Comparison of time-resolved heat transfer characteristics between laminar and turbulent convection with unsteady flow temperatures

International Journal of Heat and Mass Transfer 84 (2015) 376–389

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Comparison of time-resolved heat transfer characteristics between laminar and turbulent convection with unsteady flow temperatures Cun-liang Liu a,⇑, Jens von Wolfersdorf b, Ying-ni Zhai c a

School of Power and Energy, Northwestern Polytechnical University, You Yi Xi Lu 127, 160#mailbox, Xi’an 710072, China University of Stuttgart, Institute of Aerospace Thermodynamics, Pfaffenwaldring 31, 70569 Stuttgart, Germany c Department of Mechanical & Electrical Engineering, Xi’an University of Architectural & Technical, Yan Ta Lu 13, 3#mailbox, Xi’an 710055, China b

a r t i c l e

i n f o

Article history: Received 16 October 2013 Received in revised form 24 April 2014 Accepted 5 January 2015

Keywords: Laminar convection Turbulent convection Unsteady flow temperature Time resolved heat transfer coefficient Transient heat transfer measurement

a b s t r a c t Experimental investigations have been carried out to compare the time-resolved characteristics between laminar and turbulent heat transfer for steady boundary-layer-type flows with different unsteady flow temperatures. Time-resolved heat transfer coefficients at two locations were measured with two transient heat transfer measurement methods. One is the frequently used transient heat transfer measurement method with an assumption of time-wise constant heat transfer coefficient, and the other is the Cook–Felderman method without such assumption or restriction. Correction methods are applied for the direct measurements of time varying flow temperature and calculated surface heat flux to eliminate the influences of thermocouple thermal inertia effects. The results show that the temporal behavior of heat transfer coefficients is different between laminar and turbulent convection under unsteady flow temperatures. The change of flow temperature in step or multi-step form produces differences in the laminar heat transfer coefficient, while has very weak influence on the turbulent heat transfer coefficient. The heat transfer coefficient in a thermally transient process will be in steady state when the boundary layer is laminar and the flow temperature does not fluctuate with high frequencies and large amplitudes. However, the heat transfer coefficient varies intensely in the same frequency with the sinusoidally varying flow temperature when the boundary layer is turbulent. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Many thermal systems, such as regenerative heat exchangers, nuclear reactor fuel rods and turbomachines, are often subjected to time varying thermal boundary conditions, for instance, periodically varying flow temperatures. Knowledge of the unsteady heat transfer in a steady flow with unsteady flow temperature is of much interest and important for many engineering applications where hot and cold fluids pass in succession. A commonly accepted assumption in unsteady convective heat transfer is that the heat transfer coefficient is basically determined by the flow conditions and less influenced due to variations in flow temperature. This quasi-steady assumption prevails in analyzing thermal responses of solid surfaces interacting by steady flow with unsteady flow temperatures. For applied computations, the quasi-steady approach utilizes a steady-state heat transfer coefficient to the transient conjugated convection process. Researchers believe such an assumption is applicable to turbulent ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (C.-l. Liu). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.01.034 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

flows but questionable for the laminar case. Therefore many investigations were carried out to address the quasi-steady assumption under laminar flow conditions. In an early study of Sparrow and De Farias [1], the effect of unsteady inlet temperature on the conjugated, laminar convection of a steady slug flow was studied in a flat plate channel. Numerical results were obtained for the time dependence of the Nusselt number which identifies conditions under which the instantaneous Nusselt number is virtually timeindependent. Comparisons between the quasi-steady approach and the numerical solution showed the validation of the quasisteady approach for a range of operating conditions. Sucec [2] presented an analytical solution for the transient, conjugated, laminar convection problem consisting of a plate interacting with a steady flow whose temperature varied sinusoidally with time. Comparison of the analytical solution with the quasi-steady approach indicated acceptable agreement at some conditions and inadequate accuracy in predicting time varying wall temperature in general. Sucec [3] proposed an improved quasi-steady approach for transient, conjugated, laminar convection problems of steady flows with time varying temperature. Agreement of the improved quasi-steady approach with the finite difference solution, which

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377

Nomenclature c Dh f h Pr q Re Rex t T U V x y

specific heat capacity [J=ðkg KÞ] flow channel hydraulic diameter [m] flow temperature’s frequency [Hz] heat transfer coefficient [W=ðm2 KÞ] Prandtl number specific heat flux [W=m2 ] Reynolds number based on the hydraulic diameter (¼ qg U g Dh =lg ) local Reynolds number based on the local distance to start point of boundary layer time [s] temperature [°C] velocity [m=s] volume [m3 ] streamwise coordinate coordinate normal to the plate surface

j l d

conductivity [W=ðmKÞ] thickness [m] time point [s]

Subscripts QS transient measurement method with assumption of quasi-steady constant heat transfer coefficient C–F Cook–Felderman method g gas flow i initial time t ¼ 0 s test plate surface y ¼ 0 w test plate corr corrected data raw directly measured raw data real data of real value abs absorbed by the thermocouple pex propagating into Perspex plate

Greek symbols q density [kg=m3 ]

were generated as benchmarks, was highly satisfactory for a reasonably wide range of conditions and much better than the standard quasi-steady approach. Sucec and Sawant [4] further applied this improved quasi-steady approach in the transient, conjugated heat transfer problem of fully developed, hydrodynamic steady flow in a parallel plate duct with a sinusoidal inlet temperature variation. Recently, Hadiouche and Mansouri [5] studied the unsteady conjugated heat transfer for a fully developed laminar steady flow with periodically varying inlet temperature. The effect of the flow temperature frequency on the Nusselt number was studied by a new analytical solution. The results showed that the instantaneous Nusselt number becomes highly time-dependent under higher flow temperature frequency and the quasi-steady approach is inadequate. Other studies on the transient conjugated convection of a laminar steady flow with time varying temperature can be found in [6–11]. Those studies solved the problem with analytical or numerical methods, and the behavior of periodic responses, including amplitudes and phase lags of oscillations in the wall temperature, flow bulk temperature and heat flux, was described. Investigations on the temporal behavior of laminar heat transfer under unsteady thermal conditions have been carried out experimentally [12] and theoretically [13,14]. The transient measurement results by Butler and Baughn [12] showed that the laminar heat transfer coefficient on a flat plate could vary significantly for different upstream surface temperature conditions. Lachi et al. [13] presented an analytical/numerical approximate solution for the laminar forced convection problem with a time variation in the heat flux density over a flat plate. The unsteady behavior of the convective heat transfer coefficient was clearly put into evidence in a short initial period of a changed thermal boundary condition. Cossali [14] reported an analytical/numerical study of periodic convection in a steady laminar flow due to a periodic wall heat flux density. The instantaneous heat transfer coefficient was found to vary with a periodic heat flux, especially at relatively higher frequencies of the periodic heat flux. These investigations were performed under steady flow temperatures without addressing the conduction effects within the wall. Naveira et al. [15] developed a hybrid numerical–analytical solution methodology for transient laminar forced convection over flat plates of non-negligible thickness which is flexible to accommodate arbitrary time variations in wall heat flux imposed at the fluid–solid interface. An improved lumped approach for the conduction analysis was presented to account for the heat fluxes at the

fluid–solid interface and the heat storage within the wall at an appropriate average wall temperature which is different from the surface temperature value. For unsteady convection at steady turbulent flow with time varying temperatures, relatively fewer investigations have been presented. Some related investigations can be found in [16–21]. Kim and Ozisik [16] studied the turbulent forced convection inside a parallel-plate channel with a periodically varying flow temperature and an uniform constant wall temperature. They analyzed the variation in amplitudes and phase lag of both, fluid bulk temperature and the wall heat flux. Kakaç and Li [17] presented analytical solutions for turbulent flows with a sinusoidally varying flow temperature and compared their analysis with experimental results showing a satisfactory agreement. Santos et al. [18] and Arik et al. [19] studied unsteady forced convection of turbulent flows under periodically varying temperatures in a circular duct with theoretical and experimental methods. The effects of wall thermal capacitance and wall temperature boundary conditions were considered. Hybrid analytical–numerical solutions for the thermal response of the fluid were provided. The studies in Mansouri et al. [20,21] analyzed the transient conjugated heat transfer process in turbulent duct flows subjected to periodically varying temperature with both theoretical and experimental methods. They showed that the quasi-steady method is only able to predict the temperature behavior inside the channel for lower frequencies of the flow temperature. Mathie et al. [22,23] presented a semi-analytical model to describe one-dimensional conjugate heat transfer with a timevarying heat transfer coefficient, taking into account fluctuations in the solid and fluid temperatures. The augmentation effect caused by the fluctuating temperature differences on the heat transfer coefficient was analyzed for two fundamental nonisothermal/diabatic flow problems with this model. Although the studies introduced above, most of which were performed with analytical/numerical methods, presented some information on the temporal behavior of the laminar and turbulent convections with unsteady thermal boundary conditions, timeresolved data of heat transfer coefficient is still very scarce, especially in the experimental data. Moreover, none of the above studies compared directly the heat transfer characteristics between laminar and turbulent convections with unsteady flow temperatures. Time-resolved experimental data is important to have a deeper insight to the temporal characteristics of the

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unsteady convection. In the present work, experimental investigations have been performed to obtain time-resolved heat transfer characteristics for a steady flow over a flat plate model with unsteady flow temperatures of aperiodic and periodic types. Comparison of the time-resolved characteristics between laminar and turbulent boundary layer type flows with unsteady flow temperatures will be presented. 2. Experimental approach 2.1. Experimental facility Fig. 1(a) shows a schematic of the test facility. The ambient air is drawn in through a dust filter and then enters a mesh heater. The heater consists of six stainless steel wire meshes which are connected in series to two 9.75 kW DC power supplies. The heating power is controlled by a wave generator which can produce sinusoidal wave forms at set frequencies to produce periodically varying flow temperatures. After the heater the air flows through a converging section which provides the cross sectional change to the test channel. The outlet pass is connected by a transition section to the main pipe. This piping system is connected to a vacuum pump which operates in suction mode drawing the inflow from ambient conditions. A more detailed schematic of the test channel is shown in Fig. 1(b). The test channel is a rectangular Perspex channel with

a width of 120 mm and a height of 150 mm. Downstream of the honeycomb which is used to smooth the flow, a Perspex test plate is placed in the middle height of the channel. The density qw , conductivity kw and specific heat capacity cw of the Perspex test plate are 1190 kg=m3 , 0.19 W=ðmKÞ and 1470 J=ðkgKÞ respectively. Thus the test channel is split into two flow channels with a width of 120 mm and a height of 60 mm, resulting in a hydraulic diameter Dh of 80 mm. The test plate is a 30 mm thick flat plate with a semi-elliptic leading edge and trailing edge. The length of the leading edge is 60 mm. Two Perspex cylinders with a diameter of 15 mm are flush plugged in the test plate at position P1 and position P2 which are shown in Fig. 1(b). P1 and P2 are at 160 and 750 mm downstream of the leading edge respectively. A T-type thin-film surface thermocouple with a thickness of 0.013 mm and a head diameter of 1 mm and response time of 2–5 ms as given by the manufacturer is cemented on the top of the Perspex cylinder with OMEGABOND epoxy to measure the test plate surface temperatures at P1 and P2. The shape of the thin-film surface thermocouple is shown in Fig. 1(b). Three T-type bead thermocouples with a wire diameter of 0.076 mm are laterally placed 10 mm downstream of P1 and P2 respectively to measure the flow temperature. All the temperature data are recorded by a fast temperature data acquisition system with a sampling frequency of 200 Hz which is fast enough to resolve the regular unsteady component of the temperature in the present experiment. The flow velocity in the test flow channel is measured with a hot-wire probe.

(a) Test rig

(b) Test channel Fig. 1. Schematic of the test facility.

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(a)

f=0.25Hz

(b) f=5.5Hz Fig. 2. Comparison of the raw and the corrected data of the measured flow temperature.

2.2. Heat transfer measurement methods

Tðy ¼ 0;tÞ ¼ T s ðtÞ ¼ T i þ

Transient heat transfer measurements were performed in this experimental study. During the measurements, the flow velocity was constant providing a steady state flow condition for the laminar-to-turbulent developing boundary layer on the test surface. The thermal boundary layer was developing at the leading edge of the test plate after heating the flow. The temperature of the test plate changed in time with a slow transient due to the change in flow temperature. Two data analysis methods are used to determine the convective heat transfer coefficients. One is the frequently used transient heat transfer measurement method e.g. [24,25] based on the assumption of unsteady one-dimensional heat conduction in a semi-infinite plate with a convective boundary condition with a quasi-steady state heat transfer coefficient given as hQS . The mathematical model of this kind of transient method can be described as:

8 @Tðy;tÞ > ¼ qkwcw > > w < @t

@ 2 Tðy;tÞ @y2

; y P 0; t P 0

 @Tðy;tÞ y ¼ 0 : h ½T ðtÞ  TðtÞ ¼ k QS g w @y  > y¼0 > > : t ¼ 0 : Tðy; tÞ ¼ T i

ð1Þ

Since the flow temperature at the measurement position is a function of time, the temperature history is usually approximated by a series of step functions resulting in the solution of Eq. (1) in the following expression:

"

n X ðT g ðsi Þ i¼1

! pffiffiffiffiffiffiffiffiffiffiffi!# 2 hQS ðt  si Þ hQS t  si ð2Þ  T g ðsi1 ÞÞ 1  exp erfc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qw cw kw qw cw kw From Eq. (2), we can see that when the flow temperature history T g ðtÞ and the surface temperature history T s ðtÞ are obtained, the heat transfer coefficient hQS can be calculated, even for only a single point in time measurement for T s ðtÞ as usually done [24,25]. This method assumes that the heat transfer coefficient hQS is a quasi-steady constant in the period from the start of the measurement to the time at which hQS is determined. So hQS is a kind of mean heat transfer coefficient up to the time point of evaluation. This measurement method is not applicable for time-resolved unsteady heat transfer results. To address this problem, another transient heat transfer measurement method, which is also based on the assumption of unsteady one-dimensional heat conduction in a semi-infinite plate but with a Dirichlet boundary condition (1st kind), is used in the present study [26]. The mathematical model of this method is:

8 @Tðy;tÞ > ¼ qkwcw > > w < @t

@ 2 Tðy;tÞ ; @y2

y P 0; t P 0

y ¼ 0 : Tð0; tÞ ¼ T s ðtÞ; > > > : t ¼ 0 : Tðy; tÞ ¼ T i

  qs ðtÞ ¼ kw @Tðy;tÞ @y 

ð3Þ

y¼0

From Eq. (3) the specific heat flux at the test plate surface qs ðtÞ can be evaluated [26]. The solution is given in integral form as:

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P1 P1

P2 Fig. 3. Comparison of the raw and the filtered wall surface temperature data with varying flow temperature at a frequency of 1 Hz.

 @Tðy; tÞ @y y¼0 # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " Z qw cw kw T s ðtÞ 1 t T s ðtÞ  T s ðsÞ pffiffiffiffi pffiffi þ ¼ d s 2 0 ðt  sÞ3=2 p t

P2

qs ðtÞ ¼ kw

Fig. 4. Spectrum of the raw wall surface temperature data shown in Fig. 3.

ð4Þ

A numerical integration method is needed to calculate qs ðtÞ for arbitrary variations in flow temperatures. When the test plate surface temperature history T s ðtÞ is approximated by a series of step functions, the specific surface heat flux can be calculated with a numerical integration method as proposed by Cook and Felderman [27]:

qs ðsn Þ ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 2 qw c w kw X T s ðsi Þ  T s ðsi1 Þ pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sn  si þ sn  si1 p i¼1

ð5Þ

Afterwards the heat transfer coefficient can be determined with Newton’s convective heat transfer relationship [28]:

hCF ðsn Þ ¼

qs ðsn Þ T g ðsn Þ  T s ðsn Þ

ð6Þ

Because the approach of Cook and Felderman [27] plays a key role, this transient heat transfer measurement method is referred further

as Cook–Felderman method. As there is no assumption for the temporal behavior of the heat transfer coefficient in the Cook–Felderman method, this measurement method is well applicable for the unsteady heat transfer studies. The usually applied transient heat transfer measurement method (Eqs. (1) and (2)) referred as QS-method in this paper is also employed in the present study to compare the results from it with the data using the Cook–Felderman method. In the present study the local flow temperature history T g ðtÞ was obtained by averaging the temperature data from three bead thermocouples at each measurement position. The local surface temperature history T s ðtÞ was measured with thin-film surface thermocouples. Both T g ðtÞ and T s ðtÞ were approximated by a series of step functions and used in Eqs. (5) and (6) to calculate hCF ðtÞ. When using the QS-method, a series of T s ðsn Þ at time point sn were used respectively in Eq. (2) to gain a series of hQS ðsn Þ. For both transient heat transfer measurement methods described above, the semi-infinite wall boundary condition has

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to be ensured. According to Schultz and Jones [29], the maximum measurement times in the present study are kept below

t6

d2w qw cw ; 64kw

ð7Þ

in which dw is the thickness of the test plate, to ensure that the heat flux change at the middle surface of the test plate is less than 1% of the surface value. 2.3. Data processing methods for flow temperature and heat flux The reference flow temperature is very important for determining the heat transfer coefficient in both methods. However, the measured flow temperature is prone to be time-delayed and amplitude-damped compared to the real data due to the thermal inertia effect of the thermocouples when the flow temperature changes with time, especially for relatively large magnitudes. Terzis et al. [30] proposed a correction method based on the lumped heat capacity (LHC) assumption to correct the thermal inertia effect of thermocouples in the measurement of fast varying flow temperatures. This correction method was employed in the present study. Two cases of periodically unsteady flow temperatures with different frequencies are chosen to show the effect of the above flow temperature correction method in Fig. 2. For the two cases, the flow velocity is the same with each other. The frequencies of the heating power for the two flows are 0.25 and 5.5 Hz respectively and the amplitudes are the same. From Fig. 2(a), we can see that the raw measured flow temperature T graw ðtÞ is almost the same as the corrected temperature T gcorr ðtÞ for the 0.25 Hz case. It means that the correction has no effect when the flow temperature changes slowly with time and the response time of the thermocouple is fast enough to capture the instantaneous features accurately. When the flow temperature has a higher frequency, the thermal inertia of the thermocouple delays and damps the measured value compared to the real flow temperature to a large extent as shown in Fig. 2(b). Under this condition, the correction plays a significant role. Not only the amplitude but also the phase has to be corrected. To ensure higher accuracy, corrections have been performed for all the cases in this study. Moreover, the time delays in the measured flow temperature data caused by the thermocouples’ positions which are 10 mm downstream of P1/P2 have also been corrected based on the measured flow velocity and distance.

(a) P1

381

Essentially, the Cook–Felderman method addresses an inverse heat conduction problem [31] in which the unknown heat flux has to be determined from observable data which is the plate surface temperature history in our case. Inverse heat conduction problems are highly ill-posed [32] in the sense that any small changes in the surface temperature data can cause dramatic changes in the calculated heat flux as well as the heat transfer coefficients. Furthermore, surface temperature measurements always contain white noise. High frequency components in the surface temperature signal often play havoc on the predictions in this calculation. So it is unreasonable to use these high-frequency components for useful reconstruction of the heat flux. Low-pass filtering of temperature data provides a simple manner for reducing the effects of unwanted high-frequency components [31]. In the present research, the Butterworth low-pass filter was used to remove unwanted higher frequency components in the temperature measurement signal, especially in the surface temperature T s ðtÞ signal. The cut-off frequency of the filter was defined based on the spectral analysis and the main frequency of the temperature signal. Fig. 3 shows as an example part of the raw data of wall surface temperature T s ðtÞ under sinusoidally varying flow temperature with a frequency of 1 Hz. According to the spectrum of the raw T s ðtÞ shown in Fig. 4, a cut-off frequency of 6 Hz was defined for the Butterworth low-pass filter. The filtered wall surface temperature, which is also shown in Fig. 3, was used to calculate the heat flux at the wall surface. The Cook–Felderman method introduced in Section 2.2 only addresses the conduction process in the Perspex test plate. In the present experiment, a thin-film surface thermocouple with different thermal properties from the Perspex was cemented on the test plate surface. Therewith the thin-film surface thermocouple also absorbed some heat due to its thermal inertia. Therefore, the convection heat flux from the flow to the test plate is divided into two parts. One part is the heat flux qabs ðtÞ absorbed by the surface thermocouple to change its temperature. The other part is the conduction heat flux into the Perspex plate qpex ðtÞ. The influence of the cement layer is neglected because thermal properties of the cement are similar to the Perspex according to the data provided by the manufacturer. Lumped heat capacity (LHC) assumption for the thin-film surface thermocouple is employed here. The measured temperature of the surface thermocouple was regarded as uniform corresponding to the measured value of T s ðtÞ. The heat flux propagating into the Perspex plate qpex ðtÞ can then be calculated with Eq. (5) and the measured T s ðtÞ. The absorbed heat flux

(b) P2

Fig. 5. Time-resolved results of heat transfer coefficient for a nearly step changing flow temperature of relatively large step-change value.

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(a) P1

(b) P2

Fig. 6. Time-resolved results of heat transfer coefficient for a nearly step changing flow temperature of relatively small step-change value.

(a)

(b)

Fig. 7. Time-resolved results of specific heat flux at P1 for the two nearly step changing flow temperatures.

by the surface thermocouple can be estimated with the lumped heat capacity method:

qabs ðtÞ ¼ dTC qTC cTC

dT s ðtÞ dt

ð8Þ

where dTC is the thickness of the thin-film surface thermocouple. The real convective heat flux qreal ðtÞ is then:

qreal ðtÞ ¼ qpex ðtÞ þ qabs ðtÞ

ð9Þ

2.4. Operating conditions The Reynolds number based on the flow velocity, which was kept constant during the measurement and the same for all cases, and the hydraulic diameter of the upper flow channel was 100,000. The turbulence intensity of the main flow as measured above the test plate (outside the boundary layer as given in Fig. 1) was 1.3%. The mesh heater was controlled to produce two nearly step changing flow temperatures of different step-change values, two multi-step changing flow temperatures of different multi-step forms, five sinusoidally varying flow temperatures of frequency of 0.003, 0.25, 1, 3 and 8 Hz respectively. The mean value and

amplitude of the sinusoidally varying flow temperatures were controlled to have a close value of about 50 and 12  C respectively. Due to the variation in flow density and viscosity caused by varying flow temperature, the effect on the Reynolds number is about 3% and on the heat transfer coefficient about 1.5–2.4% which is within the overall experimental uncertainty. The uniformity of the unsteady flow temperature in the channel’s cross-section, which is perpendicular to the flow direction, was checked by comparing the temperatures measured at a series of positions with different normal and lateral coordinates locations in a cross-section. The non-uniformity was below 0.5  C. According to the local Reynolds numbers Rex at P1 and P2, which were around 210,000 and 950,000 respectively, the boundary layer should be laminar at P1 and turbulent at P2. 2.5. Uncertainty analysis A measurement uncertainty analysis for the heat transfer measurements was carried out following the method outlined by Moffat [33]. Uncertainties in heat transfer coefficient calculated with the QS method are determined by the measurement uncertainties in flow temperature T g , surface temperature T s , initial temperature

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(a)

383

(b) P1

(c)

(d) P2

Fig. 8. Time-resolved results of heat transfer coefficient for two multi-step changing flow temperatures.

T i , time t, and test plate material properties qw cw kw . The estimated measurement uncertainty with a confidence interval of 95% is 0:15 K in temperature, 0:1 s in time and 5% in material properties qw cw kw of the test plate. Those result in a total uncertainty in the hQS of about 3–8% depending on time. Uncertainties in heat transfer coefficients calculated with Cook–Felderman method are determined by the measurement uncertainties in flow temperature T g , surface temperature T s , time step Dt, and test plate material properties qw cw kw . The estimated measurement uncertainty with a confidence interval of 95% is again 0:15 K in temperature, 1% in time step and 5% in test plate material properties qw cw kw . Those result in a total uncertainty in the hCF of about 4% in most of the time except a short initial period in which the uncertainty could be up to 26%. 3. Results and discussion 3.1. Heat transfer characteristics for flow temperatures with step change Time-resolved heat transfer coefficients under two step changing flow temperatures of different step-change values were measured to investigated the effects of flow temperature on the temporal behavior of the heat transfer coefficients. Fig. 5 shows

the measured time-resolved heat transfer coefficients at P1 and P2 under a step changing flow temperature with relatively large step-change value. We can see that the heat transfer coefficient at P2 is larger than that at P1 during the whole measurement period. According to the local Reynolds number at P1 (about 210,000) and P2 (about 950,000), the boundary layer should be laminar at P1, but turbulent at P2. Fig. 5 also shows that the hQS ðtÞ data which were calculated with Eq. (2) agree well with hCF ðtÞ in the whole period at both positions P1 and P2, though small oscillations exist in the hCF ðtÞ data which are caused by the ill-posed nature of the Cook–Felderman method and the residual noise in the filtered surface temperature T s ðtÞ data. The measured results indicate that both hCF ðtÞ and hQS ðtÞ at P1 and P2 are not constant in the initial period of about 0–7 s. The hCF ðtÞ and hQS ðtÞ decrease from a larger value to the constant value with the rise of T g ðtÞ from the initial value to the constant value. This decrease phenomenon in heat transfer coefficient obtained by experimental measurement is compatible with the temporal behavior of heat transfer coefficient in a transient heat transfer process obtained by Lachi et al. [13] with analytical/numerical method. In the period when the flow temperature is constant with time, the heat transfer coefficients at P1 and P2 are nearly constant with time, and the decrease of temperature difference between T g ðtÞ and T s ðtÞ has no influence on the heat transfer coefficients.

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(a) P1

(b) P2

Fig. 9. Time-resolved results of heat transfer coefficient for a slowly varying flow temperature.

(a) Whole history

(b) Partial history P1

(c) Whole history

(d) Partial history P2

Fig. 10. Time-resolved results of heat transfer coefficient for a periodic flow temperature with frequency of 0.25 Hz.

Fig. 6 shows the measured heat transfer coefficients at P1 and P2 under the step changing flow temperature with relatively small step-change value. We can see that the oscillations in these hCF ðtÞ

data are larger than those under large flow temperature shown in Fig. 5. Fig. 7 presents the measured time-resolved specific heat fluxes at P1 under the two flow temperature conditions. As shown

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(a) Whole history

385

(b) Partial history P1

(c) Whole history

(d) Partial history P2

Fig. 11. Time-resolved results of specific heat flux for a periodic flow temperature with frequency of 0.25 Hz.

there, the values and oscillations of the qCF ðtÞ data are smaller for the case with small flow temperature changes than those for the large temperature change case. But due to the small temperature difference between T g ðtÞ and T s ðtÞ under small flow temperature, larger oscillations are caused in the hCF ðtÞ data. Except the oscillations, the basic features in the hCF ðtÞ and hQS ðtÞ data are the same for the two step changing flow temperature cases. There is a decrease in the hCF ðtÞ and hQS ðtÞ of P1 and P2 in the initial period, while they keep constant with time when the flow temperature is steady. By comparing Figs. 5 and 6, we can find that the change of flow temperature value has very weak influence on the value of the heat transfer coefficient at P2 with a turbulent boundary layer, while it has a notable influence on the heat transfer coefficient at P1 with a laminar boundary layer. The mean heat transfer coefficient at P1 is higher by about 28% under the smaller flow temperature change. This result shows that the different thermal boundary conditions have a strong effect on the laminar heat transfer coefficient as indicated in the study of Butler and Baughn [12]. In a turbulent boundary layer, the intense local exchange of momentum and energy between boundary layer and free flow caused by the turbulence make the heat transfer coefficient less sensitive to different upstream thermal boundary conditions. However, in a laminar boundary layer different upstream and time-changing thermal boundary conditions caused by different flow temperature steps

can influence the heat transfer coefficient significantly. Butler and Baughn [12] performed transient heat transfer measurements on a flat plate model for a laminar boundary layer flow with different step changing flow temperatures. They found that the heat transfer coefficient varies significantly for different non-dimensional surface temperatures defined as ðT s  T g Þ=ðT i  T g Þ. Their results also indicated that the laminar heat transfer coefficient is larger when the non-dimensional surface temperature is smaller which means the flow temperature is lower as the measurements were performed with a single coating of thermochromic liquid crystals measuring the same surface temperature at the different flow temperature conditions. . The above presented time-resolved data of hCF ðtÞ and hQS ðtÞ validates the quasi-steady assumption in the transient convection heat transfer with constant flow temperature for steady flows. However, special attention should be paid in the initial period of a transient heat transfer where the flow temperature rises rapidly. In this period, the heat transfer coefficient could be a time varying value and different from the value in the quasi-steady state. 3.2. Heat transfer characteristics for flow temperatures with multistep change To further address the effects of flow temperature with different constant values on the temporal behavior of convective heat

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(a) Whole history

(b) Partial history P1

(c) Whole history

(d) Partial history P2

Fig. 12. Time-resolved results of heat transfer coefficient for a periodic flow temperature with frequency of 1 Hz.

transfer, time-resolved heat transfer coefficients at P1 and P2 were measured under two multi-step changing flow temperatures. One is of the form of up-down-down-down-up, and the other is of the form of up-up–up-down. Fig. 8 shows the two multi-step flow temperatures and the corresponding time-resolved heat transfer coefficients at P1 and P2. The results clearly indicate that the flow temperature changes have a strong effect on the value of the laminar heat transfer coefficient. As the hCF ðtÞ data shown in Fig. 8(a) and (b), the multi-step flow temperature produces a multi-step heat transfer coefficient at P1, and the variation of heat transfer coefficient is opposite to the step in flow temperature. This is similar to the behavior of hCF ðtÞ measured under single-step flow temperatures shown in Figs. 5 and 6.When the flow temperature changes, the heat transfer coefficient at P1 changes accordingly. The hQS ðtÞ data at P1 which were calculated with Eq. (2) do not agree with the hCF ðtÞ data as well as the cases under a single-step change in flow temperature. This disagreement could be related to the assumption of a time-wise constant heat transfer coefficient until the time of evaluation for hQS ðtÞ data which thereby damps the local time-wise variations. Nevertheless the trends for hQS ðtÞ are similar to the instantaneous behavior of hCF ðtÞ. At P2, the hQS ðtÞ data agree well with hCF ðtÞ in the whole period. Contrary to the temporal behavior for the laminar heat transfer coefficient at P1, the turbulent heat transfer coefficient at P2 almost keeps a constant value, which agrees well with the value

measured under single-step change in flow temperature, in the whole period except for the initial period with a strong change in flow temperature as observed before. When the flow temperature drops from a larger value to a smaller one, the turbulent heat transfer coefficient experiences a sudden drop and then increases rapidly to the same constant value. When the flow temperature jumps from a smaller value to a larger one, the turbulent heat transfer coefficient would also have a sudden jump and then decreases rapidly to the same constant value. Of course, this could also be an artefact coming from the inverse ill-posed problem analyzed by the CF-method. From the time-resolved heat transfer coefficient results under multi-step changing flow temperatures, we can see that the laminar heat transfer coefficient in a transient heat transfer process is sensitive to the upstream thermal boundary conditions as induced by different flow temperature variations, but less sensitive to the changing process. On the contrary, the turbulent heat transfer coefficient does not vary with changes in flow temperature except for a short initial period. 3.3. Heat transfer characteristics for sinusoidally varying flow temperatures Fig. 9 shows the measured time-resolved heat transfer coefficients for a sinusoidally varying flow temperature with a very

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(a) Whole history

387

(b) Partial history P1

(c) Whole history

(d) Partial history P2

Fig. 13. Time-resolved results of heat transfer coefficient for a periodic flow temperature with frequency of 3 Hz.

low frequency of 0.003 Hz. Under this kind of slowly varying flow temperature, the temporal behaviors of hCF ðtÞ and hQS ðtÞ at P1 and P2 are very similar to those under step changing flow temperature as shown in Fig. 5, though there is a slight inverse behavior for hCF ðtÞ compared to T g ðtÞ at P1. These results indicate that the quasi-steady assumption is applicable for transient heat convection processes with steady flow velocity and slowly varying flow temperatures of moderate variation in magnitude. Fig. 10 shows the measured time-resolved heat transfer coefficients for a periodically varying flow temperature with a frequency of 0.25 Hz. We can see that the temporal behavior of laminar heat transfer coefficient at P1 differs from that of turbulent heat transfer coefficient at P2 obviously. The temporal behavior of hCF ðtÞ at P1 under periodically varying flow temperature is very close to that under step changing flow temperatures. As shown in Fig. 10, in the short initial period a decrease in hCF ðtÞ from a larger value to a constant value is noted and afterwards hCF ðtÞ shows a constant value in the whole transient heat transfer process. The small observed oscillations in hCF ðtÞ might be again caused by the illposed nature of the Cook–Felderman method and the residual noise in the filtered surface temperature T s ðtÞ data. The values of hCF ðtÞ at P1 agree well with those obtained under step changing flow temperatures with relatively large step-change value as shown in Fig. 5. Fig. 10 also shows that the hQS ðtÞ data agree well

with hCF ðtÞ at P1. The measured time-resolved results of specific heat flux qCF ðtÞ and qQS ðtÞ are shown in Fig. 11. Hereby qQS ðtÞ is calculated with hQS ðtÞ  ðT g ðtÞ  T s ðtÞÞ. We can see that qQS ðtÞ agree very well with qCF ðtÞ at P1 and the variation is in phase with the changes in T g ðtÞ. Different from the temporal features of hCF ðtÞ at P1, the heat transfer coefficient history of hCF ðtÞ at P2 exhibits notable unsteady features. The hCF ðtÞ-data at P2 vary intensely with the same frequency as T g ðtÞ. Although small variations exist also in the hQS ðtÞ-data, the hQS ðtÞ-data at P2 keep a nearly constant value in most of the time except the short initial period where there is a similar decrease process in both hCF ðtÞ and hQS ðtÞ. The values of hQS ðtÞ at P2 agree well with those obtained under step changing flow temperatures. As shown in Fig. 11, both qCF ðtÞ and qQS ðtÞ at P2 change in phase with T g ðtÞ. Thereby the amplitude of qCF ðtÞ is larger than that of qQS ðtÞ with almost the same magnitude of about 360 W=m2 in the whole history. The amplitude of the hCF ðtÞ -variations at P2 increases with time when T s ðtÞ approaches T g ðtÞ as the heat flux amplitude stays nearly constant. From the partial history figures in Fig. 10, we can obtain some detailed information about the phase relationship between hCF ðtÞ, T g ðtÞ and T s ðtÞ. The hCF ðtÞ variation is out of phase compared to the T s ðtÞ variation, but is almost in phase with T g ðtÞ while there is a little phase advance in hCF ðtÞ relative to T g ðtÞ.

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(a) Whole history

(b) Partial history P1

(c) Whole history

(d) Partial history P2

Fig. 14. Time-resolved results of heat transfer coefficient for a periodic flow temperature with frequency of 8 Hz.

(a) P1

(b) P2

Fig. 15. Time-resolved results of specific heat flux for a periodic flow temperature with frequency of 8 Hz.

Time-resolved heat transfer coefficients for periodically changing flow temperatures with higher frequencies of 1 and 3 Hz are shown in Figs. 12 and 13 respectively. The behavior of hCF ðtÞ and hQS ðtÞ under these conditions is similar to the results for the changing flow temperature with a frequency of 0.25 Hz. For the heat transfer coefficient at P2, hCF ðtÞ is well in phase with T g ðtÞ

under f = 1 Hz, but lags a little in phase with T g ðtÞ under f = 3 Hz. The heat transfer coefficient hCF ðtÞ at P1 shows random oscillations at f = 1 Hz, but exhibits regularity under f = 3 Hz and changes with T g ðtÞ in an inverse manner. This unsteady behavior for the heat transfer coefficient at P1 gets more obvious when the flow temperature changes with a frequency of 8 Hz as shown in

C.-l. Liu et al. / International Journal of Heat and Mass Transfer 84 (2015) 376–389

Fig. 14. We can see that the amplitude of hCF ðtÞ at P1 is larger than the amplitude of hCF ðtÞ under f = 3 Hz, and close to the amplitude of hCF ðtÞ at P2. In this case at both positions the determined timeresolved heat transfer coefficients change intensively with the changes in flow temperatures and increase in amplitude with time due to the decreasing temperature differences between flow and surface temperature. This is as before caused by the nearly constant behavior of the specific heat flux data as shown in Fig. 15. 4. Conclusion Time-resolved results of laminar and turbulent heat transfer have been measured for a steady flow over a flat plate with step changing and sinusoidally varying flow temperatures. Two measurement methods for the transient heat transfer process are applied. Comparisons in the temporal behavior of heat transfer coefficients under different kinds of unsteady flow temperatures and therewith different upstream thermal boundary situations show that there are obvious differences between laminar and turbulent convections. The heat transfer coefficient results from the Cook–Felderman method, which has no quasi-steady assumption on the heat transfer coefficient, indicate that the change of flow temperature value in step or multi-step form has very weak influence on the value of heat transfer coefficient with turbulent boundary layer, while it could notably influence the heat transfer coefficient with laminar boundary layer which is sensitive to the change of thermal boundary conditions. The presented timeresolved heat transfer coefficient data also validate the quasisteady assumption in the transient convection heat transfer that the laminar heat transfer coefficient will be in steady state when the flow temperature does not vary with high frequencies and large amplitudes. For the turbulent boundary layer flow, the time-resolved heat transfer coefficient is in steady state only when the flow temperature keeps constant or varies slowly with time. Otherwise, the heat transfer coefficient exhibits obvious unsteady behavior that it varying intensely with the same frequency as the periodically varying flow temperature. The time-resolved experimental results verify that the quasi-steady method cannot obtain the true unsteady heat transfer characteristics. The results presented in this paper might be used for analytical or numerical investigations addressing the characteristics of unsteady convective heat transfer under unsteady thermal boundary conditions. Conflict of interest None declared. Acknowledgments The authors acknowledge gratefully the financial support from National Natural Science Foundation of China (Grant No. 51306152), Fok Ying Tung Education Foundation of China (Grant No. 141053), the Fundamental Research Funds for the Central Universities (Grant No. 3102014JCQ01049) and Alexander von Humboldt Foundation of Germany. References [1] E.M. Sparrow, F.N. De Farias, Unsteady heat transfer in ducts with time varying inlet temperature and participating walls, Int. J. Heat Mass Transfer 11 (1968) 837–853. [2] J. Sucec, Transient heat transfer between a plate and a fluid whose temperature varies periodically with time, J. Heat Transfer – Trans. ASME 102 (1980) 126–131.

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