Heat transfer by convection may occur in a moving fluid from one region to another or to a solid surface, which can be in the form of a duct, in which the fluid flows or over which the fluid flows. Convective heat transfer may take place in boundary layers, that is, to or from the flow over a surface in the form of a boundary layer, and within ducts where the flow may be boundary-layer-like or fully-developed.

It may also occur in flows which are more complicated, such as those which are separated, for example, in the aft region of a cylinder in cross-flow or in the vicinity of a backward-facing step. The flow may give rise to convective heat transfer where it is driven by a pump and is referred to as forced convection, or arise as a consequence of temperature gradients and buoyancy, referred to as natural or free convection. Velocity and temperature profiles in boundary-layer and separated flows.

In this case, the surface is assumed to be at a higher temperature than the free-stream and the finite gradient at the wall confirms the heat transfer from the surface to the flow. It is also possible to have zero temperature gradient at the wall so there is no heat transfer to or from the surface but heat transfer within the flow. If the flow is laminar, heat transfer from the surface is given by the Fourier flux law, that is:. The same expression applies to any region of the flow and also in the case of the adiabatic wall where zero temperature gradient implies zero heat transfer.

It should be noted that the surface can be horizontal as shown, with air flow driven by a fan or a liquid flow by a pump, and that it can equally be vertical, with buoyancy providing the driving force for the flow. In the latter case, the free-stream velocity would be zero so that the corresponding profile would have zero values at the wall and far from the wall.

## Convective heat transfer in axisymmetric porous bodies | Emerald Insight

The details of flows of this type are not well-understood so it is difficult to identify the characteristics of the boundary layers and it can be imagined that the shapes of the velocity and temperature profiles— and therefore of the local heat transfer within the fluid and to the wall—will vary considerably from one location to another. It is known, for example, that the rate of heat transfer can become high at the location of reattachment of the upstream flow on to the surface of the step, as is also the case at the leading edge of a cylinder in cross-flow, but the detailed mechanisms remain incompletely understood and research continues.

With laminar flows, heat transfer to or from the wall varies with distance from the leading edge of a boundary layer. Turbulent flows can give rise to heat transfer rates which are much larger than those of laminar flows, and are caused by the manner in which the turbulent fluctuations increase mixing; they also affect the heat transfer to and from the surface, especially where the free-stream fluid is able to penetrate to the wall even for short periods of time.

The nature of the surface, for example the degree or type of roughness, usually affects heat transfer to or from it, and in some circumstances to a large extent. It is convenient, therefore, to represent the heat transfer at the wall by the expression. The two temperatures can vary with x-distance and it can be difficult to identify a free-stream temperature in some complex flows. The heat transfer coefficient is considerably greater with liquid flows and greater again with two-phase flows. It should be noted that the above equations are expressed in terms of dimensional parameters.

It is also useful to note that the heat transfer coefficient and the Nusselt number can be used to refer to local values at a location x on a surface, or to an integrated value up to the location x. The concept of dimensional analysis gives rise to several nondimensional groups, to which reference will be made in this section, and it is convenient to introduce them here. In addition to the Nusselt number, reference will be made to the following:.

These nondimensional groups may be obtained from conservation equations and are convenient in the representation of results and correlations of experimental data. It is useful to examine the equations which represent conservation of mass, momentum and energy and these are written below for rectangular Cartesian coordinates with simplification of uniform properties. The three equations representing conservation of momentum and the equation representing conservation of energy have the same form, with the terms on the left-hand sides representing convection of momentum and energy.

It should be noted that these convective terms are nonlinear, thereby presenting difficulties for any solution and that there are four individual parts of convection corresponding to variations in time and in the three directions. The terms on the right-hand side are slightly simplified forms of those representing transport by diffusion together with pressure forces and sources or sinks of thermal energy.

Terms for buoyancy may be added as shown in a following section. In later sections, these equations will be simplified to deal with convective heat transfer in steady, laminar flows of forced and free convection. It is evident from the above that there is some similarity between the equations for conservation of momentum and thermal energy so that the solutions of the two equations will have similar forms when the source terms are zero, the Prandtl number is unity and the solutions are presented in nondimensional form.

Where the surface which gives rise to the temperature difference—and therefore to the buoyant force—is not vertical, the angle of the surface to the direction of the gravitational force must be considered. This will lead to the resolution of forces so that part of the buoyancy term will appear in the first momentum equation with that in the second equation, multiplied by the sine of the angle to the vertical. This will give rise to an additional nondimensional group, the Grashof number. In the absence of convection terms, the energy equation reduces to that for heat conduction and the momentum equations are no longer relevant where the conduction takes place in a stationary material.

Many other simplifications of the above equations are possible, including those for two-dimensional flows and for boundary-layer flows, as will be seen below. Also, it is possible to integrate the equations and, in their simpler forms, this can have some merit; for example, in the integral momentum and energy equations where the dependent variable is devised so as to be represented in terms of one independent variable, and therefore solvable by simple numerical methods.

More complicated forms may also exist as discussed in the following section. All thermal parameters such as conduction, radiation and convection thermal resistances are then calculated by the program and the thermal performance calculated. This article concentrates on the formulations used to predict the convection cooling and flow within the machine. Article :. DOI: Need Help?

## Convection Heat Transfer

Another constant heat flux experimental work using vertical helical coils in air for two different groups of coils was studied by Ali [ 28 ]. He used the coil tube diameter and the coil axial length as two different characteristic lengths for analyzing his data. His obtained Nusselt number correlation, using the tube diameter as a characteristic length, showed heat flux dependence when Rayleigh number was used in the correlation. However, using the axial distance of the coil as a characteristic length in both Nusselt and Rayleigh numbers correlation made all the data to collapse on one unique curve independent of the heat flux.

Prabhanjan et al. The correlation they got was not accurate enough to be considered for a wide rage and different orientations.

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Therefore, they recommend that more study should be made to develop an accurate correlation for Nusselt numbers. On the other hand, their experiment was good in predicting the outer surface temperature of the coil. That experiment was for a range of Prandtl number of 28— For both diameter ratio and Rayleigh numbers fixed, he showed increases of the overall heat transfer coefficient with decreasing the number of coil turns.

Accordingly, keeping the diameter ratio constant and increasing the Rayleigh numbers led to increase in heat transfer coefficients. On the other hand, enhancement was obtained in heat transfer coefficients corresponding to lower diameter ratio of the coils for both constant number of turns and Rayleigh number. The overall averaged empirical correlations for coils with five and ten turns were reported, respectively, as.

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The overall average Nusselt numbers for all coils used was correlated using Rayleigh number as. He used five sets of coils; each set consisted of three coils with constant coil diameter to tube diameter ratio. Each set had two, five, and ten turns. The helical coil-to-tube diameter ratios were 30 , The coil length was used as a characteristic length in the dimensionless groups where correlation obtained Eq. Natural convection heat transfer phenomena for an inclined helical coil were investigated experimentally by Moon et al. An electroplating system was used to measure mass transfer instead of heat transfer, based on the analogy concept.

## The finite element analysis of convection heat transfer

To increase the surface area of the heat transfer, thermally interacting multiple horizontal and vertical cylinders have been investigated by different researchers. Most of the analyses for those types of geometries are circular. A pair of vertically aligned horizontal cylinders in water is investigated by Reymond et al. They observed that the heat transfer from the bottom cylinder in the array was unaffected by the upper cylinder for the used spacing between cylinders. It was also found that the presence of unheated lower cylinder had no effect on the upper cylinder heat transfer.

For that reason they did the experiment for both heated cylinders with different spacings. Their results showed that increasing the Rayleigh number increases the area-averaged heat transfer in line and the following correlation is suggested:. Their results have shown that the Nusselt number on the middle cylinder in a three-cylinder array increased compared to the lower cylinder and that of the upper cylinder increased too with no much difference than that of the middle cylinder.

Experimental investigation was performed on laminar natural convection from two cylinders arranged in vertical array with a spacing range 2 and 6 of the tube diameter by Chouikh et al. The cylinders were kept isothermal with Rayleigh numbers range 10 2 and 10 4. Their results were consistent with those in the literature. It was also observed that the upper cylinder in the array of large spacing had an enhanced numbers, but at close distances, it had a reduced Nusselt numbers.

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On the other hand, for the same spacing, a direct enhancement of heat transfer from the upper cylinder with Rayleigh number was obtained. Their observation about the heat transfer from cylinders in the array agreed with the flow field around the cylinders. Numerical work was reported on natural steady-state laminar convection heat transfer from horizontal cylinders arranged in vertical array by Corcione [ 37 ] in air. He has reported numerical correlations valid for the whole array of cylinders and also for any individual cylinder in the array.

He compared his numerical results with those in the literature, and it was found satisfactory. In addition, it was found that at any investigated Rayleigh number, degradation was generally the rule at the smaller tube spacing, while enhancement predominated at the larger ones.

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Sadeghipour and Asheghi [ 38 ] have reported an experimental study for natural convection heat transfer from horizontal isothermal cylinders in vertical arrays of 2—8 at a low Rayleigh number of , , and Their results showed that the heat transfer coefficient for the lowest cylinder is the same as that of a single horizontal cylinder. Therefore, they found a direct proportional between the number of cylinders in the array and the Nusselt numbers of the array.

Park and Chang [ 39 ] have studied numerically the interactive natural convection heat transfer from a pair of vertically separated horizontal circular cylinders of equal diameter.