CHEN-7110/7116 Homework

 

Assignment No. 1

 

Due: January 11, Fri

 

1. A tank contains 100 liters of fresh water. Two liters of salt solution, having a concentration of 1 g/L, is fed into the tank per minute, and the mixture, kept uniform by mixing, runs out at the rate of 1 L/min. What will be the exit concentration when the tank contains 150 liters of brine?

 

2. The inner and outer surfaces of a hollow sphere are maintained at the respective temperatures of T0 and T1. The inner and outer radii of the spherical shell are R0 and R1. What is the temperature distribution through the shell? Find the amount of heat lost from the sphere through the shell?

 

Assignment No. 2

 

Due: January 14, Mon

 

 

1.     Water at temperature T0 is flowing into a thin circular pipe. The temperature at the inner surface of the pipe is at Tw. The flow pattern is plug flow. The heat transfer coefficient at the wall is h. It is varying as a function of the distance from the entrance, x, such that   .

 

Determine the temperature of water as function of x. Assume temperature is uniform in radial direction. Water has properties of ρ, k, Cp representing density, conductivity, and heat capacity. The volumetric flow rate is v. Radius of the pipe is R.

 

2.      (a) A container is maintained at a constant temperature of 800°F and is fed with a pure gas A at a steady rate of 1 lb-mole/mm. the product gas stream is withdrawn from the container at the rate necessary to keep the total pressure constant at a value of 3 atm. The container contents are vigorously agitated, and the gas mixture is always well mixed. The following irreversible second-order gas-phase reaction occurs in the container:

 

2A ® B

 

At temperature of 800°F, the reaction-rate constant for the reaction has the numerical value of 1,000 ft3/ (lb-mole) (min.). Both A and B are ideal gases. If, under steady state conditions, the ‘product stream is to contain 33 and 1/3 mole % B, how large should be the volume of the reaction container?

 

(b) After the steady state of (a) has been attained the volume on the exit pipe of the isothermal vessel is abruptly closed. The feed rate is controlled so that the total tank pressure is maintained at 3 atm. How long will it take (after the instant of closing the valve) for the tank contents to be 90 mole % B?

 

 

Assignment No. 3

 

Due: January 16, Wed

 

The following elementary reactions are carried out in a batch reactor.

    1

A+S ® X

         2

X+S ® Y

 

One mole of A and two moles of S are initially added. Find the mole fraction of X remaining after half of A is consumed. Take k2/k1 = 2.

 

Hint: Look for an ODE involving dCx /dCA.

 

 

Assignment No. 4

 

Due: January 18, Fri

 

Find a complete solution:

 

1.                                                   (Method of undetermined coefficients)

 

2.                                                (Method of variation of parameters)

 

 

Assignment No. 5

 

            Due: January 23, Wed

 

1. Find a complete solution.

                       

                                   

                                     

                                      y(1) = 1, y’(1) = 0

 

                    Click on for Problems 2 and 3

           

 

Assignment No. 6

 

            Due: January 25, Fri

 

Find the infinite series solution.

 

                   1.      

 

2.      

 

 

Assignment No. 7

 

            Due: January 28, Mon

 

Find the infinite series solution for

 

                            

 

 

Assignment No. 8

 

            Due: January 30, Wed

 

Find the infinite series solution for

 

                                      

 

 

Assignment No. 9

 

Due: February 1, Fri

 

Prove:

 

1.  .  Note that

 

2.     J-n(x) = (-1)nJn(x), 

 

Hint: Let j+n = m.

Note: n is an integer, a fixed number, not a summation index,

whereas j and m are summation indexes.

Gamma function at negative integers = .

 

 

Assignment No. 10

 

Due: February 4, Mon

         

1. Prove that      

 

2. Express  in terms of J0(x) and J1(x).

 

3.     Show that 

                  

   

Assignment No. 11

 

Due: February 7, Wed

 

1. Find Bessel function solution.

         

(a)  

 

(b) 

 

(c) 

           

 

2.     Going back to Assignment 5 - Problem 3, complete part (c).

 

Note:

               

 

 

 

Assignment No. 12

 

Due: February 11, Mon

 

 

1.     Express the PDE of   using new independent variables of u and v chosen such that:

u = x – y + t

v = x – 2t

 

2.     Find the solution for

 

 

t =0, u = e2x

    

     Hint: u ® η,  η = x + a·t    (a is to be determined)

 

 

Assignment No. 13

 

Due: February 20, Wed

 

Click on Problem Statement

 

 

 

Assignment No. 14

 

Due: February 22, Friday

 

Derive the energy equation in cylindrical coordinate.

 

Assignment No. 15

 

Due: February 25, Monday

 

Problem Statement

 

(Ignore Question e.)

 

 

Assignment No. 16

 

Due: February 27, Wed

 

Solve by (a) Polymath, (b) LT method or other appropriate method.

 

Problem Statement

 

 

Assignment No. 17

 

Due: March 3, Mon

 

Solve by LT method. 

 

(a)

         

           

           

 

 

(b) 

         

           

            

 

         

Assignment No. 18

 

Due: March 5, Wed

 

Problem 1

 

Problem 2

 

 

Assignment No. 19

 

Due: March 7, Fri

 

Problem Statement

 

 

Assignment No. 20

 

Due: March 10, Mon

 

Solve by separation of variables.

 

(a)

         

           

           

 

 

(b) 

         

           

           

 

         

Assignment No. 21

 

Due: March 12, Wed

 

Find a complete solution.

 

   ,     

 

         

         

 

 

Assignment No. 22

 

Due: March 14, Fri

 

1.     A long cylinder with radius R is resting at temperature T0. The temperature at the outer surface of the cylinder was suddenly raised to T1. Derive the PDE for this situation considering that T = f (r, t), and show proper initial and boundary conditions. Identify the Eigen function and Eigen value. Complete solution is not necessary.

 

2.     Identify Eigen function and Eigen value.

 

   ,     

 

         

         

 

 

Assignment No. 23

 

Due: March 24, Mon

 

1. A spherical object with radius R is resting at temperature T0. The temperature at the outer surface of the sphere was suddenly raised to T1. Derive the PDE for this situation considering that T = f (r, t). Show proper initial and boundary conditions. Identify the Eigen function and Eigen value. Complete solution is not necessary.

 

Note:

 

2. Identify Eigen value and Eigen function.

 

    ,     

 

         

         

 

Assignment No. 24

 

Due: March 28, Mon

 

1.     Prove that sin(πx) is orthogonal to sin(2πx) with respect to weighting function r(x) =1 over the interval of (0, 1).

 

2.     Prove that a 2nd order ODE of

 

                  

 

                   may be transformed into the S-L form of

 

                  

 

with the following substitutions.

 

 

3.     Identify the weighting function for

 

(a)   x2 y” + x y’ + (ln2x2-1) y = 0

 

(b)  y” + 2y’ + ln2y = 0

 

 

Assignment No. 25

 

Due: March 30, Wed

 

1.     Express f(x) = x by an infinite series of orthogonal functions resulting from the following S-L problem.

 

 

           

           

 

 

2.     Find a complete solution for Problem 1 of Assignment 22. Also determine the average temperature of the cylinder as a function of time.

 

Note:

 

 

3.     A catalytic reaction takes place in a porous catalyst. The pore of the catalyst is in a cylindrical form (with dimension of radius = R = 1, length of the pore = L = 1). A first order reaction is taking place at the inner surface of the catalyst, rA = - k CA. At the pore mouth (z =0), the concentration of A, CA = Cb. At the end of the pore (z = 1), the circular end surface is inert. Determine CA as a function of position (r, z) under steady-state. 

 

Assignment No. 26

 

Due: April 7, Monday

 

Problem 1

 

 

          Problem 2. A thin rod of length L is resting at temperature T0. At time = 0, the temperature at one end of the rod (x = 0) was raised to T1. The other end of the rod (x =L) is insulated. Heat is lost from the surface of the rod to air which is at T0. The rod is thin enough that the temperature is uniform in radial direction. Determine the temperature of the rod as function of position (x) and time.

 

 

Assignment No. 27

 

Due: April 9, Wed

 

Problem Statement

   

Additional questions in (a):

 

(1)   Derive the PDE

(2)   Put the PDE, IC, BCs into dimensionless form.

(3)   Find the complete solution.

 

Notes in part (b):

 

§           

 

§        Use the first term only in the infinite series. Do not rely on Fig. 10-10.

 

 

Assignment No. 28

 

Due: April 11, Fri

 

 

A very thin thermocouple probe is placed into the center of a spherically shaped metal alloy for the purpose of estimating the thermal diffusivity of this material. This metal ball is placed in an oven overnight and reaches an initial uniform temperature of T0. It is then quickly placed in very fast flowing cooling water which is at T1 = 20oC. Because of high heat transfer coefficient, h, the temperature of the surface remains the same as the water temperature. The thermocouple then records the temperature at the center-point.

 

On the basis of the following data, make best estimation of the thermal diffusivity.

 

Radius of the metal ball = R = 1 cm

Initial temperature = T0 = 30oC

 

At t = 6.05 seconds, the measured temperature at the center was 20.86oC.

 

(Answer: a = 0.0527 cm2/sec)

 

Notes:  Consult Assignment 23, Problem 1.  Hint:,

 

Assignment No. 29

 

            Due: April 16, Wed

 

Problem 1. A hollow sphere with inner radius a and outer radius b is resting at temperature T0. At time = 0, the outer surface is raised to T1, the inner surface remaining at T0. Determine the temperature of the hollow sphere as function of position (r) and time (t).

 

Problem 2.

 

 

Assignment No. 30

 

            Due: April 18, Fri

 

          Assignment 26-Problem 2 is reassigned.

 

 

Assignment No. 31

 

            Due: April 21, Mon

 

 

Problem 1.

 

          Using method of Frobenius, find the infinite series solution for,

 

              ,

 

          where n is positive integer.

 

Problem 2.

         

          A solid object has a shape of half-of-sphere. It is at a thermal steady-state such that the curved external surface (top section) is at T1 and the bottom flat surface is at T0. The temperature is therefore a function of position (r, q).

 

(a)   Give the governing PDE and BCs.

(b)  Put them into dimensionless form.

(c)  Separate the PDE into two ODEs: R(r), Z(q), and introduce a new variable x = cos (q), and express the q-side ODE and BCs in terms of x.

 

 

Assignment No. 32

 

            Due: April 23, Wed

 

          Find a complete solution for Problem 2 of Assignment No. 31.

         

          Calculate the temperature:

          where r = (1/2)R, q = (1/3)π, T1 = 100oC, and T0 = 200oC.

         

          Take the first three terms only in the infinite series solution.