Multilevel Cooling Systems August Hale May 16, 2014 August Hale Multilevel Cooling Systems May 16, 2014 1 / 20
Introduction Introduction Computers all require some kind of cooling system These can be either very simple or very complex. Cell phones are essentially miniature computers; they cool themselves very simply, using the entire phone as a heat sink In contrast, More powerful devices use increasingly more intricate cooling apparatus. Desktop computers use heat sinks and fans, while high powered supercomputers use specially designed refrigeration circuits. August Hale Multilevel Cooling Systems May 16, 2014 2 / 20
Introduction Basis of Model By making the reasonable assumption that heat spreads much faster through each level of a cooling system(through a heat sink or a fluid, which would usually be turbulent) than from one layer to another(from a heat sink to a fluid, or from one fluid to another, across a barrier), A model of the heat flow in a system can be constructed based off of Newton’s law of Heating/Cooling. August Hale Multilevel Cooling Systems May 16, 2014 3 / 20
Equations Basic equations The equations being used are Newtons law of heating/ Cooling, and the simple differential equation that models the flow of a solution of different concentrations between reservoirs. Each equation (within each system of equations) is composed of some combination of two of these equations, excepting the first equation of each system, in which one of the components is a constant corresponding to the continuous heat being fed into the system. August Hale Multilevel Cooling Systems May 16, 2014 4 / 20
Equations Modifications to Equations I have collected the properties of the objects involved in Newton’s equation into two constants, one for each heat sink’s properties, and one for each boundary’s properties. Also, I have introduced an additional constant into the solution flow model in the case of the refrigeration model, in order to account for the heating and cooling effect of the compressor. August Hale Multilevel Cooling Systems May 16, 2014 5 / 20
Equations The Equations dX 1 dt = C 1 ( X 2 − X 1 ) + I B 1 dX 2 dt = C 2 ( X 3 − X 2 ) − C 1 ( X 2 − X 1 ) B 2 B 2 dX n = C n ( X n +1 − X n ) − C n − 1 ( X n − X n − 1 ) dt B n B n dX 1 dt = R ( X 2 − X 1 ) + I V 1 dX 2 dt = R ( X 1 − X 2 ) V 2 August Hale Multilevel Cooling Systems May 16, 2014 6 / 20
Equations The Equations (Continued) dX 1 dt = C 1 ( X 2 − X 1 ) + (effect from lower level) B 1 dX 2 dt = C 1 ( X 1 − X 2 ) − R ( X 2 − ( X 3 − E )) B 2 V 1 dX 3 dt = R (( X 2 + E ) − X 3 ) − C 2 ( X 3 − X 4 ) V 2 B 3 dX 4 dt = C 2 ( X 3 − X 4 ) − (effect from higher level) B 4 August Hale Multilevel Cooling Systems May 16, 2014 7 / 20
Models The models (Simple Single Level Heat Sink) dX dt = C B ( A − X ) + I var(’t,a,b,c,i’) x=function(’x’,t) de=diff(x)==c/b*(a-x)+i sol=desolve_system([de],[x],ivar=t) sol=expand(sol) latex(sol) b ) − bie ( − ct b ) b ) x (0) + a + bi − ae ( − ct + e ( − ct c c August Hale Multilevel Cooling Systems May 16, 2014 8 / 20
Models The models (Air Cooled Enclosed Case) dX 1 dt = C B ( X 2 − X 1 ) + I dX 2 dt = R V ( A − X 2 ) − C B ( X 2 − X 1 ) var(’t,a,b,c,i,r,v’) x1=function(’x1’,t) x2=function(’x2’,t) assume(4*c^2*v^2+b^2*r^2>0) des=[diff(x1)==c/b*(x2-x1)+i, diff(x2)==r/v*(a-x2)-c/b*(x2-x1)] sol=desolve_system(des,[x1,x2],ivar=t) sol (solution is ridiculous and will not fit on a slide) August Hale Multilevel Cooling Systems May 16, 2014 9 / 20
Models The models (Refrigerant Circuit With Internal Heat Sink and External Radiator) dX 1 dt = C 1 ( X 2 − X 1 ) + I B 1 dX 2 dt = C 1 ( X 1 − X 2 ) − R ( X 2 − ( X 3 − E )) B 2 V 1 dX 3 dt = R (( X 2 + E ) − X 3 ) − C 2 ( X 3 − X 4 ) V 2 B 3 dX 4 dt = C 2 ( X 3 − X 4 ) − C 3 ( X 4 − A ) B 4 B 4 no exact solution found August Hale Multilevel Cooling Systems May 16, 2014 10 / 20
Long Term Behavior Long Term Behavior Simple Single level Heat Sink t →∞ X ( t ) = A + BI / C lim Air Cooled Enclosed Case t →∞ X 1( t ) = ( CIV + ( AC + BI ) R ) / ( CR ) lim t →∞ X 2( t ) = ( AR + IV ) / R lim August Hale Multilevel Cooling Systems May 16, 2014 11 / 20
Long Term Behavior Long Term Behavior (continued) var(’x1,x2,t,x3,x4’) c1=? c2=? c3=? b1=? b2=? b3=? b4=? v1=? v2=? r=? E=?? i=?? a=298 des=[c1/b1*(x2-x1)+i, c1/b2*(x1-x2)-r/v1*(x2-(x3-E)), r/v2*((x2+E)-x3)-c2/b3*(x3-x4), c2/b4*(x3-x4)-c3/b4*(x4-a)] ics=[298,298,298,298] dvars=[x1,x2,x3,x4] times=[0, .01 .. 80] sol=desolve_odeint(des,ics,times,dvars,ivar=t) August Hale Multilevel Cooling Systems May 16, 2014 12 / 20
Long Term Behavior Long Term Behavior (continued) c1=150 c2=200 c3=75 b1=242.2 b2=626.94 b3=835.92 b4=605.5 v1=150 v2=200 r=10 August Hale Multilevel Cooling Systems May 16, 2014 13 / 20
Long Term Behavior Long Term Behavior (continued) August Hale Multilevel Cooling Systems May 16, 2014 14 / 20
Long Term Behavior Long Term Behavior (continued) sols=[] plotsol=[] for i in [1,1.2..10]: for E in [10,10.2..80]: des=[c1/b1*(x2-x1)+i, c1/b2*(x1-x2)-r/v1*(x2-(x3-E)), r/v2*((x2+E)-x3)-c2/b3*(x3-x4), c2/b4*(x3-x4)-c3/b4*(x4-a)] ics=[298,298,298,298] dvars=[x1,x2,x3,x4] times=[0.. 1000] sol=desolve_odeint(des,ics,times,dvars,ivar=t) sols.append(sol) plotsol.append((i,E,sol[999,0])) August Hale Multilevel Cooling Systems May 16, 2014 15 / 20
Long Term Behavior Long Term Behavior (continued) August Hale Multilevel Cooling Systems May 16, 2014 16 / 20
Long Term Behavior Long Term Behavior (continued) regsol=[] for j in plotsol: if abs(j[2]-298)<.1: regsol.append((j[0],j[1])) var(’a,b,x’) model=a*x+b find_fit(regsol,model,variables=[x],solution_dict=true) {b: -0.004630541522512826, a: 11.848768472829438} August Hale Multilevel Cooling Systems May 16, 2014 17 / 20
Long Term Behavior Long Term Behavior (continued) August Hale Multilevel Cooling Systems May 16, 2014 18 / 20
Long Term Behavior Long Term Behavior (continued) August Hale Multilevel Cooling Systems May 16, 2014 19 / 20
Long Term Behavior Long Term Behavior (continued) August Hale Multilevel Cooling Systems May 16, 2014 20 / 20
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