Difference between revisions of "Faraday Cup Temperature"

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Now apply the Stefan-Boltzmann Law for one Faraday cup rod
 
Now apply the Stefan-Boltzmann Law for one Faraday cup rod
  
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  <math> P = 0.924 A \sigma T^4 </math>
+
  <math> P = (0.924)\ (A)\ (\sigma)\ (T^4) </math>
  
 
Here,
 
Here,
  
−
   A is the radiated area of one rode. Assume it's 2 cm^2
+
   A is the radiated area of the rode.radiatedanradiated
 
   <math> \sigma = 5.67*10^{-8} \frac {W}{m^2 K^4} </math> is the Stefan-Boltzmann constant.   
 
   <math> \sigma = 5.67*10^{-8} \frac {W}{m^2 K^4} </math> is the Stefan-Boltzmann constant.   
  
−
 
+
Assume the all beam power goes to one FC rod and radiated pawer is 2 cm^2 (two back sides) we cac calculate the corresponding temperature:
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Solving for Temperature and taking into account the two sides of the converter we get:
 
  
 
<math> T^4 = \frac {P}{(0.924)(2A)(\sigma)} </math>
 
<math> T^4 = \frac {P}{(0.924)(2A)(\sigma)} </math>
−
 
−
where <math> \sigma </math> is the Stefan-Boltzmann constant, <math> \sigma = 5.67*10^{-8} \frac {W}{m^2 K^4} </math>.  Assume a beam spot diameter on the converter surface of 5mm, or an area of <math> A = 19.62 mm^2 = 19.62*10^{-6} m^2 </math>.
 
−
 
−
Plugging in the numbers we see that the temperature will increase <math> 217.2 K </math>.  Now, adding in the temperature of the converter at room temperature we get :
 
−
 
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<math> T = 300 + 217.2 = 517.2 K</math>
 
  
 
The melting temperature of Aluminum is <math> 933.5 K </math>.
 
The melting temperature of Aluminum is <math> 933.5 K </math>.

Revision as of 05:16, 14 October 2010

Calculating the temperature of a Faraday Cup Rod

Number of particles per second and corresponding beam power

Assume electron beam parameters at FC location are: 

Frequency:     f=300 Hz
Peak current:  I=3 Amps
Pulse width:   t= 50 ps
Beam energy:   E=45 MeV

The number of electrons per second at FC location are: 

[math] N = \frac {Q} {e} = \frac {f I ∆t} {e} = \frac {(300\ Hz)(3\ A) (50\ ps)} {1.6*10^{-19}\ C} = 0.28*10^{12} [/math]

The corresponding beam power for 45 MeV electron beam is: 

[math] P = (0.28*10^{12}\ \frac{e}{sec}) (44\ MeV) (1.6*10^{-19} \frac{J}{eV}) = 1.9\ W[/math]


Temperature calculation

Now apply the Stefan-Boltzmann Law for one Faraday cup rod 

[math] P = (0.924)\ (A)\ (\sigma)\ (T^4) [/math]

Here, 

 A is the radiated area of the rode.radiatedanradiated
 [math] \sigma = 5.67*10^{-8} \frac {W}{m^2 K^4} [/math] is the Stefan-Boltzmann constant.

Assume the all beam power goes to one FC rod and radiated pawer is 2 cm^2 (two back sides) we cac calculate the corresponding temperature:

[math] T^4 = \frac {P}{(0.924)(2A)(\sigma)} [/math]

The melting temperature of Aluminum is [math] 933.5 K [/math].

Conclusion

An Aluminum converter that is 1/2 mil thick being struck by a 44 MeV electron beam with a 50 picosecond pulse width, 300 Hz rep rate, and 50 Amp peak current is found to be safe from melting.





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