U.S. patent application number 10/093026 was filed with the patent office on 2003-09-11 for high performance liquid-cooled heat sink with twisted tape inserts for electronics cooling.
Invention is credited to Bhatti, Mohinder Singh, Joshi, Shrikant Mukund, Oddi, Frederick Vincent.
Application Number | 20030168205 10/093026 |
Document ID | / |
Family ID | 27787912 |
Filed Date | 2003-09-11 |
United States Patent
Application |
20030168205 |
Kind Code |
A1 |
Bhatti, Mohinder Singh ; et
al. |
September 11, 2003 |
HIGH PERFORMANCE LIQUID-COOLED HEAT SINK WITH TWISTED TAPE INSERTS
FOR ELECTRONICS COOLING
Abstract
A liquid-cooled heat sink for use in combination with a heat
exchanger to cool electronic or electrical devices attached to said
heat sink comprises a metal block having a plurality of circular
passageways therethrough defined by cylindrical walls. An inlet is
in fluid communication with one end of the passageways for
receiving a cooling liquid, and an outlet is in fluid communication
with an opposite end of the passageways for outputting the cooling
liquid. At least one tape insert is mounted within at least one of
the circular passageways, wherein the tape insert longitudinally
bisects the circular passageway and further wherein edges of the
tape insert abut cylindrical wall.
Inventors: |
Bhatti, Mohinder Singh;
(Amherst, NY) ; Oddi, Frederick Vincent; (Orchard
Park, NY) ; Joshi, Shrikant Mukund; (Williamsville,
NY) |
Correspondence
Address: |
PATRICK M. GRIFFIN
DELPHI TECHNOLOGIES, INC.
Legal Staff, Mail Code: 480-414-420
P.O. Box 5052
Troy
MI
48007-5052
US
|
Family ID: |
27787912 |
Appl. No.: |
10/093026 |
Filed: |
March 7, 2002 |
Current U.S.
Class: |
165/80.4 ;
165/109.1; 165/147; 257/E23.098 |
Current CPC
Class: |
H01L 2924/00 20130101;
F28F 7/02 20130101; H01L 23/473 20130101; H01L 2924/0002 20130101;
F28F 13/12 20130101; F28D 2021/0029 20130101; H01L 2924/0002
20130101 |
Class at
Publication: |
165/80.4 ;
165/109.1; 165/147 |
International
Class: |
F28F 007/00; F28F
013/12; F28F 013/08 |
Claims
We claim:
1. A liquid-cooled heat sink for use in combination with a heat
exchanger to cool electronic or electrical devices attached to said
heat sink, said heat sink comprising: a metal block having a
plurality of circular passageways therethrough defined by
cylindrical walls, an inlet in fluid communication with one end of
said passageways for receiving a cooling fluid, and an outlet in
fluid communication with an opposite end of said passageways for
outputting the cooling fluid; and at least one tape insert mounted
within at least one of said circular passageways, said at least one
tape insert longitudinally bisecting said at least one said
circular passageway and further wherein edges of said at least one
tape insert abut cylindrical wall.
2. A liquid-cooled heat sink according to claim 1 wherein said tape
insert is twisted about a longitudinal axis of said passageway.
3. A liquid-cooled heat sink according to claim 2 wherein a ratio
of a pitch of said twisted tape to a diameter of said passageway is
between the values of 2 and 5.
4. A liquid-cooled heat sink according to claim 1 wherein a ratio
of a thickness of said block to a diameter of said passageway is
between the values of 1.1 and 1.4.
5. A liquid-cooled heat sink according to claim 1 wherein a ratio
of a center-to-center distance between adjacent ones of said
passageways in said block to a diameter of said passageways is
between the values of 1.1 and 1.4.
6. A liquid-cooled heat sink according to claim 5 wherein a ratio
of a center-to-center distance between adjacent ones of said
passageways in said block to a diameter of said passageways is
generally 1.3.
7. A liquid-cooled heat sink according to claim 1 wherein said tape
insert has a thickness ranging from 0.05 mm to 0.5 mm.
8. A liquid-cooled heat sink according to claim 1 further
comprising a plurality of tape inserts wherein each of said
passageways includes one of said tape inserts mounted therein.
9. A liquid-cooled heat sink according to claim 1 wherein said
edges of said at least one tape insert abutting said wall are
affixed thereto by one of the group of adhesively bonding and
metallurgically bonding.
10. A liquid-cooled heat sink according to claim 1 wherein said at
least one tape insert includes a plurality of louvers spaced
therealong.
11. A method of cooling an electrical or electronic device, said
method comprising the steps of: providing a liquid cooled heat sink
of the type including a block having an inlet in fluid
communication with a first end of a plurality of parallel
passageways therethrough defined by cylindrical walls wherein each
passageway includes a tape insert therein and further having an
outlet in fluid communication with an opposite end of the
passageways; affixing the electrical or electronic device to a face
of the block with a heat conductive adhesive; inputting a liquid
coolant into the inlet; causing the liquid coolant to pass through
the passageways; and expelling the liquid coolant from the
outlet.
12. The method according to claim 11 wherein the tape inserts
include a plurality of louvers extending from side to side of the
tape insert.
13. The method according to claim 11 wherein the tape inserts are
twisted about a longitudinal axis of the passageways.
14. The method according to claim 13 wherein a ratio of a pitch of
the twisted tapes to a diameter of the passageways is between the
values of 2 and 5.
15. The method according to claim 11 wherein a ratio of a thickness
of the block to a diameter of the passageways is between the values
of 1.1 and 1.4.
16. The method according to claim 11 wherein a ratio of a center-to
center distance between adjacent ones of the passageways in the
block to a diameter of the passageways is between the values of 1.1
and 1.4.
17. The method according to claim 16 wherein a ratio of a
center-to-center distance between adjacent ones of the passageways
in the block to a diameter of the passageways is generally 1.3.
18. The method according to claim 11 further including the step of
affixing edges of the tape inserts to cylindrical walls by one of
the group of adhesively bonding or metallurgically bonding.
Description
TECHNICAL FIELD
[0001] The present invention is directed to heat sinks in general,
and more particularly to liquid cooled heat sinks for use in
dissipating waste heat generated by electrical or electronic
components and assemblies.
BACKGROUND OF THE INVENTION
[0002] Research activities have focused on developing heat sinks to
efficiently dissipate heat from highly concentrated heat sources
such as microprocessors and computer chips. These heat sources
typically have power densities in the range of about 5 to 35
W/cm.sup.2 (4 to 31 Btu/ft.sup.2s) and relatively small available
space for placement of fans, heat exchangers, heat sinks and the
like.
[0003] Existing heat sinks for microelectronics cooling have
generally used air to directly remove heat from the heat source.
However, air has a relatively low heat capacity. Such heat sinks
are suitable for removing heat from relatively low power heat
sources with power density in the range of 5 to 15 W/cm.sup.2 (4 to
13 Btu/ft.sup.2s). With increase in computing speed the power
density of the heat sources has increased to 20 to 35 W/cm.sup.2
(18 to 31 Btu/ft.sup.2s) requiring more effective heat sinks.
Liquid-cooled heat sinks employing high heat capacity fluids like
water and water-glycol solutions are more particularly suited to
remove heat from high power density heat sources. The cooling
liquid used in these heat sinks removes heat from the heat source
and is then transferred to a remote location where the heat can be
easily dissipated into a flowing air stream with the use of a
liquid-to-air heat exchanger. Thus, such heat sinks can be
characterized as indirect heat sinks.
[0004] A typical liquid-cooled heat sink for microelectronics
according to the prior art is shown in FIG. 1, and generally
comprises a metal block 10 with drilled passages 12. The passages
12 are connected in a serpentine pattern by means of hairpin tubes
14 to form a continuous passage. The microelectronics device 16 is
bonded to one face of the block and liquid coolant flows through
the drilled passages 12 and hairpin tubes 14. Block 10 can have one
or multiple microelectronics devices bonded to a face of the block.
Heat sinks of this type have also used a serpentine tube mounted to
one side of a block with the microelectronics device bonded onto
the other side of the block. These types of heat sinks, however,
have limitations to the density of passages 12 therethrough and
must be spaced according to the relatively large bend radii of the
hairpin tubes 14.
[0005] Therefore, these prior art heat sinks exhibit a relatively
low heat transfer capability due to wide spacing of the serpentine
flow passages, relatively low heat transfer area and low heat
transfer coefficient. The conduction losses in the solid base
construction of the heat sink further reduces the available thermal
potential for heat transfer between the wall and fluid. Also, such
heat sinks tend to be relatively heavy and thus introduce
undesirably high mechanical stresses on the electronic devices
being cooled and the circuit boards to which the heat sink is
attached. In addition, such heat sinks are limited in their ability
to extract heat at low range of heat flux through the
microelectronics device.
SUMMARY OF THE INVENTION
[0006] One aspect of the present invention is a liquid-cooled heat
sink for use in combination with a heat exchanger to cool
electronic or electrical devices attached to said heat sink
comprises a metal block having a plurality of circular passageways
therethrough defined by cylindrical walls. An inlet is in fluid
communication with one end of the passageways for receiving a
cooling liquid, and an outlet is in fluid communication with an
opposite end of the passageways for outputting the cooling liquid.
At least one tape insert is mounted within at least one of the
circular passageway, wherein the tape insert longitudinally bisects
the circular passageway and further wherein edges of the tape
insert abut cylindrical wall.
[0007] Another aspect of the present invention is a method of
cooling an electrical or electronic device comprising the steps of
providing a liquid cooled heat sink of the type including a block
having an inlet in fluid communication with a first end of a
plurality of parallel passageways therethrough defined by
cylindrical walls wherein each passageway includes a tape insert
therein and further having an outlet in fluid communication with an
opposite end of the passageways. An electrical or electronic device
is affixed to a face of the block with a heat conductive adhesive.
A liquid coolant is input into the inlet and then caused to pass
through the passageways. The liquid coolant is then expelled from
the outlet.
[0008] These and other advantages of the invention will be further
understood and appreciated by those skilled in the art by reference
to the following written specification, claims and appended
drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
[0009] FIG. 1 illustrates a prior art liquid-cooled heat sink.
[0010] FIG. 2 is a partial sectional top view of the heat sink
according to one embodiment of the present invention.
[0011] FIG. 3 is a cross sectional elevational view of the heat
sink shown in FIG. 2 taken along the line 3-3 showing placement of
the twisted tape inserts in the circular flow passages.
[0012] FIG. 4 is a plan view of a segment of one flow passage
illustrating the plain twisted tape insert in relation to the
circular flow passage.
[0013] FIG. 5 is a plan view of a segment of one flow passage
illustrating a louvered twisted tape insert in relation to the
circular flow passage.
DESCRIPTION OF THE PREFERRED EMBODIMENT
[0014] For purposes of description herein, the terms "upper",
"lower", "left", "rear", "right", "front", "vertical",
"horizontal", and derivatives thereof shall relate to the invention
as oriented in FIG. 2. However, it is to be understood that the
invention may assume various alternative orientations and step
sequences, except where expressly specified to the contrary. It is
also to be understood that the specific devices and processes
illustrated in the attached drawings, and described in the
following specification, are simply exemplary embodiments of the
inventive concepts defined in the appended claims. Hence, specific
dimensions and other physical characteristics relating to the
embodiments disclosed herein are not to be considered as limiting,
unless the claims expressly state otherwise.
[0015] Heat sinks embodying the present invention, an example of
which is shown in FIGS. 2 and 3, are especially useful in
efficiently dissipating heat from highly concentrated heat sources
such as microprocessors and computer chips operating under high
heat flux conditions. Heat sinks according to the various
embodiments disclosed herein may be used with microprocessors and
computer chips (as well as other microelectronics) having any power
density, and are particularly useful for microprocessors and
computer chips having power densities of greater than about 10
W/cm.sup.2 (9 Btu/ft.sup.2s). More importantly, heat sinks
according to the various embodiments may be advantageously used to
dissipate the higher heat flux demands of new computer chips having
power densities of 20 to 35 watts/ cm.sup.2 (31 Btu/ft.sup.2s) as
well as future chips having even higher heat flux demands.
[0016] Turning now to the drawings, FIGS. 2-3 illustrates a heat
sink 20, which is one of the preferred embodiments of the present
invention, and illustrates its various components.
[0017] Referring to FIGS. 2-3, heat sink 20 comprises a metal block
22 having a plurality of drilled circular passageways 24
therethrough. Inserted into each passageway is a twisted tape 26
having edges 27 contacting the circular passageway 24. Edges 27 may
be adhesively or metallurgically bonded to the wall 25 defining
passageway 24 at the points of contact for enhanced heat transfer.
A liquid coolant flows through passageways 24 in the direction of
the arrow 28. The metal block is further fitted with an inlet 30
and an outlet 32 for entry and exit of the liquid coolant to the
interior of the flow passages 24. The inlet 30 provides access for
the liquid coolant into inlet header 34 defined by inlet housing
33, which serves to distribute the liquid coolant across the
plurality of circular passages 24. The liquid coolant flows through
circular passages 24 in the direction of arrow 28 and gains heat
along its path. Passages 24 discharge a hotter liquid coolant into
outlet header 36 defined by outlet housing 35 which serves to
collect the liquid coolant from the circular passages 24. The
outlet 32 removes the heated liquid coolant from the outlet header
36 and transfers it to a remotely located air-cooled heat exchanger
(not shown). The heat-generating element being cooled, such as a
computer chip 38 is attached to the flat face of the metal block 22
with an intervening layer of thermal grease 23 or the like. For
increased effectiveness of the heat sink 20, two heat-generating
elements, such as electronic devices 38, may be attached to both
flat surfaces of the metal block 22.
[0018] In order to minimize thermal resistance of the metal block
22, the block thickness 40 (represented by dimension `a`) is
restricted to be no more than 1.1 to 1.4 times the flow passage
diameter 42 (represented by dimension `d`). Likewise, in order to
maintain high heat flux from the concentrated heat source, such as
electronic device 38, to the cooling fluid in the flow passages 24,
the center-to-center distance 44 (represented by dimension `c`) of
the flow passages should not exceed 1.to 1.4 times the flow passage
diameter 42, and is more preferably at 1.3 time the flow passage
diameter 42.
[0019] The components of heat sink 20 are preferably manufactured
from a material exhibiting a high degree of heat conductivity such
as aluminum or copper, and it is contemplated that the heat sink 20
may be wholly aluminum or copper, or a combination of different
metal components. The components may be joined by any known means
including metallurgical or adhesive bonding methods well known in
the art. However, furnace brazing of the assembled heat sink is the
preferred means since it reduces the contact resistance between the
twisted tape inserts and the metal block.
[0020] Referring now to FIG. 4, The details of the plain twisted
tape insert 26 in relation to the circular passage 24 housing the
tape are illustrated. The precursor material for the twisted tape
insert 26 is a plain strip of metal or plastic having a thickness
46 (represented by the dimension `.delta.`) and a width equal to
the circular passage diameter 42. The twisted tape insert 26 is
formed by twisting the strip about the circular passage axis so as
to impart desired pitch 48 (represented by the dimension `b`) to
the strip. The pitch 48 of the twisted tape 26 is simply the
distance on the twisted tape insert surface in the flow direction
28 per each 180.degree. of tape twist. As the ratio of the tape
pitch 48 to the circular passage diameter 42, called tape twist
ratio, tends to infinity, the circular passage assumes the form of
two parallel semicircular passages.
[0021] FIG. 5 shows an alternate embodiment 126 of the twisted tape
insert. Twisted tape 126 is similar to the plain twisted tape
insert 26 wherein edges 127 of twisted tape 126 are affixed to wall
25. Tape insert 126 further includes multiple louvers 150 extending
substantially from edge 127 to edge 127 of tape 126 and thereby
aligned normal to the flow direction 28 of the cooling fluid
traversing passageways 24. These louvers 150 are intended to
increase the heat transfer coefficient over and above that achieved
by the twisting of the plain tape 26. Associated with an increase
in heat transfer coefficient, louvered tape 126 also exhibits an
increase in the friction factor between the fluid and tape 126.
Since the friction factor of the plain twisted tape 26 is low, the
increase in the friction factor of tape 126 due to the inclusion of
louvers 150 does not entail a severe pressure drop penalty between
inlet 30 and outlet 32.
[0022] In addition to the louvers 150, notches may be provided in
the tape surface to promote turbulence in the cooling fluid stream
with the objective of enhancing the heat transfer rate albeit at
the expense of some pressure drop penalty.
[0023] The following relations govern design of the heat sinks of
the present invention. The prescribed rate of generation of heat
{dot over (q)} by the microelectronics heat source can be expressed
via Newton's cooling law as:
{dot over (q)}=hA(T.sub.w-T.sub.m) (1)
[0024] where h is the average heat transfer coefficient within the
flow passages of the heat sink, A is the total heat transfer area,
T.sub.w , is the wall temperature of the heat sink and T.sub.m is
the fluid mean temperature.
[0025] The average heat transfer coefficient h for the circular
passage with a plain twisted tape insert under laminar flow
conditions with uniform wall heat flux is expressible in terms of
the dimensionless Nusselt number Nu as: 1 Nu hd k = 5.172 ( 1 +
1.39 - 2.448 kd / k 1 ) 1 + 0.005484 [ Pr ( Re b / d ) 1.78 ] 0.7 (
2 )
[0026] where Pr and Re are the dimensionless Prandtl number and
Reynolds number defined as: 2 Pr = c p k ( 3 ) Re = u m d ( 4 )
[0027] The various symbols in the foregoing relations carry the
following meanings:
[0028] b is the tape pitch shown in FIG. 4
[0029] d is the tube diameter, which is equal to the tape width
shown in FIG. 4
[0030] .delta. is the tape thickness shown in FIG. 4
[0031] k.sub.t is the thermal conductivity of the tape material
[0032] k is the thermal conductivity of the fluid
[0033] .mu. is the dynamic viscosity of the fluid
[0034] .rho. is the fluid density
[0035] c.sub.p is the isobaric specific heat of the fluid
[0036] u.sub.m is the flow mean velocity through the flow
passages.
[0037] The ratio bid is referred to as the tape twist ratio. As
shown in FIG. 4, it represents the distance on the twisted tape
insert surface in the flow direction per 180.degree. tape twist. As
aforesaid, when b/d.fwdarw..infin., the passage shape assumes the
form of two semicircular straight tubes in parallel.
[0038] The ratio k.sub.t.delta./kd in Equation (2) is the thermal
conductance parameter. For relatively poor conducting tape
materials, like plastics, k.sub.t.delta.kd.fwdarw.0 while for
highly conducting tape materials, like copper and aluminum,
k.sub.t.delta./kd.fwdarw..infin..
[0039] It may be noted that in the absence of the twisted tape
insert, the Nusselt number Nu.ident.hd/k=48/11=4.3636 for the
circular passage under laminar flow conditions with uniform wall
heat flux and Nu.ident.hd/k=4.0890 for the semicircular passage
under laminar flow conditions with uniform wall heat flux.
[0040] Equation (2) is valid for laminar flow with Re .ltoreq.2,300
involving liquid or gas flow with uniform wall heat flux. For
turbulent flow with Re .gtoreq.4,000, the average heat transfer
coefficient h for the circular passage with a plain twisted tape
insert involving liquid flow with uniform wall heat flux is
expressible in terms of the dimensionless Nusselt number Nu as: 3
Nu hd k = 0.0259 ( 1 + 1.39 - 2.4486 kd / k 1 ) [ 1 + ( 2 b / d ) 2
] 2 / 5 Re 4 / 5 Pr 2 / 5 ( 5 )
[0041] where all the symbols have been previously defined.
[0042] For the transition flow regime (2,300.ltoreq.Re
.ltoreq.4,000), the Nusselt number may be assumed to vary linearly
with Re being bracketed by the values calculated using Equation (2)
at Re=2,300 and Equation (5) at Re=4,000.
[0043] The average heat transfer coefficient h for the circular
passage with a twisted tape insert with uniform wall temperature
may be estimated by reducing the calculated values from Equations
(2) and (5) by 17%.
[0044] The pressure drop .DELTA.P in the circular passage with the
twisted tape can be determined using the general relation: 4 P = 2
fl u m 2 g c d ( 6 )
[0045] where, in addition to the previously defined symbols, the
remaining symbols have the following meanings:
[0046] f is the dimensionless friction factor
[0047] l is the total passage length in the flow direction
[0048] g.sub.c=32.174 (lb.sub.m/lb.sub.f)ft/s.sup.2 is the constant
of proportionality in Newton's second law of motion.
[0049] Based on the analytical relations presented in the
reference: R. K Shah and A. L. London, Laminar Flow Forced
Convection in Ducts, pp.379-384, Academic Press, N.Y., 1978, the
friction factor f for the circular passage with the plain twisted
tape insert under laminar flow conditions (Re.ltoreq.2,300) is
expressible as: 5 f = 42.23 Re if Re b / d < 6.7 ( 7 ) f = 38.4
Re 0.95 ( b / d ) 0.05 if 6.7 Re b / d 100 ( 8 ) f = ( b d ) 0.7 [
8.8201 - 2.1193 ( b d ) + 0.2108 ( b d ) 2 - 0.0069 ( b d ) 3 ] Re
0. 7 if Re b / d > 100 ( 9 )
[0050] The dimensionless factor .PHI. in the foregoing relations is
given as: 6 = ( + 2 ) 2 ( + 2 - 2 / d - 4 / d ) 2 ( - 4 / d ) ( 10
)
[0051] where all the symbols have been previously defined.
[0052] It may be noted that in the absence of the twisted tape
insert, the friction factor f=16/Re for the circular passage and
f=15.767/Re for the semicircular passage under lamninar flow
conditions (Re.ltoreq.2,300).
[0053] By the following example, the teachings of the present
invention are illustrated to size a preferred heat sink by the use
of the foregoing relations.
[0054] EXAMPLE: A water-cooled heat sink of the present invention
comprises an array of six circular tubes each of diameter d=0.375
inch drilled through the opposing sides of a metal block of
thickness a=0.478 inch. Each tube is fitted with a plain aluminum
tape of thickness .delta.=0.005 inch and the twist ratio b/d=3 in
order to enhance the heat transfer rate in water flowing through
the tubes.
[0055] The heat sink is intended to dissipate 400 Watts (0.3792
Btu/s) of power into water flowing through each tube at the rate
{dot over (m)}=0.01 lb.sub.m/s. This requires a heat dissipation
rate {dot over (q)}=0.0632 Btu/s in each of six tube. The incoming
temperature T.sub.wi of water into the heat sink is 90.degree. F.
and the average tube wall temperature T.sub.w is 122.degree. F.
[0056] The thermal conductivity of aluminum is k.sub.t=2.22 Btu/ft
s .degree. F. and the transport properties of water are as
follows:
[0057] Density .rho.=62.34 lb.sub.m/ft.sup.3
[0058] Thermal conductivity k=0.000095 Btu/ft s .degree. F.
[0059] Isobaric specific heat c.sub.p=1.0 Btu/lb.sub.m .degree.
F.
[0060] Dynamic viscosity .mu.=0.000754 lb.sub.m/ft s
[0061] Prandtl number Pr=.mu.c.sub.p/k=7.94.
[0062] The flow mean velocity u.sub.m needed for the determination
of the heat transfer coefficient h and the friction factor f is
related to the prescribed mass flow rate {dot over (m)}, fluid
density .rho. and the free flow area A.sub.c of each flow passage
as u.sub.m={dot over (m)}/.rho.A.sub.c. Neglecting the tape
thickness 46(.delta.) shown in FIG. 4, the free flow area A.sub.c
of each tube is simply the cross sectional area of the tube given
as A.sub.c=.rho.d.sup.2/4. Introducing the tube diameter d=0.375/12
ft, the tube cross sectional area
A.sub.c=.rho.(0.375/12).sup.2/4=0.000767 ft.sup.2. Introducing this
value of A.sub.c together with the values of {dot over (m)}=0.01
lb.sub.m/s and .rho.=62.34 lb.sub.m/ft.sup.3 into the foregoing
relation, u.sub.m=0.01/(62.34.times.0.000767)=0.2091 ft/s.
[0063] Given the water density .rho.=62.34 lb.sub.m/ft.sup.3,
dynamic viscosity .mu.=0.000754 lb.sub.m/ft s, flow mean velocity
u.sub.m=0.2091 ft/s and the tube diameter d=0.375/12 ft, the
Reynolds number Re is calculated to be 541 with the use of Equation
(4). This value of Re being .ltoreq.2,300, the flow is considered
laminar.
[0064] Knowing the thermal conductivity of aluminum k.sub.t=2.22
Btu/ft s .degree. F., the thermal conductivity of water k=0.000095
Btu/ft s .degree. F., the tape thickness .delta.=0.005 inch and the
tube diameter d=0.375 inch, the value of the thermal conductance
parameter k.sub.t.delta./kd=311. Using this value of
k.sub.t.delta./kd together with the Reynolds number Re=541, the
Prandtl number Pr=7.94 and the tape twist ration b/d=3, the Nusselt
number Nu.ident.hd/k is calculated to be 23.26 with the use of
Equation (2) applicable to laminar flow.
[0065] Introducing the calculated value of Nu=23.26 together with
the thermal conductivity of water k=0.000095 Btu/ft s .degree. F.
and the hydraulic diameter d=0.375/12 ft into the defining relation
Nu.ident.hd/k, the heat transfer coefficient
h=23.26.times.0.000095/(0.37- 5/12)=0.0707 Btu/s ft.sup.2 .degree.
F.
[0066] In the absence of the twisted tape insert, the value of Nu
is given as Nu=48/11=4.3636. Thus the presence of the twisted tape
insert increases the Nusselt number and hence the heat transfer
coefficient by a factor of 23.26/4.3636=5.33.
[0067] Next, in order to determine the heat transfer area A of each
tube with the use of Equation (1), the fluid mean temperature
T.sub.m is needed. Assuming the temperature gradients in the heat
sink to be mild, the water mean temperature T.sub.m entering
Equation (1) can be taken as the arithmetic mean of the water inlet
temperature T.sub.wi=90.degree. F. and as yet undetermined water
outlet temperature T.sub.wo. This latter temperature can be
determined using the simple energy balance {dot over (q)}={dot over
(m)}c.sub.p(T.sub.wo-T.sub.wi) where the prescribed heat
dissipation rate per tube {dot over (q)}=0.0632 Btu/s, the mass
flow rate of water per tube {dot over (m)}=0.01 lb.sub.m/s, the
isobaric specific heat of water c.sub.p=1.0 Btu/lb.sub.m20 F. and
the water inlet temperature T.sub.wi=90.degree. F. Introducing
these values into the simple energy balance, T.sub.wo=96.32.degree.
F. whence T.sub.m=(90+96.32)/2=93.16.degree. F.
[0068] Using the prescribed heat dissipation rate per tube {dot
over (q)}=0.0632 Btu/s, h=0.0707 Btu/s ft.sup.2.degree. F.,
T.sub.w=122.degree. F. and T.sub.m=93.16.degree. F., the heat
transfer area of each tube is obtained as A=0.0310 ft.sup.2=4.4640
in.sup.2 with the use of Equation (1). This is practically the area
A.sub.w of the tube wall assuming that the amount of heat conducted
by the tape from the tube wall is negligible.
[0069] The tube length l in the flow direction is related to the
tube wall area A.sub.w and the tube diameter d as l=A.sub.w/.pi.d.
Introducing A.sub.w=4.4640 in.sup.2 and d=0.375 inch, the tube
length l=3.7886 inch.
[0070] Finally, in order to find the pressure drop .DELTA.P in the
heat sink, the average friction factor f needs to be determined
with the use of the applicable relation from among Equations
(7)-(9). Noting that Re/(b/d)=541/3=180, the applicable relation
for the determination of f is found to be Equation (9). Introducing
the tape thickness .delta.=0.005 inch and the tube diameter d=0.375
inch, the dimensionless factor .PHI. is calculated to be
.PHI.=1.0418 with the use of Equation (10). Introducing this value
of .PHI. together with Re=541 and b/d=3 into Equation (10),
f=0.1145.
[0071] In the absence of the twisted tape insert, the value of f is
given as f=16/Re =16/541=0.0296. Thus the presence of the twisted
tape insert increases the friction factor by a factor of
0.1145/0.0296=3.87. Compare this with an increase in the heat
transfer coefficient by a factor of 5.33.
[0072] Using the calculated value of f=0.1145 together with
l=3.7886, .rho.=62.34 lb.sub.m/ft.sup.3, u.sub.m=0.2091 ft/s,
g.sub.c=32.174 (lb.sub.m/lb.sub.f)ft/s.sup.2 and d=0.375 inch, the
pressure drop through each tube and hence through the heat sink is
found to be .DELTA.P=0.1961 lb.sub.f/ft.sup.2 via Equation (6).
This is equivalent to 0.001362 pounds per square inch or 0.0376
inch water.
[0073] While the preferred embodiment of the present invention has
been described so as to enable one skilled in the art to practice
the heat sinks disclosed, it is to be understood that variations
and modifications may be employed without departing from the
concept and intent of the present invention as defined by the
following claims. The preceding description is intended to be
exemplary and should not be read to limit the scope of the
invention. The scope of the invention should be determined only by
reference to the following claims.
* * * * *