Basic concepts-thermal considerations o In the thermally fully developed flow of a fluid with constant properties,the local convection coefficient is a constant,independent of x. h≠f(x) For the special case of uniform surface heat flux ot Axial T dT Independent of gradient Ox fd,t dx fd.t radial location q"cons tant For the case of constant surface temperature Depends on the ot (Ts -T)dTmn radial coordinate 8x\fd. (Ts-Tm)dx T,cons tant Heat Transfer #6 Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 6 Basic concepts—thermal considerations • In the thermally fully developed flow of a fluid with constant properties, the local convection coefficient is a constant, independent of x. • For the special case of uniform surface heat flux • For the case of constant surface temperature h f (x) fd t m fd t dx dT x T , , q cons t s tan Axial T gradient Independent of radial location fd t m s m s fd t dx dT T T T T x T , , ( ) ( ) T cons t s tan Depends on the radial coordinate
Example:Velocity and temperature profiles for laminar flow in a tube of radius r=10mm have the form )=0.l-(1] T(r)=344.8+75.0r/-18.8r/) with units of m/s and K,respectively.Determine the corresponding value of the mean (or bulk) temperature,,at thisaxial position KNOWN:Velocity and temperature profles for laminar flow in a tube of radius10mm FIND:Mean(r bulk)temperature,T at this axial position. SCHEMATIC: ,T0 Fluid ASSUMPTIONS:(1)Laminar incompressible flow,(2Constant propertes. Heat Transfer #7 Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 7 Example: Velocity and temperature profiles for laminar flow in a tube of radius have the form with units of m/s and K, respectively. Determine the corresponding value of the mean (or bulk) temperature, , at this axial position. ro 10mm Tm 2 4 2 ( ) 344.8 75.0 / 18.8 / ( ) 0.11 / o o o T r r r r r u r r r
ANALYSIS:The prescribed velocity and temperature profiles,(m/s and K,respectively)are ut=0.1[1-r] Tr)=344.8+75.0(rr)2-18.8(rr)4 (1,2) For incompressible flow with constant c in a circular tube,from Eq.8.27,the mean temperature and um the mean velocity,from Eq.8.8 are,respectively, aa (3,4) Substituting the velocity profile,Eq.(1),into Eq.(4)and integrating,find i)am Substituting the profiles and uinto Eq.(3),find h-oeiseaseus-iae-oos)oe n=4348t,)+75otP-18to]-3486户+750t6户-1s8t。7a(6o) Tm=4172.40+18.75-3.13]-[86.20+12.50-2.35}=367K < The velocity and temperature profiles appear as shown below.Do the values of um and Tm found above compare with their respective profiles as you thought?Is the fluid being heated or cooled? 0.1 440 0.08 420 0.06 400 0.04 380 0.02 360 340 0 0.20.40.60.8 1 0 0.20.40.60.81 Radial coordinate,r/ro Radial coordinate,r/ro Heat Transfer #8 Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 8