Governing equations in cylindrical coordinates
Velocity components are u, v, w in the r, , and z directions, respectively.
Continuity equation
v
1
1 v w
0
+u
+
+w
+ ( ru ) +
+
=
t
r r
z
r z
r r
Conservation of mass species
1 m j 1 2m j 2m j
m j

Convective Heat and Mass Transfer
Lecture Notes of Prof. A. Lavine
1 Conservation Equations
The study of convective heat and mass transfer is founded on four conservation laws: conservation
of mass, mass species, momentum, and energy. Convection implies t

2 Laminar Duct Flows
2.1 Fully Developed Poiseuille Flow
2.1.1 Fluid Flow
We next consider laminar flow in a circular pipe under fully developed conditions. Fully
developed means that the effect of the inlet is no longer felt. Near a pipe inlet, boundary

4 Laminar Boundary Layers
4.1 Boundary Layer Fluid Flow
Let us begin by considering the fluid flow. We restrict our attention to incompressible, constant
property, steady-state, two-dimensional flow of a Newtonian fluid over a solid flat plate, with
negli

6 Turbulence
6.1 Transition to Turbulence
Nothing in laminar solutions suggests that they are not valid for large Re, but in fact the laminar
flow breaks down into turbulent flow for Re greater than some critical value. Turbulent flow is
characterized by

6 Turbulence
6.1 Transition to Turbulence
Nothing in laminar solutions suggests that they are not valid for large Re, but in fact the laminar
flow breaks down into turbulent flow for Re greater than some critical value. Turbulent flow is
characterized by

2 Laminar Duct Flows
2.1 Fully Developed Velocity and Temperature
2.1.1 Fluid Flow
We next consider laminar flow in a circular pipe under fully developed conditions. Fully
developed means that the effect of the inlet is no longer felt. Near a pipe inlet,

231A Review
1. Conservation Equations
Continuity
D
r
v = 0
Dt
Incompressible means
D
r
= 0 v = 0
Dt
Momentum
r
rr r
Dv
= B
Dt
Newtonian fluid:
ij = p ij ij ,v
v v
r
ij ,v = i j v ij
x
j xi
r
rr r
Dv
= p B
Dt
If incompressible:
r
Dv
r r
= p v B
Dt

3 Nondimensionalization and Scaling of Equations
There are two ways to look at nondimensionalizing the differential equations. First, we can see it
simply as a way to write the equations in more compact form, in which the nondimensional
parameters that go

Del Operator and Indicial Notation
For indicial notation, we number the coordinates, the unit normal vectors, etc.:
x x1 y x2 z x3
i e1
j e2
k e3
A = Ax i + Ay j + Az k A1e1 + A2e2 + A3e3 = Ai ei
We also use the summation convention, that when an index is

5 Natural Convection
In natural convection, fluid motion is due to a body force rather than some external means like a
pump, fan, a moving boundary, etc. The body force could be gravitational or otherwise, but we
will consider only a gravitational body fo