Introduction to Heat Transfer

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Figure 1: One-dimensional conduction.

Heat transfer is a process whereby thermal energy is transferred in response to a temperature difference. There are three modes of heat transfer: conduction, convection, and radiation. Conduction is heat transfer across a stationary medium, either solid or fluid. For an electrically nonconducting solid, conduction is attributed to atomic activity in the form of lattice vibration, while the mechanism of conduction in an electrically-conducting solid is a combination of lattice vibration and translational motion of electrons. Heat conduction in a liquid or gas is due to the random motion and interaction of the molecules. For most engineering problems, it is impractical and unnecessary to track the motion of individual molecules and electrons, which may instead be described using the macroscopic averaged temperature. The heat transfer rate is related to the temperature gradient by Fourier’s law. For the one-dimensional heat conduction problem shown in Fig. 1, in which temperature varies along the $y$ - direction only, the heat transfer rate is obtained by Fourier’s law

${q''_y} = - k\frac{{dT}}{{dy}}\qquad \qquad(1)$

where $q'' y$ is the heat flux along the $y$ - direction, i.e., the heat transfer rate in the $y$ - direction per unit area (W/m²), and $d T / d y$ (K/m) is the temperature gradient. The proportionality constant $k$ is thermal conductivity (W/m-K) and is a property of the medium.

For heat conduction in a multidimensional isotropic system, eq. (1) can be rewritten in the following generalized form:

${\mathbf{q''}} = - k\nabla T \qquad \qquad(2)$

where both the heat flux and the temperature gradient are vectors, i.e.,

${\mathbf{q''}} = {\mathbf{i}}{q''_x} + {\mathbf{j}}{q''_y} + {\mathbf{k}}{q''_z} \qquad \qquad(3)$

While the thermal conductivity for isotropic materials does not depend on the direction, it is dependent on direction for anisotropic materials. Unlike isotropic material whose thermal conductivity is a scalar, the thermal conductivity of anisotropic material is a tensor of the second order:

${\mathbf{k}} = \left[ {\begin{array}{*{20}{c}} {{k_{xx}}} & {{k_{xy}}} & {{k_{xz}}} \\ {{k_{yx}}} & {{k_{yy}}} & {{k_{yz}}} \\ {{k_{zx}}} & {{k_{zy}}} & {{k_{zz}}} \\ \end{array}} \right] \qquad \qquad(4)$

and eq. (2) will become:

${\mathbf{q''}} = - {\mathbf{k}} \cdot \nabla T \qquad \qquad(5)$

Equation (2) or (5) is valid in a system that is uniform in all aspects except for the temperature gradient, i.e., no gradients of mass concentration or pressure. In a multicomponent system, mass transfer can also contribute to the heat flux (Curtiss and Bird, 1999).

${\mathbf{q''}} = - {\mathbf{k}} \cdot \nabla T + \sum\limits_{i = 1}^N {{h_i}{{\mathbf{J}}_i}} + c{R_u}T\sum\limits_{i = 1}^N {\sum\limits_{j = 1(j \ne i)}^N {\frac{{{x_i}{x_j}}}{{{\rho _i}}}\frac{{D_i^T}}{{{{\mathfrak{D}_{ij}}}}}} \left( {\frac{{{{\mathbf{J}}_i}}}{{{\rho _i}}} - \frac{{{{\mathbf{J}}_j}}}{{{\rho _j}}}} \right)} \qquad \qquad(6)$

where $J i$ is diffusive mass flux relative to mass-averaged velocity, which will be discussed in the latter part of this section. The second term on the right-hand side represents the interdiffusional convection term, which is not zero even though $\sum\limits_{i = 1}^N {{{\mathbf{J}}_i}} = 0$ . $h i$ is partial enthalpy (J/kg) for the i^th species. The third term on the right-hand side is the contribution of the concentration gradient to the heat flux, which is referred to as the diffusion-thermo or Dufour effect. $c$ is the molar concentration (kmol/m³) of the mixture, $R u$ is the universal gas constant, $x i$ and $x j$ are molar fractions of the i^th and j^th components respectively, $D_i^T$ is the multicomponent thermal diffusivity (m²/s) (which will be considered in the discussion of mass transfer in the latter part of this section) and $\mathfrak{D_{ij}}$ is the Maxwell-Stefan diffusivity, which is related to the multicomponent Fick diffusivity, ${\mathbb{D}_{ij}}$ (see mass transfer). A binary system gives

$\mathfrak{D_{12}} = \frac{{{x_1}{x_2}}}{{{\omega _1}{\omega _2}}}{\mathbb{D}_{12}}\qquad \qquad(7)$

where $ω 1$ and $ω 2$ are mass fractions of components 1 and 2 respectively. For a ternary system:

$\mathfrak{D_{12}} = \frac{{{x_1}{x_2}}} {{{\omega _1}{\omega _2}}}\frac{{{\mathbb{D}_{12}}{\mathbb{D}_{33}} - {\mathbb{D}_{13}}{\mathbb{D}_{23}}}} {{{\mathbb{D}_{12}} + {\mathbb{D}_{33}} - {\mathbb{D}_{13}} - {\mathbb{D}_{23}}}} \qquad \qquad(8)$

Figure 2: Forced convective heat transfer.

Typical values of mean convective heat transfer coefficients

Mode	Geometry	$\bar h{\rm{ (W/}}{{\rm{m}}^{\rm{2}}}{\rm{ - K)}}$
Forced convection	Air flows at 2 m/s over a 0.2 m square plate	12
	Air at 2 atm flowing in a 2.5 cm-diameter tube with a velocity of 10 m/s	65
	Water flowing in a 2.5 cm-diameter tube with a mass flow rate of 0.5 kg/s	3500
	Airflow across 5 cm-diameter cylinder with velocity of 50 m/s	180
Free convection $(Δ T = 20 o C)$	Vertical plate 0.3 m high in air	4.5
	Horizontal cylinder with a diameter of 2 cm in water	890
Evaporation	Falling film on a heated wall	6000-27000
Condensation of water at 1 atm	Vertical surface	4000-11300
	Outside horizontal tube	9500-25000
Boiling of water at 1 atm	Pool	2500-3500
	Forced convection	5000-100000
Natural convection- controlled melting and solidification	Melting in a rectangular enclosure	500-1500
	Solidification around a horizontal tube in a superheated liquid phase change material	1000-1500

For a system of more than four components, the Maxwell-Stefan diffusivity can be obtained using methods described in Curtiss and Bird (1999; 2001). The interdiffusional convection term represented by the second term on the right- hand side of eq. (6) is usually important for multicomponent diffusion systems. While the Dufour energy flux represented by the third term on the right-hand side of eq. (6) is negligible for many engineering problems, it may become important for cases with a very large temperature gradient.

The second mode of heat transfer is convection, which occurs between a wall at one temperature, $T w$ , and a moving fluid at another temperature, ${T_\infty }$ ; this is exemplified by forced convective heat transfer over a flat plate, as shown in Fig. 2. The mechanism of convection heat transfer is a combination of random molecular motion (conduction) and bulk motion (advection) of the fluid. Newton’s law of cooling is used to describe the rate of heat transfer:

$q'' = h({T_w} - {T_\infty }) \qquad \qquad(9)$

where $h$ is the convective heat transfer coefficient (W/m²-K), which depends on many factors including fluid properties, flow velocity, geometric configuration, and any fluid phase change that may occur as a result of heat transfer. Unlike thermal conductivity, the convective heat transfer coefficient is not a property of the fluid. Typical values of mean convective heat transfer coefficients for various heat transfer modes are listed in the table on the right. Convective heat transfer is often measured using the Nusselt number defined by:

$Nu = \frac{{hL}}{k} \qquad \qquad(10)$

where $L$ and $k$ are characteristic length and thermal conductivity of the fluid, respectively.

The third mode of heat transfer is radiation. The transmission of thermal radiation does not require the presence of a propagating medium and, therefore can occur in a vacuum. Thermal radiation is a form of energy emitted by matter at a nonzero temperature and its wavelength is primarily in the range between 0.1 to 10 μm. Emission can be from a solid surface as well as from a liquid or gas. Thermal radiation may be considered to be the propagation of electromagnetic waves or alternately as the propagation of a collection of particles, such as photons or quanta of photons. When matter is heated, some of its molecules or atoms are excited to a higher energy level. Thermal radiation occurs when these excited molecules or atoms return to lower energy states. Although thermal radiation can result from changes in the energy states of electrons, as well as changes in vibrational or rotational energy of molecules or atoms, all of these radiant energies travel at the speed of light. The wavelength $λ$ of radiative emissions is related to their frequency, $V$ by

$\lambda \nu = c \qquad \qquad(11)$

where $c$ is the speed of light and has a value of 2.998 x 10⁸ m/s in a vacuum. A quantitative description of the mechanism of thermal radiation requires quantum mechanics. An electromagnetic wave with frequency of $v$ can also be viewed as a particle –a photon– with energy of

$\varepsilon = h\nu \qquad \qquad(12)$

where $h$ = 6.626068×10^-34 m²-kg/s is Planck’s constant. Both mass and charge of a photon are zero.

For a blackbody, defined as an ideal surface that emits the maximum energy that can be emitted by any surface at the same temperature, the spectral emissive power, $E b,λ$ (W/m³), can be obtained by Planck’s law

${E_{b,\lambda }} = \frac{{{c_1}}}{{{\lambda ^5}({e^{{c_2}/(\lambda T)}} - 1)}} \qquad \qquad(13)$

where ${c_1} = 3.742 \times {10^{ - 16}}{\rm{ W - }}{{\rm{m}}^{\rm{2}}}$ and ${c_2} = 1.4388 \times {10^{ - 2}}{\rm{ m - K}}$ are radiation constants. The unit of the surface temperature $T$ in eq. (13) is $K$ .

Figure 3: Radiation heat transfer between a small surface and its surroundings.

The emissive power for a blackbody, $E b$ (W/m²), is

${E_b} = \int_0^\infty {{E_{b,\lambda }}d\lambda } \qquad \qquad(14)$

Substituting eq. (13) into eq. (14), Stefan-Boltzmann’s law is obtained

${E_b} = {\sigma _{SB}}{T^4} \qquad \qquad(15)$

where $σ S B$ = 5.67 x 10^-8W/m²-K⁴ is the Stefan-Boltzmann constant.

For a real surface, the emissive power is obtained by

$E = \varepsilon {E_b} \qquad \qquad(16)$

where ε is the emissivity, defined as the ratio of emissive power of the real surface to that of a blackbody at the same temperature. Since a blackbody is the best emitter, the emissivity of any surface must be less than or equal to 1. A simple but important case of radiation heat transfer is the radiation heat exchange between a small surface with area $A$ , emissivity ε, and temperature $T w$ , and a much larger surface surrounding the small surface. If the temperature of the surroundings is $T s u r$ , the heat transfer rate per unit area from the small object is obtained by

$q'' = \varepsilon {\sigma _{SB}}(T_w^4 - T_{sur}^4) \qquad \qquad(17)$

For a detailed treatment of radiation heat transfer, including radiation of nongray surfaces and participating media, consult Siegel and Howell (2002).

References

Curtiss, C. F., and Bird, R. B., 1999, “Multicomponent Diffusion,” Industrial and Engineering Chemistry Research, Vol. 38, pp. 2115-2522.

Curtiss, C. F., and Bird, R. B., 2001, “Errata,” Industrial and Engineering Chemistry Research, Vol. 40, p. 1791.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

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Introduction to Heat Transfer

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