1179 Chapters
Medium 9788131806135

he9-1

Mahesh M. Rathore Laxmi Publications PDF

Internal Flow

9

9.1. Flow Inside Ducts. 9.2. Hydrodynamic Considerations—Mean velocity um—Hydrodynamic entry length—Velocity profile in fully developed region—Friction factor—Pressure drop and friction factor in fully developed flow. 9.3. Thermal Considerations—The mean temperature or bulk temperature. 9.4. The Heat Transfer in Fully Developed Flow. 9.5. General Thermal Analysis—Constant surface heat flux—Constant surface temperature. 9.6. Heat Transfer in Laminar Tube Flow. 9.7. Flow Inside a Non-circular Duct. 9.8. Thermally

Developing, Hydrodynamically Developed Laminar Flow. 9.9. Heat Transfer in Turbulent Flow Inside a Circular Tube—Analogy between heat and momentum transfer in turbulent flow through tube—Correlation for turbulent flow. 9.10. Heat Transfer to Liquid Metal

Flow in Tube. 9.11. Summary—Review Questions—Problems—References and Suggested Reading.

9.1.

FLOW INSIDE DUCTS

The flow of fluid through the tubes and ducts for transporting cooling and heating fluids, etc., is of engineering importance. Most heat exchangers involve the heating or cooling of fluids flowing in the tubes. The fluid in such applications is forced to flow by a fan or pump through a tube that is sufficiently long to accomplish desired heating or cooling. Pressure drop and heat flux are associated with forced flow through the tubes and friction factor and heat transfer coefficient are used to determine the pumping power and length of tube.

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Medium 9789380386324

LAX20-2

Dr. A.J. Nair Laxmi Publications PDF

$�

PRINCIPLES

OF

BIOTECHNOLOGY

AND

GENETIC ENGINEERING

n Since the microcarrier culture is well mixed, it is easy to monitor and control different

environmental conditions such as pH, PO2, PCO2, etc. n Cell sampling is easy. n Since the beads settle down easily, cell harvesting and downstream processing of

products is easy. n Microcarrier cultures can be relatively easily scaled-up using conventional equipment

such as fermentors that have been suitably modified.

Because of the many advantages of the technique itself, it has gained great popularity.

Thus, a large variety of microcarriers are available on the market.

FIGURE 20.4 Vero cells cultured on cytodex microcarriers.

6. Fixed-bed reactors. Microcarriers, macrocarriers, or encapsulated beads could be used in fixed-bed reactors. The cells are immobilized in a matrix and the culture fluid is circulated in a closed loop. There is no agitation system. If the bed of immobilized cells is disturbed by the circulating medium, it is said to be a fluidized-bed reactor. Such a process achieves a high degree of aeration and agitation.

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Medium 9789380298818

13

Parmananda Gupta Laxmi Publications PDF

148

DIFFERENTIAL GEOMETRY AND CALCULUS OF VARIATIONS

Example 4. Find the extremal of the functional

z

a

−a

(λy +

1

2

µy″2) dx which satisfy the

boundary conditions y(– a) = 0, y′(– a) = 0, y(a) = 0, y′(a) = 0.

z FGH

IJ

K

1

µy″ 2 dx .

−a

2

1

2

Let

F(x, y, y′, y″) = λy + µy″

2 d d2

Fy ′ +

F =0

The Euler-Poisson equation is Fy − dx dx 2 y ″

Here Fy = λ, Fy′ = 0, Fy″ = µy″

Sol. Given functional is

∴ (1)

λ−

a

λy +

d d2

(0) +

(µy″ ) = 0 ⇒ dx dx 2

µ

d4 y dx 4

+λ =0 ⇒

...(1)

d4 y

λ

=–

4

µ dx

d3 y

λ

= − x + c1

3

µ dx

d2y

dx 2

=−

λ x2

+ c1x + c2

µ 2

dy

λ x3 x2

=−

+ c1

+ c2x + c3

2 dx

µ 6

λ x4 x3 x2

+ c3x + c4

+ c1

+ c2

6

2

µ 24

This is the equation of the extremals. The boundary conditions are y(– a) = 0, y′(– a) = 0, y(a) = 0, y′(a) = 0

y= −

y(– a) = 0

λa4 c1a 3 c2 a2

– c3a + c4 = 0

+

24µ

6

2

...(4)

λa4 c1 a3 c2 a2

+

+

+ c3a + c4 = 0

24µ

6

2

...(5)

λa3 c1a 2

+

+ c2a + c3 = 0

2

...(6)

y′(a) = 0 ⇒

c1 a3

– 2c3a = 0 ⇒ c1a2 + 6c3 = 0

3

(4) + (6) ⇒ c1a2 + 2c3 = 0

Solving (7) and (8), we get c1 = 0, c3 = 0

(3) – (5) ⇒

(4)

(3) ⇒

λa 3

+ 0 – c2a + 0 = 0

λa 4 a2

−0+

24µ

2

...(3)

λa 3 c1 a2

– c2a + c3 = 0

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Medium 9788131803622

5up4-6

M. Goyal Laxmi Publications PDF

232

COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES

Difference table is

∆2 f(x)

∆ f(x)

u

x

f(x)

–1

10

2854

0

14

3162

∆3 f(x)

308

74

382

1

18

3544

2

22

3992

–8

66

448

Bessel’s formula is

RS f (0) + f (1) UV + FG u − 1IJ ∆ f (0) + u (u − 1) RS ∆

2!

T 2 W H 2K

T

2

f(u) =

f (− 1) + ∆2 f (0)

2

FG

H

UV

W

(u − 1) u −

+

f (.25) =

3!

∆3 f (−1)

FG 3162 + 3544 IJ + (.25 – .5) (382) + (.25) (.25 − 1) FG 74 + 66 IJ

H 2 K

H 2 K

2

+

Hence

IJ

K

1 u

2

= 3250.875 y15 = 3250.875.

(.25 − 1) (.25 − .5) (.25)

(– 8)

6

ASSIGNMENT

1.

Apply Bessel’s formula to find the value of y2.73 given that y2.5 = 0.4938, y2.6 = 0.4953, y2.7 = 0.4965, y2.8 = 0.4974, y2.9 = 0.4981, y3.0 = 0.4987.

2. Find the value of y if x = 3.75, given that x:

2.5

3.0

3.5

4.0

4.5

5.0

y:

24.145

22.043

20.225

18.644

17.262

16.047.

Use Bessel’s formula.

3. Apply Bessel’s formula to find u62.5 from the following data:

4.

5.

x

:

60

61

62

63

64

65

ux

:

7782

7853

7924

7993

8062

8129.

Apply Bessel’s formula to find the value of f(12.2) from the following table: x :

0

5

10

15

20

25

30

f(x) :

0

0.19146

0.34634

0.43319

0.47725

0.49379

0.49865

Following table gives the values of ex for certain equidistant values of x. Find the value of ex when x = 0.644 using Bessel’s formula: x : ex

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Medium 9789380386607

ch-12

Rehana Khan Laxmi Publications PDF

LINKAGE

12

MAPPING

G ENE IN PRO

AND E UKARYOTES

AND

The frequency of crossing over, thus appears to be closely related to physical distance between genes. When one knows all the genes, linkage groups and number of linkage groups of a species, it becomes possible for him that by adopting the crossing over as a tool he may determine the relative distance between the genes in a linkage group and also their order and may give diagrammatic representation of chromosomes showing the gene as points separated by distances proportional to the amount of crossing over. Such a diagrammatic, graphical representation of relative distance between linked genes of a chromosome is called linkage or genetic map.

MAPPING GENES IN BACTERIA

The most commonly used bacterium for genetic and molecular analysis is Escherichia coli (E. coli), which is found most commonly in animal (including human) intestines. It is a good subject for study since it can be grown on a simple defined medium and can be handled with simple microbiological techniques.

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