Question

Water (inlet temperature of 30 C) flows at rate of 3.4 kg/s through a 30 cm diameter aluminium tube. You intend to heat the water to 40 C at the exit of this tube. It is recorded that the surface temperature of the tube is 98 C. Evaluate the length of tube required for this condition. Assume fully developed flow. Then, compare your calculated length with 4 m, comment on the water exit temperature if the tube length is changed to 4 m

Answer 1

Given:

• Water inlet temperature, Ti= 30°C
• Water exit temperature, To= 40°C
• Mass flow rate, = 3.4 Kg/s
• Diameter of the tube, D= 30 cm= 0.3 m
• Surface temperature of the tube, Ts= 98°C
• Type of flow: Fully developed flow.

To find:

• i) Length of the tube, L.
• ii) Water exit temperature if the length of the tube is 4 m.

Note:

i) Length of the tube, L:

We know that, bulk mean temperature, Tm= (Ti + To)/2

• Tm= (Ti + To)/2
• Tm= (30 + 40)/2
• Tm= 35°C

Properties of water at 35°C (taken from data book)

• Density, = 994.04 Kg/m^3
• Kinematic viscosity, = 0.724×10^-6 m^2/s
• Prandtl number, Pr= 4.82
• Thermal conductivity, k= 0.623 W/m.K
• Specific heat, Cp= 4179 J/Kg.K

We know that, mass flow rate, , where A= Cross sectional area of the tube(A= πD^2/4) and U= Velocity of flow of water.

• 
• 
• 
• 

We know that, Reynolds number, Re= UD/

• Re= UD/
• Re= 0.0484(0.3)/(0.724×10^-6)
• Re= 20055.25
• Since, Re= 20055.25 > 2300, the flow is turbulent

We know that, for turbulent flow (fully developed), Nusselt number, , where N= 0.4 for heating process.

• 
• 
• 
• ​​​​​​​

We know that, Nusselt number, Nu= h.D/k, where h= heat transfer coefficient.

• Nu= h.D/k
• 119.324= h(0.3)/0.623
• h= 247.8 W/m^2.K

We know that, heat transfer,

• 
• 
• 
• 

We also know that, , where = Curved surface area of the tube (= πDL)

• 
• 
• 
• 
• ​​​​​​​Length of the tube, L= 9.66 m

ii) Water exit temperature if length of the tube, L= 4m

Use the same properties of water here as in case (i).

We know that, heat transfer,  

• 
• 
• 
• 
• 

We also know that,

• 
• 
• 
• 
• 
• 

Equating both heat transfer equations,

• 
• 
• Exit temperature of water when tube length (L=4 m) is 34.33°C