Page 28 - Modelling in Transport Phenomena A Conceptual Approach

1.4. SIMPLIFICATION OF THE RATE EQUATION 9

1.4.1 Steady-State 'lkansport Without Generation
For this case Eq. (1.1-1) reduces to

Rate of input of cp = Rate of output of cp (1.41)

Equation (1.41) can also be expressed in terms of flux as

/L,. (Inlet flux of cp) dA = /Lo., (Outlet flux of cp) dA (1.42)

For constant inlet and outlet fluxes Eq. (1.42) reduces to
( Inlet flux ) ( Inlet ) = ( Outlet flux ) ( Outlet )

of cp area of cp area (1.43)

If the inlet and outlet areas are equal, then Eq. (1.43) becomes

Inlet flux of cp = Outlet flux of cp (1.44)
It is important to note that Eq. (1.44) is valid as long as the areas perpendicular
to the direction of flow at the inlet and outlet of the system are equal to each other.
The variation of the area in between does not affect this conclusion. Equation
(1.44) obviously is not valid for the transfer processes taking place in the radial
direction in cylindrical and spherical coordinate systems. In this case either Eq.
(1.42) or Eq. (1.43) should be used.

Example 1.5 Consider a solid cone of circular cross-section whose lateral surface
is well insulated as shown in Figure 1.2. The diameters at x = 0 and x = L are
25cm and 5cm, respectively. If the heat flw at x = 0 is 45W/m2 under steady
conditions, determine the heat transfer rate and the value of the heat flux at x = L.

Figure 1.2 Heat transfer through a solid circular cone.