# Friction and Heat Transfer in Membrane Distillation Channels: An Experimental Study on Conventional and Novel Spacers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Spacers

#### 2.2. Test Section

^{®}

_{,}Barrington, IL, USA) were characterized by a Red Start at 25 °C and a Colour Play of 5 °C (blue start at 30 °C).

_{c}= 4 mm, while the thickness of the hot channel is defined by the spacer thickness, and thus, it is H

_{h}= 10 mm in the presence of the overlapped or woven spacers and H

_{h}= 11 mm in the presence of the spheres spacer.

^{®}chiller (Riley Surface World, Walsall, UK) for the coarse regulation of the cold fluid temperature, a Julabo

^{®}(Seelbach, Germany) cooling thermostatic bath for the fine adjustment of the same temperature, and a second Julabo

^{®}thermostatic bath for the regulation of the hot fluid temperature. The instrumentation includes a Fuji Electric

^{®}FCX-AII differential pressure transmitter (Tokyo, Japan), six Pt100 thermoresistance probes and three Krohne Optiflux

^{®}(Duisburg, Germany) magnetic flow meters (one for the hot channel and one for each of the two cold channels, respectively).

#### 2.3. Experimental Technique

^{®}slab which constitutes the external wall of the test section. The images were then digitally processed as described in detail in [17] to obtain, based on a previous in situ calibration, the 2-D distribution of the TLC temperature, T

_{TLC}. Only a central portion of the TLC sheet including an integer number of unit cells of the spacer lattice was used for the analysis. At each point, the bulk temperatures T

_{h}and T

_{c}of the hot and cold channels were obtained as functions of the coordinate x along the flow direction by interpolating between the readings of the Pt100 probes that were located near the inlets and outlets of each channel.

_{h}to T

_{TLC}to be equal to that from T

_{TLC}to T

_{c}and using the elementary formulae for thermal resistances in a series, the following expression was obtained for h

_{h}:

_{hw}:

_{h}, q″ and T

_{hw}from the 2-D distribution of T

_{TLC}over a plane that was embedded in a solid (PC-TLC “sandwich”) is a steady-state inverse heat conduction problem, and it can be given more accurate, but also much more complex, solutions by the use of various inverse heat conduction methods, one of which is specifically adapted to the present problem (third type thermal boundary conditions on one wall), and it is based on the 3-D solutions of the corresponding direct conduction problems, and it has recently been presented by one of the present authors [18].

_{h}can be expressed in a dimensionless form as a local Nusselt number

_{eq}is the hydraulic diameter of the channel, which is conventionally identified with 2H

_{h}as in a spanwise indefinite channel, and λ

_{h}is the hot fluid thermal conductivity, which is a weak function of its bulk temperature T

_{h}.

_{h}〉 of the local Nusselt number since this latter quantity may become singular at the points where the denominator in Equation (1) vanishes.

_{c}, necessary to compute h

_{h}from Equation (1); for the assessment of entry effects and for a sensitivity analysis including the estimate of the measurement uncertainty, we direct the reader to a previous paper [17] in which these issues are thoroughly discussed.

## 3. Results and Discussion

_{eq}= 2H

_{h}as in Equation (4), $U$ is the mean velocity of the fluid and ρ and µ are its density and dynamic viscosity, respectively.

#### 3.1. Friction Coefficient

#### 3.2. Influence of Buoyancy

_{hw}) and ~8.24 (uniform q″) [20].

_{h}, the hot wall temperature T

_{hw}and the hot fluid bulk temperature T

_{h}have been defined earlier. The Rayleigh number was computed on the basis of the channel half-height, H

_{h}/2.

_{h}≈ 38 °C and T

_{hw}≈ 35.5 °C, while H

_{h}= 1.1·10

^{−2}m, g = 9.81 m/s

^{2}and, for water at ~38 °C, β = 3.21·10

^{−3}K

^{−1}, ν = 6.65·10

^{−7}m

^{2}/s and α = 1.56·10

^{−7}m

^{2}/s, thereby yielding Ra = 1.26·10

^{5}. Therefore, the Richardson number, which measures the importance of the natural convection with respect to the forced convection, is (for example) ~3.16 for Re = 200, ~0.35 for Re = 600 and ~0.039 for Re = 1800. These figures explain why the influence of natural convection (thermal buoyancy) is negligible at the high Reynolds numbers, important at the intermediate Re (e.g., 600) and dominating at a very low Re.

#### 3.3. Mean Nusselt Number in Spacer-Filled Channels

#### 3.4. Performance Comparison

_{Δp}= f·Re

^{2}(pressure number, which is proportional to the inlet-to-outlet pressure drop) and N

_{W}= f·Re

^{3}(power number, which is proportional to the required pumping power) and reporting the mean Nusselt number as a function of either parameter [21]. If the Nu(N

_{Δp}) curve for spacer A lies above that for spacer B, then spacer A will provide a better performance for any pressure drop, while, if the Nu(N

_{W}) curve for spacer A lies above that for spacer B, then spacer A will provide a better performance for any pumping power consumption.

#### 3.5. Local Heat Transfer Coefficient Distributions

_{h}around the wall-filament contact lines, while the maxima of Nu

_{h}are attained in the central region of the sides, lying above the opposite filaments (adjacent to the 90° wall), where the flow passage area is restricted and the local velocity is at its highest. The mean Nusselt number, which was computed from Equation (5), is ~16, (see Figure 8b).

_{h}around the wall-filament contact lines, which of course are now orthogonal to those that are represented for the opposite wall in graph (a). The maxima of Nu

_{h}are attained at about 1/3

^{rd}of the pitch P downstream of each wall-adjacent filament in correspondence with the reattachment of the flow separating it from the upstream wall-adjacent filament. The mean Nusselt number Nu, which was computed from Equation (5), is ~19, which is only slightly larger than that on the 0° wall, as shown also in Figure 8b, while the absolute maximum of the local Nusselt number Nu

_{h}is much larger than it is on the 0° wall.

_{h}. As indicated by the preliminary CFD simulations, the absolute and relative maxima of Nu

_{h}correspond to the regions in which the complex flow that is induced by the spacer impacts against the wall that is under consideration. The mean Nusselt number Nu, which was computed from Equation (5), is ~23, as shown also in Figure 8a, while the absolute maximum of the local Nusselt number Nu

_{h}is ~60; both of these values are the highest among all of the configurations in this figure.

_{h}on one wall of a channel that is filled with the spheres spacer (S 0–90°). Note that the pitch P is now 44 mm, which is slightly larger than it is for the previous geometry. In this case, the Nu

_{h}minima at the four corners of the unit cell, which are associated with the point-like contacts between the spacer spheres and the wall, are surrounded by horseshoe structures that are characterized by high Nu

_{h}values upstream of each sphere, which are associated with the impact of the flow against this latter. Other eye-catching structures are the low Nu

_{h}wakes that result from the downstream side of each sphere. Apart from the mentioned features, the Nu

_{h}distribution over the remaining part of the wall appears to be less structured and exhibits intermediate values. The mean Nusselt number Nu, which was computed from Equation (5), is ~15, see Figure 8a, while the absolute maximum of the local Nusselt number Nu

_{h}is ~25; therefore, the spheres geometry is that which exhibits the flattest distribution of the local heat transfer coefficient.

_{h}on the opposite walls of the hot channel are essentially identical, when we are disregarding the reflections, so it is sufficient to show a single wall. The map shows the minima of Nu

_{h}around the diagonal wall-filament contact lines, while the Nu

_{h}maxima are attained in two separate regions that are located between the filaments and presumably corresponding to the flow reattachment regions as in the O 90° case in Figure 10b. The mean Nusselt number Nu, which was computed from Equation (5), is ~14, see Figure 7, while the Nu

_{h}maxima are much higher (>36).

_{h}, are now located at the centers of the four sides of the unit cell. As in the 0–90° case (Figure 10c), an isolated and intense Nu

_{h}maximum can be observed, thus corresponding as indicated by preliminary CFD simulations, to a region in which the spacer-induced flow impinges against the wall that is under consideration. The mean Nusselt number Nu, which was obtained by calculating Equation (5), is ~26, see Figure 7, while the absolute maximum of Nu

_{h}is >64; both of these values are the highest among all of the configurations in this figure.

_{h}on one wall of a channel that is filled with the spheres spacer (S 45°). As in the 0–90° orientation (Figure 10d), the Nu

_{h}minima at the four point-like sphere-wall contacts are surrounded by horseshoe structures that are characterized by high Nu

_{h}values that are upstream of each sphere. However, unlike in the 0–90° case, the wake region of each sphere exhibits a complex structure with a minimum value immediately downstream that is followed by a maximum of the same amplitude as that in the horseshoe region, which is presumably caused by the confluence of the vortex streets issuing from the 45° filaments that connect the spheres. The mean Nusselt number Nu, which was calculated using Equation (5), is ~17, see Figure 7, while the absolute maximum of Nu

_{h}is >34.

## 4. Conclusions

^{2}, and for any pumping power, as expressed by the power number f∙Re

^{3}, the novel spheres spacer provided the highest mean Nusselt number over the whole Reynolds number range that was investigated.

^{2}of ~0.1), but it was much smaller and limited to the range Re < ~500 (Ri < ~0.5) in spacer-filled channels.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic representation of the three geometries that were tested with main nomenclature: H = channel thickness, D = sphere diameter, P = pitch, d = diameter of the connecting rods. (

**a**) overlapped; (

**b**) woven; (

**c**) spheres.

**Figure 2.**Photographs of the actual spacers tested, with their filaments oriented at 0–90° (

**top**row) or at 45° (

**bottom**row) with respect to the flow direction (arrow). (

**a**,

**d**) overlapped; (

**b**,

**e**) woven; (

**c**,

**f**) spheres. The pitch-to-height ratio P/H = 4 in all of the cases.

**Figure 3.**Exploded view of the test section, showing an overlapped spacer. (a) Transparent Plexiglas

^{®}slab; (b) rubber gasket; (c) upper PC/TLC sandwich; (d) gasket and spacer; (e) lower PC/TLC sandwich.

**Figure 4.**Schematic of heat transfer across the PC-TLC sandwich located on one of the two sides of the test section. T

_{h}= bulk temperature of the hot fluid, T

_{c}= bulk temperature of the cold fluid, T

_{cw}= cold wall temperature, T

_{hw}= hot wall temperature, T

_{TLC}= TLC temperature, h

_{h}= convective heat transfer coefficient on the hot side, h

_{c}= convective heat transfer coefficient on the cold side, q

^{″}= heat flux, s

_{PC}= thickness of the PC sheet, s

_{TLC}= thickness of the TLC foil, λ

_{PC}= thermal conductivity of the PC sheet, λ

_{TLC}= thermal conductivity of the TLC foil.

**Figure 5.**Darcy friction coefficient f as a function of the Reynolds number for all of the configurations investigated. The theoretical friction coefficient for laminar flow in a plane channel (96/Re) is also reported for comparison purposes.

**Figure 6.**(

**a**) Experimental mean Nusselt number as a function of the Reynolds number on the top and bottom walls of the test section in the case of no spacer, showing the influence of buoyancy. The shaded area represents the expected range of Nu in the absence of buoyancy (7.54–8.34). (

**b**) Schematic representation of the temperature distribution in the hot channel, with unstable stratification in its upper half and stable stratification in its lower half.

**Figure 7.**Experimental mean Nusselt number on the top and bottom walls as a function of the Reynolds number for a flow attack angle of 45° and the three spacer geometries investigated.

**Figure 8.**Experimental mean Nusselt number as a function of the Reynolds number on the top and bottom walls for a flow attack angle of 0° or 90°. (

**a**) Woven and spheres geometries; (

**b**) overlapped geometry.

**Figure 9.**Experimental mean Nusselt number (top-bottom averaged) as a function of the pressure number f·Re

^{2}(

**a**) or the power number f·Re

^{3}(

**b**) for all the spacer configurations investigated.

**Figure 10.**Distributions of the local Nusselt number on a wall for different spacer configurations at a Reynolds number of ~500. (

**a**) Overlapped spacer, flow attack angle ϕ = 0°; (

**b**) overlapped spacer, ϕ = 90°; (

**c**) woven spacer, ϕ = 0–90°; (

**d**) spheres spacer, ϕ = 0–90°. Note that for the woven and spheres geometries, there is no difference between the 0° and 90° attack angles.

**Figure 11.**Distributions of the local Nusselt number on a wall for different spacer configurations at a Reynolds number of ~500 and a flow attack angle of 45°. (

**a**) Overlapped spacer; (

**b**) woven spacer; (

**c**) spheres spacer.

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**MDPI and ACS Style**

Cancilla, N.; Tamburini, A.; Tarantino, A.; Visconti, S.; Ciofalo, M.
Friction and Heat Transfer in Membrane Distillation Channels: An Experimental Study on Conventional and Novel Spacers. *Membranes* **2022**, *12*, 1029.
https://doi.org/10.3390/membranes12111029

**AMA Style**

Cancilla N, Tamburini A, Tarantino A, Visconti S, Ciofalo M.
Friction and Heat Transfer in Membrane Distillation Channels: An Experimental Study on Conventional and Novel Spacers. *Membranes*. 2022; 12(11):1029.
https://doi.org/10.3390/membranes12111029

**Chicago/Turabian Style**

Cancilla, Nunzio, Alessandro Tamburini, Antonino Tarantino, Salvatore Visconti, and Michele Ciofalo.
2022. "Friction and Heat Transfer in Membrane Distillation Channels: An Experimental Study on Conventional and Novel Spacers" *Membranes* 12, no. 11: 1029.
https://doi.org/10.3390/membranes12111029