Time-temperature characteristics of thin skinned models as affected by thermocouple variable

Larson, M. B.

English

NASA Technical Reports Server

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FOURTH SEMIANNUAL REPORT 
of the work on 
NASA GRANT NO, NGR-35-002-003 
"Time- Temperature Characteristics of Thin- Skinned Models as Affected 
by Thermocouple Variables" 
GPO PRICE $ 
CFSTI PRICE(S) $ 
bY 
Milton B. Larson 
University of North Dakota 
Grand Forks, North Dakota ' g/ @ Ha. . m p y  (HC) 
May15, 1967 Microfiche (MF) ,65' 
ff 653 July 66 
Part of the reason this report has been delayed is to allow the 
inclusion of the thesis of Paul Nelson, which is provided as Appendix I. 
Several errors and omissions in the Third Semiannual Report have 
been noted and are mentioned throughout this report. 
Continuation of Two-wire Problem 
Inasmuch as it appears that an exact solution of the two-wire problern 
is beyond the scope of the present prajeect, i t  is desirable to see if the 
solution already obtained for the case of two semi-infinite sheets butted to 
an infinite sheet (eq. 36, 37, and 38, Third Semiannual Report) may be 
used to approximate such a solution if only to indicate the importance of 
the various terms in the solution. Such an  indication would be very helpful 
in the design of the shock tunnel models under consideration. 
Solutions for  the single simple w i r e  and the single sheet junction have 
been compared in terms of the dimensional variables. 
Single S imde Wire 
L 
where x is interpreted as  the arc length alond the skin from the junction. . -  
E 
- 
(CATEOORY) (NASA CR OR TMX OR AD NIJMBERI 
https://ntrs.nasa.gov/search.jsp?R=19670018338 2020-03-12T11:38:16+00:00Z
Single Simple Sheet 
r 
I 
-2 -  
. .  
where x is the distance in the skin from the junction. 
Comparison of these solutions indicate that they will be identical when 
the radius of the w i r e  is equal to the thickness of the plane sheet. This 
suggests that the two sheet solutions may be used to understand the two 
wi re  problem by replacing the sheet thicknesses by the radii of the wires. 
We a re  not able to prove that this procedure leads to a correct solution, 
but based upon the physics of the problem we  feel that such a solution is of 
value--at least along the line joining the two wires. 
Page 7 of the previous report was incomplete and should have been 
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- 3- 
Equation (37) on Page 9 contained an error in an exponent on K in the 
last line. (Correct by changing 3 4 2 ) .  
rewritten in the following form, allowing easier interpretation of the 
importance of each term: 
Equations 36, 37 and 38 may be 
for 0 x < d /  - -  
- 4- 
It is only recently that we have been aware of the similarity of 
these equations to those of transmission lines with multiple reflections. 
Further study of this similarity and its meaning may help in understanding 
the solutions obtained. 
Several features of these equations are of interest. 
1. The ratios B2 or  B3 a re  always less than 
2+B2  2+B3  
one and usually can be made of the order of 0 . 1  by selection of appropriate 
materials. 
2. The value of i2erfcy is 0.25 for y = 0, drops to less than 1% of 
3. Because of the occurrence of d+ in the denominator of the 
this value for y = 1 .  5, and is tabulated a s  0.000000 for y = 3.0. 
argument of i2erfcy more terms will be needed to satisfactorily represent 
the temperature as the time increases. 
A s  a consequence, if we are  interested in  the situation where x = 0 
and we a r e  willing to neglect terms which a re  approximately 1% of the 
first  correction term ( B2 4 i2erfc 
neglect any additional terms if  d/sl > 1.5J+. 
), then we may -X 
2 + B 2  2J+ 
- 5- 
B2 4i2erfc -X 
Z F j  It should be noted that the first correction term, ( 
is exactly the term which would be expected for the one sheet in the 
absence of the second. The next correction term (considering n = 1) would be 
. This is exactly the term to be d / s l  - X  B3 4i2erfc 
2 + B3 2 h  
expected a s  a contribution from the second sheet in the absence of the 
first if superposition were valid. 
The following data present the values of T/4, at x = 0 (base of the 
constantan sheet) for the case of 0.0025'' thick sheets of constantan and 
chrome1 fastened to a 0.004" thick copper skin for nondimensional times 
corresponding to approximately 0 .1  milli -second and 4.1 milliseconds after 
s tar t  of heating. The value of zero spacing was obtained from the solution 
for two wires  at  a single point, while the value at infinity was calculated 
using the single sheet equation. 
Values of Tau/PHI 
d/Sl 4 = 1.0  4, = 41.0 
0 
4 
8 
12 
16 
20 
24 
28 
32 
00 
0.87072 
0.92647 
0.92652 
0.92652 
0.92652 
0.92652 
0.92652 
0.92652 
0.92652 
0.92652 
0.87072 
0.89933 
0.91497 
0.92228 
0.92515 
0.92614 
0.92643 
0.92650 
0.92652 
0.92652 
These data should indicate the nature of Tau/Phi for w i r e s  of a radius 
equal to the thickness of the sheets. 
A solution at various values of x has been calculated for d/sl  = 16 
and is tabulated on Table I. 
in each of the summations. In no case did the third term contribute to 
This solution was made taking three terms 
the solution. 
- 6- 
II 
-8 I l l  3 c 
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F 
0 
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d 
C 
0 
0 
0 
0 
II 
3 
d 
c 
0 
0 
0 
0 
0 
d 
m m m m m 
m 
9 m 
o\ m 
d m m 
cv m 
Q 
m m 
b 
b 
d m m 
cu 
9 m m m 
m m m m m 
0 
0 
0 
8 
4 
0 
0 
0 
0 
0 
4 
-7-  
Single W i r e  with Added Capacity and Resistance 
A minus sign was omitted from the bottom equation on page 13. 
The last line of the equation at the bottom of page 16 should have read 
Equation 40 should then have been 
The  coefficients of equation (40) have been evaluated in terms of 
the parameters: L, N and B. The resulting equation has been put into 
a form containing the term 
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This is the term expressing the magnitude of the correction in the 
event of no added capacity or resistance. Thus, other terms appearing 
in the expression represent second order corrections occurring due to 
such capacity or resistance. 
I -. d1 * .  
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-0 - Fi 
/ 
liw 
A s  is pointed out by Paul Nelson in Appendix I, the above solution 
i s  valid if G and H a re  real o r  complex but unequal. M r .  Nelson has also 
formulated solutions which can be more readily used for the unequal 
complex case and for the case where G = H. Future work will attempt 
to define more clearly the probable use  of these equations in practice. 


Time-temperature characteristics of thin- skinned models as affected by thermocouple variables Semiannual report

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