lab manuel

MEM311

THERMAL & FLUID SCIENCE LABORATORY MANUAL

Edited by Brandon Terranova, Eric Wargo, Ertan Agar, Chris

Dennison and E. Caglan Kumbur

Adopted from MEM 311 Thermal & Fluid Science Laboratory

by Baktier Farouk and David Stacck

LIST OF EXPERIMENTS:

EXPERIMENT 1: FLOW MEASURING DEVICES PAGE 1

EXPERIMENT 2: CONTROL VOLUME ENERGY AND ENTROPY

ANALYSIS IN A VORTEX TUBE PAGE 10

EXPERIMENT 3: HEAT TRANSFER FROM A CIRCULAR CYLINDER PAGE 21

EXPERIMENT 4: PERFORMANCE ANALYSIS OF A STEAM TURBINE

POWER PLANT PAGE 38

Experiment 5: Lift Characteristics of an Airfoil Section PAGE 51

LIST OF APPENDICES:

APPENDIX A: MANOMETER PREPARATION AND OPERATION Page 63

APPENDIX B: FLOW MEASURING USING A ROTAMETER Page 66

APPENDIX C: ANALYSIS OF BIAS ERRORS AND EXPERIMENTAL

UNCERTAINTY Page 68

APPENDIX D: FITTING CURVES TO EXPERIMENTAL DATA Page 74

MEM311: Thermal and Fluid Science Laboratory Experiment 1: Flow Measuring Devices

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EXPERIMENT 1:

FLOW MEASURING DEVICES by Brandon Terranova 2010, adapted from Bakhtier Farouk and David Staack

A. OBJECTIVES The objective of this experiment is to determine the discharge coefficient for an orifice flow meter as a function of the Reynolds number.

B. THEORY

Review of Friction Factors in Pipe Flow

As a fluid flows through a pipe (or any other device, for that matter), area changes, friction and heat transfer affect the properties in a flow system. By evaluating the forces acting on a control volume in a pipe flow, the pressure drop for fully developed laminar pipe flow, p, is related to the wall shear stress, , by the equation

(1)

Where is the length of the pipe and is the equivalent hydraulic diameter defined as

= 4(Cross-sectional area of flow) / (Perimeter wetted by fluid). Since the wall shear stress is a complex function of the flow velocity, viscosity, density, wall surface roughness, etc., the pressure drop, p, is expressed as a product of a non-dimensional friction factor, , and

the dynamic pressure (

). Which includes the velocity and density of the fluid, V and

respectively. So the pressure drop for a horizontal pipe is given as:

(

)

(2)

Equating Eq. (1) and (2), we obtain

(3)


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Therefore, the friction factor, , is a measure of the shear stress at the wall. The friction factor, or more generally, the effects of viscosity on fluid flow, can be correlated using the flow Reynolds number (ratio of inertial forces to viscous forces) given as

(4)

where and are the density and absolute viscosity of the fluid, respectively, and V is the mean flow velocity.

Review of Bernoulli’s Equation

As explained above, friction forces induce an irreversible decrease in pressure. The

pressure can also change in a reversible way as described by Bernoulli’s equation. Because

the crux of Bernoulli’s principle is that along a streamline of flow, the increase in velocity

corresponds to drop in the static pressure of the fluid. While Bernoulli’s equation assumes

a lot of simplifications to your system (constant density (incompressible), steady flow, no

friction), it produces very accurate results compared to empirical evidence at low Mach

numbers.

Using Bernoulli’s equation, the conservation of mass and the fact that mass flow rate,

, is constant through the duct, we can write the pressure drop along the duct as a

function of only the upstream velocity and the change in area:

( ( )

)

(5)

Pipe Flowrate Meters

Equation 2 shows that the pressure drop through a pipe is a function of the velocity

of the flow through the system along with the friction factor. In fact, one method to

determine the flow rate of fluid through a piping system is to measure the pressure drop

through a device for which the friction factor and other losses are precisely known. These

are called obstruction flow meters.


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There are three basic designs used in obstruction flow meters as shown in Fig. 1. A

venturi flow meter offers the highest accuracy and the lowest overall pressure drop but is

more expensive to manufacture and accurately calibrate. Both the flow nozzle and the

orifice configurations have larger permanent pressure drops but are relatively simple to

manufacture.

When an orifice flow meter is placed in a pipe, the hole in the orifice essentially

forms a jet which expands to fill the whole pipe at some distance downstream of the plate.

Of course, frictional forces affect the pressure as the air is forced through the hole. In the

absence of viscous effects and under the assumption of a horizontal pipe, application of the

Bernoulli equation between points (1) and (2) in figure 1 gives the volumetric flow rate

through the orifice:

√ ( )

( )

(6)

Figure 1. Schematic of three typical obstruction meters


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Where V2 is the velocity of the flow immediately after the obstruction meter (in the center

of the vena contracta) and = d/D1 as labeled in figure 2.

Figure 2. Orifice meter detail

The Orifice Flow Meter

In this lab we will be examining an orifice flow meter. A typical orifice meter is constructed

by inserting between two flanges of a pipe a flat plate with a hole, as shown in figure 2. The

pressure at point (2) within the vena contracta is less than that at point (1).1 Since the vena

contracta area A2, is less than the area of the hole, Ao, and the turbulent motion near the

orifice plate introduces losses that cannot be calculated theoretically. To take these effects

into account, the orifice discharge coefficient is used. The discharge coefficient is the ratio

of the mass flow rate at the discharge end of the orifice to that of an ideal orifice which

expands an identical working fluid from the same initial conditions to the same exit

pressure.2 The following equation yields the volumetric flow rate for the orifice by

comparing the pressures on either side of the plate:

√ ( )

( )

(7)

The Inlet Nozzle

The flow pattern for the inlet nozzle used in this experiment is closer to ideal than the

orifice meter flow. There is only a slight vena contracta and the secondary flow separation

is less sever, but there are still viscous effects. These are accounted for by the use of the

nozzle discharge coefficient, Cn, where


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√ ( )

( )

(8)

with

, d being the inner diameter of the nozzle and = d/D where D is the outer

diameter of the nozzle. Practically identical to equation 7, note that the pressure drop here

is measured across the nozzle, not the orifice.

Be careful not to confuse the areas, the diameters used, the coefficients and

location of pressures used to determine the respective coefficients.

C. EQUIPMENT

An image of the Armfield F6 Air Flow facility is shown in Fig. 3. The equipment consists of

a long smooth walled pipe (diameter D = 80mm) with an orifice plate of diameter d =

50mm. One end of the pipe is connected to a centrifugal fan via a conical inlet duct while

the other end (inlet nozzle) is open to the atmosphere. The inlet nozzle has an outer

diameter of 120mm and the inner diameter is equal to the pipe diameter. The inlet nozzle

discharge coefficient was determined previously to be . Pressure taps are located

along the complete length of the pipe to allow measurement of the wall pressure as a

function of length. The centrifugal fan is mounted on a floor-standing metal frame and is

driven by a constant-speed meter. The fan discharge duct terminates is a flow control

damper and jet dispersion orifice gate, which is easily adjustable. This flow control damper

will be used to vary the airflow rate through the tube. Velocities between 0 and 35 m/s can

be obtained with this apparatus by adjusting the position of the flow damper (labeled in

figure 3).

Pressure tap Distance (cm)

1 0 2 7.5 3 31.5 4 79.5 5 137 6 148.5 7 159.5 8 183.5 9 208 10 232


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Table 1: Pressure tap distances

Figure 3. Armfield F6 flow facility with transducer array

A fourteen-tube manometer will be used to measure the pressures along the pipe. The

manometer is filled with red oil for easier reading. The specific gravity of this oil is 0.86. A

flow splitter (anti-vortex vanes) is fitted to the inlet of the pipe to prevent swirling of the

flow.

This experiment requires that the airflow rate through the pipe be determined

independently of the orifice plate. To accomplish this, a pre-calibrated inlet flow nozzle is

used. The pressure drop across the nozzle is calculated using equation 8.

In addition to the manometers, the pressure taps along the tube are also connected in

parallel to a transducer array. The pressure transducers convert the pressure force to an

electrical voltage which is then read by a signal conditioner and then displayed as inches of

water. There is one pressure transducer for every pressure tap along the tube. The first

display (labeled 1 in figure 3) is dedicated to the first pressure tap (inlet nozzle tap…starts

at gauge pressure, ie. Atmospheric pressure = 0). The second display (labeled 2 in figure 3)

reads the pressures of the remaining pressure taps as selected by the rotary switch (all

read atmospheric pressure to start, do not attempt to zero!. Use pressure taps 2-

10…disregard the #1 setting on the switch, as it is dedicated to pressure tap 11, not used in


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this experiment). The last display (labeled 3 in figure 3) can be read by toggling the switch

next to it to display the pressure difference across the orifice plate (pt 6 – pt 5). You must

have the switch toggled to the (#5, #6) position to be able to read the individual pressures

on display 2 using the rotary switch.

The pressure transducers have an uncertainty of ±1%.

D. PROCEDURE

Throughout this lab, the manometers and transducers measurements will be used to infer

pressure differences along the pipe. A description of manometer operation is given in

appendix A.

1. Turn on the fan and set a low airflow by closing the flow control damper almost all

the way (do not ever fully close the damper!). Record the level of manometer tubes and the transducer measurements for pressure taps 1 - 10 on your data sheet. Be sure to also record the approximation error in your measurements.

2. Repeat step 4 for the remaining 9 damper settings.


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E. PRE-LAB

1. A 2-in. diameter orifice plate is inserted in a 3-in. diameter pipe. If the water flowrate through the pipe is 0.90 cfs, determine the pressure difference indicated by a manometer attached to the flow meter using the figure below with the calculated Reynolds number.

2. Water flows through the orifice meter shown in the figure below at a rate of 0.10 cfs. If d = 0.1 ft, determine the value of h.


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F. DATA ANALYSIS AND REPORTING REQUIREMENTS

1. Calculate the actual orifice volumetric flow rates (m3/s) for each of the 10 conditions from the manometer and transducer measurements using equation 6.

2. Calculate the orifice discharge coefficient, Co, defined by equation 7, and the Reynolds number, defined by equation 4, for each condition as measured by the

manometer and transducers. Use the mean flow velocity,

, to calculate

the Reynolds number.

3. Write a formula for the orifice discharge coefficient Co in terms of only constants and the parameters which were directly measured. Make sure to include the measurement errors from the experiment. For the transducers, multiply the measurement error by the 1% transducer uncertainty. From this formula derive and calculate the uncertainty (equation 13 in appendix C) in Co at each flow rate condition and for both manometer and transducer measurements.

4. On the same graph, plot the orifice discharge coefficient for both manometer and transducer readings, as a function of the Reynolds number. Be sure to include the uncertainty error bars in your plots.

5. Plot the wall pressure measurements as a function of distance along the duct for your 10 flow conditions. You should have 10 plots on a single graph for both manometer and transducer measurements. In the plots, identify the curves and the location of the orifice flow meter. What does this plot tell you about the effect of an orifice flow meter on the air flow through a pipe? Why might this be an important consideration when designing a piping system? What are some explanations for the discrepancy between the manometer and transducer measurements?

G. REFERENCES

1. Munson et. al (2009). Fundamentals of Fluid Mechanics, 6th Ed., Wiley.

2. R. L. Daugherty and J. B. Franzini (1965), Fluid Mechanics, 6th Ed., McGraw-Hill.

MEM311: Thermal and Fluid Science Laboratory Experiment 2: Control Volume Energy and Entropy Analysis in a Vortex Tube

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EXPERIMENT 2:

CONTROL VOLUME ENERGY AND ENTROPY ANALYSIS IN A

VORTEX TUBE by Ertan Agar 2010, adapted from Bakhtier Farouk and David Staack

A. INTRODUCTION AND OBJECTIVES The vortex tube also known as the Ranque-Hilsch vortex tube is a unique device which

converts a flow of compressed gas into two streams – one hotter and the other colder than

the gas supply temperature. It contains no moving parts and the mechanism of its

operation is still a subject of debate, yet the usually agreed upon explanation will be given

herein. This vortex effect was discovered by G. Ranque in 1928. The United States became

focused upon the vortex tube in 1947 when R. Hilsch published a technical paper reporting

research on the device (Ref. 1). Since that time, many technical applications of vortex tubes

for cooling, air conditioning, and drying have been developed (Ref. 2).

The vortex tube is a simple mechanical device that diverts a flow of compressed gas into

two separate streams, one hot and one cold relative to the gas supply temperature. They

are commonly used to prevent thermal damage by providing spot cooling to complex

mechanical or electrical systems. Other technical applications of this technology include air

conditioning, drying, and recovering waste pressure energy from both high and low

pressure sources. The general flow distribution inside a vortex tube is depicted below in

Figure 1.

Figure 4: General Flow Pattern inside a Vortex Tube


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Figure 1-a shows high pressure compressed air entering the vortex tube. The compressed

air accelerates to a high rate of rotation due to its tangential injection point.[1] As a result, a

strong vortex flow is produced inside the tube. Figure 1-b shows hot air exiting the control

volume on the right side of the device. The remainder of the compressed gas is forced to

travel back across the high speed air stream and exit as extremely cold air as shown in

Figure 1-c. Through an energy and entropy analysis of this flow pattern, the first and

second laws of thermodynamics can be validated by experimentally determining the total

rate of entropy creation per mass of air flowing through the vortex tube.

The objective of this experiment is to apply a control volume energy and entropy analysis

to a practical engineering device, a vortex tube. The energy separation phenomenon

induced by the vortex fluid motion will be investigated and explained using basic

thermodynamic principles.

B. THEORY

Principles of Operation

On the basis of flow visualization studies, Hartnett and Eckert (Ref. 3) found that the axial

velocity component (velocity component along the length of the tube) was relatively small

over most of the radius of the tube. Therefore, the flow can be analyzed by evaluating one

plane through the vortex tube perpendicular to the tube axis as shown in Figure 3.

Hartnett and Eckert also observed that the flow consisted of a colder region in the center of

the tube that rotated as a solid body having a circumferential velocity, v = r, where is

the constant angular velocity of the fluid. In fluids, this type of rotation is called a forced

vortex because it is vortical flow, which is induced by an external force, in this case, the

outer stream. In the outer stream, the circumferential velocity is proportional to 1/r and

therefore decreases as r increases, i.e., v = K/r where K is a constant. This type of vortex is

called a free vortex, a common example of which is the vortical motion of water as it goes

down the drain in a bath tub. In a true free vortex, the circumferential velocity goes to zero

as r goes to infinity. Therefore, the outside stream only approximates a free vortex because

r must be less than or equal to rw, the radius of the tube. A flow with a forced vortex inside

and a free vortex outside is called a combined vortex and has a circumferential velocity

profile given by the following equations

v = r r ro (1a)

v = K/r r > ro (1b)


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where ro is the radius of the central flow, as shown in Figure 2. Given this velocity profile,

we can determine why the temperature separation occurs and why the hotter outer stream

surrounds the cooler inner core.

We will begin this analysis by evaluating the pressure distribution from the centerline of

the tube to the outer wall. The streamlines for the flow in the vortex tube form closed

concentric circles. Evaluating F = ma normal to a streamline, as shown in Ref. 4, Section

3.3, we find the change of pressure in a direction normal to a streamline is give by

R

V

n

p 2

(2)

where R is the local radius of curvature of the streamline and V is the velocity along the streamline.

However, the streamlines in a combined vortex form closed concentric circles, as shown in

Figure 3. Therefore, R = r and the velocity, V, is simply vq. Also, in Eq. (2), the positive n

direction points toward the “inside” of the curved streamline, i.e., opposite to the positive

direction of the radial coordinate, r. Therefore,

rn

(3)

Figure 2: Velocity profile in a vortex tube


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Substituting these results into Eq. (2), we find the radial pressure distribution in a vortex to

be

r

v

n

p2

(4)

Figure 3: Streamlines in a Vortex Tube

Substituting Eqs. (1a) and (1b), respectively, into Eq. (4), we obtain

rr

p 2

(5a)

and

3

2

r

K

r

p

(5b)

Equations (5a) and (5b) show that in both the free and forced vortex regions, the pressure

increases as r increases. Integrating these equations with respect to r, starting with a

known pressure p = p1, we find the pressure distribution in the vortex tube to be given by

1

22 pr2/1p r ro (6a)

)r/1r/1(K2/1p 22

o

2 r > ro (6b)

where po is the pressure at r = ro. The value of po is found by evaluating Eq. (6a) at r = ro to

obtain

1

22 pr2/1p (7)


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This pressure profile is plotted as a function of r in Figure 4. From this figure, we see that

the pressure is lowest at the center of the tube and increases to a maximum at the wall.

Herein lies the reason for the temperature difference between the inner and outer streams.

Recall the piston-cylinder devices that were studied extensively in basic thermodynamics.

The first law energy balance indicates that if a gas is adiabatically (no heat transfer)

compressed in a piston-cylinder, the internal energy and hence, the temperature, must

increase. As the gas enters the vortex tube, the viscosity of the fluid induces a vortical

motion which creates a forced vortex at the center of the tube. This flow produces the

pressure distribution given by Eqs. (6a) and (6b). The gas on the outside of the tube is

adiabatically compressed resulting in an increase in temperature. This gas is also at a

higher pressure so it can be drawn off at the control valve. The work to compress the outer

gas came from the gas near the centerline which is adiabatically expanded and cooled. The

cooler gas is confined to the inner core of the tube so it can be withdrawn from the

opposite end of the tube through an orifice plate. Note that even though the pressure is

lowest at the center of the tube, it is still greater than atmospheric pressure and will flow

out of the orifice. Therefore, the separation of the gas into two streams having different

temperature is caused by viscous forces in the gas which induces a pressure distribution in

the tube. The gas in the high pressure region is compressed and heated while that in the

low pressure region is expanded and cooled.

Figure 4: Radial pressure distribution

This description of the operation of a vortex tube resulted only after many experimental

observations and a detailed analysis. The first and second laws of thermodynamics,

however, present us with a simple way to evaluate any thermodynamic system to

determine whether it is thermodynamically valid. If we ever determine that a proposed

process violates either the first or second law, we know that the process is impossible. In

this experiment, you will perform a first and second law analysis of a vortex tube to

“examine” its performance.


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First and Second Law of Thermodynamics for a Control Volume

All thermodynamic analyses begin by defining the system to be evaluated. A vortex tube is

an example of a steady-state, steady-flow device and is most easily represented by a control

volume. Reference 6, Section 5-4 (Reference 5, Section 4.5) gives the general first law

energy balance for such a control volume as

∑ .

/ ∑ .

/ (8)

where is the heat, the work, the mass flow rate in and out respectively, ,

the specific enthalpy of mass in and out, , the velocity of mass in and out, the

elevation of mass in and out, and lastly, is gravitational acceleration. (The form of the first

law of thermodynamics given in Ref. 5 is slightly different than that given in Eq. 8.

Primarily, Ref. 5 defines the work done by a system as positive (ASME sign convention)

whereas this work is negative in Ref. 6 (scientific sign convention). The equations are

equivalent and you must be able to use both of them. The second law of thermodynamics is

given by the entropy generation principle for a steady-state, steady-flow control volume

(see Ref. 5, Section 6.2 or Ref. 6, Section 7-2) as

∑ ( ) ∑ ( ) ∑

(9)

where

is the total rate of entropy change for the control volume, and is the rate

of internal generation of entropy within the system (intrinsic entropy associated with

matter, for irreversible systems, for reversible systems, and system is

impossible if ). In this experiment, pressurized air is used to drive the vortex tube.

The ideal gas equations can therefore be used to evaluate Eqs. (8) and (9). See Ref. 5,

Section 2.8 and Section 5.9 (Ref. 6, Chapter 3 and Section 7-4) to review the application of

these equations to ideal gases. Pay particular attention to the evaluation of the change in

entropy for an ideal gas.


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C. EQUIPMENT

Figure 5: Experimental Setup

Figure 5 shows the experimental setup and all the components and measuring devices.

Compressed air enters the system at the top left of the picture and goes through a de-

humidifier, then a pressure control and a pressure gage. Then it passes through a mass flow

meter. See Figure 6 for a close-up of the inlet setup.

Figure 6: Inlet setup


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After the flow enters the system and has the properties desired, it then goes to the vortex

tube and separated into a hot and cold stream. Below Figure 7 is a close-up picture of the

vortex tube itself and the direction of the hot and cold streams.

Figure 7: Vortex Tube

After leaving the vortex tube both streams pass through a series of measurement devices

similar to the ones monitoring the inlet flow. For the cool stream there is a mass flow

meter, a temperature indicator connected to a thermocouple and a pressure gage, see

Figure 8. A throttle valve is used to control the cold air pressure.

Figure 8: Cold stream measuring device

The cold stream uses a pressure gauge and a temperature indicator in conjunction with a

digital mass flow meter. For the hot stream, pressure and temperature are similarly

measured, but a rotameter is used to measure mass flow rate. A rotameter has a ball in

between two tapered tracks and the air pressure flowing through the meter pushes the ball

up to a certain height and the height markings on the vertical meter correlate to a flow rate

(See Appendix B for rotameter function). A throttle valve and a muffler are also used on

both outlet air streams. See Figure 9 for the layout of the hot stream measuring devices.


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Figure 9: Hot stream measuring devices

D. PROCEDURE

1. Familiarize yourself with the general performance of the vortex tube by

manipulating the pressure regulating valve and the discharge throttling valves

found on the hot and cold ends.

2. Close the hot flow valve and open cold valve as much as possible to experimentally

determine the maximum inlet pressure for the hot flow stream. This will allow for a

complete range of on-scale flow readings for all throttling valve positions.

3. Set the inlet pressure of the main flow to 60 psi. Open the hot stream flow,

establishing a constant mass flow rate. Take 5 readings at even intervals of cold

mass flow rate values.

4. Record all temperatures, pressures, and flow rates at steady state (typically takes 4-

5 minutes) for 5 evenly spaced cold mass flow rate values.

5. Repeat steps 2-4 for the hot mass flow rate intervals.

6. Repeat all experiments at 80 psi.

E. PRE-LAB

1. Complete the assigned thermodynamic problems. These are intended to provide a

review of the control volume analysis using the First and Second Law of

Thermodynamics.

A vortex tube has an air inlet flow at 20oC, 200 kPa and two exit flows of 100 kPa, one

at 0oC and the other 40oC. The tube has no external heat transfer and no work all the


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flows are steady and have negligible kinetic energy. Find the fraction of the inlet flow

that comes out 0oC. Is this setup possible?

A Hilch (Vortex) tube has an air inlet mass flow of 50 SLPM at 20oC, 200 kPa and two

exit flows of 100 kPa one at 0oC and the other t 40oC. The tube has no external heat

transfer and no work and all the flows are at steady state and have negligible kinetic

energy. Find the fraction of the inlet flow that comes out at 0oC. Is this setup possible?

2. After reviewing the lab procedures, prepare a data sheet on which to record your

experimental measurements (pressure, temperature and flow rate for main stream, hot

stream and cold stream). Bring this sheet to the lab to record your data.


1. Calculate the volumetric and mass flow rates from the rotameter and mass flow-meters data. All volumetric flow rates (both cold plus main inlet flows) must be corrected to the actual pressure and temperature of the flowing gas to obtain an accurate volumetric flow rate. Hint: ideal gas state equation may be used.

2. Perform a mass balance for each set of operating conditions with using mass flow meters data to determine the flow rate of rotameter. Then compare the calculated value with the measured flow rate of rotameter obtained in the experiment. Use graph to show the relative error.

3. Perform an energy balance (Eq. (8)) to solve for the rate of heat transfer for each set of operating conditions from the control volume of the vortex tube. Discuss your result with drawing heat transfer rate graph for different conditions.

4. Evaluate the total rate of entropy generation (Eq. (9)) flowing through the control volume using your measurements and heat transfer rate calculated in 3 above. The property data required can be found in Ref. 5 and should be included in the sample calculations. Discuss your result with drawing rate of entropy generation graph for different conditions.

5. Do your results satisfy the Second Law of Thermodynamics (Increase-in-Entropy Principle)? What does this imply about the process that occurs in a vortex tube?


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G. REFERENCES

1. Hilsch, R., “The Use of the Expansion of Gases in a Centrifugal Field as a Cooling Process,”

Review of Scientific Instruments 18, 1947.

2. Hartnett, J.P. and Eckert, E.R.G., “Experimental Study of the Velocity and Temperature

Distribution in a High-Velocity Vortex-Type Flow,” Transactions of the ASME 79, 1957, pp.

751-758.

3. Eckert, E.R.G. and Drake, R.M., Jr., Analysis of Heat and Mass Transfer, McGraw-Hill, New

York, 1972, pp. 427-430.

4. Munson, B.R., Young, D.F., and Okiishi, T.H., Fundamentals of Fluid Mechanics, John Wiley

and Sons, 1990.

5. Black, W.Z. and Hartley, J.G., Thermodynamics (2nd edition), Harper Collins Publishers, New

York, 1991.

6. Wark, Kenneth, Jr., Thermodynamics (5th edition), McGraw-Hill, New York, 1988.

7. Holman, J.P., Experimental Methods for Engineers. 3rd ed., McGraw-Hill, 1978.

MEM311: Thermal and Fluid Science Laboratory Experiment 3: Heat Transfer from a Circular Cylinder

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EXPERIMENT 3: HEAT TRANSFER FROM A CIRCULAR

CYLINDER by Eric Wargo 2010, adapted from Bakhtier Farouk and David Staack

A. OBJECTIVES The objective of this experiment is to determine the natural convection, forced convection and radiation heat transfer (qnatural convection + qforced convection + qradiation) from an electrically heated horizontal cylinder. These values will be compared to existing heat transfer correlations provided herein. In this experiment, you will demonstrate how heat transfer from a heated surface to a quiescent environment is a combination of several mechanism of heat loss. The relative magnitudes of the natural convection, forced convection, and radiation heat transfer coefficients depend on the surface temperature and flow velocity. Radiation becomes more important as the surface temperature increases. Forced convection becomes more important as the flow velocity increase. The problem will be analyzed using a control volume under equilibrium conditions. For equilibrium, heat input to a surface must equal the heat transferred from the surface to the surroundings.

B. THEORY Natural and Forced Convection

Free convection heat transfer occurs whenever a body is placed in a fluid at a higher or lower temperature. As a result of the temperature difference, heat is transferred between the fluid and the body and causes a change in the density of the fluid layers in the vicinity of the surface. This difference in density leads to an upward flow of the lighter fluid (Figure 5). If the motion of the fluid is caused solely by differences in density resulting from temperature gradients, the associated heat transfer mechanism is called free or natural convection. If the fluid motion is enhanced using a fan or otherwise forced by some device, the heat transfer mechanism is called forced convection (Figure 6). Because the fluid velocity is usually less in free convection than in forced convection, the rate of heat transfer from a surface is also generally less. In this experiment, you will measure and compare the magnitudes of the heat transfer rates for forced convection and free convection configurations.


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Figure 5. Cross sectional view of a heated cylinder under natural convection.

Figure 6. Cross sectional view of a heated cylinder under forced convection.

The rate of heat transfer by convection (both forced and free) between a surface and a fluid may be computed using the relation ( ) (1) where is the average convective heat transfer coefficient, A is the area available for heat transfer, Ts is the surface temperature, and T∞ is the ambient temperature. The relation expressed by Equation 1 was originally proposed by the British scientist, Sir Isaac Newton in 1701. Therefore, it is sometimes referred to as ‘Newton’s law of cooling’. Even though this equation has been used for many years to evaluate convective heat transfer, it is actually more a definition of than a law of convection. If the value of is known for a certain flow configuration, the evaluation of Equation 1 to determine the rate of heat


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transfer is straightforward. However, determining the appropriate convective heat transfer coefficient is difficult because convection is a very complex phenomenon. The value of depends not only on the geometry of the surface of the object (both macroscopic and microscopic surface characteristics) but on the velocity and physical properties of the fluid, all of which affect the conditions on the boundary layer. Since these quantities are not necessarily constant over a surface, the heat transfer coefficient may vary from point to point. For this reason, we must distinguish between a local and average convective heat transfer coefficient. The local coefficient, hc, is defined by ( ) (2) while the average coefficient, , can be defined in terms of the local value by

∫ (3)

The primary problem in either forced or free convection is to determine the appropriate local or average heat transfer coefficient. Many experiments have been performed to measure these coefficients for a wide variety of geometries and flow configurations. Numerical calculations have only recently become sufficiently exact to calculate the heat transfer coefficients directly. However, heat transfer coefficients can be accurately calculated for relatively simple flow configurations. Complex configurations must still be determined experimentally. In this experiment, you will determine the heat transfer coefficient of free convection and forced convection from a cylinder placed within a range of flow conditions. Your results will be compared with existing heat transfer correlations.

Radiation Heat Transfer

Thermal radiation is heat transfer by the emission of electromagnetic waves which carry energy away from the emitting object. For ordinary temperatures (i.e. less than ‘red hot’), the radiation is in the infrared region of the electromagnetic spectrum. The relationship governing radiation from hot objects is called the Stefan-Boltzmann law. The heat transferred into or out of an object by thermal radiation is a function of several components. These include its surface emissivity, surface area, temperature, and geometric orientation with respect to other thermally participating objects. The heat loss rate caused by radiation from a heated surface to the surroundings can be calculated by (

) (4)

where σ is the Stefan-Boltzmann constant (= 5.67x10-8 W/ m2·K4), ξ is the emissivity of the surface, and Fs–a is the view factor of the surface. The surface temperature is given by Ts, and the temperature of the body receiving the radiation (the ambient environment) is given by T∞. An object's surface emissivity is a function of its surface microstructure. The view factor takes into account the geometric orientation of the surface to the external environment. If all of the radiation emitted by the surface has a direct line of sight to the


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external environment, the view factor is equal to 1. Equation 4 can also be cast in the form of Newton’s law of cooling (Equation 1) as ( ) (5) Comparing Equation 4 with Equation 5 we can define a radiative heat transfer coefficient as

(

)

( ) (6)

Natural Convection, Forced Convection, and Radiation Heat Transfer

If a surface is at a temperature above that of its surroundings and is located in stationary or moving air, heat will be transferred from the surface to the surroundings. This transfer of energy will be a combination of natural convection, forced convection (if there is a driven air flow) and radiation to the surroundings. As described above in natural convection, the motion of the fluid is caused solely by differences in density resulting from the temperature gradients. If the fluid motion is enhanced using a fan or otherwise forced by some device, the heat transfer mechanism is called forced convection. Radiation heat transfer generally becomes significant at surface temperatures well above room temperature. However, it does play a role at lower temperatures and should be accounted for, especially when comparing measurements with existing correlations for natural and forced convection. Heat loss by conduction would normally be included in the analysis of a real application. In this experiment, it is minimized by the design of the equipment and experimental procedures. The total heat lost by a surface can thus be presented as a linear superposition of the aforementioned heat losses: (7)

When there is no forced flow, the term can be neglected. When there is an air flow

typically the heat lost due to the forced convection is significantly greater than that by natural convection; thus, the term can be neglected. Equation 7 can be written in

terms of the various heat transfer coefficients:

( ) . / ( ) (8)


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Experimental Determination of the Heat Transfer Coefficients

Solving Equation 8 for the heat transfer coefficient, we obtain

( ) (9)

The heat transfer coefficient can be determined by measuring all the quantities on the right hand side of Equation 9 for a specific flow configuration and solving for . The temperatures Ts and T∞ are easily measured using thermocouples. The area, A, is simply the surface area available for convective heat transfer (sometimes called the wetted area). This can be evaluated once the experimental configuration is defined. Therefore, the problem is reduced to determining the rate of heat transfer, . Recall that the first law of thermodynamics for a control mass can be written in infinitesimal form as (10) Recall from thermodynamics that passing an electrical current, I, through an object is a form of work given as (11) where V is the applied voltage, and R is the resistance. The relations in Equation 11 are based on Ohm’s laws, V = IR and P = VI, where P is the power. Substituting Equation 11 into Equation 10, dividing by a small increment of time δt, and taking the limit as δt goes to 0, we obtain

(12)

Where is the rate of heat transfer and (V2/R) is the rate at which work is done. Since the heated cylinder can be considered to be an incompressible substance, the change in its internal energy is given as (13) where m is the mass of the cylinder, and c is the specific heat. Recall that for an incompressible substance, cp = cv = c (see Section 2.8 of Reference [1], or Sections 4-7 of Reference [3] for a review of these thermodynamic relationships). Since m and c are constant, the rate of change of internal energy is

(14)


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If the electrical current to the cylinder is controlled so that the temperature remains

constant,

, and therefore,

, is identically equal to zero. Such a situation happens in

steady state operation. For this condition, Equation 12 reduces to the following form:

(15)

The negative sign in Equation 15 indicates that the heat transfer is out of the cylinder. Substituting Equation 15 into Equation 9 yields

( )

( ) (16)

The surface temperature, TS, and the temperature of the cylinder, T, are interchangeable, because it is assumed that there are no temperature gradients inside the cylinder. This is true only if the cylinder is made of a material that conducts heat rapidly. The Biot number, defined as

(17)

provides a measure of the accuracy of this assumption. In Equation 17, L is the characteristic dimension (cylinder diameter, around which the fluid is flowing) and ks is the thermal conductivity of the material. Physically, the Biot number is the ratio of the external convective heat transfer rate to the internal conductive heat transfer rate. If Bi<<1, heat is conducted within the material much faster than it is convected away from the cylinder. The assumption of uniform temperature is then valid and Equation 16 can be used to evaluate the overall rate of heat transfer. If Bi is greater than approximately 0.1, the non-uniformities within the material must be accounted for when determining the heat transfer coefficient. Equation 16 shows that the heat transfer coefficient can be calculated by measuring the electrical voltage and current supplied to the cylinder such that the cylinder temperature of the cylinder is constant. Comparing Equation 16 and 8 we can simply write this as (18)

which simply states that in steady state the heat flow out of the cylinder is equal to the energy flow into the cylinder.

Heat Transfer Correlations

As shown by the equations above, the heat transfer coefficient is a dimensional number that depends on the area (size) of the object being evaluated and the temperature/properties of the fluid in which the object is immersed. When developing


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experimental correlations, it is convenient to non-dimensionalize so that the results are applicable to other configurations. The Nusselt number is a dimensionless heat transfer parameter defined as

(19)

where L is the characteristic dimension of the geometry being evaluated (cylinder or sphere diameter, around which the fluid is flowing), and kf is the thermal conductivity of the fluid in which the object is immersed. The Nusselt number is defined as the ratio of convective to conductive heat transfer across a boundary (the object’s surface); thus, radiation effects should not be included in its calculation. The overbar on the Nusselt number indicates that it is formed from the average heat transfer coefficient and, therefore, is a measure of the average heat transfer from the object. Other dimensionless parameters are used that represent the air flow conditions. For forced convection flows, the Reynolds number is used to correlate heat transfer data. The Reynolds number is defined as

(20)

where ρ is the density, V is the velocity, L is the characteristic length (cylinder diameter, around which the fluid is flowing), μ is the absolute viscosity, and ν is the kinematic viscosity and is equal to μ/ρ. In the case of natural convection, Nu depends on the Rayleigh number, Ra. The Rayleigh number, in turn, is frequently written in terms of two other non-dimensional numbers, namely the Grashof and the Prandtl numbers, Gr and Pr respectively. These are given by the equations:

( )

(21)

(22)

( )

(23)

The new symbols in Equations 21 – 23 are explained in Table 2. Note: D is the characteristic dimension (cylinder diameter, around which the fluid is flowing).


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Table 2. Symbol definitions for Equations 21 – 23.

Symbol Definition Value and/or Units

g gravitational acceleration 9.81 m/s2

β volume expansion coefficient, 1/Tfilm K-1

ν kinematic viscosity, µ/ρ m2/s

cp specific heat J/kg·K

ρ density kg/m3

α thermal diffusivity m2/s

k thermal conductivity W/m·K

All of the thermophysical properties given in Equations 19–23 are functions of the temperature of the fluid. Since the temperature varies from the surface to the ambient, by convention these properties are evaluated at the film temperature

(24)

Values of the thermophysical properties listed in Table 2 can be found in Appendix A of Reference [2] or any other heat transfer textbook. At the very least, a reputable source should be utilized and properly referenced in the lab report; online calculators are convenient, but their range of accuracy may be questionable. These equations form the basis for the experimental method to experimentally evaluate the heat transfer coefficient. A more detailed discussion of forced and free convective heat transfer can be found in Reference [2], Chapters 7 and 9, respectively. Specifics regarding the heat transfer from a circular cylinder in cross flow may be found in Reference [2], Section 7.4 (forced convection) and Section 9.6.3 (free convection). Reference [2] gives the following empirical correlation for the Nusselt number, Nu, for forced convection as a function of the Reynolds number: ( ) (25) where c and n are obtained from Table 3.


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Table 3. Reynolds number-Nusselt number empirical relation parameters.

Re c n

1 – 40 0.75 0.4

40 – 1000 0.51 0.5

103 – 2x105 0.26 0.6

2x105 – 106 0.076 0.7

Reference [3] gives the following empirical correlation for the Nusselt number, Nu, for natural convection as a function of the Rayleigh number: ( ) (26) where c and n are obtained from Table 4.

Table 4. Raleigh number-Nusselt number empirical relation parameters.

Ra c n

10-10 to 10-2 0.675 0.058

10-2 to 102 1.02 0.148

102 to 104 0.850 0.188

104 to 107 0.480 0.250

107 to 1012 0.125 0.333

For a given Ra, the Nu value can be computed from Equation 15 and the corresponding value from Equation 9. There are other equations that are sometimes used to calculate the natural convection heat transfer coefficient. One simplified, analytical relation that may be used to calculate the average heat transfer coefficient from a horizontal cylinder is given by

0( )

1

(27)

where D is the cylinder diameter. Equation 27 yields in W/m2·K if the

temperatures are in Kelvin and D is in meters. In the experiment, you will compare your


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experimental results with the predictions of Equations 25, 26, and 27. Note: care must be take to distinguish Ra and Re.

C. EQUIPMENT 1. Armfield HT14 Combined Convection and Radiation Accessory (see Figure 7 and Figure

8) 2. Armfield HT10X Heat Transfer Service Unit (see Figure 7 and Figure 9) 3. Computer on site or laptop for manual data entry (optional but highly recommended,

since you will be recording the surface temperature in 30 second time intervals for durations sometimes totaling over 10 minutes)

Figure 7. Experimental set-up.


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Figure 8. Schematic View of the Armfield HT14 Combined Convection and Radiation Accessory.

Figure 9. Armfield HT10X Unit.


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The HT14 accessory consists of a centrifugal fan with vertical outlet duct at the top of which is mounted a heated cylinder. The heated cylinder has an outside diameter of 10mm, a heated length of 70mm and is internally heated throughout its length by an electric heating element. The heating element is rated to produce 100 Watts nominally at 24V DC into the cylinder. The power supplied to the heated cylinder can be varied and measured on the HT10X unit. The mounting arrangement for the cylinder in the duct is designed to minimize loss of heat by conduction to the wall of the duct. The surface of the cylinder is coated with heat resistant paint which provides a consistent emissivity close to unity. A K-type thermocouple is attached to the wall of the cylinder (T10), at mid position, allowing the surface temperature to be measured under the various operating conditions. A variable throttle plate at the inlet to the fan allows the velocity of the air through the outlet duct to be varied, and a vane type anemometer within the fan outlet duct allows the air velocity in the duct to be measured over the range 0-7 meters/sec. The inside diameter of the outlet duct is 70mm (matching the length of the heated cylinder). A K-type thermocouple is also located in the air duct (T9), allowing the ambient air temperature to be measured upstream of the heated cylinder. The HT10X unit is used to control the heating of the cylinder. The voltage is controlled by a rotary knob, and the top digital display can be used to alternatively view the voltage, current, and air velocity by using the selector knob. The temperature at the thermocouples is shown on the bottom digital display, and the thermocouple selector knob determines which temperature is displayed. Connections for additional accessories and computer data acquisition and control are available on the service unit but are not used in this experiment.

D. PROCEDURE ***Before operating the equipment, please understand that the heated cylinder will reach temperatures above 500°C. Serious skin burns will result if the equipment is mishandled. Please ask an instructor if you have any questions or concerns. Equipment Set-up

Before proceeding with the exercise, ensure that the equipment has been prepared as follows: a. Locate HT14 Combined Convection and Radiation accessory alongside the HT10X Heat

Transfer Service Unit on a suitable bench.

b. Ensure that the horizontal cylinder is located at the top of the vertical metal duct with the T10 thermocouple attached.

c. Connect the thermocouple attached to the horizontal cylinder to socket T10 on the front of the service unit.

d. Connect the thermocouple located in the vertical duct to socket T9 on the service unit.


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e. Set the voltage control potentiometer to minimum (counterclockwise) and the selector switch to MANUAL.

f. Connect the power lead from the heated cylinder on the HT14 to the socket marked O/P3 at the rear of the service unit.

g. Ensure that the service unit is connected to an electrical supply.

Experimental Procedures

You will perform 2 sets of experiments: one for natural convection, and another for forced convection. For the natural convection conditions, you will measure the steady state temperature achieved on the cylinder for 5 different heating voltages (4, 7, 10, 13, and 16 volts). For the forced convection cases, you will measure the steady state temperature achieved on the cylinder for 5 different flow velocities (approximately equally spaced between 0 m/s and the maximum attainable by the device), all at a heating voltage of 10 volts. Perform the forced convection experiments first, as they achieve steady state more quickly. Record the temperatures at time intervals of 30 seconds to be able to identify when steady state has been achieved (steady state can be considered achieved when the surface temperature (T10) remains constant for 2 minutes or four, 30 second readings). 1. Prepare a data sheet on which to record the raw data. It is highly recommended that you

record your data directly into a Microsoft Excel spreadsheet, since you will be recording the surface temperature in 30 second time intervals for durations sometimes totaling over 10 minutes. The spreadsheet should have the following headings:

Table 5. Variables to record in datasheet.

Variable Symbol Units

Time t min:sec

Heater voltage V volts

Heater current I amps

Upstream air temperature, T9 T∞ °C or K

Cylinder surface temperature, T10 Ts °C or K

Upstream air velocity Ua m/s

2. Turn on the front main switch (Figure 7) (if the panel meters do not illuminate, check

the circuit breakers at the rear of the service unit). All switches at the rear should be up.


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3. Turn on the fan switch and open the throttle plate until the knob stops turning, ensuring the maximum air velocity is attained.

4. Set the heater voltage to 10 volts.

5. Record the experimental conditions as a function of time (30 second intervals) until a steady state is achieved. Steady state can be considered achieved when the surface temperature (T10) remains constant on the display for 2 minutes (four, 30 second readings).

6. When the temperature is stable, record V, I, T9, T10, and Ua. Note: the digital display outputs temperature readings in degrees Celsius.

7. Repeat steps 5 and 6 for an additional four flow velocities: 75%, 50%, 25%, 0% (letting the heater remain at 10 volts). A flow velocity of 0% is best attained by shutting the fan off. The throttle plate should be opened completely to allow for air to flow naturally through the duct during free convection.

8. The last condition for forced convection above represents the first free convection condition. Repeat steps 5 and 6 for heater voltages of 4, 7, 13, and 16 volts. In all you should measure 9 conditions, and if you deem appropriate you may change the order.

Suggestion: This lab can be completed the fastest if (after obtaining data for forced

convection at 100% – 25% flow velocity) the heated cylinder is allowed to cool back down by turning the voltage off and throttling the flow velocity back to 100% for a few minutes. Then, repeat steps 5 and 6 for heater voltages of 4, 7, 10, 13, and 16 volts with the fan off and throttle plate open completely. This ensures that the cylinder is only ever heating up to steady state for all conditions, rather than cooling down naturally (which takes far more time).

The order will be as follows:

Voltage 10 10 10 10 cool

down

4 7 10 13 16

Flow rate 100% 75% 50% 25% 0% 0% 0% 0% 0%

9. Return the voltage to 0 volts, and turn off the unit. Note: Do not set the heater voltage in excess of 16 volts when operating the cylinder in the natural convection mode (no forced airflow). The life of the heating element will be considerably reduced if operated at excessive temperature. If temperatures approach 550°C check with an instructor, and if the temperatures exceed 600°C shut off the unit.


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Use the following data for your calculations:

Diameter of horizontal cylinder D = 0.01 m Heated length of cylinder L’ = 0.07 m Emissivity of surface of the cylinder ξ = 0.75 Stefan-Boltzmann constant σ = 5.67x10-8 W/m2·K4 View Factor Fs – a = 1

E. PRE-LAB Assume an experimental setup similar to the Armfield facility and an ambient temperature of 25°C. 1. What is the proper temperature unit (°C or K) to be used in Equation 4? Why?

2. For a cylinder surface temperature of 500°C, use Equation 4 to calculate the power lost by radiative heat transfer.

3. For a cylinder surface temperature of 500°C, calculate a Raleigh number using Equation 23 and appropriate properties based on the film temperature. Use this Raleigh number to calculate a Nusselt number by Equation 26 and Table 4. Use Equation 19 and the Nusselt number to calculate a corresponding free convection heat transfer coefficient for the cylinder. Now use Equation 1 to calculate the heating power which is dissipated by natural convection at this surface temperature.

4. For a cylinder surface temperature of 500°C and a flow velocity of 5 m/s, calculate a Reynolds number using Equation 20 and appropriate properties based on the film temperature. Use this Reynolds number to calculate a Nusselt number by Equation 25 and Table 3. Use Equation 19 and the Nusselt number to calculate a corresponding forced convection heat transfer coefficient for the cylinder. Now use Equation 1 to calculate the heating power which is dissipated by forced convection at this surface temperature.

5. Sum the powers lost by radiation, forced convection and free convection to get the total heating power required to maintain this temperature. What percentage are the various mechanisms of the total power? If there were no forced convection, how do radiation and free convection compare? Briefly discuss these results.

6. After reviewing the lab procedures, prepare a data sheet to record your experimental measurements. It is highly recommended that you create a data sheet in Microsoft Excel and bring it to lab along with a laptop, so you can quickly key in temperature readings during the lab. Bring this data sheet and/or a laptop computer to the lab to record your data.


36 | P a g e

F. DATA ANALYSIS AND REPORT REQUIREMENTS 1. For all 9 test conditions, plot the measured temperature as a function of time. How long

does steady state take to attain? What conditions determine steady state? What methods could be employed to attain steady state faster?

2. For each of the five test conditions corresponding to natural convection:

a. Calculate and tabulate the following parameters using the raw data and provided theory.

Cylinder temperature K Film temperature Tfilm K Rayleigh number Ra Heat flow (power input) W Heat transferred by radiation W Heat transfer coefficient (radiation) W/m2·K Heat transferred by natural convection W

(Us Equation 18 and assume that forced convection is negligible) Heat transfer coefficient (natural convection) W/ m2·K Nusselt Number (natural convection) Nu

b. Compute the heat transfer coefficient and Nusselt number (natural convection) from

the analytical relation, Equation 27. Compute a Nusselt number and heat transfer coefficient from the calculated Raleigh number and empirical correlation, Equation 26. Use Equation 19 to switch between h and Nu.

c. Compare the analytical and empirical calculations of and obtainable from

Question 2.b with the values for and calculated only from the experimental measurements (Question 2.a) and discuss any differences in them. Plot the measured and calculated values of versus the surface temperature on a single graph. [hint: there should be 3 series on the plot]

d. On a single graph plot , , and as a function of the average surface

temperature . Comment on the relationship between and . Comment on the relative importance of natural convection and radiation heat transfer. At what temperature (if any) is = ?

3. For each of the test conditions corresponding to forced convection:

a. Calculate and tabulate the following parameters using the raw data and equations given above. (For cases where cylinder temperature distribution was not measured use the temperature measurement at the steady state condition)

Cylinder temperature K Film temperature Tfilm K Reynolds number Re Heat flow (power input) W Heat transferred by radiation W


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Heat transfer coefficient (radiation) W/ m2·K Heat transferred by forced convection W

(Us Equation 18 and assume that natural convection is negligible) Heat transfer coefficient (forced convection) W/ m2·K Nusselt Number (forced convection) Nu

b. Compute a Nusselt number and heat transfer coefficient from the calculated Reynolds

number and empirical correlation, Equation 25. Use Equation 19 to switch between h and Nu.

c. Compare the correlation calculated values of and obtainable from Question

3.b with the values for and calculated only from the experimental measurements (Question 3.a) and discuss any differences in them. Plot the measured and calculated values of versus the surface temperature on a single graph.

d. Plot as a function of the flow velocity. Comment on the relationship.

e. Plot Nu vs. Re from Equation 25 for Re from 100 to 5000. On the same graph plot Nu vs. Re attained from the experiment (Question 3.a). How close are the experimental data points to the correlation line? Comment on the non-dimensional comparison.

4. For this lab, the surface temperature of the heated cylinder was only measured at a single point. If several temperature readings were obtained from locations around the cylinder diameter, what would you expect the circumferential distribution to look like? Comment on the effects of using an average temperature versus a single point temperature measurement on the calculations made above.

G. REFERENCES [1] Holman, J. P. Experimental Methods for Engineers. 7th ed. Boston: McGraw-Hill, 2001.

Print. [2] Incropera, F. P., and D. P. DeWitt. Fundamentals of Heat and Mass Transfer. 5th ed. New

York: Wiley, 2002. Print. [3] Munson, B. R., D. F. Young, and T. H. Okiishi. Fundamentals of Fluid Mechanics. 4th ed.

New York: Wiley, 2002. Print.

MEM311: Thermal and Fluid Science Laboratory Experiment 4: Performance Analysis of a Steam Turbine Power Plant

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EXPERIMENT 4:

PERFORMANCE ANALYSIS OF A STEAM TURBINE POWER

PLANT by Chris Dennison 2010, adapted from Bakhtier Farouk and David Staack

A. OBJECTIVE

The objective of this laboratory is to offer students hands-on experience with the operation

of a functional steam turbine power plant. A comparison of real world operating characteristics to

that of the ideal Rankine power cycle will be made.

The laboratory is conducted using a miniaturized steam turbine power plant. The apparatus

is scaled for educational use and utilizes components and systems similar to full-scale industrial

facilities. Students will be able to operate and analyze this system in detail, allowing them to

determine the efficiency of the facility and suggest possible modifications for further improvement.

Figure 10. Miniature steam turbine power plant

MEM311: Thermal & Fluid Science Laboratory Experiment 4: Performance Analysis of a Steam Turbine Power Plant

39 | P a g e

B. THEORY

One of the most important ways to convert energy from fossil fuels, nuclear, and solar radiation is

through processes known as vapor power cycles. One example of the use of vapor power cycles is

electrical power plants. As engineers it is important to become familiar with these types of

systems. The first step in becoming familiar with these cycles is by studying the idealized cycles.

The ideal cycle for vapor power cycles can be modeled using the Rankine Cycle. This cycle is

composed of four components: a heater (boiler), a turbine, a condenser, and a pump. To complete

the system there must be some type of fluid flowing through the components, which is called the

working fluid. Most often the working fluid is water. As the working fluid passes through each of the

components it undergoes a process and ends up at a new state. Keeping in mind that the ideal

Rankine cycle is physically impossible, we define each process to involve no internal

irreversibilities. For the following it is necessary to number each of the states. State 1 is the state at

the boiler exit. State 2 is the turbine exit. State 3 is the condenser exit and state 4 is the pump exit.

Figure 11. Schematic of simple ideal Rankine cycle.


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Now the processes that the working fluid undergoes as it completes the cycle will be defined. First

is the heater, which in most cases is a boiler. As the fluid ends the cycle, at state 4, it is pumped into

the boiler. In the boiler, the working fluid is heated from sub-cooled liquid to saturated vapor. This

occurs at a constant pressure and is described in the following equation:

(1)

where is the rate of heat addition relative to the mass flow rate of the working fluid passing

through the boiler. The value ( ) is the difference in outlet and inlet enthalpies of the working

fluid.

Second is the turbine. Through the turbine the vapor leaving the boiler expands to the condenser

pressure. This is said to be isentropic expansion so that no heat transfer to the surroundings is

present. The equation that is used to describe this process is as follows:

(2)

where is the rate of work being done relative to the mass flow rate through the turbine.

Again the difference in inlet and exit enthalpies of the working fluid is required.

Next the working fluid enters the condenser. At this stage heat is rejected from the vapor at a

constant pressure. Ideally, this continues until all of the vapor condenses to leave nothing but

saturated liquid. The equation for this is:

(3)

where is the rate at which heat is transferred from the working fluid relative to the mass

flow rate. The value ( ) is the difference between inlet and outlet enthalpies of the condenser.

Finally, the working fluid enters the pump. The fluid goes through an isentropic

compression process to reach the boiler pressure. The equation describing this is as follows:

(4)

where is the rate of work being done relative to the mass flow rate through the pump.

Finally the difference in pump outlet enthalpy and inlet enthalpy is needed.

The efficiency of a given Rankine cycle may be computed as:

(5)


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Figure 3. P-v and T-s diagrams for the Rankine cycle.

In reality, no Rankine cycle is completely ideal. In particular, the compression of the working fluid

through the pump, and the expansion of the working fluid through the turbine are not actually

isentropic processes. Irreversibilities in these processes lead to increased power input to the pump,

and decreased power output from the turbine, both of which effectively lower the overall efficiency

of the system.

C. EXPERIMENTAL APPARATUS

The experimental hardware (RankineCycler™) consists of multiple components that make up the

necessary components for electrical power generation (utilizing water as the working fluid).

Figure 4. The RankineCycler apparatus, including the data acquisition system.


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These components include:

BOILER

A stainless steel constructed, dual pass, flame-through tube type boiler, with super heat

dome, that includes front and rear doors. Both doors are insulated and open easily to reveal

the gas fired burner, flame tubes, hot surface igniter and general boiler construction. The

boiler walls are insulated to minimize heat loss. A side mounted sight glass indicates water

level.

Figure 5. Boiler with superheat dome

TURBINE

The axial flow steam turbine is mounted on a precision-machined stainless steel shaft,

which is supported by custom manufactured bronze bearings. Two oiler ports supply

lubrication to the bearings. The turbine includes a taper lock for precise mounting and is

driven by steam that is directed by an axial flow, bladed nozzle ring. The turbine output

shaft is coupled to an AC/DC generator.

Figure 6. Axial flow steam turbine wheel


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ELECTRIC GENERATOR

The electric generator, driven by the axial flow steam turbine, is of the brushless type. It is a

custom wound, 4-pole type and exhibits a safe/low voltage and amperage output. Both AC

and DC output poles are readily available for analysis (rpm output, waveform study,

relationship between amperage, voltage and power). A variable resistor load is operator

adjustable and allows for power output adjustments.

CONDENSER TOWER

The seamless, metal-spun condenser tower features 4 stainless steel baffles and facilitates

the collection of water vapor. The condensed steam (water) is collected in the bottom of the

tower and can be easily drained for measurement/flow rate calculations.

DATA ACQUISITION

The experimental apparatus is also equipped with an integral computer data acquisition station, which utilizes National Instruments™ data acquisition software.

The fully integrated data acquisition system includes 9 sensors:

1. Boiler pressure

2. Boiler temperature

3. Turbine inlet pressure

4. Turbine inlet temperature

5. Turbine exit pressure

6. Turbine exit temperature

7. Fuel flow

8. Generator voltage output

9. Generator amperage output

The sensor outputs are conditioned and displayed in “real time” on screen. Data can be

stored and replayed. Run data can be copied off to a USB flash drive for individual student

analysis. Data can be viewed in Notepad, Excel and MSWord (all included).


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A schematic of the complete system is shown in Figure below.

Figure 7. RankineCycler schematic

*Note: When compared to figure 2 (simple ideal Rankine cycle), the steam ejected from the RankineCycler™ turbine condenses into liquid via the condenser tower and then exits the condenser into a collecting volume at the condensers base, rather than being pumped back to the boiler. This represents the main difference between the simple Rankine cycle described in figure 2, since the liquid exiting the condenser is dispensed, rather than pumped back to the boiler.

General Safety

The RankineCycler™ operates at very high pressures and temperatures. It is essential for the safety of everyone on the lab that certain safety precautions are adhered to at all times. If these guidelines are not strictly followed, SERIOUS INJURY OR DEATH MAY RESULT.

-Do not touch any of the functional components during operation. These components will be hot.

-Do not open the boiler during or immediately following operation. If the pressure gauge indicates positive pressure, the boiler must remain closed.

-Do not exceed 120 psi boiler pressure.

Burner

Boiler

LP Natural

Gas Tank

Fuel flow

Boiler

pressure

Turbine Generator

Condensate

Collection Tank

Turbine exit temperature

and pressure

Variable

Load

Current and

voltage

Condenser

Turbine inlet temperature

and pressure

Boiler

temperature


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-Be very careful working near the condensing tower. The steam exiting the tower is still extremely hot. Condensation drained from the bottom of the tower is also very hot.

-If the scent of gas is detected at any time during operation, shut the equipment down immediately.

-If any questions or concerns arise during equipment operation, notify the TA immediately.

D. EXPERIMENTAL PROCEDURE

Do not begin operation without proper supervision. Prior to beginning any operation, ensure that a trained lab technician, TA, or faculty member is present.

Prior to operation, familiarize yourself the following operator controls:

GAS VALVE

The gas valve is a simple two-position valve (On or Off). It is located on the far right

side of the slanted operator control panel. It will prevent gas flow to the burner

when in the off position- regardless of any other control positions/settings.

KEYED MASTER SWITCH

The systems electronic master switch is key operated and is located on the left side

of the operator control panel. This key switch supplies power to all electronic and

electrically operated components. A green indicator light, located directly above the

keyed master switch, will light when the master switch is selected to the on position

and power is available to the switch

BURNER SWITCH

The burner switch is labeled as such and is located next to the keyed master switch.

The burner switch powers the automatic gas valve and ignition controls. A green

indicator light, located directly above the burner switch, will light when the burner

switch is selected to the on position and power is available to the switch.

LOAD SWITCH

The load switch functions as a generator load disconnect switch.

LOAD RHEOSTAT CONTROL KNOB

The load rheostat control knob is connected in series with the load toggle switch

and generator DC output terminals. It provides a source of variable generator load.

AMP METER

The amp meter indicates generator load conditions.


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VOLTMETER

The voltmeter indicates the generator voltage output.

STEAM ADMISSION VALVE

The steam admission valve controls the steam flow rate to the steam turbine.

PRE-START

The TA will complete a pre-start procedure prior to the lab period. The pre-start

procedure includes safety and operability checks to ensure that the equipment is

functioning properly.

BOILER FILL

Fill the boiler according to the following procedure:

1. Verify that the boiler is empty

2. Fill the graduated cylinder with 6000mL of distilled water

3. Connect the fitting on the end of the plastic tube attached to the graduated cylinder to

the port on the lower, middle, back side of the boiler. The fitting should snap into place

4. Set the graduated cylinder on top of the condenser tower

5. Drain 500mL of distilled water into the boiler by operating the valve at the base of the

graduated cylinder

6. Record the water level indicated on the sight-glass attached to the boiler

7. Repeat steps 5 and 6 until all 6000mL of water have been drained into the boiler.

Record the total volume of water within the boiler, and the corresponding water level

each time.

The data recorded during this step will be used to develop a correlation relating boiler

water level to the remaining volume of water within the system. This correlation may be

found by entering the data into Excel, and obtain a curve fit of the data. The resulting

function can be used to calculate the total volume of water consumed during a steady-state

run.

If the duration of the run is known, the mass flow rate can be computed.


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where

[

]

duration of run , -

START

1. Open LP bottle gas valve

2. Turn gas valve knob CCW to "on" position

3. Turn master switch on (observe green indicator light on)

4. Turn the load switch to the "on" position.

5. Set the load rheostat to ~1/2 maximum load.

6. Turn burner switch on (observe green indicator light on)

NOTE: Combustion blower starts automatically. Wait for 30 seconds. This will allow the lines

to purge. Then turn the burner switch to the "off" position and immediately back on (this step

can be eliminated from the start procedure if the system has previously been operated using

the currently attached LP source). This resets the starting cycle and assures that the lines are

purged. After approximately 20 seconds, the automatic gas valve will open and the burner

will light.

7. Boiler pressure indication should be observed within 3 minutes of ignition.

8. Allow the boiler pressure to increase to approximately 110 psi.

NOTE: SHUT OFF BURNER SWITCH IF THE BOILER PRESSURE EXCEEDES 120 PSIG.

9. Observe the voltmeter and gently open the steam admission valve. Regulate turbine

speed to indicate 7-10 volts. This will pre-heat the turbine components and the pipes.

Close valve after 30 seconds and wait for boiler pressure to return to 110 psi. Very

small leaks may be visible due to condensation and cold turbine bearing clearances.

This is normal and will stop after normal operating temperatures are attained.

10. Repeat step 9 two more times (three times total) to completely pre-heat the system.


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DATA COLLECTION

The T/A will create a folder for you on the data acquisition computer. The T/A will also be

responsible for designating file names for the data sets from each of the experimental runs.

1. Ensure that the load switch is “on”, and the load rheostat is set to ~½ load (~12 o’clock

position).

2. Allow the boiler pressure to return to 110 psi.

3. Gently open the steam admission valve, and make fine adjustments to achieve a steady-

state condition at ~90 psi. Each group must determine their own steady-state

‘tolerance’. It is helpful to use the real-time boiler pressure display on the DAQ monitor.

4. Once a steady-state has been reached, simultaneously:

a. Record the starting water level.

b. Begin timing for ~3 minutes.

c. Begin data acquisition by clicking the ‘play’ button in the DAQ interface.

During data acquisition, you may continue to make fine adjustments to the steam

admission valve to maintain your desired steady state.

5. After 3 minutes have passed, stop the data acquisition and record the final water level.

6. Open the Excel data file to ensure that data was recorded successfully.

7. Repeat steps 1 through 6 with the rheostat set to ~¾ load. Be sure to have the T/A

rename the data file prior to data acquisition. Otherwise, all of the data from the

previous run will be overwritten.

SHUT DOWN

1. Leave the steam admission valve open.

2. Move the burner switch to the "off" position.

3. Turn gas valve off.

4. Turn LP gas bottle valve off.

5. Slowly open the steam admission valve to release the remaining pressure. Do not allow

the generator voltage to exceed 12V.

6. Move the load rheostat to the ‘no-load’ position

7. Move the load switch to the “off” position

8. Turn the master switch to the “off” position


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EMERGENCY SHUTDOWN

1. Unplug RankineCycler power cord

2. Move to a safe distance

3. If safety is not compromised: Turn burner switch off, turn load rheostat to

maximum, open steam admission valve to obtain maximum voltage.

E. PRE-LAB

Complete the following thermodynamic problem. This is intended to provide a

review of the Rankine cycle.

1. Consider a steam power plant that operates on a simple ideal Rankine cycle and has

a net power output of 30 MW. Steam enters the turbine at 7 MPa and 500ºC and is

cooled in the condenser at a pressure of 10 kPa by running cooling water from a

lake through the tubes of the condenser at a rate of 2000 kg/s. Show the cycle on a

p-v, and a T-s diagram with respect to saturation lines, and the determine (a) the

thermal efficiency of the cycle, (b) the mass flow rate of the system and (c) the

change in temperature of the cooling water.

2. Compute the Carnot efficiency of the cycle described in part 1. Discuss the reasons

for any difference between the Rankine and Carnot efficiencies. How would you

increase the efficiency of the Rankine cycle?

3. Read the entire lab procedure. Create a spreadsheet to record the data obtained in

the ‘Boiler Fill’ steps. Print this spreadsheet and bring it with you to the lab (or

bring your laptop with the spreadsheet to the lab).


1. From the time-averaged data collected by the computer, plot the state points 4, 1 and 2

(as shown in Figure 3). Why are we not asking you to plot state point 3?

2. Calculate the values of enthalpy and entropy for the state points 4, 1 and 2.

3. Compute the turbine work (Watts). The water level/volume correlation, obtained during

the “Boiler Fill” step, will yield the mass flow rate.

4. Compute the generator work from the measured voltage and current measurements.


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5. Compute the (rate of heat addition) to the boiler from the fuel flow rate and the

heating value of the fuel (propane).

6. Compute the cycle efficiency. Identify and discuss sources of inefficiency in the system.

Suggest modifications which would improve cycle efficiency.

Complete these steps at both operating conditions (½ and ¾ load).

G. REFERENCES 1. RankineCycler™ Operations Manual, Turbine Technologies, Ltd., Chetek, WI.

2. Cengel, Y. A. and Boles , M. A., Thermodynamics, An Engineering Approach (4th

Edition), McGraw-Hill, 2002

3. "Rankine Cycle." Wikipedia, the Free Encyclopedia. Web.

<http://en.wikipedia.org/wiki/Rankine_cycle>.

MEM311: Thermal and Fluid Science Laboratory Experiment 5: Lift Characteristics of an Airfoil Section

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EXPERIMENT 5:

LIFT CHARACTERISTICS OF AN AIRFOIL SECTION by Brandon Terranova 2010, adapted from Bakhtier Farouk and David Staack

A. OBJECTIVES

Determine lift force and coefficient for an airfoil section at different angles of attack for several tunnel velocities.

B. THEORY

Airfoil geometry

The wing of an airplane is its primary lifting device. Lift is the force that counteracts the aircraft weight and causes flight. The simplified cross-section of an infinite wing, an airfoil, is often tested in wind tunnels to accurately and optimally design the wing of an aircraft. The study of actual wing geometries (finite wings) is built on the understanding of the idealized airfoil aerodynamics. A schematic of the airfoil geometry with associated terminology is shown in Figure 12.

Figure 12: Airfoil geometry

This experiment uses the NACA-2415 airfoil, which is one of the NACA 4-digit series standard airfoils. The National Advisory Committee on Aeronautics (NACA) studied the characteristics of airfoils to develop a database for aeronautic engineering design.

Notice that this particular airfoil is not symmetric about the chord line. This type of airfoil is generally referred to as a “cambered airfoil.” The 4-digit series shape is mathematically identical to symmetrical airfoils with the exception that the mean camber line is bent. The first digit indicates the maximum camber height (distance from the chord to the maximum height of the mean camber line) “m” in hundredths of the chord (for NACA-2415 m = 0.02 or 2%). The second digit indicates the location of the maximum camber (as measured from the leading edge) “p” in tenths of the chord (in this case p = 0.4 or 40%). The third and

Leading edge

Trailing edge

Mean camber line

Upper surface

Lower surface Chord line


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fourth digits signify the maximum thickness “t” in percentage of the chord (in this case t = 15%). For a 4-digit cambered NACA airfoil, the following formulas are used to calculate the mean camber line:

.

/

(1)

( )

( ) .

/

(2)

Where c is the chord length and x is the position along the chord from 0 to c. The expression for the entire camber line would then be a piecewise function. The upper and lower airfoil surfaces is given by

(3)

(4)

where

( )

(5)

and

* √

.

/ .

/

.

/

.

/

+ (6)

which is the equation corresponding to the shape of a symmetrical 4-digit NACA airfoil.1

In addition to the standardization of certain airfoils, we are able to describe the arbitrary shape of any cambered airfoil mathematically. Since the shape of the camber is an arbitrary curve, we must use Fourier analysis to accurately define a function which reproduces the curve. The idea behind Fourier analysis is that any function can be created as a sum of sine and cosine curves. From airfoil theory, we note that the camber line, regardless of shape, will be a streamline of the flow around the corresponding airfoil. From this concept, an equation can be derived that is a general form for any camber line:


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( ) (

∑

) (7)

Where is the wind velocity at infinity (refers to the unperturbed conditions very far

ahead of an aerodynamic body), and is a coordinate transformed from

with α representing the angle of attack, which is the angle formed between the incoming wind and the chord line of the airfoil.

Sources of Lift, Drag and Moments

No matter how complex the airfoil geometry, or any shape for that matter, the aerodynamic forces and moments on the body are due entirely to the pressure and shear stress distributions over the body surface. As you know from fluid mechanics, the pressure difference between the upper and lower wing surfaces arises from the velocity difference between the flows. This can be visualized by noticing the spacing between the streamlines in Figure 13.

Figure 13: Streamlines over an airfoil.

The more condensed streamlines are an indication of the greater fluid velocity above the wing, and this corresponds to a decrease in pressure. The equation for dynamic pressure illuminates this, showing how pressure can change due to the velocity of the fluid alone.

( )

The Fourier coefficients are found using

(8)

∫

(9)

∫

(10)


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(11)

But wait! This equation says that pressure should INCREASE when velocity is increased. What’s going on here? Since Bernoulli’s equation involves more than just the dynamic pressure, we need some more information to understand how lift occurs. Where is the density of air, and the dynamic pressure is denoted with a q to distinguish it from the other terms in Bernoulli’s equation for incompressible flow:

(12)

Where we use to represent the total pressure, which must be constant between one region of a streamline of the fluid and another. This is the essence of Bernoulli’s equation: the fact that for a steady flow, the total pressure must be the same between one region of a fluid and another. Bernoulli’s equation as shown applies to the flow along a streamline when 1) the fluid has constant density, 2) the flow is steady, and 3) there is no friction. Figure 14a shows the pressure distribution with arises from zero angle of attack (α = 0) on a cambered airfoil. So back to the velocity vs. pressure issue. Since increasing the velocity obviously increases the dynamic pressure, where does the drop in pressure come from? Since the pressure must be constant between one region of the streamline and another, this increase in velocity must be accompanied by a decrease in pressure somewhere! The decrease happens in the static pressure. When the velocity of a fluid increases, the thermodynamic properties of a fluid changes (density changes), thus the decrease in the static pressure is what balances the equation. But you still might be wondering: “Ok, so the pressures balance out…how do we end up with a lower pressure on the wing if they are balanced??” The answer is that the static pressure is what is “felt” by the skin of the wing, thus this decrease of static pressure is what is primarily responsible for lift!

Static pressure (due to the thermodynamic

properties of the fluid)

(A) (B)

“Gravity pressure” (due to the weight of

the water on itself…needed if at reference

height, h≠0)


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Figure 14: (a) Pressure distribution. (b) Shear stress distribution.

In addition to the pressure acting on the airfoil, friction occurs as the fluid moves over the airfoil, resulting in a pulling on the airfoil which gives rise to the shear stress distribution as seen in Figure 14b. This shear stress distribution generates a moment ( ) around the body. At the same time, the pressure distribution which is induced by the difference in dynamic pressure, couples with the moment created by the shear stress, and the cumulative effect of these forces is described by a resultant vector located at a point known as the center of pressure as shown in Figure 15. This resultant vector can be broken into its component vectors, which we refer to as drag ( ) and lift ( ).

Figure 15: Lift and drag as resulting from pressure and shear stress.

Pressure and Lift Coefficients

The pressure coefficient is a very useful parameter for characterizing the flow of incompressible fluids, while the lift coefficient is directly related to wing aerodynamics (notice the inclusion of the wing area term in equation 14). The lift coefficient is related to the lift force, which as you know arises from complicated physical interactions between the moving fluid and the wing surface. In general, it is very difficult to analytically determine the value of this coefficient (as well as coefficients for drag, axial, normal force and moment coefficients) for arbitrary wing geometries, therefore it is usually determined empirically. There exist other coefficients that serve other purposes, but for this lab we can confine our interests to these four. The lift, drag and moment coefficients for an aerodynamic body can be obtained by integrated the pressure and skin friction (related to the shear stress) coefficients over the body surface from the leading to the trailing edge.2 Notice that again, all parameters are derived from only considering the pressure and shear stresses on the wing.

(13) (14)

cp = pressure coefficient

cl = lift coefficient

p = pressure at infinity

A = area of the airfoil, A = cS


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c = chord length of the airfoil S = span of the airfoil

For wind tunnel measurements, the conditions at infinity correspond to the unperturbed conditions

at the beginning of the test section.

Lift Calculation

As mentioned earlier, to calculate lift, we are primarily interested in the static pressure on the various sections of the wing. In our laboratory experiment, the static pressure ps will be measured at several locations along the lower and upper surfaces of the airfoil section. The lift force on the airfoil can be approximately calculated from the measured pressure distribution along the airfoil (lower and upper) surfaces. The pressure along the upper and lower surfaces of the airfoil is measured at several tap locations as shown in table 1. Each pressure tap is assumed to act over a “small” sectional area as shown in figure 5. The force on each sectional area is given by the product of the static pressure on that section and the sectional area. The force on the upper and lower surfaces can be obtained by summing the components of the individual forces on each section. The net lift is obtained by adding the net forces on the upper and lower surfaces. When summing the forces it should be noted that the pressure acting on the upper surface pushes downward, and the pressure acting on the lower surface pushes upward.

Tap ID X-location (in) Tap location (X/c)

c = 6

Upper Surface

1 0.18 0.029

2 0.5 0.08

3 1.07 0.172

4 1.7 0.273

5 2.55 0.409

6 3.45 0.554

7 4.3 0.69

8 5.25 0.843

Lower Surface

9 (under 1) 0.16 0.026

10 (under 2) 0.5 0.08

11 (under 3) 0.95 0.152

12 (under 4) 1.4 0.225

13 (under 5) 2.2 0.353

14 (under 6) 3.1 0.498

15 (under 7) 3.94 0.632

16 (under 8) 4.96 0.796

Table 6: Pressure tap locations along the NACA 2415 airfoil section.


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Figure 16: Schematic for determining sectional areas associated

The area associated with each pressure tap will be taken as the average distance between taps multiplied by the span of the airfoil as shown in figure 5. Starting at the leading edge and the upper surface, pressure tap #1 will be assumed to act over the sectional length

(15)

Thus the area associated with tap #1 is

(

) (16)

The sectional lift force associated with section #1 would then be

(

) (17)

For tap #2, the sectional force would be

(

) (18)

and similarly for the rest of the pressure taps. Be careful when computing the lift associated with taps close to the leading and trailing edges, as they contain terms dissimilar from the rest of the equations.

X1 X2

X3


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Wind Tunnel Design Theory

A wind tunnel is just a large Venturi tube. Wind tunnels are used in aerodynamic research to study

the effects of air moving past aerodynamic bodies. The objective of the design of a wind tunnel is to

create laminar (turbulence-free) air flow in the test section. The typical open-circuit wind tunnel

consists of three main parts which are labeled in figure 6. Air is drawn through the device by a fan

attached to the diffuser section. The compressor section is typically open to the atmosphere with the

exception of closely-spaced vertical and horizontal air vanes used to smooth out the turbulent

airflow before reaching the testing subject. The test section is where the laminar flow is

concentrated, and subsequent measurements are taken on the test subject.

The main operating principle of the wind tunnel geometry is that we can adjust the areas of the

compressor and diffuser sections to control the velocity in the test section. Flow rate can be

measured using the pressure difference between sections, as is done in a conventional venture tube

using Bernoulli’s equation and the continuity equation.

Figure 17: Open-circuit wind tunnel design

C. EQUIPMENT

Wind Tunnel

The model 1440 Flotek wind tunnel, as shown in figure 7, provides a full 12” x 12” (305 mm square) cross-section in the working area over 36” (914 mm) in length. Air is drawn through the compressor intake cone into the test section by a variable speed fan. A plastic honeycomb flow straightener attached to the mouth of the compressor assures a laminar air flow through the test section. The entrance cone has a contraction ratio of 12:1 down to the test section. The air velocity through the test section is variable up to 90 mph.


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Figure 18: The Flotek 1440 wind tunnel with detailed view of the NACA-2415 airfoil.

Airfoil

This lab uses a NACA-2415 airfoil. This airfoil has a chord length c = 6 in and a span of 11 11/16 in.

The airfoil has 16 pressure tappings, 8 on the upper surface, and 8 below.

Instrumentation

The system is connected to a computer, and includes an interface card and Labview software. Data acquisition and computer control of the wind tunnel is accomplished by using a 16 channel analog-to digital (A/D) and 2 channel digital-to-analog (D/A) converter board. The pressure taps are connected a water-filled manometer array as well as to a set of 16 pressure transducers which converts the pressure into an analog voltage. The data acquisition system works by converting this analog voltage signal to a digital signal which is read by the computer. Similarly, converting a digital signal from the computer to an analog voltage provides control for the motor which sets the angle of attack. The converter board is located in a PCI slot inside the control computer, and a cable connects it to the wind tunnel’s transducer box located directly behind the test section, underneath the exhaust section. Software based on LabVIEW’s visual programming language is used for system control and data acquisition. A screenshot of the software’s instrument panel is shown in figure 8. Slider bars (circled in red) are used to adjust the motor speed and angle of attack. The software also has the capability of setting a particular speed in the test section by toggling the velocity control (circled in blue) switch, entering the desired velocity into the “setpoint” field, and adjusting the gain (this dictates how quickly the system arrives to the set velocity). The airfoil pressure measurements are taken from the 8 pressure taps along the upper surface of the airfoil and 7 along the lower surface. The remaining pressure tap (which only has a manometer display) measures the static pressure in the wind tunnel test section.


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The static pressure in the wind tunnel test section is used to calculate the tunnel velocity using Bernoulli’s equation. This calculation is hard-coded into the LabVIEW algorithm. The total pressure which is required for that measurement is measured when the LabVIEW program is first turned on (thus to operate properly the wind tunnel fan motor should be off when starting the program). The relative pressures measured along the airfoil are shown in two plots, one for the upper and one for the lower surface. Similar to how the static pressure is calculated in the test section, the flow velocities are calculated from the pressure information at each tap on the airfoil as well. The transducer offset values which in effect determine the total pressures are measured when the program is started and are shown in the diagnostics tab (circled in green). When the record button (circled in black) is pressed, the pressure data, velocity data, motor rpm, and angle of attack are stored in a data file. This file can be opened with Excel.

Subsequent use of the record button appends a line of new data to the file. Test this feature until you are comfortable taking data.

Figure 19: Wind tunnel instrumentation panel


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D. PROCEDURE 1. Begin by starting the software located on the desktop. The fan motor should be off

when the Labview program is started.

2. Once the program is running set the angle of attack to 0, turn the key and press the green button on the Flotek fan power control (shown in Figure 18).

3. Once you are familiar with the system operation, press the “record data” button (circled

in black in Figure 19) and record data for a few moments - you should check the data output with some test velocities.

4. For steady state tunnel velocities of 60 ft/s and 75 ft/s and 90 ft/s measure the pressure distributions at -4, 0, 4, 8, 12, and 16 so that you make 18 measurements in total.

Note: Since only 15 of the 16 pressure taps on the airfoil are digitally acquired, for each condition you will need to manually write down the pressure on the 8th tap of the lower surface from the manometer board. In order to correlate the manual data with the computer acquired data you will also need to note the test section pressure (manometer tube on far left) and atmospheric pressure from the manometer board (manometer tube which is open to the environment). Be sure that your calculations match reasonably well to the transducer measurements before you are confident of the 8th pressure tap reading.

E. PRELAB 1. Consider an airfoil section with only 3 pressure taps on the upper and lower surfaces.

The pressure readings (gauge) for the taps are given below:

Upper surface Lower surface

Pressure at tap #1 -17.5 lbf/ft2 Pressure at tap #1 8.5 lbf/ft2



Assume the airfoil chord is 3 inches and the span is 6 inches. The sectional length assigned to each tap is 1 inch. Calculate the total lift on the airfoil where (hint: when working in U.S. customary units remember the conversion 1 lbf =

32.2 lbm ft/s2)

2. Using equations 1-6, find an expression for the camber line, upper surface and lower surface of the NACA-2415 airfoil. Plot all three equations on the same graph using some computer algebra program. (Note that the mean camber line equation is a piecewise


62 | P a g e

function. Also, if you get a couple discontinuities in the plots for the upper and lower surface, don’t worry about it. Your plot should look like the airfoil.)

3. a) Use the equation below to determine the center of pressure for the NACA-2415 airfoil encountering a wind speed of 90 mph (possible in our wind tunnel!). b) Explain the difference between center of pressure, center of gravity and aerodynamic center.

[

( )]

Where again is the coefficient of lift, and A1 and A2 are the Fourier coefficients. Note: Use the calculated in #1 for the lift in .

F. DATA ANALYSIS AND REPORTING REQUIREMENTS 1. Calculate and plot on the same graph the pressure profile (measured static pressures) on

the upper and lower surfaces as a function of the chord position for the -4 and the 8 angle of attack for each of the three tunnel velocities (6 plots total). Discuss how the pressure profiles change with angle of attack and velocity.

2. Calculate and plot on the same graph the pressure coefficient on the upper and lower

surfaces as a function of the chord position for -4, 0 and 8 angle of attack for 75 ft/s velocity (3 plots total).

3. Plot the lift coefficient as a function of angle of attack for the three tunnel velocities on the

same graph (1 plot total).

G. REFERENCES 1. Moran, J. (2003). An Introduction to Theoretical and Computational Aerodynamics. Dover. 2. Anderson, John D. (2007). Fundamentals of Aerodynamics, Fourth Edition. McGraw-Hill. 3. Munson et. al (2009). Fundamentals of Fluid Mechanics, Sixth Edition. Wiley.

MEM311: Thermal and Fluid Science Laboratory Appendix A

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APPENDIX A

MANOMETER PREPARATION AND OPERATION by Brandon Terranova 2010, adapted from Bakhtier Farouk and David Staack

The fluid manometer provides a relatively simple method to measure fluid pressures under steady-state

conditions. A sketch of a simple U-tube manometer is shown in Fig. A.1. In this configuration, one side

of the manometer is open to the atmosphere and the other is open to the pressure to be measured.

The theory behind manometer operation (Ref. A.1, Section 2. and Ref. A.2, Section 6-3) shows that the

difference between the pressures on each side of the manometer are given by the hydrostatic equation,

i.e.,

( ) (A.1)

where is the density of the fluid in the manometer and is the density of the fluid transporting the

unknown pressure. Typically, the density of the fluid in the manometer is much larger than the density

of the fluid transmitting the pressure so Eq. (A.1) is often written as

(A.2)

Figure A.20

Note: The distance h is measured parallel to the gravitational force and that the differential pressure is

measured at the location of the manometer. If the location of the pressure source is at a different

elevation than the manometer, there could be appreciable error in the pressure determination

depending on the density of the transmitting fluid.


64 | P a g e

The manometer can also be used to measure a pressure difference between two locations such as on

opposite sides of an orifice flow meter. In this configuration, shown in Fig. A. 1b, the manometer is

placed in a closed loop containing both the manometer fluid and the transmitting fluid. The pressure

difference between the two locations is calculated using Eq. (A.1).

Figure A.21

(A.3)

The experiments in this lab make use of manometers to measure pressure differences. A slight

difference between those used in the lab and shown in figure A.1 is that multiple manometer tubes

share the same reservoir. With reservoir manometers the cross sectional area of the reservoir is

significantly greater than that of the manometer tube. Thus a decrease in pressure causes an increase in

the height of the fluid in the tube but a negligible change in the height of the fluid in the reservoir. A

schematic of a reservoir type manometer Be sure to familiarize yourself with the configuration before

attempting to interpret your measurements.

Water manometers are simple to use once they have been properly prepared. Often, if the

apparatus has been standing for some time, air will become trapped in the pipes and in the water

columns of the manometer tubes. These air pockets will affect the pressure measurement because air

ha a different density than water. The manometer should be ready for operation when you report to

the lab. Improper use of the manometer during the experiment may cause air to become trapped in the

lines. Also, there are some experiments which, by nature, require moving the rubber tubes leading from

the manometer to various pressure taps in the system. This may result in air bubbles forming in the

manometer tubes. If this condition occurs during your experiment, you will have to remove the air

bubbles before beginning the experiment. This procedure varies depending on the exact configuration

but generally involves putting a pressure difference across the two ends of the nanometer. This will

induce a flow through the tube thereby forcing the bubble out the tube.


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Note: Extreme care must be exercised when using mercury manometers. Do not try to remove air from

a mercury manometer using the same procedures as for a water manometer. It is a much less frequent

problem because of the larger density of mercury. Because of the toxicity of mercury, spills can be

extremely hazardous. Also, never ignore or attempt to hide a mercury spill! Inform someone who can

properly clean up the spill and dispose of the mercury.

References

A.1. Munson, B.R., Young, D.F., and Okiishi, T.H., Fundamentals of Fluid Mechanics, John Wiley and

Sons, New York, 1990.

A.2. Holman, J.P., Experimental Methods for Engineers (7th edition), McGraw-Hill, New York, 2001.

MEM311: Thermal and Fluid Science Laboratory Appendix B

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APPENDIX B

FLOW MEASURING USING A ROTAMETER by Brandon Terranova 2010, adapted from Bakhtier Farouk and David Staack

Many devices and methods exist to measure the flow rate of a fluid. Several of the more popular of

these are described in Chapter 7 of Ref. B.1. An orifice flow meter and a venturi flow meter are

examples of obstruction-type flow meters because their operating principle is based on measuring

the response of the fluid to an obstruction in the flow. A rotameter is slightly different in that the

fluid flow rate is determined by measuring the response of an obstruction to the fluid motion. Flow

enters from the bottom of a rotameter, shown in Fig. B.1, and flows through a tapered vertical tube.

This flow causes a bob in the tube to move upward due to the fact that as flow increases, the area

around the float must also increase in accordance with the basic equation for volumetric flow rate

(Eq. B.1). The bob will rise to a position in the tube where the drag forces acting on the bob are just

balanced by the weight and buoyancy forces. The position of the bob is then taken as an indication

of the flow rate of the fluid. The drag force acting on the bob will be a function of not only the flow

rate of the fluid but how the tube is tapered and the shape of the bob. Therefore, each rotameter

must be calibrated to correctly account for these geometric effects.

√ (B.1)

The equations detailing the operating principle of a rotameter are presented in Ref. B.1,

Section 7-6.

*( )

+

(B.2)

where y = vertical displacement of the bob, = density of the fluid, = density of the bob, and C is

the calibration constant for the rotameter. While you may find different types of rotameters, most

are calibrated to read from the top of the bob. Since rotameters are typically used to measure the

flow rates of gases, is much larger than equation B.2 can be reduced to

√ (B.3)

where = volumetric flow rate (m3/s). Equation (B.2) shows that we must know the density of a

fluid to determine either the volumetric or mass flow rate. The calibration constant, C, is a function

not only of the geometry of the tube and bob but also of the density of the fluid used during the

calibration. Luckily, we do not need separate rotameters calibrated for every fluid density that we

may encounter during our experiment. Instead, we can correct the mass flow rate given by Eq.

(B.3) for the density of our fluid if we know the conditions at which the rotameter was calibrated

and our test conditions. Rotameters and most gas flow devices are calibrated at standard

atmospheric condition, i.e., calibrated using air at 70F and 1 atm. Calibration tables or

MEM311: Thermal and Fluid Science Laboratory Appendix B

67 | P a g e

Figure B.22

graphs give as a function of y where the subscript “s” indicates “at standard temperature and

pressure”. Therefore, the calibration constant quoted by the manufacturer can be interpreted as

√

√ (B.4)

Using the calibration constant given by the manufacturer, Eq. (B.3) becomes

(B.5)

Comparing equations B.3 and B.5:

(B.6)

For a given value of y, we can use equation B.6 to find the experimental calibration constant once a

flow measurement has been made.

References

B.1. Holman, J.P., Experimental Methods for Engineers (7th edition), McGraw-Hill, New York,

2001.

MEM311: Thermal and Fluid Science Laboratory Appendix C

68 | P a g e

APPENDIX C

ANALYSIS OF BIAS ERRORS AND EXPERIMENTAL UNCERTAINTY by Brandon Terranova 2010, adapted from Bakhtier Farouk and David Staack

A. Terminology

As in all experimental investigations, any measured data are subject to experimental errors that cannot

be eliminated completely. These errors arise because of limitations in the accuracy of one or all of the

instruments used to make the measurements or to the variations between people using the instrument.

These errors must be quantified to provide a measure of the reliability or uncertainty associated with

the data.

Various types of errors can contribute to the uncertainty in fundamental measurements. These include

accidental errors (outright mistakes), bias errors (also referred to as fixed or systematic errors), and

random errors. Accidental errors cause repeated readings to differ without apparent reason and may be

attributed to instrument friction, time lag, and personal errors. These must be identified and corrected

in any experiment. If this type of error is not removed entirely, it must at least be made to effect the

experiment the same way every time. Then it is known as a bias error. Bias errors result when data are

shifted from actual values by some characteristic of the substance being measured or the measurement

technique. Bias errors account for reductions in the accuracy of the experimental data. Random errors

may be caused by random human error, random environmental changes, or any fluctuation of the

property being measured. The magnitude of this type of error is a measure of the precision of a

measurement. The physical difference between precision and accuracy as applied to experimental

measurements is illustrated below.

Because these errors are random, they are better expressed as an uncertainty and require that

probability concepts be applied to quantify experimental errors. A concise way to describe data that

consists of both bias errors and experimental uncertainty is to specify the mean value and an

uncertainty interval, i.e.

um (xx percent uncertainty) (C.1)

Figure 23C: Precision and accuracy in measurements


69 | P a g e

where m is the mean value, u is the uncertainty interval, and xx is the percent confidence we have in the

uncertainty level. The meaning of these terms is more transparent if Eq. (C.1) is expressed in words as

“There is an xx percent probability that the true value of a property lies within u of the mean value,

m.” The factor xx is often called the confidence level or interval.

As an example, consider measuring the length of a desk using a yardstick. If the measurement was made

several times, each measurement will probably be slightly different. If every student in the class made

the same measurement, these measurements would also vary. It’s not that any single measurement was

“wrong” or “right”. The variations occur because the measurement technique and procedure (using a

yardstick) has inherent errors that limit the accuracy of the measurement. How can you express this

uncertainty about the true length of the table? The set of all the length measurements can be

represented as (x1, x2, x3, x4, … , xn) where n is the number of measurements and is assumed, for now, to

be large (greater than ~30). The following statistical quantities can then be defined calculated using this

set.

Mean:

∑

(C.2)

Deviation: (C.3)

Average Deviation:

∑

(C.4)

Variance:

∑( )

(C.5)

Standard Deviation:

√

∑( )

(C.6)

If we only have a limited set of data (less than 20 or so samples), we really don’t have sufficient data to

accurately estimate the standard deviation using Eq. (C.6). The true value of the standard deviation can

then be estimated using the following equation

√

∑( )

(C.7)


70 | P a g e

Note that the factor (n-1) is used in equation C.7 instead of n in equation C.6.

We could also plot the number of times a specific value of x was obtained for the length of the table

from the entire set of data. As n goes to infinity (an infinite number of samples), a continuous curve

would be obtained. Assuming that the errors associated with these measurements are equally likely to

be positive and negative, the curve would form a normal or Gaussian distribution about the mean

value, . This distribution is shown below.

This figure shows that the standard deviation, , is a measure of the width of the distribution.

In the above discussion, the arithmetic mean value has been assumed to be our best estimate of the

true value of a set of experimental measurements. The Gaussian distribution allowed us to estimate

how the data are distributed around the mean value. We still have a very important question to answer,

i.e., how well does this arithmetic mean approximate the true value, which is unknown? To obtain an

answer to this question experimentally, it would be necessary to repeat the entire set of measurements

and calculated a new mean value. This value would undoubtedly differ from the previous value because

of the same variations that produced the differences within any one data set. We could continue this

procedure to obtain a large number of mean values, estimate the standard deviation of the mean values

an, finally, the uncertainty in our estimate of the mean value. In other words, the mean value of all the

mean values of a large number of data sets is presumably the true value. Fortunately, this problem can

be treated with a statistical analysis that allows us to approximate the standard deviation of the mean

value using the standard deviation of a single set of data. This analysis will not be repeated here but

results in the following equation for the standard deviation of the mean, m ,

√ (C.8)

Figure C24: Normal distribution about the mean value


71 | P a g e

where is the standard deviation given by equation C.6 or C.7 and n is the number of samples in the

data set. We could then say with 95 percent confidence, for example, that the true value of x was

within m2 of our calculated mean value. An example of the application of equation C.8 to a set of

data is given in Ref. C.1, Section 3-11. If n is very small (less than 15-20) the approximation for m given

by equation 8 becomes worse, and other methods must be used to estimate the standard deviation of

the mean value. This method, known as the t distribution test, will not be discussed here but can be

found practically any book on statistics (Ref. C.2, Section 7.4 and 9.3, for example).

B. Propagation of Errors in Experiments

The analysis of uncertainties and confidence intervals discussed above were all concerned with knowing

the true value of a single measured parameter such as the fluid height is a manometer, temperature,

mass flow rate, etc. However, in all our experiments, these parameters are used to calculate additional

quantities. Bias errors and uncertainties in the individual measurements propagate through the

equation, resulting in an overall uncertainty in the calculated result. This uncertainty can be estimated if

the uncertainties of all the individual parameters are known. For instance, what is the error in

where A and B are two measured quantities with errors A and B respectively?

A first thought might be that the error in Z would be just the sum of the errors in A and B. However, this

assumes that when combined, the errors in A and B have the same sign and maximum magnitude; that

is, they always combine in the worst possible way. This could only happen if the errors in the two

variables were perfectly correlated, (i.e. if the two variables were not really independent.)

If the variables are independent (they usually are), then sometimes the error in one variable will happen

to cancel out some of the error in the other and so, on the average, the error in Z will be less than the

sum of the errors in its parts. A reasonable way to try to take this into account is to treat the

perturbations in Z produced by perturbations in its parts as if they were "perpendicular" and added

according to the Pythagorean theorem,

√( ) ( ) (C.9)

This idea can be used to derive a general rule. Suppose there are two measurements, A and B, and the

final result is ( ) for some function f. If A is perturbed by A then Z will be perturbed by

(

) (C.10)

Similarly, the perturbation in Z due to a perturbation in B is

(

) (C.11)

Combining these by the Pythagorean theorem yields


72 | P a g e

√((

) )

((

) )

(C.12)

We can then write down a general definition for the uncertainty in a variable Z by considering Z to be a

linear function of n independent variables, so that ( ). If each xi has corresponding

uncertainties , the relation for the uncertainty in Z is given as

√∑(

)

(C.13)

When equation C.13 is applied, the confidence level for UZ is the same as for the estimates of . The

value of Z is then reported as (xx percent confidence level) or equivalently

(xx percent confidence level). An example of the application of equation C.13 is given below. Several

additional examples can be found in Ref. C.1, section 3-4.

Example

Consider a pressure measurement made using an open-end U-tube water manometer. The pressure is

then given by

(C.14)

where h is the height of the fluid column, po is the atmospheric pressure measured with a barometer,

and is the density of the fluid. We find that the atmospheric pressure, po = 29.6 inches Hg and h = 24

inches H2O. During our measurements, we determine that the scale on the barometer can be read to

±0.1 inch Hg while that on the manometer can be read to ±0.25 in H2O. The density of water and the

gravitational acceleration can be taken as constant, = 62.4 lbm/ft3 and g = 32.2 ft/s2, respectively. As

always, gc = 32.2 lbm-ft/lbf-s2. The problem is to determine the pressure, p (in psia), and the uncertainty

in the measurement of p caused by the uncertainties in po and h.

Solution

First, substituting the measured values of po and h into equation C.14 and converting to psia, we find

that p = 15.41 psia. Since there was some uncertainty in reading the scale of both the barometer and

the manometer, the atmospheric pressure and the fluid height are properly expressed as po = 29.6 ±0.1

inches Hg and h = 24 ±0.25 inches H2O. For this problem, equation C.13 can be expanded as follows:


73 | P a g e

√((

) )

((

) )

(C.15)

The derivatives in equation C.15 must now be evaluated using equation C.14.

Substituting these derivatives into equation C.15, we obtain

√( ) (

)

(C.16)

Substituting the known values into equation C.16, being aware that units of psia are desired and must

be consistent in the equation, we get the calculated pressure as p = 15.41 ±0.05 psia or p = 15.41 ±0.32

%.

References

C.1. Holman, J.P. (2001), Experimental Methods for Engineers, 7th edition, McGraw-Hill, New York.

C.2. Mendenhall, W., Reinmuth, J.E. (1989), and Beaver, R., Statistics for Management and

Economics, 6th edition, PWS-Kent Publishing Company, Boston.

C.3. Taylor, John R. (1982), An Introduction to Error Analysis: The Study of Uncertainties if Physical Measurements. University Science Books.

C.4. P.V. Bork, H. Grote, D. Notz, M. Regler (1993), Data Analysis Techniques in High Energy Physics Experiments. Cambridge University Press.

C.5. H. Coleman and W. Steele, Experimentation and Uncertainty Analysis for Engineers, (2nd

edition), Wiley, New York, 1999

MEM311: Thermal and Fluid Science Laboratory Appendix D

74 | P a g e

APPENDIX D

FITTING CURVES TO EXPERIMENTAL DATA by Brandon Terranova 2010, adapted from Bakhtier Farouk and David Staack

The procedures to perform curve fits of experimental data can be some of the most useful and insightful

means of data analysis. Unfortunately, the simplicity with which most current spreadsheet and plotting

programs perform curve fits has allowed students and engineers alike to “fit” data without analyzing

which type of equation should be best based on physical grounds. For example, suppose that you just

measured the pressure and specific volume of a gas at constant temperature. If you plot your

experimental data and attempt to fit it with a curve, it might be tempting to start trying all of the curve

fits available in your plotting program to determine which gives the lowest error. However, by doing

this, you are neglecting any physics that might determine the proper form of the curve. For a gas that is

represented by the ideal gas equation, an isotherm is expressed as

(D.1)

where the constant is the quantity (nRT). Therefore, it would only make physical sense to fit this data

with a hyperbolic curve. Any other fit that gives a low error (high concentration coefficient) over a

limited range of specific volume could hide important results. What if your plotting program doesn’t

explicitly offer a hyperbolic curve fit? Practically any equation can be reduced to a linear function

provided that the variables are properly defined. This procedure is discussed in the following section.

A. General Curve Fitting

Suppose we have a set of measurement * + which we can plot on linear graph paper. However, the

relation

(D.2)

is suggested by either a trend in the data or applicable theory. To apply this type of curve fit, we must

determine a, b, and n. The simplest way to do this is to note that Eq. (D.2) can be written as

( )

Taking the natural logarithm of both sides of the equation, we obtain

( ) (D.3)


75 | P a g e

Equation (D.3) has the form of the equation of a straight line, i.e., if we define

( ), m = n, , and . Plotting ( ) vs. on linear-linear graph

paper, we obtain

Where the constant a, b, and n are thus determined. Some common functional relations and

corresponding straight-line plot are given in Ref. D.1.

B. General Curve Fitting

All plotting programs can easily perform a linear curve fit but it is beneficial to understand how these

values are being obtained. Most of these programs use a method of least squares to fit polynomial

curves. The procedure for linear and quadratic curves is discussed below. Higher-order fits generally

become cumbersome to derive but follow the same procedure

Suppose we have a set of n measurements represented by

* + (D.4)

The sum of a set of squares of numbers * + about a given number

∑( )

∑

∑

(D.5)

We want to minimize S with respect to . At any maximum or minimum, the first derivative must be

zero, i.e.

∑

(D.6)

Figure D.25


76 | P a g e

which implies

∑

(D.7)

This of course, is the definition for the mean value of * +. From this, you can see that by calculating the

mean value of an array of numbers you are actually minimizing the function S given in Eq. (D.5).

Next, consider a two-dimensional set of n measurements represented as

* + ( ) ( ) ( ) ( ) (D.8)

where the error in y is independent of the magnitude of the error in x. Suppose we have a linear

relation between y and x.

Recall from the previous section that even if the relationship is exponential, it can be expressed in linear

form by redefining the variables. Define the general linear equation

(D.9)

where is taken as the “optimum” y value to represent the data at a given x.

Then

∑( )

∑, ( )-

(D.10)

We wish to compute a and b such that S is a minimum. Therefore, S must be differentiated with respect

to both a and b and the resulting equations both set equal to zero:

∑, ( )-

(D.11)

Figure D.26


77 | P a g e

∑, ( )-( )

(D.12)

Solving for a and b leads to

∑ ∑ ∑

∑

(∑ ) (D.13)

∑

∑ ∑ ∑

∑

(∑ ) (D.14)

The equation that results from this procedure,

(D.15)

is called a regression equation. As previously stated, this method may be used for any functional relation

that can be expressed in terms of a linear relationship. Recall that we previously found that y = axb

could be written as log y = log a + b log x.

The above procedure can also be performed to determine the coefficients of the quadratic equation

that best fits a set of data. In general, a quadratic equation is given as

(D.16)

the function S (Eq. (10)) is then given as

∑( )

∑, ( )-

(D.17)

Equation (21) is then differentiated with respect to a, b, and c and all three equations set equal

to zero, e.g.,

These three equations can then be solved for the three unknowns, i.e., a, b, and c to obtain the

quadratic equation that fits the given data. Higher-order least squares methods follow a similar

development procedure but will require solution of an (N+1) X (N+1) matrix, where N is the order of the

polynomial being used to fit the data.

References

D.1. Holman, J.P., Experimental Methods for Engineers (7th edition), McGraw-Hill, New York, 2001.

D.2. H. Coleman and W. Steele, Experimentation and Uncertainty Analysis for Engineers, (2nd edition),

Wiley, New York, 1999

lab manuel

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