Convection in a pipe with a constant heat flux at the wall Water flows in an insulated pipe with an inside diameter of 0.8 cm and a length of 6.7 m. A constant heat flux of 0.7 W/cm is applied on the outside wall of the pipe under the insulation. The inlet water temperature is 15C. If the inside wall temperature must stay below 85C everywhere along the pipe, what minimum flow velocity is needed? Neglect entrance effects. Insulation Heater = 15C ter 0.8 cm 6.7 m Conservation of energy for the control volume (entire fluid volume in the pipe): Q = mcp (Te - Ti) Total heat transfer rate crossing the boundaries at the fluid-wall interface Total heat transfer rate into the fluid T= 15C Water Insulation 0.8 cm 6.7 m Heater For a uniform cross-section pipe (with diam D) with constant wall heat flux: q"TDL = p Cp (Te - Ti) - PYTD 4 Solving for average flow velocity: V= 4q"L pDCp (Te - Ti) Use relationship we found between the local surface temperature and the local average fluid temperature for the pipe with constant wall heat flux casulation Heater T= 15C Water 0.8 cm 6.7 m T(x)=T(x) qq /h(x) Apply the relationship above at the exit location: To find h let's guess flow if laminar and fully developed (problem said we can neglect entrance effects so flow is fully developed): Te = Ts NUD Aw h hD k 4.36 T= 15C Water Insulation 0.8 cm 6.7 m- Heater NUD 4.36 We need properties evaluated at bulk fluid temperature: Tm = (Ti + Te) /2. Because we do not know yet the bulk mean fluid temperature we will need to first guess it: Tm -1 (T; + Ts) (15 +85) C = 50C Next we calculate h with at the guessed average temperature: W m-K h = 4.36k 4.36 0.643 (0.8 cm) hD k Im 100 cm = 350 m.K T= 15C Water Insulation 0.8 cm 6.7 m Heater h = 4.36k D 4.36 (0.643) (0.8 cm) (100cm) T=T-2 = 85C- Using this h we calculate the exit temperature of the fluid: = 350 0.7 W) (100 cm W m.K 350- So now we can recalculate the bulk mean temperature: Tm (15+65)/2 = 40C m.K 2 65C T= 15C Water Insulation Heater 0.8 cm Tm -1 (T; +Ts) 3 (15 +85) C = 50C Difference in values is 10 degrees! Homework: reiterate properties using the last calculation of Te to find Tm for next iteration, h, new Te and new Tm until your last iteration value for Tm is within 1 degree of the previous Tm iteration Tm = (15+65)/2 = 40C But our first guess was: Next calculate the average fluid velocity using last iteration for Te value: Finally cross-check if Reynolds number is indeed for laminar flow: V= 4q"L pDcp (Te - Ti) Re= pDVm l