ABSTRACT
There are several significant problems in chemical engineering that require a
fundamental understanding of differential equations in order to fully appre-
ciate the underlying transport phenomena. In this book, differential equation means an equation containing derivatives of an unknown function to be
determined [1]. For example, Fourier’s law [2-4] for the molecular transport
of heat in a fluid or a solid can be written as a first-order differential equation
qz A ¼ d(CPT)
dz (1:1)
for constant density r and heat capacity CP. In this equation, qz/A represents the heat flux (J/s m2), a the thermal diffusivity (m2/s), and rCPT the concentration of heat (J/m3), with the subscript z indicating that energy is transferred in the z-direction. The unknown function is the temperature T(z). A second example that is familiar to chemical engineers is Fick’s law [2-4] for the
molecular transport of mass in a fluid or a solid for constant total concentra-
tion in the fluid. This fundamental transport process can be written as
JAZ ¼ DAB dCA dz
(1:2)
where JAZ is the flux of species A (kmol/s m 2), DAB is the molecular
diffusivity (m2/s) of species A in B, and CA is the concentration of A (kmol/m3). In this case, the unknown function to be determined is CA(z). A third example is Newton’s law [2-4] of viscosity, written as follows for
constant density r:
tzx ¼ g d(nxr) dz
(1:3)
where tzx is the flux of x-directed momentum in the z-direction [(kg m/s)/ s m2], g is the kinematic viscosity (m/r) or momentum diffusivity. Transport or diffusion takes place in the z-direction and m is the viscosity (kg/m s). In this equation, the unknown function to be determined is the x-component of velocity nx(z).
