Page 182 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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Hot
fluid
fluid
u Cold Hot walls CONVECTION HEAT TRANSFER
a
Figure 7.1 Flow and heat transport in a channel
7.1.1 Types of fluid-motion-assisted heat transport
The fluid-motion-assisted heat transfer (heat convection) may be classified into three differ-
ent categories. In order to explain the different types, let us consider the fluid flow through
a two-dimensional channel as shown in Figure 7.1. The inlet to the channel is at the left
side and exit is at the right. Both the top and bottom walls of the channel are at higher
temperatures than the invading fluid. The mechanism here is that the fluid, which is at a
temperature lower than the wall temperature of the channel, comes into contact with the
wall and removes heat by convection. Although this process is termed as being convective,
there are aspects of the diffusion mode of heat transfer that dominate very close to the
hot walls.
It is obvious that flow with a higher incoming velocity will transport heat at a higher
rate. The flow rate is often characterized by a quantity called the Reynolds number,which
is defined as
ρ a u a L
Re = (7.1)
µ a
where u a is the average inlet velocity, L is a characteristic dimension, for example, the
width or height of the channel, ρ a is a reference (inlet) density and µ a is a reference (inlet)
dynamic viscosity of the fluid. If the Reynolds number is small and below a certain critical
value, the flow is laminar,and if it is above this critical number, then the flow becomes
turbulent. The critical Reynolds number for pipe and channel flows, based on the diameter
or height, is approximately 2000.
In Figure 7.1, if the flow is forced into the channel by means of an external device, for
example, a pump, then the convection process is referred to as forced convection, and the
Reynolds number is normally high (Jaluria 1986; Lewis et al. 1996, 1995b; Massarotti et
al. 1998; Minkowycz et al. 1988; Patnaik et al. 2001; Srinivas et al. 1994). In such situa-
tions, the fluid motion created by the density (or temperature) difference (buoyancy-driven
motion) is negligibly small as compared to the forced motion of the fluid. However, at low
and moderate Reynolds numbers, the motion created by the local density (or temperature)
differences in the fluid is comparable to that of the forced flow. A situation in which the
forced and density difference–driven motions are equally important is called mixed con-
vection transport (Aung and Worku 1986a,b; Gowda et al. 1998). If the forced flow is
suddenly stopped and the fluid is stagnant inside the channel, then the fluid motion will
be entirely influenced by the local density (or temperature) differences until an equilibrium
state is reached, that is, no local differences in density or temperature are present. Such
a flow is often referred to as natural, free or buoyancy-driven convection (de Vahl Davis
1983; Jaluria 1986; Jaluria and Torrance 1986; Nithiarasu et al. 1998).