1. Chapter 8: Internal Forced Convection
Yoav Peles
Department of Mechanical, Aerospace and Nuclear Engineering
Rensselaer Polytechnic Institute
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

2. Objectives
When you finish studying this chapter, you should be able to:
Obtain average velocity from a knowledge of velocity profile, and average temperature from a knowledge of temperature profile in internal flow,
Have a visual understanding of different flow regions in internal flow, such as the entry and the fully developed flow regions, and calculate hydrodynamic and thermal entry lengths,
Analyze heating and cooling of a fluid flowing in a tube under constant surface temperature and constant surface heat flux conditions, and work with the logarithmic mean temperature difference,
Obtain analytic relations for the velocity profile, pressure drop, friction factor, and Nusselt number in fully developed laminar flow, and
Determine the friction factor and Nusselt number in fully developed turbulent flow using empirical relations, and calculate the pressure drop and heat transfer rate.

3. 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.

4. Average Velocity
The value of the average velocity is determined from the conservation of mass principle
For incompressible flow in a circular pipe of radius R
(8-1)
(8-2)

5. Average Temperature
It is convenient to define the value of the mean temperature Tm 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 temperature Tm
(8-3)

6. For incompressible flow in a circular pipe of radius R
The mean temperature Tm of a fluid changes during heating or cooling.
(8-4)
Idealized
Actual

7. Laminar and Turbulent Flow in Tubes
For flow in a circular tube, the Reynolds number is defined as
For flow through noncircular tubes D is replaced by the hydraulic diameter Dh.
laminar flow: Re<2300
fully turbulent: Re>10,000.
(8-5)
(8-6)

8. 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.
Boundary
layer
Irrotational
flow

9. The velocity profile in the fully developed region is
Hydrodynamic entrance region ─ the region from 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.

10. 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
(Ts-T)/(Ts-Tm)
remains unchanged.

11. Hydrodynamically fully developed:
Thermally fully developed:
(8-7)
(8-8)
(8-9)
(8-9)

12. Surface heat flux can be expressed as
For thermally fully developed region From (Eq. (8-9))
(8-10)
Fully developed flow
Fully developed flow

13. The Heat Transfer coefficient and Friction factor
Developing region
Fully developed region

14. Entry Lengths Laminar flow Turbulent flow Hydrodynamic Thermal
Thermal (approximate)
(8-11)
(8-12)
(8-13)
(8-14)

15. 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.

16. 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
The thermal conditions at the surface can usually be approximated as:
constant surface temperature, or
constant surface heat flux.
The mean fluid temperature Tm must
change during heating or cooling.
Either Ts= constant or qs = constant at the surface of a tube, but not both.
(8-15)

17. Constant Surface Heat Flux
In the case of constant heat flux, the rate of heat transfer can also be expressed as
Then the mean fluid temperature at the tube exit becomes
The surface temperature in the case of constant surface heat flux can be determined from
(8-17)
(8-18)
(8-19)

18. In the fully developed region, the surface temperature Ts 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 Tm can be determined
Noting that both the heat flux and h (for fully developed flow) are constants
(8-20)
(8-21)

19. In the fully developed region (Ts-Tm=constant)
Combining Eqs. 8–20, 8–21, and 8–22 gives
For a circular tube
(8-22)
(8-23)
(8-24)

20. Constant Surface Temperature
The energy balance on a differential control volume
Since the mean temperature of the fluid Tm increases in the flow direction the heat flux decays with x.
The surface temperature is constant (dTm=-d(Ts-Tm)) and dAs=pdx, therefore,
(8-27)
(8-28)

21. Taking the exponential of both sides and solving for Te
Integrating Eq from x=0 (tube inlet where Tm=Ti) to x=L (tube exit where Tm=Te) gives
Taking the exponential of both sides and solving for Te or
(8-29)
(8-30)

22. This dimensionless parameter is called the number of transfer
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
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.

23. Solving Eq. 8–29 for mcp gives
Substituting this into Eq. 8–15
where
DTln is the logarithmic mean temperature difference.
(8-31)
(8-32)
(8-33)

24. Laminar Flow in Tubes Assumptions: steady laminar flow,
incompressible fluid,
constant properties,
fully developed region, and
straight circular tube.
The velocity profile u(r) remains unchanged in the flow direction.
no motion in the radial direction.
no acceleration.

25. Consider a ring-shaped differential volume element.
A force balance on the volume
element in the flow direction
gives
Dividing by 2pdrdx and rearranging
(8-34)
(8-35)

26. Taking the limit as dr, dx → 0 gives
Substituting t=m(du/dr) gives
Rearranging and integrating it twice to give
Boundary Conditions:
symmetry about the centerline ∂u/∂r=0 at r=0,
no-slip condition u=0 at r=R.
(8-36)
(8-37)
(8-38)

27. Eq. 6-38 with the boundary conditions
Substituting Eq. 8–39 into Eq. 8–2, and performing the integration gives the average velocity
Combining the last two equations, the velocity profile is rewritten as
(8-39)
(8-40)
(8-41)

28. Pressure Drop
One implication from Eq is that the pressure drop gradient (dP/dx) must be constant (the left side is a function only of r, and the right side is a function only of x).
Integrating from x=x1 where the pressure is P1 to x=x1=L where the pressure is P2 gives
Substituting Eq. 8–43 into the Vavg expression in Eq. 8–40
(8-43)
(8-44)

29. Setting Eqs. 8–44 and 8–45 equal to each other and solving for f gives
A pressure drop due to viscous effects represents an irreversible pressure loss.
It is convenient to express the pressure loss for all types of fully developed internal flows in terms of the dynamic pressure and the friction factor
Setting Eqs. 8–44 and 8–45 equal to each other and solving for f gives
Circular tube, laminar:
(8-45)
(8-46)

30. Temperature Profile and the Nusselt Number
Energy is transferred by mass in the
x-direction, and by conduction in the
r-direction.
The steady flow energy balance for a
cylindrical shell element can be
expressed as
Substituting
and dividing by 2prdrdx gives, after rearranging
(8-49)
(8-50)

31. Or
Since
Eq 8-51 becomes
(8-51)
(8-52)
(8-53)

32. Constant Surface Heat Flux
Substituting Eqs and 8-41 into Eq. 8.53
(8-41)
(8-24)
(8-53)
(8-55)

33. Separating the variables and integrating twice
Boundary conditions
Symmetry at r=0:
At r=R: T(r=R)=Ts
(8-56)
C1=0 C2
(8-57)

34. The bulk mean temperature Tm 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)
(8-59)
Constant heat flux (circular tube, laminar)
(8-60)
Constant Surface temperature (circular tube, laminar)
(8-61)

35. Laminar Flow in Noncircular Tubes
The friction factor (f) and the Nusselt number relations are given in Table 8–1 for fully developed laminar flow in tubes of various cross sections.

36. 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)
For flow between isothermal parallel plates
(8-62)
(8-64)

37. 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
For fully developed turbulent flow the Nusselt number (Dittus–Boelter equation)
(8-65)
(8-68)

38. 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 tube annulus.
Original correlations are also approximately valid for:
developing Turbulent Flow in the Entrance Region,
turbulent Flow in Noncircular Tubes.