Partial differential equations (PDEs) arise naturally in chemical engineering when modelling phenomena involving multiple independent variables, such as concentration, temperature, and time. These equations describe how physical quantities vary in space and time, which is essential for understanding processes like heat transfer, mass diffusion, and fluid flow in reactors and pipelines. Mastery of PDEs enables chemical engineers in Kenya to design efficient systems and optimize industrial operations by predicting system behaviour under varying conditions.

13.1 Definition of Partial Differential Equations and Formation of First Order PDEs

Partial differential equations involve functions of two or more independent variables and their partial derivatives. Unlike ordinary differential equations that depend on a single variable, PDEs capture spatial and temporal variations simultaneously. A general first order PDE for a function \(u(x,y)\) can be written as

$$ F(x,y,u,\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}) = 0. $$

13.1.1 Concept of Partial Differential Equations

A partial differential equation relates the partial derivatives of an unknown function with respect to multiple independent variables. For chemical engineering, PDEs model spatial variations in temperature, concentration, and velocity fields. The order of the PDE is determined by the highest order derivative involved.

$$ \text{Example: } \frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2} $$

is the heat diffusion equation, a second order PDE in space and first order in time.

Worked Examples

Example 1: In a tubular reactor, temperature \(T\) varies with axial position \(x\) and time \(t\). Write the PDE representing heat conduction with thermal diffusivity \(\alpha = 1.2 \times 10^{-5} \text{ m}^2/\text{s}\).

Given: \(\alpha = 1.2 \times 10^{-5} \text{ m}^2/\text{s}\)

$$ \frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} $$

Example 2: Concentration \(C(x,y)\) varies in two spatial directions. Express the PDE of steady-state diffusion without sources.

$$ 0 = D (\frac{\partial^2 C}{\partial x^2} + \frac{\partial^2 C}{\partial y^2}) $$

Example 3: A fluid velocity \(u(x,y,t)\) depends on space and time. Write a first order PDE involving time and spatial derivatives.

$$ \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = -\frac{1}{\rho} \frac{\partial p}{\partial x} $$

13.1.2 Formation of First Order Partial Differential Equations

First order PDEs typically arise from conservation laws or constitutive relations by expressing the rate of change of a quantity with respect to variables. The formation involves identifying dependent and independent variables and their partial derivatives, leading to an equation of the form

$$ a(x,y,u) \frac{\partial u}{\partial x} + b(x,y,u) \frac{\partial u}{\partial y} = c(x,y,u). $$

Worked Examples

Example 1: Derive the first order PDE for steady-state concentration \(C(x,y)\) with velocity components \(u,v\) and no reaction.

Given: Velocity components \(u,v\)

$$ u \frac{\partial C}{\partial x} + v \frac{\partial C}{\partial y} = 0 $$

Example 2: For temperature \(T(x,t)\) in a rod with velocity \(v\) and no heat generation, form the PDE.

$$ \frac{\partial T}{\partial t} + v \frac{\partial T}{\partial x} = 0 $$

Example 3: For a chemical species concentration \(C(x,y)\) affected by diffusion coefficient \(D\) and reaction rate \(r\), write the first order PDE.

If reaction rate is zero order in \(C\),

$$ u \frac{\partial C}{\partial x} + v \frac{\partial C}{\partial y} = r $$

13.1.3 Linear and Quasi-Linear First Order PDEs

A first order PDE is linear if the dependent variable and its derivatives appear to the first power and are not multiplied together. If coefficients depend on the unknown function but the derivatives appear linearly, it is quasi-linear.

General linear PDE:

$$ a(x,y) \frac{\partial u}{\partial x} + b(x,y) \frac{\partial u}{\partial y} + c(x,y) u = d(x,y). $$

General quasi-linear PDE:

$$ a(x,y,u) \frac{\partial u}{\partial x} + b(x,y,u) \frac{\partial u}{\partial y} = c(x,y,u). $$

Worked Examples

Example 1: Identify whether the PDE \( \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = x \) is linear or quasi-linear.

Given: Coefficients independent of \(u\)

Equation is linear.

Example 2: Determine the type of PDE for \( u \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = 0 \).

Given: Coefficient depends on \(u\)

Equation is quasi-linear.

Example 3: Classify the PDE \( \sin x \frac{\partial u}{\partial x} + u \frac{\partial u}{\partial y} = e^y \).

Given: Coefficient depends on \(u\)

Equation is quasi-linear.

Chapter Summary

This chapter introduced the concept of partial differential equations (PDEs) and explained how they are formed, particularly focusing on first order PDEs. It detailed methods for solving first order PDEs, emphasizing the use of characteristics and the method of characteristics to find solutions. The chapter also covered linear and quasi-linear first order PDEs, illustrating their specific solution approaches. Moving to more complex equations, the chapter explored solution techniques for second order partial differential equations. Among these, the application of Fourier series was highlighted as a powerful tool for solving boundary value problems. Throughout, the chapter integrated theory with practical problem-solving methods to build a comprehensive understanding of PDEs in engineering contexts.