CHEMICAL ENGINEERING

210 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

to the 0.8 power of the gas velocity and the heat transfer by radiation from the walls to
the thermocouple is proportional to the temperature difference.
When the gas is flowing at 5 m/s the thermocouple reads 323 K. When it is flowing
at 10 m/s it reads 313 K, and when it is flowing at 15.0 m/s it reads 309 K. Show that
the gas temperature is about 298 K and calculate the approximate wall temperature. What
temperature will the thermocouple indicate when the gas velocity is 20 m/s?

Solution

If the gas and thermocouple temperatures areTgandTkrespectively, then the rate of
heat transfer from the thermocouple to the gas:

Q 1 DKu^0.^8 

TgT (i)

whereKis a constant anduthe gas velocity.
Similarly, the rate of heat transfer from the walls to the thermocouple is:

Q 2 Dk^0 

TwTW (ii)

wherek^0 is a constant andTwis the wall temperature.

At equilibrium: Q 1 DQ 2 andu^0.^8 D
k^0 /k
TwT/
TTg (iii)

WhenuD5m/s, TD323 K and in (iii):

50.^8 D

k^0 /k

Tw 323 /

 323 Tg D 3. 624

∴
k^0 /kD 3. 624
323 Tg/
Tw 323 (iv)

WhenuD10 m/s,TD313 K and in (iii):

100.^8 D

k^0 /k

Tw 313 /

 313 Tg D 6. 31

Substituting for
k^0 /kfrom (iv):

6. 31 D 3. 624

323 Tg

Tw 313 /[

Tw 323

313 Tg ](v)

WhenuD15 m/s,TD309 K and in (iii):

150.^8 D

k^0 /k

Tw 309 /

 309 Tg D 8. 73

Substituting for
k^0 /kfrom (iv):

8. 73 D 3. 624

323 Tg

Tw 309 /[

Tw 323

309 Tg ](vi)

IfTgD298 K, then in (v):

1. 741 D[ 

323  298 

Tw 313 ]/[

Tw 323

313  298 ]

or:
Tw 313 /
Tw 323 D 1 .045 andTwD533 K.

IfTgD298 K, then in (vi): 2. 409 D
323  298 
Tw 309 /[
Tw 323
309  298 ]

or:
Tw 309 /
Tw 323 D 1 .060 andTwD556 K