**ChBE 6100 Problem Set 1**

Due: Friday, Aug. 30, 2024 by 5 pm
 
 
**Problem 1.** Solve the equation  for the following 1-dimensional conditions:

(a)   Solid sphere of radius R, with boundary conditions of symmetry at r = 0 and convective flux at r = R
 
(b)  Solid cylinder of radius R, with symmetry at r = 0 and surface temperature maintained at TR at r = R.
 
(c)   Solid wall of thickness 2L, with one surface maintained at T1 and the other at T2.
 
**Problem 2.** (a)**** Solve the equation  for the boundary conditions CA = CAo at z = 0 and CA = 0 at z = L.
 
(b) Solve the equation  for the boundary conditions CB = 0 at both z = 0 and z = L.  This is a situation where species B is produced in a thin film by a homogeneous reaction but exists only in extreme dilution elsewhere.
 
**Problem 3.**  Solve the dimensionless temperature profile for a thin extended object (such as a fin) that is infinitely long.  Use the equation , where Pe is the Peclet number, Bi is the Biot number, q is dimensionless temperature, and z is dimensionless position, for the boundary conditions q(0) = 1 and q() = 0.  
 
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These problems are classic examples of heat and mass transfer problems, typically encountered in chemical engineering. Below are the steps and approaches to solve each problem.

### **Problem 1: Heat Conduction in Solids**

We are dealing with the heat conduction equation in one dimension, which can be expressed as:

\[
\frac{d^2 T}{dr^2} + \frac{n}{r} \frac{dT}{dr} = 0
\]

where \(n\) depends on the geometry:
- \(n = 2\) for a solid sphere
- \(n = 1\) for a solid cylinder
- \(n = 0\) for a solid wall

**(a) Solid sphere:**

- **Boundary conditions**: 
  - Symmetry at \(r = 0\): \(\frac{dT}{dr} = 0\)
  - Convective flux at \(r = R\): \(-k \frac{dT}{dr} = h(T - T_\infty)\), where \(h\) is the heat transfer coefficient and \(T_\infty\) is the ambient temperature.

**Solution Approach:**

1. Solve the differential equation for \(T(r)\).
2. Apply the boundary conditions to determine the integration constants.

**(b) Solid cylinder:**

- **Boundary conditions**: 
  - Symmetry at \(r = 0\): \(\frac{dT}{dr} = 0\)
  - Surface temperature at \(r = R\): \(T(R) = T_R\).

**Solution Approach:**

1. Solve the differential equation for \(T(r)\).
2. Apply the boundary conditions to determine the constants.

**(c) Solid wall of thickness \(2L\):**

- **Boundary conditions**: 
  - \(T(-L) = T_1\) 
  - \(T(L) = T_2\).

**Solution Approach:**

1. Solve the differential equation for \(T(x)\).
2. Apply the boundary conditions to solve for the constants.

### **Problem 2: Diffusion with Reactions**

These problems deal with diffusion and chemical reactions in a one-dimensional domain.

**(a)**

- **Equation**: 
  \[
  D\frac{d^2 C_A}{dz^2} = 0
  \]
- **Boundary conditions**: 
  - \(C_A = C_{A0}\) at \(z = 0\)
  - \(C_A = 0\) at \(z = L\).

**Solution Approach:**

1. Solve the differential equation for \(C_A(z)\).
2. Apply the boundary conditions to determine the constants.

**(b)**

- **Equation**: 
  \[
  D\frac{d^2 C_B}{dz^2} + R_B = 0
  \]
  where \(R_B\) is the reaction term.
- **Boundary conditions**: 
  - \(C_B = 0\) at \(z = 0\)
  - \(C_B = 0\) at \(z = L\).

**Solution Approach:**

1. Solve the differential equation for \(C_B(z)\), considering the reaction term \(R_B\).
2. Apply the boundary conditions to determine the constants.

### **Problem 3: Heat Conduction in a Fin**

**Given equation:**

\[
\frac{d^2 \theta}{dz^2} - \text{Pe} \frac{d\theta}{dz} - \text{Bi} \theta = 0
\]

where \(\theta\) is the dimensionless temperature, \(\text{Pe}\) is the Peclet number, \(\text{Bi}\) is the Biot number, and \(z\) is the dimensionless position.

- **Boundary conditions**: 
  - \(\theta(0) = 1\) 
  - \(\theta(\infty) = 0\).

**Solution Approach:**

1. Solve the differential equation for \(\theta(z)\) using an appropriate technique, such as assuming a solution of the form \(\theta(z) = e^{mz}\).
2. Apply the boundary conditions to solve for the constants.

### Do you want details or have any questions?

### Related Questions:

1. How does the geometry of a solid affect the temperature distribution in it?
2. What are the implications of different boundary conditions on the solution to the heat conduction equation?
3. How does the presence of a chemical reaction influence the concentration profile in a diffusion problem?
4. What is the significance of the Peclet number in convective heat transfer problems?
5. How can the Biot number affect the temperature distribution in a fin?

### Tip:

For solving differential equations with boundary conditions, always check if the problem can be simplified using symmetry or if any dimensionless groups (like Biot or Peclet numbers) can reduce the complexity of the equation.

Explore solutions to ChBE 6100 Problem Set 1 involving heat conduction in solids, diffusion with chemical reactions, and boundary value problems. Learn how to apply mathematical models to solve complex engineering scenarios.

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