# Introduction to Heat Transfer While internal energy refers to the total energy of all the molecules within the object, heat is the amount of energy flowing spontaneously from one body to another due to their temperature difference. Heat is a form of energy, but it is energy in transit. Heat is not a property of a system. However, the transfer of energy as heat occurs at the molecular level due to a temperature difference.

Consider a block of metal at high temperature that consists of atoms oscillating intensely around their average positions. At low temperatures, the atoms continue to oscillate but with less intensity. If a hotter block of metal is put in contact with a cooler block, the intensely oscillating atoms at the edge of the hotter block give off their kinetic energy to the less oscillating atoms at the edge of the cool block. In this case, there is energy transfer between these two blocks, and heat flows from the hotter to the cooler block by these random vibrations.

Thermal Energy in Microscopic Scale
Internal energy involves energy on the microscopic scale. It may be divided into microscopic potential energy, Upot, and microscopic kinetic energy, Ukin, components:

U = Upot + Ukin where the microscopic kinetic energy, Ukin, involves the motions of all the system’s particles for the center-of-mass frame. For an ideal monatomic gas, this is just the translational kinetic energy of the linear motion of the atoms. Monoatomic particles do not rotate or vibrate. The behavior of the system is well described by the kinetic theory of gases. Kinetic theory is based on the fact that during an elastic collision between a molecule with high kinetic energy and one with low kinetic energy, part of the energy will transfer to the molecule of lower kinetic energy. However, there is rotational and vibrational kinetic energy for polyatomic gases.

The microscopic potential energy, Upot, involves the chemical bonds between the atoms that make up the molecules, binding forces in the nucleus, and the physical force fields within the system (e.g., electric or magnetic fields).

A significant component of potential energy is associated with the intermolecular attractive forces in liquids and solids.

In general, when two objects are brought into thermal contact, heat will flow between them until they come into equilibrium with each other.  When a temperature difference does exist, heat flows spontaneously from the warmer system to the colder system. Heat transfer occurs by conduction or by thermal radiation. When the flow of heat stops, they are said to be at the same temperature. They are then said to be in thermal equilibrium.

As with work, the amount of heat transferred depends upon the path and not simply on the initial and final conditions of the system. There are many ways to take the gas from state i to state f.

Also, as with work, it is important to distinguish between heat added to a system from its surroundings and heat removed from a system to its surroundings. Q is positive for heat added to the system, so Q is negative if heat leaves the system. Because W in the equation is the work done by the system, then if work is done on the system, W will be negative, and Eint will increase.

The symbol q is sometimes used to indicate the heat added to or removed from a system per unit mass. It equals the total heat (Q) added or removed divided by the mass (m).

## Distinguishing Temperature, Heat, and Internal Energy

A clear distinction between these three properties can be made using the kinetic theory.

• Temperature is related to the kinetic energies of the molecules of material, and it is the average kinetic energy of individual molecules.
• Internal energy refers to the total energy of all the molecules within the object. Therefore it is an extensive property when two equal-mass hot ingots of steel may have the same temperature, but two of them have twice as much internal energy as one does.
• Finally, heat is the amount of energy flowing spontaneously from one body to another due to their temperature difference.

When a temperature difference exists, it must be added, and heat flows spontaneously from the warmer system to the colder system. Thus, if a 5 kg cube of steel at 100°C is placed in contact with a 500 kg cube of steel at 20°C, heat flows from the cube at 300°C to the cube at 20°C even though the internal energy of the 20°C cube is much greater because there is so much more of it.

A particularly important concept is thermodynamic equilibrium. In general, when two objects are brought into thermal contact, heat will flow between them until they come into equilibrium with each other.

Thermal Energy in Microscopic Scale
Internal energy involves energy on the microscopic scale. It may be divided into microscopic potential energy, Upot, and microscopic kinetic energy, Ukin, components:

U = Upot + Ukin where the microscopic kinetic energy, Ukin, involves the motions of all the system’s particles for the center-of-mass frame. For an ideal monatomic gas, this is just the translational kinetic energy of the linear motion of the atoms. Monoatomic particles do not rotate or vibrate. The behavior of the system is well described by the kinetic theory of gases. Kinetic theory is based on the fact that during an elastic collision between a molecule with high kinetic energy and one with low kinetic energy, part of the energy will transfer to the molecule of lower kinetic energy. However, there is rotational and vibrational kinetic energy for polyatomic gases.

The microscopic potential energy, Upot, involves the chemical bonds between the atoms that make up the molecules, binding forces in the nucleus, and the physical force fields within the system (e.g., electric or magnetic fields).

In liquids and solids there is significant component of potential energy associated with the intermolecular attractive forces.

## Heat Flux Density – Thermal Flux

The rate of heat transfer per unit area normal to the direction of heat transfer is called heat flux. Sometimes it is also referred to as heat flux density. In SI, its units are watts per square meter (W.m−2). It has both a direction and a magnitude, so it is a vector quantity. The average heat flux is expressed as: where A is the heat transfer area. The unit of heat flux in English units is Btu/h·ft2. Note that heat flux may vary with time and position on a surface.

In nuclear reactors, limitations of the local heat flux are of the highest importance for reactor safety. Since nuclear fuel consists of fuel rods, the heat flux is defined in units of W/cm (local linear heat flux density) or kW/rod (power per fuel rod).

References:
Heat Transfer:
1. Fundamentals of Heat and Mass Transfer, 7th Edition. Theodore L. Bergman, Adrienne S. Lavine, Frank P. Incropera. John Wiley & Sons, Incorporated, 2011. ISBN: 9781118137253.
2. Heat and Mass Transfer. Yunus A. Cengel. McGraw-Hill Education, 2011. ISBN: 9780071077866.
3. Fundamentals of Heat and Mass Transfer. C. P. Kothandaraman. New Age International, 2006, ISBN: 9788122417722.
4. U.S. Department of Energy, Thermodynamics, Heat Transfer and Fluid Flow. DOE Fundamentals Handbook, Volume 2 of 3. May 2016.

Nuclear and Reactor Physics:

1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
6. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
7. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
8. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.
9. Paul Reuss, Neutron Physics. EDP Sciences, 2008. ISBN: 978-2759800414.