HEAT TRANSFER, MODELING, AND
(4.22) [moving continuous point source, strength
(W or at = 03
In Equations 4.21 and 4.22 the y and coordinates are perpendicular to the direction. These expressions are valid provided the wake is slender, in other words, when the Peclet number is large, 1.
Equations 4.15-4.22 apply unchanged to the ure fields near con- centrated heat sinks. In such cases the numerical values of the source strengths
Q’, q”, q’, are negative.
Melting and Solidification. A semi-infinite solid that is isothermal and at the melting point melts if its surface is raised to another temperature
In the absence of the effect of convection, the liquid layer thickness increases in time according to
2 - - (4.23)
where and k are the density and thermal conductivity of the liquid, and is the latent heat of melting. Equation 4.23 is valid provided -
1, where is the specific heat of the liquid.
The solidification of a motionless pool of liquid is described by Equation 4.23, in which - is replaced by - because the liquid is saturated at and the surface temperature is lowered to In the resulting expression is the thickness of the solid layer, and k, c, and a are properties of the solid layer.
Example 4.1 The shape in which a potato is cut has an important effect on how fast each piece is cooked. Three different shapes have been proposed, each containing 5 of potato matter: (a) sphere, (b) cylinder with a length of 6 cm, (c) thin disk with a diameter of 4 cm. Each piece is initially at the temperature T, = 30°C. At time t = 0, each piece is placed in boiling water at the temperature = 100°C. The heat transfer coefficient is constant, h =
2 and the properties of potato matter are approximately =
182 HEAT TRANSFER, MODELING, AND DESIGN ANALYSIS
0.9 k = 0.6 and CY = 0.0017 For each shape, calculate the time until the volume-averaged temperature rises to 65°C. Comment on the best shape for the shortest cooking time.
Solution
MODEL
The potato matter is homogeneous and isotropic with constant proper- ties.
2. The heat transfer coefficient at the wetted surface is assumed constant, that is, independent of time.
3. The transfer of water through the potato surface and any swelling of the potato are neglected.
ANALYSIS. As shown below, the requirement of fixed mass, m = 5 de- termines the size of each piece. For example, the radius of the sphere is calculated as 1.1 cm. Calculating the Biot number for the sphere, we have
=
-
(2k
= 367
With similar calculations, we find that in each case Bi 1, and so we can rely on Figure 4.2. Accordingly, using the given temperatures, the ordinate reads (regardless of shape)
T - -
(65
(30 - = 0.5
The three curves indicate the following readings on the abscissa:
0.031 = 0.064 = 0.196
(a) In the case of the spherical shape, the radius is fixed by the mass
CONDUCTION 183
47r 3
- 1.33 0.9
= 1.1 cm
and Equation (a) pinpoints the warming time,
0.0017
= 0.031 22
(b) For a cylinder of length = 6 cm, the radius is
- 0.29
= 0.9 cm
r , = 0.54 cm and Equation (b) gives
0.0017
t = 0.064 11
(c) Finally, in the case of a disk of diameter D = 4 cm, the disk thickness L is obtained as follows:
m = p - 2 L 4
= 0.22 cm 2
0.9
with the corresponding warming time from Equation (c),
0.0017
= 0.196
COMMENT. Comparing these three time intervals (22, and 6 we see that in accord with intuition the disk shape (the potato slice) promises to cook much faster than the other two shapes.
184 HEAT TRANSFER, MODELING, AND DESIGN ANALYSIS
4.3 CONVECTION
Convection is the heat transfer process in which a flowing material (gas, fluid, solid) acts as a conveyor for the energy that it draws from (or delivers to) a solid wall, and, as a consequence, the heat transfer rate is affected greatly by the characteristics of the flow velocity distribution, turbulence). To know the flow distribution and the regime (laminar vs. turbulent) is an important prerequisite for calculating convection heat transfer rates. Also, the nature of the boundary layers (hydrodynamic and thermal) plays an important role in evaluating convection.
In this section we review the most important results of convection heat transfer together with the corresponding fluid mechanics results. It is assumed that the fluid is Newtonian, homogeneous, and isotropic and has a nearly constant density.
4.3.1 External Forced Convection
Convection is said to be external when a much larger space filled with flowing fluid (the free stream) exchanges heat with a body immersed in the fluid.
According to Equation 4.2, the objective is to determine the relation between the heat transfer rate (or the heat flux through a spot on the wall, and the wall-fluid temperature difference - The alternative is to deter- mine the convective heat transfer h, which in an external flow is defined by
(4.24)
where is the heat flux (heat transfer rate per unit area). Units for the convective heat transfer coefficient are and Figure 4.3 shows the order of magnitude of h in various cases.
Boundary Layer over a Plane Wall. When the fluid velocity is uniform and parallel to a wall of length L, the hydrodynamic boundary layer along the wall is laminar over if Re, 5 where the Reynolds number is defined by Re, = and is the kinematic viscosity. The leading edge of the wall is perpendicular to the direction of the free stream (U,). The wall shear stress in laminar flow averaged over the length L averaged) is
(4.25)
= (Re, 5 X
so that the total tangential force experienced by a plate of width and length
CONVECTION 185
vapor
nvection
, forced convection
atm, forced convection
onvection
I convection
I I I I I
1 I 03 I 05 I
1 1 I 04 I 05
h
Figure 4.3 Effect of flow configuration and fluid type on the convective heat transfer coefficient
L is F = The length L is measured in the flow direction. The thickness of the hydrodynamic boundary layer at the trailing edge of the plate is of the order of L
If the wall is isothermal at the heat transfer coefficient averaged over the flow length L is (Re, 5
(4.26)
0.664 (Pr 0.5)
(Pr 0.5)
In these expressions k and Pr are the fluid thermal conductivity and the Prandtl