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Concepts of Heat and Mass Transfer

UNIT 1 CONCEPTS OF HEAT AND MASS TRANSFER

Structure

1.1 Introduction

Objectives

1.2 Heat Transfer in Engineering

1.2.1 Aims of Studying Heat Transfer 1.2.2 Applications of Heat Transfer

1.3 Modes of Heat Transfer 1.4 Mechanism of Heat Transfer

1.4.1 Conduction Heat Transfer 1.4.2 Convection Heat Transfer 1.4.3 Radiation Heat Transfer

1.5 Mass Transfer 1.6 Temperature Field

1.7 Relationship to Thermodynamics 1.8 Units and Dimension

1.9 Summary 1.10 Key Words 1.11 Answers to SAQs

1.1 INTRODUCTION

The study of transfer phenomena has been considered as a unified discipline of

fundamental importance. Such phenomena include wide range of transfer processes such as, heat, mass, momentum and energy. Usual practice to study these phenomena is on the basis of generalized fluxes and forces. All type of transfer processes involves a flux (or force) and its effect is called current. For example,

(a) Heat flows due to the temperature gradient from higher to lower temperature.

(b) Mass transfer occurs due to a concentration gradient from higher to lower concentration.

(c) Momentum transfer occurs due to a velocity gradient from higher to lower velocity.

(d) Electric current flows due to a potential gradient.

(e) Chemical process occurs due to the flux called chemical affinity.

All such processes mentioned above are spontaneous processes and terminates once the force is withdrawn or the same diminishes to zero. It is the law of nature, which states that driving force causes the respective flux from a higher to lower potential. The transfer process indicates the tendency of a system to proceed towards the equilibrium.

Interestingly, in all such processes, driving force is linearly proportional to the gradient.

Objectives

After studying this unit, you should be able to

 define conduction, convection and radiation,

 differentiate between heat and mass transfer,

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Introduction to Heat

and Mass Transfer  know the various types of heat exchangers, and

 understand the various applications of heat transfer.

1.2 HEAT TRANSFER IN ENGINEERING

1.2.1 Aims of Studying Heat Transfer

Heat transfer is the energy interaction due to a temperature difference in a medium or between media. Heat is not a storable quantity and is defined as energy in transit due to a temperature difference. The primary aims of studying the science of heat transfer are :

(a) to understand the mechanism of heat transfer processes, and (b) to predict the rate at which heat transfer takes place.

Whenever we refer to heat transfer we actually imply the heat transfer rate. It is this

“rate” that differentiates the field of heat transfer from thermodynamics.

1.2.2 Applications of Heat Transfer

The applications of heat transfer are diverse, both in nature and in industry. Climatic changes, formation of rain and snow, heating and cooling of the earth’s surface, the origin of dew drops and fog, spreading of forest fires are some of the natural phenomena wherein heat transfer plays a dominant role. The existence of living beings is possible due to the existence of the sun.

The importance of heat transfer in industry, including medical applications, can be seen by focusing on the following classes of problems.

Designing of Energy Conversion Devices

One of the most important applications of heat transfer is in the designing of steam generators, turbine, internal combustion engines, gas turbine, jet and rocket propulsion, etc. Another important aspect is designing of waste heat recovery devices.

Heat Exchangers

There are various types of heat ex-changers with wide application in industries, such as heat wheel, shell and tube type heat ex-changers, plate type, plate-fin type, spiral type, stationary re-generator, metallic recuperators, run around coil, heat pipe, etc. (Figure 1.1). Sizing and rating problems of heat ex-changers require expertise in the subject.

(a) Ljungstorm Heat Exchanger Hot fluid

Cold fluid

Heating zone

Empty

Cooling zone

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7

Concepts of Heat and Mass Transfer

(b) Storage type of Heat Exchanger

(c) Ruthemule Heat Exchanger

Hot Air Outlet From Elevator

Ports

A

B

C Pressure

Control Connections

Cold Air Inlet

To Elevator Hot air out

Hot gas in Rotating hood

Stationary heat exchanger

Cooled out gas

Cool air in A

C D

B

In In

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8

Introduction to Heat and Mass Transfer

(d) Pebble Bed Heat Exchanger

(e) Plate type of Heat Exchanger

(f) Metallic Recuperator

(g) Run Around Coil Header

Cold fluid in Hot fluid

Cold fluid out

Hot fluid out

Cold air in

Hot air to process Waste gas

Flue Gas

tf, out

Cold air fa, in fa, out

Waste gas tf, in

Hot out

Cold in

x x

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Concepts of Heat and Mass Transfer (h) Spiral Heat Exchanger

Figure 1.1 : Different Type of Heat Exchangers

Industrial Processes

Designing of industrial furnaces, incinerators, autoclave, etc.

Thermal Insulations

In many of the heat transfer devices thermal insulators are used to reduce the heat loss from the devices. Typical examples are: Thermos flask, ice box, hot box, building walls, steam pipes, and cryogenics. In this classes, the maximum and minimum temperatures (Tmax and Tmin) experienced by a heat transfer medium are usually fixed. The main objective is to reduce the “heat loss” or “heat leak”. The thermal design involves judicious changes in the constitution of insulation (that is, its size, material, shape, structure, flow pattern) so that the heat transfer indeed decreases while Tmax and Tmin remain fixed.

Heat Transfer Enhancement (Augmentation)

The main application is in the design of heat exchangers where the total heat transfer rate (q) between the hot and cold fluid streams separated by solid surface is usually a prescribed quantity. The objective is to transfer q across a minimum temperature difference. This can be done by changing the flow patterns of the two streams with breaking of the boundary layer (causing more turbulence). There are three distinct methods for heat transfer enhancement, i.e.

(a) passive method (Figure 1.2(a)), (b) active method (Figure 1.2(b)), and

(c) combined method (i.e. combination of both the active and passive method).

Passive method does not require any external energy whereas active method involves external energy input to implement. Twisted tape, wire coil, indentation of flow surface, surface roughening, etc. are some of the passive methods of heat transfer enhancement. Picture of some twisted tapes are given in Figure 1.2(c).

Figure 1.2(a) : Twisted Tape Inserted in a Pipe for Heat Transfer Enhancement

H A Tape

Tube A Direction of

Flow

D D0

Section A-A

Direction of Flow

Tube

Reciprocating Plunger

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Introduction to Heat and Mass Transfer

Figure 1.2(b) : Active Method with a Reciprocating Plunger

Figure 1.2(c) : Picture of Some Twisted Tapes

Temperature Control

In many areas, overheating of a heat-generating body is not permissible. Examples of temperature control applications include cooling of electronic equipments such as personal computers and super computers, cooling of nuclear reactor cores, and the cooling of the outer surface of space vehicles during re-entry. Cooling of high heat flux surfaces such as electronic chips in a tightly packaged set of electronic circuits is quite challenging because of the size limitations. The temperature of the electronic chips cannot rise much above the ambient temperature, because high temperatures drastically reduce their performance. Another important application of temperature control is the film cooling of gas turbine blades by routing air through channels within the blades.

Bio-Heat Transfer

Heat transfer plays a very important role in living systems as it affects the

temperature and its spatial distribution in tissues. The primary role of temperature is the regulation of a plethors of rate processes that govern all aspects of the life process. These thermally driven rate processes define the difference between sickness and health, injury and successful therapy, comfort and pain, and accurate and limited physiological diagnosis. Typical applications of bio-heat transfer include human thermoregulation, thermal surgical procedures such as microwave, ultrasound, radio frequency and laser, cryo-preservation of living cells, and thermal burn injury.

Materials Processing

Recent years have seen surging interest among researchers to understand heat transfer aspects in various material processing systems such as solidification and melting, metal cutting, welding, rolling, extrusion, plastic and food processing, and laser cutting materials. This has led to improved designs of material processing systems.

Other areas of heat transfer applications are in power production, chemical and metallurgical industries, heating and air conditioning of buildings, design of internal combustion engines, design of electrical machinery, weather prediction and environmental pollution, oil exploration, drying, and processing of solid and liquid waste. The list is endless.

It is no wonder that J. B. Joseph Fourier, the father of the theory of heat diffusion, made this remark in 1824 : “Heat like gravity, penetrates every substance of the universe; its rays occupy all parts of space. The theory of heat will hereafter form one of the most important branches of general physics.”

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Concepts of Heat and Mass Transfer

Chemical Processes

Chemical processes involve heating and cooling of materials or fluid streams in a number of steps. For example, consider the process of manufacturing nitric acid by catalytic air oxidation of ammonia (Figure 1.3). Liquid is the raw material.

Figure 1.3 : Heat Transfer in a Chemical Process of Making Nitric Acid

It is a common practice to store ammonia at atmospheric pressure and at a

temperature of – 33oC (approx). A refrigeration unit is used to keep the liquid cool (in other words, to continuously remove the heat that enters the storage tank from the ambient air). The storage tank is properly insulated to substantially reduce the transfer of heat from the outside air to the liquid ammonia. Liquid ammonia from the tank is pumped to a vaporizer (5) where heat supplied by steam (or any other hot fluid as available) converts the liquid to a vapour.

Compressed air is used for the oxidation of ammonia. Air is usually compressed in a two stage centrifugal compressor (2) provided with a interstage cooling. When compressed, the temperature of the gas increases. Hot air leaving the first stage of the compressor is cooled in a heat exchanger by cold water. The cooled gas moves to the second stage of the compressor and exchange heat with the tail gas (4) from the absorption tower (11). Ammonia vapour is mixed with this compressed air in an appropriate proportion and fed to the reactor (6). The reaction products attain a temperature of 800oC (approx). The hot product stream, therefore, contains a huge amount of energy. This energy is recovered in a waste heat boiler, which is a part of the reactor unit, in which transfer of heat from hot gas to the boiling water occurs, and thus a large quantity of steam is generated. Even after the gas leaves the waste heat boiler, its temperature remains at about 275oC and has to be further reduced before the gas reaches the absorption tower (the gas containing the oxides of nitrogen is absorbed by water in a number of absorption towers in succession to produce nitric acid). Cooling to this gas from 275oC is carried out in a series of heat exchangers (7-10). In the first heat exchanger (7), heat transfer from the hot gas occurs to the tail gas leaving the final absorption tower. The heated tail gas is then fed to a turbine (3) that drives the air compressor. The second heat exchanger (8) acts as an economizer, and the third one (9) is used to heat boiler feed water. In this way, most of the heat energy is recovered. The gas flows through a series of absorption towers (11) after passing through a cooler condenser (10). Absorption of the gas by water gives nitric acid. The absorption process is exothermic. As a result, each absorption tower is required to be provided with a liquid cooler (i.e.

heat exchanger).

To Stack Water

Ammonia From the Storage Tank

Air 1

5

6

2 3

7 8 9 10

11

Product Nitric Acid 4

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Introduction to Heat

and Mass Transfer

1.3 MODES OF HEAT TRANSFER

Heat transfer or simply heat is the thermal energy in transit due to a temperature difference. Whenever there exists a temperature difference in a medium or between two media, heat transfer must occur. Different types of heat transfer processes may be classified in 3 (three) different modes :

(a) Conduction (b) Convection (c) Radiation

In reality, heat transfer in a system occurs in combination of these modes. However, analysis can be done for each mode of heat transfer separately. Figure 1.4 presents the 3 (three) modes of heat transfer.

Conduction through a Solid Convection from a Surface Net Radiation Heat Exchange or a Stationary Fluid to a Moving Fluid between Two Surfaces

Figure 1.4 : Different Modes of Heat Transfer

When a temperature gradient exists in a stationary medium, which may be a solid or fluid, we use the term conduction to refer to the heat transfer that will occur across the medium. For example,

(a) A solid rod is insulated along its periphery and kept the two ends open to atmosphere. If heat is added to one end of the rod, you can feel that the other end is hot after sometime. It is a case of pure conduction.

(b) Spoon used to stir a hot cup of coffee got heated at the other end is an example of a heat conduction.

If the heat transfer occurs between a surface and a moving fluid (obviously with a temperature difference between the two), the mode of heat transfer is called convection.

For example,

(a) Water heated in a pan is an example of convection.

(b) Steam flowing from boiler to turbine is an example of convection.

The third mode of heat transfer is the radiation. All surfaces at finite temperature emit energy in the form of electromagnetic wave. Hence, in absence of an intervening medium, there is net heat transfer by radiation between two surfaces at different temperatures. Some examples of radiation are,

(a) Heat received by us from sun is a case of radiation. Similar is the case when one feels hot from a fire away from him.

(b) Outside body of the air plane gets heated if flying over high. It is basically radiation heat transfer from sun. In many situations, heat transfer occur as mixed mode, i.e. conduction, convection and radiation can be prevalent in reality.

T1 T1 > T2

T2

q”

Moving Fluid, T

Ty > T

q”

T, q”1

q”2

Surface, T1

Surface, T2

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Concepts of Heat and Mass Transfer

1.4 MECHANISM OF HEAT TRANSFER

1.4.1 Conduction Heat Transfer

Conduction refers to the transfer of heat between two bodies or two parts of the same body through molecules, which are, more or less, stationary, as in the case of solids.

The physical mechanism of conduction is most easily explained by considering a gas and using ideas similar to thermodynamics.

Consider a gas in which there exists a temperature gradient and assume that there is no bulk motion. The gas may occupy the space between the two surfaces that are

maintained at different temperatures as shown in Figure 1.5. We associate the

temperature at any point with the energy of gas molecule in proximity to the point. This energy is related to the random translational motion, as well as to the internal rotational and vibrational motions of molecules.

Figure 1.5 : Mechanism of Conduction Heat Transfer

Higher temperatures are associated with higher molecular energies, and when

neighbouring molecules collide, as they are constantly doing, a transfer of energy from the more energetic to the less energetic molecules must occur. In the presence of a temperature gradient, energy transfer by conduction must then be occur in the direction of decreasing temperature. This transfer is evident from Figure 1.5. The hypothetical plane at x0 is constantly being crossed by molecules from above and below due to their random motion. However, molecules from above are associated with a larger

temperature than those from below, in which case there must be a net transfer of energy in the positive x direction. We may speak of the net transfer of energy by random molecular motion as a diffusion of energy.

The situation is much the same in liquids, although molecules are more closely spaced and molecular interactions are stronger and more frequent. Similarly, in case of a solid, conduction may be attributed to atomic activity in the form of lattice vibrations. The modern view is to ascribe the energy transfer to lattice waves induced by atomic motion.

In a non-conductor, the energy transfer is exclusively via these lattice waves, in a conductor it is also due to the translational motion of free electrons.

It is possible to quantify heat transfer processes in terms of appropriate rate equations.

These equations may be used to compute the amount of energy being transferred per unit time. For heat conduction, the rate equation is known as Fourier’s law (J. B. J. Fourier, French scientist, 1822). For the one dimensional or uni-directional plane wall

(Figure 1.6), having a temperature distribution T (x), the rate equation is expressed as :

q”x

T

T1

T(X) T2

T1 >T2

q”x

T0

T

x X0

q”x

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Introduction to Heat and Mass Transfer

Figure 1.6 : One Dimensional Conduction Heat Transfer

 

x dT

q dx . . . (1.1)

or, x = dT

q k

dx . . . (1.2)

where qxis the rate of heat flux (a vector) in W/m2, dT

dx is the temperature gradient in the direction of heat flow x and k is the constant of proportionality, which is a property of the material through which heat propagates. This property of the material is called thermal conductivity (W/MK). The negative sign is used because heat flows from a high to low temperature and the slope dT

dx is negative.

SAQ 1

(a) What are the various modes of heat transfer? Explain their differences.

(b) State the Fourier’s law of heat conduction.

Example 1.1

The wall of an industrial furnace is constructed from 0.20 m thick fireclay brick having a thermal conductivity of 1.7 W/m.K. Measurements made during steady state operation reveal temperatures of 1500 K and 1020 K at the inner and the outer surfaces, respectively. What is the rate of heat loss through a wall that is 0.6 m by 1.2 m on a side?

Solution

Known : steady state conditions with prescribed wall thickness, area, thermal conductivity and surface temperatures.

Find : Wall heat loss.

Schematic :

Assumptions :

(a) Steady state conditions.

(b) One dimensional conduction through the wall.

(c) Constant thermal conductivity.

T1 T2

k

x

q”x

w

H

qx

Wall Area, A x L

L

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Concepts of Heat and Mass Transfer

Analysis :

Since heat transfer through the wall is by conduction, the heat flux may be determined from Fourier’s law of heat conduction. Using Eq. (1.2), we have

(1500 1020) 480 2

1.7 W/m.K 1.7 4080 W/m

0.2 0.2

 

      

x

T K

q k

L m

The heat flux represents the rate of heat transfer through a section of unit area, and it is uniform (invariant) across the surface of the wall. The heat loss through the wall of area A = H  W is then

( ) (0.6 1.2) 4080 2937.6 W

x x

q  H W q   

Example 1.2

The heat flow rate through a wood board with a 5 cm thickness is 500 W/m2. Temperature difference along the direction of flow of heat between the faces of the wood board is 55oC. Calculate the thermal conductivity of the wood.

Solution

We know that conduction heat transfer across the wood board is given by

  

x T

q k L

Hence, .

  qx L

k T

Here, T = 55oC, L = 5 cm and qx = 500 W/m2 Hence, thermal conductivity of the wood is

2 o

500 W/m 0.05 m

0.45 W/m. C 55 C

o 

k

1.4.2 Convection Heat Transfer

The convection heat transfer mode is comprised of two mechanisms. In addition to energy transfer due to random molecular motion or diffusion, energy is also transferred in bulk or macroscopic, motion of the fluid. This fluid motion is associated with the fact that, at any instant, large numbers of molecules are moving collectively or as aggregates.

Such motion, in the presence of a temperature gradient, contributes to the heat transfer.

Because the molecules in the aggregate retain their random motion, the total heat transfer is then due to a superposition of energy transport by random motion of the molecules and by bulk motion of the fluid. It is customary to use the term convection when referring to the cumulative transport and the term advection when referring to transport due to bulk fluid motion.

We are specially interested in convection heat transfer, which occurs between a fluid in motion and a bounding surface when the two are at different temperatures. Consider fluid flow over the heated surface (Figure 1.7).

Fluid y u

Velocity Distribution u(y)

u(y) Heated

Surface T(y)

Temperature Distribution T(y) y T

q” Ts

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Introduction to Heat and Mass Transfer

Figure 1.7 : Development of Boundary Layer in Convection Heat Transfer

Consequence of the fluid surface interaction is the development of a region in the fluid through which the velocity varies from zero at the surface to a finite value u associated with the flow. This region of fluid is known as the hydrodynamic or velocity boundary layer. Moreover, if the surface and flow temperatures differ, there will be a region of the fluid through which the temperature varies from Ts at y = 0 to T in the outer flow. This layer is called the thermal boundary layer, may be smaller, larger or of the same size as that through which the velocity varies. In any case, if Ts > T, convection heat transfer will occur between the surface and the outer flow.

The convection heat transfer mode is sustained both by random molecular motion and by the bulk motion of fluid within the boundary layer. The contribution due to random molecular motion (diffusion) dominates near the surface where the fluid velocity is low.

In fact, at the interface between the surface and the fluid (y = 0), the fluid velocity is zero and heat is transferred by this mechanism only. The contribution of bulk fluid motion originates from the fact that the boundary layer grows as the flow progresses in the x-direction. In effect, the heat that is conducted into the layer is swept downstream and is eventually transferred to the fluid outside the boundary layer. Appreciation of the boundary layer phenomena is essential to understanding convection heat transfer. It is for this reason that the discipline of fluid mechanics will play a vital role in our later

analysis.

Convection heat transfer may be classified according to the nature of the flow. We speak of force convection when the flow is caused by external means, such as by a fan, a pump, or atmospheric winds. As an example, consider the use of a fan to provide forced

convection air-cooling of hot electrical components on a stack of printed circuit boards (Figure 1.8). In contrast, for free (or natural) convection the flow is induced by buoyancy forces, which arises from density differences caused by temperature variations in the fluid. An example is the free convection heat transfer that occurs from hot components on a vertical array of circuit boards in still air (Figure 1.8(b)). Air that makes contact with the components experiences an increase in temperature and hence a reduction in density. Since, it is now lighter than the surrounding air, buoyancy forces induce a vertical motion for which warm air ascending from the boards is replaced by an inflow of cooled ambient air.

(a) (b)

Air Forced

Flow

Hot Components On Printed Circuit Boards

Air

q”

q”

Buoyancy-driven Flow

Vapor Bubbles

Hot Plate Water

Water Droplets Moist air

Cold Water

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17

Concepts of Heat and Mass Transfer

(c) (d)

Figure 1.8 : Convection Heat Transfer Processes (a) Forced Convection and (b) Natural Convection

While we have presumed pure forced convection in Figure 1.8(a) and pure natural convection in Figures 1.8(b)-(d), conditions corresponding to mixed convection, i.e.

combined forced and natural convection may exist.

For example, if the velocities associated with the flow in Figure 1.8(b) are small and/or buoyancy forces are large, a secondary flow comparable to the imposed forced flow could be imposed. The buoyancy induced flow will be normal to the forced flow and could have a significant effect on convection heat transfer from the components. In Figure 1.8(b) mixed convection would result if a fan were used to force air upward, there by opposing the buoyancy force.

We have described the convection heat transfer mode as energy transfer occurring within fluid due to the combined effects of conduction and bulk fluid motion. Typically, the energy that is being transferred is the sensible, or internal thermal energy of the fluid.

However, there are convection processes for which there is, in addition, latent heat exchange. This latent heat exchange is due to the phase change between the liquid and vapour state of the substance. These cases are dealt in Unit 15.

Regardless of the particular nature of convection heat transfer process, the appropriate rate equation is of the form

( )

  s

q h T T . . . (1.3)

where q = Convective heat flux (W/m2), Ts = Surface temperature (K), T = Fluid temperature (K), and

h = Convective heat transfer coefficient (W/m2.K) Eq. (1.3) is also known as the Newton’s law of cooling.

Example 1.3

A hot plate of area 0.5 m2 is maintained at a temperature of 60oC by a 100 W electric heater when room temperature is 30oC. The appropriate convection coefficient is 2.15 (T)1/3 W/m2K. What fraction of the heat supplied is lost by natural convection? What happens to the rest of the heat supplied?

Solution

The convective heat transfer coefficient

1 1 1

3 3 3

2.15 2.15 (60 30) 2.15 30 6.605

       

h T

0.5 30 6.605 0.5 30 99.075

        

Q h A T h

Fraction of supplied heat lost by convection is 0.99075 99.075%

100Q  

The remaining 0.00925 or 0.925%is lost by radiation.

Example 1.4

Air at 30oC flows over a hot plate (50 cm  75 cm) maintained at 200oC with the help of an electric heater. The convection coefficient is 20 W/m2. Calculate the heat transfer.

Solution

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Introduction to Heat

and Mass Transfer From Newton’s law of cooling

( )

w

q hA T T

20 (0.5 0.75) (200 30)    = 1275 W

SAQ 2

(a) What is convection heat transfer? Why is it regarded as a mode of heat transfer?

(b) Give two examples each of natural and forced circulation.

1.4.3 Radiation Heat Transfer

Consider a solid initially at a high temperature Ts than that of its surroundings Tsur as shown in Figure 1.9. The entire medium between the solid and the surrounding is vacuum. Consideration vacuum precludes other modes of heat transfer (i.e. conduction and convection) between the solid and the surroundings. Our intuition tells us that the solid will cool and eventually achieve thermal equilibrium with its surroundings. This cooling of solid is associated with a reduction in the internal energy stored by the solid and is a direct consequence of the emission of thermal radiation from the surface. In turn, the surface of the solid will intercept and absorb radiation originating from the surroundings. If, Ts > Tsur, the net heat transfer by radiation qrad, net is from the solid surface and the same will cool until Ts reaches Tsur.

(a) (b)

Figure 1.9 : Radiative Heat Transfer

Radiation emitted by the surface originates from the thermal energy of matter bounded by the surface, and the rate at which energy is released per unit area (W/m2) is termed the surface emissive power E. There is an upper limit to the emissive power, which is prescribed by the Stefan-Boltzmann law

b s4

E  T . . . (1.4)

where Ts = absolute temperature of the surface (K), and  = Stefan-Boltzmann constant = 5.67  10– 8 W/m2K4.

In the Eq. (1.4) we have assumed the surface of the solid to be a perfect emitter. Details regarding other type of emitters are discussed in appropriate units.

Now, heat transfer between the surface and surrounding is given by

4 4

( )

  

s sur S sur

E T T . . . (1.5)

We associate thermal radiation with the rate at which energy is emitted by matter as a result of its finite temperature. At this moment thermal radiation is being emitted by all the matters that surrounds you, by the furnitures, walls of the room, etc. if you are

Gas T, h

q”con

Surface of Emissivity

, Absorptivity , and Temperature T,

Surroundings At Tsur

Surface of Emissivity

,=  Area A, and Temperature T,

Gas T, h

q”con

q”rad

G E

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19

Concepts of Heat and Mass Transfer

indoor, or by the ground, buildings, atmosphere and sun, if you are outdoors. The mechanism of emission is related to the energy released as a result of oscillations or transitions of the many electrons that constitute the matter. These oscillations are again sustained by internal energy, and therefore the temperature of the matter. Hence we associate the emission of thermal radiation with thermally excited conditions within the matter.

We know that radiation originates due to emission by matter and that its subsequent transport does not require the presence of any matter. At this point we should know the nature of transport of radiation. One theory views radiation as the propagation of a collection of particles termed photon or quanta. Alternatively, radiation may be viewed as the propagation of electromagnetic waves. In any case we wish to attribute to radiation the standard wave properties of frequency  and the wavelength . For radiation propagating in a particular medium, the two properties are related by

  c

 . . . (1.6)

where c is the speed of light in the medium. For propagation in a vacuum,

c0 = 2.998  108 m/s. The unit of wavelength is commonly the micrometer (m), where 1 m = 10– 6 m.

Example 1.5

The quantity of radiation received by earth from the sun is 1.4 kW/m2 (solar constant). Assuming that sun is an ideal radiator, calculate the surface temperature of the sun. The ratio of the radius of earth’s orbit to the radius of the sun is 216.

Solution

Total radiation from the sun 1.4 4 2

Qr   R

where Rradius of the earth’s orbit.

Total radiation emitted by the sun

2 4

r 4

Q    r T where r = radius of the sun, and

T = surface temperature of the sun.

Now,   4 r2T41.4 4 R2 or,

2 3

4 2 16 4

8

1.4 1.4 10 (216) 0.1152 10

5.67 10

T R K

r

  

         

Hence, T = 5826 K.

SAQ 3

(a) What is the mode of heat transfer in vacuum?

(b) State the Stefan-Boltzmann law of radiation.

1.5 MASS TRANSFER

Diffusional mass transfer occurs at a microscopic or molecular level which deals with the transport of one constituent of a fluid solution or gas mixture from a region of higher concentration to a region of lower concentration. Heat is transferred in a direction which

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Introduction to Heat

and Mass Transfer reduces an existing temperature gradient, and mass is transferred in a direction which reduces an existing concentration gradient.

The rate of molecular diffusion is proportional to the concentration gradient

A A

N dC

A  dy . . . (1.7)

where NA

A  diffusion rate per unit area (kg mol/m2s) of the diffusion species A, dCA

dy  concentration gradient (kg mol/m4) in the direction of diffusion Y, and CA = concentration of species A in kg mol/m3.

Now Eq. (1.7) can be written as

A = A

N dC

A D dy . . . (1.8)

where D constant of proportionality, called the diffusivity (m2/s).

Eq. (1.8) is called the Fick’s law of diffuision, which is similar to the Fourier’s law of heat conduction.

Whenever there is concentration gradient there will be mass transfer, till the concentration of the particular constituent becomes uniform.

When mass transfer occurs in a fluid at rest, the mass is transferred by purely molecular diffusion resulting from concentration gradient. The process is similar to heat diffusion resulting from temperature gradient.

When the fluid is in motion, mass transfer takes place by both molecular diffusion and convective motion of the bulk fluid. In such a case knowledge of velocity field is needed to solve the mass transfer problem. It may be noted that bulk velocities are insignificant in case of molecular diffusion. However, convective mass transfer involves bulk motion of the fluid. The convective mass transfer is given by the relation

( )

A M w

m h C C

A  

 . . . (1.9)

where Cw and C are the concentrations of the species A at the wall and free stream; hM

is the convective mass transfer coefficient (ms– 1).

For low concentration of the mass in the fluid and low mass transfer rates, the convective heat and mass transfer processes are analogous and many of the results derived in connection with convective heat transfer are applicable to convective mass transfer.

Therefore, mass transfer equations discussed in Units 3, 11-13 are obtained by analogy directly from the corresponding heat transfer equations. However, under high mass flux condition and with chemical reactions there are significant differences between the heat and mass transfer processes. Such high level situations are beyond the scope of this course.

Mass transfer processes occur in variety of applications in mechanical, chemical and aerospace engineering, physics, chemistry and biology. Typical examples are :

(a) Transpiration cooling of rocket motors and jet engines.

(b) Ablative cooling of space vehicles during reentry into the atmosphere.

(c) Mass transfer from laminar and turbulent streams onto the surface of a conduit.

(d) Evaporation and condensation on the surface of a tube or plate.

(e) Processes such as absorption, desorption, distillation, solvent extraction, drying, humidification, sublimation, etc.

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Concepts of Heat and Mass Transfer

(f) Biological applications – oxygenation of blood, food and drug assimilation, respiration, etc.

Example 1.6

Gaseous hydrogen is stored at elevated pressure in a rectangular steel container of 10 mm wall thickness. The molar concentration of hydrogen in steel at the outer surface is 2 kg mol/m3, while the concentration of hydrogen in steel at the outer surface is 0.5 kg mol/m3. The binary diffusion coefficient for hydrogen in steel is 0.26  10– 12 m2/s. What is the mass flux of hydrogen through the steel?

Solution

Assumptions

(a) Steady state, one-dimensional diffusion of hydrogen.

(b) No chemical reaction involved.

(c) Molar concentration of hydrogen is much less than that of steel (CA < < CB). So C = CA + CB is uniform

Now, NA AB CA2 CA1

A D L

  

or, 12 (2 0.5) m (kg mol/m )2 3 10 2

0.26 10 . 0.39 10 kg mol/m .s

0.01 s m

NA

A

    

Mass flux of hydrogen = NA 0.39 10 10 2 7.8 10 11 kg/m .s2 A M

     

SAQ 4

(a) State and explain the Fick’s law of diffusion.

(b) What is mass diffusivity? What is its dimension?

(c) How does diffusional mass transfer differ from convection mass transfer?

1.6 TEMPERATURE FIELD

As mentioned in the previous section, heat transfer occurs due to the temperature difference or a temperature gradient. As long as the temperature difference exists between two locations, heat will always transfer from higher to the lower temperature. A majority heat transfer problem involves calculation of temperature at a location in the domain under consideration provided the heat flux and geometry are known. A temperature field in the domain of study can indicate the direction of heat flow.

In some cases, boundary condition may appear as temperature boundary condition. In some cases, temperature gradient may be used as a boundary condition. In a domain, loci of all the points having same temperature is known as isothermal line. No heat can flow along the isothermal line. However, heat will always flow from one isotherm to the another (both isotherms being at two different temperature).

In the heat transfer study, temperature may be unsteady or steady. Unsteady temperature involves variation of temperature with time. For example, if we heat one end of a metallic rod and hold the other end with hand, we can feel the variation in warmth with time. This variation will continue till the heat transfer attains a steady state (invariant with time). Transient temperature being time variant, it can be plotted for each location with time as a variable. In case of steady state temperature, plot of temperature can be done with spatial variables (x, y and z).

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Introduction to Heat

and Mass Transfer

1.7 RELATIONSHIP TO THERMODYNAMICS

Thermodynamics essentially deals with systems in equilibrium state to another. But, thermodynamics may not be used to calculate the rate at which this change takes place, since the system is not in equilibrium during the process. Heat transfer utilizes the first and second laws of thermodynamics in addition to the rate laws such as Fourier’s law of heat conduction, Newton’s law of cooling and the Stefan-Boltzmann law of radiation.

To show the difference between heat transfer and thermodynamics, let us take a practical application such as quenching of a hot steel bar in an oil bath. Thermodynamics will help us calculate the final equilibrium temperature of the steel bar-oil combination.

However, it will not be helpful in predicting the time that will be taken by the steel bar-oil combination to reach the steady state or the rate at which the temperatures of the steel bar and oil will change with time. On the other hand, heat transfer will help us in finding out the time-temperature history of both the bar and oil.

Thermodynamics is concerned with interaction of heat and work at the boundary of the system and is concerned with the equilibrium states of the matter. Equilibrium state necessarily precludes the existence of a temperature gradient. Thermodynamics is governed by the laws of thermodynamics. Amount of heat and/or work and directions of flow during a process can be estimated by using the laws of thermodynamics.

Heat transfer is inherently a non-equilibrium process and it occurs only because of a temperature gradient. Heat transfer quantifies the rate at which the same occurs in terms of degree of thermal non-equilibrium. This is done through the rate equations for the three modes as discussed by Eqs. (1.2, 1.3, 1.4, 1.8 and 1.9).

1.8 UNITS AND DIMENSIONS

The physical quantities of heat transfer are specified in terms of dimensions, which are measured in terms of units. Four basic dimensions are required for the development of heat transfer; they are length (L), mass (M), time (t) and temperature (T). All other physical quantities of interest may be related to these four basic dimensions. In this course we will be using SI system of units throughout.

Table 1.1 : Dimensions and Units in SI System

Dimension Unit

Length (L) meter (m)

Mass (M) Kilogram (kg)

Time (t) second (s)

Temperature (T) kelvin (K)

SI base and supplementary units are given in Table 1.2

Table 1.2 : Supplementary Dimensions and Units in SI System

Dimension Unit

Length (L) meter (m)

Mass (M) kilogram (kg)

Concentration (C) mole (mol)

Time (t) second (s)

Electric current (I) ampere (A) Thermodynamic temperature (T) kelvin (K)

Plane angle () radian (rad)

Solid angle () steradian (sr)

Luminous intensity Candela (d)

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Concepts of Heat and Mass Transfer

With regard to these units note that 1 mol is the amount of substance that has as many atoms or molecules as there are atoms in 12 g of carbon-12 (12C); this is the

gramsmole (mol). Although mole has been recommended as the unit quantity of matter for the SI system, it is more consistent to work with kilogram – mol (kmol, kg-mol).

One kmol is simply the amount of substance that has as many atoms or molecules as there are atoms in 12 kg of 12C. As long as the use is consistent within a given problem, no difficulties arise in using either mol or kilogram-mole. The molecular weight of a substance is the mass associated with a mole or a kilogram-mole. For oxygen, as an example, the molecular weight M is 16 g/mol or 16 kg/kmol.

Although the SI unit of temperature is the Kelvin, use of the Celsius temperature scale remains widespread. Zero on the Celsius scale (0oC) is equivalent to 273.15 K on the thermodynamic scale in which case

( ) ( C)o 273.15

T K T  . . . (1.10)

However, temperature differences are equivalent for the two scales and may be denoted as oC or K. Also, although the SI unit of time is the second, other units of time (minute, hour, and day) are so common that their use with the SI system is generally accepted.

The SI units comprise a coherent form of the metric system. That is, all remaining units may be derived from the base units using formulas that do not involve any numerical factors. Derived units for selected quantities are given in Table 1.3.

Table 1.3 : Derived Units Under SI System

Dimension Name and Symbol Formula SI Base Unit

Force Newton (N) m.kg/s2 m.kg/s2

Pressure/stress Pascal (Pa) N/m2 kg/m.s2

Energy Joule (J) N.m m2.kg/s2

Power Watt (W) J/s m2.kg/s3

Note that force is measured in newtons, where a 1-N force will accelerate a 1-kg mass at 1 m/s2. Hence, 1 N = 1 kg.m/s2. The unit of pressure (N/m2) is often referred to as the pascal. In the SI system there is one unit of energy (thermal, mechanical, or electrical), called the joule (J), and 1 J = 1 N.m. The unit for energy rate, or power is then J/s, where one joule per second is equivalent to one watt (1 J/s = 1 W). Since it is frequently required to work with extremely large or small numbers, a set of standard prefixes has been introduced to simplify matters (Table 1.4). For example, 1 megawatt (MW) = 106 W and 1 micrometer (m) = 10– 6 m.

Table 1.4 : Multiplying Prefixes Prefix Abbreviation Multiplier

pico p 10– 12

nano n 10– 9

micro  10– 6

milli m 10– 3

centi c 10– 2

hecto h 102

kilo k 103

mega M 106

giga G 109

tera T 1012

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Introduction to Heat

and Mass Transfer SAQ 5

(a) Describe the relationship between heat transfer and thermodynamics.

(b) What are the basic units? How do you obtain force, pressure, energy and power from the basic units?

Exercise 1.1

(a) A temperature difference of 510oC is maintained across a fireclay brick 25 cm thick with thermal conductivity 3.5 W/moC. Determine heat transfer rate per square meter area.

(b) A brick wall 20 cm thick with thermal conductivity 2.45 W/moC is maintained at 32oC on one face and 220oC at the other face. Determine the heat transfer rate across 4.2 m2 surface area of the wall.

(c) The inside and outside surface temperatures of a window glass are 22oC and – 15oC, respectively. If the dimension of the glass is 82 cm  43 cm and the thickness of the glass is 1.75 cm, determine the heat loss through the glass over 4 hours. Assume the thermal conductivity of the glass

kglass = 0.778 W/moC.

(d) 250 W is transferred by conduction heat transfer through a 0.58 m2 section of a 4.2 cm thick insulating material. Determine the temperature difference across the insulating layer if the thermal conductivity is 0.12 W/moC.

Exercise 1.2

(a) Water at mean temperature of 22oC flows over a flat plate maintained at 84oC. Determine the heat transfer per square meter of the plate over 5 hours. Consider heat transfer coefficient to be 220 W/(m2.oC).

(b) Heat is supplied to a plate from its back surface at a rate of 1100 W/m2 and is removed from its front surface by air flow at 27oC. If the heat transfer coefficient between the air and the plate surface is 28 W/(m2.oC). What is the temperature of the front surface of the plate?

(c) The inside surface of an insulating layer is at 266oC, and the outside surface is dissipating heat by convection into air at 33oC. The insulating layer is 12 cm thick and has thermal conductivity k = 0.22 W/m.oC. What is the minimum value of the heat transfer coefficient at the outside surface if the outside surface temperature should not exceed 55oC?

(d) A thin metallic plate is insulated at the back surface and is exposed to the sun at the front surface. The front surface absorbs the solar radiation of 1200 W/m2 and dissipates it mainly by convection to the ambient air at 25oC. If the heat transfer coefficient between the plate and the air is 25 W/m2.oC, what is the temperature of the plate?

Exercise 1.3

(a) A thin metal plate 0.15 m by 0.1 m is placed in a large evacuated container whose walls are kept at 320 K. The bottom surface of the plate is insulated, and the top surface is maintained at 550 K as a result of electrical heating. If the emissivity of the surface of the plate is  = 0.75, what is the rate of heat exchange between the plate and the walls of the container?

Take  5.67 10 8 W/(m .K )2 4 .

(b) Two very large, perfectly black parallel plates, one maintained at 1250 K and the other at 650 K, exchange heat by radiation. Determine the heat transfer rate per square meter of the surface.

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Concepts of Heat and Mass Transfer

(c) One surface of a thin plate is exposed to a uniform flux of 600 W/m2, and the other side dissipates heat by radiation to an environment at – 8oC.

Determine the temperature of the plate assuming blackbody conditions for radiation.

(d) A pure gas is stored at elevated pressure in a rectangular stainless steel container of 20 mm wall thickness. The molar concentration of the gas in stainless steel at the outer surface is 4.5 kg mol/m3, while the concentration of the gas in stainless steel at the outer surface is 1.5 kg mol/m3. The binary diffusion coefficient for the gas in stainless steel is 0.88  10–10 m2/s. What is the mass flux of the gas through the steel?

1.9 SUMMARY

It is understood that heat and mass transfer are transport processes and they occur due to the presence of finite gradients (temperature gradient for heat transfer and concentration gradient for the mass transfer). In the present unit, importance of heat transfer is

discussed and application to many practical devices like heat exchangers are cited. A short description is provided about various modes of heat transfer. Basics of mass transfer is highlighted. Further, relationship between the heat transfer and

thermodynamics is discussed. The idea of units and dimensions are incorporated which will help in problem solving. Theory is supported with solved problems for better understanding of the unit. At the end, some unsolved problems are given which will help you to understand the topic.

1.10 KEY WORDS

Heat Transfer : Heat transfer is the process of transfering heat from a hot body to a cold body.

Conduction : It is the mode of heat transfer due to molecular diffusion.

Convection : It is the mode of heat transfer due to the motion of fluid molecules.

Radiation : It is the mode of heat transfer due to the propagation electromagnatic wave or photon.

Mass Transfer : Mass transfer is the process of transfering mass through a concentration gradient.

1.11 ANSWER TO SAQS

Refer the preceding text for all the Answers to SAQs.

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Introduction to Heat and Mass Transfer

References

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