unsteady state mass transfer · 2019-07-01 · example10 a solid slab of 5.15 wt% agar gel at 278 k...

Unsteady state mass transfer

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Steady state (สภาวะคงตัว)

ความเขม้ขน้ของสารทีต่ าแหน่งใดๆคงที ่ไม่เปลีย่นแปลงตามเวลา

Unsteady state: Transient (สภาวะไมค่งตัว)

ความเขม้ขน้ของสารทีต่ าแหน่งใดๆเปลีย่นแปลงตามเวลา


Water migration

Drying

Pickle

Shelf-life


2

2

x

CD

t

C

2

2

x

T

t

T

Second law of diffusion

Heat transfer


2

2

x

CD

t

C

Second law of diffusion

C: Concentration (kg/m3, kmol/m3)t: time (s)D: mass diffusivity (m2/s)x: distance (m)

Solution

Numerical method

Mass transfer chart


Mass transfer chart for solid object

Standard geometries

Sphere

Infinite slab

Infinite cylinder

Mass transfer chart

C: concentration at point x at time tC1: concentration of the diffusing component in the

medium surrounding the food objectC0 : the initial concentration of the diffusing component

within the food

01

1

CC

CCY

Fraction of unaccomplished change

Mass transfer chart

01

1

CC

CCY

1x

xn

1

1

xh

D

Nm

massBi

21x

DtX

Flat plate -> Fig. 5.3-5Fig. 5.3-6

Long cylinder ->Fig. 5.3-7Fig. 5.3-8

Sphere->Fig. 5.3-9Fig. 5.3-10

Mass transfer chart

1x

xn

x = distance from center (m)x1= characteristic dimension (m)

Dimensionless number in mass transfer chart

x1: characteristic dimension (m)

Infinite slab

Infinite cylinder

Sphere

x1=r

x1=r

x1=z


Mass transfer Biot number (NBi)

D

xhN mass

Bi1

D: mass diffusivity (m2/s)hmass: mass transfer coefficient (m/s)x1: characteristic dimension (m)

1

1

xh

D

Nm

massBi

Assumption: negligible resistance to mass transfer at the surface (h>> D) 1/NBi ≈ 0 m=0

Mass Transfer Fourier number (X)

21x

DtX

D: mass diffusivity (m2/s)t: time (s)x1: characteristic dimension (m)


Example10 A solid slab of 5.15 wt% agar gel at 278 K is 10.16 mm thick

and contains a uniform concentration of urea of 0.1 k mol/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 that the surface resistance can be assumed to be negligible; i.e., the convective coefficient is large. The diffusivity of urea in the agar is 4.72 x 10-10 m2/s.

1) Calculate the concentration at the midpoint of the slab and 2.54 mm from the surface after 10 h.

2) If the thickness of the slab is halved, what would be the midpoint concentration in 10 h?

Solution: Find a) n, m, Xb) Yc) C

x10.16 mm

x1=5.08 mm

?C

Example101) Calculate the concentration at the midpoint of the slab and 2.54 mm from the surface after 10 h.

1) n, m, X

01008.5

0

31

x

xn

x10.16 mm

x1=5.08 mm

Assumption: negligible resistance to mass transfer at the surface (h>> D) m=0

658.0

1008.5

6060101072.4

23

minmin10

21

sh

h

x

DtX

x=0

Y=0.275

01

1

CC

CCY


310.0

m

kmolCo

?C

31 0m

kmolC

For pure water

30275.0

1.00

0275.0

m

kmolC

C

at the midpoint of the slab or x=0


1) n, m, X

5.01008.5

1054.2

3

3

1

x

xn

x10.16 mm

x1=5.08 mm

Assumption: negligible resistance to mass transfer at the surface (h>> D) m=0

658.0

1008.5

6060101072.4

23

minmin10

21

sh

h

x

DtX

x=2.54 mm

Y=0.172

01

1

CC

CCY


310.0

m

kmolCo

?C

31 0m

kmolC

For pure water

300172.0

1.00

0172.0

m

kmolC

C

at 2.54 mm from the surface

Unsteady state mass transfer chart for mass average concentrationin three geometries (Treybal, 1968)

Cma: mass average concentrationCm: concentration of the diffusing component in the

medium surrounding the food objectCi : the initial concentration of the diffusing component

within the food

Assumption: negligible resistance to mass transfer at the surface (h>> D) 1/NBi ≈ 0

Mass transfer chartmass average concentration

mi

mma

CC

CC

Unsteady state mass transfer chart for mass average concentration in three geometries (Treybal, 1968)

Mass transfer chartmass average concentration

2

1x

Dt

Example11

Salt is being used to preserve a 4.8 mm slice of salmon muscle. The concentration of salt at the surface is 0.533 kg/kg salt free salmon (SFS), and the initial concentration is 0.012 kg/kg SFS. If the mass diffusivity, D, of salt in salmon muscle is 8.78 x 10-11 m2/s, determine the time required for the mass average concentration to reach 0.4 kg/kg SFS

Example11 255.0533.0012.0

533.04.0

mi

mma

CC

CC

2

1x

Dt

Example11

hs

D

xt

x

Dt

92.810214.3

1078.8

104.249.0

49.0

49.0

4

11

23

2

1

2

1

It takes 8.92 h for mass average salt concentration to reach 0.4 kg/kg SFS

1) จงเปรียบเทียบความเข้มข้นของยูเรีย ในอากาเจลที่มีรูปร่างดังต่อไปนี้ที่จุดก่ึงกลางของเจล เม่ือน าเจลไปแช่ในน ้าที่มีการไหลแบบ turbulent เม่ือเวลาผ่านไป 100 h ให้ความเข้มข้นเร่ิมต้นของยูเรียในเจลเป็น 0.1 k mol/m3

Diffusivity of urea in gel 4.72 x 10-10 m2/s

Thickness 4 cm Diameter 4 cm

Diameter 4 cm

Homework

2) A flat slab of wood 50.8 mm thickness

containing 30%wt moisture is being dried from

both side. The equilibrium moisture content at

the surface of the wood due to drying air blown

over it is held at 5%wt moisture. Diffusivity of

water in wood is 3.72 x 10-6 m2/h. calculate the

time for the center to reach 10% moisture.

Assume the Y value can be represented using

concentration of wt% moisture.

Homework

unsteady state mass transfer · 2019-07-01 · example10 a solid slab of 5.15 wt% agar gel at 278 k...

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