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This work focuses on an experimental investigation of convection heat transfer to a gas in a vertical tube under strongly heated conditions at high temperatures and pressures up to 943 K and 65 bar. A unique test facility for convection heat transfer experiments has been constructed, and used to obtain experimental data useful for better understanding and validation of numerical simulation models. This test facility consists of a single flow channel in a 2.7 m long, 0.11 m diameter graphite column with four 2.3 kW heaters placed symmetrically around the 16.8 mm diameter flow channel. Upward flow convection experiments with air and nitrogen were conducted for inlet Reynolds numbers from 1300 to 60,000, thus covering laminar, transition, and fully turbulent flow regimes. Experiments were performed at different flow rates (3.8 × 10−4 to 1.5 × 10−2 kg/s) and heater power up to 6 kW. Importantly, the data analysis considered the thermophysical properties of the gas and graphite changing with temperature and pressure. Nusselt number results are further compared to existing correlations. The effect of pressure and heater power on degraded heat transfer is examined. The analyses of the experimental data showed significant reductions in Reynolds number of up to 50% and Nusselt numbers of up to 90% between the gas inlet and outlet.

## Introduction

Industrial applications have attracted considerable attention to the use of high temperature gaseous coolants for their high thermodynamic efficiency, chemical inertness, additional safety benefits, and environmental acceptability. General applications include heat exchangers, fission and fusion nuclear reactors, gas turbine engines, and propulsion systems. For example, high temperature and high pressure helium will be used as a coolant in one of the generation IV reactor designs called very high temperature reactors (VHTRs), which is proposed for implementation in next generation nuclear power plants [1]. The exiting coolant gas at high temperature can be used directly in a Brayton cycle or with a steam generator in a Rankine cycle for electrical power production. A high outlet gas temperature in the Brayton cycle is advantageous since large temperature differences in a thermodynamic power cycle will lead to increased thermodynamic efficiency [2]. The high temperature helium can also be utilized for hydrogen production in another potential application [35].

This paper addresses the deteriorated turbulent heat transfer (DTHT) phenomenon, in which a strongly heated, turbulent gas flow exhibits heat transfer characteristics of a laminar flow [6]. Sometimes, it is called flow laminarization and this transition from turbulent to laminar-like flow occurs at higher Reynolds numbers than the usual critical value of ∼2300 [6], though a specific value for this transition is not fully clear. Deterioration in turbulent heat transfer could occur due to thickening of a thermal boundary layer and may result in heat transfer rates as low as 40% of those in the corresponding turbulent forced convection [7,8]. DTHT is also a direct result of decrements of turbulent heat flux caused by decreases in turbulent kinetic energy, reductions in momentum exchange near the wall, and gas property variations [9]. This would lead to a substantial reduction in the heat transfer coefficient (HTC) and an increase in wall temperature [10].

The DTHT phenomenon can be caused by strong heating of a gas, which leads to large thermophysical property variations with temperature (e.g., changes in density, viscosity, and thermal conductivity). Dynamic or kinematic viscosity increases and density decreases along the heated flow channel leading to reduced Reynolds numbers and DTHT in forced and mixed convection conditions. Under natural circulation of the gas between a heated riser and cooled downcomer, reductions in the Reynolds number could trigger flow laminarization even when the incoming flow in the riser is well above the critical Reynolds number. On the other hand, the laminar flow can transition to turbulent flow in the cooler downcomer channels as the gas is cooled and density increases while viscosity decreases, resulting in increased Reynolds numbers.

Besides the reduction in the Reynolds number, as the gas is heated, the reduction in density causes the bulk flow to accelerate upward, and in some cases the enhanced buoyancy forces can also lead to flow laminarization [6,11]. It has been proposed that for a turbulent flow in forced convection, the flow acceleration would tend to stabilize turbulence bursts from the viscous sublayer and thereby reduce turbulent transport [12,13]. Additional influences of rapid flow acceleration on turbulence include increments in viscous shear stress near the wall due to an amplified axial velocity gradient. The turbulence dissipation rate would surpass the turbulence production rate initiating a turbulence decay and a reduction in the Reynolds stress [10,14]. These conditions are accompanied by the growth of a thermal boundary layer [7], which would lead to readjustment of any previously fully developed turbulent velocity profile. The long run effect of the thermal boundary layer growth is that no truly fully established conditions are reached because the temperature increments lead to continuous axial and radial variations in thermophysical properties [11]. Ultimately, a combination of these effects, including temperature variations and substantial reductions in turbulent kinetic energy, may cause the turbulent flow to laminarize.

The criteria for the occurrence of DTHT, flow laminarization, and the heat transfer characteristics of these phenomena have been the subject of previous studies by several investigators [1520]. Shome [7] defined flow laminarization as the state in which the ratio of Reynolds shear stress to the viscous wall shear stress attenuates to 5% of the constant property value for the same value of the inlet Reynolds number. The effects of buoyancy on turbulent heat transfer due to fluid acceleration near the wall leading to velocity and shear stress profile modifications have been previously discussed by Hall and Jackson [21,22]. They suggested a buoyancy parameter, Bo*, to be used as an indicator of possible flow laminarization and an onset value of Bo* = 6 × 10−7 [13]. Another parameter called the acceleration parameter, Kv, responsible for acceleration-induced laminarization has also been proposed by McEligot and Jackson [13].

One issue with studying the flow laminarization phenomenon is that there exists a very limited knowledge base. For example, Shehata and McEligot [23,24] used air at 0.09 MPa flowing through a 27.4 mm internal diameter tube to measure turbulence using a hot wire anemometer. Some numerical simulations [11,17] were validated using their experimental data. More recently, Lee et al. [25] used air, nitrogen, helium, and CO2 gases to investigate deteriorated turbulent heat transfer in a 1.7 m long, 15.7 mm inner diameter tube at pressures up to 6.7 bar. Based on the data obtained, they proposed a heat transfer regime map indicating transitions among laminar, turbulent, and DTHT based on the Reynolds number at the inlet and a nondimensional heat flux, $qb+=q″/GcpTb$, where q″, G, cp, and Tb correspond to heat flux, mass flux, specific heat, and fluid bulk temperature, respectively.

For practical applications of gas heat transfer knowledge, more data on DTHT are needed at higher pressures well above 7 bar. To this end, a high pressure and high temperature test facility has been constructed at City College of New York for conducting forced and natural circulation heat transfer experiments using air, nitrogen, and helium. This facility consists of a pressure vessel (PV) rated at 70 bar/623 K containing a 2.7 m, 108 mm outer diameter graphite column with four 2.3 kW heater rods symmetrically placed around a central coolant channel of 16.8 mm diameter [25]. The objectives of this work are to conduct convection heat transfer experiments and obtain data under both forced flow and natural circulation conditions in high temperature, high pressure gases.

To this end, convection heat transfer data are presented in this paper for air and nitrogen at different inlet Reynolds numbers, showing significant Reynolds number reductions in both gases subjected to high heating rates. High Reynolds number flows (Re = 20,000–60,000) studied for air under low heating rates are not expected to undergo DTHT. Therefore, the experiments with air were first performed to validate the experimental methodology and provide data at moderate temperatures up to 573 K, high pressures up to 30 bar, and low to moderate heating rates. On the other hand, nitrogen experiments were conducted under strongly heated conditions. The ratio of the heating rate to mass flux can be quantified through the dimensionless heat flux, $qb+$, which was at least four times greater for nitrogen than for air in the present experiments.

As will be shown, DTHT occurrence was explored with nitrogen flowing through a graphite test section heated up to 963 K with inlet Reynolds numbers below 15,000. The convection heat transfer in the flow channel was examined using both average and local Reynolds, Nusselt, and Prandtl numbers. The experimental parameters varied include pressure, heater power level, and gas flow rate. The effect of operating temperature on the Reynolds number reduction along the heated flow channel is also addressed.

## Experimental Facility

This section first discusses the experimental facility and procedure utilizing air to collect heat transfer data. Section 2.2 describes the modification of the system for experiments involving nitrogen. The method of data analysis is also described. Additional details of the test facility and data analysis procedure can be found elsewhere [26].

### Test Section Design and Open-Loop System for Experiments With Air.

A schematic of the high pressure and high temperature gas flow loop is shown in Fig. 1(a). The test facility's main components were: (1) stainless steel pressure vessel, (2) inner insulation layer, (3) graphite test section, (4) electric heater rods, and (5) coolant channel. These main components are further highlighted in Fig. 1(b). The gas was supplied by a compressor or from a compressed gas cylinder. The graphite test section was housed in a stainless steel 304 PV, which was ASME certified at 69 bar at 623 K, and had flanges welded at the top and bottom. An inner insulation layer protected the PV wall from the high graphite temperatures, and directed heat flux into the coolant channel. This layer consisted of a 19.2 mm thick fiberglass thermal insulation (Great Lakes Textiles Inc., GB1BLTMAT, Solon, OH) and was placed between the graphite test section surface and inner wall of the PV.

The graphite test section was a 108.0 mm diameter cylindrical column made of thermally isotropic graphite G348 (GraphiteStore) with a thermal conductivity of 128 W/m K at 298 K and 83.6 W/m K at 873 K [27]. This graphite test section was 2.78 m long and had a flow channel of 16.8 mm diameter along the central axis through which compressed gas flowed upward. The coolant channel radius was chosen based on previous studies [2831]. The graphite test section also contained four 12.7 mm diameter holes symmetrically placed around the coolant channel for insertion of electric heater rods (Dheater = 12.7 mm) as shown in Fig. 1(c). These heater rods from Watlow were 3048 mm long and each rated at 2.3 kW at 240 V AC. The top 300 mm of each heater rod was an unheated section for penetration through the upper flange. Each heater rod was connected to an AC source through a power controller, a 0–240 V AC variable transformer, and circuit breaker so that the heater power could be individually controlled and recorded.

A total of 40 type-K thermocouples were installed at 10 axial locations to measure the axial and radial temperature profiles in the graphite test section. At each axial location, four thermocouples were installed at four diagonal positions and three different radial distances from the axis (Fig. 1(c)). Their azimuthal locations were rotated counterclockwise by 90 deg in each successive axial plane. Additional thermocouples were spot welded on the PV outer surface at 12 different elevations monitoring the PV's outer surface temperature. Two pressure transducers (Omega, PX313) were installed in the inlet and the outlet piping connected to the pressure vessel. A gas flow meter (Omega, FMA-878 A-V) was located in the outlet piping of the open flow loop.

All instruments were connected to a National Instruments Data Acquisition System (NI cDAQ-9188), containing six data acquisition cards, five for thermocouples (NI 9213), and one for analog signals (NI 9205) from the pressure transducers and the flow meter. The data acquisition was performed via a labview program. Additional safety systems included solenoid valves for remote operation, temperature controllers, and a water-cooled heat exchanger before the flow meter added to the open-loop system.

### Closed-Loop System for Nitrogen Heat Transfer Experiments.

In order to carry out convection heat transfer experiments with nitrogen at constant flow rates and high operating pressures, the flow loop was modified as shown in Fig. 2. To achieve a high flow rate, a gas recycling system based on a gas booster pump was added as shown in Fig. 3. The gas was provided to the test section at a constant flow rate from a bank of three high pressure gas cylinders. The gas exiting the test section then flowed into a bank of three low pressure receiver tanks. During the experiments typically lasting up to 2.5 h, the gas booster pump would pump the low pressure gas from the receiver tanks to charge the high pressure gas tanks as shown in Fig. 3 to provide a constant gas flow continuously.

## Experimental Procedure

Each experiment included the independent selection and control of three parameters: graphite midplane temperature, system pressure, and gas flow rate. While the graphite midplane temperature was controlled by the heater power, pressure and air flow rate were controlled by two manually adjusted valves. One valve was connected near the exit of the open flow loop and used to set the working pressure. The second valve was connected before the inlet of the test section to control the inlet flow rate by partially discharging the gas to the ambient for the open flow loop or directing the gas to the low pressure tanks in the closed-loop tests.

Experiments were begun by turning on and controlling the heater power with individually connected AC variable transformers. Graphite test section temperatures were monitored until a steady-state condition was reached in both the graphite and PV surface temperatures. At this point, the compressor was turned on and air was allowed to flow through the test section in the pressure vessel. Data acquisition was continued for 3.5–5 h until steady temperature readings were obtained with a change in any of the graphite temperatures of less than 1 K/h. For closed-loop tests with nitrogen at higher temperatures, a regulator on a high pressure cylinder was opened allowing nitrogen gas to flow into the test section. Data acquisition continued for a maximum of 3 h. Steady-state was declared when the change in any of the graphite temperature readings was less than 3 K/h, corresponding to a total power input of 28 W, which was less than 1% of the total power supplied by the heaters in any experiment. The data were then analyzed for this specific time period.

## Data Analysis

This section discusses the methodology involved in calculating flow and heat transfer parameters from the measurements. Having measured the volumetric flow rate of the gas through the system, it was possible to determine the local Reynolds number at any axial location, by knowing the local bulk temperature and using the below equation
$Rei=4m˙πμiD$
(1)

where the local dynamic viscosity μi was evaluated at the local bulk temperature. Viscosity was calculated as a function of the bulk temperature using Sutherland's formula for an ideal gas [32]. The mass flow rate, $m˙$, was calculated from the volumetric flow rate in standard liters per minute (SLPM), measured further downstream of the test section. The gas temperature and pressure were measured at the inlet of the mass flow meter, and appropriate corrections were applied to the measured volumetric flow rate to obtain the actual mass flow rate.

The bulk temperature profile of the gas was determined by dividing the graphite test section into 11 segments and carrying out an energy balance for each segment. The inlet and outlet bulk temperatures, Ti and Ti+1, for each segment were then determined using the below equation
$Ti+1=Ti+ΔQiCpm˙$
(2)
where ΔQi is the heat removed by the gas with a specific heat capacity of Cp (J/kg K) calculated as a function of temperature [32]. For Eq. (2), ΔQi was calculated for the ith segment as follows:
$ΔQi=Qin,i−Qout,i=∑n=1n=4Pn,i11−Qaxial,i−QHL,i$
(3)

For each segment, Qin,i was equal to the sum of the heat generated by each of the four heater rods, Pn,i, and Qout,i included heat loss from the PV surface and net axial conduction. Although the temperature in the heater rod varied axially, the heat generation rate was assumed to be axially uniform since the electrical resistance variation with temperature was small. The net axial conduction term, Qaxial,i, was the sum of the net axial heat conduction in both the stainless steel PV wall and graphite test section, which could be obtained from the axial temperature gradients measured. The rate of heat loss from the PV surface, QHL,i, was not axially uniform, however; so it was correlated to the average value of the PV surface and the graphite temperatures at each elevation. This correlation was developed from the heat loss measurements in stagnant gas tests. A typical heat loss profile and fitted polynomial for nitrogen tests are shown in Fig. 4.

The final unknown needed to perform the heat balance calculations to obtain the local convection heat transfer coefficient was the inner wall temperature of the graphite test section. Having measured the graphite temperatures at three different radial positions at any axial elevation (Fig. 1(c)), the radial temperature profile was extrapolated to obtain the inner wall temperature, Tw,i. The local convection heat transfer coefficient, hi, was then calculated using the local convection heat transfer rate in each segment, ΔQi
$hi=ΔQi/(Tw,i−Ti)Ai$
(4)
Here, Ai is the heat transfer area in segment i equal to the product of the flow channel perimeter (πD) and the length of each segment: Li = 249.4 mm. The local Nusselt number was then calculated from Eq. (5), where the thermal conductivity, ki, of the gas at the local bulk temperature, Ti, was interpolated from a table of thermal conductivities for air or nitrogen given as a function of the local bulk temperature and pressure [32]
$Nui=hiDki$
(5)
In Sec. 5, the convection heat transfer results obtained for air and nitrogen will be presented and discussed in terms of local Nusselt and Reynolds numbers as well as average Nusselt and Reynolds numbers for the entire test section representing the arithmetic mean of the eleven calculated local values. For comparison purposes, average Nusselt numbers could also be calculated using the average heat transfer coefficients calculated assuming a linear bulk temperature variation as given by the below equation
$h¯=m˙cp(Tout−Tin)(πDL)(T¯w−T¯b)$
(6)

where Tout and Tin represent the measured outlet and inlet bulk temperatures, $T¯w$ is the average of the ten wall temperatures measured, and $T¯b$ (= $(Tin+Tout)/2)$ is the average bulk temperature between the inlet and outlet.

This paper presents and discusses the results obtained for 30 convection heat transfer experiments with air and 30 with nitrogen. These experiments were systematically performed by setting a working pressure and inlet gas flow rate, and adjusting the heater power in order to achieve a 573 K maximum midplane graphite temperature for air and 873 K for nitrogen. After reaching steady-state and acquiring the data, the flow rate and heater power were changed to cover a wide range of inlet Reynolds numbers. Dimensionless heat fluxes, $qb+=q″/(GCpTb)$, lower than 5 × 10−4 are expected to induce Nusselt number reductions of less than 5% [13]. The ranges of the parameters varied in the present experiments are shown in Table 1.

The uncertainties in the experimental results were determined using the root sum square method. The measured variables (temperature, pressure, flow rate, and flow channel dimensions dimensions) and the parameters estimated from the reference property data (gas density, viscosity, and thermal conductivity) contain bias and precision errors. The systematic (bias) error is estimated based on the manufacturer specified accuracy and the instrument calibration report. The random uncertainty (precision) of the measurement is determined by capturing a large number of samples, averaging them, and multiplying the standard deviation by two so that it falls within a 95% confidence region.

Uncertainties in the measured fluid temperature, pressure, mass flow rate, and graphite temperature were 1.1%, 0.5%, 1.1%, and 0.5%, respectively. Based on these parameters and Eqs. (1)(5), the uncertainties in the calculated Reynolds and Nusselt numbers were 3.0% and 5.3%, respectively. Additional information regarding uncertainty calculations can be found elsewhere [26].

## Results and Discussion

This section presents and discusses the convection heat transfer results obtained for air and nitrogen as the working fluid. These results will include a discussion of the Reynolds number reduction along the flow channel as the gas is subjected to intense and moderate heating conditions. In addition, the local and average heat transfer parameters, and the effects of the experimental parameters listed in Table 1 which lead to heat transfer degradation, will be presented and discussed. Experimental results are also compared to existing empirical correlations for convection heat transfer.

### Wall and Bulk Temperature Profiles.

Figure 5(a) shows a typical example of the graphite temperature data obtained in each test. These results are from forced convection experiments carried out with air at 23.8 bar, 0.015 kg/s (Rein = 60,000), 473 K midplane graphite temperature, and 2.4 kW total heater power. In this run, a steady-state condition was reached after about 12,000 s. From t = 0 to approximately 1750 s, the test section was heated without any air flow. At approximately 1750 s, the air flow was turned on with the total heater power kept at 2.4 kW. This allowed the axial locations close to the exit (plane 10) to continue to heat up while the locations near the entrance (plane 1) begin to cool down.

The inset in Fig. 5(a) shows the ranges of the four radial temperatures measured at one axial location (plane 1). The radial variations in graphite temperature varied by less than 1% of the magnitude of the reading, indicating the radial temperature profile was uniform due to a high thermal conductivity of the graphite test section. The temperature data were then analyzed when steady-state was declared by averaging the last 60 s of data.

Similar results are presented in Fig. 5(b), where nitrogen was used as the coolant and under the following experimental conditions: 61 bar, 4.1 × 10−3 kg/s (Rein = 14,700), 843 K midplane graphite temperature, and 4.3 kW total heater power.

Using the near steady graphite temperatures such as those presented in Fig. 5, a local heat balance was performed over each segment to obtain the axial bulk temperature profile in each experiment. The results for nitrogen are shown in Figs. 6(a) and 6(b) for low inlet Reynolds number of 3000 and high Re = 15,400, respectively. For each segment, the local heat transfer coefficient, hi, was calculated using the rate of heat transfer to the gas, ΔQi, and the difference between the wall and bulk temperatures, (Tw,i − Tb,i), as indicated in Fig. 6. Bear in mind that although the heat input was axially uniform, wall heat flux was not due to axial variations of heat loss as shown previously in Fig. 4. It is also important to note that the wall temperature was also not constant as shown in Fig. 6. So, the present experiments are neither constant wall heat flux nor constant wall temperature experiments.

Next, the wall and bulk temperature variations for different inlet Reynolds numbers presented in Fig. 6 will be discussed. Although the axial wall temperature profiles are similar between the two cases, the bulk temperature profiles show a significant difference. For the low inlet Reynolds number case, the bulk temperature increases rapidly in the lower half of the test section and approaches the wall temperature in the upper half. The difference between the wall and bulk temperatures decreases rapidly and approaches a nearly constant value in the upper half of the test section. On the other hand, for the higher inlet Reynolds number case, the wall and bulk temperature profiles increase in a similar manner and remain nearly parallel resulting in a slightly decreasing but linear temperature difference along the test section.

Figure 6 also suggests large differences in the heat transfer coefficients that could arise when a linear bulk temperature profile is assumed and the average value between the inlet and outlet bulk temperatures is used in Eq. (6) instead of the average of all the local bulk temperatures as given by Eq. (4). For the low inlet Reynolds number of 3000, the difference in the resulting mean Nusselt numbers calculated is 51% while it is smaller at 14.7% for the higher inlet Reynolds number of Re = 15,400. The differences in the mean Reynolds numbers based on the properties evaluated using the average of the inlet and outlet bulk temperatures and average of the local bulk temperatures are 10% and 5.6% for the low and high inlet Reynolds numbers, respectively.

### Analysis of Average and Local Nusselt and Reynolds Numbers.

The average Nusselt number data for a wide range of experimental conditions summarized in Table 1 are examined covering laminar, transition, and turbulent flows. The average Nusselt number data for air under low heating rates are compared with the modified Dittus–Boelter correlation (Eq. (7)) in Fig. 7. All dimensionless numbers were calculated using the fluid properties evaluated at the average bulk temperature between the inlet and outlet, which would not be appropriate if the axial bulk temperature profile is highly nonlinear. For the air experiments under low heating rate conditions, this was not a concern unlike in the strongly heated nitrogen experiments. Figure 7 shows a good agreement with the modified Dittus–Boelter correlation with a leading coefficient of 0.021 rather than 0.023 [33]. There is an overall 12% difference for the air data for 5000 < Re < 50,000. Therefore, the present results indicate the validity of the experimental setup and data reduction procedure

$NuDittus−Boelter=0.021Re0.8Pr0.4(TwTb)−0.5$
(7)

In order to determine the leading coefficient in the Dittus–Boelter type correlation for strongly heated nitrogen data, the average Nusselt number data are plotted against Re0.8Pr0.4 in Fig. 8. When a linear bulk temperature profile is assumed and fluid properties are evaluated at the average bulk temperature between the inlet and outlet, a value of ∼0.017 is obtained for this coefficient as shown in Fig. 8(a). However, a value of 0.021 is obtained if the fluid properties are evaluated at the average of the eleven local bulk temperatures obtained from the energy balance as observed in Fig. 8(b). An explanation for this difference can be found by observing the results presented in Figs. 6(a) and 6(b). The higher the inlet Reynolds number of the flow is, the more linear its bulk temperature profile is. This is because the bulk temperature in low inlet Reynolds number increases faster in the entrance region, but at a slower rate further downstream. In other words, laminar flows have longer thermal and hydrodynamic entrance lengths than turbulent flows as well known. On average, among the 20 cases studied, the percentage differences in the average Nu and Re numbers calculated using the two different average bulk temperatures were 8% and 33%, respectively. A comparison of Figs. 8(a) and 8(b) highlights significant differences that can be obtained depending on how the average bulk temperature is evaluated when calculating the nondimensional parameters.

Next, the local Nusselt numbers obtained from the analysis described in Sec. 4 are examined. Since the gas is strongly heated in the present nitrogen experiments, there are large differences between the local wall and bulk temperatures. In such cases, some power of a wall-to-bulk temperature ratio, Tw/Tb, can be applied to turbulent convection correlations such as the Dittus–Boelter correlation and Gnielinski correlation given by the below equation [34,35]
$NuGnielinski=(f/8)(Re−1000)Pr1+12.7f/8(Pr2/3−1)(TwTb)−0.45(1+(LD)−23)$
(8)

where f is given by f = (1.82log10Re − 1.64)−2.

In Fig. 9, the local Nusselt number data are plotted against the local Reynolds number for air and nitrogen. The experimental results appear discontinuous because a single experiment could not cover the entire Reynolds number range desired; thus, the discontinuities mark the beginning or end of a new experimental data set. As a result, for a given local Reynolds number, different values of local Nusselt number were obtained depending on the local value of the wall-to-bulk temperature ratio, Tw/Tb. In Fig. 9, the local Reynolds numbers decrease from the inlet toward the exit, which causes reductions in the local Nusselt number as well. The local Nusselt number values predicted by Eqs. (7) and (8) are also shown in Fig. 9. For the air data shown in Fig. 9(a), the local Nusselt number data agree well with the modified Dittus–Boelter and Gnielinski correlations; however, the nitrogen data in Fig. 9(b) show significant deviations from the modified Dittus–Boelter and Gnielinski correlations. Since the correlations apply to fully developed flows, higher local Nusselt numbers than the values predicted by those correlations could be attributed to the entrance effects. On the other hand, significant reductions in the local Nusselt numbers toward the outlet can be attributed to a combination of the strong variations in fluid properties and DTHT effects as discussed in detail in Sec. 5.3.

### Deterioration of Forced Convection Heat Transfer.

This section discusses the impact of strong heating on turbulent gas flow and heat transfer, leading to gas property variations, buoyancy, and acceleration effects. Figure 10 displays examples of air and nitrogen flows experiencing significant reductions in the local Reynolds number between the inlet and outlet of the graphite test section. The reductions range from 22% to 50% depending on the inlet Reynolds number. If the Reynolds number at the outlet falls below 2100–2300, then the flow can be considered to have become laminar. Several factors play important roles in the Reynolds number reduction. At steady-state, as the bulk temperature increases, the gas viscosity (μ) and velocity (V) increase, while the gas density (ρ) decreases. Since the mass flux (ρv) remains constant, the Reynolds number reduction is a viscosity dominated behavior. As can be observed in Fig. 10(b), turbulent flows initially close to transitional flow traverse across the transitional flow and exit with nearly laminar Reynolds numbers.

On average, a 26% reduction in the Reynolds number over the 2.7 m test section was measured for air. This percentage was calculated from a total of 30 experimental runs. On the other hand, an average Reynolds number reduction of 41% was obtained for 30 nitrogen flow experiments, which is much higher than that for air due to higher heating rates. This clearly points out the effect of heating rate and graphite temperature on this phenomenon since the Reynolds number reductions for air were obtained at much lower graphite temperatures of ∼573 K compared to ∼773 K for nitrogen. The impact of graphite temperature will be further addressed in Sec. 5.4.

The variation of the local Nusselt number exhibits similar behavior as the local Reynolds number with reductions of up to 49% for air and 77% for nitrogen, as presented in Figs. 11(a) and 11(b), respectively. The data for nitrogen shown in Fig. 11(b) were obtained from various tests at pressures above 47.6 bar for inlet flow rates ranging from 1.4 × 10−3 kg/s to 4.1 × 10−3 kg/s. The Nusselt numbers fell more rapidly in the lower part of the test section and approached constant values near the test section outlet.

Figure 12 shows the axial variations of local parameters including the fluid properties, which are highly dependent on the bulk temperature. The Nusselt number decreased due to a continuous increase in the thermal conductivity of the gas combined with a continuously decreasing heat transfer coefficient. The wall heat flux decreased continuously due to increasing heat loss from the PV surface as the surface temperature increased with elevation. At the same time, the temperature difference, Twall − Tbulk, also decreased along the test section as previously shown in Fig. 6.

Figure 13 presents the local Nusselt number plotted against the local Reynolds number near the test section outlet. To filter any potential outlet effects, results are plotted for the ninth axial plane, which was about 35 cm below the upper flange. These results present direct evidences of degraded turbulent heat transfer (DTHT) under strongly heated conditions. Without DTHT, local Nusselt numbers of approximately 16 and 28 are expected at local Reynolds numbers of 5000 and 10,000, respectively, based on the modified Dittus–Boelter correlation. As presented in Fig. 13, the high temperature nitrogen flows experienced DTHT leading to Nusselt number reductions of over 50% from their expected values.

### Effect of Heating Rates on Turbulent Flow and Heat Transfer.

Axial variations of the local Reynolds number for nitrogen are presented in Fig. 14(a). The data were obtained under the same experimental conditions (48 bar, 1.4 × 10−3 kg/s), except for the heater power which was varied to modify the midplane graphite temperature. This was done in order to study the effect of the heating rate on deteriorated turbulent heat transfer. As can be observed in Fig. 14(a), higher heating rates lead to lower outlet Reynolds numbers for nearly the same inlet Reynolds number of about 5000. As the average bulk temperature increased by more than 200 K, the outlet Reynolds number dropped by more than 1800 for a total heater power of 1.0 kW. At higher heating rates, the Reynolds number reductions were even greater due to higher outlet bulk temperatures. The reduction in the Reynolds number increased from ∼34% to ∼45% as the average bulk temperature increased from 363 K to 583 K (see Fig. 14(a)).

From 18 nitrogen experiments conducted at two pressures (48 and 61 bar) and average gas velocities between 0.1 and 0.6 m/s, a linear relationship is found between the percent reduction in the Reynolds number from the inlet to the outlet and the outlet bulk temperature as shown in Fig. 14(b).

The mean heat transfer coefficient data are shown for two sets of data (48 bar and 1.4 × 10−3 kg/s, and 61 bar and 1.5 × 10−3 kg/s) in which the heater power was systematically increased to obtain lower mean Reynolds numbers (Fig. 14(c)) and higher mean bulk temperatures (Fig. 14(d)). For both data sets, an increased heating rate caused an increase in the mean bulk temperature (Fig. 14(d)), lower mean Reynolds number, and lower average wall-to-bulk temperature difference. Thus, the intensified heating would lead to higher mean heat transfer coefficients as shown in Fig. 14(c) for decreasing mean Reynolds numbers.

In general, increments in the heat transfer coefficient with increasing heating rate can be directly attributed to the increasing bulk temperature and thermal conductivity, as will be explained shortly. Note that these increments in heat transfer coefficient would not automatically result in increased Nusselt numbers since the thermal conductivity of the gas also increases with bulk temperature. This is consistent with the trend presented in Fig. 7, i.e., increasing average Nusselt numbers for increasing average Reynolds numbers. Once again, this highlights the importance of studying and understanding the quantitative effects of fluid property variations in strongly heated gas flows.

The heat transfer coefficient could increase while the Nusselt number decreases if the thermal conductivity increases faster as a result of the increasing gas temperature along the flow channel. The mean heat transfer coefficient results presented in Fig. 14(d) suggest that: (1) with increasing heating rates, the thermal conductivity increases faster than the heat transfer coefficient resulting in reduced Nusselt numbers and (2) the wall heat flux increases at a faster rate than the wall-to-bulk temperature difference. This is now explored in greater detail. Based on McEligot and Jackson [13], if the heat transfer coefficient for turbulent gas flows conforms well to a modified Dittus–Boelter correlation, the heat transfer coefficient can take the form
$h∼(cpμ)0.4k0.6$
(9)
For gases, one could express the following properties as functions of temperature only:
$cp∼T0.1, k∼T0.8, μ∼T0.7$
(10)
This suggests that the heat transfer coefficient would vary as
$h∼T0.24$
(11)

In Fig. 14(d), this variation is shown by a dashed line, and the experimental data for two pressures and flow rate conditions indicate the exponent to be slightly higher at 0.3–0.33.

Since Nu = hD/k, h ∼ T0.24, and k ∼ T0.8, the Nusselt number decreases with gas temperature, Nu ∼ 1/T0.56. Thus, in a strongly heated gas flow, as the outlet gas temperature increases, both the mean Reynolds and Nusselt numbers would decrease but the mean heat transfer coefficient, hmean, would increase even though DTHT occurs near the test section outlet. This result is somewhat counterintuitive as decreasing mean Reynolds and mean Nusselt numbers normally indicate deterioration in forced convection heat transfer. The present experiments with detailed examinations of mean as well as local heat transfer parameters are useful for understanding the DTHT phenomenon in strongly heated gas flows.

### Effect of Pressure on Heat Transfer and Flow Parameters.

Finally, the impact of pressure on DTHT is also examined. Nitrogen experiments were carried out under fixed conditions (1.9 × 10−3 kg/s, 863 K midplane graphite temperature), while the pressure was increased. By increasing the pressure by ∼27% from 48 to 61 bar, the mean heat transfer coefficient experienced increases of 29% and 15% for flow rates of 1.9 × 10−3 kg/s and 1.5 × 10−3 kg/s, respectively, as shown in Fig. 15(a). This is mainly due to the increased average bulk temperatures at higher pressures. Although not shown here, a direct relation exists between the average bulk temperature (as well as the measured outlet bulk temperature) and the pressure. Note that as the pressure was increased, the heater power had to be increased to maintain the desired midplane graphite temperature. This additional heat input resulted in higher Nusselt numbers and mean heat transfer coefficients.

The percent reductions in the local Reynolds numbers, however, appear to be independent of pressure but a function of bulk temperature only. Recalling a linear increase of percent reduction in Re with the outlet bulk temperature as shown in Fig. 14(b), we proceed to further explore the relation existing between these variables. If one expresses the Reynolds number as inversely proportional to viscosity ($Re∼μ−1)$, and since viscosity can in turn be expressed as ∼T0.7, one can arrive at the following expression for the Reynolds number reduction:
$Rereduction=1−ReoutRein∼(1−(TinTout)0.7)$
(12)
where Tin and Tout denote the inlet and outlet bulk temperatures, respectively. The right-hand side of Eq. (12) was divided by Tout and evaluated for 18 different nitrogen runs where the inlet velocity was varied from 0.1 m/s to 0.6 m/s while keeping the midplane graphite temperature constant at 863 K. This calculation yielded a constant value of 6.1 × 10−4 for the ratio of Reynolds number reduction to Tout as shown in Fig. 15(b). Based on this result, one could express the Reynolds number reduction for constant inlet temperature and different outlet bulk temperatures, Tout in Kelvin, by the following equation which is shown by a dashed line in Fig. 14(b):
(13)

The applicable range of Tout is given by 800 K < Tout < 900 K, and it is acknowledged that the result given by Eq. (13) is limited to nitrogen experiments under the experimental conditions covered in this work.

## Conclusions

Convection heat transfer experiments have been carried out with nitrogen and air flowing upward through a 16.8 mm diameter, 2.7 m long vertical flow channel in a graphite test section at pressures up to 65 bar. The inlet Reynolds number ranged from 1300 to 15,400 for nitrogen and 2500 to 70,000 for air, covering laminar, transition, and turbulent flows. The maximum gas temperatures at the outlet were 623 K for air and 888 K for nitrogen. The experimental data analyzed showed average reductions in the local Reynolds numbers of 26% for air and 41% for nitrogen over the 2.7 m length between the test section inlet and outlet, highlighting the impact of heating, especially for nitrogen. The average reductions in the local Nusselt numbers between the inlet and outlet for air and nitrogen were 38% and 77%, respectively. The influence of several factors leading to the reductions in Reynolds and Nusselt numbers due to strong heating was analyzed, including the changes in thermophysical gas properties, ρ, μ, and k. Increases in the viscosity and thermal conductivity with increasing bulk temperature caused large reductions in both the local and mean Reynolds and Nusselt numbers.

The average Nusselt number data for both air and nitrogen conformed well to the modified Dittus–Boelter and Gnielinski correlations applicable to fully developed turbulent flows. However, if the bulk temperature varies nonlinearly under strongly heated conditions as in the nitrogen experiments reported here, the fluid properties used in Nusselt, Reynolds, and Prandtl numbers must be calculated at the properly evaluated average bulk temperature from the local bulk temperature profile, rather than the average of the inlet and outlet bulk temperatures.

The local Nusselt numbers obtained for air under low heating rates also conformed to the modified Dittus–Boelter and Gnielinski correlations. But the local Nusselt numbers obtained in the strongly heated nitrogen flow experiments showed clear signs of degraded turbulent heat transfer (DTHT) near the test section outlet. Additionally, the enhancement of mean heat transfer coefficient in strongly heated flows has been shown to occur at higher mean bulk temperatures despite reductions in the Nusselt number due to increases in the thermal conductivity.

Future work will provide direct evidences of flow laminarization from turbulence measurements near the outlet using a hot wire anemometer in the same experimental setup. Convection heat transfer and flow laminarization results will also be provided for helium as the working fluid.

## Acknowledgment

The authors are grateful to the U.S. Department of Energy (DOE) Nuclear Energy University Program (NEUP) for financial support under a Contract No. 00119155. The participation of the following group members in the experimental part of this work is acknowledged: Jorge Pulido, Mehrrad Saadatmand, Narbeh Artoun, and Bruno Samorano. In addition, we would like to thank Dr. Manohar Sohal formerly at Idaho National Laboratory for his contribution to an extensive survey of heat transfer correlations. Useful insight into data analysis provided by Dr. Donald M. McEligot of Idaho National Laboratory is also acknowledged.

## Nomenclature

• A =

heat transfer area (m2)

•
• Bo* =

Jackson buoyancy parameter = Gr/(Re3.425Pr0.8)

•
• Cp =

specific heat capacity (J/kg K)

•
• D =

flow channel diameter (m)

•
• Dheater =

heater rod diameter (m)

•
• G =

mass flux (kg/(m2 s))

•
• Gr =

Grashof number with heat flux =  $gβq″D4/(kν2)$

•
• h =

heat transfer coefficient (W/m2 K)

•
• $k$ =

thermal conductivity (W/m K)

•
• Kv =

acceleration parameter =  $(ν/V2)(dV/dx)≈(4q+/Re)$

•
• L =

length of a flow channel segment (m)

•
• $m˙$ =

mass flow rate (kg/s)

•
• Nu =

Nusselt number $=hD/k$

•
• P =

heater power (W)

•
• Pr =

Prandtl number =  $cpμ/k$

•
• q+ =

dimensionless heat flux =  $q″/(GcpTb)$

•
• q″ =

wall heat flux (W/m2)

•
• Re =

Reynolds number $=4m˙/(πDμ)$

•
• T =

temperature (K)

•
• Tb =

bulk fluid temperature (K)

•
• Tw =

wall temperature (K)

•
• V =

axial velocity (m/s)

•
• z =

axial coordinate (m)

•
• ΔQ =

heat transfer rate (W)

### Greek Symbols

Greek Symbols

• μ =

dynamic viscosity (Pa·s)

•
• ν =

kinematic viscosity (m2/s)

•
• ρ =

fluid density (kg/m3)

Subscripts

• b =

bulk

•
• i =

segment

•
• in =

inlet

•
• o =

outlet

•
• w =

wall

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text:
## When is delta U = 0?

$\Delta U=q+w$

JeremyPark14B
Posts: 29
Joined: Fri Sep 28, 2018 12:19 am

### When is delta U = 0?

Under what conditions is delta U = 0?

Salena Chowdri 1I
Posts: 66
Joined: Fri Apr 06, 2018 11:02 am

### Re: When is delta U = 0?

For an isothermal system delta U = 0

kateminden
Posts: 63
Joined: Fri Sep 28, 2018 12:24 am

### Re: When is delta U = 0?

The first law of thermodynamics states that the internal energy (delta U) of an isolated system is constant. This means that delta U = 0 in an isolated system, because no energy can be transferred to/from the system and no work can be done on or is being done by the system.

Claudia Luong 4K
Posts: 59
Joined: Fri Sep 28, 2018 12:25 am

### Re: When is delta U = 0?

In isothermal systems, the internal energy does not change so delta U is always equal to zero.

lukezhang2C
Posts: 59
Joined: Fri Sep 28, 2018 12:18 am

### Re: When is delta U = 0?

Looking at the equation Delta U = q + w, delta U is only 0 when q = -w which means heat put into the system is equal to the heat out of the system, in other words the system is isolated with no net change in internal energy.

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In the late 70s, Cuntz and Behncke had a paper H. Behncke and J. Cuntz, Local Completeness of Operator Algebras, Proceedings of the American Mathematical Society, Vol. 62, No. 1 (Jan., 1977), pp. ...

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# American Institute of Mathematical Sciences

January  2017, 2: 1 doi: 10.1186/s41546-017-0014-7

## Stochastic global maximum principle for optimization with recursive utilities

 Zhongtai Institute of Finance, Shandong University, Jinan, Shandong 250100, People's Republic of China

Received  September 19, 2016 Revised  January 05, 2017

Fund Project: supported by NSF (No. 11671231, 11201262 and 10921101), Shandong Province (No.BS2013SF020 and ZR2014AP005), Young Scholars Program of Shandong University and the 111 Project (No. B12023).

In this paper, we study the recursive stochastic optimal control problems. The control domain does not need to be convex, and the generator of the backward stochastic differential equation can contain z. We obtain the variational equations for backward stochastic differential equations, and then obtain the maximum principle which solves completely Peng's open problem.
Citation: Mingshang Hu. Stochastic global maximum principle for optimization with recursive utilities. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 1-. doi: 10.1186/s41546-017-0014-7
##### References:
 [1] Briand, PH, Delyon, B, Hu, Y, Pardoux, E, Stoica, L:Lp solutions of backward stochastic differential equations. Stochastic Process. Appl. 108, 109-129 (2003), [2] Chen, Z, Epstein, L:Ambiguity, risk, and asset returns in continuous time. Econometrica 70, 1403-1443(2002), [3] Dokuchaev, M, Zhou, XY:Stochastic controls with terminal contingent conditions. J. Math. Anal. Appl. 238, 143-165 (1999), [4] Duffie, D, Epstein, L:Stochastic differential utility. Econometrica 60, 353-394 (1992), [5] El Karoui, N, Peng, S, Quenez, MC:Backward stochastic differential equations in finance. Math. Finance 7, 1-71 (1997), [6] El Karoui, N, Peng, S, Quenez, MC:A dynamic maximum priciple for the optimization of recursive utilities under constraints. Ann. Appl. Probab. 11, 664-693 (2001), [7] Ji, S, Zhou, XY:A maximum principle for stochastic optimal control with terminal state constrains, and its applications. Comm. Inf. Syst. 6, 321-337 (2006), [8] Kohlmann, M, Zhou, XY:Relationship between backward stochastic differential equations and stochastic controls:a linear-quadratic approach. SIAM J. Control Optim. 38, 1392-1407 (2000), [9] Lim, A, Zhou, XY:Linear-quadratic control of backward stochastic differential equations. SIAM J.Control Optim. 40, 450-474 (2001), [10] Pardoux, E, Peng, S:Adapted Solutions of Backward Stochastic Equations. Systerm Control Lett. 14, 55-61 (1990), [11] Peng, S:A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28, 966-979 (1990), [12] Peng, S:Backward stochastic differential equations and applications to optimal control. Appl. Math.Optim. 27, 125-144 (1993), [13] Peng, S:Open problems on backward stochastic differential equations. In:Chen, S, Li, X, Yong, J, Zhou, XY (eds.) Control of distributed parameter and stocastic systems, pp. 265-273, Boston:Kluwer Acad.Pub. (1998), [14] Shi, J, Wu, Z:The maximum principle for fully coupled forward-backward stochastic control system. Acta Automat. Sinica 32, 161-169 (2006), [15] Tang, S, Li, X:Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32, 1447-1475 (1994), [16] Wu, Z:Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems. Syst. Sci. Math. Sci. 11, 249-259 (1998), [17] Wu, Z:A general maximum principle for optimal control of forward-backward stochastic systems.Automatica 49, 1473-1480 (2013), [18] Xu, W:Stochastic maximum principle for optimal control problem of forward and backward system. J.Austral. Math. Soc. Ser. B 37, 172-185 (1995), [19] Yong, J:Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions. SIAM J. Control Optim. 48, 4119-4156 (2010), [20] Yong, J, Zhou, XY:Stochastic controls:Hamiltonian systems and HJB equations. Springer-Verlag, New York (1999),

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