**Heat transfer**

heat transfer

(Parte **5** de 6)

1.4 Convection

The convection mode of heat transfer actually consists of two mechanisms operating simultaneously. The first is the energy transfer due to molecular motion, that is, the conductive mode. Superimposed upon this mode is energy transfer by the macroscopic motion of fluid parcels. The fluid motion is a result of parcels of fluid, each consisting of a large number of molecules, moving by virtue of an external force. This extraneous force may be due to a density gradient, as in natural convection, or due to a pressure difference generated by a pump or a fan, or possibly to a combination of the two.

Figure 1.8 shows a plate at surface temperature Tsand a fluid at temperature flowing parallel to the plate. As a result of viscous forces the velocity of the fluid will be zero at the wall and will increase to as shown. Since the fluid is not moving at the interface, heat is transferred at that location only by conduction. If we knew the temperature gradient and the thermal conductivity at this interface, we could calculate the rate of heat transfer from Eq. (1.2):

But the temperature gradient at the interface depends on the rate at which the macroscopic as well as the microscopic motion of the fluid carries the heat away from the interface. Consequently, the temperature gradient at the fluid-plate interface depends on the nature of the flow field, particularly the free-stream velocity .Uq qc=-kfluid A0T

Uq Tq

Velocity profiley

y = 0

Heated surface

Temperature profile

FIGURE 1.8Velocity and temperature profile for convection heat transfer from a heated plate with flow over its surface.

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The situation is quite similar in natural convection. The principal difference is that in forced convection the velocity far from the surface approaches the free-stream value imposed by an external force, whereas in natural convection the velocity at first increases with increasing distance from the heat transfer surface and then decreases, as shown in Fig. 1.9. The reason for this behavior is that the action of viscosity diminishes rather rapidly with distance from the surface, while the density difference decreases more slowly. Eventually, however, the buoyant force also decreases as the fluid density approaches the value of the unheated surrounding fluid. This interaction of forces will cause the velocity to reach a maximum and then approach zero far from the heated surface. The temperature fields in natural and forced convection have similar shapes, and in both cases the heat transfer mechanism at the fluid-solid interface is conduction.

18Chapter 1Basic Modes of Heat Transfer

Velocity profile y

y = 0 qc g

Tsurface

Tfluid

Temperature profile T(y)

FIGURE 1.9Velocity and temperature distribution for natural convection over a heated flat plate inclined at angle bfrom the horizontal.

The preceding discussion indicates that convection heat transfer depends on the density, viscosity, and velocity of the fluid as well as on its thermal properties (thermal conductivity and specific heat). Whereas in forced convection the velocity is usually imposed on the system by a pump or a fan and can be directly specified, in natural convection the velocity depends on the temperature difference between the surface and the fluid, the coefficient of thermal expansion of the fluid (which determines the density change per unit temperature difference), and the body force field, which in systems located on the earth is simply the gravitational force.

In later chapters we will develop methods for relating the temperature gradient at the interface to the external flow conditions, but for the time being we shall use a simpler approach to calculate the rate of convection heat transfer, as shown below.

Irrespective of the details of the mechanism, the rate of heat transfer by convection between a surface and a fluid can be calculated from the relation qc = hcA¢T

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1.4 Convection 19

¢T=difference between the surface temperature Tsand a temperature of the fluid at some specified location (usually far way from the surface), K (°F) =average convection heat transfer coefficient over the area A(often called the surface coefficient of heat transfer or the convection heat transfer coefficient), W/m2K (Btu/h ft2°F)

The relation expressed by Eq. (1.10) was originally proposed by the British scientist Isaac Newton in 1701. Engineers have used this equation for many years, even though it is a definition of rather than a phenomenological law of convection. Evaluation of the convection heat transfer coefficient is difficult because convection is a very complex phenomenon. The methods and techniques available for a quantitative evaluation of will be presented in later chapters. At this point it is sufficient to note that the numerical value of in a system depends on the geometry of the surface, on the velocity as well as the physical properties of the fluid, and often even on the temperature difference ¢T. In view of the fact that these quantities are not necessarily constant over a surface, the convection heat transfer coefficient may also vary from point to point. For this reason, we must distinguish between a local and an average convection heat transfer coefficient. The local coefficient hcis defined by while the average coefficient can be defined in terms of the local value by

For most engineering applications, we are interested in average values. Typical values of the order of magnitude of average convection heat transfer coefficients seen in engineering practice are given in Table 1.4. Using Eq. (1.10), we can define the thermal conductance for convection heat transfer Kc as

(1.13)Kc=hc A | (W/K) |

ALLA hc dA hc dqc=hc dA(Ts-Tq) hc hc hc Tq

TABLE 1.4Order of magnitude of convection heat transfer coefficients

Convection Heat Transfer Coefficient

Air, free convection6–301–5 Superheated steam or air, forced convection30–3005–50 Oil, forced convection60–1,80010–300 Water, forced convection300–18,050–3,0 Water, boiling 3,0–60,0 500–10,0 Steam, condensing 6,0–120,0 1,0–20,0

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20Chapter 1Basic Modes of Heat Transfer and the thermal resistance to convection heat transfer Rc, which is equal to the reciprocal of the conductance, as

EXAMPLE 1.3Calculate the rate of heat transfer by natural convection between a shed roof of area and ambient air, if the roof surface temperature is 27°C, the airtemperature -3°C, and the average convection heat transfer coefficient 10 W/m2K (see Fig. 1.10).

SOLUTIONAssume that steady state exists and the direction of heat flow is from the air to the roof. The rate of heat transfer by convection from the air to the roof is then given by Eq. (1.10):

Note that in using Eq. (1.10), we initially assumed that the heat transfer would be from the air to the roof. But since the heat flow under this assumption turns out to be a negative quantity, the direction of heat flow is actually from the roof to the air. We could, of course, have deduced this at the outset by applying the second law of thermodynamics, which tells us that heat will always flow from a higher to a lower temperature if there is no external intervention. But as we shall see in a later section, thermodynamic arguments cannot always he used at the outset in heat transfer problems because in many real situations the surface temperature is not known.

=-120,0 W

20 m*20 m hc A (K/W)

Tair = 3°C

Troof = 27°C 20 m20 m

FIGURE 1.10Schematic sketch of shed for analysis of roof temperature in Example 1.3.

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1.5 Radiation 21

1.5 Radiation

The quantity of energy leaving a surface as radiant heat depends on the absolute temperature and the nature of the surface. A perfect radiator, which is referred to as a blackbody,*emits radiant energy from its surface at a rate as given by

The heat flow rate qrwill be in watts if the surface area A, is in square meters and the surface temperature T1is in kelvin; sis a dimensional constant with a value of . In the English system, the heat flow rate will be in Btu’s

per hour if the surface area is in square feet, the surface temperature is in degrees Rankine , and sis . The constant sis the Stefan- Boltzmann constant; it is named after two Austrian scientists, J. Stefan, who in 1879 discovered Eq. (1.15) experimentally, and L. Boltzmann, who in 1884 derived it theoretically.

Inspection of Eq. (1.15) shows that any blackbody surface above a temperature of absolute zero radiates heat at a rate proportional to the fourth power of the absolute temperature. While the rate of radiant heat emission is independent of the conditions of the surroundings, a nettransfer of radiant heat requires a difference in the surface temperature of any two bodies between which the exchange is taking place. If the blackbody radiates to an enclosure (see Fig. 1.1) that is also black, (that is, absorbs all the radiant energy incident upon it) the net rate of radiant heat transfer is given by

*A detailed discussion of the meaning of these terms is presented in Chapter 9.

Black body of surface area A1 at temperature qr, 1

Black enclosure at temperature T2

FIGURE 1.11Schematic diagram of radiation between body 1 and enclosure 2.

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22Chapter 1Basic Modes of Heat Transfer where T2is the surface temperature of the enclosure in kelvin. Real bodies do not meet the specifications of an ideal radiator but emit radiation at a lower rate than blackbodies. If they emit, at a temperature equal to that of a blackbody (a constant fraction of blackbody emission at each wavelength) they are called gray bod- ies. A gray body A1at T1emits radiation at the rate , and the rate of heat transfer between a gray body at a temperature T1and a surrounding black enclosure at T2is

where is the emittance of the gray surface and is equal to the ratio of the emission from the gray surface to the emission from a perfect radiator at the same temperature.

If neither of two bodies is a perfect radiator and if the two bodies have a given geometric relationship to each other, the net heat transfer by radiation between them is given by

where is a dimensionless modulus that modifies the equation for perfect radiators to account for the emittances and relative geometries of the actual bodies. Methods for calculating will be taken up in Chapter 9.

In many engineering problems, radiation is combined with other modes of heat transfer. The solution of such problems can often be simplified by using a thermal conductance Kr, or a thermal resistance Rr,for radiation. The definition of Kris similar to that of Kk, the thermal conductance for conduction. If the heat transfer by radiation is written

(1.19) the radiation conductance, by comparison with Eq. (1.12), is given by

The unit thermal radiation conductance, or radiation heat transfer coefficients, , is then

where is any convenient reference temperature, whose choice is often dictated by the convection equation, which will be discussed next. Similarly, the unit thermal resistance for radiationis

EXAMPLE 1.4A long, cylindrical electrically heated rod, 2 cm in diameter, is installed in a vacuum furnace as shown in Fig. 1.12. The surface of the heating rod has an emissivity of 0.9 and is maintained at 1000 K, while the interior walls of the furnace are black and are at 800 K. Calculate the net rate at which heat is lost from the rod per unit length and the radiation heat transfer coefficient.

K/W (°F h/Btu) hr = Kr A1

W/K (Btu/h °F)

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1.6Combined Heat Transfer Systems23

2-cm diameter

Interior walls of furnace at 800K

FIGURE 1.12Schematic diagram of vacuum furnace with heating rod for Example 1.4.

SOLUTIONAssume that steady state has been reached. Moreover, note that since the walls of the furnace completely enclose the heating rod, all the radiant energy emitted by the surface of the rod is intercepted by the furnace walls. Thus, for a black enclosure,

Eq. (1.17) applies and the net heat loss from the rod of surface A1is

Note that in order for steady state to exist, the heating rod must dissipate electrical energy at the rate of 1893 W and the rate of heat loss through the furnace walls must equal the rate of electric input to the system, that is, to the rod.

From Eq. (1.17), , and therefore the radiation heat transfer coefficient, according to its definition in Eq. (1.21), is

1.6Combined Heat Transfer Systems

In the preceding sections the three basic mechanisms of heat transfer have been treated separately. In practice, however, heat is usually transferred by several of the basic mechanisms occurring simultaneously. For example, in the winter, heat is

=151 W/m2 K

=1893 W

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24Chapter 1Basic Modes of Heat Transfer

TABLE 1.5The three modes of heat transfer

One dimensional conduction heat transfer through a stationary medium

Convection heat transfer from a surface to a moving fluid

Net radiation heat transfer from surface 1 to surface 2 hc A qc = hcA(Ts - Tq) = Ts - Tq Rc

Rk= L qk = kA

T1 T1 > T2

Ts > T∞ qc

Thermal conductivity, k

Average convection heat transfer coefficient, hc

Solid or stationary fluid

Surface 1 at T1 Surface 2 at T2

Moving fluid at T∞

Surface at Ts A

A1 qr, 1 qr, 2 T1 > T2 qk qr, net transferred from the roof of a house to the colder ambient environment not only by convection but also by radiation, while the heat transfer through the roof from the interior to the exterior surface is by conduction. Heat transfer between the panes of a double-glazed window occurs by convection and radiation acting in parallel, while the transfer through the panes of glass is by conduction with some radiation passing directly through the entire window system. In this section, we will examine combined heat transfer problems. We will set up and solve these problems by dividing the heat transfer path into sections that can be connected in series, just like an electrical circuit, with heat being transferred in each section by one or more mechanisms acting in parallel. Table 1.5 summarizes the basic relations for the rate equation of each of the three basic heat transfer mechanisms to aid in setting up the thermal circuits for solving combined heat transfer problems.

(Parte **5** de 6)