- Email: [email protected]
Impact of module design on heat transfer in membrane distillation
Journal Pre-proof Impact of module design on heat transfer in membrane distillation Alexander V. Dudchenko, Mukta Hardikar, Ruikun Xin, Shounak Joshi, Ruoyu Wang, Nikita Sharma, Meagan S. Mauter PII:
S0376-7388(19)33531-8
DOI:
https://doi.org/10.1016/j.memsci.2020.117898
Reference:
MEMSCI 117898
To appear in:
Journal of Membrane Science
Received Date: 19 November 2019 Revised Date:
21 January 2020
Accepted Date: 27 January 2020
Please cite this article as: A.V. Dudchenko, M. Hardikar, R. Xin, S. Joshi, R. Wang, N. Sharma, M.S. Mauter, Impact of module design on heat transfer in membrane distillation, Journal of Membrane Science (2020), doi: https://doi.org/10.1016/j.memsci.2020.117898. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
Alexander V. Dudchenko: Conceptualization, Methodology, Software, Formal analysis, Investigation, Writing - Original Draft, Visualization. Mukta Hardikar: Investigation, Data Curation. Ruikun Xin: Investigation, Data Curation. Shounak Joshi : Investigation, Data Curation Ruoyu Wang: Investigation, Data Curation. Nikita Sharma: Investigation. Meagan S. Mauter: Conceptualization, Supervision, Writing - Review & Editing, Funding acquisition.
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Impact of Module Design on Heat Transfer in Membrane Distillation Alexander V. Dudchenko1, Mukta Hardikar2, Ruikun Xin3, Shounak Joshi3, Ruoyu Wang3, Nikita Sharma3, and Meagan S. Mauter1,4* 1
Department of Civil & Environmental Engineering, Stanford University, Stanford, CA 94305. Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, 3 Department of Civil & Environmental Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, 4 Stanford Woods Institute for the Environmental Engineering, Stanford University, Stanford, CA 94305. 2
*Author to whom correspondence should be addressed. Email: [email protected] Keywords: heat transfer; membrane distillation; module design; Nusselt number; Sherwood number
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ABSTRACT Nusselt correlations originally developed for estimating heat transfer rates in heat exchangers poorly describe heat transfer in membrane distillation (MD) processes. In this work, we assess the impact of module design in bench-scale experiments, simplified treatment of heat transfer rates in MD models, and the effect of permeate flux on temperature polarization as sources of error in Nusselt correlation estimates of heat transfer rates. To test these effects, we systematically vary membrane structure, module sizes, temperatures, and Reynolds numbers to generate a large dataset (n=240) of MD experiments. We apply this dataset to estimate the heat transfer rate for each unique membrane/module combination and compare our predictions to the classical Sieder-Tate Nusselt correlation (Nus-t). Our results show that heat transfer rates in small modules can be up to five times higher than predicted by Nus-t. The heat transfer rate decreases with increasing module size, with heat transfer in large modules adequately described by the Sieder-Tate correlation. We demonstrate that this high heat transfer rate in small modules is a result of an entrance effect, which increases fluid mixing over the membrane area. These results validate the use of Nu correlation in large membrane modules while highlighting issues with their application in small scale systems. This work also emphasizes the importance of benchscale module design in materials evaluation and process characterization. Finally, it highlights the need for direct measurement techniques that better characterize interfacial processes in membrane systems with modeled driving forces. 1. INTRODUCTION Membrane distillation (MD) is a widely studied process for high salinity brine concentration [1]. The challenges in scaling this process for commercialization, however, is partially due to the shallow vapor pressure driving force between the warm feed and the cool permeate of traditional direct-contact MD systems. In direct-contact MD a hot flowing feed solution is separated from a cool flowing draw solution by a hydrophobic membrane, the fluid flow convectively heats and cools the membrane surface. The resulting temperature difference between membrane surfaces creates a vapor pressure driving force that promotes water evaporation from the feed and its transport across the membrane into the draw solution. Accurately describing the vapor pressure driving force across the membrane as a function of bulk feed and permeate temperatures, flow rates, and module designs will be critical to next-generation module designs that enable economically viable MD. Unfortunately, current models for relating bulk and interfacial temperatures in MD modules poorly describe heat transfer in many bench scale systems. First, conventional Nusselt (Nu) number relationships were derived for large scale systems with either calm or abrupt entrances (Figure 1A) [2,3]. These correlations do not reflect the conditions in most bench-scale experimental MD systems, where right angle entrances and exits impart a centripetal force on the fluid that increases mixing and the fluid velocity near the membrane surface (Figure 1B) [4]. In small modules, we hypothesize that these entrance and exit effects control heat transfer conditions across most of the flow channel length. As the module increases in length, the heat transfer should increasingly be controlled by overall flow channel design and not the entrance and exit effects (Figure 1C).
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We are unaware of experimental studies assessing the accuracy of heat transfer prediction of Nu correlations as a function of MD module length. Though several methods have been developed to estimate the unique heat transfer rate under a fixed set of operating conditions in a specific MD module (i.e., Schofield method, Wilson plot method), these methods are intended to estimate intrinsic membrane parameters (i.e., permeability) rather than validate Nu correlations [5–8]. Methods better adapted for estimating the accuracy of Nu correlations (e.g., Gryta et al., Phattarnawik et al., and Leitch et al.) have only been applied to one or two modules under substantially different operational conditions [9–11]. The absence of systematically collected data across a range of module sizes, operating conditions, and membrane permeabilities has made it difficult to establish the relationship between MD flow channel dimensions and heat transfer rates or the relative importance of right-angle entrance/exit effects on the accuracy of conventional Nu correlations. Second, conventional Nu correlations provide an estimate of the average heat transfer rate across flow channel length, with an implicit assumption that the average value captures the nonlinearity in heat transfer rate and temperature profile [12]. A recent computational fluid dynamic simulation study published by Lou et al. demonstrates that the heat transfer rate, temperature, and flux across the MD module length are highly non-linear and disagree with the average heat transfer rate predictions made by the Nu correlations [13]. However, we are unaware of experimental work determining the significance of errors induced by averaging the heat transfer rate over the length of the module relative to errors induced by the entrance or exit effects. Establishing the origins of inaccuracy in conventional Nu correlations will provide insight into their application for approximating heat transfer in large MD modules where the entrance effects are negligible (Figure 1C). Finally, Nu correlations were developed for systems in which there is no coupled heat and mass transfer. In MD systems, vapor transfer across the membrane inextricably couples heat and mass transfer, especially in high permeability membranes. Phattaranawik et al. derived a function that corrects the heat transfer coefficient for the compression and expansion of the thermal boundary layer due to mass transfer through the membrane (Figure 1B) [14]. While their work suggests negligible effects of mass transfer on the thickness of the thermal boundary layer, we are unaware of any experimental work validating this assumption. Establishing that the heat transfer rate does not change with intrinsic membrane permeability would provide evidence toward the prediction that mass transfer minimally impacts heat transfer in MD. In this work, we estimate the error in the commonly used laminar Seider-Tate Nu (Nus-t) correlation that was developed for laminar flow in small heat-exchangers as a function of module dimension and membrane permeability [2]. We also assess whether correcting for mass transfer and the non-linearity of the heat transfer profile reduces the errors in the average heat transfer prediction. We build an extensive MD data set (n=240) that characterizes the performance of three different MD membranes in seven MD modules under four different feed temperatures and Reynold numbers. Next, we re-formulate the Nu correction factor method developed by Leitch et al. to quantify the error in Nus-t correlations for specific module and membrane combinations. Finally, we discretize the Nus-t and MD model to account for the effect of non-linearity in heat transfer rate and surface temperature, and we apply the heat transfer correction function derived by Phattaranawik et al. to assess the effects of mass transfer on boundary layer compression.
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The results of this work provide a first systematic investigation of how Nu correlation accuracy is affected by (i) module dimensions, (ii) non-linearity in heat transfer across module length, and (iii) the coupling of mass and heat transfer.
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Figure 1: Thermal boundary formation and fluid flow through (A) classical heat exchangers modeled by Nusselt correlations, (B) bench-scale membrane distillation module with a right-angle entrance and exit, and (C) a large scale membrane distillation module.
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2. METHODS 2.1 Direct contact membrane distillation experiments 2.1.1 Module dimensions Direct contact membrane distillation experiments were performed in a fully instrumented system utilizing modules of systematically varied width and length (Figure 2, Supplemental Information 1). The modules share a common entrance/exit design and channel height of 2 mm. The first module, referred to as the 1x4 module, had a width of 1 cm and length of 4 cm. The second module, referred to as the 8x8 module, had channel width of 7.775 cm and channel length of 7.775 cm length. The third module, referred to as the 3x20 module, had channel width of 3 cm and channel length of 20 cm. Four additional modules were made by modifying the width of the 8x8 and the 3x20 modules with symmetrically positioned plastic inserts on either side of the
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channel. The channel width of 8x8 module was reduced to either 5 cm (referred to as the 5x8 module), 3 cm (referred to as the 3x8 module), or 1 cm (referred to 1x8), while the width of 3x20 module was reduced to 1 cm (referred to as the 1x20 module). The flow channel entrance and exit had a length of 5 mm and spanned the full width of the module.
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Figure 2: Dimensions of modules used in this work. Inset shows a typical module crosssection, showing dimensions of module entrance and flow channel height.
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2.1.2 Experimental conditions Experiments were performed in triplicate at four different feed temperatures and four different Reynolds numbers, resulting in 48 different experimental runs for each module. The feed temperatures was set to 40°C, 50°C, 60°C, and 70°C and the draw was maintained at 20°C. The feed solution contained 35 g/L of NaCl, while the draw solution was deionized water. Modules with a width of 1 cm were operated at Reynolds numbers of 500, 800, 1100, and 1400, while modules with a width of 3 cm, 5 cm, and 8 cm were operated at Reynolds numbers of 800, 900, 1000, and 1100. The Re values were evaluated using Equation 1, where u is mean fluid velocity in the flow channel, dh is fluid channel hydraulic diameter, µ is fluid kinematic viscosity, and ρ is fluid density estimated at bulk fluid temperature. (1)
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We selected an overlapping, but non-identical range of Reynolds numbers for the small and large modules. Higher Reynolds numbers for the 1 cm modules ensured accurate flow measurement with an analog flowmeter, while the lower Reynolds numbers used for 3 cm, 5 cm and 8 cm wide modules prevented membrane rupture. We maintained all Re values below 2,300 Re, where we assumed that the laminar flow regime changes to the transitional flow regime [15]. The transition point from laminar to transitional flow is a function of module geometry, and we did not validate that the 2,300 Re is the transition point in the used modules, as it is outside the scope of this work. 2.1.3 Measurement of entrance and exit effects We qualitatively study the effect of entrance and exit effects on permeate flux by isolating the permeable area of the membrane to a small region using Kapton tape and changing its position along the length of the 3x20 module. The permeable area is fixed at 2.55±0.1 cm in length and 3 cm in width. The position commences at either 0, 2.5 cm, 5.0 cm, 8.25 cm, or 17.5 cm down
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channel of the feed entrance. The side of the membrane with Kapton tape was always in contact with the hot feed. We never observed the tape delaminating from the membrane surface or water intrusion between the tape and the membrane. Kapton tape was about 25 microns thick and had thermal conductivity of ~0.4 W/m K [16]. 2.1.4 Membrane characteristics We used three different membranes in this work, a phase inversion polyvinylidene fluoride (PVDF) membrane, a phase inversion polypropylene (PP) membrane, and a fibrous polytetrafluoroethylene (PTFE) membrane (Table 1). The membranes were used as received. The pore size, porosity, and thickness of the PVDF and PTFE membranes are reported elsewhere [11]. The PP membrane thickness was measured using an optical microscope. We minimized the error that resulted from membrane compression during sample cutting by placing a membrane sample between two glass slides and measuring the spacing between the slides, which is equivalent to the membrane thickness. The membrane porosity (φ) was estimated from Equation 2, where polypropylene density (ρpp) of 0.86 kg/m3 was used. The membrane density (ρm) was calculated from the measured weight of a 9±0.1 cm wide and 12.5±0.1 cm long piece of membrane. The membrane sample was weighted on a precision scale with 0.1 mg accuracy, and the average membrane sample weight was 230 mg. 1− (2)
The poor structural integrity of the PP and PTFE membranes limited testing to modules of ≤ 3 cm in width and 1 cm widths, respectively. Table 1: Membrane structural parameters. Values measured in this work and prior publication [11]. Pore diameter for PP membrane provided by the manufacturer [17]. Membrane type GVHP PP PTFE
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Thickness (µm) 109±5 160±5 62±5
Porosity (%) 62±5 84±5 82±5
Pore diameter (nm) 244±387 200 2730±314
2.2 Permeability estimation via interfacial heat and mass balance model for MD 2.2.1 Average membrane distillation model We estimate permeability (βest) in MD using an interfacial energy and mass balance model that accurately captures changes in thermophysical properties of feed and draw stream. Briefly, we balance the energy transfer across the feed side thermal boundary layer, the feed side membrane/water interface, the draw side membrane/water interface, and the draw side thermal boundary layer. We use an iterative solver to find the intrinsic membrane permeability (βest) that minimizes the difference between the experimentally measured permeate flux and the permeate flux predicted by the model. We assume that bulk stream conditions are represented by the experimentally measured average bulk feed and draw temperatures and flowrates. We approximate the interfacial conditions by estimating heat transfer and mass transfer coefficients from common Seider-Tate and Sherwood correlation derived by analogy as shown in Equations 3 and 4, where Re is Reynolds number, Pr is Prandtl number, Sc is Schmidt number, dh is hydraulic diameter, L is module length, µm is viscosity of water at membrane surface, and µb is
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viscosity of bulk water [2,18]. Detailed equations and model descriptions are reported in Supplementary Information (SI) Section 2. !ℎ
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The Nus-t correlation was derived for operation below Re of 2,300 from experimentally measured heat transfer rates in a shell and tube heat exchanger [2]. The heat transfer rates were measured using five equally spaced thermocouples placed along the length of the tube side of the heat exchanger that was 157 long and had an inner diameter of 1.57 cm. The flow first flowed through a 120 cm long non-heated calming section before entering the heat exchanger tube. Due to the use of this calming section, the Nus-t correlation only accounts for a developing thermal boundary layer condition and assumes a fully developed hydrodynamic boundary layer. 2.2.2 Mass transfer correction Mass transfer through the membrane can expand or contract the thermal boundary layer. To account for this effect, we apply a correction term derived by Phattaranawik et al. to the average MD model [14]. The correction factor (φcm) (Equation 5) modifies the prediction made by Nusselt correlation (Equation 6), where Jw is the permeate flux, Cp is heat capacity of water, hnu is the heat transfer coefficient estimated by Nusselt correlation, and hcorrected is the final corrected heat transfer coefficient.
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2.2.3 Discretized membrane distillation model We discretized the MD model using a finite difference scheme to capture the effect of non-linear heat transfer along the length of the module. We estimate membrane surface temperatures on the feed and draw side, flux, and heat transfer rate at discrete points (nodes) along module length. To reduce the computational intensity, we assume that the bulk feed and draw temperature changes linearly across module length and interpolate the temperature for each node from experimentally measured inflow and outflow temperature. In addition, we use the average value of concentration polarization across the module length to approximate local concentration polarization conditions. Parallel modeling efforts suggest that the errors introduced by these two approximations are less than 5%. We account for non-linear heat transfer rates by discretizing the Nus-t correlation along the length of the module, where x is the position in the module (Equation 7). We ensure that the discretized Nusselt correlation predicts the same average heat transfer rate as the module Nus-t correlation by normalizing Nus-t,x by the average of Nus-t,x (Equation 8) over the length of the module (Equation 9). This ensures that we are correcting for the effect of non-linearity, rather than the change in average heat transfer rate. ,4
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2.3 Correction factor method The correction factor (CF) method is based on the hypothesis that trends in experimental data should follow those predicted by theory, and has been previously presented in detail be Leitch et al. [11]. Using this approach, it is possible to estimate the inaccuracy in a Nusselt correlation by comparing the derivative of permeability estimated from an MD model to the derivative of permeability estimated from the Dusty Gas Model with Field approximation (DGM-F). In this work, we have extended the CF method to use the derivative of the heat transfer ratio instead of derivatives of permeability. Theoretical permeability is weakly affected by the average membrane temperature and does not change with thermal polarization. In contrast, the theoretical heat transfer ratio is a function of average membrane temperature and temperature gradient across the membrane, which makes it a better approximator of thermal polarization in MD and heat transfer rates in MD modules. The heat transfer ratio in MD describes the ratio between latent heat transport and conductive heat transport (Equations 10 and 11), where βest is the permeability estimated by the average of the discretized MD model and βDGM-F is the permeability estimated from DGM-F theory as described by Field et al. [19] Here, Δpvap is the difference in vapor pressure across the membrane; Tm,f, Tm,d, and Tm,avg are membrane feed side surface temperature, membrane draw side surface temperature, and average membrane temperature, respectively; hvap is enthalpy of vaporization, hl is enthalpy of liquid water; km is membrane thermal conductivity; and δm is membrane thickness. ?0
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As in the initially proposed method, we take the derivative of estimated and DGM-F predicted heat transfer ratios with respect to the average membrane temperature and the average of feed and draw Reynolds number (Reavg) (Equations 12a, 12b and 13a, 13b). We find the derivative from the slope of a linear line fitted to the calculated heat transfer ratios. We estimated the 5th and 95th confidence interval in the derivatives by applying the standard Student’s t-distribution to the coefficients of the fitted line. ]?0
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We utilize a single parameter correction factor term to correct the Nus-t and Sh correlations as shown in Equations 14, 15, 16, and 17.
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306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325
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We find the CF factor using an iterative solver that minimizes the difference between the derivative of estimated heat transfer ratio and derivative of the theoretical heat transfer ratio (we utilize the differential_evolution solver built into Scipy where the CF value was bounded between 0.001 and 20) [20]. Even though all MD data in this work was acquired with high precision, there is still a substantial uncertainty that propagates into the derivatives and needs to be considered. We find that uncertainty is greatest for derivatives that are taken with respect to Reynolds number, which is a result of the low increases in permeate flux with increasing Reynolds number. To ensure that these uncertainties do not lead to a biased solution, we use the optimally weighted χ2 function to find the difference between the derivative of estimated heat transfer ratio and derivative of the theoretical heat transfer ratio. The χ2 function weighs the derivatives by their normalized uncertainty as shown by Equation 18-20, where ∆δθ is uncertainty in derivative [21]. The derivatives were found by performing a linear regression on the heat transfer ratios. The uncertainty in the slope of the linear regression was found with the standard t-student test and was used as ∆δθ. g e.f
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3. RESULTS AND DISCUSSION We evaluate the effect of membrane structural parameters, bulk operating parameters, and module dimension on measured permeate flux, and use the acquired data to understand how these effects impact heat transfer rate qualitatively. We then demonstrate how the application of Nus-t correlation can lead to inaccurate estimation of intrinsic membrane permeability. Finally, we apply the extended CF method to estimate the error in the Nus-t correlation and determine the overall contribution of entrance effects, mass transfer through the membrane, and non-linear effects to the error in Nus-t correlation. 3.1 Effect of membrane structural parameters on permeate flux For a single module at constant operation conditions, higher membrane porosity and larger pore sizes are correlated with higher membrane permeability and greater flux (Figure 3). The high porosity of the PTFE and PP membranes reduces the water molecule diffusion pathlength, while the large pore size of the PTFE membrane changes the dominant vapor transport regime from Knudsen diffusion to Ordinary Molecular Diffusion. These phenomenon are well described by past theoretical and experimental efforts [19,22]. 3.2 Effect of bulk operating parameters on permeate flux
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Increasing bulk feed temperature and Reynolds number increase flux regardless of the module dimensions (Figure 3). The nearly exponential increase in permeate flux with increasing feed temperature reflects the exponential relationship between water vapor pressure and temperature as given by Antoine correlation [5]. Increasing Reynolds number has a much smaller effect on flux (Figure 3). While high Reynolds numbers increase the heat transfer rate, thus reducing thermal polarization and increasing the vapor pressure gradient across the membrane, the absolute magnitude of heat transfer is dictated by the temperature of the bulk. Again, this is consistent with prior work [10,11,23]. 3.3 Effect of module dimensions on permeate flux Whereas past work has explored the effects of the membrane and bulk operating parameters on permeate flux in MD, there has been little work quantifying the effects of module dimensions on MD performance. Empirically derived Nusselt correlations describe a thicker thermal boundary layer and a decrease in heat transfer rate with increasing flow channel length. We observe a 40% decrease in permeate flux as the module length increases from 4 to 20 cm (Figure 3 and Supplementary Figure S2). This decrease was independent of membrane structural parameters. The permeate flux also decreased with increasing membrane area, an effect not captured by commonly used Nusselt correlations. We measured nearly identical permeate flux for modules with the same area, but different width and length (e.g. 1x20 and 3x8 module pair, and 3x20 and 8x8 module pair). We hypothesize that this decrease in permeate flux with increasing module area is the result of the decreasing contribution of the entrance effect to the total heat transfer rate in the module. In all modules, the fluid enters the module at a right angle via a round entry port (Supplementary Figure S4). This fluid entry introduces a three-dimensional distribution of mixing or unsteady fluid flow over the membrane area and increases the heat transfer rate over the affected area of the membrane. As the module size increases, the overall effect of turbulence at the entrance diminishes. Furthermore, modules with wide flow channels (e.g. 8x8 module) can have low fluid crossflow at the edges of the flow channel and in the flow channel corners (i.e. dead zones), reducing heat transfer rates. Standard Nusselt correlations do not account for entrance effects on thermal boundary layer thickness or capture the effect of possible dead zones in wide modules, thus, do not accurately capture the increase in heat transfer rate with decreasing module width and length.
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Figure 3: Measured permeate flux for the three membranes and seven modules. The bottom of the colored bar shows the permeate flux at the lowest Reynolds number, and top of the colored bar shows the flux at the highest Reynolds number for each operating feed temperature. 3.4 Effect of position along length of module on permeate flux We isolated the effects of greater mixing near the entrance and the development of the thermal boundary layer along the length of the membrane module by measuring permeate flux at five different locations along the length of the membrane. In each experiment, we isolated a small active region of the membrane by blocking the remainder of the membrane with Kapton tape (Figure 4A). Local permeate flux at the module entrance was nearly 50±5% higher than that of the average flux for the 3x20 module at all operating Reynolds numbers and feed temperatures. As we shifted the active region away from the module entrance, the permeate flux declined linearly in the first half of the module. However, the permeate flux measured at the module exit was nearly identical to the permeate flux measured at the module center. The use of Kapton tape complicates the analysis of our experimental results, as shown by the higher local permeate flux than the average permeate flux measured without Kapton tape (Figure 4A). Two mechanisms cause this higher permeate flux. First, the fluid flow stepping down from the Kapton tape onto the membrane surface increases fluid mixing and the local heat transfer rate. Second, the Kapton tape reduces the heat transfer rate upstream of the open membrane area, resulting in a reduced thermal boundary layer thickness and a higher driving force relative to operation without Kapton tape upstream. The effect of complex thermal boundary layer formation is readily captured with the simulation of experimental conditions using the discretized MD model described in Supplementary Sections S3 and S4. The change in experimentally measured permeate flux along the module position agrees well with the simulations that used a Nu correlation with an entrance effect (Figure 4B). We use the discretized MD model to isolate the effect of high heat transfer rate at module entrance and exit on measured permeate flux by simulating how the permeate flux would change along module length under a constant Nu (e.g. no entrance or exit effect), a continually decaying Nu (e.g. an entrance effect only), and a parabolic change in Nu (e.g. entrance and exit effect) (Figure 4C, Supplementary Section S3). In the first half of the module, the change in experimental permeate
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flux normalized to average permeate flux closely matched the simulation that utilized a decaying Nu. This indicates that a strong entrance effect was responsible for observed increased in permeate flux. In contrast, the linear slope of the decay of experimental permeate flux normalized to average permeate flux in the second half of the module closely matches that of a constant Nu number, suggesting that the exit effect is negligible (Figure 4B). This result contradicts part of our original hypothesis stating that both abrupt entrances and exits impact the heat transfer rate (Figure 1B). Instead, these results suggest that only the entrance is responsible for increased heat transfer rates in MD modules.
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Figure 4: Measured and simulated permeate flux at different positions in the 3x20 module. (A) The permeate flux measured at different positions in 3x20 modules. The active membrane area was reduced to 2.5±0.1 cm long and 3 cm wide with Kapton tape (shown in gold) covering inactive membrane region, the arrow with J shows where water can permeate. (B) Local permeate flux normalized to permeate flux measured without Kapton tape that was measured experimentally and simulated using discretized MD model. The simulation was performed using
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three different Nusselt Correlation. The simulation and experimental data are for conditions of 1100 Re, feed inflow temperature of 70°C and draw inflow temperature of 20°C. (C) The three local Nusselt correlations used during simulations.
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3.5 Effect of module size on permeability estimates of MD membranes Errors in heat transfer rate can lead to significant errors in the calculated permeability of MD membranes. The intrinsic permeability of an MD membrane can be described by the sum of ordinary molecular and Knudson diffusion, which depended on membrane pore size, porosity, tortuosity, and (weakly) average membrane temperature [19]. However, when applying the average MD model and Nus-t correlation, the intrinsic permeability increases as the module size decreases and feed temperature increases (Figure 5). Furthermore, the prediction of permeability in small modules (1x4, 1x8, and 1x20) exceeds the theoretical maximum of water vapor diffusion in air. The observed increase in permeability with decreasing module size is a result of increasing underestimation of heat transfer rate.
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Figure 5: Comparison of theoretical and experimentally estimated membrane permeability using the Seider-Tate correlation to describe heat transfer rates in the membrane modules. (A) GVHP, (B) PP, and (C) PTFE membranes. The top of the colored bar shows estimated permeability for the highest Reynolds number, and the bottom of the colored bar shows permeability for the lowest Reynolds number. Theoretical permeability is estimated from DGMF theory.
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3.6 Estimated permeability using corrected Nus-t correlation The application of the CF method reduces the variability in predicted membrane permeability values between modules (Figure 6A) [11]. None of the estimated permeabilities exceed the theoretical maximum water vapor diffusion rate in air and there is only a very weak correlation between permeability and feed temperature. The estimated permeability was near the DGM-F theory prediction but did not precisely match it since the CF method matches the derivative of theoretical and experimental heat transfer ratio, rather than correcting to absolute theoretical DGM-F theory values.
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3.7 Quantifying inaccuracy in the Nus-t correlation
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The degree of correction required for Sider-Tate correlation decreased with increasing module dimensions and did not appreciably change with membrane type, the addition of mass transfer correction term, or discretization of the MD model (Figure 6B). The CF values decreased with an increase in module dimensions. The small 1x4 module required a CF value of 4-5, which implies that the heat transfer rate is almost four-to-five times higher than predicted by Nus-t correlation. As modules increased in size, the CF value decreased to nearly one and eliminated the need to correct the Nus-t correlation. The observed decrease in CF value with increasing module size is consistent with the reduced contribution of the entrance effect to the total heat transfer rate in the module. Furthermore, the CF factor does not correlate well with the dh/L parameter, which is used by Nusselt correlation to scale the heat transfer rate with flow channel geometry and entrance effects (Supplementary Figure S5). This weak correlation between CF and dh/L further supports our hypothesis that the Nus-t correlation does not accurately capture the entrance effects present in MD modules.
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Our estimated CF factor was weakly affected by membrane permeability and mass transfer rate (Figure 6B). Vapor transport through the membrane results in the flow of water toward the membrane surface on the feed side and away from the membrane surface on the draw side. This fluid flow can potentially compress and expand the thermal boundary layer, increasing and decreasing the heat transfer rate, respectively. Although the CF factor was, in general, higher for low flux GVHP and PP membranes relative to the high flux PTFE membrane, the variability in CF factor prediction between the modules makes it impossible to identify a statistically significant effect.
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Inclusion of the mass transfer correction term derived by Phattaranawik et al. did not affect the magnitude of the CF factor [14]. While our experiments were not designed to isolate the effects of fluid flow to/away from the membrane surface on heat transfer rates, these results suggest that the effect is likely to be limited. Future work to address this question should use carefully designed multi-physics computation fluid dynamics studies where the variability between experiments is likely to be smaller and uncertainty in parameter estimation can be quantified explicitly. We did not observe an appreciable change in the CF factor when using a discretized MD model (Figure 6B). The discretized MD model captures the effect of non-linear heat transfer across the module length as a result of enthalpy transport through the membrane and the resulting highly non-linear flux across module length. However, the CF values computed with the discretized MD model were nearly identical to those calculated with the average MD model. The negligible change in CF factor with model discretization demonstrates that the primary error in Nus-t correlation when applied to small bench-scale modules is the result of the underestimation of the average heat transfer rate and not a result of simplifications made in the average MD model.
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Figure 6: Estimated permeability after application of correction factor and correction factor values. (A) Estimated permeability for the three membranes with corrected heat transfer correlation. (B) Estimated correction factor for each module and membrane using average MD model, average MD model with mass transfer correction, and a discretized MD model with nonlinear heat transfer rate. A correction factor of one implies that Nus-t accurately describes heat transfer rate in the module.
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4. CONCLUSION Systematically designed and rigorously executed MD experiments on seven modules, three membrane types, four feed temperatures, and four Reynolds numbers provided the first comprehensive dataset for evaluating the accuracy of standard methods for predicting heat
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transfer rate. We demonstrated that the accuracy of commonly used Nus-t correlation increases with increasing module width and length. We attribute this to abrupt entrance effects inducing fluid mixing and increasing the heat transfer rate. These entrance effects are not accounted for in the commonly used Nus-t correlation and can result in estimated heat transfer rates that are 5X too low in small bench scale modules. The contribution of the entrance effect to the overall heat transfer rate in the module decreases increasing with module width and length, thus improving the accuracy of Nus-t correlation in large modules. This work also demonstrated the validity of the previously reported Nusselt correction factor method for modules of varying size. Early work developed the CF method for a single module, raising questions about the adequacy of this method across varying module dimensions. The consistency of the estimated permeability of the three membrane types for each of the modules, temperatures, and Reynolds numbers suggests that this is a robust method that should be widely adopted by the MD community. We were unable to measure an increase or decrease in heat transfer rate with a change in mass transfer through the membrane in bench-scale MD modules. Our estimate of Nus-t correlation error (i.e. the magnitude of the CF) did not change with an increase in membrane permeability. The addition of the correction term for compression and expansion of the thermal boundary layer due to water flow to and from the membrane surface also did not change the magnitude of the CF. Finally, we have shown that accounting for non-linearity in heat transfer rate and permeate flux across module length did not change the CF. This work presents the first systematic study of how MD heat transfer rates change as a function of module size and membrane permeability. Additional work is needed to evaluate the effect of mass transfer through the membrane on heat transfer in large modules, though this work would be best accomplished with multi-physics computational fluid dynamic studies. The entrance effect can degrade the accuracy of Sherwood correlations when applied to bench-scale membrane modules, meaning that their accuracy should be studied as a function of module size. Finally, the results of this work highlight that the design of the bench-scale module can have dramatic effects on the predicted performance of a membrane process and this effect must be carefully considered when translating the results from bench-scale systems to pilot and commercial-scale systems. ACKNOWLEDGEMENTS This work was financially supported by NSF CBET-1554117 award. REFERENCES [1] P. Wang, T.-S. Chung, Recent advances in membrane distillation processes: Membrane development, configuration design and application exploring, J. Membr. Sci. 474 (2015) 39–56. https://doi.org/10.1016/j.memsci.2014.09.016. [2] E.N. Sieder, G.E. Tate, Heat transfer and pressure drop of liquids in tubes, Ind. Eng. Chem. 28 (1936) 1429–1435. [3] S. Kakaç, R.K. Shah, A.E. Bergles, North Atlantic Treaty Organization, eds., Low Reynolds number flow heat exchangers: advanced study institute book, Hemisphere Pub. Corp, Washington, 1983.
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[4] D.E. Metzger, M.K. Sahm, Heat Transfer Around Sharp 180 Degree Turns in Smooth Rectangular Channels, in: Vol. 3 Heat Transf. Electr. Power, ASME, Houston, Texas, USA, 1985: p. V003T09A020. https://doi.org/10.1115/85-GT-122. [5] R.W. Schofield, A.G. Fane, C.J.D. Fell, Heat and mass transfer in membrane distillation, J. Membr. Sci. 33 (1987) 299–313. https://doi.org/10.1016/S0376-7388(00)80287-2. [6] L. Martínez-Díez, A method to evaluate coefficients affecting flux in membrane distillation, J. Membr. Sci. 173 (2000) 225–234. https://doi.org/10.1016/S0376-7388(00)00362-8. [7] L. Li, K.K. Sirkar, Influence of microporous membrane properties on the desalination performance in direct contact membrane distillation, J. Membr. Sci. 513 (2016) 280–293. https://doi.org/10.1016/j.memsci.2016.04.015. [8] J. Vanneste, J.A. Bush, K.L. Hickenbottom, C.A. Marks, D. Jassby, C.S. Turchi, T.Y. Cath, Novel thermal efficiency-based model for determination of thermal conductivity of membrane distillation membranes, J. Membr. Sci. 548 (2018) 298–308. https://doi.org/10.1016/j.memsci.2017.11.028. [9] M. Gryta, M. Tomaszewska, A.W. Morawski, Membrane distillation with laminar flow, Sep. Purif. Technol. 11 (1997) 93–101. https://doi.org/10.1016/S1383-5866(97)00002-6. [10] J. Phattaranawik, R. Jiraratananon, A.G. Fane, Heat transport and membrane distillation coefficients in direct contact membrane distillation, J. Membr. Sci. 212 (2003) 177–193. [11] M.E. Leitch, G.V. Lowry, M.S. Mauter, Characterizing convective heat transfer coefficients in membrane distillation cassettes, J. Membr. Sci. 538 (2017) 108–121. https://doi.org/10.1016/j.memsci.2017.05.028. [12] E.M. Sparrow, Analysis of laminar forced-convection heat transfer in entrance region of flat rectangular ducts, (1955). https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930084124.pdf (accessed October 1, 2017). [13] J. Lou, J. Vanneste, S.C. DeCaluwe, T.Y. Cath, N. Tilton, Computational fluid dynamics simulations of polarization phenomena in direct contact membrane distillation, J. Membr. Sci. 591 (2019) 117150. https://doi.org/10.1016/j.memsci.2019.05.074. [14] J. Phattaranawik, R. Jiraratananon, A.G. Fane, Effects of net-type spacers on heat and mass transfer in direct contact membrane distillation and comparison with ultrafiltration studies, J. Membr. Sci. 217 (2003) 193–206. https://doi.org/10.1016/S0376-7388(03)00130-3. [15] D. Bertsche, P. Knipper, T. Wetzel, Experimental investigation on heat transfer in laminar, transitional and turbulent circular pipe flow, Int. J. Heat Mass Transf. 95 (2016) 1008–1018. https://doi.org/10.1016/j.ijheatmasstransfer.2016.01.009. [16] D.J. Benford, T.J. Powers, S.H. Moseley, Thermal conductivity of Kapton tape, Cryogenics. 39 (1999) 93–95. https://doi.org/10.1016/S0011-2275(98)00125-8. [17] Polypropylene Membrane Filters, 0.2 Micron, Sterlitech. (n.d.). https://www.sterlitech.com/polypropylene-membrane-filter-0-2-micron-254-x-3000mm-1pk.html (accessed November 13, 2019). [18] M.C. Porter, Concentration polarization with membrane ultrafiltration, Ind. Eng. Chem. Prod. Res. Dev. 11 (1972) 234–248. [19] R.W. Field, H.Y. Wu, J.J. Wu, Multiscale Modeling of Membrane Distillation: Some Theoretical Considerations, Ind. Eng. Chem. Res. 52 (2013) 8822–8828. https://doi.org/10.1021/ie302363e. [20] T.E. Oliphant, Python for Scientific Computing, Comput. Sci. Eng. 9 (2007) 10–20. https://doi.org/10.1109/MCSE.2007.58.
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[21] B.F.W. Croke, The Role of Uncertainty in Design of Objective Functions, (n.d.) 8. [22] J. Phattaranawik, Effect of pore size distribution and air flux on mass transport in direct contact membrane distillation, J. Membr. Sci. 215 (2003) 75–85. https://doi.org/10.1016/S0376-7388(02)00603-8. [23] T.Y. Cath, V.D. Adams, A.E. Childress, Experimental study of desalination using direct contact membrane distillation: a new approach to flux enhancement, J. Membr. Sci. 228 (2004) 5–16. https://doi.org/10.1016/j.memsci.2003.09.006.
1. 2. 3. 4. 5.
Study relates module design and mass transfer rate to heat transfer rate Nusselt correlations underestimate heat transfer in small modules by up to 5x Entrance effects induce mixing and increase the heat transfer rate Mass transfer does not increase the heat transfer rate in bench-scale modules Nusselt correlations more accurately describe heat transfer in large modules
S0376-7388(19)33531-8
DOI:
https://doi.org/10.1016/j.memsci.2020.117898
Reference:
MEMSCI 117898
To appear in:
Journal of Membrane Science
Received Date: 19 November 2019 Revised Date:
21 January 2020
Accepted Date: 27 January 2020
Please cite this article as: A.V. Dudchenko, M. Hardikar, R. Xin, S. Joshi, R. Wang, N. Sharma, M.S. Mauter, Impact of module design on heat transfer in membrane distillation, Journal of Membrane Science (2020), doi: https://doi.org/10.1016/j.memsci.2020.117898. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
Alexander V. Dudchenko: Conceptualization, Methodology, Software, Formal analysis, Investigation, Writing - Original Draft, Visualization. Mukta Hardikar: Investigation, Data Curation. Ruikun Xin: Investigation, Data Curation. Shounak Joshi : Investigation, Data Curation Ruoyu Wang: Investigation, Data Curation. Nikita Sharma: Investigation. Meagan S. Mauter: Conceptualization, Supervision, Writing - Review & Editing, Funding acquisition.
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Impact of Module Design on Heat Transfer in Membrane Distillation Alexander V. Dudchenko1, Mukta Hardikar2, Ruikun Xin3, Shounak Joshi3, Ruoyu Wang3, Nikita Sharma3, and Meagan S. Mauter1,4* 1
Department of Civil & Environmental Engineering, Stanford University, Stanford, CA 94305. Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, 3 Department of Civil & Environmental Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, 4 Stanford Woods Institute for the Environmental Engineering, Stanford University, Stanford, CA 94305. 2
*Author to whom correspondence should be addressed. Email: [email protected] Keywords: heat transfer; membrane distillation; module design; Nusselt number; Sherwood number
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ABSTRACT Nusselt correlations originally developed for estimating heat transfer rates in heat exchangers poorly describe heat transfer in membrane distillation (MD) processes. In this work, we assess the impact of module design in bench-scale experiments, simplified treatment of heat transfer rates in MD models, and the effect of permeate flux on temperature polarization as sources of error in Nusselt correlation estimates of heat transfer rates. To test these effects, we systematically vary membrane structure, module sizes, temperatures, and Reynolds numbers to generate a large dataset (n=240) of MD experiments. We apply this dataset to estimate the heat transfer rate for each unique membrane/module combination and compare our predictions to the classical Sieder-Tate Nusselt correlation (Nus-t). Our results show that heat transfer rates in small modules can be up to five times higher than predicted by Nus-t. The heat transfer rate decreases with increasing module size, with heat transfer in large modules adequately described by the Sieder-Tate correlation. We demonstrate that this high heat transfer rate in small modules is a result of an entrance effect, which increases fluid mixing over the membrane area. These results validate the use of Nu correlation in large membrane modules while highlighting issues with their application in small scale systems. This work also emphasizes the importance of benchscale module design in materials evaluation and process characterization. Finally, it highlights the need for direct measurement techniques that better characterize interfacial processes in membrane systems with modeled driving forces. 1. INTRODUCTION Membrane distillation (MD) is a widely studied process for high salinity brine concentration [1]. The challenges in scaling this process for commercialization, however, is partially due to the shallow vapor pressure driving force between the warm feed and the cool permeate of traditional direct-contact MD systems. In direct-contact MD a hot flowing feed solution is separated from a cool flowing draw solution by a hydrophobic membrane, the fluid flow convectively heats and cools the membrane surface. The resulting temperature difference between membrane surfaces creates a vapor pressure driving force that promotes water evaporation from the feed and its transport across the membrane into the draw solution. Accurately describing the vapor pressure driving force across the membrane as a function of bulk feed and permeate temperatures, flow rates, and module designs will be critical to next-generation module designs that enable economically viable MD. Unfortunately, current models for relating bulk and interfacial temperatures in MD modules poorly describe heat transfer in many bench scale systems. First, conventional Nusselt (Nu) number relationships were derived for large scale systems with either calm or abrupt entrances (Figure 1A) [2,3]. These correlations do not reflect the conditions in most bench-scale experimental MD systems, where right angle entrances and exits impart a centripetal force on the fluid that increases mixing and the fluid velocity near the membrane surface (Figure 1B) [4]. In small modules, we hypothesize that these entrance and exit effects control heat transfer conditions across most of the flow channel length. As the module increases in length, the heat transfer should increasingly be controlled by overall flow channel design and not the entrance and exit effects (Figure 1C).
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We are unaware of experimental studies assessing the accuracy of heat transfer prediction of Nu correlations as a function of MD module length. Though several methods have been developed to estimate the unique heat transfer rate under a fixed set of operating conditions in a specific MD module (i.e., Schofield method, Wilson plot method), these methods are intended to estimate intrinsic membrane parameters (i.e., permeability) rather than validate Nu correlations [5–8]. Methods better adapted for estimating the accuracy of Nu correlations (e.g., Gryta et al., Phattarnawik et al., and Leitch et al.) have only been applied to one or two modules under substantially different operational conditions [9–11]. The absence of systematically collected data across a range of module sizes, operating conditions, and membrane permeabilities has made it difficult to establish the relationship between MD flow channel dimensions and heat transfer rates or the relative importance of right-angle entrance/exit effects on the accuracy of conventional Nu correlations. Second, conventional Nu correlations provide an estimate of the average heat transfer rate across flow channel length, with an implicit assumption that the average value captures the nonlinearity in heat transfer rate and temperature profile [12]. A recent computational fluid dynamic simulation study published by Lou et al. demonstrates that the heat transfer rate, temperature, and flux across the MD module length are highly non-linear and disagree with the average heat transfer rate predictions made by the Nu correlations [13]. However, we are unaware of experimental work determining the significance of errors induced by averaging the heat transfer rate over the length of the module relative to errors induced by the entrance or exit effects. Establishing the origins of inaccuracy in conventional Nu correlations will provide insight into their application for approximating heat transfer in large MD modules where the entrance effects are negligible (Figure 1C). Finally, Nu correlations were developed for systems in which there is no coupled heat and mass transfer. In MD systems, vapor transfer across the membrane inextricably couples heat and mass transfer, especially in high permeability membranes. Phattaranawik et al. derived a function that corrects the heat transfer coefficient for the compression and expansion of the thermal boundary layer due to mass transfer through the membrane (Figure 1B) [14]. While their work suggests negligible effects of mass transfer on the thickness of the thermal boundary layer, we are unaware of any experimental work validating this assumption. Establishing that the heat transfer rate does not change with intrinsic membrane permeability would provide evidence toward the prediction that mass transfer minimally impacts heat transfer in MD. In this work, we estimate the error in the commonly used laminar Seider-Tate Nu (Nus-t) correlation that was developed for laminar flow in small heat-exchangers as a function of module dimension and membrane permeability [2]. We also assess whether correcting for mass transfer and the non-linearity of the heat transfer profile reduces the errors in the average heat transfer prediction. We build an extensive MD data set (n=240) that characterizes the performance of three different MD membranes in seven MD modules under four different feed temperatures and Reynold numbers. Next, we re-formulate the Nu correction factor method developed by Leitch et al. to quantify the error in Nus-t correlations for specific module and membrane combinations. Finally, we discretize the Nus-t and MD model to account for the effect of non-linearity in heat transfer rate and surface temperature, and we apply the heat transfer correction function derived by Phattaranawik et al. to assess the effects of mass transfer on boundary layer compression.
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The results of this work provide a first systematic investigation of how Nu correlation accuracy is affected by (i) module dimensions, (ii) non-linearity in heat transfer across module length, and (iii) the coupling of mass and heat transfer.
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Figure 1: Thermal boundary formation and fluid flow through (A) classical heat exchangers modeled by Nusselt correlations, (B) bench-scale membrane distillation module with a right-angle entrance and exit, and (C) a large scale membrane distillation module.
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2. METHODS 2.1 Direct contact membrane distillation experiments 2.1.1 Module dimensions Direct contact membrane distillation experiments were performed in a fully instrumented system utilizing modules of systematically varied width and length (Figure 2, Supplemental Information 1). The modules share a common entrance/exit design and channel height of 2 mm. The first module, referred to as the 1x4 module, had a width of 1 cm and length of 4 cm. The second module, referred to as the 8x8 module, had channel width of 7.775 cm and channel length of 7.775 cm length. The third module, referred to as the 3x20 module, had channel width of 3 cm and channel length of 20 cm. Four additional modules were made by modifying the width of the 8x8 and the 3x20 modules with symmetrically positioned plastic inserts on either side of the
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channel. The channel width of 8x8 module was reduced to either 5 cm (referred to as the 5x8 module), 3 cm (referred to as the 3x8 module), or 1 cm (referred to 1x8), while the width of 3x20 module was reduced to 1 cm (referred to as the 1x20 module). The flow channel entrance and exit had a length of 5 mm and spanned the full width of the module.
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Figure 2: Dimensions of modules used in this work. Inset shows a typical module crosssection, showing dimensions of module entrance and flow channel height.
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2.1.2 Experimental conditions Experiments were performed in triplicate at four different feed temperatures and four different Reynolds numbers, resulting in 48 different experimental runs for each module. The feed temperatures was set to 40°C, 50°C, 60°C, and 70°C and the draw was maintained at 20°C. The feed solution contained 35 g/L of NaCl, while the draw solution was deionized water. Modules with a width of 1 cm were operated at Reynolds numbers of 500, 800, 1100, and 1400, while modules with a width of 3 cm, 5 cm, and 8 cm were operated at Reynolds numbers of 800, 900, 1000, and 1100. The Re values were evaluated using Equation 1, where u is mean fluid velocity in the flow channel, dh is fluid channel hydraulic diameter, µ is fluid kinematic viscosity, and ρ is fluid density estimated at bulk fluid temperature. (1)
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We selected an overlapping, but non-identical range of Reynolds numbers for the small and large modules. Higher Reynolds numbers for the 1 cm modules ensured accurate flow measurement with an analog flowmeter, while the lower Reynolds numbers used for 3 cm, 5 cm and 8 cm wide modules prevented membrane rupture. We maintained all Re values below 2,300 Re, where we assumed that the laminar flow regime changes to the transitional flow regime [15]. The transition point from laminar to transitional flow is a function of module geometry, and we did not validate that the 2,300 Re is the transition point in the used modules, as it is outside the scope of this work. 2.1.3 Measurement of entrance and exit effects We qualitatively study the effect of entrance and exit effects on permeate flux by isolating the permeable area of the membrane to a small region using Kapton tape and changing its position along the length of the 3x20 module. The permeable area is fixed at 2.55±0.1 cm in length and 3 cm in width. The position commences at either 0, 2.5 cm, 5.0 cm, 8.25 cm, or 17.5 cm down
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channel of the feed entrance. The side of the membrane with Kapton tape was always in contact with the hot feed. We never observed the tape delaminating from the membrane surface or water intrusion between the tape and the membrane. Kapton tape was about 25 microns thick and had thermal conductivity of ~0.4 W/m K [16]. 2.1.4 Membrane characteristics We used three different membranes in this work, a phase inversion polyvinylidene fluoride (PVDF) membrane, a phase inversion polypropylene (PP) membrane, and a fibrous polytetrafluoroethylene (PTFE) membrane (Table 1). The membranes were used as received. The pore size, porosity, and thickness of the PVDF and PTFE membranes are reported elsewhere [11]. The PP membrane thickness was measured using an optical microscope. We minimized the error that resulted from membrane compression during sample cutting by placing a membrane sample between two glass slides and measuring the spacing between the slides, which is equivalent to the membrane thickness. The membrane porosity (φ) was estimated from Equation 2, where polypropylene density (ρpp) of 0.86 kg/m3 was used. The membrane density (ρm) was calculated from the measured weight of a 9±0.1 cm wide and 12.5±0.1 cm long piece of membrane. The membrane sample was weighted on a precision scale with 0.1 mg accuracy, and the average membrane sample weight was 230 mg. 1− (2)
The poor structural integrity of the PP and PTFE membranes limited testing to modules of ≤ 3 cm in width and 1 cm widths, respectively. Table 1: Membrane structural parameters. Values measured in this work and prior publication [11]. Pore diameter for PP membrane provided by the manufacturer [17]. Membrane type GVHP PP PTFE
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Thickness (µm) 109±5 160±5 62±5
Porosity (%) 62±5 84±5 82±5
Pore diameter (nm) 244±387 200 2730±314
2.2 Permeability estimation via interfacial heat and mass balance model for MD 2.2.1 Average membrane distillation model We estimate permeability (βest) in MD using an interfacial energy and mass balance model that accurately captures changes in thermophysical properties of feed and draw stream. Briefly, we balance the energy transfer across the feed side thermal boundary layer, the feed side membrane/water interface, the draw side membrane/water interface, and the draw side thermal boundary layer. We use an iterative solver to find the intrinsic membrane permeability (βest) that minimizes the difference between the experimentally measured permeate flux and the permeate flux predicted by the model. We assume that bulk stream conditions are represented by the experimentally measured average bulk feed and draw temperatures and flowrates. We approximate the interfacial conditions by estimating heat transfer and mass transfer coefficients from common Seider-Tate and Sherwood correlation derived by analogy as shown in Equations 3 and 4, where Re is Reynolds number, Pr is Prandtl number, Sc is Schmidt number, dh is hydraulic diameter, L is module length, µm is viscosity of water at membrane surface, and µb is
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viscosity of bulk water [2,18]. Detailed equations and model descriptions are reported in Supplementary Information (SI) Section 2. !ℎ
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2.2.3 Discretized membrane distillation model We discretized the MD model using a finite difference scheme to capture the effect of non-linear heat transfer along the length of the module. We estimate membrane surface temperatures on the feed and draw side, flux, and heat transfer rate at discrete points (nodes) along module length. To reduce the computational intensity, we assume that the bulk feed and draw temperature changes linearly across module length and interpolate the temperature for each node from experimentally measured inflow and outflow temperature. In addition, we use the average value of concentration polarization across the module length to approximate local concentration polarization conditions. Parallel modeling efforts suggest that the errors introduced by these two approximations are less than 5%. We account for non-linear heat transfer rates by discretizing the Nus-t correlation along the length of the module, where x is the position in the module (Equation 7). We ensure that the discretized Nusselt correlation predicts the same average heat transfer rate as the module Nus-t correlation by normalizing Nus-t,x by the average of Nus-t,x (Equation 8) over the length of the module (Equation 9). This ensures that we are correcting for the effect of non-linearity, rather than the change in average heat transfer rate. ,4
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2.3 Correction factor method The correction factor (CF) method is based on the hypothesis that trends in experimental data should follow those predicted by theory, and has been previously presented in detail be Leitch et al. [11]. Using this approach, it is possible to estimate the inaccuracy in a Nusselt correlation by comparing the derivative of permeability estimated from an MD model to the derivative of permeability estimated from the Dusty Gas Model with Field approximation (DGM-F). In this work, we have extended the CF method to use the derivative of the heat transfer ratio instead of derivatives of permeability. Theoretical permeability is weakly affected by the average membrane temperature and does not change with thermal polarization. In contrast, the theoretical heat transfer ratio is a function of average membrane temperature and temperature gradient across the membrane, which makes it a better approximator of thermal polarization in MD and heat transfer rates in MD modules. The heat transfer ratio in MD describes the ratio between latent heat transport and conductive heat transport (Equations 10 and 11), where βest is the permeability estimated by the average of the discretized MD model and βDGM-F is the permeability estimated from DGM-F theory as described by Field et al. [19] Here, Δpvap is the difference in vapor pressure across the membrane; Tm,f, Tm,d, and Tm,avg are membrane feed side surface temperature, membrane draw side surface temperature, and average membrane temperature, respectively; hvap is enthalpy of vaporization, hl is enthalpy of liquid water; km is membrane thermal conductivity; and δm is membrane thickness. ?0
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As in the initially proposed method, we take the derivative of estimated and DGM-F predicted heat transfer ratios with respect to the average membrane temperature and the average of feed and draw Reynolds number (Reavg) (Equations 12a, 12b and 13a, 13b). We find the derivative from the slope of a linear line fitted to the calculated heat transfer ratios. We estimated the 5th and 95th confidence interval in the derivatives by applying the standard Student’s t-distribution to the coefficients of the fitted line. ]?0
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(12 a, b) (13 a, b)
We utilize a single parameter correction factor term to correct the Nus-t and Sh correlations as shown in Equations 14, 15, 16, and 17.
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ab
c
!ℎdW
,dW
,4,dW
!ℎ ∗ ab
∗ ab ,4 ∗ ab
(14) (15) (16) (17)
We find the CF factor using an iterative solver that minimizes the difference between the derivative of estimated heat transfer ratio and derivative of the theoretical heat transfer ratio (we utilize the differential_evolution solver built into Scipy where the CF value was bounded between 0.001 and 20) [20]. Even though all MD data in this work was acquired with high precision, there is still a substantial uncertainty that propagates into the derivatives and needs to be considered. We find that uncertainty is greatest for derivatives that are taken with respect to Reynolds number, which is a result of the low increases in permeate flux with increasing Reynolds number. To ensure that these uncertainties do not lead to a biased solution, we use the optimally weighted χ2 function to find the difference between the derivative of estimated heat transfer ratio and derivative of the theoretical heat transfer ratio. The χ2 function weighs the derivatives by their normalized uncertainty as shown by Equation 18-20, where ∆δθ is uncertainty in derivative [21]. The derivatives were found by performing a linear regression on the heat transfer ratios. The uncertainty in the slope of the linear regression was found with the standard t-student test and was used as ∆δθ. g e.f
hJ
,HGP
h`0HGP
hJg
,HGP
mn mn
:^_ABC GB. i ^_XYZ[\ GB. i
g + h`0 HGP
:^_ABC GB. lA ^_XYZ[\ GB. lA
j j j DpABC GB. i jDpABC GB. i o q E^_XYZ[\ GB.i r Qnog qE^_ABC GB.i r DpXYZ[\ GB. i DpXYZ[\ GB. i
j j j DpABC GB. lA jDpABC GB. lA q E^_XYZ[\ GB. lA r Qnog qE^_ABC GB. lA r o DpXYZ[\ GB. lA DpXYZ[\ GB. lA
(18) (19)
(20)
3. RESULTS AND DISCUSSION We evaluate the effect of membrane structural parameters, bulk operating parameters, and module dimension on measured permeate flux, and use the acquired data to understand how these effects impact heat transfer rate qualitatively. We then demonstrate how the application of Nus-t correlation can lead to inaccurate estimation of intrinsic membrane permeability. Finally, we apply the extended CF method to estimate the error in the Nus-t correlation and determine the overall contribution of entrance effects, mass transfer through the membrane, and non-linear effects to the error in Nus-t correlation. 3.1 Effect of membrane structural parameters on permeate flux For a single module at constant operation conditions, higher membrane porosity and larger pore sizes are correlated with higher membrane permeability and greater flux (Figure 3). The high porosity of the PTFE and PP membranes reduces the water molecule diffusion pathlength, while the large pore size of the PTFE membrane changes the dominant vapor transport regime from Knudsen diffusion to Ordinary Molecular Diffusion. These phenomenon are well described by past theoretical and experimental efforts [19,22]. 3.2 Effect of bulk operating parameters on permeate flux
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Increasing bulk feed temperature and Reynolds number increase flux regardless of the module dimensions (Figure 3). The nearly exponential increase in permeate flux with increasing feed temperature reflects the exponential relationship between water vapor pressure and temperature as given by Antoine correlation [5]. Increasing Reynolds number has a much smaller effect on flux (Figure 3). While high Reynolds numbers increase the heat transfer rate, thus reducing thermal polarization and increasing the vapor pressure gradient across the membrane, the absolute magnitude of heat transfer is dictated by the temperature of the bulk. Again, this is consistent with prior work [10,11,23]. 3.3 Effect of module dimensions on permeate flux Whereas past work has explored the effects of the membrane and bulk operating parameters on permeate flux in MD, there has been little work quantifying the effects of module dimensions on MD performance. Empirically derived Nusselt correlations describe a thicker thermal boundary layer and a decrease in heat transfer rate with increasing flow channel length. We observe a 40% decrease in permeate flux as the module length increases from 4 to 20 cm (Figure 3 and Supplementary Figure S2). This decrease was independent of membrane structural parameters. The permeate flux also decreased with increasing membrane area, an effect not captured by commonly used Nusselt correlations. We measured nearly identical permeate flux for modules with the same area, but different width and length (e.g. 1x20 and 3x8 module pair, and 3x20 and 8x8 module pair). We hypothesize that this decrease in permeate flux with increasing module area is the result of the decreasing contribution of the entrance effect to the total heat transfer rate in the module. In all modules, the fluid enters the module at a right angle via a round entry port (Supplementary Figure S4). This fluid entry introduces a three-dimensional distribution of mixing or unsteady fluid flow over the membrane area and increases the heat transfer rate over the affected area of the membrane. As the module size increases, the overall effect of turbulence at the entrance diminishes. Furthermore, modules with wide flow channels (e.g. 8x8 module) can have low fluid crossflow at the edges of the flow channel and in the flow channel corners (i.e. dead zones), reducing heat transfer rates. Standard Nusselt correlations do not account for entrance effects on thermal boundary layer thickness or capture the effect of possible dead zones in wide modules, thus, do not accurately capture the increase in heat transfer rate with decreasing module width and length.
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Figure 3: Measured permeate flux for the three membranes and seven modules. The bottom of the colored bar shows the permeate flux at the lowest Reynolds number, and top of the colored bar shows the flux at the highest Reynolds number for each operating feed temperature. 3.4 Effect of position along length of module on permeate flux We isolated the effects of greater mixing near the entrance and the development of the thermal boundary layer along the length of the membrane module by measuring permeate flux at five different locations along the length of the membrane. In each experiment, we isolated a small active region of the membrane by blocking the remainder of the membrane with Kapton tape (Figure 4A). Local permeate flux at the module entrance was nearly 50±5% higher than that of the average flux for the 3x20 module at all operating Reynolds numbers and feed temperatures. As we shifted the active region away from the module entrance, the permeate flux declined linearly in the first half of the module. However, the permeate flux measured at the module exit was nearly identical to the permeate flux measured at the module center. The use of Kapton tape complicates the analysis of our experimental results, as shown by the higher local permeate flux than the average permeate flux measured without Kapton tape (Figure 4A). Two mechanisms cause this higher permeate flux. First, the fluid flow stepping down from the Kapton tape onto the membrane surface increases fluid mixing and the local heat transfer rate. Second, the Kapton tape reduces the heat transfer rate upstream of the open membrane area, resulting in a reduced thermal boundary layer thickness and a higher driving force relative to operation without Kapton tape upstream. The effect of complex thermal boundary layer formation is readily captured with the simulation of experimental conditions using the discretized MD model described in Supplementary Sections S3 and S4. The change in experimentally measured permeate flux along the module position agrees well with the simulations that used a Nu correlation with an entrance effect (Figure 4B). We use the discretized MD model to isolate the effect of high heat transfer rate at module entrance and exit on measured permeate flux by simulating how the permeate flux would change along module length under a constant Nu (e.g. no entrance or exit effect), a continually decaying Nu (e.g. an entrance effect only), and a parabolic change in Nu (e.g. entrance and exit effect) (Figure 4C, Supplementary Section S3). In the first half of the module, the change in experimental permeate
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flux normalized to average permeate flux closely matched the simulation that utilized a decaying Nu. This indicates that a strong entrance effect was responsible for observed increased in permeate flux. In contrast, the linear slope of the decay of experimental permeate flux normalized to average permeate flux in the second half of the module closely matches that of a constant Nu number, suggesting that the exit effect is negligible (Figure 4B). This result contradicts part of our original hypothesis stating that both abrupt entrances and exits impact the heat transfer rate (Figure 1B). Instead, these results suggest that only the entrance is responsible for increased heat transfer rates in MD modules.
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Figure 4: Measured and simulated permeate flux at different positions in the 3x20 module. (A) The permeate flux measured at different positions in 3x20 modules. The active membrane area was reduced to 2.5±0.1 cm long and 3 cm wide with Kapton tape (shown in gold) covering inactive membrane region, the arrow with J shows where water can permeate. (B) Local permeate flux normalized to permeate flux measured without Kapton tape that was measured experimentally and simulated using discretized MD model. The simulation was performed using
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three different Nusselt Correlation. The simulation and experimental data are for conditions of 1100 Re, feed inflow temperature of 70°C and draw inflow temperature of 20°C. (C) The three local Nusselt correlations used during simulations.
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3.5 Effect of module size on permeability estimates of MD membranes Errors in heat transfer rate can lead to significant errors in the calculated permeability of MD membranes. The intrinsic permeability of an MD membrane can be described by the sum of ordinary molecular and Knudson diffusion, which depended on membrane pore size, porosity, tortuosity, and (weakly) average membrane temperature [19]. However, when applying the average MD model and Nus-t correlation, the intrinsic permeability increases as the module size decreases and feed temperature increases (Figure 5). Furthermore, the prediction of permeability in small modules (1x4, 1x8, and 1x20) exceeds the theoretical maximum of water vapor diffusion in air. The observed increase in permeability with decreasing module size is a result of increasing underestimation of heat transfer rate.
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Figure 5: Comparison of theoretical and experimentally estimated membrane permeability using the Seider-Tate correlation to describe heat transfer rates in the membrane modules. (A) GVHP, (B) PP, and (C) PTFE membranes. The top of the colored bar shows estimated permeability for the highest Reynolds number, and the bottom of the colored bar shows permeability for the lowest Reynolds number. Theoretical permeability is estimated from DGMF theory.
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3.6 Estimated permeability using corrected Nus-t correlation The application of the CF method reduces the variability in predicted membrane permeability values between modules (Figure 6A) [11]. None of the estimated permeabilities exceed the theoretical maximum water vapor diffusion rate in air and there is only a very weak correlation between permeability and feed temperature. The estimated permeability was near the DGM-F theory prediction but did not precisely match it since the CF method matches the derivative of theoretical and experimental heat transfer ratio, rather than correcting to absolute theoretical DGM-F theory values.
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3.7 Quantifying inaccuracy in the Nus-t correlation
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The degree of correction required for Sider-Tate correlation decreased with increasing module dimensions and did not appreciably change with membrane type, the addition of mass transfer correction term, or discretization of the MD model (Figure 6B). The CF values decreased with an increase in module dimensions. The small 1x4 module required a CF value of 4-5, which implies that the heat transfer rate is almost four-to-five times higher than predicted by Nus-t correlation. As modules increased in size, the CF value decreased to nearly one and eliminated the need to correct the Nus-t correlation. The observed decrease in CF value with increasing module size is consistent with the reduced contribution of the entrance effect to the total heat transfer rate in the module. Furthermore, the CF factor does not correlate well with the dh/L parameter, which is used by Nusselt correlation to scale the heat transfer rate with flow channel geometry and entrance effects (Supplementary Figure S5). This weak correlation between CF and dh/L further supports our hypothesis that the Nus-t correlation does not accurately capture the entrance effects present in MD modules.
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Our estimated CF factor was weakly affected by membrane permeability and mass transfer rate (Figure 6B). Vapor transport through the membrane results in the flow of water toward the membrane surface on the feed side and away from the membrane surface on the draw side. This fluid flow can potentially compress and expand the thermal boundary layer, increasing and decreasing the heat transfer rate, respectively. Although the CF factor was, in general, higher for low flux GVHP and PP membranes relative to the high flux PTFE membrane, the variability in CF factor prediction between the modules makes it impossible to identify a statistically significant effect.
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Inclusion of the mass transfer correction term derived by Phattaranawik et al. did not affect the magnitude of the CF factor [14]. While our experiments were not designed to isolate the effects of fluid flow to/away from the membrane surface on heat transfer rates, these results suggest that the effect is likely to be limited. Future work to address this question should use carefully designed multi-physics computation fluid dynamics studies where the variability between experiments is likely to be smaller and uncertainty in parameter estimation can be quantified explicitly. We did not observe an appreciable change in the CF factor when using a discretized MD model (Figure 6B). The discretized MD model captures the effect of non-linear heat transfer across the module length as a result of enthalpy transport through the membrane and the resulting highly non-linear flux across module length. However, the CF values computed with the discretized MD model were nearly identical to those calculated with the average MD model. The negligible change in CF factor with model discretization demonstrates that the primary error in Nus-t correlation when applied to small bench-scale modules is the result of the underestimation of the average heat transfer rate and not a result of simplifications made in the average MD model.
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Figure 6: Estimated permeability after application of correction factor and correction factor values. (A) Estimated permeability for the three membranes with corrected heat transfer correlation. (B) Estimated correction factor for each module and membrane using average MD model, average MD model with mass transfer correction, and a discretized MD model with nonlinear heat transfer rate. A correction factor of one implies that Nus-t accurately describes heat transfer rate in the module.
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4. CONCLUSION Systematically designed and rigorously executed MD experiments on seven modules, three membrane types, four feed temperatures, and four Reynolds numbers provided the first comprehensive dataset for evaluating the accuracy of standard methods for predicting heat
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transfer rate. We demonstrated that the accuracy of commonly used Nus-t correlation increases with increasing module width and length. We attribute this to abrupt entrance effects inducing fluid mixing and increasing the heat transfer rate. These entrance effects are not accounted for in the commonly used Nus-t correlation and can result in estimated heat transfer rates that are 5X too low in small bench scale modules. The contribution of the entrance effect to the overall heat transfer rate in the module decreases increasing with module width and length, thus improving the accuracy of Nus-t correlation in large modules. This work also demonstrated the validity of the previously reported Nusselt correction factor method for modules of varying size. Early work developed the CF method for a single module, raising questions about the adequacy of this method across varying module dimensions. The consistency of the estimated permeability of the three membrane types for each of the modules, temperatures, and Reynolds numbers suggests that this is a robust method that should be widely adopted by the MD community. We were unable to measure an increase or decrease in heat transfer rate with a change in mass transfer through the membrane in bench-scale MD modules. Our estimate of Nus-t correlation error (i.e. the magnitude of the CF) did not change with an increase in membrane permeability. The addition of the correction term for compression and expansion of the thermal boundary layer due to water flow to and from the membrane surface also did not change the magnitude of the CF. Finally, we have shown that accounting for non-linearity in heat transfer rate and permeate flux across module length did not change the CF. This work presents the first systematic study of how MD heat transfer rates change as a function of module size and membrane permeability. Additional work is needed to evaluate the effect of mass transfer through the membrane on heat transfer in large modules, though this work would be best accomplished with multi-physics computational fluid dynamic studies. The entrance effect can degrade the accuracy of Sherwood correlations when applied to bench-scale membrane modules, meaning that their accuracy should be studied as a function of module size. Finally, the results of this work highlight that the design of the bench-scale module can have dramatic effects on the predicted performance of a membrane process and this effect must be carefully considered when translating the results from bench-scale systems to pilot and commercial-scale systems. ACKNOWLEDGEMENTS This work was financially supported by NSF CBET-1554117 award. REFERENCES [1] P. Wang, T.-S. Chung, Recent advances in membrane distillation processes: Membrane development, configuration design and application exploring, J. Membr. Sci. 474 (2015) 39–56. https://doi.org/10.1016/j.memsci.2014.09.016. [2] E.N. Sieder, G.E. Tate, Heat transfer and pressure drop of liquids in tubes, Ind. Eng. Chem. 28 (1936) 1429–1435. [3] S. Kakaç, R.K. Shah, A.E. Bergles, North Atlantic Treaty Organization, eds., Low Reynolds number flow heat exchangers: advanced study institute book, Hemisphere Pub. Corp, Washington, 1983.
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[4] D.E. Metzger, M.K. Sahm, Heat Transfer Around Sharp 180 Degree Turns in Smooth Rectangular Channels, in: Vol. 3 Heat Transf. Electr. Power, ASME, Houston, Texas, USA, 1985: p. V003T09A020. https://doi.org/10.1115/85-GT-122. [5] R.W. Schofield, A.G. Fane, C.J.D. Fell, Heat and mass transfer in membrane distillation, J. Membr. Sci. 33 (1987) 299–313. https://doi.org/10.1016/S0376-7388(00)80287-2. [6] L. Martínez-Díez, A method to evaluate coefficients affecting flux in membrane distillation, J. Membr. Sci. 173 (2000) 225–234. https://doi.org/10.1016/S0376-7388(00)00362-8. [7] L. Li, K.K. Sirkar, Influence of microporous membrane properties on the desalination performance in direct contact membrane distillation, J. Membr. Sci. 513 (2016) 280–293. https://doi.org/10.1016/j.memsci.2016.04.015. [8] J. Vanneste, J.A. Bush, K.L. Hickenbottom, C.A. Marks, D. Jassby, C.S. Turchi, T.Y. Cath, Novel thermal efficiency-based model for determination of thermal conductivity of membrane distillation membranes, J. Membr. Sci. 548 (2018) 298–308. https://doi.org/10.1016/j.memsci.2017.11.028. [9] M. Gryta, M. Tomaszewska, A.W. Morawski, Membrane distillation with laminar flow, Sep. Purif. Technol. 11 (1997) 93–101. https://doi.org/10.1016/S1383-5866(97)00002-6. [10] J. Phattaranawik, R. Jiraratananon, A.G. Fane, Heat transport and membrane distillation coefficients in direct contact membrane distillation, J. Membr. Sci. 212 (2003) 177–193. [11] M.E. Leitch, G.V. Lowry, M.S. Mauter, Characterizing convective heat transfer coefficients in membrane distillation cassettes, J. Membr. Sci. 538 (2017) 108–121. https://doi.org/10.1016/j.memsci.2017.05.028. [12] E.M. Sparrow, Analysis of laminar forced-convection heat transfer in entrance region of flat rectangular ducts, (1955). https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930084124.pdf (accessed October 1, 2017). [13] J. Lou, J. Vanneste, S.C. DeCaluwe, T.Y. Cath, N. Tilton, Computational fluid dynamics simulations of polarization phenomena in direct contact membrane distillation, J. Membr. Sci. 591 (2019) 117150. https://doi.org/10.1016/j.memsci.2019.05.074. [14] J. Phattaranawik, R. Jiraratananon, A.G. Fane, Effects of net-type spacers on heat and mass transfer in direct contact membrane distillation and comparison with ultrafiltration studies, J. Membr. Sci. 217 (2003) 193–206. https://doi.org/10.1016/S0376-7388(03)00130-3. [15] D. Bertsche, P. Knipper, T. Wetzel, Experimental investigation on heat transfer in laminar, transitional and turbulent circular pipe flow, Int. J. Heat Mass Transf. 95 (2016) 1008–1018. https://doi.org/10.1016/j.ijheatmasstransfer.2016.01.009. [16] D.J. Benford, T.J. Powers, S.H. Moseley, Thermal conductivity of Kapton tape, Cryogenics. 39 (1999) 93–95. https://doi.org/10.1016/S0011-2275(98)00125-8. [17] Polypropylene Membrane Filters, 0.2 Micron, Sterlitech. (n.d.). https://www.sterlitech.com/polypropylene-membrane-filter-0-2-micron-254-x-3000mm-1pk.html (accessed November 13, 2019). [18] M.C. Porter, Concentration polarization with membrane ultrafiltration, Ind. Eng. Chem. Prod. Res. Dev. 11 (1972) 234–248. [19] R.W. Field, H.Y. Wu, J.J. Wu, Multiscale Modeling of Membrane Distillation: Some Theoretical Considerations, Ind. Eng. Chem. Res. 52 (2013) 8822–8828. https://doi.org/10.1021/ie302363e. [20] T.E. Oliphant, Python for Scientific Computing, Comput. Sci. Eng. 9 (2007) 10–20. https://doi.org/10.1109/MCSE.2007.58.
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[21] B.F.W. Croke, The Role of Uncertainty in Design of Objective Functions, (n.d.) 8. [22] J. Phattaranawik, Effect of pore size distribution and air flux on mass transport in direct contact membrane distillation, J. Membr. Sci. 215 (2003) 75–85. https://doi.org/10.1016/S0376-7388(02)00603-8. [23] T.Y. Cath, V.D. Adams, A.E. Childress, Experimental study of desalination using direct contact membrane distillation: a new approach to flux enhancement, J. Membr. Sci. 228 (2004) 5–16. https://doi.org/10.1016/j.memsci.2003.09.006.
1. 2. 3. 4. 5.
Study relates module design and mass transfer rate to heat transfer rate Nusselt correlations underestimate heat transfer in small modules by up to 5x Entrance effects induce mixing and increase the heat transfer rate Mass transfer does not increase the heat transfer rate in bench-scale modules Nusselt correlations more accurately describe heat transfer in large modules