## Abstract

A new experiment is described on heat transfer in a Rayleigh–Bénard chamber modulated by an external flow. The results show a linear response of thermal gradients to small Reynolds numbers, here resolved with micro-Kelvin resolution at room temperature. It approaches a hyperbolic behaviour at large Reynolds numbers. This response is nowhere reminiscent of singular behaviour in boundary layer theory. It may, instead, represent a cellular model for describing heat transfer in porous media. An application to correlating residential gas-energy usage and local weather data is included.

## 1. Introduction

The phenomenology of heat transfer in moving media is surprisingly rich, especially around or above critical values for buoyancy-driven convection in closed geometries. It is perhaps best known for spontaneous pattern formation across critical points of stability and, for larger forcing, the appearance of spatio-temporal chaos (Bodenschats *et al*. 2000; Mutabazi *et al*. 2005).

In this paper, heat transfer is considered in a completely different parameter regime that has largely remained unexplored. In what appears to be a new experiment, the response is measured of internal heat transfer to perturbations produced by an external flow. The external flow passes through the Rayleigh–Bénard chamber via two apertures located symmetrically about an embedded hotplate. The response is studied in the limit of small and large excitations and expresses the results in a dimensionless Nusselt number for the induced thermal gradients in the dimensionless form. The thermal gradients are resolved down to a few micro-Kelvin (at room temperature) using a novel modulation technique (van Putten *et al*. 1994, 1995, 2002).

We report on a discovery of *linear* asymptotic behaviour at small Reynolds numbers. It stands in sharp contrast with the well-known square-root behaviour of King's law in thermal anemometry (King 1914; Lomas 1986). The excitation in a modulated Rayleigh–Bénard is therefore much more gradual than that of a hot wire or plate in open geometries. Furthermore, the observed asymptotic behaviour at large Reynolds numbers approaches a hyperbolic function. The observed response curve is *nowhere* reminiscent of King's law. Instead, the results show that subcritical Rayleigh–Bénard chambers display a remarkably soft response, described by completely *regular* (smooth) behaviour across a wide range of Reynolds numbers of the applied flow.

A modulated Rayleigh–Bénard chamber may represent a cellular model for porous media, subject to forced flow. The results reported in this paper predict a linear relation between Nusselt and Reynolds number in the presence of thermal stratification perpendicular to the direction of flow. This appears to be in reasonable agreement with recent experimental data (Zhao & Song 2001).

The smooth response of a modulated Rayleigh–Bénard chamber suggests some promise for practical use. As a contemporary topic, the experiment is applied to measure residential energy consumption by correlating natural gas usage with the local weather. This real-time picture is summarized in a two-parameter weather-sensitivity analysis. Such analysis might reveal some energy-saving opportunities which would otherwise be unnoticed.

In §2, the set-up of the experiment is described including the operational parameter settings. A Nusselt number is used to express thermal gradients in dimensionless form in §3. Experimental results for a range of Reynolds numbers are reported in §4 accompanied by analysis of the observed asymptotic behaviour at small and large Reynolds numbers. A contemporary application is discussed in §5 on residential energy–weather correlations. Some concluding remarks are included in §6.

## 2. A modulated Rayleigh–Bénard chamber

As a novel experiment, a Rayleigh–Bénard chamber is exposed to an external flow passing through two small apertures. The apertures are located symmetrically about an embedded hotplate. To facilitate control of the experiment, the external flow is extracted in the form of a small fraction of about 0.1% from a much larger flow, passing through an aluminium honeycomb structure embedded in an external duct. The pressure drop across the honeycomb structure produces the desired excitation, which scales linearly with the flow through the duct (especially at low Reynolds numbers). The pressure drop across the honeycomb structure reaches a maximal value of approximately 2 mbar at maximal flow rates.

The Rayleigh–Bénard chamber is of small dimensions. It hereby operates subcritically for all calibration fluids used and the results become a function of Reynolds number, independent of pressure. To be precise, there exists a critical value for the Rayleigh number *Ra* below which the fluid between a heated bottom plate and a cold top remains stable at rest (Chandrasekhar 1961; Chapman 1984). The Rayleigh number is given by the product of the Grashof number(2.1)of the chamber and the Prandtl number of the medium. Here *g*=9.86 m s^{−2} is the gravitational constant; *β*=Δ*T*/*T* denotes the relative difference between the temperature of the embedded hotplate and the ambient temperature (which sets the temperature of the surrounding walls), where the former is about 75°C. The hotplate is separated from the wall above by approximately mm. Furthermore, *ν*=*μ*/*ρ* is the kinematic viscosity of the medium with density *ρ*. The Prandtl number of a fluid is the ratio of the diffusion coefficients of momentum and heat. Quite generally, a vapour satisfies *Pr*=0.6–0.8, notably *Pr*=0.67 for monatomic fluids such as Ar and He, *Pr*=0.72 for nitrogen and, somewhat temperature-dependent, *Pr*=0.76 for CO_{2}. Thus, for N_{2}, Ar, He and CO_{2} we have, respectively, *Ra*≃100, 50, 0.6, 250, all below the critical value *Ra*_{c}≃1000 for the onset of buoyancy-driven convection. Nevertheless, some convection at all Rayleigh numbers is inevitable in the open space between the edges of the plate and the adjacent vertical walls. However, experiments with CO_{2} show a pressure dependence of no more than 1% bar^{−1} throughout the range 1–5 bar. This rules out any significant role by convective transport phenomena on the basis of buoyancy, consistent with the aforementioned subcritical Rayleigh numbers.

Passing an external flow through the chamber creates an asymmetry in internal heat transfer. The asymmetry can be detected in the form of thermal gradients in the embedded hotplate. To this end, the hotplate is endowed with integrated p-type resistors configured on a Wheatstone bridge in silicon technology (van Putten & Middelhoek 1974). Low-concentration boron-doped silicium (Müller 1971; Lovel *et al*. 1976; Sze 1994) introduces a large relative temperature coefficient of 5.8×10^{−3} K^{−1}. A Wheatstone bridge configuration powered by few volts hereby measures a 1 μK temperature difference with 10 nV output voltage.

Detecting thermal gradients with micro-Kelvin resolution requires full additive drift elimination in the electrical output signal of the silicon chip. This is achieved by modulating the applied flow in alternating directions and subtracting the results from two successive measurements (van Putten *et al*. 1994, 1995, 2002). The result is a near-perfect state of quiescence under no-flow conditions with a residual offset of typically 50 nV or less. The equivalent uncertainty in thermal gradients is of the order of a few micro-Kelvin *at room temperature*. The corresponding colour resolution is 10^{−8}.

## 3. A Nusselt number for thermal gradients

To quantify the response of the Rayleigh–Bénard chamber in terms of the Reynolds number of the externally applied flow, a commensurate dimensionless expression for the induced thermal gradients can be used. The latter are detected by the embedded plate, which also contains the heating elements for an elevated temperature. The plate hereby provides two independent quantities. The thermal gradient is manifest in differences in local heat flux, which are picked up by the thermistors in the integrated Wheatstone bridge. The total heat flux is determined by the sum of all electrical dissipation. A normalized expression for the thermal gradient results from the ratio of the induced asymmetry in heat flux to total dissipation (van Putten *et al*. in press). This definition is a slight departure from the common practice of normalization with respect to conductive heat flux only. The latter corresponds to heat flux in a state of quiescence, which is incommensurate with the present goal of using a pair of simultaneously measurable quantities.

The Nusselt number is dimensionless and is a function only of Reynolds number, which may include fluid-dynamical parameters such as Prandtl number (in changing to different media), but not Grashof number (2.1) in the subcritical regime. Thus, the Nusselt number is generally correlated to the Reynolds number(3.1)An equivalent set-up is achieved in seeking the response to the Peclet number, given by the product of the Reynolds and Prandtl number (Zhao & Song 2001). However, the focus here is primarily on flow-dependence, and secondarily on sensitivity to different media.

The functional form of the Nusselt number (3.1) on the basis of the output signals of the embedded sensor is determined as follows.

Power dissipated in the measurement and heater elements with typical resistance *R* satisfies and , respectively, where *V*_{m} and *V*_{h} denote the voltages across their respective bridges. The induced asymmetry in heat flux *δP*_{m} is manifest in the electrical output signal *V*_{ADM} of the measurement Wheatstone bridge. By elimination of the additive drift, it follows that *δP*_{m}=2*V*_{ADM}*V*_{m}*/R*. By this procedure, *V*_{ADM} is an odd function of the applied Reynolds number of the flow, while the simultaneously measured *P*_{h} is an even function of the same.

Measurement of the Nusselt number in terms of the sensor output signals must satisfy the *invariances* defined by (3.1). In particular, it must not explicitly depend on ambient temperature. Quite generally, the induced Nusselt numbers are small. Consider, therefore, the implicit definition(3.2)with free parameters *A*, *f* and *c*. The parameter *f* is determined by experiment to achieve the aforementioned temperature invariance over an extended flow range when *c*=1 (§4). It appears that *f* is not fundamental, but rather a tuning parameter which corrects for heat loss via the sensor holder. Thus, a fraction *f*<1 of the total energy dissipation *P*_{m}+*P*_{h} represents a correction for normalization with respect to heat flux to fluid only. This correction factor can be attributed to the details of mounting of the sensor, which includes wire-bonds and two dots of glue for a robust fixing. The quadratic term in the denominator on the right-hand side represents a nonlinear interaction, serving as an approximation in the range of the anticipated small values of *Nu*. For transparency, a scale factor *A*=1 mV(2*V*_{m})^{−1} is used in (3.2), giving(3.3)parameterized by *q*=(*p*+*f*)/*f*/*c*, *y*=*v*/*q*^{3/2}, where *p*=*P*_{h}/*P*_{m} and .

## 4. The Nusselt response curve

The Nusselt response to Reynolds number of the applied flow is determined experimentally using a calibrated flow-generator equipment.

Low-flow measurements are performed with a computerized cylinder–piston combination. It is controlled by a linear motor with feedback over a digital ruler with 2 μm resolution, corresponding to 30 μl. High-flow measurements are performed in a closed-loop wind tunnel with an Elster–Instromet rotary reference meter, complemented with pressure and temperature sensors for conversion to normal litres per minute referenced to 20°C at 1 bar. The consistency of the Elster–Instromet with the piston–cylinder combination has been verified to be 0.56%. It is in reasonable agreement with the error of 0.4% provided by Elster–Instromet, which is essentially independent of the flow rate above an initial threshold.

### (a) Low-flow measurements

Figure 1 shows an overview of the measurement results. In quiescence, the sensor output is essentially white noise with a residual mean (offset) of about 50 nV, corresponding to about 5 μK and a Nusselt number of about 10^{−5}. This behaviour is typical for all five realizations built. It should be mentioned that this resolution is achieved only by symmetrization throughout the device, including electrostatic settings in the modulation technique. A single electron on a 1 pF parasitic capacitance, for example, can cause interference of up to 160 nV. The software ensures that the sum of the voltages of the digital output ports of the microprocessor remains constant, used or unused. It effectively prevents interference with the 24 bit AD converter, which is located on the same board. The resolution of the latter is 2.5 nV. While we have not been able to identify the cause of the residual offset, it is sufficiently small for studying low-flow asymptotics.

The low-flow measurements are shown in the subsequent two windows, illustrating detection at 20 ml min^{−1} by time-averaging over 5 min. A sequence of the low-flow measurements, performed in five realizations, is shown next. The results clearly reveal a *linear* asymptotic behaviour towards quiescence (in all realizations).

### (b) High-flow measurements

High-flow measurements are shown in the middle window to the right, represented by a polynomial interpolation (continuous curve) around some 20 set-points (scattered clouds) of the closed-loop wind tunnel. These high-flow measurements have been performed around 10 and 30°C. To study temperature dependence, the inverse of (3.1) is considered, noting that the Reynolds number of the applied flow is proportional to normal litres per minute divided by dynamical viscosity (at the working ambient temperature). The resulting relationship between Reynolds and Nusselt numbers reaches temperature independence to within small error of reading by fine-tuning *f* (window in at bottom-left). This result extends over an extended range in Reynolds numbers by virtue of the quadratic nonlinearity in (3.2), and is shown for calibration fluids Ar and N_{2} (similar results are obtained for CO_{2}).

The asymptotic behaviour at high flow rates is well described by a hyperbolic tangent according to(4.1)where *κ* denotes a scale factor. This hyperbolic behaviour has been observed for all our calibration fluids, following fine-tuning of *f* for each. The bottom window to the right shows the normalized asymptotic behaviour for both N_{2} and Ar.

### (c) Sensitivity to different fluids

Temperature independence for the different calibration fluids Ar, N_{2} and CO_{2} shows the following correlations:(4.2)where *λ*_{n} denotes the thermal conductivity of nitrogen. We have not detected significant dependencies on Prandtl number, perhaps owing to the relatively similar Prandtl numbers for these calibration fluids. On this basis, the Nusselt response curve (2.1) in our realizations and extended to moderately different media satisfies(4.3)independent of Grashof and Prandtl numbers, where fine-tuning of *f* for each individual sensor is implicit.

## 5. Residential energy–weather correlations

The observed regular behaviour of the Nusselt response curve over a wide range of Reynolds numbers may be exploited in a contemporary problem of monitoring residential gas-energy usage correlated with the local weather. In a recent field test, we used it to record the gas-energy usage in an existing residence over the period from May to December 2006.

For this purpose, a compact embodiment of the modulated Rayleigh–Bénard chamber has been constructed, including software for a calibrated polynomial interpolation of the inverse *Re*=*F*^{−1}(*Nu*) on the basis of our experimental data. We neglect the variations in gas composition (*κ*=1). The Reynolds measurements can be converted to mass flow according to(5.1)where *μ* denotes the dynamical viscosity of natural gas. Features such as low-pressure drop, orientation independence (less than 1% between horizontal and vertical up), insensitivity to choice of connections (less than 0.5%) and small size of 9×9×9 cm^{3} facilitate installation in the field. A build-in Bluetooth link is used for wireless display and data logging onto a PC. The sampling rate, set by the modulation frequency of the drift elimination method, is 0.4 Hz.

Figure 2 shows a snapshot of real-time gas-energy usage. It displays common use of residential facilities: taking showers; baking eggs; and washing dishes. Use of residential facilities creates a baseline level, approximately 1 m^{3} of natural gas per day. For the best use of data, the results are correlated with the local weather. Daily mean outside temperatures are provided by the national weather service based on a nearby local weather station. The measurements cover a relatively warm period between 23 May and 22 June, which included two cold periods around 27 May and 1 June. These cold periods gave rise to appreciable heating, which becomes apparent by a strong correlation between energy usage and mean outside temperature.

The energy–weather correlation shows a smooth transition at an outside temperature *T*^{*} of approximately 15°C. Above this temperature, energy usage asymptotes to a constant baseline associated with use of residential facilities (other than heating), while below, energy usage is dominated by heating. Using a best-fit cubic polynomial *y*(*T*, {*T*_{i}}) approximation to the data on daily gas-energy usage *y* and mean outside temperatures *T*_{i}, the relative sensitivity of energy usage to room temperature is(5.2)Here, we exploit the fact that energy usage in heating depends effectively on the difference between inside and outside temperatures. (There exists scatter, largely due to winds, rain and cloudiness.) Furthermore, a choice of outside temperature *τ* well below *T*^{*} can be used to infer a measure for home energy efficiency,(5.3)The largely linear correlation between energy usage and *τ* renders the result rather insensitive to the choice of *τ* when chosen well below *T*^{*}. Below, *τ* is fixed at 10°C.

The energy–weather data of figure 3 show *α*=36% K^{−1} (May–June), and subsequent observations show *α*=0% K^{−1} (July–August) and *α*=15% K^{−1} (September–October). These strong variations reflect a seasonal change in weather. The results suggest that savings in excess of 10% can be realized by small changes in home climate temperature of less than or approximately 1°C. Upon a commensurate increase in relative humidity, the overall level of home comfort can be kept constant. The observed home energy efficiency m^{3} K^{−1} may be used to evaluate the relative merits of specific home improvements, for example, double-glazed windows and wall insulation, for long-term monitoring of home energy efficiency and for comparing the same between different homes.

## 6. Conclusions

We report on the discovery of a regular Nusselt response curve of a modulated Rayleigh–Bénard chamber to an external flow. The result is remarkable in that it features a linear asymptotic behaviour at small Reynolds numbers in complete departure from the well-known square root behaviour in King's Law for hot-wire anemometry and its hotplate equivalent. Linearity appears to be in reasonable agreement with experimental data on forced flows through porous media (Zhao & Song 2001). This suggests that the proposed experiment might be a suitable cellular model for porous media in the presence of thermal stratification orthogonal to the direction of flow.

At high flow rates, the response curve asymptotes to a hyperbolic tangent. The Nusselt response curve is hereby well defined over an exceptionally large dynamic range of Reynolds numbers of the applied flow.

This observation suggests a contemporary application to measuring residential gas-energy usage correlated with the local weather. For this purpose, a small embodiment of the experiment has been developed with wireless read-out to facilitate applications in the field. The field test is summarized in a two-parameter weather-sensitivity analysis. The data indicate that energy savings can be realized by making modest adjustments in home climate temperature, preferably combined with adjustments in relative humidity to preserve home comfort. The data also quantify home energy efficiency, which may be used to evaluate the relative merits of specific home improvements, to detect long-term trends in home energy efficiency and to compare the quality of different homes. Overall, weather-sensitivity provides a close-up view on potential opportunities for saving energy, which otherwise might be left unnoticed.

## Acknowledgments

The authors thank W. Verhoeven, E. Postma and B. Schelen for their technical assistance in the experiment and P. F. A. M. van Putten and M. J. A. M. van Putten for their stimulating discussions. This research is partially supported by the EU Craft Programme under Contract nos. BM-ST-9112 and QLK4-CT-2002.

## Footnotes

- Received March 19, 2007.
- Accepted June 18, 2007.

- © 2007 The Royal Society