when 0.5 kg of water per minute is passed through a tube of 20mm diameter it is getting heated from 20 degree c to 50 degrees c. the heating is accomplished by condensing steam on the surface of the tube which is maintained at 85 degrees c. determine the length of the tube for fully developed flow. take thremo physical properties at 60 degrees c.

 To determine the length of the tube required for fully developed flow when heating water from 20°C to 50°C with a mass flow rate of 0.5 kg/min, we follow these steps:

1. Convert mass flow rate to kg/s:

   $$\dot{m}=\frac{0.5\text{kg/min}}{60}=0.008333\text{kg/s}$$

2. Calculate the heat required to raise the water temperature:

3. Determine the log mean temperature difference (LMTD):

   $$\mathrm{\Delta }{T}_{\text{lm}}=\frac{(85-20)-(85-50)}{\ln \left(\frac{85-20}{85-50}\right)}=\frac{65-35}{\ln \left(\frac{65}{35}\right)}=\frac{30}{\ln (1.857)}\approx 48.46\text{°C}$$

4. Calculate the Reynolds number (Re) to determine the flow regime:

   $$u=\frac{\dot{m}}{\rho \cdot A}=\frac{0.008333}{983\cdot 0.000314}\approx 0.027\text{m/s}$$$$\text{Re}=\frac{\rho \cdot u\cdot D}{\mu }=\frac{983\cdot 0.027\cdot 0.02}{0.467\times 10^{-3}}\approx 1137(\text{laminar flow})$$

5. Determine the Nusselt number (Nu) for fully developed laminar flow:

   $$\text{Nu}=3.66(\text{constant wall temperature})$$

6. Calculate the convective heat transfer coefficient (h):

7. Solve for the length of the tube (L):

   $$L=\frac{q}{h\cdot \pi \cdot D\cdot \mathrm{\Delta }{T}_{\text{lm}}}=\frac{1046.25}{119\cdot \pi \cdot 0.02\cdot 48.46}\approx 2.89\text{meters}$$

Final Answer

The length of the tube required is \boxed{2.89 \text{ meters}}.