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Natural convection |
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6.1 Introduction
Natural convection is a form of convection heat transfer in which the driver of the bulk motion or advection is self-induced forces. These forces may be due to temperature or concentration gradients. In this book, we focus our attention on natural convection flow and heat transfer due to temperature gradients. In the light of the above statement, it is intuitively apparent that flow and heat transfer will be strongly coupled in natural convection. Because there is bulk motion, natural convection heat transfer will be several times more than what one would obtain in molecular conduction. Even so, because no external agency such as a pump or blower is involved in natural convection, the velocities will be small, typically of the order of cm/s or tens of cm/s as opposed to several m/s, which is typically seen in forced convection. As a consequence of this, natural convection heat transfer rates will be lower than forced convection in a particular situation, if all other controlling variables like temperature difference, geometry, and medium are the same. Recall that, traditionally, convection aided by an external agency like a pump or fan or blower has come to be known as “forced” convection. In view of this, natural convection where there is no “forcing” by an external agency, so to speak, is also known as free convection.
Consider two infinitely wide horizontal parallel plates at temperatures T1 and T2, respectively. Let the space between the two plates be occupied by a medium like air. Two possibilities exist, as shown in Fig. 6.1.
In situation (A), the top plate is hotter than the bottom plate. In view of this, as the air (or any other medium for that matter) gets heated from the top plate, it stays at the top, as heated air is less dense. So, this represents a stable arrangement in so far as convection is concerned, meaning in this case no natural convection will occur. Even so, heat transfer will occur between T1 and T2 through the air via the conduction or molecular diffusion route. Radiation may also occur, but this depends on the difference between T1 and T2, radiative properties of the surfaces, and so on. Radiative heat transfer is the subject of chapter 8 of this book.
Suffice it to say for now that Fig. 6.1A represents a “no flow,” “no natural convection” situation. The situation in Fig. 6.1B, though, is interesting. The heated plate is at the bottom, so air coming in contact with it gets heated and rises. Once the air hits the top plate, it is cooled, becomes denser, and returns to the bottom plate to get heated again and continue the cycle. This is quintessential natural convection.
Heat Transfer Engineering. http://dx.doi.org/10.1016/B978-0-12-818503-2.00006-X |
173 |
Copyright © 2021 Elsevier Inc. All rights reserved.
174 CHAPTER 6 Natural convection
FIGURE 6.1
Medium enclosed between two parallel plates with temperatures T1 and T2. (A) T1 > T2 (B) T1 < T2.
Applications of this are legion, for example, cooling of electronic equipment like transformers, heat transfer in double pane windows, solar collectors, thermal hydraulics in nuclear reactors, and so on. The list is endless. Just to reinforce a point that the above is not a trite and beaten-to-death list of applications of natural convection, we would like to draw your attention to the Fukushima Daiichi nuclear disaster that happened on March 11, 2011. The disaster first started with an earthquake and as soon as the earthquake was detected, the nuclear reactors shut down. However, due to grid problems, the electricity supply failed and the emergency diesel generator sets started to ensure that the pumps circulated the coolant through the nuclear reactor cores to remove decay heat (which does not stop immediately and follows a typical q = ae−bt kind of distribution, with a and b being known constants, t being the time, and q being the heat decay). However, the earthquake caused a nearly 50 feet high tsunami that flooded the basement of the plant, thereby paralyzing the emergency generator. This resulted in what is known as loss of coolant accident (LOCA), reactor meltdown, and radiation release to the atmosphere. The basic problem here was that the decay heat removal system was designed only for the case of forced convection, and the system was incapable of preventing LOCA, if the emergency generators failed. A decay heat removal system that would have worked even under natural convection would have involved considerable engineering effort and inclusion of chimneys and so on, but would have saved the day. Even in the ubiquitous laptop, a heat pipe removes the heat generated by the processor but the condensation of the vapor of the heat pipe itself has to be driven by natural convection and radiation from all the surfaces of the laptop. It is worthwhile to remember that eventually any heat generated has to be released to the ambient air or a nearby lake, pond, sea, or outer space. The challenge for a heat transfer specialist is to enable and engineer this pathway that is sure-shot, safe, budget friendly, environmentally benign, and meets all guidelines and legislations.
6.2 Natural convection over a flat plate 175
6.2 Natural convection over a flat plate
The vertical flat plate is a frequently encountered geometry in natural convection and also serves as an excellent baseline configuration to undertake a mathematically rigorous study of natural convection, so that we eventually get results that are simple to use and are of great practical relevance in actual engineering situations.
Before getting drowned in mathematical details, let us try to intuit about what is likely to happen if a heated vertical plate of length L, at a temperature TW is placed in quiescent (still) air at T∞ with TW > T∞ (see Fig. 6.2), with gravity acting downward.
Cold air coming near the plate gets heated and rises, thereby creating some sort of vacuum that is filled by fresh air rushing in towards the plate. In view of this, an upward current or convective flow is set up. Let x and y be the vertical and horizontal coordinates, respectively, and let the corresponding air velocities generated due to natural convection be u and v, respectively (These are indicated in Fig. 6.2). It is instructive to note that u = v = 0 everywhere if Tw = T∞. All the action is generated due to the temperature difference, ∆T = (Tw – T∞). The higher the ∆T, the stronger the natural convection.
Now, consider a horizontal section A-A as indicated in Fig. 6.2. At the wall (y = 0), the velocity u = 0. Again at a horizontal distance far away from the wall, u = 0 as the air is still. Even so, there is an upward motion due to buoyancy caused by ∆T, as already discussed. Hence, if we mark a point on A-A where the velocity is almost zero, say 0.01umax with umax being the maximum velocity in the air layer close to the plate, it stands to reason that one can expect a smooth velocity profile that is zero at y = 0 and y = δ where δ is the boundary layer thickness, with the maximum occurring somewhere in between. In fact, measurements by researchers have confirmed a velocity profile that looks like what is indicated in Fig. 6.2. From numerous measurements and flow visualization experiments, the boundary layer is seen to have a parabolic profile with increasing thickness as the height increases, as shown in Fig. 6.2. In continuation of the above arguments, it is not very difficult to
FIGURE 6.2
Natural convection from a heated vertical plate of length L.
176CHAPTER 6 Natural convection
comprehend an exponential temperature profile within the boundary layer, with T =
Tw at y = 0 and T = T∞ + 0.01 (Tw – T∞) ≈ T∞ at y = δ. The key differences between the forced convection and natural convection boundary layers are
1.In forced convection, the velocity boundary layer can exist independent of the thermal boundary layer, whereas in natural convection, it cannot, as the velocities themselves arise due to a temperature difference between the wall and the free stream.
2.The velocity profile is monotonic for forced convection with u = u∞ at y = δ, while for natural convection it has to be nonmonotonic with a peak velocity occurring within the boundary layer.
6.3 Boundary layer equations and nondimensional numbers
Consider the vertical flat plate shown in Fig. 6.2 suspended in a quiescent medium at T∞. The plate is at a uniform temperature of Tw with Tw > T∞. Consider a twodimensional, steady, incompressible (with density alone being a function of temperature) flow and heat transfer for a constant property flow. The governing equations are more or less the same as those we saw in Chapter 4, except for the addition of a body force in the x-momentum equation (please note that now the x-axis is along the height of the plate. i.e., vertical). The governing equations are
Continuity equation
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x-momentum equation |
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∂u |
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ρ u |
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+ µ |
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y-momentum equation |
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∂ v |
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Energy equation |
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∂T |
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ρCp u |
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Please note that the new term appearing in Eq. (6.2), namely, the body force term “–ρg” was not considered in forced convection. If we consider a two-dimensional control volume of size ∆x.∆y.1, the term –ρ.∆x.∆y.1.g is nothing but the weight of the medium inside the control volume. Cancelling ∆x∆y and recognizing that x is positive upward, while gravity is acting downward, this term reduces to “–ρg”, as shown in Eq. (6.2).
6.3 Boundary layer equations and nondimensional numbers 177
The big challenge before us is to now work out “ρg” in terms of the primary quantities of interest in our problem, those are, u, v, and T, if we choose to stay with the easier to solve incompressible flow formulation. The challenge is compounded by the fact that there is no free stream velocity u∞ that can be used as a reference. Outside of handling the “ρg” term, we would also like to examine the possibility of using boundary layer approximations much as in the same way as we did for forced convection.
Scale for velocity
Let the scale for velocity be uref (by velocity here we mean vertical velocity). Consider Eq. (6.1). The scales for the two terms in this equation need to be of the same order, so that the two-dimensional character of the problem is intact (Bejan, 2013).
Let the scale of u be uref. The scale for v is v and the length scales are L and δ for x and y, respectively. Substituting for these, in the continuity equation
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υ <<u |
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the y-momentum equation is |
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“axed” from the analysis. Even so, there is a key take away from the y-momentum
equation which is the following
–∂∂py = 0
p = f (x) alone
Consequent upon Eq. (6.9), the x-momentum equation reduces to
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The energy equation reduces to
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(6.8)
(6.9)
(6.10)
(6.11)
178 CHAPTER 6 Natural convection
Applying Eq. (6.10) to the free stream, i.e., the region outside the boundary layer, we have
u = 0, v = 0, |
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Eq. (6.10) becomes |
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Because of the slenderness of the boundary layer, we then get
Substituting for dpdx
p(x, y) ~ p(x) ~ p∞ (x)
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The density of the medium is a function of temperature and pressure
ρ∞ = ρ (T∞ , pref )
where pref is the reference pressure, say, at the bottom (x = 0)
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(6.13)
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Now we make an assumption that the variation of density depends more on the |
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Now we introduce a quantity called the isobaric cubic expansivity denoted by β and defined as follows
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6.3 Boundary layer equations and nondimensional numbers 179
From the definition of β it is clear that β has the units of K–1. Invoking the definition of β, we now have an expression for density ρ as
ρ = ρ∞ − ρ∞ β(T − T∞ )
ρ = ρ∞ (1 − β(T − T∞ ))
Eq. (6.14) becomes
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(6.18)
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The above simplification is frequently referred to as the Boussinesq approximation. When β(T – T∞) << 1, Eq. (6.20) simplifies to the following form
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Eq. (6.21) is a key approximation in the analysis of natural convection over a flat plate.
Finally, now we are in a position to write down the governing equations for this problem, under the boundary layer simplifications, as follows
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In Eq. (6.23), the terms on the left-hand side represent inertial forces, while the first term on the right-hand represents the frictional force and the second represents the buoyancy force. In the energy equation (Eq. 6.24), the left-hand side represents the advection terms and the right-hand side represents the conduction term. Eqs. 6.22–6.24 satisfy closure, as we have three equations in three unknowns (u, v, and T). However, Eqs. 6.23 and 6.24 are coupled, consequent upon the presence of the term with the temperature difference in the momentum equation (Eq. 6.23).
Dimensionless numbers governing natural convection
We can now carry out nondimensionalization of the governing equations, with a view to obtain the pertinent dimensionless numbers governing natural convection.
In order to normalize the velocities u and v, we need a reference velocity. In the field of natural convection, a frequently used velocity scale is ν/L (some researchers also have used α/L). Please note ν/L has the units of velocity.
180 CHAPTER 6 Natural convection
The following dimensionless quantities are introduced now
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Using the above framework, we now normalize Eqs. 6.22–6.24. The continuity equation becomes
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(6.30)
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(6.33)
Please note that all the individual terms in Eq. (6.33) are dimensionless and so is φ.
6.3 Boundary layer equations and nondimensional numbers 181
Consequent upon this and also on its own, the group |
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and a key quantity in natural convection known as the Grashof number, GrL , with
T = (Tw − T∞ ).
Hence Eq. (6.33) can be rewritten as
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where the Prandtl number is Pr = αν . The importance of the Prandtl number in forced convection flows was discussed in detail in chapter 5.
In the light of the preceding arguments, it is clear that the functional dependence of u+ and φ on the pertinent variables and parameters in the problem under consideration may be expressed as
u+ = f1 (x+ , y+ ,GrL ,φ) |
(6.37) |
and
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In our treatment of forced convection, we obtained a very important relation between the Nusselt number and the temperature gradient at the wall. Similarly, it is intuitive that
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NuL = f4 (GrL , Pr)
182 CHAPTER 6 Natural convection
In fact the product of Gr .Pr |
is known as Rayleigh number, RaL . |
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Hence, correlations for natural convection heat transfer coefficients are expressed in terms of Rayleigh number or Grashof number (and Prandtl number) and are typical power law forms that look like what is shown below
NuL |
= aRab (ς1 )c (ς2 )d |
(6.42) |
where a, b, c, and d are constants, ζ1 may typically represent the viscosity correction term, and ζ2 may be a geometric parameter like ratio of height to width of an enclosure or ratio of inner to outer radius in a cylindrical or spherical annulus. In its simplest avatar, the Nusselt number correlations look like what is given in Eq. (6.43).
NuL |
= aRab |
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Constants a and b need to be determined for the geometry under consideration. For simple geometries, a and b may be derived using one of the following methods.
1.Approximate methods like integral methods.
2.Analytical methods like similarity transformation.
3.Numerical methods like finite difference or finite element or finite volume method.
Of course, we should not forget that a and b can also be derived purely from experiments and the Rayleigh number, RaL, dependence could have also been obtained through the Buckingham pi theorem route.
Integral approach for natural convection on a vertical flat plate
The integral solution for natural convection over a flat plate has been derived by several investigators, such as Eckert and Squire (Lienhard and Lienhard, 2020). We first derive the integral forms of the momentum and energy equations that are similar to those for forced convection in the previous chapter. However, there are two key differences: (1) the presence of the buoyancy term in the x-momentum equation and (2) recognition that δt = δ as the temperature difference causes the flow and, in view of this, the two boundary layers are coupled. In fact this assumption (i.e., δt ≈ δ ) is strictly valid when the Prandtl numbers are close to one. While this assumption will also work quite well for Pr >> 1, the analysis will not be accurate for Pr << 1. In these cases, we need to rely on either experimental results or full numerical solutions to get a handle on the problem. For the sake of completeness, we present a quick overview of the derivation of the integral momentum and energy equations.