1. PATRIMONIO CULTURAL
1.3 Políticas de preservación
1.3.1 Los documentos patrimoniales
multicomponent mi ture of ideal gases b orthogonal collocation ( C = 3).
Orthogonal collocation matrices
The p e e and empe a e in he apo pha e a e
The Ma ell-S efan diff ion coefficicien a e
The leng h of he diff ion pa h i
The den i of he ga pha e follo f om he ideal ga la
Initial estimates of the flu es Initial estimates of the concentrations
2.1a. Ma - an fe coefficien in a ga ab o be .
f he he e i 2.0 c . A he e d f he e e i e , 14.32 i a e , he dia e e f he he e i
2.3a. Mass-transfer coefficients from acetone e aporation data.
I a ab a e e i e , ai a 300 K a d 1 a i b a high eed a a e he face f a
2.4b. Mass-transfer coefficients from etted- all e perimental data.
A e ed- a e e i e a e - c i f a g a i e, 50 i dia e e a d 1.0 g.
Wa e a 308 K f d he i e a . D ai e e he b f he i e a he a e f 1.04 3/ i , ea ed a 308 K a d 1 a . I ea e he e ed ec i a 308 K a d i h a e a i e h idi f 34%. Wi h he he f e a i (2-52), e i a e he a e age a - a fe c efficie ,
i h he d i i g f ce i e f a f ac i .
S i
2.7c. Ma an fe in an ann la pace.
2.8c. The Chilton-Colburn analog : flow across tube banks.
b) E i a e he a - a fe c efficie be e ec ed f e a a i f - a c h i ca b di ide f he a e ge e ica a a ge e he he ca b di ide f a a a i
e ci f 10 / a 300 K a d 1 a . The a e e f - a c h a 300 K i 2.7 Pa.
S i
P e ie f di e i e f a c h i ca b di ide a 300 K a d 1 a :
c) Za a a (Ad . Heat Transfer, , 93, 1972) ed he f i g c e a i f he hea -a fe c efficie i -a -agge ed be b-a -a -a ge e i i -a h-a died b Wi di g a d Che e :
U e he a - a fe e e i a a g e a i (2-69) e i a e he a - a fe c efficie f a b). C a e he e .
S i
2.9b. Ma an fe f om a fla pla e.
A 1-m square thin plate of solid naphthalene is oriented parallel to a stream of air flowing at 20 m/s. The air is at 310 K and 101.3 kPa. The naphthalene remains at 290 K; at this temperature the vapor pressure of naphthalene is 26 Pa. Estimate the moles of naphthalene lost from the plate per hour, if the end effects can be ignored.
Solution
2.10b. Ma an fe f om a fla pla e.
A thin plate of solid salt, NaCl, measuring 15 by 15 cm, is to be dragged through seawater at a velocity of 0.6 m/s. The 291 K seawater has a salt concentration of 0.0309 g/cm3. Estimate the rate at which the salt goes into solution if the edge effects can be ignored. Assume that the kinematic viscosity at the average liquid film conditions is 1.02 10 6 m2/s, and the diffusivity is 1.25 10 9 m2/s. The solubility of NaCl in water at 291 K is 0.35 g/cm3, and the density of the saturated solution is 1.22 g/cm3 (Perry and Chilton, 1973) .
Solution
Laminar flo
At the bulk of the solution, point 2:
At the interface, point 1:
2.11b. Ma an fe f om a fla li id face.
D i g he e e i e de c ibed i P b e 2.3, he ai e ci a ea ed a 6 / , a a e he ge ide f he a . E i a e he a - a fe c efficie edic ed b e a i (2-28) (2-29) a d c a e i he a e ea ed e e i e a . N ice ha , d e he high a i i f ace e, he a e age ace e c ce a i i he ga fi i e a i e high. The ef e, e ie
ch a de i a d i c i h d be e i a ed ca ef . The f i g da a f ace e igh be eeded: Tc = 508.1 K, Pc = 47.0 ba , M = 58, Vc = 209 c 3/ , Zc = 0.232 (Reid, e a ., 1987).
S i
A e age fi e ie :
E i a e he i c i f he i e f L ca Me h d
E i a e he diff i i f he Wi e-Lee e a i
2.12b. E aporation of a drop of ater falling in air.
Repeat E ample 2.9 for a drop of ater hich is originall 2 mm in diameter.
Solution
2.13b. Dissolution of a solid sphere into a flo ing liquid stream.
Estimate the mass-transfer coefficient for the dissolution of sodium chloride from a cast sphere, 1.5 cm in diameter, if placed in a flowing water stream. The velocity of the 291 K water stream is 1.0 m/s.
Assume that the kinematic viscosity at the average liquid film conditions is 1.02 10 6 m2/s, and the mass diffusivity is 1.25 10 9 m2/s. The solubility of NaCl in water at 291 K is 0.35 g/cm3, and the density of the saturated solution is 1.22 g/cm3 (Perry and Chilton, 1973) .
Solution
From Prob. 2.10:
2.14b. Sublimation of a solid sphere into a gas stream.
During the experiment described in Problem 2.2, the air velocity was measured at 10 m/s. Estimate the mass-transfer coefficient predicted by equation (2-36) and compare it to the value measured experimentally. The following data for naphthalene might be needed: Tb = 491.1 K, Vc = 413 cm3/mol.
2.15b. Dissolution of a solid sphere into a flo ing liquid stream.
The cr stal of Problem 1.26 is a sphere 2-cm in diameter. It is falling at terminal velocit under the influence of gravit into a big tank of water at 288 K. The densit of the cr stal is 1,464 kg/m3 (Perr and Chilton, 1973).
a) Estimate the cr stal's terminal velocit . Solution
b) Estimate the rate at which the cr stal dissolves and compare it to the answer obtained in Problem 1.26.
Solution
From Prob.1.26
From Prob.1.26:
2.16c. Mass transfer inside a circular pipe.
Water flows through a thin tube, the walls of which are lightly coated with benzoic acid (C7H6O2).
The water flows slowly, at 298 K and 0.1 cm/s. The pipe is 1-cm in diameter. Under these conditions, equation (2-63) applies.
a) Show that a material balance on a length of pipe L leads to
where v is the average fluid velocity, and cA* is the equilibrium solubility concentration.
b) What is the average concentration of benzoic acid in the water after 2 m of pipe. The solubility of benzoic acid in water at 298 K is 0.003 g/cm3, and the mass diffusivity is 1.0 10 5 cm2/s
(Cussler, 1997).
Solution
2.17b. Mass transfer in a etted- all to er.
Water flows down the inside wall of a 25-mm ID wetted-wall tower of the design of Figure 2.2, while air flows upward through the core. Dry air enters at the rate of 7 kg/m2-s. Assume the air is
everywhere at its average conditions of 309 K and 1 atm, the water at 294 K, and the mass-transfer coefficient constant. Compute the average partial pressure of water in the air leaving if the tower is 1
g.
S i
F a e a 294 K
2.18c. Ma an fe in an ann la pace.
I d i g he b i a i f a h ha e e i a ai ea , a i e iga c c ed a 3- - g a a d c . The i e i e a ade f a 25- -OD, id a h ha e e d; hi a
ded b a 50- -ID a h ha e e i e.
Ai a 289 K a d 1 a f ed h gh he a a ace a a a e age e ci f 15 / . E i a e he a ia e e f a h ha e e i he ai ea e i i g f he be. A 289 K, a h ha e e ha a a e e f 5.2 Pa, a d a diff i i i ai f 0.06 c 2/ . U e he e f P b e 2.7
e i a e he a - a fe c efficie f he i e face; a d e a i (2-47), i g he e i a e dia e e defi ed i P b e 2.7, e i a e he c efficie f he e face.
Sol ion
In hi i a ion, he e ill be a mola fl f om he inne all, NA1, i h pecific in e facial a ea, a1, and a fl f om he o e all, NA2, i h a ea a2. A ma e ial balance on a diffe en ial ol me
elemen ield :
Define:
Then:
Fo he in e io all:
For the outer all:
2.19c. Ben ene evaporation on the outside surface of a single c linder.
Be e e i e a a i g a he a e f 20 g/h e he face f a 10-c -dia e e c i de . D ai a 325 K a d 1 a f a igh a g e he a i f he c i de a a e ci f 2 / . The
i id i a a e e a e f 315 K he e i e e a a e e f 26.7 Pa. E i a e he e g h f he c i de . F be e e, Tc = 562.2 K, Pc = 48.9 ba , M = 78, Vc = 259 c 3/ , Zc = 0.271 (Reid, e a ., 1987).
S i
Ca c a e he a e age e ie f he fi
F he Wi e-Lee e a i
F he L ca Me h d
F E . 2-45:
2.20b. Ma an fe in a packed bed.
Wilke and Hougan (T an . AIChE, 41, 445, 1945) reported the mass transfer in beds of granular solids. Air was blown through a bed of porous celite pellets wetted with water, and by evaporating this water under adiabatic conditions, they reported gas-film coefficients for packed beds. In one run, the following data were reported:
With the assumption that the properties of the gas mixture are the same as those of air, calculate the gas-film mass-transfer coefficient using equation (2-55) and compare the result with the value reported by Wilke and Hougan.
Solution
From the Wilke-Lee equation
2.21b. Ma an fe and p e e d op in a packed bed.
Air at 373 K and 2 atm is passed through a bed 10-cm in diameter composed of iodine spheres 0.7-cm in diameter. The air flows at a rate of 2 m/s, based on the empt cross section of the bed.
The porosit of the bed is 40%.
a) How much iodine will evaporate from a bed 0.1 m long? The vapor pressure of iodine at 373 K is 6 kPa.
Solution
From the Wilke-Lee equation:
b) E ima e he p e e d op h o gh he bed.
Sol ion
2.22b. Volumetric mass-transfer coefficients in industrial to ers.
The interfacial surface area per unit volume, a, in many types of packing materials used in industrial towers is virtually impossible to measure. Both a and the mass-transfer coefficient depend on the physical geometry of the equipment and on the flow rates of the two contacting, inmiscible streams. Accordingly, they are normally correlated together as the volumetric mass-transfer coefficient, kca.
Empirical equations for the volumetric coefficients must be obtained experimentally for each type of mass-transfer operation. Sherwood and Holloway (Trans. AIChE, 36, 21, 39, 1940) obtained the following correlation for the liquid-film mass-transfer coefficient in packed absorption towers
The values of a and n to be used in equation (2-71) for various industrial packings are listed in the following table, when SI units are used exclusively.
a) Consider the absorption of SO2 with water at 294 K in a tower packed with 25-mm Raschig rings. If the liquid mass velocity is L' = 2.04 kg/m2-s, estimate the liquid-film mass-transfer coefficient. The diffusivity of SO2 in water at 294 K is 1.7 10 9 m2/s.
Solution
For dimensional consistency, add the constants:
b) Whitney and Vivian (Chem. Eng. Progr., 45, 323, 1949) measured rates of absorption of SO2 in water and found the following expression for 25-mm Raschig rings at 294 K
where k a is in kmole/m2-s. For the conditions described in part a), estimate the liquid-film mass-transfer coefficient using equation (2-72). Compare the results.
Solution
2.23b. Mass transfer in fluidi ed beds.
Cavatorta, et al. (AIC E J., 45, 938, 1999) studied the electrochemical reduction of ferrycianide ions, {Fe(CN)6} 3, to ferrocyanide, {Fe(CN)6} 4, in aqueous alkaline solutions. They studied different arrangements of packed columns, including fluidized beds. The fluidized bed experiments were performed in a 5-cm-ID circular column, 75-cm high. The bed was packed with 0.534-mm spherical glass beads, with a particle density of 2.612 g/cm3. The properties of the aqueous solutions were: density = 1,083 kg/m3, viscosity = 1.30 cP, diffusivity = 5.90 10 10 m2/s. They found that the porosity of the fluidized bed, e, could be correlated with the superficial liquid velocity based on the empty tube, vs, through
where vs is in cm/s.
a) Using equation (2-56), estimate the mass-transfer coefficient, kL, if the porosity of the bed is 60%.
Solution
b) Ca a o a e al. p opo ed he follo ing co ela ion o e ima e he ma - an fe coefficien fo hei fl idi ed bed e pe imen al n :
he e Re i ba ed on he emp be eloci . U ing hi co ela ion, e ima e he ma - an fe coefficien , L, if he po o i of he bed i 60%. Compa e o e l o ha of pa a).
Sol ion
2.24b. Mass transfer in a hollo -fiber boiler feed ater deaerator.
Con ide he hollo -fibe BFW deae a o de c ibed in E ample 2-13. If he a e flo a e inc ea e o 60,000 kg/h hile e e hing el e emain con an , calc la e he f ac ion of he en e ing di ol ed o gen ha can be emo ed.
Sol ion
2.25b. Mass transfer in a hollo -fiber boiler feed ater deaerator.
a) Con ide he hollo -fibe BFW deae a o de c ibed in E ample 2-13. A ming ha onl o gen diff e ac o he memb ane, calc la e he ga ol me flo a e and compo i ion a he l men o le . The a e en e he hell ide a 298 K a a ed i h a mo phe ic o gen, hich mean a di ol ed o gen concen a ion of 8.38 mg/L.
Sol ion
b) Ca c a e he a - a fe c efficie a he a e age c di i i ide he e . Neg ec he hic e f he fibe a he e i a i g he ga e ci i ide he e .
S i
Ca c a e he a e age f c di i i ide he fibe
Ca c a e he a e age ge a f ac i i he ga
F L ca e h d f e N2
F he Wi e-Lee e a i
(From E ample 2.13)
3.1a. Application of Raoult's law to a binar s stem.
Re ea E a e 3.1, b f a i id c ce a i f 0.6 e f ac i f be e e a d a e e a e f 320 K.
S i
3.2b. Application of Raoult's law to a binar s stem.
a) De e i e he c i i f he i id i e i ib i i h a a c ai i g 60 e e ce be e e-40 e e ce e e if he e e i i a e e de 1 a e e. P edic
he e i ib i e e a e.
S i
I i ia e i a e
b) De e i e he c i i f he a i e i ib i i h a i id c ai i g 60 e e ce
A e i ib i :
Ba i : 1 L a e (1 g a e )
E i ib i c ce a i , ce = 11.42 g ge /L I i ia c di i :
The i gai ge .
3.5c. Material balances combined ith equilibrium relations.
Re ea E a e 3.3, b a i g ha he a ia, ai , a d a e a e b gh i c ac i a
A : P = 1.755
b) the liquid-film coefficient, kL; Solution
Basis: 1 m3 of aqueous solution
c) the concentration on the liquid side of the interface, A,i; Solution
Initial estimates of interfacial concentrations:
d) the mass flu of A.
Solution
Check this result b calculating the gas-phase flu :
3.7b. Mass-transfer resistances during absorption.
3.8d. Absorption of ammonia b water: use of F-t pe mass-transfer coefficients.
M dif he Ma hcad g a i Fig e 3.6 e ea E a e 3.5, b i h A,G = 0.70 a d A,L = 0.10. E e hi g e e e ai c a .
Sol ion
Ini ial g e e
3.9d. Absorption of ammonia b water: use of F-t pe mass-transfer coefficients.
Modif he Ma hcad p og am in Fig e 3.6 o epea E ample 3.5, b i h FL = 0.0050 kmol/m2- . E e hing el e emain con an .
Sol ion
Ini ial g e e
3.10b. Ma - an fe e i ance d ing ab o p ion of ammonia.
I he ab i f a ia i a e f a ai -a ia i e a 300 K a d 1 a , he i di id a fi c efficie e e e i a ed be L = 6.3 c /h a d G = 1.17 / 2-h -a . The e i ib i e a i hi f e di e i f a ia i a e a 300 K a d 1 a i
De e i e he f i g a - a fe c efficie : a)
S i
b)
S i
c) Ky Solution
d) Fraction of the total resistance to mass transfer that resides in the gas phase.
Solution
3.11b. Mass-transfer resistances in hollo -fiber membrane contactors.
For mass transfer across the hollow-fiber membrane contactors described in Example 2.13, the overall mass-transfer coefficient based on the liquid concentrations, KL, is given by (Yang and Cussler, AIC E J., 32, 1910, Nov. 1986)
where kL, kM, and kc are the individual mass-transfer coefficients in the liquid, across the membrane, and in the gas, respectively; and H is Henry's law constant, the gas equilibrium concentration divided by that in the liquid. The mass-transfer coefficient across a hydrophobic membrane is from (Prasad and Sirkar, AIC E J., 34, 177, Feb. 1988)
where DAB = molecular diffusion coefficient in the gas filling the pores, eM = membrane porosity,
tM = membrane tortuosity, d = membrane thickness.
For the membrane modules of Example 2.13, eM = 0.4,tM = 2.2, and d = 25 10 6 m (Prasad and Sirkar, 1988).
a) Calculate the corresponding value of kM. Solution
For oxygen in nitrogen at 298 K and 1 atm:
b) U i g he e f a (a), E a e 2.13, a d P b e 2.25, ca c a e KL, a d e i a e ha
3.12c. Combined use of F- and k-t pe coefficients: absorption of low-solubilit gases.
D i g ab i f - bi i ga e , a a fe f a high c ce a ed ga i e
I he i id ha e:
Ini ial g e :
3.13d. Distillation of a mi ture of methanol and ater in a packed to er: use of F-t pe mass-transfer coefficients.
At a different point in the packed distillation column of Example 3.6, the methanol content of the bulk of the gas phase is 76.2 mole %; that of the bulk of the liquid phase is 60 mole %. The temperature at that point in the tower is around 343 K. The packing characteristics and flow rates at that point are such that FG = 1.542 10 3 kmol/m2-s, and FL = 8.650 10 3 kmol/m2-s.
Calculate the interfacial compositions and the local methanol flux. To calculate the latent heats of vaporization at the new temperature, modify the values given in Example 3.6 using Watson's method (Smith, et al., 1996):
Pa ame e
Ini ial e ima e
3.14b. Material balances: adsorption of ben ene vapor on activated carbon.
b) Ca c a e he i i f a e e i ed f he e e i g ac i a ed ca b ( e e be ha he e e i g ca b c ai e ad bed be e e).
S i
O he XY diag a , ca e he i (X2,Y2). Si ce he e a i g i e i ab e he e i ib i c e a d he e i ib i c e i c ca e a d , he i i e a i g i e i b ai ed b
ca i g, a he i e ec i f Y = Y1 i h he e i ib i c e, X1 a .
c) If the carbon flow rate is 20% above the minimum, what will be the concentration of ben ene adsorbed on the carbon leaving?
Solution
d) F he c di i f a (c), ca c a e he be f idea age e i ed.
S i
See e i e c c i he XY g a h
3.15b. Material balances: desorption of ben ene vapor from activated carbon.
The ac i a ed ca b ea i g he ad be f P b e 3.14 i ege e a ed b c e c e c ac
From Problem 3.14
From the XY diagram:
b) For a steam flow rate of twice the minimum, calculate the ben ene concentration in the gas mixture leaving the desorber, and the number of ideal stages required.
Solution
3.16b. Material balances: adsorption of ben ene vapor on activated carbon; cocurrent operation.
If the adsorption process described in Problem 3.14 took place cocurrentl , calculate the minimum flow rate of activated carbon required.
Solution
Fom Problem 3.14:
From the XY diagram:
3.17b. Material balances in batch processes: dr ing of soap with air.
From the XY diagram, at the e it of the fifth equilibrium stage, X = 0.06 and
3.18b. Material balances in batch processes: e traction of an aqueous nicotine solution
ith kerosene.
Nicotine in a ater solution containing 2% nicotine is to be e tracted ith kerosene at 293 K.
Water and kerosene are essentiall insoluble. Determine the percentage e traction of nicotine if 100 kg of the feed solution is e tracted in a sequence of four batch ideal e tractions using 49.0 kg of fresh, pure kerosene each. The equilibrium data are as follo s (Claffe et al., I d. E g. Che ., 42, 166, 1950):
From the XY diagram, after 4 e tractions, X = 0.00422
3.19b. Cross-flo cascade of ideal stages.
The d i g a d i id- i id e ac i e a i de c ibed i P b e 3.17 a d 3.18,
e ec i e , a e e a e f a f c fig a i ca ed a c -f ca cade. Fig e 3.27 i a che a ic diag a f a c -f ca cade f idea age . Each age i e e e ed b a ci c e, a d i hi each age a a fe cc a if i c c e f . The L ha e f f e
age he e , bei g c ac ed i each age b a f e h V ha e. If he e i ib i -di ib i c e f he c -f ca cade i e e he e aigh a d f e , i ca be h ha (T e ba , 1980)
he e S i he i i g fac , VS/LS, c a f a age , a d N i he a be f age .
S e P b e 3.18 i g e a i (3-60), a d c a e he e b ai ed b he e h d .
S i
I i ia e i a e
3.20a. Cross-flo cascade of ideal stages: nicotine e traction.
C ide he ic i e e ac i f P b e 3.18 a d 3.19. Ca c a e he be f idea age e i ed achie e a ea 95% e ac i efficie c .
S i
U e 8 idea age
3.21b. Kremser equations: absorption of h drogen sulfide.
A che e f he e a f H2S f a f f 1.0 d 3/ f a a ga b c bbi g i h
minimum flo rate is used.
Solution
at SC
b) Determine the composition of the e iting liquid.
Solution
c) Calculate the number of ideal stages required.
Solution
3.22b. Absorption ith chemical reaction: H2S scrubbing ith MEA.
A h i P b e 3-21, c bbi g f h d ge fide f a a ga i g a e i
ac ica i ce i e i e a ge a f a e d e he bi i f H2S i a e . If a 2N i f e ha a i e (MEA) i a e i ed a he ab be , h e e , he e i ed i id f a e i ed ced d a a ica beca e he MEA eac i h he ab bed H2S i he i id ha e, effec i e i c ea i g i bi i .
F hi i e g h a d a e e a e f 298 K, he bi i f H2S ca be a i a ed b (de Ne e , N., Air Poll ion Con rol Engineering, 2 d ed., McG a -Hi , B , MA, 2000):
Re ea he ca c a i f P b e 3.21, b i g a 2N e ha a i e i a ab be .
S i
3.23b. Kremser equations: absorption of sulfur dio ide.
A flue gas flows at the rate of 10 kmol/s at 298 K and 1 atm with a SO2 content of 0.15 mole %.
Ninety percent of the sulfur dioxide is to be removed by absorption with pure water at 298 K. The design water flow rate will be 50% higher than the minimum. Under these conditions, the
equilibrium line is (Ben tez, J., P oce Enginee ing and De ign fo Ai Poll ion Con ol, Prentice Hall, Englewood Cliffs, NJ, 1993):
where Xi = moles SO2/mole of water; Yi = moles SO2/mole of air.
a) Calculate the water flow rate and the SO2 concentration in the water leaving the absorber.
Solution
b) Calculate the number of ideal stages required for the specified flow rates and percentage SO2 removal.
Solution
3.24b. Kremser equations: absorption of sulfur dio ide.
An ab o be i a ailable o ea he fl e ga of P oblem 3.23 hich i e i alen o 8.5 e ilib i m age .
a) Calc la e he a e flo a e o be ed in hi ab o be if 90% of he SO2 i o be emo ed.
Calc la e al o he SO2 concen a ion in he a e lea ing he ab o be . Sol ion
Ini ial e ima e
b) Wha i he e ce age e a f SO2 ha ca be achie ed i h hi ab be if he a e f
Initial estimate:
3.26c. Co n e c en e c o -flo e ac ion.
A 1-butanol acid solution is to be e tracted ith pure ater. The butanol solution contains 4.5% (b eight) of acetic acid and flo s at the rate of 400 kg/hr. A total ater flo rate of 1005 kg/hr is used. Operation is at 298 K and 1 atm. For practical purposes, 1-butanol and
ater are inmiscible. At 298 K, the equilibrium data can be represented b YAi = 0.62 XAi, here YAi is the eight ratio of acid in the aqueous phase and XAi is the eight ratio of acid in the organic phase.
a) If the outlet butanol stream is to contain 0.10% (b eight) acid, ho man equilibrium stages are required for a countercurrent cascade?
Solution
b) If he a e i i e a a g he a e be f age , b i a c -f ca cade, ha i he e 1-b a c ce a i ( ee P b e 3.19)?
S i
3.27c. Glucose sorption on an ion e change resin.
Chi g a d R h e (AIChE S mp. Ser., 81, . 242, 1985) f d ha he e i ib i f g c e a
b) If 5 equilibrium stages are added to the cascade of part a), calculate the resin flo required to maintain the same degree of glucose sorption.
Solution
4.1a. Void fraction near the alls of packed beds.
4.2b. Void fraction near the alls of packed beds.
Beca e f he ci a a e f he id-f ac i adia a ia i f ac ed bed , he e a e a
I i ia e i a e
I i ia e i a e
I i ia e i a e
I i ia e i a e
4.3c. Void fraction near the alls of packed beds.
(a) Sh ha he adia ca i f he a i a a d i i a f he f c i de c ibed b E a i (4-1) a e he f he e a i
S i
b) F he ac ed bed f E a e 4.1, ca c a e he adia ca i f he fi fi e a i a, a d f he fi fi e i i a; ca c a e he a i de f he id f ac i ci a i a h e i .
S i
Ini ial e ima e of he oo can be ob ained f om Fig. 4.4