1

dx

dTkA - Q

k

L

)t(

A

Q

kA

L

)T(

kA

L

)t(Q

kA

L

)tt(Q ;

kA

L

)tt(Q

L

)tt(kAQ

L

)TT(kAQ

)x(

TkAQ

ttTTT

L)x(

TkA)x(Q

dTkAdxQ

kAdTQdx

2112

12

12

1212

MODES OF HEAT TRANSFER 1. Conduction: It is the transfer of heat from one part of a body to another part of the same body, or from

one body to another in physical contact with it, without appreciable displacement of the particles of the body.

2. Convection: It is the transfer of heat from one point to another point within a fluid, gas, vapor, or liquid by

the mixing of some portion of the fluid with another. A. Natural or Free Convection: the movement of the fluid is entirely caused by differences in

density resulting from temperature differences. B. Forced Convection: the motion of the fluid is accomplished by mechanical means, such as a fan

or a blower. 3. Radiation: It is the transfer of heat from one body to another, not in contact with it, by means of "wave

motion" through space. CONDUCTION:

From FOURIER'S LAW:

Note: Negative sign is used from Fourier's Equation because temperature decreases in the direction of heat flow.

2

1 2

R

Q

From Ohms Law;

)( Ohms in resistance electrical-R

(V) VoltsinEnergy -V

Amperesin currentI

R

VI

conductorelectric an forflow heat Current

Watts

K or

Watts

C

kA

LR

R

T-

R

t-Q

flow heat conductive For

where:

(-T) and(-t) - temperature potential in K or C

R - thermal resistance in C/W or K/W Q - conductive heat flow in Watts

k - thermal conductivity in W/m-C or W/m-K A - surface area in m2 L - thickness in m

THERMAL CIRCUIT DIAGRAM

3

CONVECTION:

From Newton's Law of Cooling:

W

K or

W

C ,resistance-R

K or C potential, temperatueT & t

m area, Surface - Fuid-A

K-m

W or

C-m

W e),conductanc (surface tcoefficien convective -h

Wattsflow, heat convective-Q

where

R

T

R

tQ

hA

1

T

hA

1

t

hA

1

)tt(

hA

1

)tt(Q

Watts)tt(hAQ

2

22

1221

21

RADIATION:

T1

A

Surface or

(Radiator)

T2

T1 > T2

T1 – Surface temperature, K

T2 – surrounding temperature, K

Q

4

From Stefan-Boltzmann Law: The radiant heat transfer of a blackbody is directly proportional to the product of the surface area A and its absolute temperature to the fourth power.

4

4

ATQ

AT Q

From Fig., the radiant heat flow Q from the surface (or body) to the surrounding (or to other surface or body) is equal to

K e,temperatur surface other or gsurroundin absolute -T

K e,temperatur surface absolute -T

Constant Boltzmann-Stefan K-m

W ,10 x 5.678

where

WattsTTAQ

2

1

42

8-

4

2

4

1

Black Body - a hypothetical body that absorbs the entire radiation incident upon it. Gray Body - are actual bodies or surfaces that absorbs a portion of the black body radiation, because they are not perfect radiators and absorbers. EMISSIVITY

Emissivity - is the ratio of the actual body (or surface) radiation at temperature T to the black body (or black surface) radiation at the same temperature T.

tcoefficien radiation TTTTh

T-T

T-TTTTT

T-T

TTTT

T-T

TTh

T-ThTT

T-TAh TTAQ

TT where

WattsTTAQ

to equal is

) Radiator Actualor Surface Actual(orBoby Actualanby radiated heat of rate the Therefore

T @ radiation Surface Black

T @ radiation Surface Actual

21

2

2

2

1r

21

2121

2

2

2

1

21

2

2

2

1

2

2

2

1

21

4

2

4

1r

21

4

2

4

1

21r

4

2

4

1

21

4

2

4

1

Combined Radiation and Convection heat transfer Actual surface exposed to the surrounding air involves convection and radiation simultaneously, the total heat transfer Q

ConvectionRadiationcombined

21c

hhh

)tt(AhQ

5

(Answer: Q = 8400 Watts)

Answer: 8125 W

0.50 x 0.25 m

t1 = 300C

h = 250 W/m2-K

t2 = 40C

QConvection

6

A = 0.5 m2

Q

t1 = 150C

t2 = 25C

0.8

Answer: 726 W; 547 W

4

2

4

1

4

Emitted

TTAQ

ATQ

7

1 :Assume

Example 4 The wall of a house 7 m wide and 6 m high is made from o.3 thick brick with k = 0.6 W/m-K. The surface

temperature on the inside of the wall is 16C and that on the outside is 6C. Find the heat flux through the wall and the total heat loss through it.

A = 7 x 6 m2

k

Q

1

2

L

Example 5

L

20 mm

8

Example 6

Assume the conducted is transferred to the air stream by convection and radiation to the air

stream with = 1.

9

1 :Assume

Example 7

10

8017,370,050,T

29327320T

0,841.5645,077,281-5.82T105,820,10T

0A

QThT

Th

T

0A

QThThTT

hThTTTA

Q

hThTTTA

Q

)TT(hTTA

Q

)TT(hAQc

TTAQr

QcQrQ

4

2

2

1

4

1

4

22

1

4

1

4

221

4

1

21

4

2

4

1

21

4

2

4

1

21

4

2

4

1

21

4

2

4

1

K64.20426T

426T

formulaquadratic by

00.9354507760141-82T105820105.T

s

2

s

s

22

s

11

Example 8

Answer: 1233 KW/m2

Example 9

12

;

k

L

Δt)(

A

Q ;

k

L

Δt)A(Q

k

L

A

1

Δt)(Q ;

R

Δt)(Q ;

kA

L

Δt)(Q

Ak

LR ;

Ak

LR ;

Ak

LR

k

L

k

L

k

L

)t(t

A

Q

k

L

k

L

k

L

)tA(t

k

L

k

L

k

L

A

1

)t(t

RRR

)t(t

Ak

L

Ak

L

Ak

L

)t(tQ

3

33

2

22

1

11

3

3

2

2

1

1

41

3

3

2

2

1

1

41

3

3

2

2

1

1

41

321

41

3

3

2

2

1

1

41

Q

CONDUCTION THROUGH A COMPOSITE PLANE WALL

Thermal Circuit diagram:

13

k

L

1U

k

L

t)A(-

k

L

T)A(-Q

t)UA(- T)UA(-Q

OVERALL COEFFICIENT OF HEAT TRANSFER

where: U - overall coefficient of heat transfer in W/m2-C or W/m2-K

14

k

L

h

1

Δt)(

Δt)(

kA

L

hA

1

Δt)(Q

Ak

LR ;

Ak

LR ;

Ak

LR;

Ah

1R ;

Ah

1R

;

h

1

k

L

k

L

k

L

h

1

)t(t

A

Q

h

1

k

L

k

L

k

L

h

1

)tA(tQ

h

1

k

L

k

L

k

L

h

1

A

1

)t(tQ

)t(t

RRRRR

)t(tQ ;

Ah

1

Ak

L

Ak

L

Ak

L

Ah

1

)t(tQ

3

3

3

2

2

2

1

1

1

o

o

i

i

o3

3

2

2

1

1

i

oi

o3

3

2

2

1

1

i

oi

o3

3

2

2

1

1

i

oi

oi

o321i

oi

o3

3

2

2

1

1

i

oi

A

Q

tRR ;

R

;

R

HEAT TRANSFER FROM FLUID TO FLUID SEPARATED BY A COMPOSITE PLANE WALL

L1 L2 L3

t1

t2

t3

t4k1 k2 k3

Q

Fluid

(i)

Fluid

(o)

A

hi

ti

ho

to

Thermal Circuit Diagram:

2

R1

Q

R2 R3

3 41i oRi Ro

15

k

L

h

1

1U

t)UA(-Q

k

L

h

1

t)A(-Q

OVERALL COEFFICIENT OF HEAT TRANSFER COMPOSITE PLANE WALL

k

L

h

1

R

Δt)(Q

k

L

h

1

Δt)(

A

Q

k

L

h

1

Δt)A(Q

k

L

h

1

A

1

Δt)(Q ;

kA

L

hA

1

Δt)(Q

tRR

16

)TT(kL

)TT(kLkL

kLdT)rL

21

12

2r

rQln

2r

rQln ; dT2-

r

drQ

2- r

drQ ; ,

dr

dTk(2- Q

rL2 A , dx

dTkA- Q

, dx

dTkA- Q

:Equation sFourier' From

1

2

1

22

1

2

1

is; Qsfer heat tran thedirection, radial ain flowsheat since

21tt

R

t

kL2ri

roln

tQ

CONDUCTION THROUGH CYLINDRICAL COORDINATES

L

r1

r2

1 2

k

Q

L - length of the cylinder perpendicular to the paper

resistance - R where

kL2

rr

ln

R

R

)t(tQ ;

kL2

rr

ln

)t(tQ

t- t T - T where

kL2

rr

ln

)T(TQ ;

r

rln

)TT2Q

1

2

21

1

2

21

2121

1

2

21

1

2

21

(kL

o – refers to outside

i – refers to inside

17

where: Q in Watts

k in W/m-C or W/m-K L in meters r in m

-t in C or K

R in W/C or W/K CONDUCTION THROUGH A COMPOSITE CURVED WALL

r1

r2

r3

i1 2 3

k1k2

Let L - length of the cylinder perpendicular to the paper

Lk2

rr

ln

tt Q

3 to 2 at

Lk2

rr

ln

tt Q

2 to 1 at

2

2

3

32

1

1

2

21

Lk2

rr

ln

Lk2

rr

ln

Lk2

rr

ln

)t(t Q

kL2

rr

ln

t- Q

Lk2

rr

ln

Lk2

rr

ln

Lk2

rr

ln

)t(t Q

:4 to 1 at

Lk2

rr

ln

tt Q

:4 to 3 at

3

3

4

2

2

3

1

1

2

41

i

o

3

3

4

2

2

3

1

1

2

41

3

3

4

43

k

rr

ln

)t(-2

L

Q

k

rr

ln

k

rr

ln

k

rr

ln

)t(t2

L

Q

k

rr

ln

k

rr

ln

k

rr

ln

)tL(t2 Q

k

rr

ln

k

rr

ln

k

rr

ln

L2

1

)t(t Q

i

o

3

3

4

2

2

3

1

1

2

0i

3

3

4

2

2

3

1

1

2

0i

3

3

4

2

2

3

1

1

2

0i

18

Lk2

r

rln

)t-(t Q

2

2

3

32

ii

1i

Ah

1

)tt(Q

Lk2

r

rln

)t-(t Q

3

3

4

43

HEAT TRANSFER FROM FLUID TO FLUID SEPARATED BY A COMPOSITE CURVED WALL

r1

r2

r3

io

to

hi

ti ho1 2 3

k1

k2

r4

k3

4

L - length of the cylinder perpendicular to the paper at i to 1: at 1 to 2: at 2 to 3: Q = Aihi(ti - t1) at 3 to 4: at 4 to 0: at i to 0:

003

3

4

2

2

3

1

1

2

ii

0i

hA

1

Lk2

rr

ln

Lk2

rr

ln

Lk2

rr

ln

hA

1

)tt(Q

043

3

4

2

2

3

1

1

2

i1

0i

Lhr2

1

Lk2

rr

ln

Lk2

rr

ln

Lk2

rr

ln

Lhr2

1

)tt(Q

Lk2

r

rln

)t-(tQ

1

1

2

21

Lr2A

hA

1

)tt(Q

)tt(hAQ

40

00

04

0400

Lr2A1i

19

043

3

4

2

2

3

1

1

2

i1

0i

hr

1

k

rr

ln

k

rr

ln

k

rr

ln

hr

1

L2

1

)tt(Q

043

3

4

2

2

3

1

1

2

i1

0i

hr

1

k

r

rln

k

r

rln

k

r

rln

hr

1

)tt(L2Q

043

3

4

2

2

3

1

1

2

i1

0i

hr

1

k

r

rln

k

r

rln

k

r

rln

hr

1

)tt(2

L

Q

General Equation:

kL2

r

rln

Ah

1

)t(Q

i

o

;

R

)t(Q

; where

kL2

r

rln

Ah

1R i

o

o - refers to outside i - refers to inside

k

r

rln

rh

1

L2

1

)t(Q

i

o

k

r

rln

rh

1

)t(L2Q

i

o

k

r

rln

rh

1

)t(2

L

Q

i

o

20

21

22

23

24

25

OVERALL COEFFICIENT OF HEAT TRANSFER

Q = UA(-t) A. For a composite Plane Wall

k

L

h

1

1 U

kA

L

hA

1

k

L

h

1

Δt)A(Q

Rt

Rt

t

B. For a Composite Curved Wall

kL2

rr

ln

hA

1UA

kL2

rr

ln

hA

1

kL2

rr

ln

hA

1

Δt- Q

i

o

i

o

i

o

tR

Rt

t

UA = UiAi = UoAo

where:

Ui - overall coefficient of heat transfer based on

inside surface, W/m2-C or W/m2-K Uo - overall coefficient of heat transfer based on

outside surface, W/m2-C or W/m2-K

26

HEAT EXCHANGER OR HEAT TRANSFER EQUIPMENT

TYPES OF HEAT EXCHANGERS

1. DIRECT CONTACT TYPE: The same fluid at two different states is mixed. 2. SHELL AND TUBE TYPE: One fluid flows inside the tubes and the other one on the outside.

mh, hh

mc, hc

m, h

Transfer Heat Total )hh(m)h(hmQ

)hh(m)h(hm

hmhmhmhm

h)mm(hmhm

hmhmhm

)negligible are PEΔ and KEΔ( BalanceEnergy By

mmm

Balance Mass By

cchh

cchh

chcchh

chcchh

cchh

ch

Example:

27

Shell and Tube Type Heat Exchanger

By energy balance:

Heat Rejected by hot fluid = Heat Absorbed by cold fluid

Fluid Cold )tt(CmQ

Fluid Hot )tt(CmQ

QQ

1c2cpccc

2h1hphhh

ch

28

2 AMTD

ln

LMTD

12

2

12

1

difference eTemperatur Meanc Arithmetion Based )AMTD(UAQ

difference eTemperatur Mean Log on Based )LMTD(UAQ

QQQ ch

Where: A – total heat transfer area, m2

LMTD - Log Mean Temperature Difference, C or K

AMTD - Arithmetic Mean Temperature Difference, C or K

U – Overall Coefficient of Heat Transfer, Km

KW or

Cm

KW22

where: - terminal temperature difference

29

30

If the design of the heat exchanger is more complex, the LMTD is modified by a correction factor F.

factor correction - F

:Where

)LMTD(UAFQ

31

passes. of number theby tubes of number the

multiply and passes of numberby length the divide passes, tube Multiple For

tables or charts from determine be can F factor correction - F

meters sheet, tube of thickness - t

2t L L

tube of length actual - L

meters tube, of length efdfective - L

tubes of number total - n

area) surface inside on based U (For diameter tube inside - d

dLnπA

area) surface outside on based U for ( tube of diameter outside - D

m DLnπA

sec

m tubes of Number velocity x x Area V

t

t

2

3

flow

32

Example No. 1

Exhaust gases flowing through a tubular heat exchanger at the rate of 0.3 kg/sec are cooled from 400 to 120C by water

initially at 10C. The specific heat capacities of exhaust gases and water may be taken as 1.13 and 4.19 KJ/kg-K

respectively, and the overall heat transfer coefficient from gases to water is 140 W/m2-K. Calculate the surface area required when the cooling water flow is 0.4 kg/sec; a. for parallel flow (4.01 m2) b. for counter flow (3.37 m2)

C3.5366.7-120t-tθ

C39010-400t-tθ

Flow Parallel For

C7.66t

)10t)(19.4(4.0)120400)(13.1(3.0

QcQh

C10t

C120t ; C400t

C-KJ/kg 1.13C ;kg/sec 0.3m

C-KJ/kg 4.19C ;kg/sec 4.0m

Given

c2h21

c1h12

2c

2c

c1

h2h1

phh

pcc

)LMTD(U

QA

)LMTD(UAQ

C43.201

110

3.333ln

1103.333LMTD

C11010120ttθ

C3.3337.66400t-1tθ

Flow Counter For

C2.169

3.53

390ln

3.53390

θ

θln

θ-θLMTD

1c2h1

c2h2

1

2

12

2

2

m 4.3)43.201(140

920,94A

m 01.4)2.169(140

920,94A

Watts94920Q

KW 92.94)120400)(13.1(3.0Q

33

SAMPLE PROBLEMS 1. A composite wall is made up of an external thickness of brickwork 11 cm thick, inside which is a layer of fiber glass 7.5 cm thick. The fiber glass is faced internally by an insulating board 2.5 cm thick. The coefficient of thermal conductivity for the three are as follows;

Brickwork - 1.5 W/m-C

Fiber Glass -0.04 W/m-C

Insulating Board - 0.06 W/m-C

The surface transfer coefficient of the inside wall is 3.1 Wm2-C while that of the outside

wall is 2.5 Wm2-C. Determine the overall coefficient of heat transfer and the heat loss

through such a wall 6 m high and 10 m long. Take the internal ambient temperature as 10C

and the external temperature as 27C.

2. A furnace is constructed with 20 cm of firebrick, k = 1.36 W/m-K, 10 cm of insulating

brick, k = 0.26 W/m-K, and 20 cm of building brick, k = 0.69 W/m-K. The inside surface

temperature is 650C and the outside air temperature is 32 C. The heat loss from the furnace wall is 0.56 W/m2. Determine

a. the unit convective coefficient for the air W/m2-K (3.545)

b. the temperature at 25 cm from the outside surface in C. (460 C) 3. A furnace wall consists of 20 mm of refractory fireclay brick, 100 mm of sil-ocel brick, and

6 mm of steel plate. The fire side of the refractory is at 1150C and the outside of the

steel is at 30C. An accurate heat balance over the furnace shows the heat loss from the wall to be 300 W/m2. It is known that there may be thin layers of air between the layers of brick and steel. to how many mm of sil-o-cel are these air layers equivalent. (400 mm)

k = 1.52 W/m-C (fireclay)

k = 0.138 W/m-C (sil-ocel)

k = 4.5 W/m-C (steel) 4. A composite plane wall consisting of two layers of materials, 38 mm steel and 51 mm

aluminum, separates a hot gas at ti = 93 C; hi = 11.4 W/m2-C, from a cold gas at to = 27C;

ho = 28.4 W/m2-C. If the hot fluid is on the aluminum side, Find

a. the transmittance U in W/m2-C (8)

b. the total resistance R in C -m2/W (0.124)

c. the interface temperature at the junction of two metals in C (45) d. the heat through 9.3 m2 of the surface under steady-state conditions. (4937) 5. A heat exchanger is to be designed to the following specifications:

Hot gas temperature = 1145C

Cold gas temperature = 45C

Unit surface conductance on the hot side = 230 W/m2-K

Unit surface conductance on the cold side = 290 W/m2-K

Thermal conductivity of the metal wall = 115 W/m-K Find the maximum thickness of metal wall between the hot gas and the cold gas, so that the

maximum temperature of the wall does not exceed 545C. (20.115 mm) 6. A composite furnace wall is made up of a 300 mm lining of magnesite refractory brick, a 130 mm thickness of 85% magnesia and a steel 2.54 mm thick. Flue gas temperature is

1205 C and the boiler is at 27 C. Gas side film coefficient is 85 W/m2-C and the air side

is 23 W/m2-C. Determine: a. the thermal current Q/A in W/m2

b. the interface temperatures in C c. effect on thermal current and the inside refractory wall temperature if the magnesia insulation were doubled.

k for magnesite = 296 W/m-C

k for 85% magnesia = 0.0692 W/m-C

k for steel = 43.3 W/m-C

34

7. Determine the thermal conductivity of a wood that is used in a 1.5 m2 test panel, 25 mm thick, if during a 4 hours test period there are conducted 190 KJ through the panel with a

temperature differential of 6C between the surfaces. Express answer in W/m-C. (0.0244)

8. The walls of a cold storage are composed of an insulating material (k = 0.065 W/m-C)

10.16 cm thick held between two layers of concrete ( k = 1.04 W/m-C) each 10.16 cm thick.

The film coefficients are 22.7 W/m2-C on the outside and 11.4 W/m2-C on the inside. Cold

storage temperature is -7C and the ambient temperature is 32C. Determine the heat transmitted in KW through an area of 56 m2. (1.2) 9. A 12 in thick furnace wall with a dimensions of 5 m x 2 m has temperature difference of

60C. The wall has a thermal conductivity of 0.75 BTU/hr-ft-F. Calculate the heat transmitted across the wall. (2554 W) 10. A 15 cm thick wall has a thermal conductivity of 5 W/m-K. If the inside and outside surface

temperature of the wall are 200C and 30C, respectively. Determine the heat transmitted. ( 5.67 W/m-K)

11. Two walls of cold storage plant are composed on an insulating material (k = 0.07 W/m-K),

100 mm thick at the outer layer and material (k = 0.97 W/m-K), 15 cm thick at the inner

layer. If the surface temperature of the cold side is 30C and hot side is 250C, find the heat transmitted in W/m2. (138) 12. An insulated steam pipe runs through a dark warehouse room.The pipe outside diameter is 60 mm and its surface temperature and emissivity are 165 C and 0.95, respectively. The

warehouse and air is kept at 5C. If the coefficient of heat transfer by natural convection

from the outside surface to the air is 11.4 w/m2-C and the pipe surface maybe treated as a gray body, what is the rate of heat loss from the surface per meter of pipe length. ( 657 W/m)

13. Saturated steam at 500 K flows in a 20 cm ID; 21 cm OD pipe. The pipe is covered with 8

cm of insulation with a thermal conductivity of 0.10 w/m-C. The pipes conductivity is 52

W/m-C. the ambient temperature is 300 K. The unit convective coefficient are hi = 18 000

W/m2-C and ho = 12 W/m2-C. Determine the heat loss from 4 m of pipe. Calculate the

overall coefficient of heat transfer base on the outside area. (822 W; 0.9 W/m2-C) 14. A tube 60 mm OD is lagged with a 50 mm layer of asbestos for which the conductivity is

0.21 W/m-C, followed with a 40 mm layer of cork with a conductivity of 0.05 W/m-C. If

the temperature of the outer surface of the pipe is 150C and the temperature of the

outer surface of the cork is 30C, Calculate the heat loss in Watts per meter length of pipe.(59)

15. An economizer receives hot gas (Cp = 1.13o6 KJ/kg-K) and water in the ratio 1.5 kg gas/kg

water. The gas enters at 455C and leaves at 180C; the water enters at 50C. Find the exit temperature of the water and the LMTD:

a. for parallel flow ( 125.425C)

b. for counter flow (200.81C) Assume no energy losses external to the system. 16. A flat furnace wall is constructed of a 11 cm layer of sil-o-cel brick with a thermal

conductivity of 0.14 W/m-C, backed by a 23 cm layer of common brick of conductivity 1.4

W/m-C. The temperature of the inner face of the wall is 760 C, and that of the outer

face is 77C. a. What is the heat loss through the wall b. What is the temperature of the interface between the refractory brick and common brick c. Supposing that the contact between the two brick layers is poor and that a

contact resistance of 0.95 C/W is present, what would be the heat loss.

35

17. Steam initially saturated at 2.05 MPa, passes through a 10.10 cm standard steel pipe for a total distance of 152 m. the steam line is insulated with a 5.08 cm thickness of 85%

magnesia. For an ambient temperature of 22C, what is the quality of the steam which arises at its destination if the mass flow rate is 0.125 kg steam/sec. (x = 93%) At 2.05 MPa: hf = 914.52 KJ/kg; hfg = 1885.5 KJ/kg; hg = 2800 KJ/kg 18. The hot combustion gases of a furnace are separated from the ambient air and its

surrounding which are at 25C, by a brick wall 15 cm thick.The brick has a thermal

conductivity of 1.2 W/m-C and a surface emissivity of 0.8. Under steady-state

conditions and outer surface temperature of 100C is measured. Free convection heat transfer to the air adjoining this surface is characterized by a convection coefficient of

20 W/m2-C. what is the brick inner surface temperature inC. (352.5C) 19. A counter flow heat exchanger is designed to heat fuel oil from 28 C to 90 C, while the

heating fluid enters at 138C and leaves at 105C. The fuel has a specific gravity of

21API, a specific heat of 2.1 KJ/kg-K and enters the heat at the rate of 3 000 L/hr. Determine the required heating surface area in m2 if the overall coefficient of heat

transfer is 465.2 W/m2-K. ((3.5 m2)

20. Brine enters a circulating brine cooler at the rate of 5.7 m3/hr at -10C and leaves at

-16C.Specific heat of the brine is 1.072 KJ/kg-C and the specific gravity is 1.10. The refrigerant evaporates at -25 C. What is the required heat transfer area if U = 454 W/m2-C. (2.1 m2)

21. A steel pipe 100 mm bore and 7 mm wall thickness, carrying steam at 260C is insulated with 40 mm of a moulded high-temperature diatomaceous earth covering, this covering in

turn insulated with 60 mm of asbestos felt. The atmospheric temperature is 15C. The

heat transfer coefficients for the inside and outside surfaces are 550 and 15 W/m2-K, respectively and the thermal conductivities of steel, diatomaceous earth and asbestos felt

are 50, 0.09, and 0.07 W/m-K respectively. Calculate: a. the rate of heat loss by the steam per unit length of pipe (116 W)

b. the temperature of the outside surface (22.8C) 22. A furnace wall consists of 125 mm wide refractory brick and a25 mm wide insulating firebrick separated by an air gap. The outside wall is covered with a 12 mm thickness of

plaster. The inner surface of the wall is at 1100C and the room temperature is 25C. The heat transfer coefficient from the outside wall surface to the air in the room is

17 W/m2-K and the resistance to heat flow of the air gap is 0.16 K/W. The thermal conductivities of refractory brick, insulating brick, and plaster are 1.6, 0.3, and

0.14 W/m-K, respectively. Calculate: a. the rate of heat loss per unit area of wall surface (1344 W) b. the temperature of each interface throughout the wall

(995, 780, 220, 104 C)

c. the temperature at the outside surface of the wall ( 104.1C) 23. Exhaust gases flowing through a tubular heat exchanger at the rate of 0.3 kg/sec are

cooled from 400 to 120C by water initially at 10C. The specific heat capacities of

exhaust gases and water may be taken as 1.13 and 4.19 KJ/kg-K respectively, and the

overall heat transfer coefficient from gases to water is 140 W/m2-K. Calculate the surface area required when the cooling water flow is 0.4 kg/sec; a. for parallel flow (4.01 m2) b. for counter flow (3.37 m2)

24. A properly designed steam heated tubular pre heater is heating 5.7 kg/sec of air from 21C

to 77C when using steam at 0.14 MPa. It is proposed to double the rate of air flow

through the heater and yet heat the air from 21C to 77C; this is to be accomplished by increasing the steam pressure. Calculate the new steam pressure required to meet the change condition expressed in KPa. (661.5 KPa)

36

25. A 404.34 m2 heating surface counter-flow economizer is used in conjunction with a 72,000

kg/hr boiler. the inlet and outlet water temperature are 100C and 155C. The inlet and

outlet gas temperature are 340C and 192C. Find the overall coefficient in W/m2-C.

(85.6 w/M2-C)

26. A boiler tube for steam at 8.2 MPa (ts = 296.79C; h = 2574.82 KJ/kg) is 9 cm OD and 7 cm

ID. assume an internal film coefficient of 11,350 W/m2-C and a thermal flow of 157,640 W/m2 based on the outside area, calculate a) Outside tube temperature

b) Allowable boiler scale thickness if the metal is not to exceed 482C. Assume k for

steel = 43.26 W/m-C anf for scale k = 0.52 W/m-C.

27. An 8" steel pipeline (OD = 22 cm; ID = 19 cm) carries steam at 232C. An 85% magnesia

(k = 0.07 W/m-C) pipe covering is to be applied of such a thickness so as to limit the

surface temperature to 50C with room temperature of 16C. Assume inside and outside

coefficients of 1700 W/m2-C and 0.011 W/m2-C and k for steel = 41 W/m-C, calculate the magnesia thickness in cm,

28.Calculate the energy transfer rate across 6 in wall of firebrick with a temperature difference across the wall

of 50C. The thermal conductivity of the firebrick is 0.65 BTU/hr-ft-F at the temperature interest.

C

. m-

W 731

F-ft-hr

BTU

k = 0.65(1.73) = 1.1245 W/m-C L = 0.6" = 0.01524 m

2W/m 33689.

L

)tΔ(k

A

Q

29. A carpenter builds an outer house wall with a layer of wood (k = 0.080 W/m-K) 2 cm thick on the outside and a layer of styrofoam (k = 0.01 w/m-K) insulation 3.5 cm thick as the inside wall surface. What is the

temperature at the plane where the wood meets the Styrofoam? Interior temperature is 19C; exterior

temperature is -10C. L1 L2 1 2 3 Q k1 k2

1

1

21

2

2

1

1

31

k

L

)tt(

k

L

k

L

)tt(

A

Q

t2 = - 8C Exams

1. An 8" steel pipeline (OD = 22 cm; ID = 19 cm) carries steam at 232C. An 85% magnesia (k = 0.07 W/m-C) pipe

covering is to be applied of such a thickness so as to limit the surface temperature to 50C with room temperature of

16C. Assume inside and outside coefficients of 1700 W/m2-C and 0.011 W/m2-C and k for steel = 41 W/m-C, calculate the magnesia thickness in cm,

2. The emissivity of tungsten is 0.35. A tungsten sphere with a radius of 1.5 cm is suspended within a large enclosure

whose walls are at 290K. What power input is required to maintain the sphere at a temperature of 3000K if heat

conduction along the supports is neglected? (Area of sphere = 4r2)

3. A copper cylinder is initially at 20C. At what temperature will be its volume be 0.150%larger than it is at 20C.

Coefficient of linear expansion of copper is 1.7 x 10-5.

t1 = 19C

t3 = -10C k1 = 0.01 k2 = 0.08 L1 = 0.035 m L2 = 0.02 m