Natural Convection Over a Cylinder



Natural Convection from a Vertical Surface

"Hot Air Rises"

Since the density of most fluids vary with temperature, temperature gradients within a fluid medium will give rise to density gradients. If these density gradients are such that the fluid is in an unstable situation, heavy fluid on top of light fluid, the fluid will begin to move. This motion is termed natural convection.

Physical Background

When heat is transported from a solid surface by a moving fluid we call it convection. It is important to note that there are three primary conditions for convective heat transfer.

(i) presence of a solid surface (hence a free shear jet

would not be applicable to convective heat transfer)

(ii) presence of a "moving" fluid (will have to be careful

in defining a moving fluid)

(iii) presence of a temperature difference between the

surface and the fluid (otherwise there will be no

tendency for heat to flow)

Convective heat transfer is modeled by Newton's law of cooling which states that the convective heat flux is proportional to the temperature difference between the fluid and the surface.

[pic] (1)

where we have defined the heat flux as positive leaving the surface. Now introducing the constant of proportionality

[pic] (2)

where hc is called the convective heat transfer coefficient.

It is clear that the major obstacle in utilizing Eq. (2) for convective heat transfer calculations is the evaluation of the convective heat transfer coefficient. There are three standard methods used to evaluate hc. The first involves a mathematical solution to the conservation equations in differential form. For problems where these equations are too complicated to be solved analytically, we can employ the second method, a computational solution. Finally for problems that are so complicated that we cannot even write the appropriate conservation equations, we must go into the laboratory and make measurements in employing an experimental solution. Before we contemplate employing one of these solution methods, it is useful to use our intuition to figure out upon what the convective heat transfer coefficient depends. Our intuition tells us that the three major factors associated with the calculation of hc should be

(i) fluid mechanics

(ii) fluid transport properties

(iii) geometry

Starting with the fluid mechanics, we recognize there are a variety of ways to characterize the flow. First, we consider what is driving the flow.

Forced Convection: The fluid is moved by some external agent such as a pump or a fan. It is normally recognized by the specification of some fluid velocity, so that, for example, wind is considered a forced convective process.

Natural Convection: The fluid moves due to density differences and gravity. The observation that hot air rises is evidence of natural convection. It should also be clear that if the fluid and surface are at different temperatures natural convection will always occur, so that the air is never really still.

Next, we consider how many boundaries the fluid flow interacts with

External Flow: The fluid is constrained by only one boundary. Mathematically we would call this a semi-infinite domain. A good example of this would be an airplane wing.

Internal Flow: The fluid is completely constrained by boundaries. Mathematically we would call this a finite domain. Air in a room and water in a pipe are both examples of internal flows.

The difference between internal flows and external flows is often one of perspective. For example, the air in the room is fully constrained by the walls of the room. Yet, if we were only concerned about the heat transfer from one wall of the room, we might model this wall as an external flow.

A third way to characterize the flow is by the presence or absence of turbulence.

Laminar Flow: The fluid flows in smooth streamline paths.

Turbulent Flow: The fluid flows is a chaotic fashion with vortices flying around in many different directions.

An important consideration in the handling of convective heat transfer coefficients is the notion of dynamic similarity. It is found that certain systems in fluid mechanics or heat transfer are found to have similar behaviors even though the physical situations may be quite different. Recall the fluid mechanics of flow in a pipe. What we are able to do is to take data as shown in Fig. 1 for different fluids and pipe diameters and by appropriately scaling collapse these curves into one curve.

Figure 1. Dynamic Similarity for Pipe Flow

[pic]

[pic]

In convective heat transfer we may apply dynamics scaling to make a parallel transformation.

Figure 2. Dynamic Similarity for Convective Heat Transfer

[pic]

[pic]

We have defined a dimensionless convective heat transfer coefficient called the Nusselt number as

[pic] (3)

where

hc: convective heat transfer coefficient

L: characteristic length

k: thermal conductivity of the fluid

The characteristic length is chosen as the system length that most affects the fluid flow.

Another important feature introduced by the fluid mechanics is the local nature of the convective heat transfer coefficient. If as the fluid flows over different regions of the surface, the fluid mechanics change, then the convective heat transfer coefficient will change. For natural convection over a vertical flat plate we will have the boundary layer flow shown in Fig. 3.

Figure 3. Flat Plate Boundary Layer Development

As the boundary layer thickness grows, the fluid mechanics change significantly, so that the convective heat transfer coefficient will vary along the length of plate and we will have a local convective heat transfer coefficient, hc(x). Then we may also define local Nusselt and Rayleigh numbers as

[pic] (6)

[pic] (7)

Though local heat transfer conditions can be extremely important, an average heat transfer coefficient over the entire surface length is often desirable. By definition we have

[pic] (8)

with an average Nusselt Number given as

[pic] (9)

On a physical basis we find that the Nusselt number is

[pic] (5)

Hence, the Nusselt number must always be greater than one and is never negative. For natural convection the dimensionless parameter which represents the fluid mechanics is the Rayleigh number, defined by

[pic] (6)

where

ΔT: temperature difference between the fluid and the surface

L: characteristic length

g: acceleration due to gravity

β: fluid thermal expansion coefficient

ν: fluid kinematic viscosity

α: fluid thermal diffusivity

On a physical basis,

[pic] (7)

The dimensionless parameter which is used to represent the affect of fluid properties is the Prandtl number

[pic] (8)

or on a physical basis

[pic] (9)

Sometimes there will be a significant temperature difference between the surface and the fluid. Since the fluid properties are temperature dependent this large temperature range may give rise to a large range in the values of the fluid properties. This is often accounted for by introducing another fluid property dimensionless parameter which is the ration of the property at two temperatures. For example, we might have

[pic] (10)

The influence of geometry may be seen in a couple of ways. First, for those configurations that have two length dimensions, such as our cylinder example above, we introduce a dimensionless geometric parameter

[pic] (11)

The second way in which we see geometrical influences is through the functional form of the Nusselt number. In general we may write

Nu = fn(Ra,Pr,M,X) for natural convection (12)

For simple situations these may often be written as power law relationships

[pic] (13)

where the constants a, m, and n will change for different geometries.

Experimental Background

Since there are physical limits as to what we can measure in the laboratory, we find that the convective heat transfer coefficient is not measured directly, but rather is derived from measurements of the temperature and heat transfer rate or using Newton's law of cooling

[pic] (14)

The heat transfer rate may be measured by using a lumped capacitance transient approach, a heat meter as described in the radiation experiment, or by measuring the voltage and current for electrical heating. For electrical heating the heat transfer rate is simply

[pic] (15)

In measuring the temperatures. it is important to note that it is the temperature difference that we desire. By being clever we can measure this directly and hence reduce our experimental error. We take a thermocouple with a measurement junction at one end and the reference junction at the other end. We then place the measurement junction on the surface to measure Ts, while the reference junction is placed in the fluid. Hence the reading from the thermocouple will correspond to the temperature difference Ts - Tf.

Before moving on to the details of the specific experiment, we need to discuss some aspects of the data processing. Recall that we have measured temperature difference, heat transfer rate, and in some cases velocity or flow rate. We can directly calculate the convective heat transfer coefficient from these measurements. However, to be most useful, we have seen that it is best to express our results in terms of a Nusselt number and either a Reynolds number or a Rayleigh number. To accomplish this we must obtain certain fluid properties. We can do this by referring to tables in the back of your text book. But when we look at these tables we find that for air there is not just one value of the thermal conductivity, but several values. In fact, what we see is the temperature dependence of these properties. Thus, we must chose a temperature at which to evaluate the properties. The standard is to chose the film or average temperature between the surface and the fluid,

[pic] (16)

One aspect of this approach is that during our experiment we will need to measure either Ts or Tf independently.

In this experiment the student will develop the relationship among the Nusselt number and other dimensionless parameters for natural convection from a vertical flat surface. The students will determine local heat transfer coefficients which are indicative of a boundary layer phenomena. These local measurements will then be averaged and compared to published correlations.

Natural convection heat transfer from a vertical surface will be investigated using an electrically heated flat plate. A schematic of the apparatus is shown in Figure 4.

Figure 4. Schematic of Experimental Apparatus

[pic]

Since we wish to establish a relationship between the Nusselt number and the Rayleigh number, we must vary the Rayleigh number. We can accomplish this by varying the power. Hence, the experiment begins at a low power level and then the power is incremented so as to produce a range of Rayleigh numbers. The primary result of your experiment will be the development of a Nusselt number Rayleigh number relationship, which you will show graphically.

Now let’s practice some data processing with the following exercise.

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x

[pic]

dð

δ

................
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