Basic concepts-thermal considerations In the thermally fully developed flow of a fluid with constant properties, the local convection coefficient is a constant, independent ofx h≠f(x) For the special case of uniform surface heat flux aT d Axial t Independent of gradient fa, t dx radial location fa, t cons tan t For the case of constant surface temperature Depends on the aT (T-T)dmn radial coordinate Ox\d, (T。-Tn)ax T=cons tan t Heat Transfer 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 f d t m s m s f d 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 o10mm have the form ()=0.-(/) 7(r)=3448+750(/)-18.8(/rn with units of m/s and K, respectively. Determine the corresponding value of the mean(or bulk) temperature,, at this axial position KNOWN: Velocity and temperature profiles for laminar flow in a tube of radius ro=10mm FIND: Mean(or bulk) temperature, Tm, at this axial position. SCHEMATIC u(n,7(0 Fluid ASSUMPTIONS: (1) Laminar incompressible flow, (2)Constant properties Heat Transfer 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 u(r)=0.11(rn) T(r)=3448+75.0(rr02-18.8(r (1,2) For incompressible flow with constant c, in a circular tube, from Eq. 8. 27, the mean temperature and un the mean velocity, from Eq. 8.8 are, respectively 25u()r()r u(r).rdr Substituting the velocity profile, Eq (1), into Eq (4)and integrating, find Substituting the profiles and um into Eq. (3), find 4{38()+088(+080) Tn=417240+1875-3131820+1250-235}=367K The velocity and temperature profiles appear as shown below. Do the values of um and Tm found above with th ive profiles as you thought? Is the fluid being heated or cooled? 008 E> 004 002 0 002040608 002040608 Radial coordinate riro Radial coordinate r/ro Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 8