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FKKKSA Chem. Eng. Dept

UNSTEADY-STATE HEAT TRANSFER

•  before steady-state conditions •  important due to large number of heating and cooling problems in industry eg. In metallurgical and food processing • Time dependent eg. Hot Metal slab is removed from a furnace and exposed to a cool stream or surrounding.

FKKKSA Chem. Eng. Dept

BIOT NUMBER

Biot number NBi – compares the relative values of internal conduction resistance & surface convective resistance

NBi =resistance to internal heat flowresistance to external heat flow

=hx1

k

Sphere

where x1 = characteristic dimension of the body = V/A

Long Cylinder Long Square Rod

x1 = r/3 x1 = r/2

x1 = x/2 where x = ½ thickness

NBi < 0.1 internal temperature gradients = small

Lumped thermal capacity – single mass averaged temperature

5/31/15

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FKKKSA Chem. Eng. Dept

HEAT TRANSFER DIFFUSION EQUATION

•  conduction in one direction in a solid

Rate of heat Input + rate of Generation = rate of heat Output + rate of heat Accumulation

y

x

Δx

Δz qx|x qx|x+!x

qy|yqz |z+!z

qz |z

Δy

FKKKSA Chem. Eng. Dept

rate of heat input : qx|x = !k("y."z)#T#x x

rate of heat ouput: qx|x+!x = "k(!y.!z)#T#x x+!x

rate of heat accumulation:

!

("x"y"z)#cp$T$x

rate of heat generation:

!

("x"y"z) ˙ q

Rate of heat Input + rate of Generation = rate of heat Output + rate of heat Accumulation

!

˙ q ="k

#T#x x +$x

"#T#x x +$x

%

& '

(

) *

$x= +cp

#T#x

HEAT TRANSFER DIFFUSION EQUATION

5/31/15

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FKKKSA Chem. Eng. Dept

!

"T"x

=k#cp

" 2T"x 2 +

˙ q #c p

= $" 2T"x 2 +

˙ q #cp

•  conduction in one direction in a solid

!

"T"x

= #" 2T"x 2 +

" 2T"y 2 +

" 2T"z2

$

% &

'

( ) +

˙ q *c p

•  conduction in 3 dimensions:

HEAT TRANSFER DIFFUSION EQUATION

FKKKSA Chem. Eng. Dept

!

"# 2T#x 2 +

# 2T#y 2 +

# 2T#z2

$

% &

'

( ) +

˙ q *c p

=#T#t

Cartesian coordinates ;

!

1r""r

r "T"r

#

$ %

&

' ( +

1r2" 2T") 2 +

" 2T"z2 +

˙ q k

=1*"T"t

!

1r2

""r

r2 "T"r

#

$ %

&

' ( +

1r2 sin)

"")

sin) "T")

*

+ , -

. / +

1r2 sin2)

" 2T"0 2 +

˙ q k

=11"T"t

k = a + bT

Cylindrical coordinates:

Spherical coordinates :

HEAT TRANSFER DIFFUSION EQUATION

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FKKKSA Chem. Eng. Dept

UNSTEADY-STATE HEAT TRANSFER Lumped capacity /Newtonian heating or cooling method

T∞ = sudden change in ambient temperature at t = 0

Negligible/ very low internal conductive resistance (NBi < 0.1) :

where

T = average temperature of object at t s To = temperature of object at t = 0 (K)

A = surface area of object (m2) V = volume of object (m3) ρ = density of object (kg/m3)

t = time (s)

tρVpchA

oeTT

TT ⎟⎠⎞

⎜⎝⎛

=−−

−

∞

∞

]ρV)t(hA/c

e)[1TρV(TcQ pop

−−−=

∞

Q = total amount of heat transferred from the object

FKKKSA Chem. Eng. Dept

... ... ... ... ... ... ...

EXAMPLE 5.2-1 & 2

A steel ball having a radius of 25.4 mm is at a uniform temperature of 699.9 K. It is suddenly plunged into a medium whose temperature is held constant at 394.3 K. Assuming a convective coefficient of h = 11.36 W/m2.K, calculate the temperature of the ball and the total amount of heat removed after 1 h. The average physical properties are k = 43.3 W/m.K, ρ = 7849 kg/m3 and cp = 0.4606 kJ/kg.K. Ans: 474.9 K, 5.589 x 104 J

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FKKKSA Chem. Eng. Dept

FLAT PLATE WITH NEGLIGIBLE SURFACE RESISTANCE Very large heat transfer coefficient at the surface or a relatively large conductive resistance in the object

T1 or c1 = sudden change in ambient temperature or concentration at t = 0

No convective resistance and unsteady-state conduction in the x-direction only

where T = T0 and c = c0 at t =0

T or c = temperature or concentration in solid at t = t

FKKKSA Chem. Eng. Dept

RELATION BETWEEN HEAT AND MASS TRANSFER

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FKKKSA Chem. Eng. Dept

RELATION BETWEEN HEAT AND MASS TRANSFER

FKKKSA Chem. Eng. Dept

UNSTEADY-STATE CONDUCTION IN A SEMI-INFINITE SOLID

where

Semi-infinte solid:

01

0

cKccc−

−

tD2xAB

tDDKk

ABAB

c

t = time (s)

α = thermal diffusivity = k/ρcp (m2/s)

K = equilibrium distribution coefficient = cLi/ci

h, kc = convective transfer coefficient

DAB = mass diffusivity

Surface resistance

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FKKKSA Chem. Eng. Dept

... ... ... ... ... ... ...

EXAMPLE 5.3-1

The depth in the soil of the earth at which freezing temperatures penetrate is often of importance in agriculture and construction. On a certain fall day, the temperature in the earth is constant at 15.6 oC to a depth of several meters. A cold wave suddenly reduces the air temperature from 15.6oC to -17.8 oC. The soil convective coefficient above the soil is 11.36 W/m2.K. The soil properties can be assumed as α = 4.65 x 10-7 m2/s and k = 0.865 W/m.K. Neglect any latent heat effects. a) What is the surface temperature after 5 h? b) To what depth in the soil will the freezing temperature of 0oCpenetrate in 5h? Ans: 5.24oC, 0.0293 m

FKKKSA Chem. Eng. Dept

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EXAMPLE 7.1-2

A very thick slab has a uniform concentration of solute A of co = 1.0 x 10-2 kgmol A/m3. Suddenly, the front face of the slab is exposed to a flowing fluid having a concentration c1 = 0.10 kgmol A/m3 and a convective coefficient kc = 2 x 10-7 m/s. The equilibrium distribution coefficient K = cLi/ci = 2.0. assuming that the slab is a semi-infinite solid, calculate the concentration in the solid at the surface and x = 0.01 m from the surface after t = 3 x 10-4 s. The diffusivity in the solid is DAB = 4 x 10-9 m2/s Ans: 6.96 x 10-2 kgmol/m3, 2.04 x 10-2 kgmol/m3

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FKKKSA Chem. Eng. Dept

UNSTEADY-STATE HEAT AND MASS TRANSFER

Large flat plate:

where

n = relative position

m = relative resistance

01

1

cKc

cKc

−

−

21

ABxtD

1c

ABxKk

Dm=

1xxn=

1hxkm=

X = relative time Y = unacccomplished change, a dimensionless ratio

FKKKSA Chem. Eng. Dept

UNSTEADY-STATE HEAT AND MASS TRANSFER

Temperature (or concentration) at center of a large flate plate:

01

1

cKc

cKc

−

−

21

ABxtD

1c

ABxKk

Dm=

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FKKKSA Chem. Eng. Dept

UNSTEADY-STATE HEAT AND MASS TRANSFER Long cylinder:

01

1

cKc

cKc

−

−

21

ABxtD

1c

ABxKk

Dm=

1xxn=

1hxkm=

where

n = relative position

m = relative resistance

FKKKSA Chem. Eng. Dept

UNSTEADY-STATE HEAT AND MASS TRANSFER Temperature (or concentration) at center of a long cylinder:

01

1

cKc

cKc

−

−

21

ABxtD

1c

ABxKk

Dm=

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FKKKSA Chem. Eng. Dept

UNSTEADY-STATE HEAT AND MASS TRANSFER Sphere:

01

1

cKc

cKc

−

−

21

ABxtD

1c

ABxKk

Dm=

!

n= xx1

1hxkm=

where

n = relative position

m = relative resistance

FKKKSA Chem. Eng. Dept

UNSTEADY-STATE HEAT AND MASS TRANSFER Temperature (or concentration) at the center of a sphere:

01

1

cKc

cKc

−

−

21

ABxtD

1c

ABxKk

Dm=

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FKKKSA Chem. Eng. Dept

... ... ... ... ... ... ...

EXAMPLE 7.1-1

A solid slab of 5.15 wt. % agar gel at 278K is 10.16 mm thick and contains a uniform concentrations of urea of 0.1 kgmol/m3. Diffusion is only in the x direction through two parallel flat surfaces 10.16 mm apart. The slab is suddenly immersed in pure turbulent water, so the surface resistance can be assumed to be negligible: that is, the convective coefficient kc is very large. The diffusivity of urea in the agar is 4.72 x 10-10 m2/s. a)  Calculate the concentration at the midpoint of the slab and 2.54 mm from the

surface after 10 h. b)  If the thickness of the slab is halved, what would be the midpoint

concentration in 10 h? Ans: 0.0172 kgmol/m3,2.0 x 10-4 kgmol/m3

FKKKSA Chem. Eng. Dept

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EXAMPLE 5.3-2

A rectangular slab of butter which is 46.2 mm thick at a temperature of 277.6 K in a cooler is removed and placed in an environment at 297.1K. The sides and bottom of the butter container can be considered to be insulated by the container side walls. The flat top surface of the butter is exposed to the environment. The convective coefficient is constant at 8.52 W/m2.K. Calculate the temperature in the butter at the surface, at 25.4 mm below the surface, and at 46.2 mm below the surface at the insulated bottom after 5h of exposure. Ans: 19oC, 15.1oC, 14.2 oC

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FKKKSA Chem. Eng. Dept

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EXAMPLE 5.3-3

A cylindrical can of pea puree has a diameter of 68.1 mm and a height of 101.6 mm and is initially at a uniform temperature of 29.4 oC. The cans are stacked vertically in a retort and steam at 115.6 oC is admitted. For a heating time of 0.75 h at 115.6 oC, calculate the temperature at the center of the can. Assume that the can is in the center of a vertical stack of cans and that is is insulated on its two ends by the other cans. The heat capacity of the metal wall of the can will be neglected. The heat transfer coefficient of the steam is estimated as 4540 W/m2.K. Physical properties of puree are k = 0.830 W/m.K and α = 2.007 x 10-7 m2/s. Ans: 104.4oC

FKKKSA Chem. Eng. Dept

UNSTEADY-STATE HEAT AND MASS TRANSFER 2-dimensional systems:

Transfer in x & y direction:

01

x1x TT

TTY−−

=

!

Yr =T1"TyT1"T0

01

x1

x cKc

cKc

Y−

−=

!

Yr =c1K "cyc1K "c0

Yx,r = (Yx)(Yr) = 01

yx,1TTTT−−

01

yx,1

cKccK

c

−

−=

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FKKKSA Chem. Eng. Dept

UNSTEADY-STATE HEAT AND MASS TRANSFER 3-dimensional systems:

Transfer in x, y, & z direction:

01

x1x TT

TTY−−

=

01

y1y TT

TTY

−−

=

01

z1z TT

TTY−−

=

01

x1

x cKc

cKc

Y−

−=

01

y1

y cKc

cKc

Y−

−=

01

z1

z cKc

cKc

Y−

−=

Yx,y,z = (Yx)(Yy)(Yz) = 01

zy,x,1TTTT−−

01

zy,x,1

cKccK

c

−

−=

FKKKSA Chem. Eng. Dept

... ... ... ... ... ... ...

EXAMPLE 5.3-4

A cylindrical can of pea puree has a diameter of 68.1 mm and a height of 101.6 mm and is initially at a uniform temperature of 29.4 oC. The cans are stacked vertically in a retort and steam at 115.6 oC is admitted. For a heating time of 0.75 h at 115.6 oC, calculate the temperature at the center of the can. Assume that the can is in the center of a vertical stack of cans and conduction also occurs from the two flat ends. The heat capacity of the metal wall of the can will be neglected. The heat transfer coefficient of the steam is estimated as 4540 W/m2.K. Physical properties of puree are k = 0.830 W/m.K and α = 2.007 x 10-7 m2/s. Ans: 106.6oC

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FKKKSA Chem. Eng. Dept

UNSTEADY-STATE HEAT AND MASS TRANSFER Example:

In a manufacturing process stainless steel cylinders (AISI 304) initially at 600 K are quenched by submersion in an oil bath maintained at 300 K with h = 500 W/m2 K. Each cylinder is of length 2L = 60 mm and diameter D = 80 mm. Consider a time 3 min into the cooling process and determine temperatures at the center of the cylinder, at the center of a circular face, and at the mid-height of the side.