Convection Heat Transfer Coefficient

Progettazione Ottica Roncati [email protected] www.progettazioneottica.it

Iterative calculation of the heat transfer coefficient D.Roncati Progettazione Ottica Roncati, via Panfilio, 17 – 44121 Ferrara

Aim The plate temperature of a cooling heat sink is an important parameter that has to be determined with accuracy. The estimated value depends on its geometrical shape, on the total amount of energy to be dispersed, and on the air flow. The heat transfer coefficient, h, is the most difficult parameter to be settled. In this report it is shown a fast and easy iterative method to calculate the h value and later, the temperature of cooling for heat sink.

Introduction The heat transfer coefficient or convective coefficient (h), is used in thermodynamics to calculate the heat transfer typically occurring by convection.  simple way to calculate h is to define it through the classical formula for convection, and compare it with a different definition of h, through dimensionless parameters. !nfortunately, even if defined by means of different parameters, both the environment and the heat sink temperature are important to estimate h. n iterative iterative method is then re"uired, by setting an initial value of the Tp.

Convection heat transfer coefficient The formula for heat transfer is#

 = ℎ ∗  ∗  −  

($)

%here#

− − − − −

& 'heat transferred, s ' % h ' heat transfer coefficient, %(m * +)  ' transfer surface, m * Tp ' -late temperature, + Ta ' ir temperature, +

or convection we use the convection heat transfer coefficient h c, %(m* +).  different approach is to define h through the /usselt number /u, which is the ratio between the convective and the conductive heat transfer#

     =  = (ℎ ∗ ) )!   %here#

− − − −

/u ' /usselt number hc ' convective heat transfer coefficient k ' thermal conductivity, %m+ 0 ' characteristic length, m

(*)


The convection heat transfer coefficient is then defined as following#

ℎ = "∗# $

(1)

The Nusselt number depends on the geometrical shape of the heat sink and on the air flow. or natural convection on flat isothermal plate the formula of /a is given in table $. Table $# /usselt number formula. 2ertical fins 0aminar flow  = %&' ∗ *+&,Turbulent flow  = %&/. ∗ *+&00

3orizontal fins !pward laminar flow 4ownward laminar flow Turbulent flow

 = %&'. ∗ *+&, = %&12 ∗ *+&, = %&/. ∗ *+&00

%here#

* = 34 ∗ 54 is the Rayleigh number defined in terms of -randtl number (-r) and 6rashof number (6r) . If 7a 8 heat flow is laminar, while if 7a 9 /%6the flow is turbulent.

(5)

/%6the

The Grashof number, 6r is defined as following#

34 =

7∗$8 ∗9∗:; <: >?

(:)

%here# *

− − − − − −

g ' acceleration of gravity ' ;.<$, ms 0 ' longer side of the fin, m = ' air thermal e>pansion coefficient. or gases, is the reciprocal of the temperature in +elvin# @ = :A , $+  Tp ' -late temperature, ?. Ta ' ir temperature, ? A ' air kinematic viscosity, is /&'- at *B ?. /&B- at 1B ?.

or plate temperature, Tp, set a e>pected value. inally, the Prandtl number, -r is defined as# 54 = C∗ # %here#

− − −

D ' air dynamic viscosity, is /&D/- at *B ?. /&DB- at 1B ?. cp ' air specific heat ' $BB: (+gE+) for dry air k ' air thermal conductivity ' B.B*C % (mE+) at *F ?

(C)


Radiant heat transfer coefficient The total amount of energy leaving a surface as radiant heat depends on the absolute temperature and on the nature of the surface.  nonGblackbody, emits radiant energy from its surface at a rate that is given by#

E = F ∗ G ∗  ∗ H I = F ∗ G ∗  ∗ I − I  %here F

(F)

= '&B2 ∗ /%
from B (for non emitting body) to $ (for blackbody). 2alues of emissivities for aluminum is presented in Table *. 7eplacing "of J". F in J".$ we o btain#

F ∗ G ∗  ∗ I − I  = ℎ ∗  ∗  −  

(<)

nd solving for h obtain the formula for radiant heat transfer coefficient, h r# O∗P∗:;N <:N  ℎ = : <:  ; 

(;)

Table *. Jmissivities of different aluminum surface treatment luminum -olished K>idized *5GT weathered nodized (at $BBB?)

1$B + B.B5 B.$$ B.5B B.;5

:1B + B.B: B.$* B.1* B.5*

or not negligible radiation the overall heat transfer coeffic ient is#

ℎ = ℎ Q ℎ

($B)

In igure $ are reported the values of hr as a a function of Tp for emissivity of B.< and Ta ' *B ?.

ig. $# 7adiant heat transfer coefficient for varies temperature plate with ambient temperature *B ? and emissivity B.<.


Iterative method n initial guess value for Tp is set in J". : to determine 6r. Knce calculated the dimensionless parameters 6r, -r, 7a and /u,the value of h is obtained through the J". 1 . Hy introducing the value of h in J". $ , a new value of Tp is given by#

R  =  Q ∗S

(F)

and the obtained Tp can be used to estimate iteratively the proper value of 6r for the particular e>perimental setGup by using it in the J". :. The obtained value of h is then used again to estimate the 6r parameters, in an iterative way, that rapidly converges to the correct estimation of h.

Example 1.  0J4 is assembled on a 5B > 5B > C mm aluminum heat sink with four fins 1B mm long and 5 mm large. The 0J4 generate 1 % and the room temperature is *B ?. %hich is the value of the heat sink temperatureL

Fig. 2 scheme showing the assembled heat sink and energy flow.

$. 4efine 6rashof number with 0 ' 5B mm, the greater side of fins and assuming a plate temperature of $BB ?#

/ TD/ ∗ %T%.0 ∗ 12UQ1% ∗ (/%%−1%) 34 = = 2&B1 ∗ /%, (/T'∗/% ) *. 4efine -randtl number#

/TD/ ∗ /%- ∗ /%%' 54 = = %&2 %T%1B 1. 4efine the 7ayleigh number#

* = 2&B1 ∗ /%- ∗ %&2 = '&UU ∗ /%- V / ∗ /%6

so is laminar flow.


or laminar flow, horizontal fins (flat and isothermal with good appro>imation) and heat upward flow, from Table $ the value of /usselt number /u is determined by using the relation#

 = %&'. ∗ *+&,- = %&'. ∗ ('&UU∗/%- )+T,- = /.&' 5. inally, the heat transfer coefficient turns out to be#

ℎ =

/.&' ∗ %T%1B W = &' , %T%. X ∗Y

:. The cooling area is the sum of lateral fins surface and the plane surface, as shown in ig. 1.

Fig. 3. J>planation of the dispersion surfaces

The total lateral fins surface is# .% ∗ U% ∗ D = B%% XX , = %T%%B X ,, the plan surface is /B%% XX, = %T%%/B X, for a total cooling surface of %T%//1 X, .

.% ∗ .% =

Hy using the value of the cooling surface, it is possible to check the correctness of the Tp that was initially set to calculate a starting value for 6r (J". :).

 = 1% Z[ Q

UW &' X,W∗ Y ∗ %T%//1 X ,

= .D Z[

s evident, the set and the estimated values of Tp are inconsistent. The calculated value of Tp is used in J". : to estimate a more appropriate value of 6r. %ith Tp '5< ?, the new parameters are#

34 = 1&B2 ∗ /%-, * = /&D2 ∗ /%-,  = //&11, ℎ = 2&U

K . L? ∗M

-r is always B,F./ow with a check the plate temperature result to be urther iterations gets the following values# $. ℎ = 2&D *. ℎ = 2&2

 = '2 Z[ .

K and  = '. Z[ . L? ∗M K and  = '' Z[ . L? ∗M

The last iteration give a values of Tp for both convection heat formula and via /usselt number. The value of ℎ = 2&2 LK ? ∗M can be considered.


If also the flat side of the heat sink is in free air we must add this other convection which has a different h value. In the J>ample * such effect is also considered.

Example 2. onsider the heat sink in e>ample $ with heat generated by a 5 > 5 mm 0J4 die set in the middle of the plane as show in figure 5. 4efine the heat sink temperature, Tp, considering the two different heat transfer coefficients.

Fig.4. !am"le #ith u"#ard and do#n#ard con$ection

.% ∗ .% = /B%% XX , . /B%% \ /B = /'D. XX , = %&%%/'D. X ,. The flat surface is

or the cooling area we must subtract the die area#

The /usselt formula for horizontal flat plate in natural air f low and downward heat flow is#  = %&12 ∗ *+&,%e have already defined the value for 6r, and 7a with 0 ' 5B mm and Tp ' :: ? and can calculate /u and h#

34 = U&UU ∗ /%-M

* = 1&UU ∗ /%-,

 = '&UM

ℎ = U&

K . L? ∗M

%ith two different convection coefficient the J".$ becomes#

 = (ℎA ∗ A Q ℎ, ∗ , ) ∗  −  

(<)

%here the subscript $ is for upward convection and * for downward. olving for  the e".F becomes# R ] ∗S] ^? ∗S?

 =  Q  ubstituting the values we get#

 = 1% Q

U = '1 Z[ 2&2 ∗ %&%//1%% Q U& ∗ %&%%/'D.

or the iterative method we set Tp ' :* ? in e".: to reGcalculate /u and h c for upward and downward convection and set the new /u and h value in e".;. $. ℎA = 2&'

K K , ℎ, = U&D ? ,  = 'U Z[ ? L ∗M L ∗M

(;)


*. ℎA = 2&B

K K , ℎ, = U&D ? ,  = 'U Z[ ? L ∗M L ∗M

The last iteration give a values of Tp for both convection heat formula and via /usselt number. The value of K K . ℎA = 2&B ? and ℎ, = U&D ? can be considered. L ∗M L ∗M

Validation of the method The iterative method was applied to define the heat sink temperature of the 0J4 floodlight I7IK*5 (a new product developed by -rogettazione Kttica 7oncati and -aolo olombani 4esign, see igure :). The results of the iterative method were compare with both the thermal simulation (performed with the finite element analysis software 0I <.B.B) and the laboratory tests.

Fig.%. &' floodlight ()R)O24

The I7IK*5 heat sink is a black anodized aluminum $;< > $1* >C mm plate with *B fins $1* > 15 > 1 mm. The heat flu> transferred from 0J4s to the heat sink i s $C %att. The laboratory temperature is $: ? and the proNector is placed with the 0J4s on the downside. Input parameters for the iterative method are# − 3eat power, & ' $C % − mbient temperature, ta ' $: ? − 6reater fin dimension, 0 ' $1* mm − air kinematic viscosity, A ' $.5:E$B G: − air dynamic viscosity, D '$.F

Fig.*. &' floodlight ()R)O24 3' model for thermal analysis created #ith &)(+ .-.-

t first, a 14 model of the heat sink was created with 0I <.B.B.  $C % flow rate was then applied on the K plane surfaces as well as a convection for cooling the surfaces with a coefficient ℎ = .&D ? . The result L ∗M of the simulation are shown in ig. F in which the estimated temperature varies from 1B.;B ? to 1$.:C ? while the average temperature is 1$.*1 ?. The analyticiterative method based on the /usseltOs formula for isothermal fin appro>imation was also compared with the previous results (ee Table 1).

Fig.. &' floodlight ()R)O24 thermal analysis done #ith &)(+ .-.-

The previous results were compared with a laboratory test of heat sink temperature ($: ? ambient temperature, horizontal fins with upward heat flow) using an infrared thermal gun with precision :P, resolution B.$ ?. The e>perimental results agree with the simulations within the accuracy of the testing device (see Table 1). /able 3. I7IK*5 heat sink temperature results.

Iterative method 1$.* ?

J with 0I <.B.B 1$.* ?

0aboratory 1*.$ ?


Conclusions. Iterative method and thermal analysis done with finite element software 0I given values that match with difference of B.$P with a heat sink 14 model precision of :P. The difference of B.; ? (*.plained in terms of uncertainty in the determination of &. Indeed, the heat power 0J4 generated is calculated as difference between power absorption and radiant power. The latter is estimated from comple> computation. The iterative method can be used to evaluate accurately the temperature value for heat sink or to define with better precision the heat transfer convection coefficient for finite elements software for thermal analysis like 0I. It is worth noting that the /usselt formula tabulated are fo r rectangular plate and natural air flow (free convection) and is not valid for other cases like for heat sink with different shape or for forced convection.

About the Author 4ario 7oncati is an Kptical 4esigner specialized in 0J4 systems. 3e got his Qaster degree in -hysics and dvanced -hysical Technologies at the 4epartment of -hysics and Jarth cience of the !niversity of errara. %orked for different company as Kptical 4esigner in railway and airfield light sectors. /ow as owner of -rogettazione Kttica 7oncati realize optical analysis and design for 0J4 and solar concentrator applications.

Convection Heat Transfer Coefficient

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