributes to the overall fatigue of the object. One such
method is Palmgren-Miner’s damage rule, where accumulation of damage is assumed to be
linear.{Miner 1945 MINER1945 /id}
m
ni
N
i 1
1
(11)
i
where ni is the number of cycles of damage and Ni is the life at an i-th level of amplitude. The
fatigue failure is expected to occur when the sum of the damage becomes 1.0. The reciprocal of
Ni provides a measure of the damage at an i-th level of loading amplitude.
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Appendix B: SUPPLEMENTAL MATERIALS AND METHODS
A. Heat Flux Calculations
The fluid flow in the cooling channels can be quite complex. Therefore, to get a good
estimate of G, the mass flow rate, (see Eqs. 1 and 3), we first modeled fluid flow in the cooling
tubes using ANSYS 5.6. Due to license limitations our software could only model the cooling
tubes and insufficient nodes were available to model the rest of the megavoltage target assembly.
The fluid velocity across the surface of the inlet of the cooling tubes was assigned to the finite
elements assuming that the bulk fluid flow rate is 60 ml/s (1 gallon/minute{Varian Medical
Systems 1999 VARIANMEDICALSY1999 /id}). This is a minimum value and so should
represent a conservative estimate of the flow. The inlet flow was assumed to be uniform across
the opening except for the elements right at the wall of the tube where the flow was assumed to
be zero. Since, fully developed turbulent flow occurs quite rapidly in the model,{Rohsenow 1973
ROHSENOW1973 /id} and our flow model has a long introductory inlet, the boundary
conditions have negligible effects on the final results. The result of the calculations is the flow
rate at all locations in the cooling tube. The average flow rate (flow velocity) times the density of
the fluid gives G, the mass flow rate. This numerical value has been used to calculate the heat
flux for that region of the cooling tube. In one of our models, the flow rate on one side of the
cross-over tube is different from the flow rate on the other side. For this situation the mass flow
rate for each half of the tube has been calculated independently.
From the G values, different curves for the total heat flux as a function of wall
temperature can be derived. The cooling tubes in each thermal model have been divided into four
regions depending on the mass flow rate. Each region has a different table of total heat flux
versus wall temperature assigned to it, which in finite element calculations is called the
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convection heat transfer coefficient. At the transitions between different regions the convection
heat transfer coefficient has been assigned to give a conservative estimate of the heat transfer, in
W/m2, at that location in the model.
B. 3D Models
Four materials - oxygen-free, high-conductivity (OFHC) copper, tungsten, tungstenrhenium, and the braze material have been included in the models. The thermal properties (e.g.
thermal conductivity, heat capacity, thermal expansion, density) and mechanical properties
(Young’s modulus, yield stress, Poisson’s ratio, ductility, ultimate strength) of these materials
have been found from the literature{White & Minges 1997 WHITE1997 /id}{Yih & Wang 2001
YIH2001 /id}{Upthegrove & Burghoff 1953 UPTHEGROVE1953 /id} or provided by the
manufacturer. In the case of the braze material many of the above quantities were only known at
room temperature. Since the braze material is a copper alloy, we assumed the braze material
exhibited the same temperature dependence as OFHC copper.
Since the calculation speed depends on the number of elements used in the models, it is
beneficial to minimize the number of elements to increase the calculation speed. We tested three
thermal models where the linear dimensions of elements were scaled by 50 per cent. Thus, the
number of elements in each model increased by 30~70 per cent as the scale was reduced. We
found essentially no difference in the temperature distributions of the three models and so we
used the intermediate model for all subsequent calculations. This struck a balance between
calculation time and accuracy. Interactions with the manufacturer suggested that our elements
were smaller than those typically used for similar calculations.
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C. Simulation procedures
Heat loading was applied gradually using small sub-steps. The temperature distribution
was calculated in each step and material properties were re-evaluated for the next step. In every
heat loading step, the standard deviation of the relative temperature change (known as the L2
norm) for each element was used to judge the convergence of the thermal simulation. The
relative temperature change was the temperature change between successive iterations divided
by the temperature of the current iteration. The calculation was assumed to have converged
when the L2 norm was less than 0.1 percent.
The maximum temperature of key locations in the target assembly as a function of input
power and source size were calculated. The input power ranged between 500 – 1000 Watts for
the megavoltage target and 600 – 1800 Watts for Target 1 and Target 2. After thermal analysis
was complete, the thermal elements were converted to structural elements for the stress-strain
(thermal deformation) analysis. The temperature distributions, which were the result of the
thermal analysis, were used as the input load in the structural models. The resulting output from
ANSYS 5.6 was linear stress/strain distributions.
Many structural constraints were assumed in the thermo-mechanical calculations. The
plane where cooling pipes were brazed was assumed to be securely fixed by the pipes. We also
assumed that there was no residual stress after brazing and that no residual stress existed at a
temperature of 40oC. The structural analysis generally required much finer sub-steps than the
thermal analysis to make the simulation converge. The calculation was assumed converge when
the L2 norm was less than 0.5% for force and 5% for displacement (the default values in ANSYS
5.6).
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There are two ways to estimate cyclic stress/strain properties. The first way is to use the
non-linearity material features, such as plastic deformation, provided by ANSYS 5.6. In this
way, the cyclic stress/strain relation can be considered as one of the material non-linearities.
However, ANSYS 5.6 is unable to generate the exact shape of cyclic stress/strain curve and the
calculations take a prohibitive amount of time. Therefore, we used Matlab 5.3, release 11 to
convert the ANSYS 5.6 linear stress/strain results into cyclic stress/strain using Eqs. 6-11. The
linear stress/strain, temperature, and the material type for all the elements in the model were
exported into the Matlab 5.3 environment in the form of a text file. Material properties such as
Young’s modulus, ductility, and ultimate strength as a function of temperature were used for the
fatigue analysis. Cyclic stress/strain data were calculated from the linear stress/strain data using
Neuber’s rule (see Appendix A for theoretical details). The strain range and amplitude
calculations were then performed using the Masing assumption. The Nelder-Mead simplex
method, which is one of the general optimization functions provided in Matlab, was used for the
strain range and amplitude calculations. Fatigue life or damage was also calculated using the
Nelder-Mead simplex method. Strain life curves were generated using the modified universal
slope method based on the assumption that all the materials were fully annealed during the
brazing process and contained no residual stress/strain at a temperature of 40 OC.
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APPENDIX C: SUPPLEMENTAL RESULTS
A. Heat Flux Calculations
The results of the flow calculations for Target 1, from which the heat flux across the
solid/fluid boundary can be estimated, are shown in Fig. C1(a). While the flow is complex, it can
be seen that there are four distinct flow regions: the inlet and outlet regions; the cross-tube,
which has two different flow regions - on either side of the tube; and the end tube. Each of these
regions has been labeled R1-R4 in Fig. C1(a). The corresponding heat flux values as a function
of wall temperature are shown in Fig. C1(b). As the wall temperature increases, the heat flux
increases, as different phenomena contribute to the heat exchange. At wall temperatures below
170 oC, forced convection dominates, then nucleate boiling and fully developed boiling occur,
which greatly increase the heat transfer across the boundary. Finally, critical heat flux occurs,
where no further increase in heat transfer occurs. There are a number of methods of estimating
the magnitude of heat transfer when critical heat flux occurs and we performed thermal
calculations using a number of different estimates for critical heat flux. The results of the thermal
calculations did not depend upon the values used for critical heat flux because very few (or even
none) of the boundary elements reached such high temperatures (> 200 oC).
B. Kilovoltage Stress-Strain Analysis
Fig. C2 (a) shows cyclic stress/strain relationship of the materials forming the target
assembly - generated using the modified universal slope method [Eq. (6)] - while Fig. C2 (b)
shows the fatigue properties of the materials, which are known as strain life curves [Eq. (10)].
Note how the same stress in the tungsten rhenium alloy (W-25Re) results in much less strain
(i.e., deformation) than in pure tungsten [Fig. C2 (a)].
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(a)
12
(b)
R1
R1
R3
R2
R4
[m/s]
(c)
R1
(d)
R1
R2
R4
R3
[m/s]
Fig. C1
Heat transfer at the cooling tube of the target assembly. The fluid velocity
in the cooling path of the existing target is shown in (a). The maximum fluid velocity in the
existing cooling path is ~10 m/s. The cooling surface has been divided into four regions, R 1, R2,
R3 and R4, depending on the flow rate, one of the most important parameters which determines
the heat transfer at the surface. Figure (b) shows heat transfer in each region - as a function of the
wall temperature of the cooling tube. Figure (c) shows the fluid velocity in the Target 2 design
where another cooling channel has been added to the target assembly. The shape of cooling tube
is also designed to reduce flow resistance and to balance the flow rate at each channel. The
overall flow rate in Target 2 is increased by ~60 percent compared to that in Target 1. The heat
transfer for each region of Target 2 is shown in (d).
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(a)
(b)
Fig. C2
Cyclic stress-strain curves and total strain-life curves of the target
materials - generated using the modified universal slope method. The target materials are
assumed to be fully annealed during the brazing process. The yield stress of copper (solid line) is
increased under the cyclic load due to cyclic hardening effects while the yield stress of tungsten
(dashed line)/ tungsten alloy (dashed dot line) are decreased due to cyclic softening effects, as
shown in (a). Figure (b) shows allowable the total strain of each material as a function of number
of loading to failure.
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