Solution manual for heat and mass transfer 2nd edition by rolle

A boundary is the surface across which heat, energy, and mass flow and is the primary location for predicting these transfers.. Heat transfer is rate at which heat or energy flows across

Heat and Mass Transfer Solutions Manual Second Edition

This solutions manual sets down the answers and solutions for the Discussion Questions, Class Quiz Questions, and Practice Problems There will likely be variations of answers to the discussion questions as well as the class quiz questions For the practice problems there will likely be some divergence of solutions, depending on the interpretation of the processes, material behaviors, and rigor in the mathematics It is the author’s responsibility to provide accurate and clear answers If you find errors please let the author know of them at

rolle@uwplatt.edu

Chapter 1

Discussion Questions Section 1-3

1 What is meant by a system? By a body?

A volume in space having a boundary surface separating it from its surroundings A body is a system that typically has a mass and in the study of mechanics, is subjected to external forces and body forces

2 What is the importance of a boundary in heat and mass transfer?

A boundary is the surface across which heat, energy, and mass flow and is the primary location for predicting these transfers

3 What is the difference between a boundary and a surface?

A surface is a three dimensional geometric construct but which has no thickness A boundary is a surface but a surface may not necessarily be a boundary

4 What is meant by system properties?

These are the quantities that describe the action or reaction of a system to its surroundings Examples of properties are pressure, temperature, volume, mass, and energy

5 What is steady state?

Steady State is the situation when a system does not change with elapsing time

6 What is a control volume?

A boundary containing a system and defining the system volume A control volume allows for mass to pass through the boundary in both ways

7 What is heat transfer?

Heat transfer is rate at which heat or energy flows across a systems boundary

8 What is meant by reversible heat transfer?

Reversible heat transfer is an approximation to allow for a predictable answer to heat transfer Rigorously, reversible heat transfer must only occur over a

differential temperature or differential temperature gradient and must proceed over an infinitesimally small time interval so that it needs to be fast

Section 1-4

9 What are the three modes of heat transfer?

The three modes are conduction, convection, and radiation

10 What is significant about the temperature difference when considering radiation heat

transfer?

The temperature difference is a difference of temperature, each to the fourth power

11 What mode of heat transfer does not satisfy the thermodynamic definition of heat

transfer and why?

Convection heat transfer does not rigorously satisfy the thermodynamic concept of heat or heat transfer because there is not a precise boundary across which heat can flow and there is mass transfer mingled with the process

Section 1-5

12 What are causes of diffusional mass transfer?

Diffusional mass transfer occurs due to a pressure difference or a pressure gradient, due to a concentration or density difference or gradient, due to a temperature difference or gradient, due to body forces on the mixture components, due to forced or free convection currents, due to turbulent flow,

or due to phase change phenomena

13 What is the difference between a concentration gradient and a density gradient?

Concentration is the number of moles in a prescribed volume while density is the amount of mass in a prescribed volume The gradients just reflect these two definitions

Class Quiz Questions

1 What is meant by an ideal or perfect gas?

A substance or material which behaves according to the equation
 = 

2 What is meant by an incompressible material?

A material which does not change its volume when subjected to different pressures or temperatures is called an incompressible material

3 Define specific heat at constant volume

Specific heat at constant volume,  is  

4 Define specific heat at constant pressure

Specific heat at constant pressure,  , is  



5 What is the first law of thermodynamics for a system?

The first law of thermodynamics, in words, is heat minus work equals energy change of the system

6 What is flow work?

Flow work is the work done against a pressure when pushing a fluid, causing it

to move or flow As an equation flow work is


7 What is thermal conductivity?

The defining equation for thermal conductivity  is Fourier’s law of conduction;

,=  

8 What is the convective heat transfer coefficient?

Newton’s law of heating or cooling is the defining equation for the convective heat transfer coefficient ℎ; = ℎ |"− $|

9 What is the relative humidity of air when it is at its dew point?

The relative humidity is 100 %

Practice Problems Section 1-3

1 Ten lbm of carbon dioxide gas are cooled from 1500F to 800F The specific heat at constant volume is = 0.323 − )*++ -./*01 where T is in Rankine degrees and  is in

Btu/lbm·0F Determine the internal energy change for the cooling process

Solution The internal energy change for the cooling process is

Using some integral calculus we have

540

2 610

148 32045 0.323

540 2 610 0

32045 0.323 148 ln

113.8

T

Btu

=

2 Two kg of water are heated from 150C to 3000C at 101 kPa Determine the amount of

heat added during the process Assume c P is 4.18 kJ/kg·K for liquid water and 1.86

kJ/kg·K for steam Assume the heat of vaporization is 2258 kJ/kg at 101 kPa

Solution

For the heat we have

and then

3 One hundred lbm of air in a pressure tank is cooled from 1800F and 160 psia to 800F when the surroundings are at 700F Determine the tank pressure at 800F

Solution Assume the tank is rigid so that the volume is constant and that the air behaves as an ideal gas Then

and

4 Steam is heated from 4 MPa, 4800C to 4 MPa, 6400C Determine the heat added to the steam per unit mass

Solution The heat added will be the increase in the steam’s enthalpy, The enthalpies may be read from a superheat steam table from a thermodynamics book

or approximated from Appendix Chart C-2

Q= ∆Hn= ∆m hn

5970.6

kg

1

540

640

q=hn −hn

1 2

3390 /

3766 /

=

=

and then

5 Three hundred kg of water are heated from 200C to 800C and at constant pressure

Determine the enthalpy change and the heat

Solution The enthalpy change is the heat so

and then

6 Mercury is heated from 1000F to vapor at 8000F Assume hnfg is 122 Btu/lbm at 6760F and use a cP of 0.032 Btu/lbm ·0F for liquid mercury and 0.015 for the vapor Determine the enthalpy change per unit mass and the heat per unit mass

Solution The enthalpy change will be the heat so that

And then

7 A refrigerator condenser is a heat exchanger that converts a refrigerant from a vapor to

a liquid at constant pressure and approximately constant temperature If ammonia is used as a refrigerant and 10 kg/s is condensed at 250C when the surrounding

temperature is 200C, determine the heat transfer from the refrigerant using data from the Appendix Chart C-7

Solution The heat transfer is determined from the equation

( 2 1)

P

Q = ∆ m hn = mc T − T

(300kg)(4.18kJ kg/ K)(80 20K) 75, 240kJ

The enthalpies may be approximated from Chart C-7 or read from a thermodynamic table of saturation properties of ammonia

and then

8 Using the psychrometric chart Appendix Chart C-1, determine the vapor pressure of

water in air at 800F, 40% relative humidity Also determine the humidity ratio, the enthalpy, the dew point temperature and the wet bulb temperature

Solution

For the air-water vapor mixture at 800F, 40% relative humidity, the dry bulb temperature is 800F Then, reading from the Chart C-1E

9 Ten m3/s of air at 00C, 80% relative humidity is heated to 250C Determine the amount of heat transfer required and the final relative humidity of the air Determine the amount

of water required to increase the final relative humidity to 50% at 250C How much additional heat is then required to accomplish this humidification?

Solution

From the psychrometric Chart C-1

fg

Qi = Hni = ∆ m hni = m hni

11, 670 / 11.67

0 0

0.2

54 64

v

dp

wb

grains Btu lbmdryair

ϖ

≈

≈

≈

≈

≈

The mass flow is just

And the volume flow rate, , is 10 m3

/s The specific volume is read from Chart C-1 as about 0.775 m3/kg so that the heat transfer is

At 50% relative humidity and 250C the humidity ratio is about 10 g/kg from Chart C-1 so that the amount of water needed to be added to the air is

And the heat required for the humidification is

10 Air at 1000F, 70% relative humidity is to be conditioned to 750F and 60% relative humidity Determine the partial pressure of the water vapor in the air at both states, the amount of water removed per lbm of dry air, and the lowest temperature to which the air must be cooled to accomplish this conditioning process

Solution

From the psychrometric Chart C-1E the following properties are read:

1 2 2

3.2 /

17%

g kgdryair

hn kJ kgdryair

hn kJ kgdryair

β

≈

≈

≈

=

/

mi =Vi ρ=V vi

3

hn

∆

10

10 3.2 87.7 / 0.775

w

V

 

 

i i

Qi =m hni −hn ≈ kg s kJ kg− kJ kg = kW

The amount of water removed is the difference in humidity ratios, or 205 - 78 = 127 grains/lbm dry air, or 0.18 lbm/lbm dry air The lowest temperature to which the air must be cooled is the dew point temperature, namely about 610F

11 One hundred kg of dry air is humidified by mixing with 1 kg steam at 300C Determine the partial pressure of the dry air and the steam if the total pressure is 100 kPa

Solution

The humidity ratio is

Solving for the partial pressures,

P v = 1.6 kPa and p da = 98.4 kPa

12 For a real substance the enthalpy is function of temperature and pressure Write an

integral equation that expresses the enthalpy change per unit mass of the substance

when the temperature and pressure change from T 1 to T 2 and p 1 to p 2

Solution

We have

And

1 1 2 1 2

0.64 0.26

v v

grains lbmdryair

grains lbmdryair

ω ω

≈

≈

≈

≈

≈

100

−

( , )

hn f T p

=

Section 1-4

13 A Styrofoam ice chest is 2 cm thick If the inside of the chest is at 00C and outside the chest it is 250C, estimate the heat transfer by conduction through the Styrofoam per unit area

Solution

Where the thermal conductivity for Styrofoam is read from Table B.2

14 A concrete nuclear reactor containment wall has a temperature distribution given by

the equation

where x is in feet Determine the heat transfer through the wall per unit area due to conduction at the center of the wall, where x = 0.5 ft

Solution

Using Fourier’s law of conduction

Reading the thermal conductivity for reinforced concrete from Table B-2E and at x = 0.5

ft we find

15 A large 8 ft by 4 ft thermopane glass window loses 300 Btu/hr of heat when the inside

temperature is 700F If the thermopane has an average thermal conductivity of 0.032 Btu/hr·ft·0F and is ½ inch thick, estimate the outside temperature of the window

Solution

Using Fourier’s law of conduction we have

(25 0 0)

q

i

i

2

ft

A

T

i

0

A

q

⋅ ⋅

i

i

16 A cast iron frying pan ¼ inch thick is used to prepare some food on a stove top If the

lower surface of the pan is at 6000F and the upper surface is 5000F, estimate the heat transfer through the pan per unit area

Solution

Using Fourier’s law of conduction and reading the value for thermal conductivity for cast iron from Table B.2E,

17 Wind blows at 50 m/s around a 5 cm diameter electric power line when the air

temperature is -100C Estimate the heat loss of the power line per unit length if the surface temperature of the power line is 50C

Solution

Using Newton’s law of cooling and an approximate value from Table 1-4 for the convective heat transfer coefficient,

18 Water at 600F flows through a copper tube of 1 inch inside diameter (ID) at aspproximately 1 ft/s Determine the heat transfer to the water per foot of tube length

if the inside surface temperature of the tube is 1800F

Solution

Using Newton’s law of cooling and an approximate value from Table 1-4 for the convective heat transfer coefficient,

0

0

70

1/ 24

i

0 0

100

1/ 48

A

q

i

2 0

l

⋅

i

2 0

1

12

l

i

19 An automobile is moving on a highway at 100 km/hr Estimate the heat loss per unit

area from the auto’s roof, which is at 300C, if the air temperature is 150C

Solution

Using Newton’s law of cooling and an approximate value from Table 1-4 for the convective heat transfer coefficient for 30 m/s,

20 Estimate the heat loss per unit area of a vertical south-facing wall of a large office

building when the air temperature is -100F and the wall temperature is 50F

Solution

Using Newton’s law of cooling and an approximate value from Table 1-4 for the convective heat transfer coefficient for still air,

21 A well-clothed person walks into a large auditorium that is empty If the auditorium

walls are at an average temperature of 550F and the average surface temperature of the person’s clothes is 850F, estimate the net radiation heat transfer between the person and the auditorium walls An average person can be assumed to have a surface area of 19.4 ft2 (1.8 m2)

Solution

Assuming black body radiation between the person and the auditorium walls,

22 A mercury-in-glass thermometer reads an outside temperature of 200C If the sky and surroundings of the thermometer have an average surface temperature of 50C, estimate the net radiation per unit area from or to the thermometer

A

⋅

i

A

i

i

Solution

Assuming black body radiation

The radiation is from the thermometer to the sky and surroundings

23 The surface of the sun seems to be about 10,0000F What would you guess the rate of heat emission from the sun to be for an area of the sun that measures 1 ft2?

Solution

24 A radiation pyrometer is a device that uses radiant heat to measure the temperature of

a surface Assume that a pyrometer has a surface area of 5 cm2 and is at a temperature

of 200C when directed towards a furnace opening having a temperature of 11000C Estimate the net rate of heat transfer towards the pyrometer if black body radiation is assumed

Solution

Section 1-5

25 Predict the concentration of ammonia in air 5 mm from an interface of vapor ammonia

and air if the interface area is 1500 mm2 and the evaporation rate is found to be 0.002 g/min Assume the ammonia and air are at 200C

therm sky A

i

A

i

100.5

W

⋅

= i i

Solution

Assume the evaporation rate of ammonia occurs where the ammonia is a liquid and its concentration is 100% Using Table B-3 the density is 602 kg/m3 The diffusivity from Table 1-5 is about 0.236 cm2/s Then, using Fick’s law

 = −2 3456

26 Liquid mercury is contained in a lead beaker as shown in Figure 1-14 Estimate the

amount of mercury that migrates by diffusion into the beaker after 48 hours if the concentration of mercury is 2% at a distance 0.01 cm into the beaker wall from the inside surface Assume that the system is at 200C and neglect evaporation to air

Solution

Assume the mercury has 100% concentration at the surface of the beaker Then the density of diffusing mercury is 12,816 kg/m3 The approximate value for the diffusivity of mercury into lead is given in Table 1-5 Using Fick’s law

 = −2 345Δ8

Δ9

3

1

0.99992 99.9992%

i

Solution The enthalpy change will be the heat so that... and the wall temperature is 50F

Solution

Using Newton’s law of cooling and an approximate value from Table 1-4 for the convective heat transfer coefficient for. .. and 1.86

kJ/kg·K for steam Assume the heat of vaporization is 2258 kJ/kg at 101 kPa

Solution

For the heat we have

and then

3