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Exercices supplémentaires de phénomènes de transfert Matériaux 4ième semestre

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IIIIII EExxeerrcciicceess ssuupppplléémmeennttaaiirreess ddee pphhéénnoommèènneess ddee ttrraannssffeerrtt (Copy of Transport Phenomena in Materials Processing E.J, D.R Poirier)

Exercise 1.1 Determine the thermal conductivity of a test panel 150 x 150 mm and 12mm thick, if during a two-hour period 8.4 x 104 [J] are conducted trough the panel when the two faces are at 290K and 300K. Exercise 1.2 At steady state, the temperature profile in a laminated system appears thus: Determine the thermal conductivity of II if the steady-sate heats flux is 12.6 x 103 [W/m2] and the conductivity of I is 52 [W/mK] Exercise 1.3 The thermal conductivity of helium at 400K is 0.176 [W/mK]. Knowing only this datum, estimate the thermal conductivity of helium at 800K. Exercise 1.4 Repeat problem 1.3 but use the following equation to estimate the thermal conductivity of helium at 800K.

154

RkM

η= ⋅

Where R is the gas constant, M is the molecular weight and η is the viscosity.

x

0.3m

T2

T3 = 310K

0.5m

T1 = 530K

0

Material II Material I


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Exercise 1.5

a) Calculate the thermal conductivity of a gas containing 40-mol % He, 40-mol % H2 and 20-mol % N2 at 1000K

b) Assume that the concentration of He is constant but that the concentration of H2 and N2 vary as much as ± 5 % in a process. What is the variation in the thermal conductivity of the gas?

Exercise 1.6 Refer to Fig 7.9 and explain the variation of the thermal conductivity of MgO (magnesia) with temperature. Exercise 1.7 Use the Maxwell-Eucken equation and predict the thermal conductivity of a two phase solid (A plus B) as a function of composition (wt. pct.). A and B are insoluble in each other, and the

following data apply: 3 33 313 ; 7 ; 4 10 ; 3 10A B A B

W W kg kgk kmK mK m m

ρ ρ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= = = ⋅ = ⋅⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Exercise 1.8 A flat heater is sandwiched between two solids of equal areas (0.1m2) with different thermal conductivities and thicknesses. The heater operates at a uniform temperature and provides a constant power of 290 W. The external surface temperature of each solid is 300 K, and there is a perfect thermal contact at each internal interface.

a) Calculate the heat flux through each solid. b) What is the operating temperature of the heater?

Solid Thermal conductivity

[W/mK]

Thickness [mm] A 35 60 B 9 30

Exercise 2.1 The surface temperature of a vertical plate is maintained at 390K. At 0.24 [m] from the bottom of the plate, calculate the heat transfer coefficient to:

a) Air at 290K b) Helium at 290K

x e2 0 e1

T

T2 T1

Tc

T1 = T2 = 300K


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Exercise 2.2

a) Determine an expression that gives the heat flow P [W] through a solid spherical shell with inside and outside radii of r1 and r2 respectively.

b) Examine the results regarding what happens as the shell thickness becomes larger compared with the inside radius.

Exercise 2.3 Consider heat conduction trough a plane wall of thickness ∆x, and T1 and T2 are the surface temperatures. Derive the steady-state heat flux in terms of T1, T2 and ∆x if the thermal conductivity varies according to ( )0 1k k a T= + ⋅ where 0 andk a are constants. Exercise 3.1 A sheet of glass (1m length) is cooled from an initial temperature of 1250K by blowing air at 300K parallel to the surface of the glass. The free stream velocity of the air is 30 [m/s]. Calculate the initial heat transfer coefficient and when the glass has cooled to 400K. Exercise 3.2 A long cylindrical bar of steel (30 mm diameter) is heated in a tempering furnace to 810K. It is then cooled in a cross-stream of moving air at 300K with a free stream velocity of 30 [m/s]. Calculate the heat transfer coefficient that applies when the bar begins to cool.


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Exercise 3.3 Calculate the initial rate of loosing energy [W] of aluminium plate (1.2m x 1.2m x 10mm) and heated uniformly to 370K when it is:

a) Cooled in a horizontal position by a stream of air at 290K with a velocity of 2 [m/s]; b) Suspend vertically in stagnant air at the same temperature.

Exercise 3.4 A copper mould is used to make a casting of a nickel-base alloy. During most of the solidification period, the temperature profile in the copper is linear, with the surface on the casting side maintained constant at 1000K. The other side of the mould contacts the water at 40°C (313K). What is the surface temperature on the waterside of the mould?

Eau T2 = 40°C

T2

T'2 = ?

T1= 1000K

10mm

Cu

k (Cu) fct de T

355

360

365

370

600 700 800 900 1000

T [K]

k (C

u)

coefficient h (T)

0

5000

10000

15000

20000

25000

30000

400 500 600 700 800 900 1000

T [K]

h [W

/m2K

]

Ces graphiques sont des estimations : Les valeurs sont à utiliser seulement pour cet exercice. (Ils manquent de précision car il y a seulement trois points !)


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Exercise 4.1 A furnace wall is constructed of 7 in. of fire brick [ ]( )0.60k Btu h ft F= ⋅ ⋅ ° , 4 in. of red

brick ( )k 0.40= , 1 in. of glass-wool insulation ( k = 0.4 ), and 1/8 in. steel plate ( k = 26 ) on the outside. The heat transfer coefficients on the inside and outside surfaces are 9 and 3 [Btu/h ft2 °F], respectively. The gas temperature inside the furnace is 2500°F, and the outside air temperature is 90°F.

a) Calculate the heat-transfer rate through the wall [Btu/h ft2]. b) Determine the temperatures at all interfaces.

Exercise 4.2 Consider the flow of heat through a spherical shell. For steady-state conditions, the inside surface ( r = R1 ) is at temperature T1, and the outside surface ( r = R2 ) is at T2.

a) Write the pertinent differential energy equation that applies. b) Write the boundary conditions and develop an expression for the temperature

distribution in the shell c) Develop an expression for the heat flow ( P[W] ) through the shell. d) Determine the thermal resistance of the spherical shell.

Exercise 4.3 The wall of a blast furnace is water-cooled. Given the inside- and outside- surface temperature (2400°F and 180°F), what is the heat transfer coefficient for the water? The water, itself, is at 80°F. Assume steady-state conditions.

0.05ft 1ft

2400°F

180°F

Water 80°F

H2O

Steel k = 20 [Btu/h ft2°F]

Refractory k = 0.5 [Btu/h ft2°F]

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