Massoud M. Engineering Thermofluids: Thermodynamics, Fluid Mechanics, and Heat Transfer

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892 VIe. Applications: Nuclear Heat Generation

36. Consider the slab of Problem 34. However, in this case the localized planar

source is replaced with a uniformly distributed neutron source emitting S

neu-

trons/s cm

. Find the flux in the slab. [Ans.: {1 – (cosh x/L)/cosh a/L}(S

/Σ

)].

37. Albedo, or the reflection coefficient (

), is defined as the ratio of the reflected

to the incident current,

= J

out

. Derive the albedo expression for a slab of

thickness a. [Ans.:

= (1 – b)/(1 + b) where b = (2D/L)coth(a/L)]. [Hint: Use

Equations VIe.2.3 and VIe.2.4].

out

38. Consider an infinite medium through which monoenergetic sources of neutron

emitting S

neutrons/s cm

are uniformly distributed. Find neutron flux in this

medium. [Ans.:

= S

/Σ

39. An isotropic point source emitting S

neutrons/s is placed in the center of a

bare sphere of radius R. The sphere is made up of carbon (L

= D/Σ

). Show that

the general solution for flux inside the sphere is given by:

LrLr

)( +=

−

where C

and C

are constants of integration. Apply the boundary conditions and

find the neutron flux anywhere at r = R/2. [Hint: Start with Equation VIe.2.14

and make a change of function;

/r].

∆

Point Source

Problem 39 Problem 40 Problem 41

40. Solve problem 39 considering the sphere is located in an infinite medium

made up of water.

Questions and Problems 893

41. Solve problem 40 considering the thickness of the water region is ∆ and be-

yond 2R +

∆ is vacuum.

42. Start with the one-group neutron diffusion equation and derive the relation for

neutron flux in a bare critical parallelepiped reactor. Find the maximum to aver-

age flux for this reactor.

43. Start with the one-group neutron diffusion equation and derive the relation for

neutron flux in a bare critical spherical reactor. Find the maximum to average flux

for this reactor.

44. Categorize the type of neutron source used in problems 32 through 43. For

this purpose, use the flow chart as shown below and find the box that best matches

the source of neutron used in the above problems. In this figure, 1-D for example,

stands for neutron diffusion equation in a one dimensional medium.

Neutron Source

Flux-dependent

(Fuel)

Flux-independent

(Moderator)

Neutron decay Distributed source

Plane source

in slab

Line source

in cylinder

Point source

in sphere

Parallelepiped Cylinder Sphere

Distributed source

(Fission)

Localized source

1-D 2-D 3-DInfinite

1-D 2-D 3-DInfinite 1-D 2-D 3-DInfinite

1-D 2-D 3-DInfinite

45. Determine the maximum linear heat generation rate to limit the average exit

void fraction of a BWR to 0.60. Use the power profile of

)/sin()(

max

Lzqzq

′

′ 

where z is the distance from the assembly inlet and L is

the assembly length. Use both homogeneous and drift flux models for void frac-

tion. H = 12 ft, P = 1000 psia, T

f,in

= 530 F, (A

Flow

)

Assembly

= 15 in

, =m



68E6

lbm/h.

46. Water flows in a uniformly heated tube. At the entrance to the tube, water is

subcooled at system pressure. Water leaves the tube as a saturated two-phase mix-

ture. The figure below shows the heated tube and the plots of temperature versus

quality. Match a) numbers with the lower case letters and b) temperatures with

the upper case letters.

894 VIe. Applications: Nuclear Heat Generation

0.0

Flowing Quality

c. Single phase

a. Bulk boiling

b. Nucleat boiling

d. Subcooled boiling

C. Bulk liquid temperature

A. Wall temperature

B. Incipient boiling temperature

D. Saturation temperature

47. Water enters a BWR channel. Use the following data and find the location of

the incipient boiling and the clad temperature corresponding to the incipient boil-

ing.

Data: Pellet diameter: 0.45 in, Rod diameter: 0.55 in, Square array pitch: 0.8 in,

core height = 12 ft, flow velocity: 8.5 ft/s, water inlet temperature: 530 F, system

pressure: 1035 psia,

7E25.1

max

′′′



Btu/h·ft

[Ans.: Nu = 1239, h = 5355 Btu/h·ft

·F, Y = 0.7, z

= –36.2 in, and T

545.6 F].

48. Water enters a BWR channel. Use the following data and find the location of

the incipient boiling and the clad temperature corresponding to the incipient boil-

ing.

Data: Pellet diameter: 0.5 in, Rod diameter: 0.6 in, Square array pitch: 0.9 in, core

height = 12 ft, flow velocity: 9 ft/s, water inlet temperature: 550 F, system pres-

sure: 1050 psia,

7E3.1

max

′′′



Btu/h·ft

[Ans.: z

= –63.72 in and T

= 553.77 F].

49. In this problem we want to find the bulk temperature corresponding to the in-

cipient boiling temperature of the surface. Use the data of Example VIe.3.5 and

find T

). [Ans.: T

) = 528.3 F].

50. Find the bulk temperature corresponding to the incipient boiling temperature

of problem 45.

51. Use the Bowring correlation for the calculation of CHF and solve Exam-

ple VIe.4.1. Plot the results and find the MDNBR. [Ans.: CHF in MBtu/h·ft

for

various nodes: 1.85, 1.388, 0.11, 0.9251, 0.7929, 0.6937, 0.6166, 0.5549, 0.504,

0.4624, 0.4268. The MDNBR = 1.5].

52. Find the required number of rods for a PWR producing 1200 MWe having an

efficiency of 30%. Other pertinent data at steady state operation are as follows:

MDNBR = 2.5, F = 2.3, z

DNB

= 25” from the core mid-plane, H = 12 ft, d

= 0.45

in,

6E5.1)( =

′′′

DNBCHF



Btu/h·ft

. [Ans.: N

Rod

= 31,638].

Questions and Problems 895

53. We would like to load a core with fuel rods in which fuel pellet enrichment is

such that the neutron flux increases linearly along the vertical axis as shown in the

figure. Water is used as coolant. Use the given data

Central

Rod in the

central

channel

′′′

)(

max

qzq



Pellet

Gap

Clad

Coolant

z = 0

z = H

′′′

Gap

Clad

Fuel

and find a) water temperature at the exit of the channel, b) peak temperatures of

clad outside, clad inside, fuel surface, and fuel center, c) the incipient boiling tem-

perature, and d) critical heat flux at the channel exit.

Data:

(in): 0.38 P (psia): 1000

(in) 0.39 T

(F): 500

(in): 0.44 m



(lbm/s): 1.00 (per rod)

p (in): 0.59

′′′



(Btu/h ft

): 0.01E6

54. The fuel assembly shown in the left figure consists of periodic arrays of annu-

lar bare fuel rods, which are cooled by passing water through the center of the rods

as well as over the outer surface. We want to analyze the thermal performance of

the fuel rods by dividing the assembly into a number of unit cells (control vol-

umes) and evaluating the performance of a cell, as shown in the right figure.

896 VIe. Applications: Nuclear Heat Generation

Fuel

Inner

coolant

channel

Outer

coolant

channel

a) Find the ratio of the average coolant velocities in the inner and outer channels at

the axial level z. At this level, the pressure drop per unit length is the same in both

channels and the bulk coolant temperature is 293 C. Assume that the flow is tur-

bulent and fully developed in both channels.

b) Find the maximum temperature in the rod, and the radius at which the maxi-

mum temperature occurs, for a particular axial location where the inner and outer

surface temperatures of the rod are 371 C. At that level the volumetric heat gen-

eration rate may be assumed to be uniform and equal to 0.1 MW/m

. Assume

constant fuel conductivity for the fuel.

c) Find the mean temperature of the coolant at the core exit (i.e., the mixture of

the coolant passing through the inner channel and that passing through the outer

channel). The fuel rod is 4.3 m long and the axial power profile along the channel

is give by:

()

′′′

3.4

cos

qzq



where z is in m and 52.0=

′′′



MW/m

. Other pertinent data: Inlet water

temperature = 293 C, water pressure at the outlet = 14 MPa, water flow rate per

unit cell = 18.37 kg/s, unit cell side (L) = 6.35 cm, fuel inner radius (r

) = 1.27 cm,

and fuel outer radius (r

) = 2.54 cm.

55. A reactor that has been operating at nominal power level of 2700 MWth for 2

years is shut down. The decay power from this reactor, for any time after shut-

down, can be fairly well estimated from:

26.0

nominal

095.0

)(

−

= t



where t in this formula is the time after reactor shutdown in seconds. Find a) the

power obtained from the reactor 1 day after shutdown, and b) the amount of en-

ergy produced by the decay power in a period of 24 hours. [Ans.: 13.35 MWth

and 1.56E6 MJ].

56. A curve fit to the rate of the decay heat data of a typical PWR fuel cycle re-

sulted in the following equation:

Questions and Problems 897

nominal

)(



where )(tQ



is the decay power at time t,

nominal



is the nominal reactor power,

and t is the time after reactor shutdown. Coefficients A and B

are: A =

0.3826033E-2, B

= 276.6013, B

= -5,124,569, B

= 0.6344872E11, B

= -

0.4427653E15, B

= 0.1551979E19, and B

= -0.2086165E22. Evaluate the ac-

uuracy of the formula given in Problem 55 by plotting both equations and compar-

ing the results.

57. This problem deals with a CE-designed 2 × 4 PWR (i.e., 2 hot legs and 4 cold

legs as shown in Figures I.6.2(b), I.6.4(CE), I.6.5, I.6.6(a), and I.6.6(b)). Use the

given data to find the answers to the questions that follow the set of data.

Primary side data (BU - SI)

Core power (Btu/h - MWth): 9.2124E9 - 2700

Pressure in lower plenum (psia - MPa): 2595 - 18

Core pressure drop (psia - kPa): 14 - 100

Vessel pressure drop (psia - kPa) 37.1 - 256

Cold leg temperature (F - C): 550 - 288

Mass flow rate through core (lbm/h - kg/s): 138.5E6 - 17451

Number of fuel assemblies: 217

Number of rods per assembly: 14 × 14 (square array)

Fuel rod (Core) length (ft – m): 12 – 3.657

Fuel rod outside diameter (in - cm): 0.44 - 1.12

Fuel rod inside diameter (in - cm): 0.388 - 0.98

Fuel pellet diameter (in - cm): 0.377 - 0.96

Fuel pellet length (in - cm): 0.45 - 1.14

Fuel rod pitch (in - cm) 0.58 - 1.473

Gap heat transfer coefficient

(Btu/h·ft

·F - W/m

·K): 1000 - 5678

Thermal conductivity of fuel pellet, UO

(Btu/h·ft·F - W/m·K): 1.5 - 2.6

Thermal conductivity of Zircaloy

(Btu/h·ft·F - W/m·K): 3.0 - 5.2

Hot leg inside diameter (ft - m): 3.5 - 1.067

Cold leg inside diameter (ft - m): 2.5 - 0.762

Volume of one hot leg (ft

- m

): 138 - 3.9078

Volume of one cold leg (ft

- m

): 224 - 6.343

Total peaking factor: 1.15

Steam generator (SG) data (BU - SI)

Number of steam generators: 2

Number of tubes per SG: 8485

Tube inside diameter (in - cm): 0.654 - 1.66

898 VIe. Applications: Nuclear Heat Generation

Tube outside diameter (in - cm): 0.750 - 1.905

Volume of the SG inlet plenum (ft

- m

): 250 -7.079

Volume of the SG outlet plenum (ft

- m

): 250 - 7.079

Inlet plenum hydraulic diameter (ft - m): 7.6 - 2.316

Outlet plenum hydraulic diameter (ft - m): 7.6 - 2.316

Steam dome pressure (psia - MPa): 880 - 6.067

Feedwater pressure (psia - MPa): 1095 - 7.5

Feedwater temperature (F - C): 440 - 227

Tube material: Stainless steel

Boiling heat transfer coefficient

(Btu/h·ft

·F - W/m

·K): 6400 - 36340

Pressurizer data (BU - SI)

Geometry: Circular cylinder

Steam dome Pressure (psia - MPa): 2250 - 15.51

Volume (ft

- m

): 1500 - 42.477

Height (ft - m): 30 - 9.15

Water level (ft - m): 18 - 5.486

Wall thickness (in - cm): 4.5 - 11.43

Insulation thickness (in - cm): 0.00

Wall material: Carbon Steel

Number of Relief Valves: 2

Relief valve flow area (ft

- cm

): 0.01 - 0.01

Relief valve discharge coefficient 0.61

Ambient temperature (F - C): 90 - 32

Ambient pressure (psia - kPa): 14.7 - 101.35

Balance of plant data (BU - SI)

Total pumping power (condensate, booster,

heater, and feewater) (MW): 190

Condenser Pressure (in Hg - mm Hg): 26 - 660

Circulating water inlet temperature (F - C): 60

Circulating water flow rate (lbm/h - kg/s): 5.2E8 - 65,520

s = 0.58 in

D = 0.44 in

Find the answer to the following questions:

1. The centerline temperature in an average fuel rod

2. The centerline temperature in the fuel rod located in the hot assembly

Questions and Problems 899

3. Total power developed by the turbine (MW). Assume

turbine

= 100%

4. The core total

∆P due to skin friction and the core total loss coefficient

(K =

ΣK

)

5. The overall heat transfer coefficient in SG (a: assume feedwater is saturated,

b: fouling = 0)

6. The average tube length of one SG (for questions 5 and 6 use Equations

VIa.1.1 and VIa.2.12)

7. Pressure drop across the primary side of the SG (i.e., from the hot leg inlet to

the outlet to cold leg)

8. The

∆P across the reactor coolant pump and the total power used by a reactor

coolant pump

9. The temperature rise of the circulating water

10. We open one of the pressurizer relief valves. Find the maximum flow rate

through the valve if one of the pressurizer relief valves is lifted.

11. Find the steam mass flow rate through the relief valve and the pressurizer

pressure versus time if the relief valve is stuck open for one minute. Assume

that the pressurizer is isolated from the rest of the reactor.

VII

. Engineering Mathematics

The purpose of this chapter is to discuss the commonly used mathematical con-

cepts and formulae in mathematical physics and engineering applications. This in-

cludes such topics as differential equations, mathematical functions, vector and

matrix operations, and numerical analysis. Many of these topics are applied in

various chapters of this book.

VIIa. Fundamentals

This chapter deals exclusively with the definitions of terminologies pertinent to

engineering mathematics.

1. Definition of Terms

Independent variable is a quantity that may be equal to any one of a specified

set of values.

Dependent variables are variables that denote values of a function. A function

is a relationship between two variables such that each value of the independent

variable corresponds exactly to one value of the dependent variable. For example,

x in equation

xyedxdy

sin/ =+ is the independent variable and y is the function

or the dependent variable.

Domain includes the collection of all values assumed by the independent vari-

able.

Range is the collection of all values assumed by the dependent variable. For

example, the domain and range of the function

1−= xy is determined as fol-

lows. Since the radical should be positive for y to be defined, then the domain of x

∞<≤ x1 and the range of y is

∞<≤ y0

Coordinate systems. As described in Section 1 of Chapter VIIc.

Explicit and implicit functions. A dependent variable is an explicit function

of an independent variable if the function can be expressed in terms of the inde-

pendent variable. For example, y = 2x + sin x + ln x is an explicit function of x

and the y in

01sin

=−++ xyxyy is an implicit function of x.

902 VIIa. Engineering Mathematics: Fundamentals

Continuous function. A function in the interval (a, b) is continuous if for any

positive number

there is a positive number

so that when

δδ

+<<−

xxx

we have

<− )()(

xfxf . For example, f(x) in Figure VIIa.1.1 is not continu-

ous at x

because for h

– h

there is no

to satisfy the condition for conti-

nuity.

f(x)

Figure VIIa.1.1. Example of a discontinuous function

Periodic functions. Function f(x) is considered periodic if for any fixed num-

ber such as P, we could have the following relation f(x + P) = f(x). If P is the

smallest number for which the periodic condition exists, then x is called the period

of the function.

Harmonic function. A harmonic function is a solution to the Laplace partial

differential equation (defined in Section 2.2) which has continuous second-order

partial derivatives.

Homogeneous function. A function f(x, y) is considered to be homogeneous

of degree n if for any number s and constant n, we have f(sx, sy) = s

f(x, y). For

example, x

+ xy

is homogeneous of degree 3 because (sx)(sx)(sx)+(sx)(sy)(sy) =

+ xy

) but x

+ y is not homogeneous.

Power series is defined as

∞→

−

xxC

])([lim

for those values of x where the limit exists. For such values of x, the series is said

to converge. An important aspect of a power series is that convergent power se-

ries can be treated as polynomials. We now consider function f(x) which, along

with all its derivatives, is continuous in an interval including x = x

. This function

can be expressed as a finite Taylor series plus a residual as follows:

])(

)(

[])(

)(

[)(

xfd

xf −+−=

−

where

is a point in the interval (x

, x). If x

= 0, the Taylor series becomes a

Maclaurin series: