This paper is intended to promote two (2) concepts, advection, and heat flux. Advection is the transport of thermal energy via a moving (working) fluid and heat flux is the heat flow divided by the heat transfer surface area.

1. MathCAD - Advection-convection Heat Transfer.xmcd
Thermal Balance on a Fluid Flow Conduit
by Julio C. Banks, PE
Reference
"Heat Transfer" 10th Ed. J. P. Holman
IISBN 978–960–'9607–96352936–963 or MHID 0–9607–96352936–962.
Pp. 21-23.
The intent of this paper is to show the modeling of a heat balance in a conduit with
convection heat transfer and the moving of the transferred energy via advection,
Note: The author has found it to be practical to distinguish Q, and q as "Total heat
transfer flow" and "Heat transfer flux", respectively. This clarification is very important
in the modeling of convection with advection, Some authors fail to name the transfer
of thermal energy via a moving (working) fluid as qa wCp ΔTB=
The energy transfer expressed by Equation (1-8) is used for evaluating the convection
loss for flow over an external surface. Of equal importance is the convection gain or loss
resulting from a fluid flowing inside a channel or tube as shown in Figure 1-8. In this
case, the heated wall at Tw loses heat to the cooler fluid, which consequently rises in
temperature as it flows through the conduit from inlet conditions at Ti to exit conditions
at Te. Using the symbol i to designate enthalpy (to avoid confusion with h, the
convection coefficient), the energy balance on the fluid is
Figure 1-8 Convection and advection in a channel of length L
Qw w ie ii =
Where w is the is the fluid mass flow rate. For many single-phase liquids and gases
operating over reasonable temperature Δi CpΔTB= ranges and e have
Qw w Cp TBe TBi = TBe TBi 1( )
Qh Ah Tw TB = Tw TB 2( )
Equation 2 uses the mean temperature of the wall and that of the bulk fluid.
Julio C. Banks, PE Sell-A-Vision@Outlook.com page 1 of 2

2. MathCAD - Advection-convection Heat Transfer.xmcd
Equating A heat balance is obtained by equation Eq. 1 to Eq. 2 as follows
Qw Qh=
w Cp TBe TBi  Ah Tw TB = 1 8a( )
Introduce the concept of "Heat flux", q
Q
A
= , into Eq. 1-8a by dividing it by the heat
transfer surface area, A.
w
A






Cp TBe TBi  h Tw TB = 1 8b( )
Let G
w
A
= 1 8c( )
GCp TBe TBi  h Tw TB = 1 8d( )
We must be careful to distinguish between the surface area for convection that is
employed in convection Equation (1-8) and the cross-sectional area that is used
to calculate the flow rate from
w ρuAc= 3( )
For a circular conduit, the fluid flow area and the heat transfer surface area are
Ac
π
4
d
2
= 4( )
A πdL= 5( )
Julio C. Banks, PE Sell-A-Vision@Outlook.com page 2 of 2