11. Experimental Heat Transfer = - Heat and Mass Transfer, 2nd Edition

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11 Experimental Heat Trans= fer

= 11.1 INTRODUCTION

Experiments in heat transfer = are used for the validation of equipment and process design. Estimation of = heat transfer rate, heat transfer coefficients, and thermal conductivity of= materials are the main objectives of laboratory experiments. Many times sm= aller models of the actual equipments are built and studied in the laborato= ry for their heat transfer performance. The variables to study a heat trans= fer process are measured using suitable instrumentation and compared with t= he desired set values. If there is a difference between them, then it may b= e corrected either manually or with the help of an automatic controller as = shown in Fig. 11.1

Figure 11.1 Control System

For accurate maintenance of = the set values of variables, automatic control systems are preferred in lab= oratory experiments. To measure temperature, pressure, flow, conductivity, = and emissivity, measuring instruments with best level of accuracy are requi= red. Though mechanical and pneumatic instruments are simple to maintain, hi= ghly reliable and independent of external power, electrical, and electronic= devices provide higher accuracy and sensitivity. All instruments have to b= e calibrated before using them for measurements to estimate the error invol= ved in the measured data. Digital devices are used f= or data measurement and storage. Softwares are available for data analysis = with the help of computers.

= 11.2 HEAT TRANSFER EXPERIMENTS

Experimental methods used for= measurement of thermal conductivity, heat transfer coefficient, heat flux,= and emissivity are discussed in this section.

11.2.1= Thermal Conductivity Measurement of Solids

Measurement of thermal conduc= tivity of solids and fluids using different methods is discussed in this se= ction.

11.2.1.1 Thermal conductivity me= asurement in a plate

Measurement of thermal conduc= tivity is based on Fourier's law of heat conduction. If in a thin plate of = material as shown in Fig. 11.2= heat transfer rate (Q), material thickness (=E2=88=86x),= and difference in temperature (T₁ =E2= =88=92 T₂) can be measured, then the th= ermal conductivity k of the plate is

Figure 11.2 Thermal Conductivity Measurement

Heat may be supplied from on= e side of the plate using an electric heater and may be removed from the ot= her side by a cooled plate. Temperature of the plate on each side can be me= asured using thermocouples or thermistors.

The limitation of this metho= d in determining the thermal conductivity of a plate is that heat escapes f= rom the edges of the plate. If the edges are covered with insulation, then = the two-dimensional temperature profile that results can cause an error. Th= is problem can be avoided by using side heaters.

11.2.1.2 Insulated apparatus for= thermal conductivity measurement

Another method for the measur= ement of thermal conductivity of metals is shown in Fig. 11.3. A metal rod of known thermal conductivity = is in contact with another metal whose thermal condu= ctivity is to be measured. Heat source and sink are placed to the ends of t= he composite metals. Both the metal plates are covered with an insulating m= aterial to minimize heat loss and to ensure one-dimensional heat flow throu= gh the metals. Thermocouples are used to measure temperatures at various lo= cations. The temperature of the known metal is measured and the heat flow i= s determined. This heat flow is then used to calculate the thermal conducti= vity of the unknown metal using Equati= on (11.1).

Figure 11.3 Thermal Conductivity Measurement of Metals

11.2.2= Thermal Conductivity Measurement of Liquids and Gases

11.2.2.1 Thermal conductivity me= asurement of liquids =E2=80=93 hot-plate method

The hot-plate method is used = for determining thermal conductivity of liquids as shown in Fig. 11.4. An annular arrangement is used for= the determination of liquid thermal conductivities and the thickness of th= e liquid layer must be thin enough to minimize thermal convection currents.= Thermal conductivity of the liquid is determined using Equation (11.1) by measuring heat flow, temperatures,= thickness, and area of cross-section.

Figure 11.4 Guarded Hot Plate for Measurement of Thermal Conduct= ivity of Liquids

11.2.2.2 Thermal conductivity me= asurement of gases

A concentric cylinder arrange= ment, as shown in Fig. 11.5, i= s used for the measurement of the thermal conductivity of gases such as oxy= gen, nitrogen, and other gases. The inner and outer cylinders are both cons= tructed of silver and the gap is used for filling the gas. The electrical h= eater serves as the heat source. A temperature difference of 5=E2=80=9310= =C2=B0C is maintained between the heat sink and the source. The heat transf= er rate is measured by determining the electric power input, while thermoco= uples installed on the surface of the heater and rec= eiver measures the temperature difference. Equation (11.1) is used to find out the thermal conductivity of th= e gas.

Figure 11.5 Thermal Conductivity Measuring System for Gases at H= igh Temperatures

11.2.3= Convection Heat Transfer Measurement

Free and forced convection he= at transfer experiments are discussed in this section.

11.2.3.1 Free convection heat tr= ansfer

Free convection heat transfer= coefficient on a horizontal rod is measured using the experimental set up = as shown in Fig. 11.6. The hor= izontal rod has three sections: the center section is the test element with= its surface chrome or nickel plated and the two end sections are guard hea= ters to prevent end losses from the center section. The rod is electrically= heated and thermocouples are used to measure the surface temperatures. The= current to the guard heaters is maintained in such a way that their temper= ature is equal to the temperature of the center section.

Figure 11.6 Free-Convection Heat-Transfer Coefficient Measuremen= t Setup

The heat supplied to the cen= ter section is dissipated to the surrounding air and the heat loss is

Q =3D hA(T_s =E2=88=92 T_a) + =CF= =83=CE=B5A (T_s⁴ =E2=88=92 T_a⁴)

where h is the free-= convection heat transfer coefficient, T_s<=
/em> is the test section surface temperature, =CE=B5 is the = surface emissivity, =CF=83 is the Stefan=E2=80=93Boltzman's consta= nt, T_a is the surrounding air temperatu= re, A is the surface area of the test section, and Q is t= he electric power input to the test section. To find the value of h, other factors should be measured.

11.2.3.2 Forced convection heat = transfer
Figure 11.7 show the experimental setup used to find the fo= rced convection heat transfer coefficient. An electric heater is used on th= e tube which is made up of stainless steel. Thermocouples are welded on the= surface of the tube to measure the temperatures. Water temperature at inle= t and outlet of the heated section are measured usin= g thermometers. Electric power supplied to the tube can be measured using a= wattmeter, which is considered as the heat input. A venturi or orifice met= er can be used to measure the water flow rate. The heat transfer coefficien= t is calculated using

Figure 11.7 Forced Convection Heat Transfer Coefficients Measure= ment

Q =3D hA(T_=
s =E2=88=92 T_f)

where A is the inner= surface area of the tube, T_s and T= _f are the wall surface and bulk fluid tempe= ratures, and Q is the heat transfer rate. The outer surface of the= tube is insulated to ensure that all the electric energy is dissipated in = to the water as heat.

11.2.4= Heat Flux Measurement

Heat flux measurements using = different heat flux meters are discussed in this section.

11.2.4.1 Slug-type heat flux met= er

Direct measurement of heat fl= ux can be carried out using a slug sensor as shown in Fig. 11.8 with a slug of high thermal conductivity = installed on the wall with a layer of insulation. The slug is exposed to he= at flux at the surface and its temperature will rise and measuring it will = give the magnitude of the heat flux. Temperature gradient, =E2=88=86T, of the slug is maintained uniform with respect to time =E2=80=98t<= /em>=E2=80=99 and the energy balance gives

Figure 11.8 Slug-Type Heat-Flux Meter

where q is the heat = flux, m is the mass of the slug, c is the specific heat o= f the slug, U is the coefficient for the conduction loss to the su= rrounding wall, T_s is the temp= erature of the slug, and T_w is the surr= ounding wall temperature.

The s= lug temperature gradient =E2=88=86T is a function of the heat flux= and the properties of the slug, and hence the measurement of T_s is an indication of the heat flux.

11.2.4.2 Foil-type heat flux met= er

A foil-type heat flux meter i= s shown in Fig. 11.9. A thin c= onstantan disc is mounted on a heat sink, which is installed in the wall ma= terial. A copper wire is attached to the center of the disc and it is connected to the copper sink. Thermocouples are used to m= easure the temperature difference between the center of the disc and the si= des. Heat flux absorbed by the disc is conducted radially outward and that = creates a temperature difference. Radiation loss takes place from the back = side of the disc to the copper heat sink and through proper calibration thi= s can be accounted for. A copper disc along with a positively doped bismuth= =E2=80=93telluride center connection increases sensitivity.

Figure 11.9 Foil-Type Heat-Flux Meter

11.2.5= Measurement of Emissivity

11.2.5.1 Thermopile radiometer
Measurement of total normal e= missivity is carried out using a thermopile radiometer as shown in Fig. 11.10. An electric heater maint= ains the temperature of the sample and the temperatures are measured using = thermocouples. Junctions are made up of silver constantan. Two blackened al= uminum-foil strips are attached to the junctions. A slot in rear shield exp= oses the hot-junction strip to the radiant flux from the sample and the col= d-junction strip is exposed to the temperature of the thermopile enclosure.= The temperature difference between the hot and cold junctions represents t= he emissivity of the sample. A conical shield on the front of the device co= ncentrates the energy from the sample. The temperatures of the hot and cold= junctions are assumed to be same and equal to the inside temperature of th= e radiometer. If the sample is assumed gray, then the total normal emissivi= ty of the sample is

<= /a>Figure 11.10 Thermopile Radiometer

where E is the volta= ge output of the thermopile in mV (millivolt) and T_R and T_s are radi= ometer and sample surface temperatures. The shape factor is F =3D = R²/(R^{2 + L²), where R is the radi=
us and L is the length.}

11.2.5.2 = Radiation pyrometer

The radiation pyrometer is us= ed to measure surface emissivities. A heat source is used on the surface an= d surface temperature is measured using a thermocouple. The apparent temper= ature is measured with the radiometer by assuming unit emissivity and the a= pparent emissivity is given by

where T_a and T_s are the apparent= and surface temperatures. The radiometer is to be set for the apparent emi= ssivity value and the radiation measurement of temperature is repeated till= the thermocouple and radiation measurements are equal. The set value of =CE=B5_a is acceptable. If they d= o not converge, then it may be due to the reflected radiation from surround= ing surfaces.

11.2.6= Reflectivity and Transmissivity Measurements

The properties of thermal rad= iation such as emissivity, reflectivity, and transmissivity are measured us= ing an integrating sphere reflectometer. A monochromator produces monochrom= atic radiation using narrowband optical filters. The fraction of incident r= adiation that is reflected or transmitted by the surface is determined usin= g the integrating sphere reflectometer as shown in Fig. 11.11. The monochromatic radiation is chopped to= produce an AC source and a detector amplifies it. The pivoted mirror allow= s divergence of the monochromatic beam directly to the specimen placed insi= de the sphere. The inner surface of the sphere is coated with magnesium oxi= de and produces a very high and diffuse reflectance. The radiation falling = on the surface will be diffusely reflected throughout the sphere and unifor= m irradiation is produced from the inner surface. Detectors are used to sen= se the reflected radiation and the signal for direct impingement of the ref= erence beam is compared with that which results from reflected radiation fr= om the sample. The ratio of these two signals is the reflectivity of the sa= mple. The measurement of transmissivity is made by keeping a sample over th= e input port of the sphere. The signal is compared after transmission throu= gh the specimen to that obtained from direct impingement on the inner surfa= ce of the sphere.

<= /a>Figure 11.11 Integrating-Sphere Reflectometer

= 11.3 TEMPERATURE MEASUREMENT

Temperature variation is rela= ted to changes in pressure, volume, electrical resistance, and expansion co= efficients of fluids and metals. These changes are used to measure temperat= ure.

11.3.1= Thermometers

A thermometer, shown in Fig. 11.12, is the most commonly u= sed instrument for temperature measurement. The liquid in the bulb of the t= hermometer expands with heat and rises in the calibrated glass tube. A bulb= is also placed at the top of tube for safety if the liquid expansion excee= ds thermometer limit. Alcohol is the commonly used thermometer liquid and h= as higher coefficient of expansion than the other widely used medium mercur= y. Its use is limited to low-temperature measurements below its boiling tem= perature. The temperature measurement range decides the size of the glass t= ube, bulb, and liquid.

<= /a>Figure 11.12 Glass Thermometer

The thermometer is to be cal= ibrated for a certain specified depth of immersion to account for the heat = transfer to the bulb from the environment and heat conducted into the bulb = from the stem. Thermometers have accuracies of =C2=B10.05=C2=B0C and mercur= y thermometers are used up to 530=C2=B0C by filling the space above the mer= cury with gases such as nitrogen to increase the pressure on the mercury by= raising its boiling point.

11.3.2= Fluid Expansion Thermometer

Fluid expansion thermometer h= as a bulb with a liquid, gas, or vapor connected through a capillary tube t= o a pressure-measuring device such as a bourdon gage as shown in Fig. 11.13. The bulb will be submerged = in the environment whose temperature is to be measured. Temperature expands= the fluid inside the bulb and pressure increases and it is measured as an = indication of the temperature. To avoid the influence of temperature on cap= illary tube and a mixture of liquid and vapor is used in the bulb. A small = bulb connected to an electric-pressure transducer through a short capillary= gives the highest response. It is less costly, stable in operation, and ac= curate within =C2=B11=C2=B0C.

<= /a>Figure 11.13 Fluid-Expansion Thermometer

11.3.3 Temperature Measurement Using Electrical Methods
Temperature measurement by el= ectrical methods is convenient as they provide signals that are easily dete= cted, amplified, and used for control purpose. Also, they are accurate when= properly calibrated and compensated.

11.3.3.1 Electrical-resistance t= hermometer

Electrical-resistance thermom= eter has a resistive element that is exposed to the medium whose temperatur= e to be measured as shown in Fig.= 11.14. The change in resistance of that element is taken as the measur= e of temperature. The temperature coefficient of resistance =CE=B1 of the e= lement is linear (=CE=B1 =3D R₂ =E2=80= =93 R₁ /(R₁ (T₂ =E2=80=93 T₁)), where R₁ and = R₂ are the resistances of the element a= t temperatures T₁ and T₂). This relationship can be used for a narrow temperat= ure range and for wider temperature ranges; the resistance of the material = can be expressed as R =3D R_R (= 1 + aT + bT²) where R= is the resistance at temperature T and R_R is the resistance at reference temperat= ure T_R. The resistance of the element s= hould be free of mechanical stresses and should not be exposed to moisture.= Hence, the platinum element is covered by glass or ceramic and sealed with= molten glass. The accuracy of thermometer is within =C2=B10.1 =C2=B0C.

<= /a>Figure 11.14 Electrical-Resistance Thermometer

11.3.3.2 thermistor

Thermistor is a sensitive sem= iconductor device with negative temperature coefficient of resistance that = varies exponentially with temperature as

where the value of resistance= R_R is with reference to temperature T_R and =CE=B2 is an experimental= ly determined constant, which depends on the thermistor material and temper= ature. The accuracy of the thermistor is within 0.01=C2=B0C. The resistance= of the thermistor changes rapidly with temperature in a nonlinear way and = hence a data acquisition system can give a direct temperature readout from = the resistance measurement.

The thermistor usually deter= iorates at high temperatures and can be generally used for temperature meas= urements below 300=C2=B0C only. The amplifier circuit used with a thermisto= r is shown in Fig. 11.15. Res= istance is expressed as R_T and, for lin= ear operation, R_T =E2=89=AB R_l and R_B is a= pproximately R₁/10. The output voltage = of the circuit is

<= /a>Figure 11.15 Amplifier circuit

where =CE=B1 is the temperatu= re coefficient of resistance and =CE=94T is the temperature difference from= balanced conditions.

11.3.3.3 thermocouple

Temperature measurement using= a thermocouple is another electrical method in practice. Two dissimilar me= tals are joined together as thermocouple as shown in Fig. 11.16. The emf generated in the thermocouple d= epends on the junction temperatures and this phenomenon is known as Seebeck= effect. The Seebeck emf is important as it depends on the junction tempera= ture and the emf produced at the junction is measured as a function of temp= erature. Any thermocouple circuit should have atleast two junctions. If the temperature of one junction is known, then the temp= erature of the other junction can be calculated. A known temperature such a= s the ice or atmospheric temperature is used as the reference temperature a= t one junction.

<= /a>Figure 11.16 Iron-Constantan Thermocouple

The output voltage relations= hip of a thermocouple is

where T is the tempe= rature in celsius and E is based on the reference junction tempera= ture. The constants A, B, and C depend on the th= ermocouple material.

Thermocouples with precision= -grade wire will have an accuracy of =C2=B10.5=C2=B0C and a =E2=80=9Ccommer= cial=E2=80=9D grade will have =C2=B12=C2=B0C. The output of thermocouples w= ill be in millivolt range. The complete set of thermocouple wires and milli= voltmeter should be calibrated directly for accurate temperature measuremen= t. The thermocouple output can also be measured using a potentiometer circu= it or by a microvolt potentiometer within 1 =CE=BCV. The resistanc= e of the wires has no influence on accuracy when a potentiometer is used as= the current flow is zero in the thermocouple circuit.

11.3.= 4 Temperature Measurement Using Thermal Radiation

Temperature of any medium can= be measured by measuring its thermal radiation and the commonly used metho= ds are optical pyrometry and emittance determination. Thermal radiation emi= tted by a body is distinguished from radio waves and X-rays and is in the w= avelength region of about 0.1=E2=80=93100 =CE=BCm. The total thermal radiat= ion emitted by a blackbody is

E_b = =3D =CF=83T⁴

where =CF=83 the is Stefan=E2= =80=93Boltzmann constant (5.669 =C3=97 10^{=E2=88=92 ⁸ W /m².K⁴), E_b is the emissiv=
e power, and T is the absolute temperature.}

The measurement of temperatu= re through the use of optical pyrometry is by identification of the tempera= ture of a surface with the color of the radiation emitted. When a surface i= s heated, it becomes red, orange, and finally white. Wien's law gives the w= avelength =CE=BB_max as

=CE=BB_maxT =3D 2897.6 =CE=BCmK

where T is the tempe= rature. Radiation from the source is viewed through the lens and filter as = shown in Fig. 11.17. An absor= ption filter is used to reduce the intensity of the incoming radiation. The= lamp placed in the optical path is adjusted so that the color of the filam= ent matches with the color of the incoming radiation. The red filter instal= led in the eyepiece ensures that comparisons are made for monochromatic rad= iation.

<= /a>Figure 11.17 Optical Pyrometer

In the balance conditions, t= he filament will seem to disappear in the total incoming radiation field. T= emperature is measured in terms of the lamp heating current.

11.3.= 5 High-speed Gas Flow Temperature Measurement

A stationary probe in the hig= h-speed gas flow temperature measuring device as shown in Fig. 11.18, measures the temperature of the ga= s that is brought to rest at its surface. The stagnation temperature, when = the gas velocity is reduced to zero, is

<= /a>Figure 11.18 High-Speed Gas Flow Temperature Measuring Device

where T_: is the free-stream temperature and U is the flow v= elocity. The stagnation temperature of the gas is also expressed in terms o= f the Mach number as

The stagnation temperature i= s same as the static temperature for low-speed flow. In a high-speed flow, = the temperature measured by the probe is not exactly equal to the stagnatio= n temperature and is known as the recovery temperature (T_r) which depends on the probe configuration. The actual s= tagnation temperature T₀ can be which e= stimated from

Where= the recovery factor R can be determined from the calibration of t= he probe at different flow conditions and is in the range of 0.8 to 0.9.
11.3.= 6 Solar Radiation

The temperature of the Sun is= generally in billions of Kelvins and its radiation spectrum has a shape si= milar to that of a blackbody with a range of 0.2 and 2.0 =CE=BCm g= enerally. The peak value may reach about 0.5 =CE=BCm. As per Wien'= s law, the blackbody temperature is

Solar radiation is measured = using a pyrheliometer. It collimates the radiation to find the beam intensi= ty as a function of incidence angle. It has two concentric rings serving as= hot and cold junctions and these rings are coated black and white to produ= ce a temperature difference when exposed to solar radiation. Barium sulfate= is the whitening agent, and the whole assembly is covered with a glass cov= er of hemispherical shape to transmit radiation.

Pyranometer that measures th= e total hemispheric solar irradiation uses the principle of differential ex= pansion of bimetallic strips when exposed to solar radiation. A simple mech= anical linkage is used to measure radiation and electric-power is not requi= red. Photovoltaic cells can also be used for solar radiation measurements. = Cadmium sulfide, silicon. and selenium cells may also be used for many appl= ications.

= 11.4 DIMENSIONAL ANALYSIS

Dimensional analysis is a met= hod to combine several variables into dimensionless groups, which facilitat= e the interpretation of experimental data and extend the range of applicati= on. Heat transfer coefficients are usually calculated from the empirical eq= uations developed by correlating experimental data using dimensional analys= is. The limitation of dimensional analysis is that it does not give informa= tion on the nature of a phenomenon and it is necessary to know in advance a= bout the variables that influence the phenomenon and the success or failure= of dimensional analysis depends on the proper selec= tion of these variables. A preliminary theory or a thorough physical unders= tanding of a phenomenon is needed before dimensional analysis can be perfor= med.

11.4.= 1 Primary dimensions

Selection of primary dimensio= ns is arbitrary and the dimensional formula of all variables can be express= ed in terms of them. The dimensional formula of any physical quantity is de= rived from the definition or physical law. The primary dimensions of some p= hysical quantities used in heat transfer problems are given in Table 11.1.

T= able 11.1 Primary dimensions

Physical quantities Symbols Dimensions

Length
L

L

Density
=CF=81

M/L³

Mass
M

M

Time
t

T

Force
F

ML/t²

Velocity
u, u, U_=E2=88=9E

L/t

Acceleration
a, g

L/t²

Work
W

ML²/t²⁼

pressure
=CF=81

M/t²L

Temperature
T

T

Heat
Q

ML²/t²⁼

Internal energy
e

L²/t^2<=
/sup>

Enthalpy
i

L²/t^2<=
/sup>

Specific heat
c

L²/t^2<=
/sup>T

Thermal conductivity
k

ML/t³T

Thermal diffusivity
=CE=B1

L²/t

Thermal resistance
R

Tt³/ML^=
2

coefficient of expansion
=CE=B2

1/T

absolute viscosity
=CE=BC

M/Lt

Kinematic viscosity
v (=3D =CE=BC/p)

L²/t

Surface tension
=CF=83

M/t²

Heat transfer coefficient
h

M/t³T

mass flow rate
m

M/t

Shear stress
=CF=84

M/Lt²

11.4.= 2 forces

The forces considered for the= dynamic similarity in fluid flow and heat transfer problems are

Force by Pressure =3D Pressure =C3= =97 Area =3D P. L²

Force= by Inertia =3D Mass =C3=97 Acceleration =3D

Force by Gravity =3D Mass = =C3=97 Fravitational acceleration =3D =CF=81L³g

Viscous force =3D Shear st= ress =C3=97 Area

Surface tension =3D =CF=83= L, where =CF=83 is the coefficient of surface tension and has the = dimension of force per unit length.

Elasticity force =3D E= _v.L², where = E_v is the bulk modulus of elasticity of the= fluid.

11.4.= 3 Significance of Non-dimensional Numbers

11.4.3.1 Reynolds number

Reynolds number is a measure = of the relative magnitude of the inertia force to the viscous force in a fl= uid flow. Higher the Reynolds number, greater will be the relative contribu= tion of inertia effect and smaller its value greater will be the relative m= agnitude of the viscous stresses.

11.4.3.2 Prandtl number

Prandtl number is the ratio o= f kinematic viscosity to thermal diffusivity.

where v indicates th= e impulse transport through molecular friction and =CE=B1 indicates the hea= t energy transport through conduction.

11.4.3.3 Nusselt number

Nusselt number represents the= dimensionless heat transfer coefficient and is defined as the ratio of the= heat transfer rate to the heat conducted within the fluid. The ratio of ch= aracteristic length L to the thickness =CE=94x of a stati= onary fluid layer, conducting the heat at the same rate and same temperatur= e difference as in the case of convection process, can also be defined as t= he Nusselt number.

When conduction heat transfe= r rate is equal to convection heat transfer rate

The Nusselt number is expres= sed as

11.4.3.4 = Stanton number

Stanton number is the ratio o= f the heat transfer coefficient h to the heat capacity of the flui= d.

Stanton number,

11.4.3.5 Peclet number

Peclet number is the ratio of= heat capacity of the fluid passing through unit cross-sectional area per u= nit time to the heat flow rate by conduction under a unit temperature gradi= ent and through a thickness L.

Peclet number,

11.4.3.6 Graetz number

It is defined as the ratio of= heat capacity per unit length of the fluid flowing through the pipe to the= conductivity of the pipe.

where D is the diame= ter of the pipe and L is the length of the pipe.

11.4.3.7 Grashof number

It is defined as the ratio of= product of inertia force and buoyancy force to the square of viscous force= in natural convection heat transfer problems.

Where =CF=85 is the velocity = of fluid caused due to buoyancy force and (=CF=81=CE=B2g = =CE=94=CE=B8.L³) is the buoyan= cy force for the total volume.

11.4.= 4 Model Experiments and criteria for Similitude

The complexity of the basic l= aws makes it impossible to solve analytically all practical problems of hea= t transfer. It becomes necessary in many cases to do experimentation on phy= sical models for predicting the behavior of the actual system. Information = on heat transfer coefficient and pressure drop is required for the design o= f heat transfer experiments and equipments. The design is made using empiri= cal relationships developed from experimental results. Such empirical estim= ations often leads to discrepancies with the actuals. The reason for such d= eviation is due to the difference between the condit= ions of fluid flow and heat transfer in actual heat transfer equipments and= in the experiments.

The similarities required fo= r the heat transfer models and their testing are

Geometric similarity

Kinematic similarity

Dynamic similarity

Similarity to fluid entry conditions, = and

Similarity of boundary temperature= field.

Knowledge of these laws of si= militude is essential for planning and analysis of model experiments.

11.4.4.1 Geometrical similarity<= /h4>
Two models are geometrically = similar if the ratios of all homologous dimensions are equal and this simil= arity involves only similarity in form.

11.4.4.2 Kinematic similarity
Two models are kinematically = similar if the pattern or paths of motion are similar and if the ratios of = the velocities of the various homogenous particles involved in the motion a= re equal.

11.4.4.3 Dynamic similarity

Two models are dynamatically = similar if they are kinematically similar and if the ratios of the respecti= ve forces in the system are same.

As force is proportional to = the mass and acceleration, in systems having geometric and dynamic similari= ty, kinematic similarity will be achieved if the ratio of masses at points = in the two systems is same. Hence, geometric and dynamic similarities are e= ssential for complete similarity.

As per Newton's second law o= f motion, for complete dynamic similarity, the vector sum of all active for= ces in the system must be equal to the inertia force. Two systems are said = to be completely similar if there is geometric similarity and ratios of the= forces for the model and the proto type are same i.e., Reynolds, Froude, W= eber, and Mach number are same for both systems. It is impossible to get pe= rfect similarity by satisfying all these conditions. If important ratios ar= e satisfied, then suitable corrections can be made to satisfy other conditi= ons. The gravity force, surface tension force, and the elastic forces can b= e neglected if they are balanced or insignificant.

The fulfillment of all these= conditions is very difficult and may be possible under some conditions of = stability and automodellity. Stability is defined as the capacity of a flow= ing viscous fluid for definite distribution of velocity. This distribution = depends on the Reynolds number, shape of the channel, and relative length o= f the curved path. The profiles of velocities will be similar if these fact= ors are identical. The distribution of velocity for the given channels vari= es with Reynolds numbers. Automodellity is defined as the independence of t= he nature of flow from the Reynold number. The condition for automodellity = is that, the friction factor should be constant.

The geometric condition is f= ulfilled by constructing an exact scale model of the actual equipment. The = similarity of entry condition is ensured by making the inlet section of the= model geometrically similar to that of the actual equipment. As it is not = possible to maintain the similarity of all physical parameters in the scale= model, combination of parameters are maintained constant. It is also neces= sary that the physical parameters in the model and actual equipment must ch= ange in similar way to satisfy the condition of similarity.

In the case of forced convec= tion, the similarity condition of model and actual equipment are ensured if= the Reynolds number in both cases is maintained same. This condition may b= e omitted if automodellity sets in the conditions of model testing.

In th= e case of free convection, the similarity condition of model and actual equ= ipment are ensured if the product of Grashof and Prandtl numbers in both ca= ses is maintained same. A model must be used for studying the free convecti= on phenomenon only if G_n P_n > 2 =C3=97 10⁷ and for cases = in which G_nP_n < 2 =C3=97 10⁷, model testing is not useful= .

It is expensive to make full= -size prototype equipments for design verification and testing. Hence, smal= l scale models are used to study flow and heat transfer phenomena which cou= ld not be easily solved using mathematics. Test models are used to study th= e performance of hydraulic machines and structures, thermal systems and als= o used to verify the design of systems such as aircrafts, rockets, and miss= iles using wind tunnels. Data on drag forces and wake patterns can be gener= ated for the design of automobiles and ships by conducting the experiments = on small models. Though the model studies do not provide solutions to all q= uestions, the results will help the engineers to understand and design prot= otypes. Knowledge of principles of model similitude is essential for design= , construction, and operation of the model. In the case of models used for = testing, geometric similarity, kinematic similarity, and dynamic similarity= between the model and prototype are essential.

11.4.4.4 Correlation of experime= ntal data

If experiments were conducted= with air flowing over a pipe at different velocities and the heat transfer= coefficient is measured experimentally, then the data may be correlated as= shown in Fig. 11.19. This cu= rve can be used for determination of heat transfer coefficient at any veloc= ity for the pipe used in the tests. However it cannot be used to find the h= eat transfer coefficients for pipes that are different in diameter than the= one used in the tests. Also, it cannot be used if the air pressure and den= sity are different from that used in the tests. It is necessary then to con= duct separate experiments for every pipe diameter and density.

<= /a>Figure 11.19 Variation of Heat Transfer Coefficient with Veloci= ty

Dimensional analysis, howeve= r, helps to apply the results of one series of tests to other problems by r= eplotting in the data in terms of dimensionless numbers such as Reynolds nu= mber (velocity) and Nusselt number (heat transfer coefficient) as shown in = Fig. 11.20. The correlation o= f these numbers is used for the evaluation of the convection heat transfer = coefficient for air flowing over any size of pipe if the Reynolds number of= the system is within the range of the experiment and the systems are geome= trically similar.

<= /a>Figure 11.20 Variation of Dimensionless Numbers for flow of Air= Over a Pipe

11.4.5 Buckingham =CF=80 Iheorem

Buckingham =CF=80 th= eorem is used to find the number independent dimensionless groups needed to= obtain a relation for describing a physical phenomenon. The number of inde= pendent dimensionless groups that can be formed by combining the physical v= ariables is equal to the total number of these physical quantities =E2=80= =9Cn=E2=80=9D minus the number of primary dimensions =E2=80=9C= m=E2=80=9D required to express the dimensional formulas of the =E2=80= =9Cn=E2=80=9D physical quantities. If these groups are referred as= =CF=80₁,=CF=80₂, =E2=80=A6, etc, the relationship is

f (=CF=80₁, =CF=80₂ =E2=80=9A= =CF=80₃ =E2=80=9A =E2=80=A6) =3D 0.<= /div>

If there are five physical q= uantities in a problem with three primary dimensions, then two dimensionles= s groups can be formed:

f(=CF=80₁=E2=80=9A =CF=80₂) =3D 0.

Experimental data for such a= problem can be analysed by plotting =CF=80₁ aga= inst =CF=80₂ and the curve will show the functio= nal relationship between =CF=80₁ and =CF=80₂.

For problems with three dime= nsionless groups

f (=CF=80₁=E2=80=9A=CF=80₂=E2=80=9A=CF=80₃) =3D 0

and it may also be written as=

=CF=80₁ = =3D f (=CF=80₁=E2=80=9A=CF=80₃)=C2=B7

For such problem, the experi= mental data may be analyzed by plotting =CF=80₁ = against =CF=80₂ for various values of =CF=803 or with other combinations. It is also possible to= combine any two of the =CF=80 groups by addition, subtraction, multiplicat= ion, or division and to plot this value against the remaining =CF=80 as a s= ingle curve.

= 11.5 CONCLUSIONS

Estimation of heat transfer r= ate, heat transfer coefficients and thermal conductivity of materials are t= he main objectives of laboratory experiments. To measure temperature, press= ure, flow, conductivity and emissivity, measuring instruments with best lev= el of accuracy are required. Though mechanical and pneumatic instruments ar= e simple to maintain, highly reliable and independent from external power, = electrical and electronic devices provide higher accuracy and sensitivity.<= /p>
Measurement of thermal condu= ctivity is based on Fourier's law of heat conduction.

Direct measurement of heat f= lux can be carried out using a slug sensor. Measurement of total normal emi= ssivity is carried out using a thermopile radiometer. The radiation pyromet= er is used to measure surface emissivities. The properties of thermal radia= tion such as emissivity, reflectivity and transmissivity are measured using= an integrating sphere reflectometer. Temperature variation is related to c= hanges in pressure, volume, electrical resistance and expansion coefficient= s of fluids and metals and these changes are used to measure temperature. T= emperature measurement by electrical methods are convenient as they provide= signals that are easily detected, amplified and used for control purpose. = Temperature of any medium can be measured by measuring its thermal radiatio= n and the commonly used methods are optical pyrometry and emittance determi= nation. Solar radiation is measured using a pyrheliometer.

Pyranometer that measures th= e total Solar irradiation uses the principle of differential expansion of b= imetallic strips when exposed to solar radiation.

Dimensional analysis is a me= thod to combine several variables into dimensionless groups, which facilita= te the interpretation of experimental data and extend the range of applicat= ion.

Two models are geometrically= similar if the ratios of all homologous dimensions are equal and this simi= larity involves only similarity in form

Two models are kinematically= similar if the pattern or paths of motion are similar and if the ratios of= the velocities of the various homogenous particles involved in the motion = are equal.

Two models are dynamatically= similar if they are kinematically similar and if the ratios of the respect= ive forces in the system are same.

Buckingham =CF=80 theorem is= used to find the number independent dimensionless groups needed to obtain = a relation for describing a physical phenomenon.

KEY CONCEPTS

Temperature is an objective (exact) measure of heat.

Words like hot and cold are subjective (relative to one another).

Since heat is also known as vibration, the heat amount of vibration= possible is to be perfectly still.

Perfectly is absolute zero degrees (on the Kelvin scale).

There is no way to have =E2=80=98negative=E2=80=99 vibration.

There is no imaginable limit to how hard something can vibrate.
=

This would be in the millions of degrees like the sun.

= EXAMPLE PROBLEMS

11.1. Find the pos= sible dimensionless groups from the experimental convection heat transfer d= ata for a fluid flowing across a heated tube. The heat transfer depends on = the diameter of the tube (D), fluid velocity (u), density (=CF=81), dynamic= vicosity (=CE=BC), specific heat (c_p) and heat = transfer coefficient h.

Out of seven physical qu= antities, four have all primary dimensions among them. Three dimensionless = groups will be required to correlate the data.

Four physical variables = D, K, =CF=81 and =CE=BC are selected as= repeated variables and among them they have all the four primary dimension= s. They can be grouped with each one of the remaining variables to form thr= ee =CF=80 groups.

Forced convection heat t= ransfer is a function of seven variables and with the help of dimensional a= nalysis, these seven variables are combined into three dimensionless groups= . The functional relationship between them may be written as

=CF=80₃ = =3D f(=CF=80₁=E2=80=9A=CF=80₂)

Nu =3D f(Re= , Pr)

The experimental data ca= n be correlated in terms of these three dimensionless numbers instead of th= e seven variables and the relationship will be where C₁, C_{2 and C₃ are constants.}

11.2. A fluid whic= h is heated by a hot vertical plate and forced up due to buoyancy has seven= variables that influence the heat transfer. They are heat transfer coeffic= ient (h), fluid density (=CF=81), plate length (L), fluid dynamic viscosity= (=CE=BC), thermal conductivity (K), specific heat (c_p=
) and buoyancy force (=CE=B2g=CE=94T). Find the possible dimensionles= s groups from these variables to estimate the heat transfer coefficient.

Let the heat transfer co= efficient is a function of the physical factors as

h =3D f (= =CF=81, L, =CE=BC, c_p= , K, =CE=B2g=CE=94T)

The buoyancy force, (=CE=B2g=CE=94T) will be considered as one physical factor which is ca= using the flow.

f [=CF=81, L<= /em>, =CE=BC, K, h, c_p<=
/sub>, (=CE=B2g=CE=94T)] =3D 0

T= his problem is controlled by seven physical quantities containing four fund= amental dimensions and using the Buckingham theorem, three dimensionless gr= oups can be formed.

According to the Bucking= ham theorem, four variables =CF=81, L, =CE=BC an= d K which contain all four fundamental dimensions may be selected = as the repeating variables and they form three non-dimensional groups with = each of the remaining variable.

Therefore,

=CF=80₁ = =3D h (=CF=81)^a (= L)^b (=CE=BC)^c (k)^d = =3D M⁰L^=
0t⁰T⁰

(MT^=
=E2=88=923 T^=E2=88=921)(ML^=E2=88=923)^{a(L)^b(ML^=E2=88=921 T^=E2=88=921=
)^c(MLT^=E2=
=88=923 T^=E2=88=921)^d =3D M⁰ L⁰ t⁰ =
T⁰}

Equating the powers of <= em>M, L, t and T on both sides,

a + c + d + 1 =3D 0<= /div>
=E2=88=923a + b =E2=88=92 c + d= =3D 0

      = ;    =E2=88=92c =E2=88=923d =E2=88=92= 3 =3D 0

      = ;           =E2= =88=92d=E2=88=921 =3D 0

a =3D 0, b =3D 1,= c =3D 0, d =3D =E2=88=92 1

Similarly,

=CF=80₂ =3D (=CF=81)^a1 (L= )^b1 (=CE=BC)^c1(k)^d1= C_p =3D M⁰L⁰ t⁰T⁰

Therefore,

(ML^=
=E2=88=923)^a1 (L)^bl (ML^=E2=88=921=t^=E2=88=921)^=
c1 (MLT^=E2=88=923 T<= /em>^=E2=88=921)^{d1<=
/sup> (L²t^=E2=88=922T^=E2=88=92l)=3D M<=
/em>⁰L⁰t=
⁰T⁰

Equating the powers of <=
em>M, L, t and T on both sides,

      =
;         a₁ + c₁ + d1 =3D 0
=E2=88=923a₁ + b₁ =E2=88=92c<=
sub class=3D"EBsub">1 + d₁ + 2 =
=3D 0
      =
;            &n=
bsp;=E2=88=92c₁ =E2=88=923d1 =E2=88=922 =3D 0
      =
;            &n=
bsp;       =E2=88=92d₁ =E2=88=921 =3D 0
a_{1 =3D 0, b₁ =3D 0, c₁ =3D 1, d₁ =3D =E2=
=88=921}

Therefore,

Similarly,}
=CF=80₃ =3D (=CF=81)^a² (L)^b2(=CE=BC)^c2(k)^d2 (=CE=B2g=CE=94T= ) =3D M⁰L⁰t⁰T⁰

That is,

(ML^=
=E2=88=923 (L)^b2 (ML= ^=E2=88=921t^=
=E2=88=921)^c2 (MLT^=E2=88=923T^{=E2=88=921)^d2(LT^=
=E2=88=922) =3D M⁰L⁰t⁰T0}

Equating powers of M= , L, t and T on both sides,

      = ;       a₂₌ + c₂ + d₂ =3D 0

=E2=88=923a₂ + b₂ =E2=88=92 c= ₂ + d₂+ 1 = =3D 0

      = ;           =E2=88= =92c₂ =E2=88=923d₂ =E2=88=922 =3D 0

      = ;            &n= bsp;          =E2=88=92d₂ =3D 0

a_{2 =3D 2, b₂ =3D 3, c₂ =3D =E2=88=92 2 and d_{2 =3D 0}}

Dimensionless groups may= be related as =CF=80₁ =3D f(<= em>=CF=80₂. =CF=80₃).

T= herefore,

where C, n<= /em> and m are the constants.

= REVIEW QUESTIONS

What is the role of the control s= ystem in heat transfer experiments?

How will you measure thermal conductiv= ity of a plate?

Explain a method for the measurement o= f thermal conductivity of

a liquid and

a gas

Discuss the experimental procedure to = determine the free convection heat transfer coefficient.

Explain the experimental procedure to = determine the forced convection heat transfer coefficient of a fluid.

How will you determine the surface emi= ssivity of a material using a radiation pyrometer?

What are the limitations in using a th= ermometer to measure very high temperatures?

Explain the procedure to measure tempe= rature using electrical=E2=80=93resistance thermometer.

In what way is a thermistor differ fro= m other temperature measuring instruments?

List the different combinations of= thermocouples and compare their capacity to measure temperature.

= OBJECTIVE QUESTIONS

Automatic control systems are pre= ferred in laboratory experiments for

accurate maintenance of the set v= alues of variables

better accuracy of the set values of v= ariables

better error analysis of the set value= s of variables

all of the above

Measurement of thermal conductivity is= based on

Newton's law

Fourier's law

Stefan=E2=80=93Boltzman's law

all of the above

Water flow rate can be measured using = a

thermometer

thermocouple

venturimeter

all of the above

A slug type heat flux meter is used to= measure

thermal conductivity

heat flux

flow

emissivity

A thermopile radiometer is used to mea= sure

reflectivity

transmissivity

emissivity

conductivity

Solar radiation is measured using

pyroheliometer

gas flow meter

optical pyrometer

all of the above

The primary dimensions of pressure are= given by the expression

M/t^2L²

M/tL²

M/t²L

M²/= tL

Force by inertia is a product of mass = and

shear stress

area

acceleration

pressure

Prandtl number is the ratio of kinemat= ic viscosity to

thermal diffusivity

friction

force

pressure

For complete similarity between mo= del and prototype, it is essential to have

geometric similarity

dynamic similarity

both a and b

kinematic similarity

Answers:

a

b

c

b

c

a

c

c

a

c

= EXERCISE PROBLEMS

Find the dimensions of the quanti= ties given below: (i) angular acceleration, (ii) discharge, (iii) force, (i= v) specific weight and (v) dynamic viscosity. (Ans. T<= /em>^=E2=88=922, L³⁼T^=E2=88=921 MLT², MLT^=E2=88=92²T², ML=E2=88=921 T^=E2=88=
=92l)

The pendulum time period (t) = depends upon the length of the pendulum (L) and acceleration due t= o gravity (g). Using Rayleigh's method, show that the expression f= or the time period is .

The drag force of a smooth sphere depe= nds on its diameter, D, moving velocity V, fluid density,= =CF=81, and dynamic viscosity, =CE=BC. Derive the expres= sion for drag force F as C=CF=81D²V² =3D f= (=CE=BC/=CF=81VD).

Derive an expression for the power, P, of a pump which depends upon the head, H, the discharge,= Q, and specific weight, w, of the fluid. (Ans.P =3D CQH)

Find the efficiency of a fan, n, which depends on the density, =CF=81, the dynamic viscosity, <= em>=CF=81 of the fluid, the angular velocity, =CE=BC, diamete= r, D, of the rotor and the discharge, Q.

Derive an expression for the resisting= force, F, of an aircraft which depends on its length, L, velocity= , V, air density, =CF=81, and bulk modulus of air, K.=

The resistance force, F, to t= he motion of a partially submerged body towed in water depends on the densi= ty, =CF=81, water viscosity, =CE=BC, length, L, = of the body, velocity, v, of the body and the acceleration due to = gravity, g. Express the relationship between these variables.

Derive an expression for the efficienc= y, =CE=B7, of a fan which depends on density, =CF=81, flu= id dynamic viscosity, =CE=BC, angular velocity, =CF=89, d= iameter, D, of the rotor and the discharge, Q.

Derive an expression for the velocity = through a circular orifice as
where H is the head = causing the flow, D is the diameter of the orifice, =CE=BC is the coefficient of viscosity, =CF=81 is the mass density and = g is the acceleration due to gravity.

Derive an expression for the pressure = difference =CE=94P for a turbulent flow in a pipe of diameter = D, length L, velocity V, viscosity =CE=BC, = density =CF=81 and roughness k.

Derive an expression for the pressure = difference =CE=94P for a viscous flow in a pipe of diameter D<= /em>, length L, velocity V, viscosity =CE=BC and= density =CF=81.

Derive an expression for the thrust P developed by a propeller which depends on the angular velocity, =CF=89, speed of advance V, diameter D, dynamic vi= scosity =CE=BC mass density =CF=81, and elasticity of the= fluid medium denoted by the speed of sound in the medium C.

Using Buckingham's theorem, show that = the frictional torque, T, of a disc of diameter D rotatin= g at a speed N in a fluid of viscosity =CE=BC and density= =CF=81 in a turbulent flow is

Show that the discharge Q con= sumed by an oil ring is given by

where D is the inter= nal diameter of the ring, N is its rotational speed, =CF=81 is the density, =CE=BC is the viscosity, =CF=81 is the= surface tension and =CF=89 is the specific weight of oil.

Oil of specific gravity 0.90 and visco= sity 3 =C3=97 10^=E2=88=922 poise is moved at the= rate of 3000 L/s through a pipe of diameter 1.5 m. Model tests were conduc= ted on a pipe of diameter 15 cm, using water at 20=C2=B0C in the laboratory= to simulate the flow. Find the velocity and rate of= flow in the model. Viscosity of water at 20=C2=B0C =3D 0.01 poise. (Ans. 5 m/s, 90 L/s)

In a heat exchanger water flows throug= h a pipe of diameter 0.3 m at a velocity of 240 m/min. Estimate the velocit= y of oil flowing in another pipe of diameter 0.1 m, if there is dynamic sim= ilarity between the two pipes. Assume that the viscosities of water and oil= are 0.01 poise and 0.025 poise. The specific gravity of oil =3D 0.8. (Ans. 37.5 m/s)

If the ratio of lengths of a submarine= and its model is 60:2, and the speed of the submarine (prototype) is 600 m= /min, find the speed of air in wind tunnel. Assume kinematic viscosities of= sea water and air to be 0.012 stokes and 0.016 stokes, respectively. The d= ensity of sea-water and air are 1030 kg/m³ and 1= .24 kg/m³, respectively. (Ans. = 24 km/min)

An aircraft model is tested in water. = The actual aircraft is 10 times bigger than the model. The pressure drop in= an aircraft model is 160 N/m². What will be the= corresponding pressure drop in the actual aircraft? Assume the density of = air to be 1.24 kg/m³, the viscosity of water to = be 0.01 poise and the viscosity of air to be 0.00018 poise. (Ans. 0.026 N/m²)

A 1:15 model of a flying boat is towed= through water. A boat that moves in sea-water of density 1024 kg/m³ at a velocity of 72 km/h is to be tested in the labora= tory with water having density of 1000 kg/m³. Fi= nd the corresponding speed of the model. (Ans. 18.6 km/hr)=

A vessel model is pulled through f= resh water in lab condition at 7.2 km/hr with a total measured drag of 12 N= . The drag coefficient, f, for the model and prototype are 0.03 an= d 0.002, respectively, in the equation R_f =3D fAV². The wetted surface area of= the model is 25 m². Find the total drag on the = prototype. Assume that the density of sea-water is 1030 kg/m³ and that the prototype is 40 times bigger than the model. (Ans. 606.1 kN)

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=20 =20 =20 =20 Previous Chapter
10. Modeling and Ana= lysis
=20 =20

=20 =20 =20 =20 Next Chapter
12. Mass Transter =20 =20
=20

Physical quantities	Symbols	Dimensions
Length	L	L
Density	=CF=81	M/L³
Mass	M	M
Time	t	T
Force	F	ML/t²
Velocity	u, u, U_=E2=88=9E	L/t
Acceleration	a, g	L/t²
Work	W	ML²/t²⁼
pressure	=CF=81	M/t²L
Temperature	T	T
Heat	Q	ML²/t²⁼
Internal energy	e	L²/t^2<= /sup>
Enthalpy	i	L²/t^2<= /sup>
Specific heat	c	L²/t^2<= /sup>T
Thermal conductivity	k	ML/t³T
Thermal diffusivity	=CE=B1	L²/t
Thermal resistance	R	Tt³/ML^= 2
coefficient of expansion	=CE=B2	1/T
absolute viscosity	=CE=BC	M/Lt
Kinematic viscosity	v (=3D =CE=BC/p)	L²/t
Surface tension	=CF=83	M/t²
Heat transfer coefficient	h	M/t³T
mass flow rate	m	M/t
Shear stress	=CF=84	M/Lt²

Table= of Contents for =20 Heat and Mass Transfer, 2nd Edition =20

11

Experimental Heat Trans= fer

= 11.1 INTRODUCTION

= 11.2 HEAT TRANSFER EXPERIMENTS

11.2.1= Thermal Conductivity Measurement of Solids

11.2.1.1 Thermal conductivity me= asurement in a plate

11.2.1.2 Insulated apparatus for= thermal conductivity measurement

11.2.2= Thermal Conductivity Measurement of Liquids and Gases

11.2.2.1 Thermal conductivity me= asurement of liquids =E2=80=93 hot-plate method

11.2.2.2 Thermal conductivity me= asurement of gases

11.2.3= Convection Heat Transfer Measurement

11.2.3.1 Free convection heat tr= ansfer

11.2.3.2 Forced convection heat = transfer

11.2.4= Heat Flux Measurement

11.2.4.1 Slug-type heat flux met= er

11.2.4.2 Foil-type heat flux met= er

11.2.5= Measurement of Emissivity

11.2.5.2 = Radiation pyrometer

11.2.6= Reflectivity and Transmissivity Measurements

= 11.3 TEMPERATURE MEASUREMENT

11.3.1= Thermometers

11.3.2= Fluid Expansion Thermometer

11.3.3 Temperature Measurement Using Electrical Methods Temperature measurement by el= ectrical methods is convenient as they provide signals that are easily dete= cted, amplified, and used for control purpose. Also, they are accurate when= properly calibrated and compensated.

11.3.3.1 Electrical-resistance t= hermometer

11.3.3.2 thermistor

11.3.3.3 thermocouple

11.3.= 4 Temperature Measurement Using Thermal Radiation

11.3.= 5 High-speed Gas Flow Temperature Measurement

11.3.= 6 Solar Radiation

= 11.4 DIMENSIONAL ANALYSIS

11.4.= 1 Primary dimensions

11.4.= 2 forces

11.4.= 3 Significance of Non-dimensional Numbers

11.4.3.1 Reynolds number

11.4.3.2 Prandtl number

11.4.3.3 Nusselt number

11.4.3.4 = Stanton number

11.4.3.5 Peclet number

11.4.3.6 Graetz number

11.4.3.7 Grashof number

11.4.= 4 Model Experiments and criteria for Similitude

11.4.4.1 Geometrical similarity<= /h4> Two models are geometrically = similar if the ratios of all homologous dimensions are equal and this simil= arity involves only similarity in form.

11.4.4.2 Kinematic similarity Two models are kinematically = similar if the pattern or paths of motion are similar and if the ratios of = the velocities of the various homogenous particles involved in the motion a= re equal.

11.4.4.3 Dynamic similarity

11.4.4.4 Correlation of experime= ntal data

11.4.5 Buckingham =CF=80 Iheorem

= 11.5 CONCLUSIONS

KEY CONCEPTS

= EXAMPLE PROBLEMS

= REVIEW QUESTIONS

= OBJECTIVE QUESTIONS

Answers:

= EXERCISE PROBLEMS

Highlight

Highlight

Table= of Contents for
=20 Heat and Mass Transfer, 2nd Edition =20

11.3.3 Temperature Measurement Using Electrical Methods
Temperature measurement by el= ectrical methods is convenient as they provide signals that are easily dete= cted, amplified, and used for control purpose. Also, they are accurate when= properly calibrated and compensated.

11.4.4.1 Geometrical similarity<= /h4>
Two models are geometrically = similar if the ratios of all homologous dimensions are equal and this simil= arity involves only similarity in form.

11.4.4.2 Kinematic similarity
Two models are kinematically = similar if the pattern or paths of motion are similar and if the ratios of = the velocities of the various homogenous particles involved in the motion a= re equal.