EXERCISES: 3
3
1. A tank contains 100 ft of fresh water; 2 ft of brine, having a concentration of 1 3 pcf (1 lb/ft ) of salt, is run into the tank per minute, and the mixture, kept uniform by 3 mixing, runs out at the rate of 1 ft /min. What will be the exit brine concentration co ncentration 3 when the tank contains 150 ft of brine. 3
2. Three tanks of 25 m capacity are each arranged so that when water is fed into the first an equal quantity of solution overflows from the first to the second tank, likewise from the second to the third, and from the third to some point out of the system. Agitators keep the contents of each tank uniform in concentration. To start, 3 let each of the tank be b e full of a salt solution of concentration 100 10 0 kg/m . Run water 3 into the first tank at 0.2 m /min , and let the overflows functions as described abo ve. Calculate the time required to reduce the salt concentration in the first tank to 10 3 kg/m . Calculate the concentrations in the other two tanks at this time. 3. The consecutive, second order, irreversible irreversible reaction are carried out in a batch reactor: k X A S k X S Y 1
2
One mole of A and two moles of S are initially added. Find the mole-fraction of X remaining in solution after half the A is consumed. Take k 2/k 1=2. 3
4. A tank tank 30 m in volume contains CO2 at pressure of 1000 kPa and temperature of 310 K. Suddenly, there is a small hole (leakage) in the tank. Gas flow rate through the hole at that time is 0.2 kgmole/hr. Then, the gas flow rate through the hole can be obtained by the following equation, F
k
P P atm kgmoles/hr. Find the
pressure in the tank 15
minutes after the leakage occurs. 5. Hemispherical tank of 1-meter 1-meter diameter is initially full of volatile liquid. Vaporization rate of the liquid is proportional to liquid surface area. From the observation, it is known that the time required to decrease the liquid surface level height of 5 cm is 30 minutes. Derive an equation relating liquid volume in the tank and time. 6. The reversible set of reactions represented by k 1
k 3
A B C k 2 k 4
is carried out in a batch reactor under unde r conditions of constant volume and temperature. Only one mole of A is present p resent initially, and any time t the moles are NA,NB,NC. The net rate of disappearance of A is given by dN A k 1 N A k 2 N B dt and for B, it is dN B (k 2 k 3 ) N B k 1 N A k 4 N C dt and for all times, the stoichiometry must be obeyed
NA+NB+NC=1 a) Show that the behavior of NA(t) is described by the second order ODE 2 d N A dN (k 1 k 2 k 3 k 4 ) A (k 1k 3 k 2 k 4 k 1k 4 ) N A k 2 k 4 2 dt dt b) One initial condition for the second order equation in part (a) is NA(0)=1;what is the second necessary initial condition? c) Find the complete solution for NA(t), using the condition in part (b) to evaluate the arbitrary constants of integration. 7. Solid, stubby, cylindrical metal rods (length-to-diameter ratio =3) are used as heat o promoters on the exterior of a hot surface with surface temperature of 700 C. The o ambient air flowing around the rod-promoters has a temperature of 30 C. The metal conductivity(k) takes a value of 0.247 cal/(sec.cm.K). The heat transfer 2 o coefficient(h) around the surface of the promoter is constant at 3.6 Kcal/(m .hr. C). a) Anayze a single rod of 4-mm diameter and show that the steady-state differential balace yields the following differential equation 2 d T 2h T T A 0; 2 R diameter 2 dx Rk for the case when metal temperature change mainly in the x direction( x is directed outword from the hot surface, and rod radius is R). b) Find the characteristic roots for the ODE in part (a). What are the physical units of these roots? c) Find the solution of part (a) using the boundary conditions: T T H , x 0 (hot surface ) dT
hT T A , x L (exposed flat tip) dx and show that the temperature profile is represented by L 2 tanh 2 Bi Bi T T A x L x L B D cosh2 sinh 2 Bi i L T H T A L D L D 2 B tanh 2 Bi i D k
where Bi=hD/k (Biot number, dimensionless; the ratio of film to metal transfer rate). d) Use the definition of total heat flow and find the effectiveness factor for the present promoter Q Qma x where Q is the actual heat flow from the promoter to the surrounding air and o Qmax is the heat flow assuming uniform promoter temperature of 700 C, and Show that the general expression for is
L 1 tanh 2 B B i i 1 1 D 2 L 1 1 L 2 B 1 B tanh 2 Bi i i D 4 2 D e) For small arguments, 2( L / D) Bi 1, show that the effectiveness factor becomes approximately 1 L 1 . Bi D
f) Compute for the present promoter. 8. In an experimental study of the saponification of methyl acetate by sodium hydroxide, it is found that 25% of the ester is converted to alcohol in 12 min when the initial concentration of both ester and c austic are 0.01 m. What coversion of ester would be obtained in 1 hour if the initial ester concentration were 0.025 m and the initial caustic concentration 0.015 m? 9*. The dried gas from an ammonia oxidation catalyst chamber contains 9 % NO, 9 % o oxygen, and 82% nitrogen (by volume). This gas is passed at 25 C and 1 atm into a vertical wetted-wall absorption column, the wallof which are wet with a dilute solution of sodium hydroxyde. The tower is 200 cm tall, 5 cm ID, and the inlet gas velocity is 97.1 cm/sec. What is the percentage recovery of nitrogen oxides in the column?. For purpose of calculation, the following assumtions may be made: i)The concentrations at any point will be based on total moles taken as constant and equal to the arithmatic mean of initial value and the final value corresponding to complete absorption. ii)The NO2 concentration will be so low that N2O4 formation may be neglected. iii)The NO is not absorbed. iv)The NO2 is absorbed as such, with gas film controlling and no partial pressure of NO2 over the solution. . -7 2 The gas-side mass transfer coefficient k G is 3.1 10 g moles/(sec.cm .mmHg). The NO oxidation reaction is homogeneous and third order. The rate equation is
k NO2 O2 and
r NO2 .
12
6
o
the value of the reaction rate constant at 25 C is k=
2
1.77 10 cm (g mole) /min. 10*. A simple adiabatic converter for the oxidation of SO2 to SO3 is to operate upon raw o gas entering at 400 C and 1.7 fps and containing 0.6% SO3, 10.1% SO2, 10.0% O2, and 79.3% N2. Using the following data, estimate the thickness of catalyst mass necessary to convert 57% of the entering SO2 to SO3. Pressure = 1 atm. The net rate of SO2 oxidation may be calculated from the equation r x 2 r SO2 k 0.2 ln e r y
where, r SO2 x y t
moles SO2 reacted per second per 100 moles entering gas. = moles SO2 at time t per 100 moles entering gas = moles SO3 at time t per 100 moles entering gas = time of contact,second ( based on catalyst bulk volume and superficial gas velocity = temperature of gas after time of cntact t = molal ratio of SO3 to SO2 at equilibrium at temperature T
T r e
k
= molal ratio of SO3 to SO2 at time t = reaction rate constant at temperature T
K
= equilibrium constant = pSO3 / pSO2
r
o
T( C) K
400 6.6
425 13.6
450 5.2
pO2 ( p in atm)
475 2.7
500 2.0
RT ln K=22,600-21.36 T T in K 1 SO 2 O 2 SO3 H 22,500 gcal/ gmole 2 o MCp for N2 = 6.8 gcal/(gmole. C) o Mcp for O2 = 6.8 gcal/(gmole. C) o MCp for SO2 = 11.0 gcal/(gmole. C) o MCp for SO3 = 14.4 gcal/(gmole. C) 11*. Your task as a design engineer in a chemical company is to model a fixed bed reactor packed with the company proprietary catalyst of spherical shape. The catalyst is specific for the removal of a toxic gas at very low concentration in air, and the information provided from the catalytic division is that the reaction is first order with respect to the toxic gas concentration. The reaction rate has units of moles of toxic gas removed per mass of catalyst per time. The other informations you get are :the catalyst density p = 2300 kg/m3, bed porosity =0.4, bed length L=2 m, supervisial gas velocity u0=15 m/sec, and entrance toxic gas concentration, C0=0.0001 kgmole/m3. Your first attempt is to model the reactor so that you can develop some intuition about the system. a)Develop a differential equation to model the reactor by assuming the steady state condition, isotermal condition, and neglecting diffusion inside the catal yst. Write down the appropriate boundary equations and solve the equation using the k value 3 of 0.5 m /(kg.sec) and estimate the exit concentration of the toxic gas. b)Estimate the exit concentration of the toxic gas by assumung steady state and isotherma condition and neglecting axial diffusion. Use the same value of k as a) 12* When gas is injected into a column of water, a liquid circulation pattern develops. 3 Thus, upflow at a rate Qu (m /s) rises in the central core and down flow occurs at a rate Qd in the annulus. If liquid of composition C 0 is also injected at the column base at a rate Q0 with out flow at the same rate, then Qu=Qd+Q0 (if density is constant). * 3 a) The injected gas contains a soluble component (with solubility C moles/m ) so
that mass transfer occurs by way of a constant volumetric mass transfer coefficient denoted as k ca. There is also an exchange of solute between upflowing and downflowing liquidat a rate per unit height equal to K E(Cu-Cd). If the flow areas for upflow and downflow are equal (A), perform a material balance and show that dC u k c aA(C* C u ) K E (C u C d ) Qu dz dC Q d d k c aA(C* C d ) K E (C u C d ) dz where z is distance from column base. b) Define new variables to simplify matters, u C u C* , d C d C*
z(k c aA K E ) / Q 0
, qu
Qu / Q0 ,
qd
Qd / Q0
and show that the coupled relations are d u u d qu d
q d
d d d
d u
where K E /( K E k c aA) (c) Obtain the general solution of Coupled ODE in (b) (d) Apply the saturation condition d , u 0 as and a material balance at the entrance Qd C d (0) Q0 C 0 Qu C u (0) to evaluate the constants of integration and thereby obtain relations to predict composition profiles along the axis. (e) Deduce asymptotic solutions for the case when Qd 0 and when K E Note:
( 1); this corresponds to the plug-flow,nonrecirculating result.
Problem with superscript * is assigned to a group of students
