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 because of a temperature difference. 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 moving enthalpy from a cold body to a hotter one, we will have to do extra work, as in the case of a refrigerator. This invariably involves some other process, such as mechanical work, and cooling by expansion of gases, but within the overall activity heat transfer always goes from the hotter 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.

Temperature

Temperature is an intensive property: that is it does not depend on the amount of substance. Thus one kilogram of copper at 80 °C and 12 kg of copper at 80 °C both have the same temperature. 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.

Enthalpy

Enthalpy is an extensive property: it depends on the amount of substance. 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, to define the enthalpy of something we have to agree on reference conditions. Data is commonly given for a fixed mass (usually 1 kg, on older books perhaps 1 lb or 1 gram) of a substance (properly called the specific enthalpy) at various temperatures. For example, tables are available for the specific enthalpy of water. Thus at 80 °C, liquid water 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.

However, note that these tables are relative. In this case, it is the enthalpy which would have to be put into the same amount of water to raise it from a reference temperature and a standard state. In this case the reference temperature is 0 °C, and the standard state for water is liquid. 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.

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 energy to give up if it condenses.

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.

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.

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.

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. 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.

A simple calculation

Suppose 15 kg of copper 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.

Rate of Heat transfer is denoted as $\dot{Q}$. 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 ${\dot{Q}}^{\prime }$, ${\dot{Q}}^{″}$, and ${\dot{Q}}^{‴}$ 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 $\dot{q}$.

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

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

or

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

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 $\frac{dT}{dx}$ 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. If the fluid is not in motion (its Nusselt number is 1), the problem can be classified as Conduction. Convection is governed by two phenomenon. The movement of energy due to molecular vibrations and 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.

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

$\dot{Q}=hA(T_s-T_{\mathrm{\infty }}),$

where $h$ is the heat transfer coefficient, $T_s$ is the surface temperature of the solid, and $T_{\mathrm{\infty }}$ 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.

Radiation

Radiation is the transfer of thermal energy between two objects through electro-magnetic waves. Unlike conduction and convection, radiation does not require a medium. In general, gasses do not take part in radiation heat transfer.

Radiation is based on the fact that all objects of finite temperature, i.e. not absolute zero, emit radiation in the form of electro-magnetic waves. These waves travel until they impinge another object. The second object in turn either absorbs, reflects, or transmits the energy. It should be noted that if the second object is of a finite temperature, it is also emitting radiation.

A basic fact of radiation is that the heat of radiation is proportional to the fourth power of the temperature of the radiating source. The heat loss is related to the emissivity ε of the material by the equation:

$\dot{Q}=Aϵ\sigma T^4$

An idealized material called a black body has an emissivity of 1. A is the surface area of the radiating object and sigma is a constant $5.670x10^{-8}W/(m^2{K}^4)$