heat project ii

8/17/2019 Heat Project II

1/13

University of Puerto Rico Campus of Mayagüez

Department of Mechanical Engineering

Heat & Mass Transfer Proect !"

#ame

$ection

Heat & Mass Transfer

Professor


2/13

Project 1

The governing equations for ow over a at plate are:

For the ow conditions over the at plate

ℜ=ú L

v =200,000

which is laminar ow. Figures 1.1 & 1.2 are respectivel the temperature and

velocit pro!les o"tained from ##$%. Figures 1. & 1.' are the numericall

o"tained temperature and velocit pro!les. These were o"tained " solving the

appropriate equations using the 'th order (unge)*utta method. The solutions

o"tained from the mentioned method can "e found in !gures 1.+ & 1.,. The pro!leso"tained in the pro"lem are similar to each other and therefore we conclude that

the (unge)*utta method was a success.

Project 2

The cooler simulation was done two times- one where the uid was air and

second case where the uid was water. To achieve this- the simulation was ept with

all the same conditions "ut one changes the properties of the air to match the

properties of water. This is due to the fact that the most important properties in

convection are dnamic viscosit- thermal conductivit- densit- and speci!c heat-

as well as the uid velocit of the ow. /ince can change these values from one

su"stance to another that0s all one has to do to change the environment of the

simulation.

To o"tain the theoretical heat transfer coecient one !rst calculates the heat

transfer with ##$% and uses the initial temperatures Ts 3 1445# & Tinf 3 45#6

along with the surface area of the !ns to calculate to o"tain an average value.

For water we have that:

Q=h A s (T s−T ∞ )=106W

h= Q

A s (T s−T ∞ )=246.23

W

m2 K

ε= Lc3/2

√ h

k A p=0.52


3/13

7ith these values we can go to the ecienc curves and o"tain that the magnitude

for the ecienc is:

n=0.85

8nd therefore that the theoretical heat transfer is:

Q=

h (T b−

T ∞ ) ( Auf +

n A f )=91.14W

8nd an error of 1+.19 is o"tained when compared to the numerical value of 14,7.

7hen the uid is air we instead have that:

Q=h As (T s−T ∞ )=14.3W

h= Q

A s (T s−T ∞ )=33.22

W

m2 K

ε= Lc3/2

√ h

k A p=0.19

Then the ecienc is:

n=0.96

"taining a theoretical heat transfer rate of:

Q=h (T b−T ∞ ) ( Auf +n A f )=13.77W

This value gives us an error of .;9 with the numerical heat transfer rate of 1'.7.

Project

/ince (e 3 144 this is laminar ow with uniform heat u


4/13

T mo=T mi+ q s

' ' πDL

ú π

4 D

2C p

T mo=

T mi+ 4q s

'' L

ú DC p=548.76 K

8fter calculating the mean temperature one determines the heat transfer coecient

Nu=hD

k =4.36

h=4.36 k

D=6.54

W

m2 K

8nd lastl the ma


5/13

$8T=8> code to solve the di?erential equations

clc, clear all, close all;

dn = 0.05; vN = 0:dn:30;

%Initial Conditions

Y0 = 0; Y1 = 0; Y2 = 0.332;

%Function Vectors

vY0 = Y0;

vY1 = Y1;

vY2 = Y2;

or i = 2:len!t"#vN$

%Calculate net vY0 usin! Y1

&1 = Y1;

&2 = Y1 ' dn(2)&1;

&3 = Y1 ' dn(2)&2;

&* = Y1 ' dn)&3;

dY0 = dn(+)#&1'2)&2'2)&3'&*$;

vY0#i$ = vY0#i1$ ' dY0;

%Calculate net vY1 usin! Y2

&1 = Y2;

&2 = Y2 ' dn(2)&1;

&3 = Y2 ' dn(2)&2;

&* = Y2 ' dn)&3;

dY1 = dn(+)#&1'2)&2'2)&3'&*$;

vY1#i$ = vY1#i1$ ' dY1;

%Calculate net vY2 usin! Y3 = Y0)Y2(2

&1 = Y0)Y2(2;

&2 = Y0)Y2(2 ' dn(2)&1;

&3 = Y0)Y2(2 ' dn(2)&2;

&* = Y0)Y2(2 ' dn)&3;

dY2 = dn(+)#&1'2)&2'2)&3'&*$;

vY2#i$ = vY2#i1$ ' dY2;

%-et ne values

Y0 = vY0#i$;

Y1 = vY1#i$;

Y2 = vY2#i$;

end

i!ure; /lot#vN,vY0;vY1;vY2$; le!end#,,$;

%Constants

r = 1;

%Initial Conditions

40 = 0;

41 = 0.332 ) r#1(3$;


6/13

%Function Vectors

v40 = 40;

v41 = 41;

or i = 2:len!t"#vN$

%Calculate net v40 usin! 41 &1 = 41;

&2 = 41 ' dn(2)&1;

&3 = 41 ' dn(2)&2;

&* = 41 ' dn)&3;

d40 = dn(+)#&1'2)&2'2)&3'&*$;

v40#i$ = v40#i1$ ' d40;

%Calculate net v41 usin! 42 = r(2 41

&1 = r(2)vY0#i$)41;

&2 = r(2)vY0#i$)41 ' dn(2)&1;

&3 = r(2)vY0#i$)41 ' dn(2)&2;

&* = r(2)vY0#i$)41 ' dn)&3;

d41 = dn(+)#&1'2)&2'2)&3'&*$;

v41#i$ = v41#i1$ ' d41;

%-et ne values

40 = v40#i$;

41 = v41#i$;

end

i!ure; /lot#vN,v40;v41$; le!end#6),6)$;

%Constants

78a = 2; % 8(s

= 0.5; % 8

V&in = 105; % 82(s

n9l/"a = srt#78a ( V&in ( $;

6in = 300; %

6s = *00; %

%lot Velocit< roile

vVelocit< = 78a ) vY1;

vY = vN ( n9l/"a;

i!ure;

/lot#vVelocit


7/13

Figure 1.1

Temperature Pro!le o"tained from ##$%

Figure 1.2

@elocit Pro!le o"tained from ##$%


8/13

280 300 320 340 360 380 400 4200

0.005

0.01

0.015

0.02

0.025

0.03

Temperature [K]

Y

P o s i t i o n [ m ]

Figure 1.

Temperature Pro!le o"tained analticall

0 0.5 1 1.5 2 2.50

0.005

0.01

0.015

0.02

0.025

0.03

Velocity [m/s]

Y

P o s i t i o n [ m ]

Figure 1.'

@elocit Pro!le o"tained analticall


9/13

0 5 10 15 20 25 300

5

10

15

20

25

30

f

f'

f''

Figure 1.+

(unge)*utta solution for the >lasius Aquation

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

T*

T*'

Figure 1.,

(unge)*utta solution for the Temperature Aquation


10/13

Figure 2.18ir Temperature in a /ection Plane

Figure 2.2

8ir @elocit in a /ection Plane


11/13

Figure 2.7ater Temperature in a /ection Plane

Figure 2.'

7ater @elocit in a /ection Plane


12/13

Figure 2.+

Beat Transfer from the #ooler in 8ir

Figure 2.,

Beat Transfer from the #ooler in 7ater

Figure .1

Temperature across the whole tu"e

Figure .2

@elocit Pro!les in the Antrance (egion


13/13

Figure .

Temperature at the Tu"e /urface

Figure .'

$ean Temperature at the end of the Beated (egion

heat project ii

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