1. 高等輸送二 — 熱傳 Lecture 10 Fundamentals of Heat Transfer
郭修伯 助理教授

2. Heat transfer Three modes of energy transfer Conduction Convection
Fourier’s law for conduction
Energy transport through a thin film
Energy transport in a semiinfinite slab
Convection
Radiation

3. Conduction Two ways Molecular interaction: By “free” electrons:
greater motion of a molecule at a higher energy level (temperature) imparts energy to adjacent molecules at lower energy lever.
By “free” electrons:
The ability of solids to conduct heat varies directly with the concentration of free electrons.

4. Conduction - Molecular phenomenon
Fourier (1807):
Where qx is the heat-transfer rate in the x direction (Watts or Btu/hr); A is the area normal to the direction of heat flow (m2 or ft2), dT/dx is the temperature gradient in the x direction (K/m or °F/ft) and k is the thermal conductivity (W/mK or Btu/hr ft °F)
More general form:
Fourier’s first law of heat conduction
Primarily a function of temperature, vary significantly with pressure only in the case of gases subjected to high pressures

5. Steady heat conduction across a thin film
On each side of the film is a well-mixed solution of one solute, T0 > Tl T0 Tl z l z
Energy balance in the layer z
Energy conducted out of the layer at z + z
Energy conducted into the layer at z
Energy accumulation =
s.s.

6. l T0 Tl z Dividing A z B.C. z = 0, T = T0 z = l, T = Tl
linear concentration profile
z  0 T0 Tl z l z
Since the system is in s.s., the flux is a constant.

7. Questions
How are the results changed if the fluid at z = 0 and T0 is replaced by a different liquid that is at the same temperature?
There is no change as long as the interfacial temperature is constant.
What will the temperature profile look like across two thin slabs of different materials that are clamped together?
In steady state, the heat flux is constant. Thus the temperature drop across the poorly conducting slab will be larger than that across the better conductor.

8. Imagine that for the system the fluid at z = l has a small volume, V, but the fluid at z = 0 has a very large volume. How will Tl change with time?
An energy balance on the fluid at z = l:
Specific heat capacity of the liquid
Mass of fluid located at z = l
B.C.
t = 0, Tl = Tl0
The temperature rises to a limit of T0.

9. Unsteady heat conduction into a thick slab
Any heat conduction problem will behave as if the slab is infinitely thick at short enough times.
At time zero, the temperature at z = 0 suddenly increases to T0 T0
Energy balance on the thin layer Az
Energy conducted out of the layer at z + z
Energy conducted into the layer at z
Energy accumulation =
time
Too
position z

10. Boundary conditions
Dividing
z  0
Thermal diffusivity
The heat conduction equation

11. Questions
To what depth does the temperature change penetrate in a steel slab?
equals unity. For steel, α ~ 0.15 cm2/sec; t = 10 min, z = 15 cm
How does the flux vary with physical properties for the thick slab as compared with the thin film?
Doubling the temperature difference doubles the heat flux in both cases. Doubling the thermal conductivity increases the flux by 2^0.5 for the thick slab and by 2 for the thin film. Doubling the heat capacity decreases the flux by 2^0.5 for the thick slab, but has no effect for the thin film.

12. Imagine a well-insulated pipe used to transport saturated steam
Imagine a well-insulated pipe used to transport saturated steam. How much will the heat loss through the pipe’s walls be reduced if the insulation thickness is doubled? Assume that the thermal conductivity of the pipe’s walls is much higher than that of the insulation.
Steady-state energy balance on cylindrical shell of insulation of volume 2πrΔrL:
Energy conducted out of the layer at r + r
Energy conducted into the layer at r
Energy accumulation =
integration

13. General energy balances
Energy balance can be difficult because energy and work can take so many different forms.
Internal, kinetic, potential, chemical and surface energies are all important.
Work can involve forces of pressure, gravity, and electrical potential.
As a result, a truly general balance is extraordinary complicated (Slattery, 1978).

14. Energy balances for a single pure component that has internal and kinetic energy:
Energy accumulation
Energy convection in minus that out
conduction =
Work by gravity
Work by pressure forces
Work by viscous forces
Subtracting the mechanical energy balance
Energy accumulation
Energy convection in minus that out
conduction =
Reversible work
Irreversible work

15. Batch system and restricted to changes in internal energy only
Steady-state open system of fixed volume

16. Conduction in a thin film at steady state:
One dimensional
Steady state
No energy convection
No flow work

17. Conduction in a thick film at unsteady state:
One dimensional
No energy convection
No flow work

18. Heating a flowing solution
A viscous solution is flowing laminarly through a narrow pipe. At a known distance along the pipe, the pipe’s wall is heated with condensing steam. Find a differential equation from which the temperature distribution in the pipe can be calculated.
Steady state
no reversible work
heating due to viscous dissipation is small
Energy transfer along the pipe axis is largely by convection
Energy transfer in the radial direction is largely by conduction

19. Heat transfer coefficient
Fourier’s law of heat conduction
useful for heat conduction in solids
difficult to use in fluid systems, especially when heat is transferred across phase boundaries.
Heat transfer across interfaces:
isothermal in the separated phases.
the temperature gradients are close to the interface
heat flux:
overall heat transfer coefficient

20. Common choice of temperature difference at some position z:
Another choice(for full size industrial equipment):
Solid wall T1
T1i
T3i T3
Hot fluid
Cold fluid
The heat flux:
???

21. Harmonic average
The overall heat transfer coefficient, U, Vs. the overall mass transfer coefficient, k :
U is simpler than k
the hot face at the wall = the temperature of the solid wall in contact with the hot fluid (c.f. mass transfer)
U: a sum of resistance
k: involve weighting factors

22. Finding the overall heat transfer coefficient
A total of 1.8 x 104 liters/hr crude oil flows in a heat exchanger with forty tubes 5 cm in diameter and 2.8 m long. The oil, which has a heat capacity of 0.43 cal/g-°C and a specific gravity of 0.9 g/cm3, is heated with 240°C steam from 20°C to 140°C. The steam is condensed at 240 °C but is not cooled much below this temperature. What is the overall heat transfer coefficient based on the local temperature difference? What is it when based on the average temperature difference?
Fig
Energy balance:
Energy conducted through walls
Energy in minus energy out by conduction
Energy accumulation =

23. Based on the average temp.
Dividing by
B.C.
Based on the local temperature difference

24. The time for tank cooling
A 100-gallon tank filled with water initially at 80°F sits outside in air at 10°F. The overall heat transfer coefficient for heat lost from the water-containing tank is 3.6 Btu/hr-ft2-°F, and the tank’s area is 27 ft2. How long can we wait before the water in the tank starts to freeze?
Energy balance on the tank:
Energy accumulation
Heat lost from tank =
i.c.

25. The effect of insulation
Insulation advertisements claim that we can save 40% on our heating bills by installing 10 inches of glass wool as insulation. The glass wool has a thermal conductivity of about 0.03 Btu/hr-ft2-°F; the average winter temperature is 15°F and the house temperature is 68°F. If the advertisements are true, and if heat loss from doors and windows is minor, how much can we save with 2 ft of insulation?
Heat loss in our current home:
Adding 10 inches of glass wool:

26. Heat loss from a bar
In 1804, Biot carried out an experimental investigation of the conductivity of metal bars by maintaining one end at a high known temperature and taking readings of thermometers places in holes along the bar. He found that the steady state temperature decreased exponentially along the bar. Why?
Energy in minus energy out by conduction T0
Energy accumulation
Energy lost to the surroundings
T(z) =
Air at Too
Dividing by
B.C.

27. Thermal conductivity, thermal diffusivity, and heat transfer coefficient
The thermal conductivity k, the thermal diffusivity α, and the heat transfer coefficient h:
Values for gases can be predicted from kinetic theory
Thermal conductivities of gases:
Values for liquids and solids are found by experiment
Molecular weight
Of order 1, a weak function of
Collision diameter A
Table and Table
(Hirschfelder, Curtiss, and Bird, 1954)

28. Values of k, α and h
Typical values of k and α in gases, liquid and solids are in Table
Thermal conductivities of metals are much higher than those of liquids or gases.
Thermal conductivities of nonmetallic solids and liquids are comparable.
The effective thermal conductivity of composite materials tends to be dominated by the continuous phase.
Common correlations of h are given in Table
All refer to heat transfer across a solid-fluid interface.

29. Dimensionless groups in heat transfer
Nusselt number
c.f. Sherwood number for mass transfer
Prandtl number
c.f. Schmidt number for mass transfer
viscosity - thermal conductivity

30. The overall heat transfer coefficient of a heat exchanger
As part of a chemical process, we plan to use a shell-tube heat exchanger of 20 banks of 5 cm outside-diameter steel tubes with 0.3 cm walls. Outside the tubes, we plan to use 400°C flue gas; inside, we expect to be heating aromatics like benzene and toluene at around 30°C. The gas flow will be 17 m/s, and the liquid flow will be 2.7 m/s. Determine the overall heat transfer coefficient.
hot flue gas
Table , flow over tube banks:
steel wall
liquid
Table , liquid inside the tube: