# Heat transfer air at 1 atm and 200 degrees Celsius flows through a tube of 2.54 cm at a mean...

## Question:

Heat transfer air at 1 atm and 200 degrees Celsius flows through a tube of 2.54 cm at a mean velocity of 10 m/s.

a) Calculate the heat transfer per unit length of the tube if a constant heat flux condition is maintained at the wall temperature is 20 degrees Celsius above the air bulk temperature all along the tube length.

b) How much would the bulk temperature increase over the 3-m length of the tube?

## Heat transfer

Heat transfer deals with temperatures. It changes the internal energy of both systems involved according to the First Law of Thermodynamics. This happens when there is a heat that's being transferred from a lower temperature object to a higher temperature object. Nusselt number is a dimensionless variable used in calculations of heat transfer between a moving fluid and a solid body.

Given:

{eq}T = 200^0 C \\ T_s = 220^0 C \\ v = 10 m/s \\ d = 2.54 cm = 0.0254 m \\ l = 1 m \\ \\ {/eq}

PART A.

Condition 1, Constant heat flux condition:

Nusselt number - {eq}Nu = 4.364 {/eq}

{eq}\frac {hd}{k} = 4.364 \\ \frac {h (0.0254)}{0.0386} = 4.364 \\ h = 6.6391 w/m^2 k {/eq}

Heat transfer per unit length:

{eq}Q = hA (T_s - T ) \\ Q = h (\pi DL ) (T_s - T ) \\ Q = 6.6319\pi (0.0254) (220-200) \\ Q = 10.584 w/m \\ {/eq}

PART B. Fully developed turbulent flow

Reynold's number: {eq}\frac {vd}{v} = \frac {10 (0.0254)}{17.21 \times 10 ^\text {-6}} \\ Re = 14758.86113 {/eq}

{eq}Nu_d = 0.023 (Re_d)^\text {0.8} (P_0)^\text {0.4} \\ Nu_d = 0.023 (14758.86113 )^\text {0.8} (0.681)^\text {0.4} \\ Nu_d = 42.68044 {/eq}

{eq}Nu_d = \frac {hd}{k }\\ \frac {h (0.0254)}{ 0.0386} = 42.68044 \\ h = 64.86083 w/m^2 k {/eq}

Heat transfer for length L=3m

{eq}Q = hA (T_s - T ) \\ Q = h (\pi DL ) (T_s - T ) \\ Q = 64.86083 \pi (0.0254) (220-200) \\ Q = 310.5398 \\ {/eq}

Therefore the heat transfer rate in fully developed turbulent condition is {eq}Q = 310.5398 {/eq} watts