1 Million+ Step-by-step solutions

Derive the form of the Longmuir isotherm for the complete dissociative adsorption of SO2, i.e.:

Diagrams included in the Solution

Diagrams included in the Solution

Take the data at 363 K from problem 5.2. Can you distinguish which of the three isotherms – Langmuir, Freundlich, or Temkin – is the best obeyed? If so, which one?

Derive the form of the Langmuir isotherm when an adsorbing species occupies two surface sites, i.e:

Diagrams included in the Solution

Diagrams included in the Solution

The following data have been reported by Shen and Smith for benzene (Bz) adsorption on silica gel [16]:

a) Do these data better fit a single site or a dual site Langmuir isotherm? Why?

b) Assuming single site adsorption, calculate the enthalpy and entropy of adsorption for benzene on SiO2

c) What is saturation coverage at each temperature?

a) Do these data better fit a single site or a dual site Langmuir isotherm? Why?

b) Assuming single site adsorption, calculate the enthalpy and entropy of adsorption for benzene on SiO2

c) What is saturation coverage at each temperature?

A boron-doped carbon was prepared by adding 0.1 wt % B to a graphitized carbon black (Monarch 700, Cabot Corp.) and then heat treating this material, designated BC-1, at 2773 K under Ar [71,72]. Its surface area was determined by measuring N2 adsorption at 80 K and using the BET equation (Eqa. 3.4). The equilibrium N2 uptakes versus the pressure are listed in the table below. What is the surface area of this carbon? If the heat of liquefaction for N2 is 1:34 kcal mole-1, what is the estimated heat of adsorption in the first monolayer? The Po value for N2 at the actual temperature measured was Po = 732

Torr (760 Torr = 1 atom).

N2 physisorption on carbon (from ref. 73)

Torr (760 Torr = 1 atom).

N2 physisorption on carbon (from ref. 73)

In a study of iron catalysts, the BET surface areas of a number of materials were determined using N2 or Ar physisorption near 80 K [71–73]. The Po values vary slightly because the bath temperature varied slightly around 77–80 K. What were the surface areas of the following solids based on the data provided below? What were the C values and the heats of adsorption for the initial monolayer of adsorbate? The heat of liquefaction for N2 is 1:34 kcal mole – 1

a) Silica (SiO2) (Cab-O-Sil, Grade 5, Cabot Corp) – N2 adsorption, assume Po = 725 Torr (760 Torr = 1 atm). (Ref. 73)

a) Silica (SiO2) (Cab-O-Sil, Grade 5, Cabot Corp) – N2 adsorption, assume Po = 725 Torr (760 Torr = 1 atm). (Ref. 73)

The following H2 chemisorptions results are reported for a 1.5% Rh/Al2O3 catalyst. What is the amount of hydrogen chemisorbed on Rh? What is the dispersion (fraction exposed) of the Rh?

H2 pressure (Torr) H2 Uptake (μmole=g catalyst)

50 ___________ 40.0

75 ___________ 50.0

100 ___________ 60.0

150 ___________ 65.0

200 ___________ 70.0

250 ___________ 75.0

300 ___________ 80.0

350 ___________ 85.0

400 ___________ 90.0

H2 pressure (Torr) H2 Uptake (μmole=g catalyst)

50 ___________ 40.0

75 ___________ 50.0

100 ___________ 60.0

150 ___________ 65.0

200 ___________ 70.0

250 ___________ 75.0

300 ___________ 80.0

350 ___________ 85.0

400 ___________ 90.0

An O2 adsorption isotherm was obtained at 443 K for a 2.43% Ag/SiO2 catalyst and the uptake results are given below. No irreversible adsorption occurs on the silica. What is the dispersion of the silver?:

Tables included in Solution

Tables included in Solution

The following observed rate data have been reported for Pt-catalyzed oxidation of SO2 at 763K obtained in a differential fixed-bed reactor at atmospheric pressure and containing a catalyst with a bed density of 0:8g cm–3 [2]. The catalyst pellets were 3.2 by 3.2 mm cylinders, and the Pt was superficially deposited upon the external surface. If the void volume of the catalyst bed is 1/2, the density of the catalyst itself will be 1:6g cm–3.

Tables included in Solution

Tables included in Solution

A commercial cumene cracking catalyst is in the form of pellets with a diameter of 0.35 cm which have a surface area, Am, of 420m2 g –1and a void volume, Vm, of 0:42 cm3 g–1. The pellet density is 1.14 g cm–3. The measured 1st-order rate constant for this reaction at 685K was 1:49 cm3 s–1 g–1. Assume that Knudsen diffusion dominates and the path length is determined by the pore diameter, dp. An average pore radius can be estimated from the relationship rp = 2Vm / Am if the pores are modeled as non interconnected cylinders (see equation 4.94). Assuming isothermal operation, calculate the Thiele modulus and determine the effectiveness factor, h, under these conditions.

A 1.0% Pd/SiO2 catalyst for SO2 oxidation is being studied using a stoichiometric O2/SO2 feed ratio at a total pressure of 2 atm. At a temperature of 673 K, a rate of 2:0 mole SO2 s-1 L cat-1 occurs. The average velocity for SO2 molecules is 3 x 104 cm s-1, and the average pore diameter in the alumina is 120 A ˚ (10 A = 1 nm). Assume the Pd is uniformly distributed throughout the catalyst particles. What is the largest particle size (diameter in cm) one can use in the reactor and still be assured that you have no significant diffusion effects?

Assume that Pt was dispersed throughout the pore structure of the entire pellet in Problem 4.1 and apply the Weisz-Prater criterion to determine if mass transport limitations are expected. Do only one calculation using the lowest observed rate. Assume that the average pore diameter in the catalyst is 100 A˚ (10 A˚ = 1 nm), that Knudsen diffusion dominates, and that no external transport limitations occur (Cs = Co).

Vapor-phase benzene (Bz) hydrogenation over carbon-supported Pd catalysts has been studied [65, 66]. A 2.1% Pd/C catalyst prepared with a carbon black cleaned in H2 at 1223 K had a surface-weighted Pd crystallite size of 21 nm, giving a Pd dispersion of 5%, based on TEM. The carbon itself had an average mesopore diameter of 25 nm, while the average microspore diameter was 0.9 nm; thus the majority of the Pd resided in the mesopore. The highest activity of this catalyst at 413 K and 50 Torr Bz (Total P = 1 atm, balance H2) was 1:99 μ mole Bz s-1 g -1. The density of the catalyst was 0:60 g cm -3. The catalyst particle size distribution ranged from 10–500 microns (1 m = 10-6 m).
(a) Assuming all the Pd is in the mesopore, are any mass transfer limitations expected based on the W-P criterion?
(b) If, instead, this catalyst had all the Pd in the microspores and it gave this performance, would mass transfer limitations exist? Why?

The reforming of CH4 with CO2 to produce CO and H2 has been examined over a number of dispersed metal catalysts [67, 68]. At 723K, a CO2 partial pressure of 200 Torr, PCH4 = 200 Torr, and a total pressure of 1 atm (1 atm = 760 Torr), the catalysts listed below produced the rates listed for CO2 consumption. The catalysts used were sieved to a 120/70 mesh fraction, thus the largest particles had a diameter of 0.020 cm. Assume a catalyst density of 1g cm-3, which provides a conservative overestimate. The pore diameter, δ, 4. Acquisition and Evaluation of Reaction Rate Data was obtained from pore size distribution measurements. Determine the W-P number for:

a) Ni/SiO2.

b) Pt/ZrO2,

c) Rh/TiO2,

d) Ru/Al2O3.

Should one be concerned about pore diffusion limitations with any of these catalysts? Why?

a) Ni/SiO2.

b) Pt/ZrO2,

c) Rh/TiO2,

d) Ru/Al2O3.

Should one be concerned about pore diffusion limitations with any of these catalysts? Why?

Utilize the data given in Problem 4.5, assume the benzene hydrogenation reaction is zero order in benzene and first order in H2, and then calculate the Thiele modulus for the largest and smallest catalyst particles assuming all the Pd is in the mesopore. What is the Thiele modulus for the largest (500 μ) particles if all the Pd were dispersed in only the micropores? Should there be any concern about pore diffusion limitations for any of the three possibilities? Why?

The rate of formaldehyde (CH2O) oxidation over a Ag=SiO2 catalyst at 493 K and a CH2O pressure of 9 Torr in air was 1:4 x 10 –7 mole CH2Os –1 cm –3. The catalyst particles passed through a 100-mesh sieve (149 micron opening), and the average pore diameter of the SiO2 was 60 A˚. Can the rate be considered to be free of mass transport limitations?

What is the optimum fractional coverage of a nonuniform catalyst surface if the transfer coefficient a is 2/3 rather than 1/2; in other words, if equation 8.57 utilizes a = 2/3?

The Temkin rate equation for NH3 synthesis is based on the 2-step sequence provided in Illustration 8.1 and it is given by equation 2 in that Illustration. With the information given there, i.e., m = a = 1/2, verify that equation 3 can be derived and that the listed values of k * and k are obtained.

Derive the rate expression for an enzyme-catalyzed unimolecular (single substrate) reaction, such as that shown in steps 9.1 and 9.2, assuming that the decomposition of the reactive intermediate to give the product is reversible, rather than irreversible as indicated in step 9.2. Can the initial rate in the forward direction and the initial rate in the reverse direction be expressed in the form of a Michaelis-Menten rate equation? If so how? If not, why?

Initial rates of an enzyme-catalyzed reaction for various substrate (reactant) concentrations are listed in the table below. Evaluate rmax and Km by a Lineweaver-Burke plot.

Derive an expression for the reaction rate, r, in terms of S, E and the constants shown for the following reaction sequence, which includes substrate inhibition:

Multiple complexes can be involved in some enzyme-catalyzed reactions. For the reaction sequence shown below, develop suitable rate expressions using:

(a) The Michaelis equilibrium approach and

(b) The steady-state approximation for the complexes.

(a) The Michaelis equilibrium approach and

(b) The steady-state approximation for the complexes.

The following first-order rate constants were obtained for the thermal decomposition of ethane:

Calculate the activation energy and the pre-exponential factor.

Calculate the activation energy and the pre-exponential factor.

Using statistical thermodynamics, determine the temperature dependence of the pre-exponential factor for:

a) The gas-phase reaction of two N atoms to form N2.

b) The bimolecular gas-phase reaction of H2O + CO to form H2 and CO2.

c) The bimolecular gas-phase reaction of H2 + Cl2 to form HCl.

a) The gas-phase reaction of two N atoms to form N2.

b) The bimolecular gas-phase reaction of H2O + CO to form H2 and CO2.

c) The bimolecular gas-phase reaction of H2 + Cl2 to form HCl.

Use the steady-state approximation and derive the rate expression for the following sequence of elementary steps describing a chain reaction. Define your rate clearly.

The series of elementary steps below has been proposed to describe acetaldehyde pyrolysis to methane and CO. Derive the steady-state rate expression making the usual long-chain approximation; i.e., that the ‘‘chain length’’ is very high. What comprises the apparent activation energy?

Using the BOC-MP method, calculate the heat of adsorption for the following four species. All the information and values needed are contained in the various tables in section 6.3 of this chapter. State the appropriate adsorption mode, i.e., (תx μy).

(a) Nitric oxide on Fe (110) – what end of NO should bond to Fe? What is the preferred coordination, i.e., is y = 1 or 2? Why?

(b) H2O on Pd (100)

(c) Acetylene (HC ≡ CH) on Pt (111)

(d) The methylene radical (-CH2) on Pt (100)

(a) Nitric oxide on Fe (110) – what end of NO should bond to Fe? What is the preferred coordination, i.e., is y = 1 or 2? Why?

(b) H2O on Pd (100)

(c) Acetylene (HC ≡ CH) on Pt (111)

(d) The methylene radical (-CH2) on Pt (100)

Based on the BOC-MP method, calculate the heat of adsorption for the following species. All information and values needed are contained in the various tables in Section 6.3 of this chapter.

(a) NO on Ni (100) (ת1 μ2).

(b) C2H4 on Pd (111) (ת2 μ2)

(c) NH3 on Fe (110) (ת1 μ1) (Check ref. 25 for DAB value)

(d) NH2 and NH on Fe (110) (Check ref. 25 for DAB value)

(a) NO on Ni (100) (ת1 μ2).

(b) C2H4 on Pd (111) (ת2 μ2)

(c) NH3 on Fe (110) (ת1 μ1) (Check ref. 25 for DAB value)

(d) NH2 and NH on Fe (110) (Check ref. 25 for DAB value)

The kinetics of acetone hydrogenation on Pt catalysts has been studied by Sen and Vannice (See Problem 7.9) [35]. To determine which bonding mode is more probable, the heat of adsorption for an on-top ת1 μ2 adsorbed acetone species was compared to that for a di-s-bonded ת2 μ2 species which had a C atom and the O atom interacting with a close-packed Pt (111) surface. What are these two values? Which species is favored?

Derive Guideline 3 in Table 6.10 for a monatomic species of 30 amu:

a) Using collision theory and assuming a 2-dimensional gas.

b) Using absolute rate theory and assuming immobile adsorption.

c) Using absolute rate theory and assuming a 2-dimensional gas.

a) Using collision theory and assuming a 2-dimensional gas.

b) Using absolute rate theory and assuming immobile adsorption.

c) Using absolute rate theory and assuming a 2-dimensional gas.

Kircher and Hougen studied NO oxidation, 2NO + O2 → 2NO2, over activated carbon and SiO2 in the presence of water and proposed a Rideal-Eley mechanism between O2 and (N2O2) ad, which gave a rate expression of

Derive a rate expression for each of the following single reactions taking place through a sequence of steps as indicated. Define your rate clearly. S represents an active site.

Derive the form of the rate expression based on the elementary steps below, where S is an active site: a) Assuming [B_S] is the MARI, and b) Assuming [A_S] is the MARI.

Consider the following sequence of steps to describe the high-temperature kinetics of NO reduction by CH4 on La2O3 and Sr = La2O3 in the absence of O2:

(a) Provide any necessary stoichiometric coefficient for each step to balance the overall reaction, i.e., no active centers are contained in the overall stoichiometry;

(b) Derive a kinetic rate expression from this sequence assuming NO* and CH2* are the only two significant surface species;

(c) Does the final rate expression have the capability of fitting the rate data reported in the paper by Vannice et al. [4], which gave reaction orders of 0.19–0.26 for CH4 and 0.73–0.98 for NO? Why?

(a) Provide any necessary stoichiometric coefficient for each step to balance the overall reaction, i.e., no active centers are contained in the overall stoichiometry;

(b) Derive a kinetic rate expression from this sequence assuming NO* and CH2* are the only two significant surface species;

(c) Does the final rate expression have the capability of fitting the rate data reported in the paper by Vannice et al. [4], which gave reaction orders of 0.19–0.26 for CH4 and 0.73–0.98 for NO? Why?

A series of elementary steps to describe the water gas shift reaction over copper has been proposed as shown below. Define your reaction rate and derive a rate expression consistent with this model, where * indicates an active site on Cu, assuming that the surface concentration of O atoms is negligible.

Derive the rate expression in Table 7.10:

a) For a bimolecular reaction with nondissociative adsorption as the RDS;

b) For a bimolecular reaction with nondissociative adsorption, but now product desorption is the RDS;

c) For a bimolecular reaction with dissociative adsorption of one reactant as the RDS;

d) For a bimolecular reaction with dissociative adsorption of one reactant and with desorption of the product as the RDS.

a) For a bimolecular reaction with nondissociative adsorption as the RDS;

b) For a bimolecular reaction with nondissociative adsorption, but now product desorption is the RDS;

c) For a bimolecular reaction with dissociative adsorption of one reactant as the RDS;

d) For a bimolecular reaction with dissociative adsorption of one reactant and with desorption of the product as the RDS.

Sinfelt has studied ethane hydrogenolysis, C2H6 + H2 → 2CH4, over the Group VIII metals, and he has proposed the following sequence of steps on cobalt [55]:

The decomposition of N2O over a 4.56% Cu/ZSM-5 catalyst (ZSM-5 is a zeolite discovered by Socony Mobil, hence the letter designation) has been investigated by Dandekar and Vannice [56]. Temperatures were varied from 623 to 673 K and partial pressure dependencies were determined for N2O, O2 and N2 at three different temperatures under approximately differential reaction conditions. The rate data are given in Table 1. The N2

N2O decomposition has also been investigated on a 4.9% Cu=ת ─ Al2O3 catalyst in which, after a pretreatment in H2 at 573 K, the Cu was highly dispersed and present almost completely as zero-valent Cu at the surface [56]. These small Cu crystallites exhibited significant deactivation, which was attributed to strong oxygen adsorption at the Cu surface, except at high temperatures above 823 K. Additional studies indeed showed that rapid O2 desorption required much higher temperatures than with the Cu/ZSM-5 catalysts; therefore, O2 adsorption/desorption was assumed to be reversible, rather than quasi-equilibrated, and no RDS existed, as shown in the sequence of steps proposed below, where * represents an active site:

The kinetics of acetone hydrogenation over 5.0% Pt/SiO2 and 1.9% Pt/TiO2 catalysts, previously characterized by H2 chemisorption, were studied by Sen and Vannice [57]. The dispersion of Pt was 0.31 in the former catalyst and in the latter catalyst, which exhibited SMI behavior, DPt ¼ 0:75. The kinetic parameters at 303 K and 1 atm from a power rate law, TOFIPA (s-1) = Ae-E/RT PXAce PYH2, are given in the table below. Because of the uncertainty in Y, consider it to be unity. Propose a model that yields a derived rate expression consistent with these results knowing that dissociative H2 adsorption occurs on Pt. If more than one reaction model can be proposed which gives acceptable rate expressions, can you evaluate which might be rejected? How regarding the heat of adsorption of an on-top (ת1μ1) acetone species vs. that for a di-s-bonded (ת2μ2) species which has both a C atom and the O atom interacting with a surface metal atom.)

As an alternative to the model proposed in Illustration 7.3, consider instead the following H-W-type reaction mechanism for NO decomposition, in which unimolecular decomposition occurs with an additional active site, *, and N2 desorption is the rate determining step

Methane combustion on La2O3-based catalysts has been studied by Toops et al. [46]. With a 4% Sr-promoted La2O3 catalyst (2:5m2g-1) operating between 773 and 973 K, 0:5 -5 Torr CH4 and 3 _ 23 Torr O2 (760 Torr 1 atm) under differential reactor conditions, an apparent activation energy of 29 kcal mole-1 for CO2 formation was observed. Near 900 K, selectivity to CO2, rather than CO, was about 75% or higher, and the reaction orders from a power rate law are given in Table 1. Propose a L-Htype model for CO2 formation with a sequence of elementary steps that results in a derived rate expression consistent with these results. It can be assumed that only the adsorbed reactants and products need be included in the site balance, and dissociative O2 adsorption occurs. Under low-conversion conditions, the surface concentrations of the products can be ignored, so what is the form of the rate equation? Fitting this latter equation to the data produced the optimized rate parameters listed in Table 2, where k0 is the lumped apparent rate constant. Evaluate them to determine if they are consistent and state why.

The reduction of NO by H2 on La2O3 and Sr-promoted La2O3 was examined by Huang et al. [59]. Both N2 and N2O were observed as products. The sequence of elementary steps proposed for the catalytic cycle was the following, which invoked one type of site (S) to adsorb and activate H2 while the second type (*) interacted with the oxygen-containing reaction intermediates. Stoichiometric numbers for an elementary step are included when needed:

It was mentioned in Chapter 7.1 that if more than one active site is required in a unimolecular decomposition reaction, the denominator must be raised to a power greater than unity. Now, if a RDS does not exist, the derivation can become more complicated. An example of this is provided by a study of ammonia decomposition on a 4.8% Ru/carbon catalyst with a dispersion near unity [60]. Using a power rate law of the form rm = kPa NH3 Pb H2 , values of a ¼ 0:75 and 0.69 were obtained at 643 and 663 K, respectively, while values of b were _2.0 and _1.6 at 643 and 663 K, respectively. The elementary steps proposed for the catalytic cycle are given below with their stoichiometric numbers, where * represents an active site

As stated in this chapter, a reaction sequence need not contain a RDS. A study of N2O reduction with CO over a 4.9% Cu=Al2O3 catalyst provides an example of this [56]. This reaction around 523 K was much more rapid than N2O decomposition, and the following sequence of elementary steps was proposed to describe it, where * is an active site

The catalytic reforming of methane with carbon dioxide, i.e., CH4 + CO2 ≡ 2 H2 +2 CO

is a complex high-temperature reaction. It has been studied on nickel catalysts between 673 and 823 K by Bradford and Vannice [61], and the following catalytic cycle was proposed, where * represents an active site:

is a complex high-temperature reaction. It has been studied on nickel catalysts between 673 and 823 K by Bradford and Vannice [61], and the following catalytic cycle was proposed, where * represents an active site:

As another alternative to the model proposed in Illustration 7.3, consider a sequence with no RDS on the surface and with the formation of an adsorbed (NO) 2 species which then decomposes to give adsorbed N2O and oxygen. For example, with only NO and O2 adsorption quasi-equilibrated:

Examine the reaction kinetics in Illustration 7.5 again. Focus in particular on steps 10–14. If a H-W-type model is chosen that assumes irreversible toluene desorption as the RDS, is the derived rate expression consistent with the data? Why?

In a study of formaldehyde oxidation over Ag catalysts, the power rate law over Ag powder was r = kPo2 and over supported Ag it was r = kP0:3O2 P0:3H2CO [62]. The following sequence of steps was proposed

Estimation of dense-gas viscosity. Estimate the viscosity of nitrogen at 68oF and 1000 psig by means of Fig. 1.3-1, using the critical viscosity from Table E.1. Give the result in units of lbm/ft ∙ s. For the meaning of "psig," see Table F.3-2.

Estimation of the viscosity of methyl fluoride. Use Figure 1.3-1 to find the viscosity in Pa ∙ s of CH3F at 370oC and 120 atm. Use the following values 1 for the critical constants: Tc = 4.55oC, Pc = 58.0 atm, pc = 0.300 g/cm3.

Computation of the viscosities of gases at low density. Predict the viscosities of molecular oxygen, nitrogen, and methane at 20oC and atmospheric pressure, and express the results in mPa. s. Compare the results with experimental data given in this chapter.

Gas-mixture viscosities at low density. The following data2 are available for the viscosities of mixtures of hydrogen and Freon-12 (dichlorodifluoromethane) at 25oC and 1 atm:

Mole fraction of H2: 0.00 0.25 0.50 0.75 1.00

μ x 106 (poise): 124.0 128.1 131.9 135.1 88.4

Use the viscosities of the pure components to calculate the viscosities at the three

intermediate compositions by means of Eqs. 1.4-15 and 16.

Viscosities of chlorine-air mixtures at low density. Predict the viscosities (in cp) of chlorine-air mixtures

at 75oF and 1 atm, for the following mole fractions of chlorine: 0.00, 0.25, 0.50, 0.75, 1.00. Consider air as a singlie component and use Eqs. 1.4-14 to 16.

Estimation of liquid viscosity. Estimate the viscosity of saturated liquid water at 0oC and at 100oC by means of

(a) Eq. 1.5-9, with ∆Uvap = 897.5 Btu/lbm at 100oC, and

(b) Eq. 1.5-11. Compare the results with the values in Table 1.1-1.

Molecular velocity and mean free path. Compute the mean molecular velocity u (in cm/s) and the mean free path λ (in cm) for oxygen at 1 atm and 273.2 K. A reasonable value for d is 3 A. What is the ratio of the mean free path to the molecular diameter under these conditions? What would be the order of magnitude of the corresponding ratio in the liquid state?

Velocity profiles and the stress components τij. For each of the following velocity distributions, draw a meaningful sketch showing the flow pattern. Then find all the components of 'r and pvv for the Newtonian fluid. The parameter b is a constant.

(a) vx = by, vy = 0, vz = 0

(b) vx = by, vy = bx, vz = 0

(c) vx = -by, vy = bx, vz = 0

(d) vx = – ½bx, vy = – ½by, vz = bz

A fluid in a state of rigid rotation

(a) Verify that the velocity distribution (c) in Problem 1B.1 describes a fluid in a state of pure rotation; that is, the fluid is rotating like a rigid body. What is the angular velocity of rotation?

(b) For that flow pattern evaluate the symmetric and anti symmetric combinations of velocity derivatives:

(c) Discuss the results of (b) in connection with the development in S1.2.

Viscosity of suspensions, data of V and3 for suspensions of small glass spheres in aqueous glycerol solutions of ZnI2 can be represented up to about Ф = 0.5 by the semi empirical expression Compare this result with the Mooney equation. .

Some consequences of the Maxwell-Boltzmann distribution, in the simplified kinetic theory in S1.4, several statements concerning the equilibrium behavior of a gas were made without proof. In this problem and the next, some of these statements are shown to be exact consequences of the Maxwell-Boltzmann velocity distribution. The Maxwell-Boltzmann distribution of molecular velocities in an ideal gas at rest is *f*(u_{x}, u_{y}, u_{z}) = *n*(m/2πkT)^{3/2} exp(–mu^{2}/2kT) (1C.1-1) in which u is the molecular velocity, n is the number density, and *f*(u_{x}, u_{y}, u_{z})du_{x }du_{y }du_{z} is the number of molecules per unit volume that is expected to have velocities between u_{x} and u_{x} + du_{x}, u_{y} and u_{y} + du_{y}, u_{z} and u_{z} + du_{z}. It follows from this equation that the distribution of the molecular speed u is *f*(u) = 4πnu^{2}(m/2πkT)^{3/2} exp(–mu^{2} /2kT)

(a) Verify Eq. 1.4-1 by obtaining the expression for the mean speed *u* from

(b) Obtain the mean values of the velocity components u_{x}, u_{y}, and u_{z}. The first of these is obtained from what can one conclude from the results?

(c) Obtain the mean kinetic energy per molecule by the correct result is ½ mu^{2} = 3/2kT.

The wall collision frequency, it is desired to find the frequency Z with which the molecules in an ideal gas strike a unit area of a wall from one side only. The gas is at rest and at equilibrium with a temperature T and the number density of the molecules is n. All molecules have a mass m. All molecules in the region x 0 will hit an area S in the yz-plane in a short time ∆t if they are in the volume Sux∆t. The number of wall collisions per unit area per unit time will be verify the above development.

Pressure of an ideal gas.4 It is desired to get the pressure exerted by an ideal gas on a wall by accounting for the rate of momentum transfer from the molecules to the wall.

Force on a surface of arbitrary orientation? (Figure 1D.2) Consider the material within an element of volume OABC that is in a state of equilibrium, so that the sum of the forces acting on the triangular faces ∆OBC, ∆OCA, ∆OAB, and ∆ABC must be zero. Let the area of ∆ABC be dS, and the force per unit area acting from the minus to the plus side of dS be the vector πn. Show that πn = [n ∙ π].

(a) Show that the area of ∆OBC is the same as the area of the projection ∆ABC on the yz-plane; this is (n ∙ δx) dS write similar expressions for the areas of ∆OCA and ∆OAB.

(b) Show that according to Table 1.2-1 the force per unit area on ∆OBC is δx πxλ, + δy πxy + δz πxz. Write similar force expressions for ∆OCA and ∆OAB.

(c) Show that the force balance for the volume element OABC gives in which the indices i, j take on the values x, y, z. The double sum in the last expression is the stress tensor π written as a sum of products of unit dyads and components. Fig. 1D.2 Element of volume OABC over which a force balance is made. The vector πn = [n ∙ π] is the force per unit area exerted by the minus material (material inside OABC) on the plus material (material outside OABC). The vector n is the outwardly directed unit normal vector on face ABC.

Thickness of a falling film, water at 20oC is flowing down a vertical wall with Re = 10. Calculate

(a) The flow rate, in gallons per hour per foot of wall length, and

(b) The film thickness in inches.

Determination of capillary radius by flow measurement, one method for determining the radius of a capillary tube is by measuring the rate of flow of a Newtonian liquid through the tube. Find the radius of a capillary from the following flow data:

Length of capillary tube 50.02 cm

Kinematic viscosity of liquid 4.03 x 10^{–5} m^{2}/s

Density of liquid 0.9552 x 10^{3} kg/m^{3}

Pressure drop in the horizontal tube 4.829 x l0^{5} Pa

Mass rate of flow through tube 2.997 x 10^{–3} kg/s

What difficulties may be encountered in this method? Suggest some other methods for determining the radii of capillary tubes.

Volume flow rate through an annulus, a horizontal annulus, 27 ft in length, has an inner radius of 0.495 in. and an outer radius of 1.1 in. A 60% aqueous solution of sucrose (C12H22O11) is to be pumped through the annulus at 20oC. At this temperature the solution density is 80.3 lb/ft3 and the viscosity is 136.8lbm/ft ∙ hr. What is the volume flow rate when the impressed pressure difference is 5.39 psi?

Loss of catalyst particles in stack gas

(a) Estimate the maximum diameter of micro spherical catalyst particles that could be lost in the stack gas of a fluid cracking unit under the following conditions:

Gas velocity at axis of stack = 1.0 ft/s (vertically upward)

Gas viscosity = 0.026 cp

Gas density = 0.045 lb / ft^{3}

Density of a catalyst particle = 1.2 g/cm^{3}

Express the result in microns (1 micron = 10^{–6} = 1/μm).

(b) Is it permissible to use Stokes' law in (a)?

Different choice of coordinates for the falling film problem, Re-derive the velocity profile and the average velocity in S2.2, by replacing x by a coordinate x measured away from the wall; that is, x = 0 is the wall surface, and x = 8 is the liquid-gas interface. Show that the velocity distribution is then given by and then use this to get the average velocity. Show how one can get Eq. 2B.1-1 from Eq. 2.2-18 by making a change of variable.

Alternate procedure for solving flow problems, in this chapter we have used the following procedure: (i) derive an equation for the momentum flux, (ii) integrate this equation, (iii) insert Newton's law to get a first-order differential equation for the velocity, (iv) integrate the latter to get the velocity distribution. Another method is: (i) derive an equation for the momentum flux, (ii) insert Newton's law to get a second-order differential equation for the velocity profile, (iii) integrate the latter to get the velocity distribution. Apply this second method to the falling film problem by substituting Eq. 2.2-14 into Eq. 2.2-10 and continuing as directed until the velocity distribution has been obtained and the integration constants evaluated.

Laminar flow in a narrow slit (see Fig. 2B.3).

(a) A Newtonian fluid is in laminar flow in a narrow slit formed by two parallel walls a distance 2B apart. It is understood that B < < W, so that "edge effects" are unimportant. Make a differential momentum balance, and obtain the following expressions for the momentum-flux and velocity distributions: In these expressions P = *p* + *pgh* = *p* – *pgz*.

(b) What is the ratio of the average velocity to the maximum velocity for this flow?

(c) Obtain the slit analog of the Hagen-Poiseuille equation.

(d) Draw a meaningful sketch to show why the above analysis is inapplicable if B = W.

Laminar slit flow with a moving wall ("plane Couette flow") Extend Problem 2B.3 by allowing the wall at x = B to move in the positive z direction at a steady speed v0. Obtain

(a) The shear-stress distribution and

(b) The velocity distribution. Draw carefully labeled sketches of these functions.

Interrelation of slit and annulus formulas, when an annulus is very thin, it may, to a good approximation, is considered as a thin slit. Then the results of Problem 2B.3 can be taken over with suitable modifications. For example, the mass rate of flow in an annulus with outer wall of radius R and inner wall of radius (1 - ε) R, where ε is small, may be obtained from Problem 2B.3 by replacing 2B by εR, and W by 2π (1 â€“ Â½ ε) R. In this way we get for the mass rate of flow: Show that this same result may be obtained from Eq. 2.4-17 by setting K equal to 1 â€“ ε everywhere in the formula and then expanding the expression for w in powers of ε. This requires using the Taylor series (see SC.2) and then performing a long division. The first term in the resulting series will be Eq. 2B.5-1. Caution: In the derivation it is necessary to use the first four terms of the Taylor series in Eq. 2B.5-2.

Flow of a film on the outside of a circular tube (see Fig. 2B.6), in a gas absorption experiment a viscous fluid flows upward through a small circular tube and then downward in laminar flow on the outside. Set up a momentum balance over a shell of thickness Ar in the film, as shown in Fig. 2B.6. Note that the "momentum in" and "momentum out" arrows are always taken in the positive coordinate direction, even though in this problem the momentum is flowing through the cylindrical surfaces in the negative r direction.

(a) Show that the velocity distribution in the falling film (neglecting end effects) is

(b) Obtain an expression for the mass rate of flow in the film.

(c) Show that the result in (b) simplifies to Eq. 2.2-21 if the film thickness is verysmall

Annular flow with inner cylinder moving axially (see Fig. 2B.7), a cylindrical rod of diameter KR moves axially with velocity v0 along the axis of a cylindrical cavity of radius R as seen in the figure, the pressure at both ends of the cavity is the same, so that the fluid moves through the annular region solely because of the rod motion. Fig. 2B.7 annular flow with the inner cylinder moving axially

(a) Find the velocity distribution in the narrow annular region.

(b) Find the mass rate of flow through the annular region.

(c) Obtain the viscous force acting on the rod over the length L.

(d) Show that the result in (c) can be written as a "plane slit" formula multiplied by a "curvature correction." Problems of this kind arise in studying the performance of wire-coatingdies.1

Analysis of a capillary flow meter (see Fig. 2B.8). Determine the rate of flow (in lb/hr) through the capillary flow meter shown in the figure. The fluid flowing in the inclined tube is water at 20oC, and the manometer fluid is carbon tetrachloride (CC14) with density 1.594 g/cm3. The capillary diameter is 0.010 in. Note: Measurements of H and L are sufficient to calculate the flow rate; θ need not be measured. Why?

Low-density phenomena in compressible tube flow2,3 (Fig. 2B.9). As the pressure is decreased in the system studied in Example 2.3-2, deviations from Eqs. 2.3-28 and 2.3-29 arise. The gas behaves as if it slips at the tube wall. It is conventional2 to replace the customary "no-slip" boundary condition that vz = 0 at the tube wall by in which ζ is the slip coefficient. Repeat the derivation in Example 2.3-2 using Eq. 2B.9-1 as the boundary condition. Also make use of the experimental fact that the slip coefficient varies inversely with the pressure ζ = ζ0/p, in which ζ0 is a constant. Show that the mass rate of flow is in which Pavg = Â½(p0 + p1) When the pressure is decreased further, a flow regime is reached in which the mean free path of the gas molecules is large with respect to the tube radius (Knudsen flow). In that regime3 in which m is the molecular mass and K is the Boltzmann constant. In the derivation of this result it is assumed that all collisions of the molecules with the solid surfaces are diffuse and not specular, the results in Eqs. 2.3-29, 2B.9-2, and 2B.9-3 are summarized in Fig. 2B.9.

Incompressible flow in a slightly tapered tube, an incompressible fluid flows through a tube of circular cross section, for which the tube radius changes linearly from R0 at the tube entrance to a slightly smaller value RL at the tube exit. Assume that the Hagen-Poiseuille equation is approximately valid over a differential length, dz, of the tube so that the mass flow rate is this is a differential equation for P as a function of z, but, when the explicit expression for R(z) is inserted, it is not easily solved.

(a) Write down the expression for R as a function of z.

(b) Change the independent variable in the above equation to R, so that the equation becomes

(c) Integrate the equation, and then show that the solution can be rearranged to give

Interpret the result. The approximation used here that a flow between nonparallel surfaces can be regarded locally as flow between parallel surfaces is sometimes referred to as the lubrication approximation and is widely used in the theory of lubrication. By making a careful order-of-magnitude analysis, it can be shown that, for this problem, the lubrication approximation is valid as longas4

The cone-and-plate viscometer (see Fig. 2B.11) a cone-and-plate viscometer consists of a stationary flat plate and an inverted cone, whose apex just contacts the plate. The liquid whose viscosity is to be measured is placed in the gap between the cone and plate. The cone is rotated at a known angular velocity Ω, and the torque Tz required to turn the cone is measured. Find an expression for the viscosity of the fluid in terms of Ω, Tz, and the angle ψ0 between the cone and plate. For commercial instruments ψ0 is about 1 degree.

(a) Assume that locally the velocity distribution in the gap can be very closely approximated by that for flow between parallel plates, the upper one moving with a constant speed. Verify that this leads to the approximate velocity distribution (in spherical coordinates) this approximation should be rather good, because ψ0 is so small.

(b) From the velocity distribution in Eq. 2B.11-1 and Appendix B.1, show that a reasonable expression for the shear stress is this result shows that the shear stress is uniform throughout the gap. It is this fact that makes the cone-and-plate viscometer quite attractive. The instrument is widely used, particularly in the polymer industry.

(c) Show that the torque required to turn the cone is given by This is the standard formula for calculating the viscosity from measurements of the torque and angular velocity for a cone-plate assembly with known R and ψ0.

(d) For a cone-and-plate instrument with radius 10 cm and angle ψ0 equal to 0.5 degree, what torque (in dyn ∙ cm) is required to turn the cone at an angular velocity of 10 radians per minute if the fluid viscosity is 100 cp?

Flow of a fluid in a network of tubes (Fig. 2B.12), a fluid is flowing in laminar flow from A to B through a network of tubes, as depicted in the figure. Obtain an expression for the mass flow rate w of the fluid entering at A (or leaving at B) as a function of the modified pressure drop PA-PB. Neglect the disturbances at the various tube junctions.

Performance of an electric dust collector (see Fig. 2C.1)^{5}

(a) A dust precipitator consists of a pair of oppositely charged plates between which dust-laden gases flow. It is desired to establish a criterion for the minimum length of the precipita-tor in terms of the charge on the particle e, the electric field strength 2, the pressure difference (P_{0} - P_{L}), the particle mass m, and the gas viscosity μ,. That is, for what length L will the smallest particle present (mass *m*) reach the bottom just before it has a chance to be swept out of the channel? Assume that the flow between the plates is laminar so that the velocity distribution is described by Eq. 2B.3-2. Assume also that the particle velocity in the z direction is the same as the fluid velocity in the z direction. Assume further that the Stokes drag on the sphere as well as the gravity force acting on the sphere as it is accelerated in the negative x direction can be neglected.

(b) Rework the problem neglecting acceleration in the x direction, but including the Stokes drag.

(c) Compare the usefulness of the solutions in (a) and (b), considering that stable aerosol particles have effective diameters of about 1-10 microns and densities of about 1 g/cm^{3}.

Residence time distribution in tube flow, define the residence time function F(t) to be that fraction of the fluid flowing in a conduit which flows completely through the conduit in a time interval t. Also define the mean residence time t_{m} by the relation

(a) An incompressible Newtonian liquid is flowing in a circular tube of length L and radius R, and the average flow velocity is (v_{z},}. Show that

F(t) = 0 for t __<__ (L/2(v_{z}>)

F(t) = 1 – (L/2(*v*_{z},>t)^{2} for t __>__ (L/2(v_{z>})

(b) Show that t_{m} = (L/<v_{z}>).

Velocity distribution in a tube, you have received a manuscript to referee for a technical journal. The paper deals with heat transfer in tube flow. The authors state that, because they are concerned with non-isothermal flow, they must have a "general" expression for the velocity distribution, one that can be used even when the viscosity of the fluid is a function of temperature (and hence position). The authors state that a "general expression for the velocity distribution for flow in a tube" is in which y = r/R. The authors give no derivation, nor do they give a literature citation. As the referee you feel obliged to derive the formula and list any restrictions implied.

Falling-cylinder viscometer (see Fig. 2C.4)^{6} A falling-cylinder viscometer consists of a long vertical cylindrical container (radius R), capped at both ends, with a solid cylindrical slug (radius KR). The slug is equipped with fins so that its axis is coincident with that of the tube. One can observe the rate of descent of the slug in the cylindrical container when the latter is filled with fluid. Find an equation that gives the viscosity of the fluid in terms of the terminal velocity *v*_{0} of the slug and the various geometric quantities shown in the figure.

(a) Show that the velocity distribution in the annular slit is given by in which s c = r/R is a dimensionless radial coordinate.

(b) Make a force balance on the cylindrical slug and obtain in which p and P0 are the densities of the fluid and the slug, respectively.

(c) Show that, for small slit widths, the result in (b) may be expanded in powers of ε = 1 – k to give See SC.2 for information on expansions in Taylor series.

Falling-cylinder viscometer (see Fig. 2C.4)^{6} A falling-cylinder viscometer consists of a long vertical cylindrical container (radius R), capped at both ends, with a solid cylindrical slug (radius KR). The slug is equipped with fins so that its axis is coincident with that of the tube. One can observe the rate of descent of the slug in the cylindrical container when the latter is filled with fluid. Find an equation that gives the viscosity of the fluid in terms of the terminal velocity *v*_{0} of the slug and the various geometric quantities shown in the figure.

(a) Show that the velocity distribution in the annular slit is given by in which s c = r/R is a dimensionless radial coordinate.

(b) Make a force balance on the cylindrical slug and obtain in which p and P0 are the densities of the fluid and the slug, respectively.

(c) Show that, for small slit widths, the result in (b) may be expanded in powers of ε = 1 – k to give See SC.2 for information on expansions in Taylor series.

Rotating cone pump (see Fig. 2C.6), find the mass rate of flow through this pump as a function of the gravitational acceleration, the impressed pressure difference, the angular velocity of the cone, the fluid viscosity and density, the cone angle, and other geometrical quantities labeled in the figure.

(a) Begin by analyzing the system without the rotation of the cone. Assume that it is possible to apply the results of Problem 2B.3 locally. That is, adapt the solution for the mass flow rate from that problem by making the following replacements:

Replace (P_{0} – P_{L}) L by – dP/dz

Replace W by 2πr = 2πz sin β thereby obtaining

The mass flow rate w is a constant over the range of z. Hence this equation can be integrated to give

(b) Next, modify the above result to account for the fact that the cone is rotating with angular velocity Ω. The mean centrifugal force per unit volume acting on the fluid in the slit will have a z-component approximately given by what is the value of K? Incorporate this as an additional force tending to drive the fluid through the channel. Show that this leads to the following expression for the mass rate of flow:

Here P_{1}, = p_{1} + pgL_{1}, cos β.

A simple rate-of-climb indicator (see Fig. 2C.7) under the proper circumstances the simple apparatus shown in the figure can be used to measure the rate of climb of an airplane. The gauge pressure inside the Bourdon element is taken as proportional to the rate of climb. For the purposes of this problem the apparatus may be assumed to have the following properties: (i) the capillary tube (of radius R and length L, with R

(a) Develop an expression for the change of air pressure with altitude, neglecting temperature changes, and considering air to be an ideal gas of constant composition.

(b) By making a mass balance over the gauge, develop an approximate relation between gauge pressure p, - Po and rate of climb vz for a long continued constant-rate climb. Neglect change of air viscosity, and assume changes in air density to be small.

(c) Develop an approximate expression for the "relaxation time" trel of the indicator that is, the time required for the gauge pressure to drop to 1/e of its initial value when the external pressure is suddenly changed from zero (relative to the interior of the gauge) to some different constant value, and maintained indefinitely at this new value.

(d) Discuss the usefulness of this type of indicator for small aircraft.

(e) Justify the plus and minus signs in the figure.

Rolling-ball viscometer, an approximate analysis of the rolling-ball experiment has been given, in which the results of Problem 2B.3 are used? Read the original paper and verify the results.

Drainage of liquids^{9} (see Fig. 2D.2) how much liquid clings to the inside surface of a large vessel when it is drained? As shown in the figure there is a thin film of liquid left behind on the wall as the liquid level in the vessel falls. The local film thickness is a function of both z (the distance down from the initial liquid level) and t (the elapsed time).

(a) Make an unsteady-state mass balance on a portion of the film between z and z + ∆_{z} to get

(b) Use Eq. 2.2-18 and a quasi-steady-assumption to obtain the following first-order partial differential equation for δ(z, t);

(c) Solve this equation to get what restrictions have to be placed on this result?

Torque required to turn a friction bearing (Fig. 3A.1). Calculate the required torque in lbs- ft and power consumption in horsepower to turn the shaft in the friction bearing shown in the figure. The length of the bearing surface on the shaft is 2 in, and the shaft is rotating at 200 rpm. The viscosity of the lubricant is 200 cp, and its density is 50lbm/ft3. Neglect the effect of eccentricity.

Friction loss in bearings) each of two screws on a large motor-ship is driven by a 4000-hp engine. The shaft that connects the motor and the screw is 16 in. in diameter and rests in a series of sleeve bearings that give a 0.005 in clearance. The shaft rotates at 50 rpm, the lubricant has a viscosity of 5000 cp, and there are 20 bearings, each I ft in length. Estimate the fraction of engine power expended in rotating the shafts in their bearings. Neglect the effect of the eccentricity.

Effect of altitude on air pressure, when standing at the mouth of the Ontonagon River on the south shore of Lake Superior (602 ft above mean sea level), your portable barometer indicates a pressure of 750 mm Hg. Use the equation of motion to estimate the barometric pressure at the top of Government Peak (2023 ft above mean sea level) in the nearby Porcupine Mountains. Assume that the temperature at lake level is 70oF and that the temperature decreases with increasing altitude at a steady rate of 3°F per 1000 feet. The gravitational acceleration at the south shore of Lake Superior is about 32.19 ft/s, and its variation with altitude may be neglected in this problem.

Viscosity determination with a rotating-cylinder viscometer, it is desired to measure the viscosities of sucrose solutions of about 60% concentration by weight at about 20°C with a rotating-cylinder viscometer such as that shown in Fig. 3.5-1. This instrument has an inner cylinder 4.000 cm in diameter surrounded by a rotating concentric cylinder 4.500 cm in diameter. The length L is 4.00 cm. The viscosity of a 60% sucrose solution at 20°C is about 57 cp, and its density is about 1.29 g/cm^{3}. On the basis of past experience it seems possible that end effects will be important, and it is therefore decided to calibrate the viscometer by measurements on some known solutions of approximately the same viscosity as those of the unknown sucrose solutions. Determine a reasonable value for the applied torque to be used in calibration if the torque measurements are reliable within 100 dyne/cm and the angular velocity can be measured within 0.5%. What will be the resultant angular velocity?

Fabrication of a parabolic mirror, it is proposed to make a backing for a parabolic mirror, by rotating a pan of slow-hardening plastic resin at constant speed until it hardens. Calculate the rotational speed required to produce a mirror of focal length f = 100 cm. The focal length is one-half the radius of curvature at the axis, which in turn is given by

Scale-up of an agitated tank, experiments with a small-scale agitated tank are to be used to design a geometrically similar installation with linear dimensions 10 times as large. The fluid in the large tank will be a heavy oil with μ = 13.5 cp and p = 0.9 g/cm^{3}. The large tank is to have an impeller speed of 120 rpm.

(a) Determine the impeller speed for the small-scale model, in accordance with the criteria for scale-up given in Example 3.7-2.

(b) Determine the operating temperature for the model if water is to be used as the stirred fluid.

Air entrainment in a draining tank (Figure 3A.7) a molasses storage tank 60 ft in diameter is to be built with a draw-off line 1 ft in diameter, 4 ft from the sidewall of the tank and extending vertically upward I ft from the tank bottom. It is known from experience that, as molasses is withdrawn from the tank, a vortex will form, and, as the liquid level drops, this vortex will ultimately reach the draw-off pipe, allowing air to be sucked into the molasses. This is to be avoided. It is proposed to predict the minimum liquid level at which this entrainment can be avoided, at a draw-off rate of 800 gal/min, by a model study using a smaller tank. For convenience, water at 68^{o}F is to be used for the fluid in the model study.

Determine the proper tank dimensions and operating conditions for the model if the density of the molasses is 1.286 g/cm^{3} and its viscosity is 56.7 cp. It may be assumed that, in either the full-size tank or the model, the vortex shape is dependent only on the amount of the liquid in the tank and the draw-off rate; that is, the vortex establishes it-self very rapidly.

Flow between coaxial cylinders and concentric spheres.

(a) The space between two coaxial cylinders is filled with an incompressible fluid at constant temperature. The radii of the inner and outer wetted surfaces are KR and R, respectively. The angular velocities of rotation of the inner and outer cylinders are Ω, and Ω_{0}. Determine the velocity distribution in the fluid and the torques on the two cylinders needed to maintain the motion.

(b) Repeat part (a) for two concentric spheres.

Laminar flow in a triangular duct (Figure 3B.2)^{2} one type of compact heat exchanger is shown in Figure 3B.2 (a). In order to analyze the performance of such an apparatus, it is necessary to understand the flow in a duct whose cross section is an equilateral triangle. This is done most easily by installing a coordinate system as shown in Fig. 3B.2 (b).

(a) Verify that the velocity distribution for the laminar flow of a Newtonian fluid in a duct of this type is given by Fig. 3B.2. (a) Compact heat-exchanger element, showing channels of a triangular cross section; (b) coordinate system for an equilateral-triangular duct

(b) From Eq. 3B.2-1 find the average velocity, maximum velocity, and mass flow rate.

Laminar flow in a square duct

(a) A straight duct extends in the z direction for a length L and has a square cross section, bordered by the lines x = ± B and y = ± B. A colleague has told you that the velocity distribution is given by since this colleague has occasionally given you wrong advice in the past, you feel obliged to check the result. Does it satisfy the relevant boundary conditions and the relevant differential equation?

(b) According to the review article by Berker,^{3} the mass rate of flow in a square duct is given by Compare the coefficient in this expression with the coefficient that one obtains from Eq. 3B.3-1.

Creeping flow between two concentric spheres (Fig. 3B.4). A very viscous Newtonian fluid flows in the space between two concentric spheres, as shown in the figure. It is desired to find the rate of flow in the system as a function of the imposed pressure difference. Neglect end effects and postulate that v_{θ} depends only on r and θ with the other velocity components zero.

(a) Using the equation of continuity, show that v_{θ} sin θ = *u*(r), where *u*(r) is a function of r to be determined.

(b) Write the θ-component of the equation of motion for this system, assuming the flow to be slow enough that the [v ∙ ∆v] term is negligible. Show that this gives

(c) Separate this into two equations where B is the separation constant, and solve the two equations to get where P_{1} and P_{2} are the values of the modified pressure at θ = ε and θ = π – ε, respectively.

(d) Use the results above to get the mass rate of flow

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