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

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

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.

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

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.

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.

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.

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.

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

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

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.

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

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).

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

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

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

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

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

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

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: ???

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

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 19.3.2 Energy balance: Energy conducted through walls Energy in minus energy out by conduction Energy accumulation =

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

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.

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:

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.

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 5.1-2 and Table 19.4-2 (Hirschfelder, Curtiss, and Bird, 1954)

Values of k, α and h Typical values of k and α in gases, liquid and solids are in Table 19.4-2. 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 19.4-3. All refer to heat transfer across a solid-fluid interface.

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

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 19.4-3, flow over tube banks: steel wall liquid Table 19.4-3, liquid inside the tube: