- Theory of Heat Transfer with Forced Convection Film Flows
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- Convection From a Rectangular Plate

Mixed convection is a mixture of natural and forced convection occurring at low fluid speeds. Natural upward convection is produced by an upward-facing plate which is warmer than the fluid, or a downward-facing plate which is colder than the fluid. Natural downward convection is produced by an upward-facing plate which is cooler than the fluid, or a downward-facing plate which is warmer than the fluid. A correlation approximate equation relating dimensionless quantities can model an isothermal surface or a constant heat flux through the surface.

The correlations for these two regimes usually differ only in coefficients and additive constants. This article addresses convection from isothermal plates. A forced convection correlation can be for local heat flow or for heat flow averaged across the surface. Section " Laminar-Turbulent Progression " gives the derivation for average heat flow from the local heat flow which appears in several texts. While it produces the customary averaged correlations for purely laminar or purely turbulent flow, the correlation it produces for transitional flows requires a Reynolds-number threshold from measurement of the specific configuration under investigation, which limits its predictive ability.

Convection correlations can be derived from theory, numerical simulation, or experiment empirical. The accepted correlations for natural convection are empircal. At the plate surface, the fluid velocity is near zero. At some distance from the plate surface the fluid velocity approaches the bulk fluid velocity. In between is the boundary layer. Flow in the boundary layer is laminar or turbulent. The h values for turbulent regions of the plate boundary layer are larger than h values for laminar regions.

Natural convection is strongly affected by the inclination of a plate. The natural convective surface conductivity from a vertical plate is between the conductivities of upward facing and downward facing level plates of the same size. Upward convection has the largest heat flow. Forced convection is insensitive to inclination.

Forced convection is greater for rough surfaces than for smooth ones. Natural convection is insensitive to roughness whose mean height is much less than the dimensions of the plate. About Fig. The graph is based on [84] which gives the dimensions of the plate as a 0. Without a quantitative relation between surface conductance and plate size, application to surfaces as large as roofs is uncertain.

Two of the lines are bent: "Clear pine" and "Glass and white paint on pine". The formulas I fit for glass and pine are:. The first task is to separate the surface conductivity due to convection from that due to radiative transfer. Measurements of convective surface conductance take place in a large enclosure whose temperature matches the fluid air inside it. To the right is plotted the difference in blackbody radiance between the plate and enclosure divided by the temperature difference.

If the surface and test apparatus both have emissivity of 0. But the suite of materials tested do not necessarily have the same emissivity. There are multiple books and web-pages offering tables of emissivity for building materials.

Of the five excerpted below, those that cite a source cite handbooks or textbooks, so they are not original data. None of these five sources provided a value for stucco. Source [70] gives emissivities which are less than those given by the other sources; so I didn't use it in finding consensus values. For six of the seven materials the spread in emissivity values is small enough for consensus values. Although non-metallic paints are given emissivity values from 0. The emissivity for plaster. Thus this is a plot of surface conductivities due only to convection, ie. These two points, with radiative conductance subtracted out, are marked with dots on the graph.

They are closest to the curve for clear pine. Rowley, A. Algren, and J. Blackshaw is the source for Fig. Their test plate was inserted flush with interior into a. Rowley at al take and report measurements with the flow shut off. In such a long duct with no forced airflow, equilibrium must involve conduction through the walls of the insulated duct. As shown to the right, natural convective surface conductance is much more sensitive to temperature difference than mean temperature average of surface and air temperatures ; but only mean temperatures are revealed in their graphs.

Two of its rows are for still air. Here they are converted to metric units. The surface conductance column includes both the convective and radiative surface conductances. The radiative surface conductance depends on the inner wall temperature of the duct, which is not provided. Rowley et al write:. So how would convection from a vertical plate in a horizontal duct compare with convection from an open-air vertical plate?

Inside the duct heated rising air is obstructed and so accumulates in the duct. With processes driving h in opposite directions, these still-air measurements can be justified as neither upper nor lower bounds for natural convective surface conductance. Particularly disappointing is that the relationship between sub-millimeter surface roughness and natural convection from a flat vertical plate remains unquantified the lack of measured emissivities also contributes uncertainty.

Surface roughness is treated only for forced convection inside pipes and ducts.

For external plates only horizontal and vertical orientations are covered. It has no example calculations of unobstructed convection from a plate. It appears that with the move to a more theoretical treatment, Fundamentals lost practical applicability to roofs. The intent of this article is to develop a theory sufficient to predict convection from rough inclined rectangular roofs. For dry air, Kadoya, Matsunaga, and Nagashima[93] is the authoritative source for viscosity and thermal conductivity. The example calculations in this text were performed before the model incorporated humidity.

The moist air values of these properties are computed by combining the values for dry air and water vapor in proportion to their presence in the moist air mixture, in some cases with correction factors. Both sources contain errors; the obvious errors don't occur in the corresponding quantities. Wexler[94] is the authoritative source for water-vapor partial pressure versus temperature.

So humidity will not be a major influence on air's properties at outdoor temperatures. Model moist air as a mixture of ideal gases. P sat is the partial pressure of saturated water vapor at temperature T F Kelvins :. But P sat must be evaluated at the bulk fluid temperature T F because the amount of water-vapor in the fluid doesn't change when heated to intermediate temperature T. If T S is colder than T F , then condensation may occur. Simulation of roofs can side-step the issue by constraining the roof temperature to not drop below the ambient dew-point temperature.

Because of water's high latent heat, this treatment should not result in large errors. The specific heat at constant pressure c p for dry air and water vapor each vary little over our temperature range. But the mixture at a given relative humidity is sensitive to temperature. These formulas from Tsilingiris[91] take temperature t in degrees Celsius.

The "trunc" traces are with the two higher order terms of c pa t dropped. Both [91] and [92] give formulas for the viscosity of dry air. The formula from [91] is:. A linear fit to [93] data over the range of interest is:.

## Theory of Heat Transfer with Forced Convection Film Flows

The viscosity of saturated water vapor is less studied. Morvay and Gvozdenac[92] give a formula for the viscosity of water vapor which matches Tsilingiris' graph well. Morvay and Gvozdenac introduce a parameter they call absolute humidity , the ratio of masses of water vapor and dry air:.

The authoritative formula for the thermal-conductivity of dry air is from Kadoya, Matsunaga, and Nagashima[93] :. Over the roof range of interest this is hardly different from:. Tsilingiris[91] gives a formula for the thermal conductivity of water vapor:. Morvay and Gvozdenac[92] also give a formula for the thermal conductivity of water vapor:. They diverge mostly at the dry end of the curve, where k v has insignificant effect on the moist air thermal-conductivity k m. Over the roof range of interest the latter curve is hardly different from:. For an ideal gas with pressure held constant, the volumetric thermal expansivity i.

For natural convection from a horizontal plate it is:. For a flat rectangular plate, L c is the length of the side parallel to the direction of flow. Below are formulas for dimensionless quantities governing convection along with ranges for air under the conditions:. Ranges for vertical plates, where L c is the height, are marked with a subscripted V.

Ranges for horizontal plates, where L c is the ratio of area to perimeter, are marked with a subscripted H. The Prandtl number is insensitive to L c , depending only on fluid air properties. The L c for Reynolds and Nusselt numbers in forced convection is the length of the plate in the direction of flow. Re will be zero when the wind-speed is zero. It is worth noting that a larger h value doesn't necessarily correspond to a larger Nu value because Nu gets divided by different L values. The range for T9. About T9. Roofs aren't vertical. But this formula is a component for modeling the cases of roofs which are neither horizontal nor vertical.

On this subject their paper says only:. The maximum relative error for Nu in T9. Formulas T9. A function dashed line constructed like T9. Despite being cited in footnote b , there is no support for T9. Whitaker [71] also perpetuates T9. With the L c values which generate curves matching Fig. In Natural convection heat transfer below downward facing horizontal surfaces [77] , Schulenberg develops correlations for downward convection for an infinite strip and circular plate I added the equivalent T9. In Schulenberg's derivation, the characteristic-length used for computing the Rayleigh-number Ra is R :.

The graph to the left shows what the various correlations predict for a smooth square plate with side length from 0. The middle trace is for a vertical plate h T9. The bottom three traces are for a horizontal downward facing hot plate or upward facing cold plate. As discussed above , h T9. The upper left plot shows that all of the conductances grow with decreasing length of the square's side; so simply adjusting the square's dimensions will not improve the fit.

Would non-square rectangular plates bring these points closer to their targets? For the vertical plate, the width does not affect its surface conductance h T9. So lets leave the height fixed and adjust the width. Therefore downward facing surface conductance cannot be reduced by changing the aspect-ratio from 1 square. The plot of surface conductance is linked to a larger, high-resolution version. And that is the behavior of h UHT3. As fluid heated at the bottom of a vertical plate rises, it stays near the plate, creating a thermal boundary layer which is wider at the top of the plate than at the bottom.

As the plate tilts up, buoyancy of heated fluid pulls parcels away from the plate, increasing convective transport and surface conductance. Because the layer of heated fluid is thick at the upper end of a vertical plate, as the plate is tilted downward, the further thickening of the boundary layer does not much affect convection. Further downward-tilting reduces the buoyancy of fluid parcels traveling along the plate.

A correspondent wonders what the convection from both sides of a solar-cell panel is. For natural convection, the double-sided panel shows much less variation with angle than the single-sided case. Natural convection correlations for inclined rectangular plates involve characteristic-lengths derived from L H and L W. The derivation of L eff is in the section Natural Off-Axis Convection Each correlation is computed using its own characteristic-length.

These equations assume that T9. The section " Natural Convection from a Rough Plate " addresses the effects of surface roughness on T9. For a flat plate there is much less experimental data available for forced convection than for natural convection. To accurately measure forced convection requires all the equipment necessary for measuring natural convection incorporated in a very uniform, laminar wind-tunnel.

Wind speed and direction add two independent variables, multiplying the number of measurements which must be taken. The common correlations for forced convection are due to Blasius ' mathematical analysis. Recalling Dimensionless Quantities , we introduce local versions of Nu and Re:. The plate orientation is not specified. L is the length of the plate [assumed in the direction of fluid flow]. No citations are given for these 5 correlations. Re c does not appear in the text. Local Re changes with distance x from the leading edge of the plate. Casting T8. Integrate along the direction of flow from leading edge to L c ; then divide by L c to obtain the average value of h for the plate.

FH can be cast as a dimensionless correlation by reversing the previous substitutions:. For a characteristic-length L c of 0. If the impinging flow were laminar, the Reynolds numbers for this size smooth plate are low enough that the boundary layer would not progress to turbulence. If the data is valid, then turbulence must have been present confirmed in Convection in a Duct. Perhaps Re c has a smaller value in this situation. In the abstract for A comprehensive correlating equation for forced convection from flat plates [86] , Stuart Churchill writes:. On Fig.

Correlation T8. It is troubling that, although there are many points on the turbulent asymptote, there are only nine measured points on the laminar asymptote. Those nine points are reported by only two of the sources: Parmelee and Huebscher; and Jakob and Dow. The lack of points on the laminar asymptote from the other two experimental sources one has one close point could indicate that their apparatus was producing only turbulent, not laminar flow. That could have been caused by turbulence in the impinging flow, surface roughness, dullness of the leading edge, or vibration [85].

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With the small number of laminar points, there doesn't seem to be enough evidence in Fig. The FC curves cyan and green stay under the turbulent asymptote because they model part of the plate being in laminar convection contributing lower Nu values to the average. This stands in support of FC over Churchill's formula Given how difficult it seems to be to produce high Re c values, for engineering purposes the pointwise maximum of T8. Note that Churchill's data points are for smooth plates.

Transition to turbulence seems to be a phenomena like supercooling. When pure water lacks nucleation particles, it can be cooled below its freezing point. In view of the insensitivity of the results to the angles of attack and yaw, a single formula can be given which represents all the data to an accuracy of 2. The formula is. L c is the side of the square, 7. V ranged from 4. Thus they tested only laminar flow. Correlation FCLY computes the same convective surface conductance as Sparrow and Tien do for their example application 1.

If that is correct, then FCLY won't apply to the large plates because natural convection from them is turbulent. Given T8. Sparrow and Tien measured mass transfer, not heat transfer, and presented their data only in terms of dimensionless quantities. The information in their paper seems insufficient to resolve this discrepancy.

An additional problem with this paper is detailed below. Sparrow and Tien's claim that the sensitivity of laminar convection to flow angle is less than 2. The worst case variation should be in comparing an edge-parallel flow h p to a diagonal flow h d. Nothing they report can be tested against established correlations. They claim:. The pipe and duct correlations assume that a laminar flow enters the test section from open space.

Their 5. At the minimum non-zero flow they used for measurements, 3. So the flow is certainly turbulent. Moreover, flowing through 5. The duct correlations assume that the test section is heated on all sides; but only one side is heated in the apparatus of Rowley et al. Unlike natural convection, forced flow is not much affected by heat convection. So the surface conductance for one heated side should be close to that for four sides.

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However, forced flow is affected by the surface roughness. So the surface-conductance values for rough four-sided test segments may differ from rough one-sided segments with three smooth unheated sides. The three roughest surfaces have nearly straight trajectories, as expected from the Moody Chart. As cautioned earlier, the Fig. There appears to be no published theory or measurements relating mean-height-of-roughness to forced convective heat transfer of open plates Rowley et al [84] measured a duct.

Someone trying to compute forced surface conductance of a rough plate has only the smooth plate conductance as a lower bound; the rest will be guesswork.

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- Convection From a Rectangular Plate!

As mentioned earlier , measuring convective heat transfer from flat plates is difficult. Measuring forced convective heat transfer from flow inside pipes and ducts is comparatively easy; theory and experimental data are abundant. Can forced convection pipe correlations be adapted to flat plates? Combining equation T8. In the Colburn analogy, Nu is proportional to the product of the friction factor and Re. Thus horizontal sections of the friction factor curve produce Nu and h values which increase linearly with Re. DF5 is shown in red on the Moody chart.

It is a good match for the middle of the smooth pipe curve. Conversely, when the wall roughness is too small to disrupt the viscous sublayer, the convective surface conductance should be the same as for a smooth plate under the same conditions. At the scale of the viscous sublayer, the inner radius curvature being perpendicular to the flow in a large diameter pipe should be indistinguishable from a flat plate.

Because most of the convective thermal resistance occurs close to the surface of a pipe or plate, it is reasonable to expect the turbulent correlations for pipe and plate to be related. Consider the similarity of Colburns's analogy T8. DF11 in blue is parallel to DF5 in red because Re has the same exponent in both.

DF9 in cyan looks like it may be parallel to the asymptote of the low Re end of the solutions of the Colebrook-White equation. If D and Re in the Colebrook-White equation were scaled to move D5 to D11 then this scaled Colebrook-White equation would yield smooth flat plate correlations using the Colburn analogy. Although it must have the correct relation to dimensions, the scale of characteristic-length is otherwise a matter of convention.

The forced-convection correlations are usually monomials in Re and scale easily. With their characteristic-lengths being perpendicular, the geometry of pipes and flat plates are so different from each other that it is not unreasonable that scaling may be necessary in order to make correspondences between them. Scaling the characteristic-length and Reynolds number by 9. Energy conservation in petrochemical Industries. Pollution control in petrochemical industries. New trends in petrochemical industry. Planning and commissioning of a petrochemicals complex.

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## Convection From a Rectangular Plate

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