1  

Problem  1    (a)      

                                                               

   

   

!"

#"

$"

%"

&!%"

&#%"

&$%"

!"

#"

$"

%"

&!'"

&#'"

&$'"

'"&%'"

&()*+"

!"

#"$"

%!$"

%&'()"

%#$" %&'()"

  2  

Node  4    !x = !y

q14 = "kT4

p "T1p

!y#!x2#d

q54 = "kT4

p "T5p

!x# !y #d

q74 = "kT4

p "T7p

!y#!x2#d

!Egen = !q #V = !q # !x2# !y #d

!Estore =mCp$T$t

= !!x2# !y #d

%

&'

(

)*Cp

T4p+1 "T4

p

!t

!Est = !Ein " !Eout + !Egen + !Eout = 0 and !Ein = q14 + q54 + q74

!!x2# !y #1

%

&'

(

)*cp

T4P+1 "T4

P

!t%

&'

(

)*= k

!x2#1

%

&'

(

)*T1

P "T4P

!y%

&'

(

)*+ k

!x2#1

%

&'

(

)*T7

P "T4P

!y%

&'

(

)*+ k !y #1( ) T5

P "T4P

!y%

&'

(

)*+ !q

!x2# !y #1

%

&'

(

)*

Fo = !!t!x2

=k!t

"cp!x2 + T4

P+1 = T4P +Fo T1

P +T7P + 2T5

P " 4T4P{ }+ !q!t

!cp    

 

T4P+1 = T4

P +Fo T1P +T7

P + 2T5P ! 4T4

P{ }+ !q"t!cp  

     

  3  

Node  5    

q25 = !kT5

p !T2p

"y# "x #d

q45 = !kT5

p !T4p

"x# "y #d

q65 = !kT5

p !T6p

"x#"y2#d

q85 = !kT5

p !T8p

"y#"x2#d

qconv = h T$ !T5p( ) # "x2 #d + h T$ !T5

p( ) # "y2 #d

!Egen = !q #V = !q # 3"x"y4

#d

!Estore =mCp%T%t

= !3"x"y4

#d&

'(

)

*+Cp

T5p+1 !T5

p

"t

!Est = !Ein ! !Eout + !Egen , !Eout = 0 and !Ein = q25 + q45 + q65 + q85 + qconv

!34"x # "y #1

&

'(

)

*+cp

T5P+1 !T5

P

"t&

'(

)

*+= k"y

T4P !T5

P

"x&

'(

)

*++ k"x

T2P !T5

P

"y&

'(

)

*++ k

"x2

T8P !T5

P

"y&

'(

)

*++ k

"y2

T6P !T5

P

"x&

'(

)

*+

+ h "x2+"y2

&

'(

)

*+ T$ !T5

P( )+ !q 34"x # "y #1

&

'(

)

*+

Fo = !"t"x2

=k"t

"cp"x2 , T5

P+1 = T5P +23Fo T8

P +T6P + 2T4

P + 2T2P ! 6T5

P{ }+ 43h"t!cP"x

T$ !T5P( )+ !q"t

!cp

     

T5P+1 = T5

P +23Fo T8

P +T6P + 2T4

P + 2T2P ! 6T5

P{ }+ 43h"t!cP"x

T# !T5P( )+ !q"t

!cp  

   

  4  

Node  6    

q36 = !kT6

p !T3p

"y#"x2#d

q56 = !kT6

p !T5p

"x#"y2#d

qconv = h T$ !T6p( ) # "x2 #d + h T$ !T6

p( ) # "y2 #d

!Egen = !q #V = !q # "x"y4

#d

!Estore =mCp%T%t

= !"x"y4

#d&

'(

)

*+Cp

T6p+1 !T6

p

"t

!Est = !Ein ! !Eout + !Egen , !Eout = 0 and !Ein = q36 + q56 + qconv

!"x # "y4

&

'(

)

*+cp

T6P+1 !T6

P

"t&

'(

)

*+= k

"y2

T5P !T6

P

"x&

'(

)

*++

k"x2

T3P !T6

P

"y&

'(

)

*++ h

"x2+"y2

&

'(

)

*+ T$ !T6

P( )+ !q "x # "y4

&

'(

)

*+

Fo = !"t"x2

=k"t

"cp"x2 , T6

P+1 = T6P +Fo 2T5

P + 2T3P ! 4T6

P{ }+ 4h"t!cP"x

T$ !T6P( )+ !q"t

!cp

 

   

T6P+1 = T6

P +Fo 2T5P + 2T3

P ! 4T6P{ }+ 4h"t

!cP"xT# !T6

P( )+ !q"t!cp  

   

  5  

(b)    

Stability limit for 2-D explicit method: !t "!x( )2

4!

!x( )2

4!=

!x( )2

4 k!Cp

=0.01m( )2

4 #100W /m$K( )

2000 kg /m3( ) 300 J / kg$K( )

= 0.15 sec

%!t " 0.15 sec

 

   (c)    T1

P = T2P = T3

P = T4P = T5

P = T6P = 300 K and T7

P = T8P = 400 K

T! = 500 K

"x = "y = 0.01m and "t = 0.1s

Fo = k"t!cP"x

2 =100W /m#K( ) 0.1 s( )

2000 kg /m3( ) 300 J / kg#K( ) 0.01m( )2= 0.167

!q =106 W m3 and h =100W m2K

 

   

T4P+1 = 316.8 K

T5P+1 = 311.7 K

T6P+1 = 301.5 K

     

Δt  =  0.1  sec  is  suitable  because  the  stability  limit  for  the  given  situation  is  Δt  ≤  0.15  sec.  

  6  

Problem  2    (a)    1r!!r

k r !T!r

"

#$

%

&'+1r2

!!!

k !T!!

"

#$

%

&'+

!!z

k !T!z

"

#$

%

&'+ !q = !Cp

!T!t  

 Assuming  1-­‐D  conduction  and  steady  state  condition,    1r!!r

k r !T!r

"

#$

%

&'+ !q = 0 (

ddr

k r dTdr

"

#$

%

&'= ) !q r

k r dTdr

= )!q r2

2+C1 (

dTdr

= )!q r2k

+C1k r

BC 1: dTdr r=0

= 0 ( *C1 = 0

dTdr

= )!q r2k

( dT+ = )!q r2k

dr+

T = ) !q2k

r2

2+C2 ( T (r) = ) !q r

2

4k+C2

BC2 : Tr=0 = Tc =C2 ( *C2 = Tc

*T (r) = Tc )!q r2

4k

T1 = Tr=r0 = Tc )!q r0

2

4k

 

   

T1 = Tc !!q r0

2

4k  

   

  7  

(b)    

     

   

 Assuming  the  length  of  the  rod,  L  =  1m,    

q = !q !V = !q ! !D12

4!L = 20000W /m3( ) ! ! (0.05m)

2

4! 1m( ) = 39.27W

Eb,1 "Eb,2

1"!1A1!1

+1

A1F12+1"!2A2!2

=" T1

4 "T24( )

1" 0.2! 0.05m( ) 1m( ) 0.2( )

+1

! 0.05m( ) 1m( ) 1( )+

1" 0.5! 0.06 m( ) 1m( ) 0.5( )

=5.67#10"8W /m2 "K 4( ) T14 " 5004( )K 4

1" 0.2! 0.05m( ) 1m( ) 0.2( )

+1

! 0.05m( ) 1m( ) 1( )+

1" 0.5! 0.06 m( ) 1m( ) 0.5( )

= 39.27W $ T1 = 544.99 K

T1 = Tc "!q r0

2

4k= Tc "

20000W /m3( )(0.025m)24 ! 15W /m"K( )

= 544.99 K $ Tc = 545.20 K

 

     

Tc = 545.20 K and T1 = 544.99 K      

!"#$#%&&#'#("#$#&)%#

!*#$#+#'#(*#$#&)"#

!"#$%&% '$% '(% !"#(%

1!!1A1!1

1A1F12

1!!2A2!2

  8  

(c)    !Ein ! !Eout + !Egen = !Estore

!Ein = !Estore = 0

" !Eout = !Egen

!Egen' = !q # !D1

2

4= 20000W /m3( ) # ! (0.05m)

2

4= 39.27W /m

" !Eout' = q1

' = !Egen' = 39.27W /m

 

 

!q1' = 39.27W /m

     

  9  

Problem  3    (a)    

     

By  neglecting  the  thickness  of  the  container  wall,    

   

!!q A = qrad =! Tc

4 "Ts4( )

1"!c!cA

+1AFcs

+1"!s!sA

; Fcs =1 # !!q =" Tc

4 "Ts4( )

1"!c!c

+1+1"!s!s

Tc =!!q"1"!c!c

+1+1"!s!s

$

%&

'

()+Ts

4*

+,

-

./

14

= 372 K

 

   

!Tc = 372 K      

!"#$#%#&#'"#$#()*#

+,-..#/#$#()0*#1#

!2344#

!2#$#50*#&#'2#$#()*#

67"331

#

89#$#:*(#;<10#

!=#$#5((#&#

!"#$%&%'%&()%

*$% *+% !"#+%

1!!cA!c

1AFcs

1!!sA!s

  10  

(b)    !!qconv = h Ts "T#( )

Ra =g! Ts "T#( )H 3

!"= 3.01$107

h =Nu kH=kH0.825+ 0.387Ra

16

1+ 0.492Pr

%

&'

(

)*

916%

&''

(

)**

827

%

&

''''''

(

)

******

2

+ h = 4.54W m2 "K

!!qconv = 4.54W m2 "K( ) 325 K "300 K( ) =114W m2

 

 

! ""qconv =114W m2  

   (c)    !!q =150W /m2 = !!qconv + !!qrad " !!qrad = !!q # !!qconv = !s" Ts

4 #Tsurr4( )

Tsurr = Ts4 #

!!q # !!qconv!s"

$

%&

'

()

14

= 315 K

 

   

!Tsurr = 315 K      

  11  

Problem  4    (a)    

     

(b)    

Nu = 0.3+ 0.62Re1 2 Pr1 3

1+ 0.4 Pr( )2 3!"

#$1 4 % 1+

Re282000&

'(

)

*+5 8!

",,

#

$--

4 5

Re = !uDµ

=0.94 kg m3( ) 5 m s( ) 0.2 m( )

218.10/7 N % s m2 = 4311.93

Nu = 0.3+0.62 4311.93( )1 2 0.7( )3

1+ 0.4 0.7( )2 3!"

#$1 4 % 1+ 4311.93

282000&

'(

)

*+5 8!

",,

#

$--

4 5

= 33.858 = h %Dk

=h % 0.02 m( )0.031W mK( )

0h = 52.48W m2K

qconv = h T1 /Th( )Ah; Ah = !DhL

qconv = h T1 /Th( )Ah = 52.48W m2K( ) 375 K /350 K( )! 0.02 m( ) 0.1m( )= 8.24353W

 

 

!qconv = 8.24 W (into the hotdog)      

!"#$%

&'(!"#'%

&"% !"#"%

1!!hAh!h

1AbFbh

1!!bAb!b

1AhFhR

1AbFbR

  12  

(c)    !Ein ! !Eout + !Egen = !Estore

!Egen = !Eout = 0

!Estore =mCpdTdt

= 0.03 kg( ) 2500 J / kg!K( ) 0.25 K / s( ) =18.75W

!Ein = qconv + qrad

" !Estore = !Ein = qconv + qrad # qrad = !Estore ! qconv =Eb,b !Eb,h

Rtot=! Tb

4 !Th4( )

Rtot

Rtot =1!!b!bAb

+1!!h!hAh

+ AbFbh +1

AbFbR+

1AhFhR

$

%&

'

()

!1*

+,,

-

.//

!1

FbR =1!Fbh = 0.93

Fhb =AbAhFbh = 0.23

FhR =1!Fhb = 0.77

"Rtot = 558.21m2

Tb =!Estore ! qconv( ) 0Rtot

!+Th

4*

+,,

-

.//

14

= 586.6 K

 

   

!Tb = 586.6 K    

  13  

Problem  5    (a)    

x fdD

!

"#

$

%&thermal

= 0.05RePr =0.05 20( ) 1.0( ) =1 Pr =1.0( )0.05 20( ) 10( ) =10 Pr =10( )

'()

*)

x fd,thermal =D Pr =1.0( )10D Pr =10( )

'()

*)

Tm = Tm,x=0 +++q P!mcp

x = Tm,x=0 + constant( ) , x - Linear Line

 

   

   

     

!"#

$#%#&#

'(#)#*+#

'(#)#*,+#-,./#

01#

$#%#

'(#)#*,+#2#*+#013$)+#

013$)%#

  14  

(b)    

Tm =!ucpT dAcAc

!!mcp

=!ucp T dAcAc

!!uAccp

Ac = y "d and dAcdy

= d # dAc = d "dy

Tm =T "d dy

$H

H!2H "d

=

%%q2k f H

y2 $H 2( )+Tw dy$H

H!

2H=12H

%%q2k f H

13y3 $H 2y

&

'(

)

*++Twy

,

-..

/

011$H

H

=12H

%%q2k f H

13H 3 $H 3&

'(

)

*++TwH

,

-..

/

011$

%%q2k f H

$13H 3 +H 3&

'(

)

*+$TwH

,

-..

/

011

234

54

674

84

=12H

%%q2k f H

$23H 3&

'(

)

*++TwH $

%%q2k f H

23H 3&

'(

)

*++TwH

,

-..

/

011

=12H

2TwH $%%q

2k f H43H 3&

'(

)

*+

,

-..

/

011=12H

2TwH $%%qkf

23H 2&

'(

)

*+

,

-..

/

011=12H

2TwH $2 %%q3k f

H 2,

-..

/

011

= Tw $%%q

3k fH # Tm = Tw $

%%q3k f

H

%%q = h Tw $Tm( ) # h = %%qTw $Tm

=%%q

Tw $ Tw $%%q

3k fH

&

'((

)

*++

=%%q%%q

3k fH=3k fH

 

   

!h =3k fH