Heat Transfer in Double Pipe Heat Exchangers With Small Tube Spacing

Havlík, Jan; Dlouhý, Tomáš; Krempaský, Jakub

doi:10.1115/1.4066930

Abstract

In this work, the heat transfer in double pipe heat exchangers (DPHEs) with small tube spacing is analyzed. This type of heat transfer is generally described in many literary sources; however, in laminar flow, the resulting values of the theoretical equations differ. Moreover, the small spacing of the pipes can affect the fluid flow and subsequent heat transfer. The suitability of using available theoretical methods for the design of double pipe heat exchangers (the Stephan, VDI, and Baehr methods) was evaluated. A series of experiments was carried out on a double pipe heat exchanger with a tube spacing of 1.4 mm. The experimentally determined value of the heat transfer coefficient (HTC) was up to two times higher in comparison with the values calculated according to theoretical methods. The relative increase was more significant for the lower Reynolds number values. At the same time, the difference of the wall temperature along the circumference of the outer tube was detected. This phenomenon was theoretically analyzed and can be explained by tube misalignment in the small tube spacing. The procedure of quantification of this effect was proposed. This effect may cause an inhomogeneity of the media flow and temperature distribution and, as a result, increase the performance of the heat exchanger by tens of percent.

Introduction

Double pipe heat exchangers (DPHEs) are a widely used configuration in heating systems as well as in industrial systems [1]. DPHEs can be used in the chemical, food, oil and gas industries, solar and geothermal energy applications, air-conditioning applications, reheating, preheating, and digester heating processes [1–3]. Moreover, DPHE is still widely used for the research and development of experimental facilities in energy engineering laboratories [2]. One advantage is the low cost of design and maintenance [1]. This type of heat exchanger is simple in structure and easy to install, clean, maintain, and modify, which allows a significant extension of life and service [1,2]. Reducing the size and cost of heat exchangers, combined with efforts to improve their efficiency, requires an increase in heat transfer intensity. It has a good possibility to intensify heat transfer by heat-transfer enhancement [4]. Thermoeconomic optimization of heat transfer may be obtained by an appropriate ratio of inner and outer pipe diameters [5].

In industrial applications, there is often a requirement to preheat the media to the highest possible temperature in a heat exchanger. When preheating the medium, the requirement for the highest possible outlet temperature leads to a decrease in its flow rate and velocity; therefore, a small intertube spacing is necessary. For these purposes, it is appropriate to use a DPHE when the heated medium is placed in the annulus with a relatively small cross section area and the heating medium (e.g., steam, hot water) flows in the inner pipe. Even so, laminar flow can often occur in the annulus [6]. Furthermore, tube misalignment in a close-packed tubular heat exchanger may occur [7,8]. An eccentric annular duct may influence the heat exchanger's operation. The local Reynolds number varies between narrow and wider parts, which may influence friction factors and fluid flow [7]. There is less fluid velocity in the narrower gap, and the local Reynolds number is significantly lower than that in the wide gap. The mean temperature gradient in the narrow gap averages up to three times larger than in the wide gap [8,9]. The heat transfer is affected by temperature changes in the flow in the passages of DPHE [10]. The heat transfer rate is higher in an eccentric annulus than in a concentric annulus and a single tube; therefore, the heat exchanger performance is also higher [11,12]. As the Reynolds number increases, the effect of eccentricity on the convection heat transfer coefficient (HTC) becomes negligible [11].

An overview of suitable equations for the calculation of the heat transfer coefficient in the annulus is described by Omidi et al. [1] and Dirker and Meyer [13]. However, a validity range of the Reynolds number is often not specified for the given equations, which can lead to varying results. The calculation of the heat transfer coefficient inside a pipe is described for different cases and conditions, e.g., the Dittus–Boelter equation for turbulent fluid flow, the Nusselt equation for condensation of water vapor in a vertical pipe, or other equations according to the mode of flow inside the pipe [14–16].

This article aims to verify the suitability of using available theoretical methods for the design of a DPHE with a small tube spacing and low velocities corresponding to a laminar type of flow of a heat transfer medium in the annulus. In a case of a difference in a comparison with the experimental results, the possible cause will be analyzed, the effect on the heat exchanger performance will be quantified and a recommendation for the design of such a DPHE will be proposed.

Material and Methods

Heat Transfer in Double Pipe Heat Exchangers.

In DPHE (Fig. 1), heat is transferred from the medium in the inner tube to the medium in the annulus through the inner tube wall. Presented equations are valid for heat transferred to the inner tube, while the outer tube is insulated to maintain a constant wall temperature with a hydrodynamically and thermally developing flow.

Fig. 1

View large Download slide

Cross section of a double pipe heat exchanger

The characteristic dimension

d_{h}

for determining the heat flow transferred between the wall and fluid flowing in the annulus is given by

d_{h} = D_{i} - d_{o}

(1)

The geometrical similarity of the annulus depends on the diameter ratio a, where

d_{o}

is the outer diameter of the inner tube and

D_{i}

is the inner diameter of the outer tube

a = \frac{d_{o}}{D_{i}}

(2)

The physical properties of the fluid flowing in the annulus are calculated for the mean fluid temperature

T_{m} = (\frac{T_{in} + T_{out}}{2})

(3)

The Nusselt and Reynolds numbers are provided by the usual equations

Nu = \frac{h \cdot d_{h}}{k}

(4)

Re = \frac{u \cdot d_{h}}{ν}

(5)

The theoretical methods for the calculation of the Nusselt number for laminar flow are described below.

Calculation by VDI Heat Atlas (VDI Method).

The calculation of the HTC for medium flow in the annulus is described in section, G2 Heat transfer in concentric annular ducts in VDI Heat atlas [14]. The HTC of cooling water flowing in the annulus is defined by the Nusselt number specified for each region according to the Reynolds number. The characteristic equation for the mean Nusselt number for laminar flow in the region of Re < 2300 can be calculated by

{Nu}_{lam} = \sqrt[3]{({Nu}_{1}^{3} + {Nu}_{2}^{3})}

(6)

where Nu₁ and Nu₂ are given for the outer tube insulated by

{Nu}_{1} = (3.66 + 1.2 {(\frac{d_{o}}{D_{i}})}^{- 0.8})

(7)

{Nu}_{2} = f_{g} \sqrt[3]{RePr \frac{d_{h}}{L}}

(8)

Coefficient f_g depends on the boundary condition for heat transfer in an annular gap. Heat transferred at the inner tube f_g is given by

f_{g} = 1.615 \cdot (1 + 0.14 \cdot {(\frac{d_{o}}{D_{i}})}^{- 1 / 2})

(9)

The effect of temperature dependent physical properties is considered by the equation

Nu = {Nu}_{lam} \cdot {(\frac{\Pr}{\Pr_{w}})}^{0.11}

(10)

where Pr is the Prandtl number at the mean temperature T_m, and Pr_w is the Prandtl number at the wall temperature.

Calculation by the Baehr and Stephan (Baehr Method).

Baehr and Stephan [17] offer an alternative approach on how to calculate the mean Nusselt number of the fluid flowing in the annulus. This approach utilizes an equation for calculating the Nusselt number of the fluid flowing in the tube. For a hydrodynamically fully developed laminar flow, the mean Nusselt number

{Nu}_{m}

is calculated by the equation

\frac{{Nu}_{m} - {Nu}_{\infty}}{{Nu}_{m, tube} - 3.657} = f (\frac{d_{o}}{D_{i}})

(11)

where

{Nu}_{\infty} = 3.657 + 1.2 {(\frac{d_{o}}{D_{i}})}^{- 0.8}

(12)

and

f (\frac{d_{o}}{D_{i}}) = 1 + 0.14 {(\frac{d_{o}}{D_{i}})}^{- 0.5}

(13)

Equations and are valid for

0 \leq X^{+} \leq \infty; Re \leq 2300

⁠; and 0 ≤

(d_{o} / D_{i})

≤ 1. The mean Nusselt number for the fluid flowing inside the tube is calculated by the equation

{Nu}_{m, tube} = \frac{3.657}{\tanh (2.264 X^{+ \frac{1}{3}} + 1.7 X^{+ \frac{2}{3}})} + \frac{0.0499}{X^{+}} \tanh X^{+}

(14)

Equation is valid for

0 \leq X^{+} \leq \infty; Re \leq 2300

with

X^{+} = \frac{L}{d_{h} \cdot Pe}

(15a)

Pe = \frac{u \cdot d_{h} \cdot ρ \cdot c_{p}}{k}

(15b)

Calculation by Stephan (Stephan Method).

Stephan developed a heat transfer correlation for hydrodynamically developed laminar flow; the outer wall of the annulus is insulated, described in Ref. [18] as

Nu = {Nu}_{\infty} + [1 + 0.14 {(\frac{d_{o}}{D_{i}})}^{- 1 / 2}] \cdot \frac{0.19 {(Pe \cdot d_{h} / L)}^{0.8}}{1 + 0.117 {(Pe \cdot d_{h} / L)}^{0.467}}

(16)

where

{Nu}_{\infty}

is the Nusselt number for fully developed flow

{Nu}_{\infty} = 3.66 + 1.2 {(\frac{d_{o}}{D_{i}})}^{- 1 / 2}

(17)

These equation may be used for laminar follow in the range of $0.1 < (Pe \cdot d_{h} / L) < 10^{4}$ and $Re < 2300.$

The Experimental Setup

A schematic diagram of the experimental apparatus is shown in Fig. 2. The DPHE was designed as a vertical heat exchanger consisting of two concentric stainless tubes. The inner tube of the heat exchanger is 2000 mm long with an inner diameter of 23.7 mm and a wall thickness of 1.6 mm. The outer tube is 1500 mm long with an inner diameter of 29.7 mm and a wall thickness of 2 mm. The material of the tubes is stainless steel 1.4301 (AISI 304). The heat exchanger is covered with microfiber insulation. The annulus, made from concentric tubes, is 1.4 mm wide. Stainless pins are used as spacers at three circumferential positions to keep the annulus concentric. The heat exchanger is heated by steam which enters at the top and condenses in the inner tube. The cooling water countercurrently flows in the annulus section.

Fig. 2

View large Download slide

Experimental setup

The steam temperature is controlled so that it is in a saturated or slightly superheated state before it enters the heat exchanger. The condensate flowing out of the tube is collected in a tank and its production is determined by weighing. Any excessive steam in the heat exchanger outlet is released into the ambient. The position of probes for the temperature, pressure, weight, and flow measurements are shown in Fig. 2. Temperatures are measured by T-type thermocouples which are calibrated to within 0.3 °C. The cooling water flow is measured by a flowmeter FLONET FN2010.1 (Elis Pilsen Ltd., Pilsen, Czech Republic) with current output calibrated with a deviation of 0.6%. A laboratory balance with unit weighing 1 g is used to determine the amount of condensate. The datalogger DAS 240 BAT (SEFRAM, Saint-Etienne, France) with the daslab software is used for data collection.

Experiments' Evaluation Procedure.

The calculation of the HTC is based on the heat balance of the heat exchanger [16,17]. The heat transfer rate released by condensation

Q_{c}

has to be equal to the heat transfer rate to the cooling water

Q_{w}

and heat loss of the condenser

Q_{loss}

Q = {Q_{c} = Q}_{w} + Q_{loss}

(18)

{Q_{c} = M}_{c} \cdot h_{fg}^{'}

(19)

where

M_{c}

is the condensate flow rate, and

h_{fg}^{'}

is the specific latent heat of condensation. The heat transfer rate of side water side

Q_{w}

(the heat performance of DPHE) is given by the equation

Q_{w} = M_{w} \cdot c_{w} \cdot (T_{w, out} - T_{w, in})

(20)

where

M_{w}

is the cooling water flow,

c_{w}

is the specific heat capacity of water,

T_{w, out}

is the outlet cooling water temperature, and

T_{w, in}

is the inlet cooling water temperature. Heat loss

Q_{loss}

was determined by measuring the surface temperature of the heat exchanger; in all measurements, it achieved a maximum of 0.3% of the heat performance, so it has a negligible effect on the calculation. The heat transfer rate for the evaluation was taken from the water side. The heat transfer rate from the amount of condensate was also determined and used for control. The difference in all measurements was up to 2%. The overall HTC U is given by the equation

U = \frac{Q}{A \cdot {Δ T}_{\log}}

(21)

where A is the heat transfer surface, and

{Δ T}_{\log}

is the logarithmic mean temperature difference

{Δ T}_{\log} = \frac{(T_{v} - T_{w, in}) - (T_{v} - T_{w, out})}{\ln \frac{(T_{v} - T_{w, in})}{(T_{v} - T_{w, out})}}

(22)

The inner tube-side HTC

h_{c}

describing vapor condensation may be calculated by the formula determining the overall HTC

U = \frac{\frac{1}{d_{o}}}{\frac{1}{d_{i} \cdot h_{c}} + \frac{1}{2 k} l n (\frac{d_{o}}{d_{i}}) + \frac{1}{d_{o} \cdot h_{w}}}

(23)

where $d_{i}$ is the inside diameter of the tube, k is the thermal conductivity of the tube, $h_{w}$ is the waterside HTC, and $d_{o}$ is the outside diameter of the tube.

Because the steam inside the pipe flows at a high velocity, it is necessary to take shear stress into account when calculating the heat transfer coefficient [19]. Heat transfer is determined from the condensate film thickness, which depends on the local mass flow rate and interfacial shear stress. According to Blangetti et al. [20], Kageyama et al. [21], and Toman et al. [22], the local mass flow rate can be expressed as

Г = \dot{\frac{m}{π \cdot d_{i}}} = ρ_{L} \frac{g (ρ_{L} - ρ_{g}) δ_{L}^{3}}{{3 μ}_{L}} + ρ_{L} \frac{τ_{g}}{{2 μ}_{L}} δ_{L}^{2}

(24)

The shear stress

τ_{g}

exerted by flowing gas on the condensate film may be determined according to the following relationship:

τ_{g} = \frac{1}{2} ρ {u_{g}}^{2} f

(25)

where

u_{g}

is the velocity of the flowing gas, and f is the coefficient of the frictional losses, its determination is described, e.g., in Ref. [15]

f = 0.079 \cdot {Re}^{- \frac{1}{4}}

(26)

For a given condensate flow, it is possible to determine the thickness of the laminar film

δ_{L}^{*}

by introducing the characteristic length L, Reynolds number of condensate

{Re}_{L}

⁠, and dimensionless shear stress

τ_{g}^{*}

L = {(\frac{μ_{L}^{2}}{ρ_{L}^{2} \cdot g})}^{\frac{1}{3}}, {Re}_{L} = \frac{4 \cdot Г}{μ_{L}}, τ_{g}^{*} = \frac{τ_{g}}{g \cdot ρ_{L} \cdot (1 - \frac{ρ_{g}}{ρ_{L}}) \cdot L}, δ_{L}^{*} = \frac{δ_{L}}{L}

(27)

Combining these parameters and Eq. , the expression for calculating the laminar film thickness is obtained

\frac{{Re}_{L} / 4}{1 - ρ_{g} / ρ_{L}} = \frac{δ_{L}^{* 3}}{3} + \frac{τ_{g}^{*} \cdot δ_{L}^{* 2}}{2}

(28)

The HTC is expressed from the local Nusselt number at location x which is obtained as

N u_{x} = \frac{h_{c} \cdot L}{k} = {({Nu}_{x, lam}^{4} + {Nu}_{x, turb}^{4})}^{1 / 4}

(29)

where the local Nusselt number for the laminar film

N u_{x, lam}

is given

N u_{x, lam} = \frac{1}{δ_{L}^{*}}

(30)

and the local Nusselt number for the turbulent film

N u_{x, turb}

is given

N u_{x, turb} = a \cdot {Re}_{L}^{b} \cdot \Pr^{c} \cdot [1 + e \cdot τ_{g}^{* f}]

(31)

The coefficients a, b, c, e, and f were tabulated depending on the parameter

τ_{g}^{*}

[21]. The heat transfer coefficient at location x along the tube is determined as

h_{x} = \frac{Nu \cdot k}{d_{h}}

(32)

The resulting value

h_{c}

is then calculated as the average of the

h_{x}

along the length of the tube. This procedure was experimentally verified with the mean deviation of HTC values of 5%. The HTC of cooling water in the annulus

h_{w}

is determined as

h_{w} = \frac{\frac{1}{d_{o}}}{\frac{1}{d_{o} \cdot U} - \frac{1}{d_{i} \cdot h_{c}} + \frac{1}{2 k} l n (\frac{d_{o}}{d_{i}})}

(33)

where the overall heat transfer coefficient U is taken from Eq. (23).

The results of the HTC of cooling water in the annulus are sensitive to the theoretically determined condensation HTC. Although using methods as the parameter estimated approach [23,24] is possible to determine the correlations for both sides independently, the problem is a complex equation of the dependences on the condensation side and on the annulus side compared to the assumptions for turbulent liquid flow without phase change used in frequent applications of this method. It was taken into account in the design of the experiments. The experiments were carried out in individual series, maintaining the measurement conditions on the annulus side (mainly the water flow rate) and varying the steam flow rate on the condensing side, in order to verify the theoretical model used for determining the condensation side HTC (see Fig. 3). While adjusting the parameters, the mean temperature of the cooling water changed slightly, but this effect on the result is relatively small.

Fig. 3

View large Download slide

The experientially determined HTC of cooling water in the annulus for various velocities of condensing steam

Moreover, another theoretical model for calculating the condensation HTC in shear dominated flows described in Ref. [14] was used for comparison. The results of the condensation HTC were in agreement with the used model with a deviation of up to 7%.

For the comparison of theoretical methods, the error range for each method was determined as

error = \frac{(calculated value - experimental value)}{experimental value} \cdot 100 %

(34)

Uncertainty Analysis.

The uncertainty analysis of experimental results was performed according to the procedure described in Refs. [25] and [26]. The maximum expanded uncertainty based on a 95% confidence level is 3.8–8.6% for the HTC in the annulus according to the experimental conditions. Higher uncertainty is achieved at lower values of the cooling water Reynolds number and condenser heat performance, and vice versa.

Results and Discussion

The measurement was carried out at various cooling water flow rates. The pressure and temperature of steam were constant for all measurements, the steam flow rate was varied in a series of measurements. The HTC in the annulus for the Reynolds number of cooling water in a range from 700 to 2000, which corresponds with a laminar flow of cooling water, was analyzed.

Experimental Results.

The experimental results of the HTC determination for DPHE consisting of two coaxial tubes with a small gap are shown in Table 1. The condensation HTC is evaluated according to the model described above.

Table 1

Summary of experimental conditions

Cooling water inlet temperature (°C)

13.4–15.2

Cooling water outlet temperature (°C)

58.9–93.6

Cooling water mean Re

723–1894

Steam inlet velocity (m/s)

16.5–9.4

Steam outlet velocity (m/s)

9.0–1.3

Heat performance of DPHE (kW)

5.5–11.4

Overall HTC (W/m² K)

1369–1493

Condensation HTC (W/m² K)

6140–7993

It can be seen from Fig. 3 that the used model sufficiently takes into account changes in the velocity of the condensing steam while maintaining the conditions on the side of the HTC annulus, since its value is similar for different steam flow rates. It can be stated that the methodology used to determine condensation HTC has been verified with acceptable accuracy.

The sensitivity analysis of the accuracy of the determination of the condensation HTC to the resulting value of the HTC in the annulus is shown in Fig. 4. An increase in condensation HTC of +50% was considered as the maximum value, while the minimum value corresponded to the use of the standard Nusselt model of steam condensation without flow on the vertical wall. It can be seen that even a larger deviation in the determination of the condensation HTC has a relatively small effect on the resulting values of the HTC in the annulus.

Fig. 4

View large Download slide

Sensitivity of the determination of the HTC in the annulus to the value of the condensation HTC

The thermal resistance of the cooling water side is significantly higher than the value of the thermal resistance of the condensation side. Therefore, the error in the HTC determination is given mainly by the accuracy of the measurement of cooling water temperatures and its flow rate.

The comparison of the theoretically calculated value of the HTC of the cooling side and the experimentally determined value is shown in Fig. 5. The error range of theoretical values compared to experimental values are −38.2% to 30.2%, −44.0% to −28.3%, and −50.4% to −44.2% for the Stephan, VDI, and Baehr methods, respectively. The values calculated by the VDI and Baehr methods have a similar trend and reach equally lower values compared to experimental values.

Fig. 5

View large Download slide

Comparison of the theoretical and experimental HTC of cooling water in the annulus in its dependence on the Reynolds number of cooling water

In the range of lower Reynolds numbers, the difference between the theoretical and experimental values increases. Heat transfer in the DPHE can also be influenced by the narrow width of the annulus. The proposed condenser was designed with an annulus width of 1.4 mm. An annulus with such a narrow width approaches the scale of microchannels where heat transfer increases significantly and can reach values up to 790 W/cm² [27,28]. A review of the issues related to heat transfer in microchannels was made by Kakac and co-workers [27]. It is generally considered that a microchannel is any channel with a hydraulic diameter within the range of a micrometer, i.e., less than 1 mm. Therefore, it can be assumed that the flow of cooling water in the annulus of the proposed condenser is relatively close to this phenomenon, and it can therefore influence the HTC value of the cooling side of the condenser.

The difference between the theoretical and experimental results of the cooling water HTC is shown in Fig. 6, where the ratio to their dependence on the cooling water Reynolds number is analyzed. Experimental values are 1.8–2.0 times higher than those obtained by calculation according to the Baehr method. The values obtained by calculation according to the Stephan and VDI methods are closer to the experimental results. For the Stephan method, experimental values are 1.45–1.6 times higher. For the VDI method, there is a larger difference in the area of the lower Reynolds numbers (up to 75%) and, on the contrary, a lower difference as the increasing value of the Reynolds number goes down to 40%.

Fig. 6

View large Download slide

Ratio of the experimental and theoretical values of the HTC in the annulus

The dependencies of the overall HTC on the cooling water Reynolds number are shown in Fig. 7. Experimental values are higher compared to theoretical values in the range of 22–59%.

Fig. 7

View large Download slide

Comparison of the theoretical and experimental overall HTCs

The difference in the heat exchanger performance for experimental results in comparison with the theoretical results is shown in Fig. 8. The differences are more pronounced in the area range of the lower Reynolds numbers, where experimental values are up to 35%, 60%, and 45% higher than theoretical values for the Stephan method, the Baehr method, and the VDI method, respectively. It can be seen that the results of the Stephan and VDI methods are closer to the experimental results, but it can be stated that the results of both methods lie in the safety aspect for the design of heat exchangers.

Fig. 8

View large Download slide

The differences between theoretical and experimental heat exchanger performances

Temperature Distribution on the Outer Surface of the Heat Exchanger.

Longitudinal and circumferential temperature distribution at the outer wall of the condenser was measured during experiments with surface thermocouples. From the measured values, it was observed that the distribution of temperatures along the outer tube of the condenser significantly varied. This means that an irregular flow of cooling water appeared in the annulus of the condenser, and the heat flux from the inner tube did not transfer equally around its circumference to the cooling water. The measured temperature distribution was verified with FLIX i7 thermal-camera (Tedelyne FLIR LLC, Wilsonville, OR). The measurement conditions were maintained, but the thermal insulation was removed. The surface of the tubes was coated with a matt paint to eliminate reflectivity. The photos taken by the camera with the temperature profile of the outer tube of condenser are shown in Figs. 9(a)–9(c). In those pictures, several of the points where the temperature was measured are marked 100 mm under the exit of cooling water from the condenser. It is clear that the temperature of the wall differs around the circumference. The temperatures measured by the thermo-camera may not correspond directly (without insulation, heat loss increases), but the relative difference between the measured temperatures on the surface of the tube is investigated here. The maximum temperature difference around the tube circumference reached approximately 16 °C and varied according to the place of measurement. The approximate schematic distribution of temperatures is shown in Fig. 9(d).

Fig. 9

View large Download slide

The temperature profile of the outer surface of the heat exchanger for measurement number 2: (a)front view, (b) right view, (c) rear view, and (d) temperature distribution

There are several possible reasons why the temperature profile varied around the tube circuit:

The eccentricity of the annulus occurred due to the inaccurate manufacturing of the condenser, or temperature dilatation and deflection of the inner tube.
The possible influence of pins, which should keep the annulus concentric, on the cooling water flow.
The uneven distribution of cooling water flow from its inlet.

It is difficult to assess the influence of the final two reasons; however, it can be assumed that it is less significant. Theoretical analysis regarding the influence of the eccentricity of annulus or the inner tube dilatation on the flow and temperature profile of cooling water is described below.

The Effect of Tube Misalignment.

Nikitin et al. [8] made a statistical analysis of turbulent flow and heat transfer in an eccentric annulus. In their work, they mention that the Reynolds number can be several times smaller in the narrow gap of annulus than in a wider gap. This means that inner tube shift and annulus eccentricity can influence the velocity profile of a flowing fluid, the value of local HTC, and the temperature distribution of cooling water over the tube circumference. Puranik et al. [29] concerned the mathematical modeling of eccentricity; the conclusion is that with increasing eccentricity, the velocity of the flow increases, but its temperature decreases.

In practice, there is the possibility that due to manufacturing inaccuracy or temperature dilatation, the center of the inner tube shifts radially off the center of the outer tube, creating varying gaps between both tubes. The annulus of the condenser can be divided into two parts, and it may be assumed that the inner tube is radially shifting into the center of one gap of the annulus (Fig. 10). This causes two main effects. First, the area of the inner tube extending into the wider gap of the annulus decreases as its deflection increases, thereby reducing the overall heat flux transferred from the inner tube to the cooling water in the wider gap of the annulus. Second, the theoretical mass flows of cooling water in the two gaps of the divided annulus vary.

Fig. 10

View large Download slide

A diagram of the division of the annulus cross flow area into areas A and B

Both aspects have the same influence on the temperature of the cooling water flowing in each part of the annulus. A larger inner tube displacement off the annulus center increases the heat transfer to a narrower gap and reduces the cooling water flow here. This causes the heating of the cooling water in the narrow gap of the annulus to be higher than in the wider one. The ratio of the wide and narrow area of annulus $A_{cf A, B}$ depends on the inner tube radial shift and is shown in Fig. 11.

Fig. 11

View large Download slide

Effect of tube displacement on the distribution of the annulus areas

A theoretical analysis was conducted in order to evaluate the actual influence of the radial shift of the inner tube on the temperature distribution of the cooling water flowing in the annulus. This analysis was made with certain assumptions which simplify the calculation:

The exchanger is divided by a longitudinal section passing through the axis of the outer tube into two parts with a wide and a narrow gap according to Fig. 10, each part is balanced separately.
The heat flow from the inner tube is equally distributed over the inner tube circuit.
The defined parameters are considered as mean values.
The mass flow of cooling water is divided according to calculated velocities into both gaps.
There is no secondary flow, nor mixing of cooling water, between the wide and narrow gap.

As discussed above, the annulus is divided into two gaps where the mean velocity, and therefore the mean Reynolds number, are both calculated for the cooling water flowing in each gap. The mean velocity of cooling water can be calculated by the assumption that the friction losses of fluid are uniform in both gaps (below indexed A and B). Therefore, from Eq. for friction losses

p_{z}

⁠, the dependence of velocities on the hydraulic diameter of the gap can be derived

p_{z} = λ_{A} \cdot \frac{L}{d_{h A}} \cdot ρ_{w} \cdot \frac{{u_{A}}^{2}}{2} = λ_{B} \cdot \frac{L}{d_{h B}} \cdot ρ_{w} \cdot \frac{{u_{B}}^{2}}{2}

(35)

It can be seen that the variety in velocities is dependent on the friction factor and hydraulic diameter

u_{A, B} = f (λ_{A, B}; d_{h A, B})

(36)

For laminar flow, the friction factor can be expressed as [30]

λ = \frac{64}{Re}

(37)

The hydraulic diameter

d_{h}

is a commonly used term when handling flow in noncircular tubes and channels. By using this term, one can calculate many things in the same way as for a round tube. When the cross section is uniform along the tube or channel length, it is defined as

d_{h} = \frac{4 \cdot A}{P}

(38)

where A is the cross-sectional area of the flow, and P is the wetted perimeter of the cross section (see Fig. 10). These parameters must be determined individually for each case of displacement. By substituting the Reynolds number according to Eq. and subtracting the term from Eq.

\frac{64}{\frac{d_{h A} \cdot u_{A}}{ν_{w}}} \cdot \frac{L}{d_{h A}} \cdot ρ_{w} \cdot \frac{u_{A}^{2}}{2} = \frac{64}{\frac{d_{h B} \cdot u_{B}}{ν_{w}}} \cdot \frac{L}{d_{h B}} \cdot ρ_{w} \cdot \frac{u_{B}^{2}}{2}

(39)

assuming constant values of density and viscosity and shortening the equation, the dependence between the velocities and hydraulic diameters is in the form

\frac{u_{A}}{{d_{h A}}^{2}} = \frac{u_{B}}{{d_{h B}}^{2}}

(40)

The total mass flow of cooling water flow

M_{w}

is divided into the mass flows

M_{w A}

and

M_{w B}

in the gaps according to the equations

{M_{w} = M}_{w A} + M_{w B}

(41)

M_{w} = u_{A} \cdot A_{cf A} \cdot ρ_{w} + u_{B} \cdot A_{cf B} \cdot ρ_{w}

(42)

The flow velocity in area B is expressed by modifying Eq.

u_{B} = \frac{u_{A}}{{d_{h A}}^{2}} \cdot {d_{h B}}^{2}

(43)

The flow velocity in area A is then obtained by substituting Eq. into Eq. and making the adjustment

u_{A} = \frac{M_{w}}{A_{cf A} \cdot ρ_{w} + \frac{{d_{h A}}^{2}}{{d_{h B}}^{2}} \cdot A_{cf B} \cdot ρ_{w}}

(44)

The flow rates

M_{w A}

and

M_{w B}

could be given from the corresponding calculated velocities

M_{w A, B} = u_{A, B} \cdot A_{cf A, B} \cdot ρ_{w}

(45)

The temperature of the heated water in the gaps

T_{w 2 A}

and

T_{w 2 B}

is derived by iterative calculation from the equation for the heat performance

Q_{A, B} = M_{w A, B} \cdot c_{w} \cdot (T_{w 2 A, B} - T_{w, in}) = U_{A, B} \cdot A \cdot \frac{(T_{v} - T_{w, in}) - (T_{v} - T_{w 2 A, B})}{\ln \frac{(T_{v} - T_{w, in})}{(T_{v} - T_{w 2 A, B})}}

(46)

The overall HTC U_A,B is obtained by the values deducted from the experimental data for the corresponding Reynolds number. Temperature difference between the wide and narrow gap

Δ T_{A B} = T_{w 2 B} - T_{w 2 A}

(47)

The calculated mean cooling water outlet temperatures $T_{w 2}$ from the wide and narrow gap were compared, and their mean difference is shown in Fig. 12 as a function of inner tube radial shift for five measurements with Reynolds numbers from 1894 to 723. The upper line with triangles corresponds to the conditions in the photos from Fig. 9 for the measurement with Re = 1894 and cooling water heating from 13.4 °C to 59.8 °C. The mean temperature difference of the cooling water in the examined gaps depends on the radial shift of the inner tube. As the radial shift of the inner tube increases, the temperature difference between the cooling water in narrow and wide gaps also increases. The magnitude of the temperature difference for each radial shift value depends mostly on the Re number of the cooling water. The lower cooling water Re number corresponds to a lower temperature difference of the cooling water for each radial shift.

Fig. 12

View large Download slide

The influence of the radial shift of the inner tube on the temperature difference of the cooling water flowing in narrow and wide gaps for various Reynolds numbers

From Fig. 12, it can be concluded that the maximum temperature difference of 16 °C for measurement with Re = 1894 corresponds to a circa 0.2 mm radial shift of the inner tube.

Nevertheless, the actual shift of the inner tube is less than 0.2 mm because the mean temperature difference is less than the maximal one shown in Fig. 9 and corresponds to a lower radial shift of the inner tube. These results are in agreement with the conclusions of Refs. [8] and [9], that the mean temperature gradient in the narrow gap is significantly larger than it is in the wide gap.

Conclusion

Heat transfer in DPHE for laminar flow of cooling water in the annulus was experimentally analyzed. A series of experiments was carried out on a double pipe condenser with a tube spacing of 1.4 mm. The HTC in the annulus for the Reynolds number of the cooling water in a range from 700 to 2000 was analyzed. The suitability of using three theoretical methods for the thermal design of DPHE was evaluated. The values calculated by the VDI, Stephan, and Baehr methods have a similar trend and reach equally lower values compared to experimental values. The experimentally determined value of the HTC is about 75% for the Reynolds number of 700 and 40% for the Reynolds number of 2000 higher than theoretically determined according to the VDI method, respectively, about 60% and 45% for the Stephan method, and about 100% and 80% for the Baehr method. An increase in the HTC value can be caused by the formation of microturbulence in the annulus with a reduced width of 1.4 mm, similar to the microchannels.

During the experiments, the wall temperature difference along the circumference of the outer tube was ascertained, which was further explained by the eccentricity of the double pipes due to manufacturing inaccuracy or temperature dilatation. The eccentricity of the tubes results in uneven heat transfer and cooling water flow in the annulus and their differing heat. A more detailed analysis and the proposed procedure of quantification of this effect indicated that, for a measured temperature difference of 16 °C in radial distribution of cooling water temperature, the pipe shift would be at a maximum of 0.2 mm. The heat performance in this case is higher than about 20% of that calculated according to the VDI method, about 30% of that calculated according to the Stephan method, and about 50% of that calculated according to the Baehr method.

In practice, the designed concentric tubes may not always be perfectly concentric. Providing a general correlation is an experimentally very demanding task. In general, however, it can be stated that when using a DPHE with a small tube spacing and small media flows in the annulus, even a small manufacturing inaccuracy has a large effect on the change in the performance of the heat exchanger. The use of available theoretical methods is of limited use. Therefore, when designing such a heat exchanger, it is always necessary to begin with control experiments. As a result, the heat exchanger performance may increase, and it would be possible to produce a smaller exchanger with sufficient heat performance and thus to save the material and the build-up space.

In applications where uniform temperature distribution is required, the tube misalignment effect can be eliminated by using spacers (spikes) which are placed along the length of the inner tube to fix its position relative to the outer tube.

Funding Data

Ministry of Education, Youth and Sports under OP RDE (Grant No. CZ.02.1.01/0.0/0.0/16_019/0000753 “Research centre for low-carbon energy technologies”; Funder ID: 10.13039/501100001823).

Data Availability Statement

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

Nomenclature

a =: diameter ratio
A =: heat transfer area, m²
c =: specific heat capacity, J/kg K
d =: inner tube diameter, m
D =: characteristic diameter, m
D =: outer tube diameter, m
d_h =: hydraulic diameter, m
f_g =: geometrical correction factor
h =: heat transfer coefficient, W/m² K
$h_{fg}^{'}$ =: latent heat of condensation, kJ/kg
k =: thermal conductivity, W/m K
L =: tube length, m
M =: mass flow rate, kg/s
Nu =: Nusselt number
Pe =: Péclet number
Q =: heat transfer rate, W
T =: temperature, K
ΔT_log =: logarithmic mean temperature difference, K
u =: velocity, m/s
U =: overall heat transfer coefficient, W/m² K
λ =: friction factor
μ =: dynamic viscosity, Pa·s
ν =: kinematic viscosity, m²/s
τ =: shear stress, Pa

ann =: annulus
c =: condensation
cf =: cross flow
i =: inner
in =: inlet
L =: liquid
o =: outer
out =: outlet
v =: water vapor
w =: water

References

1.

Omidi

,

M.

,

Farhadi

,

M.

, and

Jafari

,

M.

,

2017

, “

A Comprehensive Review on Double Pipe Heat Exchangers

,”

Appl. Therm. Eng.

,

110

, pp.

1075

–

1090

.10.1016/j.applthermaleng.2016.09.027

Google Scholar

Crossref

2.

Zhang

,

Y.

,

Hangi

,

M.

,

Wang

,

X.

, and

Rahbari

,

A.

,

2023

, “

A Comparative Evaluation of Double-Pipe Heat Exchangers With Enhanced Mixing

,”

Appl. Therm. Eng.

,

230

, p.

120793

.10.1016/j.applthermaleng.2023.120793

Google Scholar

Crossref

3.

Ghani

,

S.

,

Gamaledin

,

S. M. A.

,

Rashwan

,

M. M.

, and

Atieh

,

M. A.

,

2018

, “

Experimental Investigation of Double-Pipe Heat Exchangers in Air Conditioning Applications

,”

Energy Build.

,

158

, pp.

801

–

811

.10.1016/j.enbuild.2017.10.051

Google Scholar

Crossref

4.

Gorman

,

J. M.

,

Krautbauer

,

K. R.

, and

Sparrow

,

E. M.

,

2016

, “

Thermal and Fluid Flow First-Principles Numerical Design of an Enhanced Double Pipe Heat Exchanger

,”

Appl. Therm. Eng.

,

107

, pp.

194

–

206

.10.1016/j.applthermaleng.2016.06.134

Google Scholar

Crossref

5.

Söylemez

,

M. S.

,

2004

, “

Thermoeconomical Optimization of Double-Pipe Heat Exchanger for Waste Heat Recovery

,”

Thermophys. Heat Transfer

,

18

(

4

), pp.

559

–

563

.10.2514/1.6149

Google Scholar

Crossref

6.

El-Shaarawi

,

M. A. I.

,

Abualhamayel

,

H. I.

, and

Mokheimer

,

E. M. A.

,

1997

, “Developing Laminar Flow in Eccentric Annuli,”

ASME. J. Fluids Eng.

, 119(3), pp.

724

–

728

.10.1115/1.2819307

7.

Usui

,

H.

, and

Tsuruta

,

K.

,

1980

, “

Analysis of Fully Developed Turbulent Flow in an Eccentric Annulus

,”

Chem. Eng. Jpn.

,

16

, pp.

445

–

450

.10.1252/jcej.13.445

8.

Nikitin

,

N.

,

Wang

,

H.

, and

Chernyshenko

,

S.

,

2009

, “

Turbulent Flow and Heat Transfer in Eccentric Annulus

,”

J. Fluid Mech.

,

638

, pp.

95

–

116

.10.1017/S002211200900812X

Google Scholar

Crossref

9.

Jonsson

,

V. K.

, and

Sparrow

,

E. M.

,

1966

, “

Experiments on Turbulent-Flow Phenomena in Eccentric Annular Ducts

,”

J. Fluid Mech.

,

25

(

1

), pp.

65

–

86

.10.1017/S0022112066000053

Google Scholar

Crossref

10.

Dezfoli

,

A. R.

, and

Ansari

,

M. A.

,

2009

, “

The Overall Heat Transfer Characteristics of a Double-Pipe Heat Exchanger

,”

Heat Transfer Res.

,

40

(

6

), pp.

555

–

570

.10.1615/HeatTransRes.v40.i6.40

Google Scholar

Crossref

11.

Hosseini

,

R.

,

Ramezani

,

M.

, and

Mazaheri

,

M. R.

,

2009

, “

Experimental Study of Turbulent Forced Convection in Vertical Eccentric Annulus

,”

Energy Convers. Manage.

,

50

(

9

), pp.

2266

–

2274

.10.1016/j.enconman.2009.05.002

Google Scholar

Crossref

12.

Hosseini

,

R.

,

Heyrani-Nobari

,

M. R.

, and

Hatam

,

M.

,

2005

, “

An Experimental Study of Heat Transfer in an Open-Ended Vertical Eccentric Annulus With Insulated and Constant Heat Flux Boundaries

,”

Appl. Therm. Eng.

,

25

(

8–9

), pp.

1247

–

1257

.10.1016/j.applthermaleng.2004.08.016

13.

Dirker

,

J.

, and

Meyer

,

J. P.

,

2005

, “

Convective Heat Transfer Coefficients in Concentric Annuli

,”

Heat Transfer Eng.

,

26

(

2

), pp.

38

–

44

.10.1080/01457630590897097

Google Scholar

Crossref

14.

VDI-GVC, 2010

, VDI Heat Atlas, Springer, Berlin, Germany.

15.

Hewitt

,

G. F.

,

Shires

,

G. L.

, and

Bott

,

T. R.

,

1994

, Process Heat Transfer, Begell House, New York.

16.

Cengel

,

Y. A.

,

2003

, Heat Transfer, McGraw-Hill, New York.

17.

Baehr

,

H. D.

, and

Stephan

,

K.

,

2011

, Heat and Mass-Transfer, Springer, Berlin, Germany.

18.

Kakaç

,

S.

,

Liu

,

H.

, and

Pramuanjaroenkij

,

A.

,

2002

, Heat Exchangers: Selection, Rating, and Thermal Design, 2nd ed., CRC Press, Boca Raton, FL, pp.

84

–

88

.

19.

Kracík

,

P.

,

Toman

,

F.

, and

Pospíšil

,

J.

,

2020

, “

Effect of the Flow Velocity of Gas on Liquid Film Flow in a Vertical Tube

,”

Chem. Eng. Trans.

,

81

, pp.

811

–

816

.10.3303/CET2081136

20.

Blangetti

,

F.

,

Krebs

,

R.

, and

Schlunder

,

E. U.

,

1982

, “

Condensation in Vertical Tubes—Experimental Results and Modelling

,”

Chem. Eng. Sci.

,

1

, pp.

20

–

63

.

21.

Kageyama

,

T.

,

Peterson

,

P. F.

, and

Schrock

,

V. E.

,

1993

, “

Diffusion Layer Modeling for Condensation in Vertical With Noncondensable Gases

,”

Nucl. Eng. Des.

,

141

(

1–2

), pp.

289

–

302

.10.1016/0029-5493(93)90107-K

22.

Toman

,

F.

,

Kracík

,

P.

, and

Pospíšil

,

J.

,

2022

, “

Heat Transfer During Condensation of Water Vapor in the Presence of Non-Condensable Gas in Vertical Tube of Small Diameter

,”

Case Stud. Therm. Eng.

,

40

, p.

102519

.10.1016/j.csite.2022.102519

Google Scholar

Crossref

23.

Vocale

,

P.

,

Bozzoli

,

F.

,

Mocerino

,

A.

,

Navickaitė

,

K.

, and

Rainieri

,

S.

,

2020

, “Application of an Improved Parameter Estimation Approach to Characterize Enhanced Heat Exchangers,”

Int. J. Heat Mass Transfer

, 147, p.

118886

.10.1016/j.ijheatmasstransfer.2019.118886

24.

Rainieri

,

S.

,

Bozzoli

,

F.

,

Cattani

,

L.

, and

Vocale

,

P.

,

2014

, “Parameter Estimation Applied to the Heat Transfer Characterization of Scraped Surface Heat Exchangers for Food Applications,”

J. Food Eng.

, 125, pp.

147

–

156

.10.1016/j.jfoodeng.2013.10.031

25.

Farrance

,

I.

, and

Frenkel

,

R.

,

2012

, “

Uncertainty of Measurement: A Review of the Rules for Calculating Uncertainty Components Through Functional Relationships

,”

Clin. Biochem. Rev.

,

33

, pp.

49

–

75

.https://pmc.ncbi.nlm.nih.gov/articles/PMC3387884/

Google Scholar

PubMed

26.

Joint Committee for Guides in Metrology (JCGM)

,

2008

, “

Evaluation of Measurement Data—Guide to the Expression of Uncertainty in Measurement

,” Bureau International des Poids et Mesures, Sèvres, France, accessed date Nov. 8, 2024, p.

134

, https://www.bipm.org/documents/20126/2071204/JCGM_100_2008_E.pdf/cb0ef43f-baa5-11cf-3f85-4dcd86f77bd6

27.

Sun

,

W.

,

Kakac

,

S.

, and

Yazicioglu

,

A. G.

,

2007

, “

A Numerical Study of Single-Phase Convective Heat Transfer in Microtubes for Slip Flow

,”

Int. J. Therm. Sci.

,

46

(

11

), pp.

1084

–

1094

.10.1016/j.ijthermalsci.2007.01.020

Google Scholar

Crossref

28.

Palm

,

B.

,

2001

, “

Heat Transfer in Microchannels

,”

Microscale Thermophys. Eng.

,

5

(

3

), pp.

155

–

175

.10.1080/108939501753222850

Google Scholar

Crossref

29.

Puranik

,

S. M.

,

Indira

,

R.

, and

Sreegowrav

,

K. R.

,

2021

, “

Flow and Heat Transfer in Eccentric Annulus

,”

Eng. Math.

,

127

(

1

), p.

21

.10.1007/s10665-021-10103-9

Google Scholar

Crossref

30.

Jones

,

O. C.

, Jr., and

Leung

,

J. C. M.

,

1981

, “

An Improvement in the Calculation of Turbulent Friction in Smooth Concentric Annuli

,”

ASME J. Fluids Eng.

,

103

(

4

), pp.

615

–

623

.10.1115/1.3241781

Google Scholar

Crossref

Heat Transfer in Double Pipe Heat Exchangers With Small Tube Spacing

Abstract

Introduction

Material and Methods

Heat Transfer in Double Pipe Heat Exchangers.

Calculation by VDI Heat Atlas (VDI Method).

Calculation by the Baehr and Stephan (Baehr Method).

Calculation by Stephan (Stephan Method).

The Experimental Setup

Experiments' Evaluation Procedure.

Uncertainty Analysis.

Results and Discussion

Experimental Results.

Temperature Distribution on the Outer Surface of the Heat Exchanger.

The Effect of Tube Misalignment.

Conclusion

Funding Data

Data Availability Statement

Nomenclature

References

Get Email Alerts

Cited By

ASME Journals

ASME Conference Proceedings

ASME eBooks

Resources

Opportunities

Heat Transfer in Double Pipe Heat Exchangers With Small Tube Spacing

Abstract

Introduction

Material and Methods

Heat Transfer in Double Pipe Heat Exchangers.

Calculation by VDI Heat Atlas (VDI Method).

Calculation by the Baehr and Stephan (Baehr Method).

Calculation by Stephan (Stephan Method).

The Experimental Setup

Experiments' Evaluation Procedure.

Uncertainty Analysis.

Results and Discussion

Experimental Results.

Temperature Distribution on the Outer Surface of the Heat Exchanger.

The Effect of Tube Misalignment.

Conclusion

Funding Data

Data Availability Statement

Nomenclature

References

Get Email Alerts

Cited By

Related Articles

Related Proceedings Papers

Related Chapters

ASME Journals

ASME Conference Proceedings

ASME eBooks

Resources

Opportunities

This Feature Is Available To Subscribers Only