TEM Clamping Documentation
Fastener Design:
Fastener design for clamping the thermoelectric device is accomplished using
equations from Walsh’s Electromechanical Design Handbook. The relevant
equations are included in an Excel spreadsheet for ease of modification of design
parameters.
When the material of the externally threaded part (in our case a steel stud) is
different from that of internally threaded part (in our case the aluminum heat sink
baseplate) a ratio called the relative strength must be calculated. This is shown in
Equation 1 below:
𝑅=
𝐴𝑠𝑒 𝑆𝑒
Eq. 1
𝐴𝑠𝑖 𝑆𝑖
Where R = relative strength factor
Asi = shear area of internal thread,[in2]
Ase = shear area of external thread [in2]
Si = tensile strength of internal thread material [psi]
Se = tensile strength of external thread material [psi]
The table of where each of these values is located in the spreadsheet is shown
below:
R
H13
Asi [in2]:
E27
Ase [in2]:
D20
σult stud [ksi]
B22
σult plate [ksi]
B25
Next the engagement length is calculated using Equation 2 below:
𝐸𝐿 =
2𝐴𝑡𝑠
1 𝑛(𝑝𝑑 −𝐷𝑚 )
𝜋𝐷𝑚 { +[
]}
2
√3
Eq. 2
where EL = length of engagement of the thread [in]
Dm = maximum minor diameter of internal threads [in]
n = number of threads per in
Ats = tensile stress area of screw thread as given by Aux. Eq. 1 [in2]
Pd = minimum pitch diameter of external thread [in]
𝜋
0.9743 2
𝐴𝑡𝑠 = 4 (𝐷 − 𝑛 )
where D = basic major diameter of thread [in]
Aux. Eq. 1
The table of where each of these values is located in the spreadsheet is shown
below:
EL [in]:
C20
Dm int [in]:
n [in-1]:
Ats [in2]:
pd ext [in]:
D
C25
E18
B27
C18
E25
The R value serves as a modifier for the calculated engagement length for the stud. If
the R value is less than 1 the engagement length calculated by standard formulas
shown below is appropriate. If the R value is greater than the engagement length,
the engagement must be multiplied by R to obtain a new, greater engagement
length. The new engagement length required is then calculated using Equation 3
shown below:
Eq. 3
𝐺 = 𝐸𝐿 𝑅
Where G = engagement length to prevent external thread strip [in]
The new engagement length is calculated in cell I13.
The auxillary equations for calculating shear area are shown below:
1
𝐴𝑠𝑒 = 𝜋𝑛𝐸𝐿 𝐷𝑚 [2𝑛 +
1
𝐴𝑠𝑖 = 𝜋𝑛𝐸𝐿 𝐷𝑀 [2𝑛 +
𝑝𝑑 −𝐷𝑚
√3
𝐷𝑀 −𝐷𝑝
√3
]
Aux. Eq. 2
]
Aux. Eq. 3
Where Dp = maximum pitch diameter of internal thread [in]
DM = minimum major diameter of external thread [in]
And other variables have been previously defined.
The table of where each of these values is located in the spreadsheet is shown
below:
Dp [in]:
B25
DM ext [in]: D18
To calculate the fatigue factor of safety for the studs Equation 8-38 from Shigley’s
Mechanical Engineering Design was used. It is shown as Equation X below:
𝑆𝑒 (𝑆𝑢𝑡 −𝜎𝑖 )
𝑢𝑡 𝜎𝑎 +𝑆𝑒 (𝜎𝑚 −𝜎𝑖 )
Eq. 4
𝑛𝑓 = 𝑆
Where Se = endurance strength of the stud
Sut = ultimate tensile strength of the stud
σa = alternating stress of the stud
σm = mean stress of the stud
σi = initial stress of the stud
The table of where each of these values is located in the spreadsheet is shown
below:
Se [ksi]
D22
σult stud [ksi]
σa [ksi]
σm [ksi]
B22
M9
L9
σi [ksi]
K9
n
N9
Auxiliary equations are required for the σ terms and these are shown below:
𝐶(𝑃𝑚𝑎𝑥 −𝑃𝑚𝑖𝑛 )
σ𝑎 =
σ𝑚 =
2𝐴𝑡
𝐶(𝑃𝑚𝑎𝑥 −𝑃𝑚𝑖𝑛 )
2𝐴𝑡
𝐹
Aux. Eq. 4
𝐹
+ 𝐴𝑖
𝑡
σ𝑖 = 𝐴𝑖
Aux. Eq. 6
𝑡
𝐶=𝑘
Aux. Eq. 5
𝑘𝑏
Aux. Eq. 7
𝑏 +𝑘𝑚
Where Pmax = Maximum external tensile load per stud [lbf]
Pmin = Minimum external tensile load per stud [lbf]
At = Tensile area of stud [in2]
Fi = Stud preload [lbf]
C = Joint stiffness constant
The table of where each of these values is located in the spreadsheet is shown
below:
F thermal [lbf]
F57
F preload [lbf]




G57
Pmax is equivalent to the Thermal Expansion Force imparted on the stud by
the clamped members. Thus Pmax is equal to Fthermal.
P min is the condition where the clamping assembly is at ambient
temperature. At this condition Pmin = 0 lbf.
Fi is the preload, specified in the spreadsheet as Fpreload.
C is conservatively assumed to be 1, corresponding to the case where 100%
of the studing load is taken by the studs.
Bending Analysis:
To analyze the structural bending caused by clamping forces solid models were first
created in Solidworks, and then analyzed using Creo 2.0. Figure 1.1 below shows the
three dimensional enclosure model while Figure 1.2 shows the schematic drawing
of the enclosure.
Figure 1.1: 3D model of enclosure
Figure 1.2: Enclosure schematic
The Solidworks model was then
imported into Creo 2.0 for analysis. 150
lbf loads were applied to each of the 6
clamping stud holes. The loads were
applied to the interior curved surface of
each hole with the 150 lbf force
representing a clamping pressure of
121 psi, which is greater than the
desired pressure of 110 psi. The
anterior sides of the heat sink baseplate
and pressure plate (in the XX plane)
were constrained in the X and X
directions. The loading and constraints
imposed on the Creo model are shown
in Figure 1.3 at right.
Figure 1.3: Simulation Constraints & Loading
The static analysis simulation shows a maximum deflection of 0.004”, which occurs
in the pressure plate. Figure 1.4 shows a view of the bending contour of the entire
assembly, while Figure 1.5 shows a close up of the pressure plate and Figure 1.6
shows a close up of the heat sink baseplate.
The maximum deflection of the
baseplate is 6.4E-5”. This is
much less than the 0.001”
specified by Custom
Thermoelectric for acceptable
mounting surfaces. (And less
than surface roughness?)
Figure 1.4: Bending contour of clamping assembly
Figure 1.5: Bending in pressure plate
Figure 1.6: Bending in heat sink baseplate
Belleville Washer Design:
To effectively deal with the thermal expansion of the clamped members,
Belleville Washers must be employed. Doctor Stevens has conducted lab
experiments which indicated that for thermoelectric modules clamped with thermal
grease as an interface material, the benefits of increased clamping pressure
effectively stopped increasing above about 110 psi. Because the thermal expansion
analysis is based on a large number of simplifying assumptions the pressure from
thermal expansion is less predictable than the pressure applied during preload. In
addition, the preload pressure applied can be verified using pressure film, while the
high temperatures of a full “hot” run make the pressure film unsuitable for verifying
the thermal expansion pressure. Due to this, the assembly is designed to achieve 90
psi of the desired 110 psi of clamping pressure during preload. The thermal
expansion pressure above 110 psi will be minimized to protect the TEMs, but the
modules should easily withstand up to 200 psi if correct mounting procedures are
followed.
In order to fit the #8 studs specified in the stud design, #8 washers must be
used. For this analysis, washers from Precision Specialties are used. Total thermal
deflection of the clamped members during heated system operation is calculated
using the linear form of the thermal deflection equation, shown in Equation 5 below.
Eq. 5
𝛥𝐿 = 𝐿𝛼𝐿 𝛥𝑇
The thermal expansion of the clamped members contributes to the thermal force
exerted on the studs and washers, while the thermal expansion of the stud itself
works to ease the thermal expansion force exerted on the studs and washers. By
subtracting the expansion of the stud from that of the clamped members, we can
calculate a relative thermal expansion figure which will be useful in our analysis.
Those calculations are shown below. It should be noted that the temperatures used
in these calculations are obtained from our Thermal Circuit spreadsheet.
Thermal Expansion of Heat Spreader
alpha spreader [1/oF]
delta T [oF]
E spreader [psi]
Height [in]
δ [in]
9.30E-06
626
1.60E+07
0.984251969
5.73E-03
Thermal Expansion of TEM
alpha TEM [1/oF]
delta T [oF]
Estud [psi]
Length [in]
δ [in]
9.30E-06
392
0.118110236
4.31E-04
Thermal Expansion of Clamping Insulation
alpha ins [1/oF]
delta T [oF]
Estud [psi]
Length [in]
δ [in]
0.5
alpha plate
0.001
Thermal Expansion of Pressure Plate
delta T [oF]
Estud [psi]
Length [in]
δ [in]
1.20E-05
309
0.125
0.0004635
Thermal Expansion of Stud
[1/oF]
alpha stud [1/oF]
delta T [oF]
Estud [psi]
Length [in]
7.30E-06
212
2.90E+07
δ [in]
2
3.10E-03
From these values we obtain a relative thermal expansion of 0.00453”. This is the
deflection that will be imposed on the Belleville washers due to thermal expansion
of the clamping assembly. Precision Specialties lists a deflected height for each of
their washers, and the load at that deflected height. For a #8 sized washer
(specifically the PSB W0 343 019S washer), the initial washer height is 0.028”, while
the deflected height is 0.023” and the load at that height is between 43 and 58 lbf.
Assuming a linear spring constant, we can back out a spring constant for this washer
using Equation 6 below.
𝐹
𝑘 = ∆𝑥
Eq. 6
By averaging the two load extremes to a 50.5 lbf load for our deflection, we arrive at
a k of 10,100 lbf/in for the case where only one of these washers is used per stud.
Multiplying k by our relative thermal expansion results in a thermal expansion force
of 45.7 lbf per stud, which corresponds to a thermal expansion induced pressure of
about 37 psi over the 7.44 in2 clamping area. Adding this pressure to our desired 90
psi of preload pressure results in a total clamping pressure of 127 psi, which is just
slightly over the optimal value of 110 psi. Finally, total washer deflection must be
calculated. The total force per stud including preload and thermal expansion for this
case is 157 lbf. Dividing this by the spring constant results in a total washer
deflection of 0.016”. The bottoming-out deflection of these studs, however, is only
0.01”! This indicates that using one washer alone is not an appropriate solution.
The best arrangement seems to be a group of 3 washers in series put in parallel with
another group of 3 washers in series. This type of arrangement is shown in Figure
1.7. For this arrangement, the spring constant is increased by a factor of 1.5 to
15150 lbf/in, and the bottoming-out deflection of the washer assembly is doubled to
0.02”. Running our calculations again for this configuration, we obtain a thermal
load per stud of 68.6 lbf and a stud preload of 111.6 lbf. These values correspond to
a thermal clamping pressure of 55 psi and a preload clamping pressure of 90 psi.
This yields a total clamping pressure of 145 psi. Finally, the deflection for this
arrangement is 0.012” , which is just slightly over of our bottoming-out deflection of
0.02”. This indicates a safe operating point for the washers.