Chemical Engineering Mathematics
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Presentation Contents
 Partial Differentiation Equation in long non-uniform rod
 Partial Differentiation Equation Diffusion equation in one dimension
 Newton's Law of Cooling and Applications
 Applications of Separation of Variables
 Growth and Decay
 Laplace transform and Applications
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Partial Differential Equation
A partial differential equation is a differential equation that contains unknown
multivariable functions and their partial derivatives.
A relatively simple PDE (partial differential equation) is
𝜕𝑢
𝑥, 𝑦 = 0
𝜕𝑥
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Partial Differentiation Equation in long non-uniform
rod
In a metal rod with non-uniform temperature, heat (thermal energy) is transferred from regions
of higher temperature to regions of lower temperature. Three physical principles are used here.
1. Heat (or thermal) energy of a body with uniform properties:
𝐻𝑒𝑎𝑡 𝐸𝑛𝑒𝑟𝑔𝑦 = 𝑐𝑚𝑢
where m is the body mass, u is the temperature, c is the specific heat, units c is the energy
required to raise a unit mass of the substance 1 unit in temperature
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Contd….
2. Fourier’s law of heat transfer
rate of heat transfer proportional to negative temperature gradient,
𝑅𝑎𝑡𝑒 𝑜𝑓 ℎ𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟
𝜕𝑦
= −𝑘
𝐴𝑟𝑒𝑎
𝜕𝑥
where K0 is the thermal conductivity,. In other words, heat is transferred from areas of
high temp to low temp.
3. Conservation of energy
Consider a uniform rod of length l with non-uniform temperature lying on the x-axis
from x = 0 to x = l. By uniform rod, we mean the density ρ, specific heat c, thermal
conductivity K0, cross-sectional area A are ALL constant. Assume the sides of the rod
are insulated and only the ends may be exposed
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Contd….
of the rod are insulated and only the ends may be exposed. Also assume there is no
heat source within the rod. Consider an arbitrary thin slice of the rod of width Δx
between x and x+ Δx. The slice is so thin that the temperature throughout the slice
is u (x,t).
Fig 1 taken from
https://en.wikiversity.org/wiki
/Nonlinear_finite_elements/A
xial_bar_strong_form
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Diffusion equation in one dimension
The one-dimensional diffusion equation, for a function P(x,t), is
𝜕𝑃(𝑋,𝑡)
𝜕2 𝑃(𝑋,𝑡)
=D
𝜕𝑡
𝜕𝑋 2
where D is the diffusion constant. The larger the value of D the faster is diffusion. D
has units of length squared over time
P could be, for example, the concentration of say sugar molecules in water. Thermal
energy also diffuses, so P could also be temperature.
We know the solution will be a function of two variables: x and t, P(x,t). We assume
it has a specific form, namely that it is the product of a function only of x, X(x),
times a function only of t, T(t).
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Ordinary Differential Equation (ODE)
An ordinary differential equation (ODE) is a differential equation containing one or
more functions of one independent variable and the derivatives of those functions
Application
1. Cooling/Warming Law
2. Population Growth and Decay
3. Radio-Active Decay and Carbon Dating
4. Mixture of Two Salt Solutions
5. Series Circuits
6. Draining a tank
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Newton's Law of Cooling
Newton's Law of Cooling states that the rate of change of the temperature of an
object is proportional to the difference between its own temperature and the ambient
temperature.
𝑑𝑇
= −𝑘(𝑇 − 𝑇𝑠 )
𝑑𝑡
𝑇𝑠 is the ambient temperature and
K is the experimental constant
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Applications Newton's Law of Cooling
1. To predict how long it takes for a hot object to cool down at a certain temperature.
2. To find the temperature of a soda placed in a refrigerator by a certain amount of time.
3. It helps to indicate the time of death given the probable body temperature at the time of death
and current body temperature.
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Applications of Separation of Variables
the rate at which a chemical reaction occurs is determined by a differential
equation called a rate equation. For a reaction with a single reactant, the
concentration C of the reactant obeys a rate equation of the form
𝑑𝑐
= −𝑟𝑐 𝑛
𝑑𝑡
where r is a constant called the rate constant, and n is a constant called the order of
the reaction
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Growth and Decay
In many natural phenomena, quantities grow or decay at a rate proportional to their size
In general, if y(t) is the value of a quantity y at time t and if the rate of change of y with
respect to t is proportional to its size y(t) at any time, then
𝑑𝑦
= 𝑘𝑦
𝑑𝑡
where k is a constant, and is sometimes called the law of natural growth (if k > 0) or
the law of natural decay (if k < 0). Thus, the law of Exponential Growth and Decay can
be written as
𝑦 = 𝑐𝑒 𝑘𝑡
c is the initial value and can be found from the initial condition y(t0) = y
k is the constant of proportionality
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Laplace Transform
Definition of Laplace Transform
Suppose that, f is a real or complex valued function of the variable t > 0 and s is a real or
complex parameter. We define the Laplace transform of f as
−∞ −𝑠𝑡
F(s) = L{f(t)} = 0 𝑒 𝑓 𝑡 𝑑𝑡
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Applications Laplace Transform
• To solve the problem related to communication and network analysis.
• To make a equation in simple form from hard equation like vibration of spring.
• To solve Mixing Problem Involving Two Tanks
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