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Date: 12 Apr 2014
Thermodynamics Mini Project
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 a flat 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# day# the fan of the computer malfunctioned and and stopped orking. 'o '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 the replacement fan be if e ant to keep the maximum temperature of the processor chip at 0,-+ Assumptions: 1. Temperature 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. 10mm
5mm
Figure 1. Structure of the computer processor
a( The thermal conducti*ity# k# of %i is 1/0 &m. hile i ts thermal diffusi*ity# # is 310 ! 2 m s. 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.!310 1/ m/ Area of the exposed surface 6 710310 /( 37!310/( 6 !310! m2 −13 hV ( 10)( 4.5 × 10 ) − 10 B i= = ≪ 0.1 −5 = 6.92 × 10 kA ( 130 )( 5 × 10 ) %ince the 8i *alue is less than 0.1# e can treat this as a lumped parameter case. Doing some energy balance# e get dT q´ −hA ( T −T ∞ ) = ρ C p V dt 9et be
T −T ∞
dθ q´ hA ´ − B ´θ = − θ= A dT ρ C p V ρC ρ C p V
here
´= A
q´ ρ C p V and
´= B
hA ρC ρ C p V
θ
1
∫ A´ − B´ θ
t
dθ=∫ dt
0
´= A
0
´ − B ´θ −1 A ln ´ ´ B A
(
)
q´ q´ 0.04 = = =54701 ρ C p V k 130 −13 ( 4.5 × 10 ) V −5 α 8 × 10
´ = h A = h A = B ρC p V
t =
t =
−1 683.76
ln
k V α
(
−5
10 ( 5 × 10 130
) − 13
(
− 5 4.5 × 10
8 × 10
=683.76 )
54701− ( 683.76 ) ( 100 −28 ) 54701
)
−3
=3.37 × 10 s
b( To make sure -;< does not o*erheat# a *ertically orientated fan as considered for cooling the -;<. Therefore# the heat balance has to be reached: q´ −h A ∆ T =0 # here ∆ T =T s− T ∞ =80 ℃−28 ℃ =52 ℃ . This means re=uired h is 1!.4 &m 2..
Nu L =
'eat transfer coefficient h is related to >usselt number:
1/ 2
?eynolds number and ;randtl number:
Nu L =0.664 ℜ L Pr
Air properties ere found at film temperature:
T f /2@
ρ
μ
1.00! kgm /
The re=uired uniform *elocity $or 9 6 10mm#
U ∞
T s + T ∞ 2
U L
∞ ℜ = L # here ν
= 80
ν
1."@x10! ;a.s
U ∞
T f =
1/ 3
hL k # hich depends on
1./x10! m2s
℃ + 28 ℃ 2
.
=54 ℃ .
;r
k
0.@01
2./1x10 2 &m.
as found for both orientations of -;<.
6 0.00!!/ ms. And for 96!mm#
U ∞
6 0.00@2" ms. Thus# 9 6
10mm as used. To find the re=uired air pressure# drag on the flat plate as calculated: 3 ¿ w =0.664 √ μρLU ∞ # and its relationship to the pressure difference re=uired: H is
KD =∆ p A f!"w= ∆ p w#
# here
approximate thickness of *elocity field and as assumed to be similar to boundary layer thickness# hich as around /cm. K is a constant that represents the other components that disturb the flo and increase pressure drop# it as assumed to be !00. Assuming the efficiency $ is a =uite lo 1B# the poer re=uired as found
P
=
%∆p $
2
=
& D U ∞ 4$
2
∆p
=
& D U ∞
0.664
4$
#
√ 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.
