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Thermodynamics Mini Project Date: 12 Apr 2014

A computer is kept on all the time to monitor financial stock exchanges. The core functions of 

the computer are mainly carried out by its processor chip. The shape of the processor is like aflat slab ith dimensions of 10mm by !mm by "nm# as shon in $ig. 1 belo.

%ome of the electrical energy used by the processor is lost as heat# leading to heat generation

ithin the processor. The rate at hich heat is generated ithin the processor chip of the

computer is 0.04&. 'ence# there is a fan in the computer hich blos air past the processor

chip to cool it don.

a( )ne day# the fan of the computer malfunctioned and stopped orking. 'o long ill it

take for the computer to o*erheat and spoil+ Assume that the computer ill fail once the

 processor chip reaches a temperature of 100,-.

 b( &hat poer rating should the replacement fan be if e ant to keep the maximum

temperature of the processor chip at 0,-+

Assumptions:1. Temperature of air is constant at 2,-.

2. The processor chip is made of pure silicon.

/. )nly top side of the processor chip is exposed to air# as illustrated in $ig. 1.

4. A circular fan is used# and the diameter of the fan is /cm.

!. Density of air is constant.

Figure 1. Structure of the computer processor

a( The thermal conducti*ity# k# of %i is 1/0 &m. hile its thermal diffusi*ity# # is 310

!m2s.

The con*ecti*e heattransfer coefficient of air can be taken to be 10 &m 2.

5olume of the processor 6 710310/( 37!310/( 37"310"( 6 4.!3101/ m/

Area of the exposed surface 6 710310 /( 37!310/( 6 !310! m2 

B i=hV kA = (10)(4.5×10

−13

)(130)(5×10−5)

 =6.92×10−10≪0.1  

%ince the 8i *alue is less than 0.1# e can treat this as a lumped parameter case.

Doing some energy balance# e get

q́−hA (T −T ∞ )= ρ C  p V  dT 

dt   

9et beT −T ∞  

d θ

dT 

=  q́

 ρ C  p V 

 −  h A

 ρC  p V 

 θ= ´ A−B́ θ

here

´ A=  q́

 ρ C  p V   and

B́=  h A

 ρC  p V   

10mm

5mm

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∫0

θ1

´ A−B́ θdθ=∫

0

t 

dt    t =−1B́

ln( ´ A− ́B θ´ A   )´ A=

  q́ ρ C  p V 

 =  q́k 

α  V 

=  0.04

130

8×10−5 (4.5×10

−13

)

=54701

B́=  h A

 ρC  p V  =

 h A

k 

α  V 

=  10(5×10−5)130

8×10−5 (4.5×10

−13)=683.76

t =  −1

683.76ln( 54701− (683.76 ) (100−28 )54701   )=3.37×10−3 s

 b( To make sure -;< does not o*erheat# a *ertically orientated fan as considered forcooling the -;

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 P=% ∆ p

$=

& D2

U ∞

4$∆ p=

& D2

U ∞

4$

0.664

 #   √  μρLU ∞

3=4 μ ' 

Figure 2. Raspberry pi micro fan (D1!mm". #t uses P$.$%&'.')'.2* and it is !'+'''times more

po,erfu- than the considered design in part (b"

9imitations to our calculations are due to our assumptions abo*e as ell as assuming that

heat loss is only due to con*ection. Cn reality# a microprocessor is made up of *arious other

materials and does not really ha*e a solid slab structure# and heat can be lost in any direction

and not restricted to con*ection.