## Abstract

An experiment was carried out on a bayonet tube that was kept at a constant temperature using condensing steam. In contrast, cold water was permitted to enter the central tube and discharge via an annular portion. The water flow rate was varied, covering laminar, transition, and turbulent regimes. The inner part of the bayonet tube is CPVC (chlorinated polyvinyl chloride, k = 0.136 W · m−1 · K−1), which reduces short-circuit heat transfer across the tube. Temperatures were recorded at different points in the tube. From the results of experiments on total heat transfer and short-circuit heat transfer, the Nusselt number can be calculated. The pressure drop across a bayonet tube determined the friction factor. In examining a range of Reynolds numbers, Effectiveness and figure of merit have been resolved. It has been observed that as the Reynolds number increases, the Nusselt number increases while the friction factor decreases. Both Effectiveness and Figure of Merit decrease with the addition of the Reynolds number, and it is observed that the maximum effective value is 0.86 for a 75 Reynolds number, which is suitable for bayonet solar collectors, and the minimum effective value is 0.2 for an 8062 Reynolds number, which is suitable for bayonet heat exchangers. It serves as reference work for bayonet tubes for designing a parabolic solar collector and heat exchanger.

## 1 Introduction

A bayonet tube consists of two concentric tubes, the inner tube is connected to the annulus, and the outer tube of the bayonet tube is closed at one end. A pump circulates a working fluid, which can be gas, liquid, or phase-changing substance, through a bayonet tube, based on the application. Fluid flows into the inner portion of the bayonet tube, which is laminar, transitional, and turbulent flow [1,2]. It returns into the annulus tube through the confined clearance. In the bayonet tubes, energy is exchanged predominantly through two surfaces: (i) A space between the inner and outer portions of the bayonet tube; and (ii) A space between the annulus and inner portions of the bayonet tube [1,3,4]. The bayonet tube can be used under high-temperature conditions and in highly corrosive environments when suitable materials are used [5,6]. The bayonet tube has a wide range of uses. The bayonet tube was used extensively in underground mines as a freezing element [7,8] and in tunnel construction [9,10] to create a cooling zone around the construction site. A direct heat exchanger (bayonet tube) is used in liquid metal cooled reactors [1115] to remove waste heat from the primary coolants in liquid metal reactors [16,17]. as well as in lead-cooled fast reactors to generate steam. In addition to being used as col-lectors for molten salt in photovoltaic power plants [3,4], as [18] decomposition reactors for hydrogen production, and evaporators for alkali metal thermoelectric converters, bayonet tubes are also used in alkali metal thermoelectric converters. Furthermore, Bayonet tubing can also be used as heat exchangers for external combustion combined cycle power plants, commonly found in fluidized bed coal furnaces and gasifiers [19,20]. Bayonet tubes are used as freezers in medicine because their cold tip works like one, typically in cryosurgery operations, to Eliminate aberrant or diseased tissue in specific tiny places [21]. Gas can also be stored underground in salt caves using the bayonet tube technique [22]. In addition, it is also used to estimate the effect of a magnetic field on ferrofluid flow in a cylindrical annulus [23], the flow stability of eccentric annuli [24], and the impact of a porous wall on solute dispersion in concentric annuli [25,26]. The bayonet tube is also used to produce hydrogen with the decomposition of sulfuric acid in a high-temperature gas-cooled reactor [27]. Despite all of these uses, there are no worldwide standards to oversee the fabrication of bayonet tubes [28,29], necessitating more research and development in this field. Conduction and convection are the primary modes of heat transmission in a bayonet tube [30]. However, in ultrahigh temperature applications where the temperatures of the direct and indirect fluids exceed 550 °C [31], radiation heat transfer should be considered. Heat loss from radiation is quite [6] significant when fluid temperatures exceed 800 °C [32]. Solar thermal power plants use bayonet tubes to generate electricity using molten salt as the working fluid [33]. Future aerospace thermal management systems may utilize bayonet tubes with dual compensation chamber loops [34].

When designing a bayonet tube, geometry is one of the most critical factors. So far, only a few studies [6,3544] have gone into depth on this subject. Specifically, researchers have examined pressure drop, friction factor, Nusselt number, and heat transfer rates at constant heat flux conditions based on bayonet tube length, clearance length [45], and tube cross-sectional area ratio. Till now, there has been no work found in bayonet tube with different tube materials, the concept of short circuit heat transfer, and constant wall temperature conditions under different flow regions such as Laminar, transition, and turbulent. This study aims to look into the heat transfer and pressure drop parameters of a single-phase flow in a bayonet tube. So, this Investigation seeks to establish how the Reynolds number influences friction factor, Nusselt number, short circuit heat transfer rate, total heat transfer, and bayonet tube performance caused by constant wall temperature conditions for novel work. In this study, With the help of a pressurized steam supply, a steady wall temperature condition is established. Depending on the Reynolds number, a different constant wall temperature is chosen. As the wall temperature is changed then corresponding fluid properties are also changed, for experimental calculation all properties are chosen at bulk fluid temperature. Bulk fluid temperature is the mean of inlet and outlet temperature. Short-circuit heat transfer rate is the heat transfer rate from an inner tube to an outer tube in the normal direction of flow. To further reduce the rate of short-circuit heat transfer (SCHTR), CPVC was utilized as the inner tube, because the thermal conductivity of CPVC tube is low.

## 2 Model Development

The experimental Investigation used a bayonet tube with two concentric tubes. During fluid passage into the inner tube, the outer tube is wrapped around the tube at one end to deflect the fluid into the annulus direction between the inner and outer tubes. The bayonet tube is placed in a glass chamber covered by polyurethane (PUF) foam insulating material. To maintain a constant wall temperature condition, Autoclave is used to create pressurized steam. Pressurized steam transfers heat to the outer tube, and the outer tube transfers heat to the annular space. Due to the lower thermal conductivity of the inner tube, there is less heat transfer through the annular space to the inner tube. Figure 1 illustrates the experimental setup schematically. An experimental setup consists of the following components:

Fig. 1
Fig. 1
Close modal
1. Bayonet tube

2. Autoclave (constant wall temperature Source)

3. Flowmeter

4. Thermocouple and Datalogger

5. Submersible pump

6. Manometer

7. Storage tank

Bayonet tube: The outer tube of the bayonet tube is made of galvanized iron (GI), whose thermal conductivity is 52 W · m−1 · K−1 [55]. The inner tube of the bayonet tube is made of chlorinated polyvinyl chloride (CPVC), whose thermal conductivity is 0.136 W · m−1 · K−1 [56]. The constant cross-sectional area technique calculates the clearance length, which is 4.4 mm. The bayonet tuber is inserted into the borosilicate glass chamber to circulate pressurized steam around it. The glass chamber is covered by polyurethane (PUF) foam as an insulating material to minimize heat transfer to the surroundings. Cold water flows through the inner tube with the help of a pump, and then it comes out through the annulus portion of the tube.

The geometrical parameters and properties of the bayonet tube and working fluid are shown in Fig. 2 and are mentioned in Table 1.

Fig. 2
Fig. 2
Close modal
Table 1

Bayonet tube dimensions

S. no.Variable parameterSymbolValueUnit
1.Length of the GI tubeL21.0044m
2.Length of the CPVC tubeL11.0m
3.Clearance length of the bayonet tubeLc4.4mm
4.Inner diameter of GI piped1,i27.3mm
5.Outer diameter of GI piped1,o31.5mm
6.Inner diameter of CPVC piped2,i17.63mm
7.Outer diameter of CPVC piped2,o20.85mm
8.Hydraulic diameter of the bayonet tubedh6.45mm
9.Thermal conductivity of GI pipeK252W · m−1 · K−1
10.Thermal conductivity of CPVC pipeK10.136W · m−1 · K−1
11.Specific heat of waterCp4186J · kg−1 · K−1
12.Density of waterρ998.2kg.m−3
S. no.Variable parameterSymbolValueUnit
1.Length of the GI tubeL21.0044m
2.Length of the CPVC tubeL11.0m
3.Clearance length of the bayonet tubeLc4.4mm
4.Inner diameter of GI piped1,i27.3mm
5.Outer diameter of GI piped1,o31.5mm
6.Inner diameter of CPVC piped2,i17.63mm
7.Outer diameter of CPVC piped2,o20.85mm
8.Hydraulic diameter of the bayonet tubedh6.45mm
9.Thermal conductivity of GI pipeK252W · m−1 · K−1
10.Thermal conductivity of CPVC pipeK10.136W · m−1 · K−1
11.Specific heat of waterCp4186J · kg−1 · K−1
12.Density of waterρ998.2kg.m−3

Autoclave: An autoclave is mainly used to generate high-temperature steam with the help of a 2 KW heating coil inside the Autoclave. A pressure gauge on top of the Autoclave maintains the steam temperature constantly. The pressurized steam is transferred through the glass chamber with the help of a hose pipe and check valve. The construction material of the Autoclave is stainless steel, which is situated on a tripod stand. The Autoclave has a 20-liter storage capacity and produces pressurized steam at a pressure of up to 30 PSI.

Flowmeter: A flowmeter is mainly used to measure the volume flow rate of working fluid inside the tube. This research uses a digital flowmeter with a 0–30 L/min flow range with ±0.15 L/min accuracy. This digital flow meter's operating pressure and temperature ranges are 0–3.2 MPa and 0–140 °C, respectively.

Thermocouple and data logger: In this research work, 3-wire, PT100 thermocouple is used, where PT denotes that the sensor is made from platinum and 100 indicates that at 0 °C the sensor has a resistance of 100 ohms. The temperature range of the PT100 type thermocouple is −200 to 850 °C with ±0.2 °C accuracy. When PT100 thermocouple ends are exposed to a temperature difference, the Seebeck effect occurs, which results in electricity being generated between the thermocouples [54,57]. All thermocouples are calibrated with the help of a microcontroller-based temperature calibrator using MTCCAL3.0 software. Thermocouples are placed at eight different positions, which are the inlet of the inner tube, the outlet of the inner tube, the outlet of the annulus portion, the storage tank, atmosphere, and three thermocouples are placed at the outer surface of the bayonet tube to measure the temperature of the wall. A data logger scans temperature data every 1–59 s across all eight channels and can store 5000 records of each of the eight channels. The input of this data logger is 8 PT100 sensors and 230 V/50 Hz alternating current with an accuracy of ±0.1 °C of the full scale. It uses 21CFR software for calibrating data.

Submersible pump: The water circulates along the bayonet tube with the help of a submersible pump. This pump operates at 220–240 V/50 Hz and requires 50 watts, which is good enough to produce a laminar, transition, and turbulent flow inside the bayonet tube. The pump's maximum flow rate is 35 L/min, and the efficiency of this submersible pump is 75%.

Manometer: The bayonet tube's pressure is measured with a U-tube manometer. It has a two-quarter inch hose nipple and a 0–300 mm pressure head range with an accuracy of ±0.127 mm.

Storage tank: The plastic storage tank has a 27.65 mm inner diameter and a 750 mm height. It has a 45-liter storage capacity and is covered with glass wool insulating material. Operating parameters are shown in Table 2.

Table 2

Operating parameter working range

S. no.Variable parameterSymbolValueUnit
1Pressurized steam rangeSP1–5MPa
2Wall temperatureTw333–370K
3Fluid inlet temperatureT1,i293–299K
4Reynolds numberRe75–8062
S. no.Variable parameterSymbolValueUnit
1Pressurized steam rangeSP1–5MPa
2Wall temperatureTw333–370K
3Fluid inlet temperatureT1,i293–299K
4Reynolds numberRe75–8062

## 3 Experimental Setup and Procedure

All components of the experimental setup are shown in Fig. 3. This bayonet tube is primarily used to heat rather than boil water. In this experiment, cold water is stored in a storage tank (at atmospheric temperatures) and then pumped through the inner portion of the bayonet tube until it emerges through the annulus portion using a pump. The bayonet tube's pressure is measured with a U-tube manometer. The flow rate through the bayonet tube is measured by a digital flowmeter. A voltage regulator and a check valve control the mass flow rate of cold water. The compressed steam from the Autoclave heats the water running through the annulus part and create constant wall temperature condition.

Fig. 3
Fig. 3
Close modal

Wait until the steady-state is not achieved, then note down all the readings corresponding to all the thermocouples. In this bayonet tube, the same procedure was repeated for different Reynolds numbers, and the effects of Reynolds numbers were examined. For this research, Reynolds numbers from 75 to 8062 were considered. A digital temperature data logger was used to measure the temperature with the help of calibrated PT-100 thermocouples. An average wall temperature was determined by considering the position of the outer wall surface thermocouple at 5, 500, and 1000 mm from the bottom of the bayonet tube, as shown in Fig. 1. During each experiment, all parameters, such as outlet temperature, end temperature, wall temperature, pressure, and mass flow rate, were recorded at a predetermined Reynolds number. Each experiment obtains four readings for each Reynolds number, and the mean is utilized. The bayonet tube performance was assessed based on all the data collected.

## 4 Methodology for Calculation

Reynolds number (Re): It is a dimensionless number that relates fluid momentum and viscous shear force. It also indicates whether fluid flow past a body or through a duct is regular or turbulent. The Reynolds number is carried out systematically
$Re=ρw·v·dhμw$
(1)
where the hydraulic diameter is determined by the bayonet tube diameters.
$dh=d2,i−d1,o$
Pressure drop (ΔP): It is the pressure difference along the bayonet tube
$ΔP = f · ρw · v2 · LF 2 · d1,i$
(2)

For friction factor (f), length (LF) is the fluid travel distance inside the bayonet tube.

Mass flow rate ($m˙$): It is the total amount of water (in kg) flowing during a specific time period (in sec)
$m˙ = ρw ·A1,i ·v$
(3)
Heat transfer: Heat transfer occurs in the form of sensible heat to raise the fluid's temperature, and it is computed using Eqs. (4) and (5)
$QT=m˙Cp(T2,o−T1,i)$
(4)
$Qs=m˙Cp(T1,o−T1,i)$
(5)
Effectiveness (ɛ): The ratio between the maximum heat transfer and the actual heat transfer is the Effectiveness of a heat exchanger (ɛ) [46]
$ɛ= QT Qmax = m˙ Cp(T2,o − T1,i )Cmin (T2,w − T1,i ) = (T2,o − T1,i )(T2,w − T1,i )$
(6)

Nusselt number (Nu): In order to improve heat transfer, a Nusselt number is essential. Nusselt number is a nondimensional heat transfer coefficient. This measure determines whether heat is transmitted via conduction or convection [47]

Overall Nusselt Number
$Nuo=QT (d2,i−d1,o) π K d2,i L (T2,w− Tf)$
(7)

where Tf = bulk fluid temperature $Tf = (T2,o + T1,i)2·$

Pump work (Ppump): It is determined by the flow rate and the pressure drop over the bayonet tube [48]where, $η$ is the efficiency of the pump.
$Ppump=1η·V˙·ΔP$
(8)
Figure of merit (FOM): This parameter is used to quantify the total heat transfer rate through a Bayonet tube concerning the power consumed by the flowing fluid [49]. FOM is a performance parameter, and it is expressed as
$FOM = QT Ppump$
(9)

## 5 Results and Discussion

This section discusses the effects of different Reynolds numbers or mass flow rates at constant wall temperature conditions on flowing fluid, i.e., the outlet temperature of both tubes, pressure drop, short circuit heat transfer rate, total heat transfer rate, Nusselt number, friction factor, figure of merit, and Effectiveness.

As the constant wall temperature is changed corresponding to the Reynolds number the fluid properties are also changed. In Fig. 4, the variation of specific heat, density, dynamic viscosity, and thermal conductivity of water is shown with constant wall temperature conditions. Using a DSC equipment, the water's specific heat was calculated [50]. The density of water is estimated using the Archimedes principle [51]. The KD-2 Protool measures the thermal conductivity of water. The viscosity of water was measured using an A&D vibro-viscometer. Fluid properties are taken at bulk fluid temperature to find all the resulting parameters.

Fig. 4
Fig. 4
Close modal

As illustrated in Fig. 5, when the Reynolds number of the flowing fluid is increased, the temperature difference between the inlet and outlet of the CPVC tube and GI tube is decreased. The bayonet tube's inner CPVC tube has a low thermal conductivity (0.135 W · m−1 · K−1), so the temperature gradient across the tube is low so that there is little heat transfer rate across the tube (short circuit heat transfer rate). Due to the low mass flow rate in Reynolds number 75, heat is transferred axially into the inner tube against the fluid flow length, resulting in a high-temperature fluid observed in the inner CPVC tube for this mass flow rate. As for the other case, the temperature rise in the inner CPVC tube is low due to the high mass flow rate. Due to the high thermal conductivity of the GI tube, there is a significant temperature rise of fluid when compared to the CPVC inner tube (52 W · m−1 · K−1). The bulk mean temperature of this flowing fluid is used to calculate all fluid properties.

Fig. 5
Fig. 5
Close modal

The Reynolds number is classified into three categories: laminar transition, and turbulent. Figure 6 shows fluid outlet temperature (T2o), inner CPVC outlet temperature (T1o), and wall surface temperature (Tw) variation with different mass flow rates. The constant wall temperature will also differ depending on the mass flow rate. All calculations are done on that particular constant wall temperature so that the Tw line shows the variation in the graph. When the mass flow rate of the fluid is increased, the velocity of the fluid particle also increases because the cross section area of the bayonet tube is constant, so heat transferred or carried away by the fluid particle also increases with the mass flow rate. It is observed that the inner CPVC tube outlet temperature (T1o) and Gi outlet temperature (T2o) decrease as the mass flow rate increases. This flow behavior is observed under specific constant wall temperature circumstances in laminar, transition and turbulent flows.

Fig. 6
Fig. 6
Close modal

As shown in Fig. 7, a constant cross-sectional area of the bayonet tube increases fluid velocity as mass flow increases at different wall temperature conditions. Based on the pressure drop relationship, the pressure drop increases as the mass flow rate increases, which shows that the pressure drop is directly proportionate to the square of the velocity and frictional shear force.

Fig. 7
Fig. 7
Close modal

One of the bayonet tube's primary purposes is to reduce short-circuit heat transfer. As shown in Fig. 8, short-circuit heat transfer gradually increases, attains a maximum value in the transition region, and then gradually decreases while total heat transfer continuously increases. The lower mass flow rate would lead to low QSCHTR and low QTotal for the fluid in the pipe. Initially, in the laminar and semitransition regions, QTotal and QSCHTR increase with mass flow rate from 0.0016 kg/sec to 0.07 kg/sec because, in this range, heat conduction across the tube is the dominant factor in comparison to heat convection along the tube due to low forced convection along the tube. It does not mean that the higher mass flow rate would increase continuously QSCHTR. As the mass flow rate increases from 0.07 kg/sec to 0.18 kg/sec, QSCHTR continuously decreases because, in the remaining semitransition and turbulent region, heat convection along the tube is higher due to high forced convection than heat conduction across the tube [58]. The mass flow rate considerably influences total heat transfer through a bayonet tube. Hence, incremental heat transfer increases as the mass flow rate rises. That means this bayonet tube effectively works in laminar and turbulent regions.

Fig. 8
Fig. 8
Close modal

As shown in Fig. 9, According to Nusselt theory, the Nusselt number is dependent on the Reynolds number and increases as the Reynolds number increases. The Nusselt number has a minimum value at the lowest Reynolds number in the laminar region and a maximum value in the turbulent region for this experiment. In the laminar region, the velocity gradient is shallow, so the energy transfer rate is low compared to transition and turbulent flow. In the turbulent region, high-velocity fluid is carried near the tube wall surface by mixing inside the boundary. During this process, slower-moving fluid is transferred to free flow. A lot of mixing is also aided by flow velocity swirls called streaks, which occasionally form along the wall and quickly increase and disappear. This trip may occur at speeds more significant than the inlet velocity. They behave nonlinearly, resulting in unpredictable circumstances that define the turbulent flow. This unexpected velocity profile affects the heat transfer within the boundary layer. As mass flow rates increase, fluid mixing also increases, which significantly affects convective heat and mass transfer rates. Convective heat and mass transfer rates are impacted because fluid mixing reduces the role of conduction and diffusion in establishing boundary layer thickness. In turbulent flow, the velocity, temperature, and species boundary layers are much thicker than in laminar flow [59]. Long tubes with constant wall temperatures have a constant Nusselt number, while short tubes have a variable Nusselt number. Figure 9 illustrates that, As the kinetic energy of the fluid particles increases with an increased Reynolds number, the friction factor varies at constant wall temperatures. As a result, the layer thickness of shear resistance decreases as the Reynolds number increases, and therefore, the friction factor decreases.

Fig. 9
Fig. 9
Close modal

As exemplified in Fig. 10, constant wall temperatures and Reynolds numbers play an essential role in FOM. FOM is determined by the rate at which heat is transferred through the bayonet tube and the pumping work done. According to the above discussion, the rate of total heat transfer and assignment performed by the pump increase with the Reynolds number, but the FOM falls because of Reynolds number has a more significant impact on pump work. In pump work, the volume flow rate and pressure drop are directly related to the Reynolds number. Although the mass flow rate is only associated with the Reynolds number in the overall heat transfer rate, the temperature difference is inversely related. With the help of the figure of merit, we quickly evaluate the optimum mass flow rate in different regions for maximum performance parameters.

Fig. 10
Fig. 10
Close modal

As illustrated in Fig. 10, Effectiveness frequently decreases with the Reynolds number at constant wall temperature conditions. The highest Effectiveness is obtained in the laminar region, which is 0.86 at a 75 Reynolds number, implying that outlet water temperature attains its maximum temperature at this Reynolds number. The lowest Effectiveness obtained in this experimental work is found in the turbulent region, which is 0.27 at 8062 Reynolds number, implying that outlet water temperature attains a minimum value at this Reynolds number.

The optimum points are obtained for two different parameters for this research work.

1. Maximum outlet temperature

2. Maximum heat transfer rate

The maximum outlet temperature (for example, in an application of a steam generation plant or solar water heater) is achieved by reducing the mass flow rate. While temperatures differ most when Reynolds numbers are low, other parameters, such as short-circuit heat transfer, pressure drop, total heat transfer, effectiveness, and the figure of merit, can also be important. By considering these parameters, the optimum points for this desired condition are mentioned in Table 3.

Table 3

Optimum parameter at constant wall temperature (338 K) for maximum outlet temperature

S. no.Optimum parameterValueUnit
1.Reynolds number562
2.Mass flow rate0.0123kg · sec−1
3.Outlet temperature322K
4.Nusselt number10.4
5.Pressure drop64.2Pa
6.Effectiveness0.62
7.Figure of merit1.5* 106
S. no.Optimum parameterValueUnit
1.Reynolds number562
2.Mass flow rate0.0123kg · sec−1
3.Outlet temperature322K
4.Nusselt number10.4
5.Pressure drop64.2Pa
6.Effectiveness0.62
7.Figure of merit1.5* 106

By increasing the mass flow rate, the maximum heat transfer rate (for example, heat exchanger) can be estimated, according to the results of this study. The highest heat transfer rate is obtained through a bayonet tube at high Reynolds numbers, but other factors like short-circuit heat transfer, effectiveness, pressure drop, total heat transfer, and figure of merit are also important. By considering these parameters the optimum points for this desired condition are mentioned in Table 4.

Table 4

Optimum parameter at constant wall temperature (346 K) for maximum heat transfer

S. no.Optimum parameterValueUnit
1.Reynolds number8062
2.Mass flow rate0.18kg · sec−1
3.Outlet temperature304K
4.Nusselt number130
5.Pressure drop2292Pa
6.Effectiveness0.20
7.Figure of merit14356
S. no.Optimum parameterValueUnit
1.Reynolds number8062
2.Mass flow rate0.18kg · sec−1
3.Outlet temperature304K
4.Nusselt number130
5.Pressure drop2292Pa
6.Effectiveness0.20
7.Figure of merit14356

### 5.1 Uncertainty in Experimental Work.

Regardless of how careful you are, there is always the possibility of uncertainties in experimental work measurement. As a result, it is critical to assess the system to identify the most significant possible uncertainty and the validity of experimental measurements. The experimental data recorded during Investigation often differ from the actual data due to many unaccountable factors while performing experiments. This deviation of the recorded data from existing data is called uncertainty. There is a strategy for calculating the uncertainty in data measurement. The uncertainties associated with various parts and devices have been estimated and summarized in Table 5 using the method of Kline and McClintock [5254] for determining uncertainties. Table 6 shows the measurement uncertainties of several different parameters.

Table 5

Uncertainty in various measuring parts and devices

S. no.Parts/DevicesInstrumentLeast countUncertainties
1.Tube lengthLinear meter Scale1 mm±1 mm
2.Inner tube diameterCaliper Scale0.05 mm±0.04 mm
3.Outer tube diameterCaliper Scale0.05 mm±0.07 mm
4.TemperaturePT100 thermocouple0.1 °C±0.2 °C
5.Data loggerDigital PT100 logger0.1 °C±0.1 °C
6.Pressure drop across the tubeU-tube manometer0.05 mm of Hg±0.127 mm of Hg
7.Volume flow rateDigital flow meter±0.15 L/min
S. no.Parts/DevicesInstrumentLeast countUncertainties
1.Tube lengthLinear meter Scale1 mm±1 mm
2.Inner tube diameterCaliper Scale0.05 mm±0.04 mm
3.Outer tube diameterCaliper Scale0.05 mm±0.07 mm
4.TemperaturePT100 thermocouple0.1 °C±0.2 °C
5.Data loggerDigital PT100 logger0.1 °C±0.1 °C
6.Pressure drop across the tubeU-tube manometer0.05 mm of Hg±0.127 mm of Hg
7.Volume flow rateDigital flow meter±0.15 L/min
Table 6

Uncertainty in various parameter

S. no.ParameterUncertainty (%)
1.Cross-sectional area of inner tube±0.38
2.Cross sectional Area of outer tube±0.44
3.Hydraulic diameter±0.31
4.Temperature±0.41
5.Pressure drop±1.64
6.Volume flow rate±1.70
7.Mass flow rate±1.70
8.Heat transfer rate±1.78
9.Reynolds number±1.67
10.Nusselt number±1.71
11.Friction factor±2.42
S. no.ParameterUncertainty (%)
1.Cross-sectional area of inner tube±0.38
2.Cross sectional Area of outer tube±0.44
3.Hydraulic diameter±0.31
4.Temperature±0.41
5.Pressure drop±1.64
6.Volume flow rate±1.70
7.Mass flow rate±1.70
8.Heat transfer rate±1.78
9.Reynolds number±1.67
10.Nusselt number±1.71
11.Friction factor±2.42

According to Kline and McClintock uncertainty, the following procedure should be used when evaluating uncertainty:

Let a parameter be calculated using specific measured quantities as
$y=y(x1,x2,x3,…⋯,xn)$
Then uncertainty in the measurement of y is given as follows:
$[(δyδx1δx1)2+(δyδx2δx2)2+(δyδx2δx2)2+⋯⋯⋯⋯⋯+(δyδxnδxn)2]12$

where $δx1,δx2,δx3……………δxn$ are the possible uncertainties in measurements of $x1,x2,x3…xn, δy$ is absolute uncertainty and $δyy$ is relative uncertainty. All the uncertainty calculations are mentioned in Appendix  A.

### 5.2 Correlation.

It has been discovered that an average Nusselt number correlates with various Reynolds numbers, as follows:
$NuCorrelation=−423.3 +543.1 log 10(Re)−229.4(log 10(Re))2+32.39(log 10(Re))3$
(10)

A correlation has been created for 75 ≤ Re ≤ 8062 and Nusselt number. It is observed that the cubic logarithmic curve is fitted with experimental values with 99.63% R-Square values and 99.56% R-Square (Adj) values. The above-predicted and experimental results are shown in Fig. 11. This figure shows that all points lie inside the ±5% dashed–dotted uncertainty line. As a result, the correlation-derived value of the average Nusselt number agrees with the experimental results.

Fig. 11
Fig. 11
Close modal
It has been discovered that a friction factor correlates with various Reynolds numbers, as follows:
$log 10(f) =3.780−3.065 log 10(Re) +0.6821(log 10(Re))2−0.06388(log 10(Re))3$
(11)

A correlation between 75 ≤ Re ≤ 8062 and the friction factor has been created. It is observed that the cubic logarithmic curve is fitted with experimental values with 99.81% R-Square values and 99.89% R-Square (Adj) values. As a result, the correlation-derived value of the friction factor agrees with the experimental results. A dotted line indicates an uncertainty band of ±1.5%, and a dashed–dotted line indicates a ±2.5% uncertainty band. This figure shows that all points are inside of the ±1.5% uncertainty band. It can be observed that no point is within the uncertainty band of ±2.5%. As a result, the correlation-derived value of the friction factor agrees with the experimental results (Fig. 12).

Fig. 12
Fig. 12
Close modal

## 6 Conclusion

This research was conducted on bayonet tubes with varying Reynolds numbers at varying constant wall temperatures. The results conclude that effectiveness, figure of merit, heat transfer, pressure drop, and friction factor are strongly related to design parameters and Reynolds number. This bayonet tube can be used for heating the cold water under the laminar region to achieve maximum Effectiveness and figure of merit. It is also used in heat exchangers at high Reynolds numbers to attain maximum energy transfer. The following are recommended conclusions derived from this research work:

1. Increasing Nusselt number is intensely related to wall temperature conditions and flow characteristics.

2. The maximum effective value is 0.86 for a 75 Reynolds number best, and the minimum effective value is 0.2 for an 8062 Reynolds number. With increasing Reynolds numbers, Effectiveness decreases.

3. With increasing Reynolds number, other parameters such as outlet water temperature, friction factor, and figure of merit fall, implying that at high mass flow rates, the ratio of overall heat transfer to power consumption reduces.

4. The overall heat transfer rate increases as the Reynolds number rises while the bayonet tube's Effectiveness drops.

5. The short circuit heat transfer rate increases with a rising Reynolds number and reaches its maximum value in the transitional zone. Following that, it drops as the Reynolds number rises.

6. A correlation is derived in terms of the Nusselt number and friction factor. This means that derived Eqs. (10) and (11) are compatible with experimental results since all points are within the ±5% uncertainty band for the Nusselt number and inside the ±2.5% uncertainty band for the friction factor.

## Funding Data

• National Institute of Technology, Jamshedpur, India (Grant No. 15-1/2016-PN; Funder ID: AACAN7221M).

## Competing Interest

According to the authors, the research presented here was not influenced by competing financial interests or personal ties.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

A =

tube surface area (m2)

cp =

specific heat capacity (J · kg−1 · K−1)

Cp =

specific heat of the fluid (KJ · kg−1 · K−1)

d =

diameter (m)

F =

fluid

f =

friction factor

FOM =

figure of merit subscripts

h =

hydraulic

i =

inner surface/inlet

k =

thermal conductivity (W · m−1 · K−1)

L =

length of the bayonet tube (m)

Lc =

clearance length of the bayonet tube (m)

$m˙$ =

mass flow rate of the flowing fluid (kg · s−1)

m =

mean

Nuo =

over all Nusselt number

o =

outer surface/outlet

P =

pressure (Pa)

Ppump =

pumping power (W)

Qmax =

rate of maximum possible heat transfer (W)

Qs =

rate of short circuit heat transfer (W)

QT =

rate of total heat transfer (W)

Re =

Reynolds number

T =

temperature (K)

TW =

wall temperature (K)

v =

velocity of fluid (m · s−1)

w =

water

$V˙$ =

flow Rate of volume (m3 · s−1)

Δ =

difference

μ =

dynamic viscosity of the fluid (Pa · Sec)

ɛ =

effectiveness of the bayonet tube

ρ =

density (kg · m−3)

1 =

inner tube (CPVC tube)

2 =

outer or annulus tube (GI tube)

### Appendix A

##### A Measurement Uncertainty.

Regardless of how careful you are, there is always the possibility of errors in experimental work measurement. As a result, it is critical to assess the system to identify the most significant possible uncertainty and the validity of experimental measurements. The experimental data recorded during Investigation often differ from the actual data due to many unaccountable factors while performing experiments. This deviation of the recorded data from existing data is called uncertainty. There is a strategy for calculating the uncertainty in data measurement. The uncertainties associated with various parts and devices have been estimated and summarized in Table 5 using the method of Kline and McClintock [5254] for determining uncertainties. Table 6 shows the measurement uncertainty of several different parameters.

According to Kline and McClintock uncertainty, the following procedure should be used when evaluating uncertainty:

Let a parameter be calculated using specific measured quantities as
$y=y(x1,x2,x3,……,xn)$
Then uncertainty in the measurement of y is given as follows:
$[(δyδx1δx1)2+(δyδx2δx2)2+(δyδx2δx2)2+⋯⋯⋯⋯⋯+(δyδxnδxn)2]12$

Where $δx1,δx2,δx3……………δxn$ are the possible errors in measurements of $x1,x2,x3…xn⋅$

$δy$ is absolute uncertainty and $δyy$ is relative uncertainty.

1. The cross-sectional area of the inner tube
$δAiAi=[(πDi×δDi2)πDi24]0.5=[2δDiDi]=[2×(±0.04)20.85]=±0.003837=±0.3837 %$
2. The cross-sectional area of outer tube
$δAoAo=[(πDo×δDo2)πD024]0.5=[2δDoDo]=[2×(±0.07)31.5]=±0.0044=±0.44 %$
3. Hydraulic diameter
$δdhdh=[δDhDh]=[±0.026.45]=±0.0031=±0.31%$
4. Temperature
$δTT=[±0.249]=±0.004081=±0.4081%$
5. Pressure drop
$δPP=δhh=[±0.1277.7]=±0.01649=±1.649%$
6. Volume flow rate
$δV˙V˙=[±0.15 8.8]=±0.0170=±1.70 %$
7. Mass flow rate
$m˙=ρ(A·v)=ρ·V˙δm˙m˙=[(δρρ)2+(δV˙V˙)2]0.5=[(0)2+(±0.0170)2]0.5=±0.0170=±1.70%$
8. Reynolds Number
$Re=ρw· v· dhμw=4·ρw· V˙π·μw· dhδReRe=[(δρρ)2+(δV˙V˙)2−(δdhdh)2]0.5=[(0)2+(±0.0170)2 −(±0.0031)2 ]0.5=±0.0167=±1.67%$
9. Nusselt number
$Nuo = QT (d2,i−d1,o) π K d2,i L (T2,w− Tf)δNuoNuo=[(δQT QT )2+(δdhdh)2−(δd2,id2,i)2−(δLL)2−(δTT)2]0.5=[(±0.01748)2+(±0.0031)2−(±0.00227)2−(±0.000996)2−(±0.004081)2]0.5=±0.017098=±1.71 %$
10. Heat transfer rate
$δQQ=[(δm˙m˙)2+(δΔTΔT)2]0.5δQQ=[(±0.017)2+(±0.004081)2]0.5=±0.01748=±1.748 %$
11. Friction factor
$f α ΔP · d1,i ρw · v2 · Lf α ΔP · d1,i5 ρw · V˙2 · Lfδff=[(δPP)2+(5*δd1,id1,i)2−(2*δV˙V˙)2−(δLfLf)2]0.5δff=[(±0.01649)2+(±0.05681)2−(±0.054)2−(±0.000996)2]0.5=±0.02415=±2.415 %$

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