Transport Phenomena - Heat Transfer Problem :
Forced convection heat transfer for plug flow in plane slit
- constant wall heat flux in thermally fully developed flow region
Problem.
A very thick paste flows in a plane narrow slit formed by two parallel walls a distance 2B apart. The length L and width W of the slit are such that B << W << L, so end effects can be neglected.
Since the paste flows nearly as a solid plug, the velocity profile is approximately flat over the slit cross-section and vz = V (constant) may be assumed. For z < 0, the fluid temperature is uniform at the inlet temperature T1. For z > 0, heat is added at a uniform constant flux q0 through both the slit walls. The heat conduction in the z-direction and the viscous dissipation effect may be neglected. The thermal conductivity k and thermal diffusivity
a may be assumed constant.
Figure. Plug flow in plane narrow slit being heated by a uniform heat flux.
a) Show that the temperature profile T(x, z) far downstream in the thermally fully developed region (i.e., for large z) is given by the following dimensionless asymptotic solution:
where
b) Show that the limiting local Nusselt number far downstream for plug flow in a plane slit with constant wall heat flux is Nu = 12.
Solution.
Click here for stepwise solution
Step. Differential equation and boundary conditions
For forced convection heat transfer, the equation of energy (on neglecting the viscous dissipation effect) simplifies to
(1)
In Cartesian coordinates for flow in a plane narrow slit, the above equation gives
(2)
where
is the thermal diffusivity. Note that the heat conduction in the z-direction is neglected because it is typically small compared to the heat convection in the z-direction. The solution of the above partial differential equation gives the temperature profile T(x, z).
For plug flow, vz = V (constant) and the above equation in dimensionless form is
(3)
where the dimensionless quantities are defined as
(4)
The dimensionless quantities are typically chosen such that the number of parameters in the problem is minimized. In this problem, the choice of dimensionless x-coordinate s
and the dimensionless z-coordinate z
naturally follow from the differential equation, while the choice of the dimensionless temperature Q
logically follows from the boundary conditions given below.
The boundary conditions are given by
(5)
(6)
(7)
The first two boundary conditions state that heat is added at a uniform constant flux through the two walls, i.e., qx =
-q0 at x = +B and qx = q0 at x =
-B with qx in accordance with Fourier's law.
Step. Temperature profile in thermally fully developed region
The asymptotic solution far downstream in the thermally fully developed region (i.e., large z)
is postulated to be of the form:
(8)
where C0 is a constant to be determined. The above form suggests that the constant wall heat flux will cause the fluid temperature to rise linearly with z
once the fluid is far downstream from the start of the heated section. Furthermore, the shape of the temperature profile T(x) will eventually remain the same for increasing values of z.
On substituting the above form for
Q
in the partial differential equation (3),
the following ordinary differential equation is obtained.
(9)
On integrating twice, we get
(10)
Thus, the dimensionless temperature profile is of the form:
(11)
On imposing the boundary conditions, BC 1 gives C0 + C1 = 1 and BC 2 gives
-C0 + C1 =
-1. Thus, C0 = 1 and C1 = 0. However, it is not possible to satisfy BC 3 because the postulated function is the asymptotic solution (and not the complete solution); therefore, BC 3 is replaced by the following condition:
(12)
The above condition states the heat entering through both the walls over the distance z equals the heat leaving with the fluid at z (with T1 as the reference temperature at z = 0). On substituting the temperature profile and integrating, the condition gives
(13)
On substituting C0 = 1, C1 = 0 and C2 =
-1/6, the asymptotic solution for the dimensionless temperature profile in the thermally fully developed region is finally obtained as
(14)
Step. Calculation of bulk temperature
The bulk temperature is defined as the temperature obtained if the fluid flowing at z were collected in a vessel and completely mixed. This average temperature is therefore also called the flow-average temperature or the cup-mixing temperature. Thus, on noting that the velocity and temperature profiles are symmetric about the mid-plane of the slit, the bulk temperature is mathematically defined by
(15)
On substituting the temperature and velocity distributions, integration gives the bulk temperature as
(16)
Step. Expression for Nusselt number
The heat flux is the product of the heat transfer coefficient h and the local heat transfer driving force (which is the difference between the wall temperature and the bulk temperature at an axial distance z). Thus,
(17)
Thus the Nusselt number for plug flow in a plane slit far downstream for constant wall heat flux is
(18)
Note that the characteristic length used above for noncircular conduits is the equivalent diameter = 4 (mean hydraulic radius), where the mean hydraulic radius Rh is the ratio of the flow cross-sectional area to the wetted perimeter. For the plane slit, Rh = 2BW / (4B + 2W). Since B << W, the hydraulic radius is approximately given by Rh = 2BW / (2W) = B. Thus, the equivalent diameter for the plane slit is 4B.
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