# Heat Transfer/Introduction

Introduction to Heat Transfer

This book deals with heat transfer in the engineering context, particularly for chemical and mechanical engineers. It includes the basic physics and technology which is used for heating and cooling in industry. Of course, the principles may be applied in other fields if appropriate, and engineers may deal with new technology quite unlike traditional ones. It is intended as a beginning text for first or second year engineering degree students.

If you add to or amend this (and you are most welcome) please do so either by careful reference to an authoritative textbook, or on the basis of your trustworthy professional experience, if you have this.

Here is a quick run through some basics, which will be covered in more detail in subsequent chapters.

### Basic Concepts

Heat transfer in engineering consists of the transfer of enthalpy(Rate of movement) because of a temperature difference(Difference in potential between two points of movement). Enthalpy is the name for heat energy, to distinguish it from other sorts, such as kinetic energy, pressure energy, useful work. There has to be a temperature difference, or no heat transfer occurs.

(If we insist on a change in enthalpy from a cold body to a warm one, we will have to do more work, as in the case of a refrigerator(We are moving up or against the current[defined as natural flow]). This invariably involves some other process, such as mechanical work via outside forces, and cooling by expansion of gases/internal forces, but within the overall activity heat transfer always goes from the warmer to the cooler.)

The temperature difference is called the driving force. Other things being equal, a greater temperature difference will give a greater rate of heat transfer.

Enthalpy - The Rate of Movement within a system

System - Defined as a Vector between two separate Forces

Force - Defined as a Basic non-functional unit

Non-Functional Unit - Heat or Cold

Enthalpy thus defines Heat as a Non-Functional Unit within a System, its location, and intensity

### Temperature

Temperature is an intensive property: that is it does not depend on the amount of substance. On the Atomic Level, Heat is completely dependent upon the Volume of an Atom, thus: As the volume of an atom increases, so does its capacity to hold heat, as a function of the same as mass can hold cold(the inverse of Enthalpy). Thus one kilogram of copper at 80 °C and 12 kg of copper at 80 °C both have the same Enthalpy. Note that unless we are dealing with radiated heat, it is not normally necessary to change these values to the Absolute Temperature scale. The Celsius temperature is simply defined as the number of kelvin above 273.15 K. If we wish to calculate heat transfer from these blocks of copper to water at 20 °C, it is quite adequate to say the temperature difference is 80 °C - 20 °C = 60 K. We get the same answer with more effort by saying it is 353 – 293 = 60 K. (As I am working to the nearest degree, I have omitted the 0.15 K). Temperatures may be given on the Absolute or Celsius temperature scales, but temperature differences should be given in Kelvin. Temperature is also defined as the degree of hotness. It plays an important role in the subject of thermodynamics and heat transfer (i.e.in Thermal energy). Temperature in Algebra can be defined on the number line as the difference in potential between positive and negative infinity, with infinity itself being related to Zero Kelvin, thus it is understood why concepts such as Negative Kelvin, SuperConductors, and other phenomena are able to exist in the Universe. Hence we are also able to infer why Zero Kelvin and SuperConductors, often go hand in hand. (Further reading on Wetting, Cohesiveness, and Adhesiveness in Chemistry and NanoSciences)

### Enthalpy

Enthalpy is a measure of the total energy(or the Rate of Movement) stored in a thermodynamic(^needs to be defined) system. It includes the internal energy, which is a function of temperature, and it's inverse, which is the amount of energy required to make room for it by displacing the environment and establishing its respective volume and pressure(see Partial Gas Pressure, Partial Volume).

example:

${\displaystyle V_{x}=V_{tot}\times {\frac {p_{x}}{p_{tot}}}=V_{tot}\times {\frac {n_{x}}{n_{tot}}}}$  (1.3)

Enthalpy is the preferred expression of a System of measurement as energy changes in many chemical, biological, and physical ways, because it tends to simplify certain descriptions of energy transfer. This is because a change in Enthalpy takes account of energy transferred to the environment through the expansion of an observed System.

Enthalpy is a State Property: the Enthalpy of a System depends upon measurable properties(Completeness) of the system, but not on the incompleteness of the system.

Enthalpy is an extensive property: it depends on the amount of material. Thus 12 kg of copper at 80 °C will have 12 times the enthalpy of one kilogram of the same substance at the same temperature. However, we generally express enthalpy (more properly specific enthalpy) per unit mass. The unit of measurement for enthalpy in the International System of Units (SI) is the joule, but other historical, conventional units are still in use, such as the British thermal unit and the calorie. The specific enthalpy hence has units of J/kg, or BTU/lb. Enthalpy is theorized to be based entirely upon the volumetric structure of an atom. At such a scale, vacuum of course is present within an the System we would call an Atom, where Kinetic Energy, and other Laws of Physics will of course apply. At this Level of Existence (LOG), Radiation, and other forms of Electromagnetic Radiation, such as Heat, can either be contained or expelled based on the structure of an Atom, which is determined by the most simple equation, E=mv. The Rate of Mass to Volume, will determine the behavior of an atom, while the amount of Mass will determine the radioactivity (due to the ability of a mass to control objects in movement around it (Enthalpy)). If Equillibrium in an atom is lost, then we will observe a Heat Transfer from the internal structure of an Atom, to the External World (Our world). This is due to the Event Horizon of an Atom, and the amount of Total Heat within the Atom. As the Total Heat reaches Equillibrium, and past it (Meaning it spills out the atom quite literally), the Atom will change in Shape and Structure, as a function of the same.

The total enthalpy, H, of a system cannot be measured directly. Thus, Change in Enthalpy, ΔH, is a more useful quantity than its absolute value. ΔH of a system is equal to the sum total of all non-mechanical work done on it and the heat supplied to it. If a body passes from a thermodynamic state A to a thermodynamic state B at the same pressure as A, the heat transferred to the environment B (and thus it's pressure as well) is given by: ${\displaystyle Q=\Delta H=H_{end}-H_{start}}$  Coupled systems, where heat transfer produces changes in pressure or volume (and vice versa) will be treated later in the text.

Tables and graphs are available listing the specific enthalpy of many materials at various thermodynamic states.

For each table, a reference state is chosen. The given Enthalpy can be understood as the amount of energy which would have to be put into the system to raise it from a reference temperature (more precisely, a reference state). For water, a common reference state is 0 °C, atmospheric pressure, with all the water in the liquid phase. A more precise reference state will refer to the neutral state of an atom. Zero Kelvin, Zero Pressure, Equal amounts of Mass and Volume.

The standard state for copper is solid. The standard state for oxygen is gas. As a guide, the standard state is the phase the material would have at normal laboratory conditions of temperature and pressure. However, various engineering disciplines have their own conventions. This of course is changing as we are approaching the Interstellar Era, where most programs will begin to be conducted during Flight, where atmospheric pressures are less of an issue.

At 80 °C, water (at atmospheric pressure) has a specific enthalpy of 391.7 kJ/kg. Therefore one kilogram of liquid water at 80 °C would have an enthalpy of 391.7 kJ, and 7.3 kg would have an enthalpy of 7.3 x 391.7 = 2584 kJ.

There are two components to enthalpy, one due to the temperature, another to the phase. For example, from the above table, liquid water at 100 °C has a specific enthalpy of 419.1 kJ/kg, but steam at 100 °C has a specific enthalpy of 2675.4 kJ/kg – quite a lot more! The difference, 2257.9 kJ/kg is the enthalpy which has to be put into water to change it from a liquid to a vapour. This quantity is called the enthalpy of vaporization of water, or the latent heat of steam. “Latent” means hidden, because the steam is not hotter than the water, but has all this hidden internal energy which spills out from the inside of the atom to the outside. Thus determining the Event Horizon(or the point at which internal becomes external) of an atom would be based upon the Enthalpy of Vaporization as a function of space, possibly volume, while it's mass would be based upon the amount of movement of the whole system of the atom itself in space. Essentially how far an atom moves in space, versus it's change in state and the Enthalpy required to achieve such a physical reaction. This equation is difficult to reproduce however since particle wave behavior and it's conversion to number theory is still incorrect. Numbers would have to be understood in terms of Quantum Behavior before such an equation could be solved. (Note - The relationship between ration and irrational numbers where everything before the decimal point is rational and everything after is irrational meaning that rational numbers represent particles and irrational numbers represent waves, where the rational number in quantity exceeds the quality of the irrational numbers a state change could occur)

Note that this phenomenon is not restricted to boiling. Water at a temperature of 37 °C has a specific enthalpy of vaporization of 2414 kJ/kg: this heat is taken up if it evaporates at that temperature, which is why sweating cools you down. A heat exchange occurs similar to gases and liquids (refer to equation 1.3) where gases within a liquid are free to ionize into a gaseous state, after which an equilibrium is reached, then begin to return to a non-ionized and soluble state as a solid once more. With Enthalpy, the heat is transferred into the liquid, (presumably from a solid form of water, possible within a cell, specifically the nucleus) after which point it reaches its radioactive state(heat when outside of an atom), whereafter an equilibrium is reached, then begins to return back to its non-radioactive or internal state from an external environment.

Similarly, when ice at 0 °C melts to water at 0 °C it requires the input of heat to match its latent heat of melting, or enthalpy of crystallization.

Thus (according to this convention) liquid water at 0 °C has zero enthalpy. Ice at the same temperature has a negative enthalpy. Other tables may give enthalpy relative to Absolute Zero, 0.0 K, or to a laboratory temperature of 298 K. In terms of Radiation, Enthalpy would technically still be present within an atom until it again reaches a gaseous state after becoming a solid[meaning it was a plasma for a split second])This would only be possible if a perfect lattice of the atomic structure were to form, and would be dependent upon the surroundings of the material in question).

Chemical engineers sometimes use the ambient temperature (i.e. the temperature of the surroundings) as a reference condition – perhaps 10 °C in a cold country, 30 °C in a hot country. This means that all material stored or added to a system at ambient temperature has zero enthalpy and they only have to worry about things that are hotter or colder. This can simplify the energy balance. This is only useful on Earth of course. Any program of Mars would require a complete overhaul of this past system. Atoms since their natural state is always preserved despite the external world which we live in,(meaning essentially an atom's internal enthalpy and entropy remain constant despite outside influences [unless somehow it is radioactive]) it would be much more intelligent to use such a system as the new system of constants for measuring enthalpy, and other useful tidbits which will definitely be required in Space Flight and on Mars.

### Heat capacity or specific heat

If we have to heat something up (without a phase change), for example 12 kg of copper from 20 °C to 80 °C, the amount of enthalpy we have to put in depends on three things.

(1) The temperature difference to be achieved, in this case 60 K.

(2) The mass, in this case 12 kg.

(3) A property of the substance called specific heat capacity, which is a measure of how much energy is required to raise the temperature of 1 kg by 1 K.

So, we have: ${\displaystyle H=mc_{p}\Delta T}$

The subscript ${\displaystyle _{p}}$  remembers that the value of specific heat capacity is valid only if the transformation takes place at constant pressure. In practice, only for gases there's a relevant difference between constant-pressure specific heat and specific heat for other transformations (e.g. constant volume, polythropics...). Solids and liquids have only one value of specific heat capacity. For example, copper has a specific heat capacity of 0.383 kilojoules per kilogram per kelvin (0.383 kJ kg-1 K-1). Therefore we have to put in 0.383 x 12 x 60 = 276 kJ.

If on the other hand, we had to heat up 12 kg of water from 20 °C to 80 °C, we would use the specific heat capacity of water, 4.184 kJ kg-1 K-1, and our calculation would be: 4.184 x 12 x 60 = 3012 kJ.

Note that these terms tend to be used loosely. What is properly the specific heat capacity is often referred to as the specific heat or the heat capacity. If in doubt, look at the units. Technically the heat capacity refers to the whole body, the specific heat capacity to a mass – in the SI system one kilogram. In thermodynamic tables, data is sometimes given per mole or kilomole instead of per kilogram, especially for gases. You may also come across older data in which the obsolete unit the calorie (= 4.184 J) is used and the mass is one gram. Sorry, but you will have to convert. Always look at the units.

The definition of specific heat capacity allows to write, for a pressure-constant transformation: ${\displaystyle Q=\Delta H=mc_{p}\Delta T}$

Pressure-constant transformations in heat transfer problems, are often heat exchanges between a fluid and other fluids or solids, e.g. in a heat exchanger. In these cases, mass ${\displaystyle m}$  is not constant, because it's flowing. So, we have to refer not to mass, but to mass flow rate ${\displaystyle {\dot {m}}}$ , and not to energy ${\displaystyle Q}$  but to power ${\displaystyle {\dot {Q}}}$ . Hence:

${\displaystyle {\dot {Q}}={\dot {m}}c_{p}\Delta T}$

#### A simple calculation

Suppose 15 kg of cu at 80 °C is put into a bath of 25 kg of water at 20 °C, and there are no heat losses to the surrounding. What will be the final condition?

Answer Both the copper and the water will have the same temperature, somewhere between 20 °C and 80 °C. The total enthalpy will be unchanged.

Let us take the reference condition as 20 °C. Thus the water has zero enthalpy, and the copper has 15 x (80-20) x 0.383 = 344.7 kJ. This is the enthalpy of the system.

Now the total heat capacity of the system is (mass x specific heat capacity of copper) + (mass x specific heat capacity of water) = (15 x 0.383) + (25 x 4.184) = 5.75 + 104.6 = 110.4 kJ K-1

In other words, it would take 110.4 kJ of enthalpy to raise the temperature of the whole system by 1 K (= 1°C).

Therefore adding 344.7 kJ of enthalpy would raise the temperature of the system by 344.7 ÷ 110.4 = 3.1 K, so the final temperature would be 23.1 °C.

Looking at this a different way, we can see that the specific heat capacity of water is 4.184 ÷ 0.383 = 10.92 times greater. Thus 15 kg of copper has the heat capacity of only 15 ÷ 10.92 = 1.37 kg of water. Thus adding this amount of water to 25 kg would dilute the 60 K temperature difference as 60 x 1.37 ÷ 26.37 = 3.1 K.

### Heat Transfer Mechanisms

There are three modes of Heat Transfer: Conduction, Convection, and Radiation. Conduction is concerned with the transfer of thermal energy through a material without bulk motion of the material. This phenomenon is fundamentally a diffusion process that occurs at the microscopic level. Convection is concerned with the transfer of thermal energy in a moving fluid (liquid or gas). Convection is characterized by two physical principles, conduction (diffusion) and bulk fluid motion (advection). The bulk fluid motion can be caused by an external force, for example, a fan, or may be due to buoyancy effects. Finally, Radiation is the transfer of thermal energy through electro-magnetic waves (or photons). It is interesting to note that Radiation requires no medium.

#### Conduction

Conduction is the diffusion of thermal energy, i.e., the movement of thermal energy from regions of higher temperature to regions of lower temperature. On a microscopic level, this occurs due to the passing energy through molecular vibrations. The reason diffusion occurs is due to the nature of equilibrium. Heat contains energy and as it progresses towards equilibrium, especially in excess amounts, it then continues to proceed past equilibrium where it now considered to be a Joule. In other words, it is like throwing a ball into the ocean, it has excess energy, and as it it goes into the water the excess energy is delivered to the water, and as soon as all this energy has dissipated (or rather reached a specific quantity or heat you could say), then it again begins to rise to the surface of the water, again towards equilibrium (in this case the difference between the water and air).

Rate of Heat transfer is denoted as ${\displaystyle {\dot {Q}}}$ . The units of heat transfer rate are watts. It should be noted that heat transfer rate is a vector quantity. It is often convenient to describe heat transfer rate in terms of the geometry being studied. Thus we define ${\displaystyle {\dot {Q}}'}$ , ${\displaystyle {\dot {Q}}''}$ , and ${\displaystyle {\dot {Q}}'''}$  as the heat transfer rate per unit length, area (a.k.a. heat flux), and volume, respectively. It is useful to note that different conventions are often used with notation, and heat flux (heat transfer rate per unit area) is often denoted by ${\displaystyle {\dot {q}}}$ .

The governing rate equation for conduction is given by Fourier's Law. For one dimension, Fourier's law is expressed as:

${\displaystyle {\dot {q}}=-k{\frac {dT}{dx}}}$

or

${\displaystyle {\dot {Q}}=-kA{\frac {dT}{dx}}}$

Where x is the direction of interest, A is the cross-sectional area normal to x, k is a proportionality constant known as thermal conductivity and ${\displaystyle {\frac {dT}{dx}}}$  is the temperature gradient at the location of interest. The negative sign indicates that heat is transferred in the direction of decreasing temperature.

The thermal conductivity is a measure of how readily a material conducts heat. Materials with high conductivity, such as metals, will readily conduct heat even at low temperature gradients. Materials with low conductivity, such as asbestos, will resist heat transfer and are often referred to as insulators.

#### Convection

Convection is the transfer of thermal energy between a solid and a moving fluid(sometimes magnetism). If the fluid is not in motion (its Nusselt number is 1), the problem can be classified as Conduction. Convection is governed by two phenomena: the movement of energy due to molecular vibrations within an object and the large-scale motion of fluid particles. In general, Convection is of two types, Forced Convection and Free Convection.

Forced Convection occurs when a fluid is forced to flow. For example, a fan blowing air over a heat exchanger is an example of Forced Convection. In Free Convection, the bulk fluid motion is due to buoyancy effects. For example, a vertical heated plate surrounded by quiescent air causes the air surrounding it to be heated. Because hot air has a lower density than cold air, the hot air rises. The void is filled by cold air and the cycle continues.

In terms of Electromagnetic Convection of a Solid, it is the attempt of a solid to form a perfect lattice, thus we are able to understand why a solid melts under convection via a high voltage coil. Thus the convection of a gas can yield a liquid. Very useful for Chemical Engineers.

The governing rate equation for Convection is given by Newton's Law of Cooling:

${\displaystyle {\dot {Q}}=hA(T_{s}-T_{\infty }),}$

where ${\displaystyle h}$  is the heat transfer co-efficient, ${\displaystyle T_{s}}$  is the surface temperature of the solid, ${\displaystyle A}$  is the area and ${\displaystyle T_{\infty }}$  is the temperature of the fluid far from the surface. This expression, in spite of its name, is not law. Rather, it is an empirical expression of proportionality of the heat flux and the temperature difference between the solid and the fluid. The heat transfer coefficient is typically determined by experiment. Correlations for heat transfer coefficient for various kinds of flows have been determined and are documented in literature.

${\displaystyle {\dot {Q}}=A\epsilon \sigma T^{4}}$
An idealized material called a black body has an emissivity of 1. A is the surface area of the radiating object and sigma, σ, is known as the Stefan-Boltzmann constant ${\displaystyle 5.670\cdot 10^{-8}W/(m^{2}K^{4})}$