Internal Forced Convection Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Introduction Pipe circular cross section. Duct noncircular cross section. Tubes small-diameter pipes. The fluid velocity changes from zero at the surface (no-slip) to a maximum at the pipe center. It is convenient to work with an average velocity, which remains constant in incompressible flow when the cross-sectional area is constant.
Average Velocity The value of the average velocity is determined from the conservation of mass principle & (8-1) m ρv A ρu r da = = ( ) avg C C A A c For incompressible flow in a circular pipe of radius R ρu( r) da R C ρu( r) 2πrdr A 0 2 c V = = = u r rdr avg R 2 2 ρ AC ρπ R R 0 ( ) (8-2)
Average Temperature It is convenient to define the value of the mean temperature T m from the conservation of energy principle. The energy transported by the fluid through a cross section in actual flow must be equal to the energy that would be transported through the same cross section if the fluid were at a constant t temperature t T m E & = & = ( ) & = ( ) ( ) fluid mct p m ct p r δm ρct p r u rvdac (8-3) m& A c
For incompressible flow in a circular pipe of radius R T m ct r ( ) δ m& ( ) ( p ρ )2 p m& = = A c ct r u r πrdr ( 2 π ) & (8-4) mc ρv R c p avg p R = 2 2 V avg R 0 T r u r rdr ( ) ( ) The mean temperature T m of a fluid changes during heating or cooling. Idealized Actual
Laminar and Turbulent Flow in Tubes For flow in a circular tube, the Reynolds number is defined as ρvavg D Vavg D (8-5) Re = = μ ν For flow through noncircular tubes D is replaced by the hydraulic diameter D h. 4A D = c h P (8-6) laminar flow: Re<2300 fully turbulent: Re>10,000.
The Entrance Region Consider a fluid entering a circular pipe at a uniform velocity. Because of the no-slip condition a velocity gradient develops along the pipe. The flow in a pipe is divided into two regions: the boundary layer region, and the and the irrotational (core) flow region. The thickness of this boundary layer increases in the flow direction until it reaches the pipe center. Irrotational flow Boundary layer
Hydrodynamic entrance region the region from y y g g the pipe inlet to the point at which the boundary layer merges at the centerline. Hydrodynamically fully developed region the region beyond the entrance region in which the velocity profile is fully developed and remains unchanged. The velocity profile in the fully developed region is parabolic in laminar flow, and somewhat flatter or fuller in turbulent flow.
Thermal Entrance Region Consider a fluid at a uniform temperature entering a circular tube whose surface is maintained at a different temperature. Thermal boundary layer along the tube is developing. The thickness of this boundary layer increases in the flow direction until the boundary layer reaches the tube center. Thermal entrance region. Thermally fully developed region the region beyond the thermal entrance region in which the dimensionless temperature profile expressed as (T s -T)/(T s -T m ) remains unchanged.
Surface heat flux can be expressed as T k( T r) r = q& ( ) R s = hx Ts Tm = k hx = r r=r Ts Tm For thermally fully developed region From (Eq. (8-9)) (8-10) T r ( ) T s T m r= R f x ( ) h f x x ( ) Fully developed flow h x = constant Fully developed flow
The Heat Transfer coefficient and Friction factor Developing region Fully developed region
Hydrodynamic Entry Lengths Laminar flow L 0.05Re D (8-11) h,laminar Thermal L 005RePr 0.05Re D = Pr L (8-12) t,laminar Turbulent flow h,laminar Hydrodynamic 14 L = 1.359D Re (8-13) h, turbulent Thermal (approximate) L L 10 D (8-14) h, turbulent t, turbulent 10
Turbulent flow Nusselt Number The Nusselt numbers are much higher in the entrance region. The Nusselt number reaches a constant value at a distance of less than 10 diameters. The Nusselt numbers for the uniform surface temperature and uniform surface heat flux conditions are identical in the fully developed regions, and nearly identical in the entrance regions. Nusselt number is insensitive to the type of thermal boundary condition.
General Thermal Analysis In the absence of any work interactions, the conservation of energy equation for the steady flow of a fluid in a tube Q& = mc & T T (8-15) ( ) (W) p e i The thermal conditions at the surface can usually be approximated as: constant surface temperature, or constant surface heat flux. The mean fluid temperature T m must change during heating or cooling. Either T s = constant or q s = constant at the surface of a Either T s constant or q s constant at the surface of a tube, but not both.
Constant Surface Heat Flux In the case of constant heat flux, the rate of heat transfer can also be expressed as Q& = q& A = mc & T T (8-17) s s p e i ( ) (W) Then the mean fluid temperature t at the tube exit becomes qa s s T = & e Ti + (8-18) mc & The surface temperature in the case of constant surface heat flux can be determined from q s q& = ht ( T) T = T + & (8-19) s s m s m h p
In the fully developed region, the surface temperature T s will also increase linearly in the flow direction Applying the steady-flow energy balance to a tube slice of thickness dx, the slope of the mean fluid temperature T m can be determined m dt q& p mc & dt = & p m q s pdx = = dx mc & p Noting that both the heat flux and h (for fully developed flow) are constants dtm dts = (8-21) dx dx ( m s p ) constant (8-20)
In the fully developed region (T s -T m =constant) T T 1 T T T dt = 0 = 0 = x T T T T x x s m s m x dx s s s Combining Eqs. 8 20, 8 21, and 8 22 gives (8-22) For a circular tube T dt s dtm qp & = = = s = constant x dx dx mc & T dt 2 & s dtm q = = = s = constant x dx dx ρv c R avg p p (8-23) (8-24)
Constant Surface Temperature The energy balance on a differential control volume δ Q & = mc & dt = h T T da (8-27) ( ) p m s m s Since the mean temperature of the fluid T m increases in the flow direction the heat flux decays with x. The surface temperature is constant (dt m=-d(t s-t m)) and da s =pdx, therefore, d ( T ) s Tm hp dx = T T mc & s m p (8-28)
Integrating geq. 6-28 from x=0 (tube inlet where T m =T i ) to x=l (tube exit where T m =T e ) gives T T ha = (8-29) T T mc & ln s e s s i p Taking the exponential of both sides and solving for T e or T = T T T hpl mc & ( ) exp( ) e s s i p T x T T T hpx mc & ( ) = ( ) exp ( ) m s s i p (8-30)
The temperature difference between the fluid and the surface decays exponentially in the flow direction, and the rate of decay depends on the magnitude of the exponent has mc & p This dimensionless parameter is called the number of transfer units (NTU). Large NTU value increasing tube length marginally increases heat transfer rate. Small NTU value heat transfer increases significantly with increasing tube length.
Solving Eq. 8 29 for mc p gives mc & p = ha ln Ts Te Ts Ti s ( ) ( ) Substituting this into Eq. 8 15 where p s ln (8-31) Q& = mc & = haδt (8-32) Ti Te Δ Te Δ Ti Δ Tln = = ln Ts Te Ts Ti ln ΔTe ΔT ( ) ( ) [ ] i (8-33) ΔΤ ln is the logarithmic mean temperature difference.
Laminar Flow in Tubes Assumptions: steady laminar flow, incompressible fluid, constant properties, fully developed region, and straight circular tube. The velocity yp profile u(r) ( ) remains unchanged in the flow direction. no motion in the radial direction. no acceleration.
The bulk mean temperature T m is determined by substituting the velocity and temperature profile relations (Eqs. 8 41 and 8 57) into Eq. 8 4 and performing the integration (8-58) T m = T s 11 24 qr & s k q& = h T T ( ) s s m = 24 k 48 k 436 4.36 k 11 R = 11 D = D (8-59) h (8 59) Constant heat flux (circular tube, laminar) hd Nu = = 4.36 (8-60) k Constant Surface temperature (circular tube, laminar) hd Nu = = 3.66 (8-61) k
Laminar Flow in Noncircular Tubes The friction factor (f) and the Nusselt number relations are given in Table 8 1 1forfully fully developed laminar flow in tubes of various cross sections.
Developing Laminar Flow in the Entrance Region For a circular tube of length L subjected to constant surface temperature, the average Nusselt number for the thermal entrance region (hydrodynamically developed flow) 0065 0.065( DL) Re Pr Nu = 3.66 + (8-62) 23 1+ 0.04 ( D L ) Re Pr For flow between isothermal parallel plates Nu 0.03( D ) h L Re Pr = 7.54 + (8-64) 23 1+ 0.016 ( D ) h L Re Pr
Turbulent flow in Tubes Most correlations for the friction and heat transfer coefficients in turbulent flow are based on experimental studies. For smooth tubes, the friction factor in turbulent flow can be determined from the explicit first Petukhov equation f 2 = 0.79ln Re 1.64 3000<Re<5 10 (8-65) ( ) 2 6 For fully developed dturbulent flow the Nusselt number (Dittus Boelter equation) 0.8 n Re > 10,000 000 n = 04 0.4 heating Nu = 0.023Re Pr 0.7 Pr 160 n = 0.3 cooling (8-68)
Modified correlations are available for/due to : liquid metals (Pr<<1), large variation in fluid properties due to a large temperature difference, surface roughness, flow through h tube annulus. Original correlations are also approximately valid for: developing Turbulent Flow in the Entrance Region, turbulent Flow in Noncircular Tubes.