Multilevel Cooling Systems August Hale May 16, 2014 August Hale - - PowerPoint PPT Presentation

Multilevel Cooling Systems

August Hale May 16, 2014

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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.

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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.

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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.

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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.

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Equations

The Equations

dX1 dt = C1 B1 (X2 − X1) + I dX2 dt = C2 B2 (X3 − X2) − C1 B2 (X2 − X1) dXn dt = Cn Bn (Xn+1 − Xn) − Cn−1 Bn (Xn − Xn−1) dX1 dt = R V1 (X2 − X1) + I dX2 dt = R V2 (X1 − X2)

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Equations

The Equations (Continued)

dX1 dt = C1 B1 (X2 − X1) + (effect from lower level) dX2 dt = C1 B2 (X1 − X2) − R V1 (X2 − (X3 − E)) dX3 dt = R V2 ((X2 + E) − X3) − C2 B3 (X3 − X4) dX4 dt = C2 B4 (X3 − X4) − (effect from higher level)

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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) −ae(− ct

b ) − bie(− ct b )

c + e(− ct

b )x (0) + a + bi

c

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Models

The models (Air Cooled Enclosed Case)

dX1 dt = C B (X2 − X1) + I dX2 dt = R V (A − X2) − C B (X2 − X1) 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)

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Models

The models (Refrigerant Circuit With Internal Heat Sink and External Radiator)

dX1 dt = C1 B1 (X2 − X1) + I dX2 dt = C1 B2 (X1 − X2) − R V1 (X2 − (X3 − E)) dX3 dt = R V2 ((X2 + E) − X3) − C2 B3 (X3 − X4) dX4 dt = C2 B4 (X3 − X4) − C3 B4 (X4 − A) no exact solution found

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Long Term Behavior

Long Term Behavior

Simple Single level Heat Sink lim

t→∞ X(t) = A + BI/C

Air Cooled Enclosed Case lim

t→∞ X1(t) = (CIV + (AC + BI)R)/(CR)

lim

t→∞ X2(t) = (AR + IV )/R

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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)

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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

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Long Term Behavior

Long Term Behavior (continued)

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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]))

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Long Term Behavior

Long Term Behavior (continued)

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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}

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Long Term Behavior

Long Term Behavior (continued)

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Long Term Behavior

Long Term Behavior (continued)

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Long Term Behavior

Long Term Behavior (continued)

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