4.10. INTRODUCTION TO RADIATION HEAT TRANSFER

4.10A. Introduction and Basic Equation for Radiation

1. Nature of radiant heat transfer

In the preceding sections of this chapter we have studied conduction and convection heat transfer. In conduction, heat is transferred from one part of a body to another, and the intervening material is heated. In convection, heat is transferred by the actual mixing of materials and by conduction. In radiant heat transfer, the medium through which the heat is transferred usually is not heated. Radiation heat transfer is the transfer of heat by electromagnetic radiation.

Thermal radiation is a form of electromagnetic radiation similar to X rays, light waves, gamma rays, and so on, differing only in wavelength. It obeys the same laws as light: It travels in straight lines, can be transmitted through space and vacuum, and so on. It is an important mode of heat transfer and is especially important where large temperature differences occur, as, for example, in a furnace with boiler tubes, in radiant dryers, or in an oven baking food. Radiation often occurs in combination with conduction and convection. An elementary discussion of radiant heat transfer will be given here, with a more advanced and comprehensive discussion being given in Section 4.11.

In an elementary sense the mechanism of radiant heat transfer is composed of three distinct steps or phases:

2. Absorptivity and black bodies

When thermal radiation (such as light waves) falls upon a body, part is absorbed by the body in the form of heat, part is reflected back into space, and part may actually be transmitted through the body. For most cases in process engineering, bodies are opaque to transmission, so this will be neglected. Hence, for opaque bodies,

Equation 4.10-1

where α is absorptivity or fraction absorbed and ρ is reflectivity or fraction reflected.

A black body is defined as one that absorbs all radiant energy and reflects none. Hence, ρ = 0 and α = 1.0 for a black body. Actually, in practice there are no perfect black bodies, but a close approximation is a small hole in a hollow body, as shown in Fig. 4.10-1. The inside surface of the hollow body is blackened by charcoal. The radiation enters the hole and impinges on the rear wall; part is absorbed there and part is reflected in all directions. The reflected rays impinge again, part is absorbed, and the process continues. Hence, essentially all of the energy entering is absorbed and the area of the hole acts as a perfect black body. The surface of the inside walls is "rough" and rays are scattered in all directions, unlike a mirror, where they are reflected at a definite angle.

As stated previously, a black body absorbs all radiant energy falling on it and reflects none. Such a black body also emits radiation, depending on its temperature, and does not reflect any. The ratio of the emissive power of a surface to that of a black body is called emissivity ε and is 1.0 for a black body. Kirchhoff's law states that at the same temperature T₁, α₁ and ε₁ of a given surface are the same, or

Equation 4.10-2

Equation (4.10-2) holds for any black or nonblack solid surface.

3. Radiation from a body and emissivity

The basic equation for heat transfer by radiation from a perfect black body with an emissivity ε = 1.0 is

Equation 4.10-3

where q is heat flow in W, A is m² surface area of body, σ is a constant 5.676 × 10^-8 W/m² · K⁴ (0.1714 X 10^-8 btu/h · ft² · °R⁴), and T is temperature of the black body in K (°R).

For a body that is not a black body and has an emissivity ε < 1.0, the emissive power is reduced by ε, or

Equation 4.10-4

Substances that have emissivities of less than 1.0 are called gray bodies when the emissivity is independent of the wavelength. All real materials have an emissivity ε < 1.

Since the emissivity ε and absorptivity α of a body are equal at the same temperature, the emissivity, like absorptivity, is low for polished metal surfaces and high for oxidized metal surfaces. Typical values are given in Table 4.10-1 but do vary some with temperature. Most non-metallic substances have high values. Additional data are tabulated in Appendix A.3.

Table 4.10-1. Total Emissivity, ε, of Various Surfaces

Surface	T(K)	T(°F)	Emissivity, ε
Polished aluminum	500 850	440 1070	0.039 0.057
Polished iron	450	350	0.052
Oxidized iron	373	212	0.74
Polished copper	353	176	0.018
Asbestos board	296	74	0.96
Oil paints, all colors	373	212	0.92–0.96
Water	273	32	0.95

4.10B. Radiation to a Small Object from Surroundings

In the case of a small gray object of area A₁ m² at temperature T₁ in a large enclosure at a higher temperature T₂, there is a net radiation to the small object. The small body emits an amount of radiation to the enclosure given by Eq. (4.10-4) as . The emissivity ε₁ of this body is taken at T₁. The small body also absorbs an amount of energy from the surroundings at T₂ given by . The α₁₂ is the absorptivity of body 1 for radiation from the enclosure at T₂. The value of α₁₂ is approximately the same as the emissivity of this body at T₂. The net heat of absorption is then, by the Stefan-Boltzmann equation,

Equation 4.10-5

A further simplification of Eq. (4.10-5) is usually made for engineering purposes by using only one emissivity for the small body, at temperature T₂. Thus,

Equation 4.10-6

EXAMPLE 4.10-1. Radiation to a Metal Tube A small oxidized horizontal metal tube with an OD of 0.0254 m (1 in.), 0.61 m (2 ft) long, and with a surface temperature at 588 K (600°F) is in a very large furnace enclosure with fire-brick walls and the surrounding air at 1088 K (1500°F). The emissivity of the metal tube is 0.60 at 1088 K and 0.46 at 588 K. Calculate the heat transfer to the tube by radiation using SI and English units. Solution: Since the large-furnace surroundings are very large compared to the small enclosed tube, the surroundings, even if gray, when viewed from the position of the small body appear black, and Eq. (4.10-6) is applicable. Substituting given values into Eq. (4.10-6) with an ε of 0.6 at 1088 K, Other examples of small objects in large enclosures occurring in the process industries are a loaf of bread in an oven receiving radiation from the walls around it, a package of meat or food radiating heat to the walls of a freezing enclosure, a hot ingot of solid iron cooling and radiating heat in a large room, and a thermometer measuring the temperature in a large duct.

4.10C. Combined Radiation and Convection Heat Transfer

When radiation heat transfer occurs from a surface, it is usually accompanied by convective heat transfer, unless the surface is in a vacuum. When the radiating surface is at a uniform temperature, we can calculate the heat transfer for natural or forced convection using the methods described in the previous sections of this chapter. The radiation heat transfer is calculated by the Stefan-Boltzmann equation (4.10-6). Then the total rate of heat transfer is the sum of convection plus radiation.

As discussed before, the heat-transfer rate by convection and the convective coefficient are given by

Equation 4.10-7

where q_conv is the heat-transfer rate by convection in W, h_c the natural or forced convection coefficient in W/m² · K, T₁ the temperature of the surface, and T₂ the temperature of the air and the enclosure. A radiation heat-transfer coefficient h_r in W/m² · K can be defined as

Equation 4.10-8

where q_rad is the heat-transfer rate by radiation in W. The total heat transfer is the sum of Eqs. (4.10-7) and (4.10-8),

Equation 4.10-9

To obtain an expression for h_r, we equate Eq. (4.10-6) to (4.10-8) and solve for h_r:

Equation 4.10-10

A convenient chart giving values of h_r in English units calculated from Eq. (4.10-10) with ε = 1.0 is given in Fig. 4.10-2. To use values from this figure, the value obtained from the figure should be multiplied by ε to give the value of h_r to use in Eq. (4.10-9). If the air temperature is not the same as T₂ of the enclosure, Eqs. (4.10-7) and (4.10-8) must be used separately and not combined together as in (4.10-9).

EXAMPLE 4.10-2. Combined Convection Plus Radiation from a Tube Recalculate Example 4.10-1 for combined radiation plus natural convection to the horizontal 0.0254-m tube. Solution: The area A of the tube = π(0.0254)(0.61) = 0.0487 m2. For the natural convection coefficient to the 0.0254-m horizontal tube, the simplified equation from Table 4.7-2 will be used as an approximation even though the film temperature is quite high: Substituting the known values, Using Eq. (4.10-10) and ε = 0.6, Substituting into Eq. (4.10-9), Hence, the heat loss of -2130 W for radiation is increased to only -2507 W when natural convection is also considered. In this case, because of the large temperature difference, radiation is the most important factor. Perry and Green (P3, pp. 10-14) give a convenient table of natural convection plus radiation coefficients (hc + hr) from single horizontal oxidized steel pipes as a function of the outside diameter and temperature difference. The coefficients for insulated pipes are about the same as those for a bare pipe (except that lower surface temperatures are involved for the insulated pipes), since the emissivity of cloth insulation wrapping is about that of oxidized steel, approximately 0.8. A more detailed discussion of radiation will be given in Section 4.11.

4.10D. Effect of Radiation on Temperature Measurement of a Gas

When a temperature sensor or probe (thermometer, thermocouple, etc.) is used to measure the temperature of a gas flowing in an enclosure, significant errors can occur. Radiation heat exchange will take place between the sensor and the wall and convection heat transfer between the sensor and the gas. The sensor will indicate a temperature between the true gas and wall surface temperatures. This is shown in Fig. 4.10-3, where the wall temperature T_w is less than the true gas temperature T_g.

The equations for the heat transfer q_c by convection to the probe and radiation q_r from the probe to the wall are as follows for T_w < T_g:

Equation 4.10-11

where A_p is the area of the tube in m² and ε is the emissivity of the probe.

EXAMPLE 4.10-3. Effect of Radiation on Temperature Measurement in a Gas A thermocouple is measuring the temperature of hot air flowing in a pipe whose walls are at Tw = 400 K (260°F). The true gas temperature Tg = 465 K (377°F). Calculate the temperature Tp indicated by the thermocouple. The emissivity of the probe is assumed as ε = 0.6 and the convection heat-transfer coefficient hc = 40 W/m2 · K. Solution: Substituting into Eq. (4.10-11) for convection, qc, and for radiation, qr, Equating qc = qr, canceling out the term Ap, and solving by trial and error, Tp = 451.4 K. Hence, the thermocouple reading of Tp = 451.4 K (352.5°F) is 13.6 K (24.5°F) lower than the true gas temperature of 465 K (377°F).

Probes with a radiation shield as shown in Fig. 4.10-3 are often used to reduce radiation errors. The shield will have a temperature which is closer to the gas temperature than is the wall. Since the probe now radiates heat to a surface which is closer to the gas temperature, the radiation loss is less. It can be shown that with one shield, the radiation heat loss will be halved. Multiple shields can be used to further reduce the error. Using a polished surface on the probe to reduce the emissivity lowers the radiation heat loss. This also reduces the measurement error.