ME 4310 Heat Transfer Summer II, 2013 Example Problems

Dr. Bade Shrestha G-218, Department of Mechanical and

Aerospace Engineering

Example 1 (Conduction)

One face of a copper plate 3 cm thick is

maintained at 400o C, and the other face is

maintained at 100o C. How much heat is

transferred through the plate?

Example 2 (Convection)

Air at 20o C blows over a hot plate 50 by

75 cm maintained at 250o C. The

convection heat transfer coefficient is 25

W/(m2 .oC). Calculate the heat transfer.

Example 3 (Multimode)

Assuming that the plate in Example 2 is

made of carbon steel 2 cm thick and that

300 W is lost from the plate surface by

radiation, calculate the inside plate

temperature.

Example 4 (Plane Wall)

Consider a large plane wall of thickness L=0.2 m, thermal conductivity k= 1.2W/m K and surface area A = 15 m2. The two side of wall are maintained at constant temperature of T1=120 o C and T2 = 50 o C, respectively. Determine a) the variations of temperature within the wall and the value of temperature at x=0.1m and b) the rate of heat conduction through the wall under steady conditions.

Example 5 (k(T))

Consider a 2 m high and 0.7 m wide bronze

plate whose thickness is 0.1 m. One side of the

plate is maintained at a constant temperature of

600 K while the other side is maintained at 400

K. The thermal conductively of the bronze plate

can be assumed to vary linearly in that

temperature range as k(T)=ko(1+T) where

ko=38 W/mK and = 9.21 X 10 -4 K-1. Determine

the rate of heat conduction through the plate

assuming steady state conditions.

Example 6 (heat generation)

A plane wall of thickness 0.1m and

thermal conductivity 25 W/mK having

uniform volumetric heat generation of

0.3MW/m3 is insulated on one side, while

the other side is exposed to a fluid at 92 o

C. The convection heat transfer coefficient

between the wall and the fluid is 500

W/m2K. Determine the maximum

temperature in the wall.

Example 7 (Multilayer)

A thermo pane window consists of two pieces of glass 7 mm thick that enclose an air space 7 mm thick. The window separates room air at 20 o C from outside ambient air at -10 C. The convection coefficient associated with the inner (room-side) surface is 10 W/m2K. If the convection coefficient associated with the outer (ambient) air is 80 W/M2k, what is the heat loss through a window that is 0.8m long by 0.5 m wide? Neglect radiation, and assume the air enclosed between the glasses to be stagnant.

Example 8 (Over all heat transfer)

Two by four wood studs have actual

dimensions of 4.13X9.21 cm and a

thermal conductivity of 0.1 W/m2K. A

typical wall of a house is constructed as

shown in the figure. Calculate the over all

heat transfer coefficient and R value of the

wall.

Example 9 (Fin)

Compare the temperature distributions in a

straight cylindrical rod having a diameter

of 2 cm and a length of 10 cm and

exposed to a convection environment with

h = 25 W/m2K for three fin materials:

copper (k=385W/m2K), stainless steel

(k=17W/mK) and glass (k=0.8 W/mK).

Also compare the relative heat flows and

fin efficiencies.

Example 10 (Fin)

An aluminum fin (K=200 W/mK) 3.0 mm

thick and 7.5 cm long protrudes from a

wall. The base is maintained at 300 o C,

and the ambient temperature is 50 o C with

h=10W/m2K. Calculate the heat loss from

the fin per unit depth of material.

Example 12 (Unsteady)

A steel ball (c= 0.46 kJ/kg K, k=35W/mK)

5 cm in diameter and initially at a uniform

temperature of 450 C is suddenly placed

in a control environment in which the

temperature is maintained at 100 C. The

convective heat coefficient is 10 W/m2K.

Calculate the time required for the ball to

attain a temperature of 150 C.

Example 13

A large block of steel (k=45W/mK, =1.4

10-4 m2/s) is initially at a uniform temperature

of 35oC. The surface is exposed to heat flux (a)

by suddenly raising the surface temperature to

250o C and (b) through a constant surface heat

flux of 3.2x10 5 W/m2. Calculate the

temperature at a depth of 2.5 cm after a time of

0.5 min for the both these cases.

Example 14 (boundary layer)

Air at 27 C and 1 atm flows over a flat

plate at a speed of 2 m/s. Calculate the

boundary layer thicknesses at distances of

20 and 40 cm from the leading edge of the

plate. Calculate the mass flow that enters

the boundary layer between x=20 and

x=40 cm.

Example 14a (Laminar)

Air at 27 C and 1 atm flows over a flat plate at a

speed of 2 m/s. Calculate the boundary layer

thicknesses at distances of 20 and 40 cm from

the leading edge of the plate. Calculate the

mass flow that enters the boundary layer

between x=20 and x=40 cm. And assuming that

the plate is heated over its entire length to a

temperature of 60 C, calculate the heat

transferred in the first 20 cm of the plate and the

first 40 cm of the plate.

Example 14b contd.

For the flow system in example 14, calculate

the drag force exerted on the first 40 cm of

the plate using the analogy between fluid

friction and heat transfer.

Example 14c

The leading edge of a wing is to be

heated to a constant temperature of 3o C

to prevent ice formation. How much heat

must be supplied to the heating system

per meter of wing span? (length of the

heating edge is 10 cm, stream velocity is

200 Km per hour and ambient temperature

is -15oC)

Example 15

Assuming a transition Reynolds number of

5X 105, determine the distance from the

leading edge of a flat plate at which the

transition will occur for the following fluids

when u = 1 m/s and temperature = 27 oC;

atmospheric air, engine oil, water and

mercury.

Example 16 (Tub. Heat)

Air at 20 oC and 1 atm flows over a flat plate at

35 m/s. The plate is 75 cm long and is

maintained at 60 C. Assuming unit depth:

a) calculate the heat transfer from the plate

b) critical distance from the leading edge when

the flow becomes turbulent.

c) and thickness of the boundary layers at the

critical distance and the end of the plate.

Example 17 (turb. H. T.)

A flat plate of width 1m is maintained at a uniform surface temperature of Tw = 150 oC by using independently controlled, heat generating rectangular modules of thickness a = 10 mm and length b = 50 mm. Each module is insulated from its neighbors, as well as on its back side. Atmospheric air flows at 25 o C over the plate at a velocity of 30 m/s.

Find the required power generation (W/m3), in a module positioned at a distance 700 mm from the leading edge.

Find the maximum temperature in the heat-generating module.

(Take k =5.2 W/mK; cp = 320 j/kg K and = 2300 kg/m3 for the module).

Example 18 (cylinder)

Air at 1 atm and 35 o C flows across a 5

cm diameter cylinder at a velocity of 50

m/s. The cylinder surface is maintained at

a temperature of 150 o C. Calculate the

heat loss per unit length of the cylinder.

Example 19 (sphere)

Air at 1 atm and 27 o C blows across a 12

mm diameter sphere at a free stream

velocity of 4 m/s. A small heater inside the

sphere maintains the surface temperature

at 77 o C. Calculate the heat lost by the

sphere.

Example 20 (Lam. Pipe)

Water at 60 C enters a tube of 2.54 cm

diameter at a mean flow velocity of 2 cm/s.

Calculate the exit water temperature if the

tube is 3 m long and the wall temperature

is constant at 80 C. (neglect the entrance

effect).

Example 21 (Entrance)

Water at 60 C enters a tube of 2.54 cm

diameter at a mean flow velocity of 2 cm/s.

Calculate the exit water temperature if the

tube is 3 m long and the wall temperature

is constant at 80 C. (including the

entrance effect).

Example 22 (Free Conv.)

A large vertical plate 4 m high is

maintained at 60 C and exposed to

atmosphere air at 10 C. Calculate the heat

transfer if the plate is 10 m wide. Find the

location where boundary layer becomes

turbulent. And maximum velocity in the

boundary layer at this location and position

of maximum. Find the boundary layer

thickness at this position.

Example 23 (LMTD)

Water at the rate of 68 Kg/min is heated

from 35 to 75 oC by an oil having a specific

hate of 1.9 kJ/kg oC. The fluids are used in

a counter flow double pipe heat

exchanger, and the oil enter the

exchanger at 110 oC and leaves at 75 oC.

Calculate the overall heat transfer

coefficient if inner diameter of the pipe is

30 mm and the outer annulus diameter is

50 mm, and the length of the heat

exchanger needed.

Example contd.

If the overall heat-transfer coefficient is

320 W/m2 oC, and instead of the double

pipe heat exchanger of the previous

example, it is desired to use a shell and

tube exchanger with water making one

shell pass and the oil making two tube

pass,calculate the new area.

Example 24 (Overall HT coefficient)

Hot oil be cooled in a double tube counter flow heat exchanger. The copper inner tubes have a

diameter of 2 cm and negligible thickness. The

inner diameter of the outer tube (the shell) is 3

cm. Water flows through the tube at a rate of 0.5

kg/s, and the oil through the shell at a rate of 0.8

kg/s. Taking the average temperatures of water

and oil to be 45 oC and 80 oC, respectively,

determine the overall heat coefficient of this

heat exchanger.

Example 25 (Fouling) A double pipe (shell-and-tube) heat exchanger is

constructed of a stainless steel (k=15.1 W/mK) inner

diameter Di =1.5 cm and outer diameter Do =1.9 cm and

an outer shell of inner diameter 3.2 cm. The convective

heat coefficient is given to be hi= 800 W/m2k on the inner

surface of the tube and ho= 1200 W/m2K on the outer

surface. For a fouling factor of Rfi =0.0004 m2.K/W on the

inside tube and Rfo = 0.0001 m2.K/W on the shell side,

determine:

a) the thermal resistance of the heat exchanger per unit

length.

b) the overall heat transfer coefficient Ui and Uo base on

the inner and outer surface areas of the tube

respectively.

Example 26 (Radiator) A test is conducted to determine the overall heat

transfer coefficient in an automotive radiator that

is a compact cross-flow water-to-air heat

exchanger with both fluids unmixed. The radiator

has 40 tubes of internal diameter 0.5 cm and

length 65 cm in a closely spaced plate-fin matrix.

Hot water enters the tubes 90 oC at a rate of 0.6

Kg/s and leaves at 65 oC. Air flows across the

radiator through the inter-fin spaces and is

heated from 20 oC and 40oC. Determine the

overall heat transfer coefficient, Ui of this

radiator based on the inner surfaces area of the

tubes.

Example 27 (NTU)

A couter flow double tube heat exchanger is

to used to heat water from 20 oC to 80 oC

at a rate of 1.2 Kg/s. The heating is to be

accomplished by geothermal water

available at 160 oC at a mass flow rate of

2 kg/s. The inner tube is thin-walled and

has a diameter of 1.5 cm. The overall heat

transfer coefficient of the heat exchanger

is 640 W/m2K. Using the NTU method

determine the length of the exchanger.

Example 28 (NTU)

Hot oil is to be cooled by water in a 1-shell-pass and 8-tube-passes heat exchanger. The tubes are thin-walled

and are made of copper with an internal diameter of 1.4

cm. The length of each tube pass in the heat exchanger

is 5 m, and the overall heat transfer coefficient is 310

W/m2K. Water flows through the tubes at a rate of 0.2

kg/s, and the oil through the shell at a rate of 0.3 kg/s.

The water and oil enter at temperatures of 20 oC and

150 oC, respectively. Determine the rate of heat transfer

in the heat exchanger and the outlet temperatures of

water and the oil.

Example 29 (NTU)

Hot oil at 100 oC is used to heat in a shell-

and-tube heat exchanger. The oil makes

six tube passes and the air makes one

shell pass; 2.0 kg/s of air are to be heated

from 20 to 80 oC. The specific heat of the

oil is 2100 J/kg oC an its flow rate is 3.0

kg/s. Calculate the area required for the

heat exchanger for U = 200 W/m2 oC.

Example 30 (NTU)

A counter flow double pipe heat exchanger is used to heat 1.25 kg/s of water from 35o to 80o C by cooling and oil ( cp=2 kJ/kg oC) from 150o to 85o C. The overall heat transfer coefficient is 850 W/m2 C. A similar arrangement is to be built at another plant location, but it is desired to compare the performance of the single counter flow heat exchanger with two smaller counter flow heat exchangers connected with series on the water side and in parallel on the oil side. The oil flow is split equally between two exchangers, and it may be assumed that the overall heat transfer coefficient for the smaller exchangers is the same as for the large exchanger. If the smaller exchangers cost 20 % more per unit of surface area, which would be the most economical arrangement- the one large exchanger or two equal sized small exchangers?