606 BELL SYSTEM TECHNICAL JOURNAL 



Here A is nearly constant for given tube geometry and materials. In 

 radiation the power lost varies as the area, which varies as / , and as the 

 temperature to the fourth power. 



The power lost by end cooling, for the large and the small tube, P«o and P.i 

 will be given by 



P«o = BloT^ (13.19) 



Pei = BhTi. (13.20) 



Here B is another constant. These relations express the fact that the 

 power lost by end cooling (thermal conduction) varies as cross sectional 

 area divided by length and hence as / and as temperature difiference, taken 

 as proportional to T. 



Now, in scaling the tube the power to be dissipated has been kept constant. 

 Further, in making the tube small, the hottest point of the grid cannot be 

 run hotter than the melting point of the wire; in fact, it cannot be run nearly 

 this hot without unreasonable evaporation of metal. Suppose we let the 

 grid in the smaller tube attain the maximum allowable temperature Tm 

 and let the power the wire must dissipate be P. Then for the large tube 



P = Pro-\- Peo = {AloTl + B)loTo (13.21) 



and for the smaller tube 



P --= Pri+ Pel = (AhTl + B)hT,„. (13.22) 



Hence, the smallest value h can have without running the grid too hot is 

 given by the equation 



To (AIqTI + B) 

 Tm{AhTl+B) 



We see that if /o is very small. 



^1-^0^" ',::z z - (13.23) 



AloT:n«B 



AhT^n « B 



(13.24) 



Numerical examples show that this is so for a tube such as the 2K50. This 

 means that nearly all of the power dissipated by the grid is lost through end 

 cooling, not radiation.^* Further, in the 2K50 the grid is already operating 

 near the maximum allowable temperature. Hence, nearly. To = Tm and 

 the smallest ratio in which the tube can be scaled down without overheating 

 the grids is approximately unity. This means that in making a tube for 

 .625 cm. the grid wire cannot be made half the diameter of the wire used in 



"The fact that one kind of dissipation predominates in both cases justifies the as- 

 sumption of the same temperature distriliution in both cases. 



