which is a function only of geometry and single fin efficiency. To get the heat rate from a fin array, we write

Qarray = Kh (Tb - T~) A where A is the total area exposed.

Transient Conduction. Heating and cooling problems involve the solution of the time-dependent conduction equation. Most problems of industrial significance occur when a body at a known initial temperature is suddenly exposed to a fluid at a different temperature. The temperature behavior for such unsteady problems can be characterized by two dimensionless quantities, the Biot number, Bi = hL/k, and the Fourier modulus, Fo = ax/L2. The Biot number is a measure of the effectiveness of conduction within the body. The Fourier modulus is simply a dimensionless time.

If Bi is a small, say Bi < 0.1, the body undergoing the temperature change can be assumed to be at a uniform temperature at any time. For this case,

Fig. I.5 Fins of various shapes. (a) Rectangular, (b) Trapezoidal, (c) Arbitrary profile, (d ) Circumferential.

where T/ and T( are the fluid temperature and initial body temperature, respectively. The term (pCV/hA) takes on the characteristics of a time constant.

If Bi > 0.1, the conduction equation must be solved in terms of position and time. Heisler4 solved the equation for infinite slabs, infinite cylinders, and spheres. For convenience he plotted the results so that the temperature at any point within the body and the amount of heat transferred can be quickly found in terms of Bi and Fo. Figures I.7 to I.10 show the Heisler charts for slabs and cylinders. These can be used if h and the properties of the material are constant.

I.3.2 Convection Heat Transfer

Convective heat transfer is considerably more complicated than conduction because motion of the medium is involved. In contrast to conduction, where many geometrical configurations can be solved analytically, there are only limited cases where theory alone will give convective heat-transfer relationships. Consequently, convection is largely what we call a semi-empirical science. That is, actual equations for heat transfer are based strongly on the results of experimentation.

Convection Modes. Convection can be split into several subcategories. For example, forced convection refers to the case where the velocity of the fluid is completely independent of the temperature of the fluid. On the other hand, natural (or free) convection occurs when the temperature field actually causes the fluid motion through buoyancy effects.

We can further separate convection by geometry into external and internal flows. Internal refers to channel, duct, and pipe flow and external refers to unbounded fluid flow cases. There are other specialized forms of convection, for example the change-of-phase phenomena: boiling, condensation, melting, freezing, and so on. Change-of-phase heat transfer is difficult to predict analytically. Tongs5 gives many of the correlations for boiling and two-phase flow.

Dimensional Heat-Transfer Parameters.

Because experimentation has been required to develop appropriate correlations for convective heat transfer, the use of generalized dimension-less quantities in these correlations is preferred. In this way, the applicability of experimental data covers a wider range of conditions and fluids. Some of these parameters, which we generally call "numbers," are given below:

Prandtl number: Pr =

ratio of momentum transport to heat-transport characteristics for a fluid: it is important in all convective cases, and is a material property

Grashof number: Gr =

serves in natural convection the same role as Re in forced convection: that is, it controls the character of the flow h

Stanton number: St =

p uCp

Nusselt number: Nu =

where k is the fluid conductivity and L is measured along the appropriate boundary between liquid and solid; the Nu is a nondimensional heat-transfer coefficient.

Reynolds number: Re =

defined in Section 1.4: it controls the character of the flow Fig. I.6 (a) Efficiencies of rectangular and triangular fins, (b) Ef ficiencies of circumferential fins of rectangular profile.


Fig. 1.7 Midplane temperature for an infinite plate of thickness 2L. (From Ref. 4.)


Fig. 1.7 Midplane temperature for an infinite plate of thickness 2L. (From Ref. 4.)


Fig. I.8 Axis temperature for an infinite cylinder of radius ro. (From Ref. 4.)

also a nondimensional heat-transfer coefficient: it is very useful in pipe flow heat transfer.

In general, we attempt to correlate data by using relationships between dimensionless numbers: for example, in many convection cases, we could write Nu = Nu(Re, Pr) as a functional relationship. Then it is possible either from analysis, experimentation, or both, to write an equation that can be used for design calculations. These are generally called working formulas.

Forced Convection Past Plane Surfaces. The average heat-transfer coefficient for a plate of length L may be calculated from

if the flow is laminar (i.e., if ReL < 4,000). For this case the fluid properties should be evaluated at the mean film temperature Tm, which is simply the arithmetic

Fig. 1.9 Temperature as a function of center temperature in an infinite plate of thickness 2L. (From Ref. 4.)

Fig. 1.9 Temperature as a function of center temperature in an infinite plate of thickness 2L. (From Ref. 4.)

average of the fluid and the surface temperature.

For turbulent flow, there are several acceptable correlations. Perhaps the most useful includes both laminar leading edge effects and turbulent effects. It is


Fig. I.10 Temperature as a function of axis temperature in an infinite cylinder of radius ro. (From Ref. 4.)

if the properties of the flow are constant.

Sieder and Tate developed the following more convenient empirical formula for short tubes:

where the transition Re is 4,000.

Forced Convection Inside Cylindrical Pipes or Tubes. This particular type of convective heat transfer is of special engineering significance. Fluid flows through pipes, tubes, and ducts are very prevalent, both in laminar and turbulent flow situations. For example, most heat exchangers involve the cooling or heating of fluids in tubes. Single pipes and/or tubes are also used to transport hot or cold liquids in industrial processes. Most of the formulas listed here are for the 0.5 < Pr < 100 range.

Laminar Flow. For the case where Ref < 2300, Nusselt showed that Nuf = 3.66 for long tubes at a constant tube-wall temperature. For forced convection cases (laminar and turbulent) the fluid properties are evaluated at the bulk temperature T^. This temperature, also called the mixing-cup temperature, is defined by

f J0

uTr dr f J0

ur dr

The fluid properties are to be evaluated at Tb except for the quantity which is the dynamic viscosity evaluated at the temperature of the wall.

Turbulent Flow. McAdams suggests the empirical relation

where n = 0.4 for heating and n = 0.3 for cooling. Equation 1.7 applies as long as the difference between the pipe surface temperature and the bulk fluid temperature is not greater than 10°F for liquids or 100°F for gases.

For temperature differences greater then the limits specified for equation I.7 or for fluids more viscous than water, the following expression from Sieder and Tate will give better results:

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