10 Energy Heat Transfer

M K Nanda

1. Learning Outcomes
2. Introduction
3. Conduction
3.1. Conduction in earth-atmosphere system
3.2. Heat flow in soil
3.3. Effect of moisture content on thermal properties of soil
4. Convection
4.1. Convection vs advection
4.2. Nature of fluid flow: laminar vs turbulent flow
4.3. Development of boundary layer
4.4. Heat transfer in boundary layer
4.4.1. Heat transfer in forced convection
4.4.2. Heat transfer in free convection
4.4.3. Criteria for free and forced convections
4.5. Measurement of convective heat flux
4.6. Latent heat transfer
5. Radiation
5.1. Wavelength of thermal radiation
5.2. Temperature and quantity of thermal radiation
6. Energy budget of earth-atmosphere system
7. Summary

1. Learning outcomes

After studying this module, you shall be able to:have a clear understanding about different modes of heat transfer understand the process of heat flow in soil know about the heat transfer in turbulent and laminar boundary layer understand the the concept of free and forced convection know about the radiative heat transfer in earth-atmosphere system know about the energy budget of earth and atmosphere

2. Introduction

The term heat in engineering context is taken as synonymous to thermal energy. This usage has its origin in the historical interpretation of heat as a fluid (caloric) that can be transferred through various processes. Thus the heat transfer is the exchange of thermal energy between physical systems. The rate of heat transfer is dependent on the temperatures of the systems and the properties of the intervening medium through which the heat is transferred. The amount of heat transfer is calculated from driving force for the flow of heat and the heat transfer coefficient. The amount of heat transfer is called heat flux which is the quantitative, vectorial representation of heat-flow through a surface.

The rate of transport of heat energy is usually expressed as the product of a proportionality factor and a driving force which is expressed in terms of Fourier ‘s law for heat transport in the simplest form as:

where, H is the heat flux density (expressed as W m-2), in a substance with thermal conductivity, k (W.m-1.K-1) and temperature gradient, dT/dz (K.cm-1). The negative sign is a reminder that heat moves in the direction of decreasing temperature.

The three fundamental modes of heat transfer are conduction, convection and radiation.

3. Conduction

Conduction is a form of heat transfer produced by the exchange of translational, rotational, and vibrational energy among the molecules. In microscopic scale, heat conduction occurs as hot, rapidly moving or vibrating atoms and molecules interact with neighboring atoms and molecules, transferring some of their energy (heat) to these neighboring particles. In other words, heat is transferred by conduction when adjacent atoms vibrate against one another, or as electrons move from one atom to another. Thermal conduction is also called diffusion.

Heat spontaneously flows from a hotter to a colder body. In the absence of an external driving energy source to the contrary, within a body or between bodies, temperature differences decay over time, and thermal equilibrium is approached, temperature becoming more uniform.

Steady state conduction is a form of conduction that happens when the temperature difference driving the conduction is constant, so that after an equilibration time, the spatial distribution of temperatures in the conducting object does not change any further. In steady state conduction, the amount of heat entering a section is equal to amount of heat coming out of it.

Transient conduction occurs when the temperature within an object changes as a function of time. Analysis of transient systems is more complex and often calls for the application of approximation theories or complex numerical analysis.

3.1. Conduction in earth-atmosphere system

Conduction is the most significant means of heat transfer within a solid or between solid objects in thermal contact. Fluids, especially gases are less conductive. This is due to the large distance between atoms in a gas and fewer chances of collisions between atoms. However, the conductivity of gases increases with temperature. Conductivity also increases with increasing pressure up to a critical point at which the density of the gas is such that the gas molecules will have higher chances to collide with each other and transfer heat from one surface to another. After this point, conductivity increases only slightly with increasing pressure and density.

In fact, the layers of motionless gas provide excellent insulation. In earth-atmospheric system, conduction is important for heat transfer in the soil, in other solid object and in fluid at rest, but not in the free atmosphere where the effects of molecular diffusion are trivial in relation to mixing by turbulence. The conductivity of still air is four orders of magnitude less than the values for metals like, copper and silver.

If the temperature gradient in a solid or motionless fluid is dT/dz, the rate of heat transfer is expressed by Fourier’s law as described in equation (1). For a steady flow of heat between two parallel surfaces at temperature T2 and T1, separated by a uniform slab of material with thickness t, integration of equation (1) gives:

This equation is used to describe the vertical flow of heat in soil.

3.2. Heat flow in soil

Conduction is the dominant process of energy transfer in soil. The heat transfer in soil is an example of transient conduction where the temperature of soil changes as a function of time. The thermal properties of soil which regulate the heat transfer in soil are briefly explained in this section.

• Heat capacity: It is the ratio of heat absorbed to the change in temperature of any substance. [Unit: J.K-1]
• Mass specific heat (C): It is the heat required to raise the temperature of unit mass (1 kg) of a substance through 1 K. [Unit: J.kg-1.K-1]
• Volume specific heat (Cv): It is the heat required to raise the temperature of unit volume (1 m3) of a substance through 1 K. [Unit: J.m-3.K-1] In many literatures volume specific heat is termed as heat capacity per unit volume or simply heat capacity. . C = Cv
•  Thermal conductivity (k ) : It is the quantity of heat flowing through unit cross section of area per unit time in response to unit temperature gradient. [Unit: W.m-1.K-1]

Soil heat flux (Heat flow rate per unit cross sectional area) is given by equation –

The negative sign is used when the flux is outward from the soil

The thermal conductivity (k) mainly depends upon porosity, moisture content and organic/mineral constituents of the soil. thermal conductivity increases with increase in soil moisture content decreases from fine sand to silt and clay with increase in porosity thermal conductivity of organic matter is less than that of mineral matter.

Thermal conductivity of soil determines the rate of heat transfer. However the rate of change of temperature experienced by the body as a result of heat transfer will vary with heat capacity. Therefore, a parameter which considers heat capacity (volume sp. heat) more useful.

3.3. Effect of moisture content on thermal properties of soil

Addition of water increases the thermal conductivity of dry soil at faster rate initially because very high amount of heat is transmitted through evaporation and condensation of water (latent heat) through the soil pores. Gradually as water content increases the rate of increases in conductivity becomes much slower because the diffusion of vapour is much restricted due to filling of water molecules in the pore space in soil.

Addition of water increases the thermal conductivity more rapidly than volume specific heat ( C ) which leads to increase in diffusivity. But in later stage increase in conductivity is less as compared to the volume specific heat and consequently the thermal diffusivity decreases.

4. Convection

Convective heat transfer, or convection, is the transfer of heat from one place to another by the movement of fluids, a process that is essentially the transfer of heat via mass transfer. Convection is the dominant mode of heat transfer process in a fluid medium. Convection is generally described as the combined effect of heat conduction within the fluid (diffusion) and heat transference by bulk fluid flow streaming.

4.1. Convection vs advection

The bulk motion of fluid enhances heat transfer in many physical situations, such as (for example) between a solid surface and the fluid. In meteorology, convection usually refers to atmospheric motions that are predominantly vertical, such as rising air currents due to surface heating. On the other hand, the process of transport by fluid streaming is known as advection. In earth- atmospheric system the wind and ocean current are accompanied by large scale transport of heat energy through advection. Like convection, advection in a fluid is always also accompanied by transport via heat diffusion (also known as heat conduction).

4.2. Nature of fluid flow: laminar vs turbulent flow

Since transport of heat by convection is intimately related to fluid flow along a surface it is important to understand the properties of fluid flow. Fluid dynamists recognize two primary types of flow in fluids, laminar flow and turbulent flow. Laminar flow is a streamlined flow where the flow takes place in layers, i.e., each layer slides past the adjacent layers. In a laminar boundary layer any exchange of heat, mass or momentum takes place only between adjacent layers in a microscopic scale which is not visible to the eye. Consequently molecular viscosity is able predict the shear stress associated.

Turbulent flow is a flow regime in fluid dynamics characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow regime, which occurs when a fluid flows in parallel layers, with no disruption between those layers.

It can be generalized as transport within a gas by “carriers,” which may be molecules or particles or eddies, capable of transporting units of a property P such as heat, water vapor, or a gas. Even when the carriers are moving randomly, net transport may occur in any direction provided that the concentration of P decreases with distance in that direction. The fluctuations, or eddies, in the atmosphere are, in a sense, like molecules in a gas. They bounce about with random motion, but are carried along with the wind. It is these fluctuations that transport heat, water, momentum, etc. in the atmosphere.

4.3. Development of boundary layer

Since the process of heat transfer in earth atmospheric system is analogous to a system where a surface is immersed in a fluid in motion the understanding of boundary layer is essential for interpretation of convective heat transfer.

When the streamlines of flow are almost parallel to the surface (laminar flow) the flow of momentum across it takes place by the momentum exchange between individual molecules. Consequently, the flow becomes unstable and breaks down to a chaotic pattern of swirling motions called a turbulent boundary layer. A second laminar layer of restricted depth—the laminar sublayer—forms immediately above the surface and below the turbulent layer.

Free vs forced convection: The process of convection is of two types – Free convection and Forced convection. Free convection (or natural, convection) occurs when bulk fluid motions (streams and currents) are caused by buoyancy forces that result from density variations due to variations of temperature in the fluid. The buoyancy forces occur when thermal energy expands the fluid (for example in a fire plume), thus influencing its own transfer. Forced convection is a term used when the streams and currents in the fluid are induced by mechanical forces such as deflection by a large scale surface irregularity, turbulent flow caused by friction at the boundary of a fluid, or motion caused by any applied pressure gradient.

Free and forced convection are not necessarily exclusive processes. On a windy day with overcast sky, the heat exchange between ground and air is an example of forced convection. On a sunny day with a little wind where the ground temperature rises, both kinds of convection take place. Both in forced and in free convection, the magnitude of the Nusselt number depends on the character of flow in the boundary layer—laminar or turbulent. In turn, this depends partly on the turbulence in the upwind airstream and partly on the roughness of the surface which tends to generate turbulence.

4.4.1. Heat transfer in forced convection

In forced convection the boundary layer of a surface is exposed to air stream. The rate of transfer depends on the velocity of flow. Hence, the Nusselt’s number (Nu) is assumed to be a function of Reynold’s number (Re) which determines the flow pattern. It is also a function of a ratio related to momentum and heat transfer in boundary layer which is expressed as “Prandtl number (Pr)”

4.4.2. Heat transfer in free convection

In free convection the circulation around the surface object is governed by density gradient, created and maintained by the temperature gradient between the surface and the surrounding fluid. In this case Nu is a function of another non-dimensional constant called “Grashof Number (Gr)”. Grashof number is a function of –

a) Coefficient of thermal expansion of fluid (a)

b) Acceleration by gravity (g)

c) Characteristic dimension (d)

d) Temperature difference between the surface and the surrounding (Ts – T)

Coefficient of kinematic viscosity (v ) Mathematically

4.4.3. Criteria for free and forced convections

For determination of whether heat transfer process is free convection or forced convection or a combination of both, the comparison of Gr with Re2 should be considered. The ratio Gr/Re2 indicates the dominance of buoyancy or inertia force. As a thumb rule –

• If                    Gr >  16 Re2                      free convection is the rule
• If                    Gr <     0.01 Re2 forced convection is the mode

For intermediate values, Nu should be calculated for both free and forced convection and larger number should be used to estimate the rate of heat transfer.

The onset of turbulence in free convection occurs when Gr exceeds 108 which is quite unusual situation in micrometeorology (The surface temperature of the object has to exceed from the ambient air temperature by 30 0C).

4.5. Measurement of convective heat flux

The flux (mass/energy/momentum flow per unit area per unit time) can be measured by averaging the product of fluctuations of temperature, horizontal wind, or mass, and vertical wind. This method of measuring fluxes is called eddy correlation or eddy covariance.

In mathematical terms, “eddy flux” is computed as a covariance between instantaneous deviation in vertical wind speed (w’) from the mean value (w) and instantaneous deviation in any property (energy concentration) from its mean value, multiplied by mean air density (ρa). Using Reynolds decomposition the equation is simplified as-

where, H = Convective heat flux, ρ = density, Cp = specific heat of the air media, T = Temperature, over bar represents mean value and the inverted comma represents fluctuation

Considering the rapidity in mixing process and sharp fluctuation in properties (heat, gas concentration, wind speed etc in the turbulent layer, this method requires fast responding measurement systems like sonic anemometer and infra red gas analyzer.

The heat flux (H) can also be determined by using profile-gradient approach which follows the simple principle that the flow is directly proportional to the potential gradient and inversely proportional to the resistance offered by the medium through the flow occurs.

where, ρ = density, Cp = specific heat of the air media (J kg-1 °C-1), T = Temperature, z = the distance across which the temperature is measured. KH = heat transfer coefficient

The heat transfer coefficient is a function of aerodynamic resistance and thus depends upon the horizontal wind speed, surface roughness and also the stability factor which is being explained in Monin-Obukhov’s Similirity Theory (MOST).

4.6. Latent heat transfer

Transformations between solid, liquid and gaseous phases of matter are accompanied by either the release or the absorption of heat, referred to generally as latent heat. Thus, Latent heat flux is the flux of heat from the Earth’s  surface to the atmosphere that is associated with evaporation of water at the surface and subsequent condensation of water vapor in the troposphere. Latent heat fluxes (e.g. evaporation) are driven by difference in vapor pressure between surface and atmosphere and turbulence of atmospheric layer. Latent heat flux is represented by a resistance network similar to sensible heat in which The latent heat flux over a typical moist surface is represented by

where, λ = latent heat of vaporization, ρ = density,  — (19)

= Psychrometric constant (66.5 Pa °C-1),

Cp = specific heat of the air media (J kg-1 °C-1), z = the distance across which the temperature is measured. Kv = vapour transfer coefficient and e = vapour pressure (Pa)

Latent heat flux has significant contribution to energy balance of earth-atmospheric system considering that the water makes up about 71 percent of earth surface. In the prevalent temperature condition of earth atmospheric system, water is present in all three phases (solid, liquid and vapour) and there is continuous transformation among the three phases – evapotranspiration from the surface, cooling, condensation and cloud formation in air followed by precipitation. A typical summertime evapotranspiration at 5 mm of water per day is equivalent to a heat loss of 141 W m-2.

5. Radiation

Radiation is the emission or transmission of energy in the form of waves or particles through space or through a material medium. Radiative exchange requires no intervening molecules to transfer energy from one surface to another. Thus, by thermal radiation heat is transmitted through a vacuum or any transparent medium.

Electromagnetic radiation is a form of energy derived from oscillating magnetic and electrostatic fields perpendicular to each other and perpendicular to the direction radiative flow. The velocity of electromagnetic radiation is c = 3.0 × 108 m.s−1. The frequency of oscillation, ν is related to the wavelength λ by the standard wave equation c = λν and the wave number 1/λ = ν/c is sometimes used as an index of frequency.

The radiative energy is derived from the vibration and rotation of individual atoms within the molecular structure. Since these atoms and molecules are composed of charged particles (protons and electrons), their movement results in the emission of electromagnetic radiation, which carries energy away from the surface.

5.1. Wavelength of thermal radiation

Thermal radiation can be emitted from objects at any wavelength. However, the infrared radiation at wavelength 3-15 µm is popularly known as “heat radiation”. The popular association of infrared radiation with thermal radiation is only a coincidence based on typical temperatures often found near the surface of planet Earth. At very high temperatures like that of sun such radiations are associated with spectra far above the

infrared, extending into visible, ultraviolet, and even X-ray regions. The relationship between temperature and wavelength of emitted radiation is demonstrated by Wein’s displacement law as follows.

where, λmax = Wavelength of maximum radiation and T = Surface temperature (K)

Considering the earth and terrestrial objects at a temperature in the range of 300 K, the wavelength of major part of the terrestrial radiation is obtained in the range of 3-15 µm (λmax= 9.7 μm) that encompasses mid-wave infrared (3–8 µm) and longwave infrared (8–15 µm). The sun on the other hand emits in short wave range 0.15 – 3 µm (λmax = 0.48 μm) due to its surface temperature 6000 K.

5.2. Temperature and quantity of thermal radiation

Any object at a temperature of more than 0 K emits radiation. The quantity of radiation is described in terms of radiant flux density which is defined as the rate of energy transfer per unit area per unit time, expressed as J m-2 s-1 (W.m-2) or Cal.cm-2 .min -1 or Langley min-1. The radiant flux density depends upon the energy state of the object defined by its surface temperature and emissivity as explained by Stefan-Boltzmann equation.

R = ε σ T4 W.m-2,…………(20)

Where, R = Radiant flux (quantity of energy transferred per unit area per unit time)

              ε = Emissivity associated with changes in the energy state of the constituents

            T = Surface temperature (K)

Thermal radiation contributes substantially to the Earth’s energy budget. Different natural processes of geogenic origin as well as anthropogenic perturbations in the earth-atmospheric system are responsible for a positive radiative forcing which results in global warming by reducing the net longwave radiation loss to space.

6. Energy budget of earth-atmosphere system

The thermal radiation has substantial contribution to the energy balance of earth-atmosphere system. About 69% of the energy flow into the atmosphere is being contributed by the terrestrial radiation, whereas only 14% of the energy is being contributed by the direct solar radiation. The remaining amount (about 17%) is received by the atmosphere in form of latent and sensible heat flux (Figure 2). Similarly, the earth-atmospheric system emits about 239 W m-2 of longwave radiation into the outer space to maintain the energy balance of the system.

There exists wide spatial variation in the net energy budget over the global surface. An excess of incoming shortwave radiation is being observed between 35 °S and 40 °N and a deficit at higher latitudes compared with the outgoing longwave radiation budget. To maintained latitudinal energy balance heat energy is transported from low latitudes poleward through oceanic and atmospheric circulation.

7. Summary

• Heat transfer is the exchange of thermal energy between physical systems. The rate of heat transfer (heat flux density in W m^-2) is dependent on the temperatures of the systems and the properties of the intervening medium through which the heat is transferred. The three fundamental modes of heat transfer are conduction, convection and radiation.
• Conduction (or diffusion) is caused by the exchange of translational, rotational, and vibrational energy among the molecules. Conduction is the most significant means of heat transfer within a solid or between solid objects in thermal contact. In earth-atmospheric system, conduction is important for heat transfer in the soil. The thermal properties which regulate the heat transfer in soil include specific heat and thermal conductivity as controlled by porosity, moisture content and organic constituents of the soil.
• Convection is the transfer of heat by the movement of fluids. This is dominant mode of heat transfer process in earth- atmospheric system through wind and ocean current. In meteorology, convection usually refers to the motions that are predominantly vertical e.g. ascent or descent of air mass, whereas the horizontal transport like ocean current, surface winds etc is known as advection.
• Convective heat transfer can take place in laminar flow where heat transfer between adjacent layers in a microscopic scale or turbulent flow characterized by chaotic changes in pressure and flow velocity (eddies) that act as a carrier of heat energy in the convective process, the pattern of flow determined by Reynold’s number.
• The convection can be free or forced convection. Free or natural convection is dominated by buoyancy force resulted from steep thermal gradient whereas forced convection is induced by frictional force at the boundary of a fluid, or by applied pressure gradient. The convective heat flux is measured by eddy covariance approach or by profile-gradient approach where the aerodynamic resistance is estimated from wind speed and roughness parameters.
• Latent heat flux is the flux in atmosphere of involves evaporation and condensation at the surface and atmosphere. Latent heat fluxes are driven by difference in vapor pressure between surface and atmosphere and turbulence of atmospheric layer.
• Radiative exchange occurs in from of electromagnetic waves through space or through a material medium and requires no intervening molecules. Thermal radiation occurs in infrared band at wavelength 3-15 µm.
• The relationship between temperature and wavelength of emitted radiation is demonstrated by Wein’s displacement law whereas the quantity of radiation (radiant flux density) is defined by surface temperature and emissivity in accordance with Stefan-Boltzmann equation. The thermal radiation has substantial contribution to the energy balance of earth-atmosphere system.