1. Convection: Internal Flow (8.1-8.6)
Chapter 8
Convection: Internal Flow ( )
Internal Flow

2. Introduction
In Chapter 7 we obtained a non-dimensional form for the heat transfer coefficient, applicable for problems involving external flow:
Calculation of fluid properties was done at surface temperature, bulk temperature of the fluid, or film temperature
In this chapter we will obtain convection coefficients for geometries involving internal flow, such as flow in tubes
Recall Newton’s law of cooling:
For flow inside a tube we cannot define
Must know how temperature evolves inside the pipe and find alternative expressions for calculating heat flux due to convection. .
Internal Flow

3. Flow Conditions for Internal Flow
Onset of turbulent flow at
Hydrodynamic entry length:
Laminar flow
Turbulent flow
Internal Flow

4. Mean Velocity
Velocity inside a tube varies over the cross section. For every differential area dAc:
(8.1)
Overall rate of mass transfer through a tube with cross section Ac:
where um is the mean (average) velocity
and
Combining with (8.1):
(8.2)
Can determine average temperature at any axial location (along the x-direction), from knowledge of the velocity profile
Internal Flow

5. Velocity Profile in a pipe
For laminar flow of an incompressible, constant property fluid in the fully developed region of a circular tube (pipe):
(8.3a)
(8.4)
(8.3b)
Internal Flow

6. Thermal Considerations: Mean Temperature
We can write Newton’s law of cooling inside a tube, by considering a mean temperature, instead of
(8.5)
The internal energy per unit mass for a differential area is:
Integrating over the entire cross section:
(8.6)
Overall rate of energy transfer :
where Tm is the mean (average) velocity
and
Combining
with (8.6):
(8.7)
Internal Flow

7. Example 8.1
For flow of a liquid metal through a circular tube, the velocity and temperature profiles at a particular axial location may be approximated as being uniform and parabolic respectively. That is, u(r)=C1 and T(r)-Ts=C2[1-(r/ro)2], where C1 and C2 are constants and Ts the temperature at the surface of the tube. What is the value of the Nusselt number, NuD at this location? ro
Ans. NuD=8
Internal Flow

8. Fully Developed Conditions
Can we claim that
For internal flows, the temperature, T(r), as well as the mean temperature, Tm always vary in the x-direction, ie.
Internal Flow

9. Fully Developed Conditions
Although T(r) changes with x, the relative shape of the temperature profile remains the same: Flow is thermally fully developed.
A fully developed thermally region is possible, if one of two possible surface conditions exist :
Uniform temperature (Ts=constant)
Uniform heat flux (qx”=const)
Thermal Entry Length :
Internal Flow

10. Fully Developed Conditions
It can be proven that for fully developed conditions, the local convection coefficient is a constant, independent of x:
Internal Flow

11. Mean temperature variation along a tube
We are still left with the problem of knowing how the mean temperature Tm(x), varies as a function of distance, so that we can use it in Newton’s law of cooling to estimate convection heat transfer.
Consider an energy balance on a differential control volume inside the tube:
Main contributions are due to internal energy changes [= ], convection heat transfer and flow work [=pu], needed to move fluid.
P=surface perimeter
The rate of convection heat transfer to the fluid is equal to the rate at which the fluid thermal energy increases, plus the net rate at which is work is done in moving the fluid through the control volume
(8.8a)
Internal Flow

12. Mean temperature variation along a tube
Considering perfect gas, or incompressible liquid:
(8.8b)
By integrating:
(8.9)
qconv is related to mean temperatures at inlet and outlet.
Combining equations 8.5 and 8.8b:
(8.10)
where P=surface perimeter =
pD for circular tube, width for flat plate
Integration of this equation will result in an expression for the variation of Tm as a function of x.
Internal Flow

13. Case 1: Constant Heat Flux
where P=surface perimeter
=pD for circular tube,
=width for flat plate
Integrating equation (8.10):
(8.11)
Internal Flow

14. Example (Problem 8.15)
A flat-plate solar collector is used to heat atmospheric air flowing through a rectangular channel. The bottom surface of the channel is well insulated, while the top surface is subjected to a uniform heat flux, which is due to the net effect of solar radiation absorption and heat exchange between the absorber and cover plates.
For inlet conditions of mass flow rate=0.1 kg/s and Tm,i=40°C, what is the air outlet temperature, if L=3 m, w=1 m and the heat flux is 700 W/m2? The specific heat of air is cp=1008 J/kg.K
Ans: Tm,o=60.8°C
Internal Flow

15. Case 2: Constant Surface Temperature, Ts=constant
From eq.(8.10) with Ts-Tm=DT:
Integrating from x to any downstream location:
For the entire length of the tube:
(8.12)
(8.13)
where
(8.14)
As is the tube surface area, As=P.L=pDL
Internal Flow

16. Example 8.3
Steam condensing on the outer surface of a thin-walled circular tube of 50 mm diameter and 6-m length maintains a uniform surface temperature of 100°C. Water flows through the tube at a mass flow rate of 0.25 kg/s and its inlet and outlet temperatures are Tm,1=15°C and Tm,o=57°C. What is the average convection coefficient associated with the water flow?
Ts=100°C
D=50 mm
Tm,i=15°C
Tm,o=57°C
L=6 m
Internal Flow

17. Case 3: Uniform External Temperature
Replace Ts by and by (the overall heat transfer coefficient, which includes contributions due to convection at the tube inner and outer surfaces, and due to conduction across the tube wall)
(8.15)
(8.16)
Internal Flow

18. Summary ( )
We discussed fully developed flow conditions for cases involving internal flows, and we defined mean velocities and temperatures
We wrote Newton’s law of cooling using the mean temperature, instead of
Based on an overall energy balance, we obtained an alternative expression to calculate convection heat transfer as a function of mean temperatures at inlet and outlet.
We obtained relations to express the variation of Tm with length, for cases involving constant heat flux and constant wall temperature
(8.9)
Internal Flow

19. Summary ( )
We used these definitions, to obtain appropriate versions of Newton’s law of cooling, for internal flows, for cases involving constant wall temperature and constant surrounding fluid temperature
( )
We can combine equations ( ) with (8.9) to obtain values of the heat transfer coefficient (see solution of Example 8.3)
In the rest of the chapter we will focus on obtaining values of the heat transfer coefficient h, needed to solve the above equations
Internal Flow