3. Internal Flow General Concepts:

Transcription

1 3. Internal Flow General Concet: ρ u u 4 & Re Re, cr 2300 μ ν π μ Re < 2300 lainar 2300 < Re < 4000 tranitional Flow Regie : Re > 4000 turbulent Re > 10,000 fully turbulent (d) 1

2 (e) Figure 1 Boundary layer develoent for lainar flow in a circular tube: (a) The hydrodynaic boundary layer and velocity rofile. (b) The theral boundary layer and teerature rofile for urface theral condition: contant teerature, T. (The fluid in the tube i being wared.) (c) Velocity and teerature rofile for deterining the ean (average) teerature at a location. (d) The develoent of the theral boundary layer in a tube. (The fluid in the tube i being cooled.) (e) Velocity rofile in turbulent flow. - Hydrodynaic and theral entry length: For lainar flow: ( ) 0.05Re For turbulent flow: ( fd, h fd, t ) la la ( 0.05Re 10 ( fd, t ) turb fd, h ) ( r r ( turb fd, h 60 ) 10 turb fd, h ) la Figure 2Variation of the friction factor and the convection heat tranfer coefficient in the flow direction for flow in a tube (r>1) 2

3 Figure 3 Variation of local Nuelt nuber along a tube in turbulent flow for both unifor urface teerature and unifor urface heat flu [eiler (1953)]. Note: Nuelt nuber i inenitive to the tye of theral boundary condition in turbulent flow, and the turbulent flow correlation can be ued for either tye of boundary condition. The Mean Teerature: Figure 4 Actual and idealized teerature rofile for flow in a tube (the rate at which energy i tranorted with the fluid i the ae for both cae). Note: Unlike the ean velocity, the ean teerature T will change in the flow direction whenever the fluid i heated or cooled. - Conervation of energy rincile: C & T C T δ & C T ( ρ V da ) T C T δ & C & ρ C 3 ρ C Aue: 1. Contant denity and ecific heat 2 2. A circular ie of radiu R: A π R and da 2π r dr - The bulk ean fluid teerature: c T c T V da V A 2 T r V r r dr R V (, ) (, ) 2 T T b + T 2, i, e c c c

4 Figure 5 Therally fully develoed flow characteritic for contant urface teerature Heating: Relative hae of the teerature rofile reain unchanged in the flow direction ( 2 > 1 ). Hydrodynaically fully develoed: Therally fully develoed: V ( r, ) 0 V V ( r) T ( ) (, ) T r 0 T T Theral Analyi: Figure 6 The heat tranfer to a fluid flowing in a tube. Overall Tube Energy Balance: Newton Law of Cooling: Q& C & q& ( T, e T, i h ( T T ) ) Figure 7 Energy interaction for a differential control volue in a tube. Energy Balance on a ifferential Control Volue: δ Q & C & dt 4

5 δ Q& q& da q& ( d) dt d q& C & C & h ( T T ) 1. Contant Surface Heat Flu ( q& contant ): By integration: dt d T q& C & T, i contant q& + C & q& Δ T T T h Figure 8 Variation of the tube urface and the ean fluid teerature along the tube for the cae of contant urface heat flu. 1. Contant Surface Teerature ( T contant ): dt d ΔT d d C & By integration: ΔT ln ΔT Nuber of Tranfer Unit (NTU): Note: For NTU > 5, e T T i C & Δ T T T h h NTU ΔT d ΔT ΔT C & T T h A e T T i C &, h A C & h d 5

6 Figure 9 The variation of the ean fluid teerature along the tube for the cae of contant teerature. Q& h A ΔT l Log Mean Teerature ifference (LMT): ΔTo ΔTi ΔTl ΔTo ln ΔT Convection Correlation for Tube: Fully eveloed Region Table 1 Suary of Forced Convection Heat Tranfer Correlation for Internal Flow in Sooth Circular Tube i Note: In any alication the tube length will eceed the theral entry length, it i often reaonable to aue that the average Nuelt nuber for the entire tube i equal to the value aociated with the fully develoed region: N u Nu, fd Flow in Noncircular Tube Hydraulic iaeter: Where i wetted erieter. h 4A c 6

7 Note: 1. Hydraulic diaeter that hould be ued in calculating the Reynold and Nuelt nuber. 2. The aroach can be ued for both lainar and turbulent flow. Table 2 Nuelt Nuber for Fully eveloed Lainar Flow in Noncircular Tube 7

8 Eale 1: 8

9 Eale 2: 9

10 10

11 Eale 3: 11

12 12

13 Eale 4: 13

14 14