radial conduction lab report

MCHE 3450 Mechanical Engineering Laboratory

Radial Conduction

Experiment Performed 01/14/15 Report Submitted 01/28/15 Group 2

By: Damon Dunwody And team

I certify that all the writing here is our own and not acquired from external sources. sources. We have cited sources sources appropriately and paraphrased correctly. correctly. We have not shared our writing writing with students outside our group, nor have we acquired any written portion of this document from past or present students.

Trusted by over 1 million members

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.





Abstract:

This lab report a!e students a "a# to appl# te$hni$al %no"lede from E&GR '150( )n the pro$edure* an insulated metalli$ dis% "as heated at the inner radius and temperatures "ere ta%en at six different radial distan$es( The resistan$e heater+s ener# ex$hane "as set to 10,* '0,* and -0,* for three trials( .o"e!er* due to the $apabilities of the resistan$e heater* the trial usin -0, $ould not be performed( The trial "ith the heater set to 10, left temperatures !ar#in from 20(8  to 28(4 ( The trial "ith the heater set to '0, a!e results measurin from 24(-  to 45(- ( The boundar# $onditions "ere used at points ST1 and ST-* for the 10, trial* in order to find $oeffi$ients to the anal#ti$al euation for heat transfer in the metalli$ dis%( The boundar# $onditions for ST2 ST2 and ST5 "ere also used to find the $oeffi$ients for the heat transfer euation as "ell( The t"o sets of $oeffi$ients "ere a!eraed in order to a$hie!e a more a$$urate euation( This "as repeated for the '0, trial( The anal#ti$al euations #ielded !alues !er# $lose to that of the experimental data and both data plots follo"ed a similar trend* $lose to a loarithmi$ fun$tion( &ext the $oeffi$ients* $oeffi$ients* found from the anal#ti$al anal#ti$al solution* "ere used to to find the heat transfer at the the inner and outer radius of the dis%( The students found the per$ent differen$e of the t"o heat transfers to be 0( The per$ent differen$e is 0 for both euations* "hi$h $onfirms that the experiment* at both 10, and '0,* "as in the stead# state( 3rom this lab* students learned ho" radial $ondu$tion $an be found in a metalli$ dis%* ho" to de!elop an anal#ti$al euation usin boundar# $onditions* $onditions* and ho" to $al$ulate the the heat transfer of the dis% at a $ertain radial lo$ation( lo$ation(

Introduction:

ondu$tion is the transfer of ener# from the more enereti$ parti$les of a substan$e to





inner radius( Probes $onne$ted to the $omputer "ill read the temperatures of the plate at different distan$es from the inner radius( The metalli$ dis% "as also insulated on the top and bottom surfa$e "ith near perfe$t insulation as "ell as usin a $oolin "ater a$%et around the outer radius to absorb the heat transfer from the dis%( This laborator# tea$hes students to de!elop boundar# $onditions for heat transfer problems as "ell as an anal#ti$al solution for radial $ondu$tion throuh a metalli$ dis%( 3rom the data $olle$ted in the experiment* students then $an $ompare their anal#ti$s results "ith the experimental data* sho"in the a$$ura$# of their euations( Some other $on$epts pra$ti$ed in this experiment in!ol!e ener# balan$e as "ell as expressin heat transfer in $#lindri$al $ #lindri$al $oordinates( $oordinates(

Technical ac!ground

The foundation that of our %no"lede of heat transfer throuh $ondu$tion oriinates from the first la" of thermod#nami$s "hi$h states that ener# $an neither be $reated nor destro#ed durin a pro$ess6 it onl# $hanes form* or also %no"n as the $onser!ation of ener# prin$iple* "hi$h $an be expressed as

(

total total energy energy enteri entering ng the system

)( −

Total energy leaving leaving the system

)( =

Change ∈the total total ener energy of the syste system m

)

( 1− 9 )

Ener# transfer $an happen in forms of heat* "or% and mass transfer( ,hen the ener# $hane of a s#stem is 7ero* as lon as the state of the s#stem doesn+t $hane durin the pro$ess* the pro$ess is stead#(

´ = E ´ E out ( 1− 11 ) ( )f there are no external effe$ts* the total ener# of a s#stem durin ¿





Q¿ −Qout + Egen =∆ Ethermal ,system [ J ] ( 1−13 )

.eat transfer is dri!en b# a temperature differen$e and happens throuh $ondu$tion* $ondu$tion* $on!e$tion* radiation or a $ombination of all three and is either in stead# state or transient flo"( ein in stead# state* there is no $hane in temperature "ith respe$t to time( 9no"in mass is ne$essar# in findin heat transfer*

´ * mass must Q * but "hen $al$ulatin the rate of heat transfer* Q

be$ome mass flo" rate*

´ = ρV A c m

[ ]( kg s

m ´ * and the heat balan$e euation is then altered as su$h

1 −16 )

[ ](

´ =m Q ´ ∆ h =m ´ c p ∆ T

kJ s

1− 18 )

)n the pro$ess of stead# state $ondu$tion* the rate of heat transfer "as dis$o!ered throuh experiments "ith a plane "all( The experiments sho"ed that the rate of heat transfer !ia $ondu$tion depends on its eometr#* thi$%ness* material of the medium and the temperature differen$e and ho" these !ariables "ere proportional to heat transfer( Then Then :ean aptiste :oseph 3ourier* a 3ren$h mathemati$ian and ph#si$ist* $ame alon and further de!eloped this into "hat is no" %no"n as the 3ourier+s la" of heat $ondu$tion* "hi$h is

´ Q cond =

kA (T 1−T 2 ) ∆x

=−kA

dT [  ] ] ( 1−22 ) dx





proportional to

∆ x ( ;nd in stead# state* sin$e there is no $hane in temperature "ith respe$t

to time* the rate of heat transfer into the $#linder $ #linder must eual the rate of heat transfer out of it*

´ cond Q cond , cyl= constant ( The ne" euation is ´ Q cond cond , cyl= 2 !"k

T 1−T 2 ln

[  ] ] ( 3 −39 )

() r2 r1

"rocedure

1( Re$ord Re$ord the eometr# eometr# of the outer outer dis% dis% and turn on on the po"er po"er for the TST TST $ontroll $ontroller( er( 2( TR?( >TR?( '( ;t the prompt prompt "indo"* "indo"* $li$% >ST;RT >ST;RT?? in the top left $orner( The real time temperature plot is displa#ed displa#ed at the bottom and $an be modified modified b# $li$%in on on the i$ons of temperature sensors in the bottom left $orner* so that the desired readin $an be sho"n onl#( 4( =pen the the tap "hi$h "hi$h is $onne$ted $onne$ted to the the $old "ater "ater influent influent tube tube and adust adust the the flo" so that S2 is approximatel# 2(0 hours? >hours? as the time unit for the sensor plot( @( ST=P? at the top top left $orner and $lose the soft"are( soft"are(




Table 1 Distan$

Temp  10,

Temp  '0,

;nal#ti$al Data A10,B

;nal#ti$al Data A'0,B

e

A ℃¿

A ℃¿

A ℃¿

A ℃¿

8

28(4

45(-

28(4-2'@'45

45(52-@4C-@

1-

25(-

'@(8

25(51-4C@C'

'@(-45---2'

24

24(1

'4

2'(@C'2@122

''(0'552@C5

'2

2'

'0

22(5@0-2241

2C(@-4582@C

40

21(@

2@(@

21(-222-2'2

2@(22@440-1

48

20(8

24(-

20(84@'C5@

25(15444451

3iure 1

Radial Location (mm)







Table 2 .eater Po"er

Rate of .eat Transfer at

Rate of .eat Transfer at

Per$ent

Settin F,

)nner Radius F,

=uter Radius F,

Differen$e

10(4

4(44

4(44

0

'2

11(8@

11(8@

0





%iscussion and Results

The experiment a!e data for the temperatures at the radial distan$es on the dis% "hen the resistan$e heater "as set to 10, and '0,( '0,( The experimental data for -0, $ould not be obtained due to the restri$tions of the resistan$e heater( 3rom the data the $ondu$tion euation* the rate of heat transfer at the inner radius* and the rate of heat transfer at the outer radius "ere $al$ulated for both 10, and '0,( The $ondu$tion euation "as found usin t"o boundar# $onditions and the euationH

T ( r )= C 1 ln ( r ) + C 2

,ith the heater po"er set to 10, the $ondu$tion euation "as $al$ulated to be

T ( r )=−4.25 4.25 ln ( r ) + 37.3 usin the a!erae of the boundar# $onditions of ST1 and ST- and the boundar# $onditions ST2 and ST5( ,ith the heater po"er set to '0, the $ondu$tion euation "as $al$ulated to be

T ( r )=−11.37 ln ( r )+ 69.17 usin the a!erae of the boundar#

$onditions of ST1 and ST- and the boundar# $onditions ST2 and ST5( The rate of heat transfer "as $al$ulated next usin the euationsH

( )





"as found to be 4(44, and the rate of heat transfer at the outer radius "as found to be 4(44,( 4(44,( The per$ent differen$e bet"een the rate of heat transfer at the inner radius and the rate of heat transfer at the outer radius "as found to be 0( ,ith the heater po"er set to '0, the rate of heat transfer at the inner radius "as f ound to be 11(8@, and the rate of heat transfer at the outer radius "as found to be 11(8@,( The per$ent differen$e bet"een the rate of heat transfer at the inner radius and the rate of heat transfer at the outer radius "as found to be 0(

The $omplete solution of the differential $ondu$tion euation for 10(4, is as follo"sH

oundar# $onditionsH ST1H

T ( 8 ) =28.4 and ST-H

T ( 48 )=20.8

T ( r )= C 1 ln ( r ) + C 2

28.4 =C 1 ln ( 8 ) + C 2 20.8=C 1 ln ( 48 ) + C 2

28.4 =C 1 ln ( 8 ) + C 2





oundar# $onditionsH ST2H

T ( 16 ) =25.6 and ST5H

T ( 40 )=21.7

25.6= C 1 ln ( 16 ) + C 2 21.7 =C 1 ln ( 40 ) + C 2

C 1 and

C 2 $an be found the same "as as abo!e

C 1 =−4.26 C 2 =37.4

The

C 1 and

C 2 from ea$h $al$ulation are a!eraed to obtain the final $ondu$tion euationH

T ( r )=− 4.25 4.25 ln ( r ) + 37.3

The oundar# $onditions used in the solution "ere temperatures found at ST1 A8mmB and ST-





seen abo!eB( )n both 3iures 1 and 2 the experimental data fallo" a !er# similar $ur!e to the anal#ti$al data( )n both 3iures the experimental data follo"s a shallo"er $ure than the anal#ti$al data( The data presented in Table Table 2 is the rate of heat transfer at the inner and outer radius( The per$ent differen$e bet"een the rate of heat transfer at the inner radius and the rate of heat transfer at the outer radius "as 0 for both the 10, and the '0,( This means that the $ondu$tion in the dis% is eual to the $ondu$tion at the outer radius of the dis%(

Conclusion

)n this experiment the data "as used to sol!e a onedimensional $ondu$tion euation( Due to the restri$tions of the resistan$e heater* the data for -0, $ould not be obtained( The $ondu$tion euation "as sol!ed usin boundar# $onditions found in the experiment at $ertain radial temperatures i!en( Rate of heat transfer "as $al$ulated for in this experiment b# usin data found and the $oeffi$ients found in the $ondu$tion euations( )t "as $on$luded that s#stem "as in a stead# state be$ause the $ondu$tion at the inner radius of the dis% "as eual to the $ondu$tion at the outer radius of the dis%( This "as found "hen the per$ent differen$e bet"een the rate of heat transfer at the inner radius and the outer radius "as 0( This "as true "hen the resistan$e heater "as set to 10, and '0, both(

radial conduction lab report

Recommend Documents