Numerical study of heat transfer performance of helical coiled tubes for heating high-solids slurry in household biogas digester

Journal Pre-proofs Numerical study of heat transfer performance of helical coiled tubes for heating high-solids slurry in household biogas digester Yaowen Chen, Yanfeng Liu, Dengjia Wang, Tao Li, Yingying Wang, Yong Li PII: DOI: Reference:

S1359-4311(19)31618-7 https://doi.org/10.1016/j.applthermaleng.2019.114666 ATE 114666

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Applied Thermal Engineering

Received Date: Revised Date: Accepted Date:

12 March 2019 11 October 2019 10 November 2019

Please cite this article as: Y. Chen, Y. Liu, D. Wang, T. Li, Y. Wang, Y. Li, Numerical study of heat transfer performance of helical coiled tubes for heating high-solids slurry in household biogas digester, Applied Thermal Engineering (2019), doi: https://doi.org/10.1016/j.applthermaleng.2019.114666

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Numerical study of heat transfer performance of helical coiled tubes for heating high-solids slurry in household biogas digester Yaowen Chena, b, Yanfeng Liua, b, *, Dengjia Wanga, b, **, Tao Lia, b, Yingying Wanga, b, a

Yong Lia, b

State Key Laboratory of Green Building in Western China, Xi'an University of

Architecture and Technology, NO.13 Yanta Road, Xi'an, 710055, China b

School of Building Services Science and Engineering, Xi'an University of

Architecture and Technology, NO.13 Yanta Road, Xi'an, 710055, China

Corresponding author: Yanfeng Liu, Professor. State Key Laboratory of Green Building in Western China School of Building Services Science and Engineering Xi'an University of Architecture and Technology No.13 Yanta road, Beilin district, Xi'an 710055, China E-mail: [email protected] Tel: +86-29-82202506 Mobile: +86-13909261178 Dengjia Wang, Professor. State Key Laboratory of Green Building in Western China School of Building Services Science and Engineering Xi'an University of Architecture and Technology No.13 Yanta road, Beilin district, Xi'an 710055, China E-mail: [email protected] Mobile: +86-13279455510

ABSTRACT:

A computational fluid dynamics model to simulate conjugate heat transfer between a helical coiled tube and high-solids fermentation slurry in a household biogas digester was developed. The model simulation results were then verified using field experiment results. The effect of total solids concentration, pitch, helical diameter, and tube materials on the thermal performance of the helical coiled tubes were analysed. The results show that the accuracy of the numerical model that was based on the rheological and thermal parameters of the fermentation slurry was significantly improved compared with that based on the corresponding water parameters. As the total solids concentration increased, the zero-shear viscosity of the slurry significantly increased and the heat transfer performance of the helical coiled tubes was notably weakened. In the high-solids fermentation slurry, the influence of pitch on the heat transfer coefficient of the coiled tubes was greater than that of the helical diameter. The difference in the rates of total thermal resistance between cross-linked polyethylene (PEX) and metal (stainless steel and copper) coiled tubes was small for the high-solids fermentation slurry compared to that for the water. Therefore, low-cost PEX coiled tubes are more suitable for high-solids fermentation slurry than metal tubes in household biogas digesters.

Keywords: Household biogas digester, High-solids fermentation slurry, Helical coiled tube, Heat transfer performance, Computational fluid dynamics

Nomenclature

H

height, m

Pr

D

diameter, m

Gk

dc

diameter of the coiled tube, mm

Gb

P

pitch of the coil, mm

YM

U

velocity vector, ms-1

C1, C2

p

local pressure, Pa

Tgs

g

acceleration of gravity, ms-2

Ags

T cp

temperature, °C specific heat, Jkg-1K-1

ST

viscous dissipation term

Re

Reynolds number

Recr

critical Reynolds number flow velocity of heat medium, u ms-1 Ra Rayleigh number F body force vector weight fractions of the major xi components i Greek symbols δ wall thickness of the coil, mm

T0 γ

Prandt number the turbulence items due to the average velocity gradient the turbulence items due to the buoyancy the effect of compressible turbulent flow pulsation expansion on the total dissipation rate the calculation coefficient of the main components annual average temperature of ground surface, °C annual temperature fluctuation of ground surface, °C annual fluctuation period, h soil thermal diffusivity, m2s-1 convective heat transfer coefficient of the ground surface, Wm-2K-1 total solids concentrations of slurry, % reference temperature, °C shear rate, s-1

n '

power law index second viscosity coefficient

μτ

τ0

turbulent viscosity Prandtl numbers of the turbulent kinetic energy Prandtl numbers of turbulent dissipation rate initial time, s

β

volume expansion coefficient

fs hm gs a ref

fermentation slurry heat medium ground surface outdoor air reference

k a hsur TS

ρ

density, kgm-3

σk

τ

time, s

σε

dynamic viscosity coefficient, Nsm-2 η apparent viscosity, kgm-1s-1 λ thermal conductivity, Wm-1K-1 Subscripts s soil column d biogas digester c coil pipe w wall of coil pipe in inlet of coil pipe Abbreviations BD biogas digester HBDs household biogas digesters FS fermentation slurry μ

ZSV CSV TR

zero-shear viscosity critical-shear viscosity temperature rise

HCTs HTP PEX UDF

helical coiled tubes heat transfer performance cross-linked polyethylene user defined function

HF AHTC TS

heat flux average heat transfer coefficient total solids concentrations

1 Introduction Since 2000, China's household biogas digesters (HBDs) have developed rapidly and the number of HBDs has increased from less than 10 million to nearly 40 million [1-3]. However, the majority of existing HBDs are not heated and stirred, and the slurry temperature varies with ambient temperature and solar radiation [4]. There are obvious geographical differences in the popularity of HBDs in China, which are more common in the southwest and central regions than in the northwest and northeast [5]. Chen [6] analysed the regional suitability of HBDs in China and found that most of the northern areas were more suitable based on the criteria of fermentation raw materials. Nevertheless, the suitability of the northern areas was poor based on the criterial of ambient temperature because most of these areas experience cold and sever cold climates [7]. Therefore, a low ambient temperature is one of the most important limitations for the popularity of HBDs in cold regions. To solve the problem of low productivity or even lack of gas production of HBDs in the cold environments, a series of technologies has been proposed for heating biogas digester (BD) using renewable energy, including solar collectors [8, 9], heat pumps [10] and biogas boilers [11]. Among these, solar heating technology is the most extensively studied. In the northwest of China, the winter is cold and long but there are abundant solar energy resources. Accordingly, a solar heated BD system is an effective method to promote the development of HBDs in north western China. Several researchers have established experimental systems and mathematical models for solar-heated BDs, and they have analysed and evaluated the overall operating performances of these systems [12-14]. In the solar heated BD system, in

addition to the collectors, pumps and controllers, the heat exchanger is a vital component affecting the operating performance of the system. Two types of heating modes are commonly used when operating a BD heating system: external heating with an outside heat exchanger (such as a plate heat exchanger, shell-and-tube heat exchanger or scraped-surface heat exchanger) and internal heating with a coiled tube. The outside heat exchanger is advantageous as a result of its high heat transfer efficiency; however, the initial investment and operating cost are much higher than the coiled tube. Thus, it is usually applied to the heating and wasted heat recovery of slurry in large biogas plants. In comparison, internal coiled tubes are usually used for HBD heating systems due to their low investment and simple structure [15]. Helical coiled tubes (HCTs) are usually used for heat exchangers in air conditioning systems, nuclear power plants and many other industrial scenarios owing to their compact structure and high heat transfer efficiency [16]. At present, they are also used in the heating system of BDs. The heating process of fermentation slurry (FS) heated by HCTs in BDs is essentially a conjugate heat transfer process among the coiled tube inner and outer fluids. Initial research into the heat transfer performance for HCTs often used prespecified boundary conditions (constant heat flux or constant temperature) to simplify the model [17]. Jayakumar et al. compared the simulated and measured values of helical tube exchangers under different boundary conditions and concluded that conjugate heat transfer was a more applicable boundary condition than both a constant wall temperature and constant wall heat flow [18]. Mirgolbabaei studied the thermal performance of vertical helical coiled tube heat exchangers with forced convection outside the tube [19]. In that study, a conjugate thermal boundary condition of the tube wall fluid-to-fluid heat transfer mechanism was considered. Bahrehmand and Abbassi

studied conjugate heat transfer of nanofluids in a helical coiled tube-shell heat exchanger under turbulent conditions [20]. The heat transfer process of the shell and helical tube heat exchanger was characterized by forced convection of the fluid inside and outside the tube. The conjugate heat transfer progress of HCTs in BDs is similar to that in hot water storage tanks is characterized by forced convection in the tube and natural convection outside the tube [21-22]. The main difference between the heat transfer of HCTS in BDs and hot water storage tanks is that the physical properties of the fluid outside the tubes, which results in differences in the thermal performance of the coiled tubes and the rate of temperature increase process of the fluid outside the tubes. Some studies have revealed that the total solids concentration (TS) is an important factor affecting the physical properties of FS [23, 24]. When TS is low (TS<2%), its physical properties are close to water, but the difference of physical properties between the FS and water gradually increases with the increase of TS [25, 26]. According to the rheological properties, FS is classified as a pseudoplastic non-Newtonian fluid [27]. Under high TS conditions, the viscosity (apparent viscosity) of FS is very different from that of water. This difference in viscosity will have an important influence on the natural convection heat transfer of fluids outside the coiled tube. Moreover, high-solids anaerobic fermentation is usually applied in a large number of rural HBDs [28]. Consequently, studying the transfer performance of HCTs in FS with high TS has practical significance for the design and popularization of HBD heating systems. However, the FS was replaced by water to test and simulate the heat transfer performance of helical coiled tube in several studies [29-31]. Chen et al. established a numerical model and analysed the thermal performance for a wasted heat recovery scraped-surface heat exchanger outside the BDs based on the rheological properties of

FS. Forced convection facilitated the flow of water and FS in the heat exchanger, which was different from the natural convection of FS in the BDs [32]. Wu et al. and Yu et al. studied the mixing performance of a stirrer in the high-solids FS using the computational fluid dynamics (CFD) numerical simulation method with the rheological property parameters of the slurry [33-35]. These studies have built the foundation for further research on the heat transfer performance (HTP)of HCTs in high-solids HBDs. At present, there are few studies focusing on the HTP of HCTs for heating the highsolids FS in HBDs or the development of corresponding conjugate heat transfer numerical models. In this study, a conjugate heat transfer numerical model for simulating the heat transfer of HCTs in HBDs was developed based on the rheological and thermal properties of high-solids FS. The simulation results were compared with our previous field test data to verify the reliability of the model. To analyse the heat transfer weakening mechanism of helical coiled tube in high-solids FS and the influence of coiled tube structure parameters and tube material, the overall fluid temperature, heat flux (HF), outlet temperature, apparent viscosity, heat transfer coefficient and thermal resistance were numerically calculated. This study provides useful information for the design and application of high-solids HBD heating systems. 2. Methodology 2.1 Geometry model In previous work we established an experimental system for heating a BD with an HCT. The experimental device, parameters of the measuring instrument, locations of the measuring points and experimental are thoroughly introduced in Liu et al [36]. The temperature of the FS in the digester; the inlet and outlet temperature; and the flow rate of heat medium in the coils were monitored. Photos of the BD and HCT in the

experiment are shown in Fig. 1 (a) and (b). The digester material was fiberglassreinforced plastic with a wall thickness of 10 mm and the top cover was movable. The HBDs are usually built underground to reduce heat loss [37]. In this study, the digester was built underground at a depth of 1.0 m from the bottom of the digester to the surface. The digester was insulated with 30 mm of rubber-plastic cotton. A cross-linked polyethylene (PEX) HCT was installed inside the digester. The physical model was composed of soil, a BD and a coiled tube heat exchanger (Fig. 2 (c)). To solve the model, it was simplified as follows: (1) the heat dissipation of the digester has a limited influence on the soil temperature over a short period of time, thus the semi-infinite soil was simplified into a soil column with a limited height and diameter [38]. (2) the thermal resistance of the digester wall is much smaller than that of the insulation layer, and thus the digester wall was ignored in the physical model. (3) the thickness of the insulation layer was much thinner than the digester diameter and the soil column diameter, thus the insulation layer was ignored in the geometry and only considered for its effect in heat conduction [39]. (4) considering the large thermal resistance of the air layer and the insulation layer above the slurry surface, the upper surface of the slurry was simplified as an adiabatic boundary and the air layer was ignored. The height (Hs) and diameter (Ds) of the soil column were both 5.0 m; the height (Hd) and diameter (Dd) of the digester were 0.8 m and 1.0 m, respectively; the helical diameter (Dc), tube diameter (dc), wall thickness (δ) and pitch (P) of the HCT were 0.5 m, 16 mm, 2 mm, and 75 mm, respectively.

(a)

(b)

Hb

Hs

Helical coil pipe

P

Biogas digester

Soil column

Db

Dc

Ds

(c) Fig. 1. Photos and physical geometry model: (a) BD and cattle manure slurry, (b) PEX helical coiled tube, (c) geometry model of soil column, digester and helical coiled tube. 2.2 Mathematical model Several assumptions were made when developing the heat transfer model: (1) The physical properties of the soil, HCT and thermal insulation material were considered to be constant. (2) The FS exhibited non-Newtonian pseudoplastic fluid behaviour when TS was greater than 2.5% [26]. (3) The physical properties of water and FS in the digester were presumed to be constant, except for the water density variations in the buoyancy terms with

Boussinesq approximation, the density of FS changed with temperature and the apparent viscosity of FS changed with temperature and shear rate [32]. (4) The slurry was thoroughly stirred before the experiment, so it is assumed that the temperature distribution of the slurry was uniform at the initial time. 2.2.1 Governing equations The governing equations (continuity (Eq. (1)), momentum (Eq. (2)), and energy (Eq. (3)) for the fluid region are normally expressed as the following: (1) Continuity equation

   U   0 

(1)

(2) Momentum equation

U  U U   p    U TU   F  U   ' U  





(2)

(3) Energy equation

    T     UT     T   ST     cp 

(3)

where U is the velocity vector, ms-1; ρ is the density, kgm-3; τ is the time, s; p is the local pressure, Pa; μ is the viscosity coefficient of fluid, Nsm-2; F is the body force vector of microelement that arise from gravitational or somebody force field, kgm-2s-2; 2  ' is the second viscosity coefficient, usually  '    , Nsm-2; T is the temperature, 3

°C; λ is the thermal conductivity, Wm-1K-1; cp is the constant pressure specific heat, Jkg-1K-1; and ST is the viscous dissipation term. The expressions of the Reynolds number (Re) and the critical Reynolds number (Recr) of the heat medium in the HCT were as follows [40]:

Re 

du i



(4)

 2d Recr  2 104  i D  p

  

0.32

(5)

where u is the flow velocity of heat medium, ms-1 and ν is the kinematic viscosity coefficient, m2s-1. The Rayleigh number (Ra) was used to determine the fluid flow state of natural convection outside the coiled tube [41]. R aL 

g  L3 Pr (Tw  Tfs )

2

(6)

where Pr is the Prandtl number; Tw is the wall temperature of the coil, °C; and Tfs is the FS temperature, °C. According to the calculation, the Reynolds number (Re) of the heat medium in the tube was approximately 2.6×104, which is greater than the critical Reynolds number (Recr) (about 7.6×103), hence the flow state of the heat medium in the tube was considered turbulent flow. The Rayleigh number (RaL) was approximately 1015 when the water is heated, whereas the Rayleigh number was less than 1010 when heating the slurry. Ali [41] illustrated that the flow state of the fluid is turbulent natural convection when the Rayleigh number range is from 4.8 × 1011 to 4 × 1015 during the process of heating water by a vertical HCT. Accordingly, the turbulence model was adopted. The Realizable k–ε turbulence model can accurately predict the bending and rotating flows, while providing higher computational time performance. The respective equations of k and ε are given as follows [42]:    ( k )+ ( ku i )=   xi x j

   ( )+ ( kui )=  xi x j

  t  k       G k  Gb    YM  S k   k  x j   

 t     

(7)

   2   C1 C3 Gb  S (8)    C1 S    C2  k k    x j 

The turbulent kinematic viscosity is:

t   C

k2



(9)

The empirical constants for the model are:

   C1  max  0.43,   5  

C 2   1 . 9;  k  1 . 0;  k  1 . 2

(10) (11)

where μt is the turbulent viscosity; σk and σε are the Prandtl numbers of the turbulent kinetic energy and turbulent dissipation rate, respectively; Gk and Gb are the turbulence items due to the average velocity gradient and buoyancy, respectively; and YM is the effect of compressible turbulent flow pulsation expansion on the total dissipation rate. Boussinesq approximation assumes that the fluid density is a linear function of temperature. Only the density variation of the momentum equation in the natural convection was considered. This model is applicable to the flow with little change in fluid density [22].

(  ref )g  ref  (T -Tref )g

(12)

where β is the volume expansion coefficient of fluid and g is the acceleration of gravity, ms-2. Since only the thermal conduction process is considered in the solid region, the energy equation is simplified to the heat conduction differential equation in the following form:

T  2  T   c p

(13)

2.2.2 Initial and boundary conditions (1) Initial conditions The slurry temperature field for the initial condition was assumed to be uniform. The coiled tube was immersed in the slurry, and thus the initial temperature of the heat

medium was considered to be the same as the slurry. The soil initial temperature field can be described by the temperature distribution of semi-infinite objects under periodically varying boundary conditions [43]. The expression for the initial condition is as follows:

Tfs,i  Thm,i  11.5 C

(14)

    2   Ts ( z, 0 )  Tgs  Aw exp   z  cos   0  z  ak   k ak  

(15)

where Tfs,i is the initial temperature of the slurry, °C; Thm,i is the initial temperature of the heat medium, °C; Tgs is the annual average temperature of ground surface, °C; Aw is the annual temperature fluctuation of ground surface, °C; k is the annual fluctuation period, h; a is the soil thermal diffusivity, m2s-1; and τ0 is the initial time, s. (2) Boundary conditions For the soil column model, the boundary condition of the upper surface was convective while that of the other surfaces were adiabatic. The boundary conditions for soil columns are as follows:  T  hsur [T   z z 0  T 0   r r 2.5  T 0   z z 5

z 0

( )  Ta ( )]

Ta ( )  2.34104  4.83108 2 1.411012 3  271.33 (R2 =0.9987)

(16)

(17)

where Ta is the outdoor air temperature, °C; and hsur is the convective heat transfer coefficient of the ground surface, Wm-2K-1. The boundary conditions of BD and the HCP were as follows:

T z

0 z 0.2, r 

(18)

Tin  60 C   u  1.2 ms-1  in

(19)

where Tinlet is the inlet temperature of the coil, °C; and Vinlet is the inlet flow velocity of the coil, ms-1. 2.2.3 Physical and rheological properties (1) Density Gerber and Schneider proposed a formula for calculating the density of biogas slurry based on organic composition and temperature [44]. The weight fractions of main components of cattle manure slurry were found from the book of Coprology of Domestic Animals [45].

 (T , x)=   (xi /i (T )) 

1

i (T )=C1  C2 (T -273.15)

(20) (21)

where xi is weight fractions of the major components i (water, crude protein, crude fat, crude fiber, Nfe, crude ash), ρi the density of different components, kgm-3, C1 and C2 are the calculation coefficient of the main components. The fitting quadratic polynomial functions of density and temperature of biogas slurry for different TS are as follows:

TS=7.5%:  fs =1024.85-0.01(Tfs -273.15)-0.0037(Tfs -273.15) 2

(22)

TS=9.1%:  fs =1030.95-0.013(Tfs -273.15)-0.0036(Tfs -273.15) 2

(23)

TS=9.5%:  fs =1032.48-0.014(Tfs -273.15)-0.0036(Tfs -273.15) 2

(24)

TS=12.1%:  fs =1042.59-0.019(Tfs -273.15)-0.0036(Tfs -273.15) 2

(25)

(2) Specific heat capacity and thermal conductivity

The specific heat capacity, and thermal conductivity of the FS were calculated by equations (26), (27) [25], [46].

c p ,fs =4190  27.5TS

fs =0 .6173  0 .0069TS

(26) (27)

where ρfs is the density of slurry, kgm-3; cp,fs is the specific heat capacity of slurry, Jkg1K-1;

λfs is the thermal conductivity of slurry, Wm-1K-1; and TS is the total solids

concentrations of slurry, %. (3) Apparent viscosity Cattle manure slurry is a pseudoplastic non-Newtonian fluid whose apparent viscosity can be expressed as a power law formula, as shown in equation (28) [33].

 = k  n 1exp T

0

/ Tfs

(28)

where η is the apparent viscosity, kgm-1s-1; T0 is the reference temperature, °C, γ is the shear rate, s-1; and n is the power law index. Chen and Shetler proposed the apparent viscosity calculation formula of cattle manure slurry, which combines the effect of temperature and shear rate [23].

 =2.927 106 (TS )1.822 (0.01 )n1exp(1659/Tfs ) n =0.746exp(  0.0381TS )

(29) (30)

For pseudoplastic non-Newtonian fluids, the power exponent n is less than 1. The apparent viscosity η reaches a constant, which is usually defined as zero shear viscosity (ZSV) instead of infinity when the shear rate approaches zero. In contrast, when the shear rate approaches infinity, the apparent viscosity η reaches a constant, which is usually defined as the critical shear viscosity (CSV) [47]. Chen et al. provided the polynomial equations of ZSV and CSV for different TS (7.5%, 9.1%, and 12.1%) of FS [32]:

 ZSV=  6.33  106 Tfs 3  6.26  103 Tfs 2  2.08Tfs  230.33  TS=7.5%:  3 2 6 3  CSV=  4.59  10 Tfs  4.45  10 Tfs  1.44Tfs  155.25

(31)

 ZSV=  14.24  106 Tfs 3  14.17  103 Tfs 2  4.71Tfs  524.45  TS=9.1%:  (32) 6 3 3 2  CSV=  8.97  10 Tfs  8.69  10 Tfs  2.81Tfs  302.69  ZSV=  191.25 106 Tfs 3  189.59 103 Tfs 2  62.79Tfs  6950.98 TS=12.1%:  (33) 6 3 3 2 CSV=  9.84 10 Tfs  10.70 10 Tfs  3.85Tfs  457.75 Through the method of the linear interpolation and polynomial fitting, the ZSV and CSV calculation equation of the FS (TS=9.5%) in the experiment was obtained: 3 2 6 3   ZSV  37.84 10 Tfs  37.56 10 Tfs  12.46Tfs  1381.32 TS=9.5%:  (34) 3 2 6 3  CSV  9.08 10 Tfs  8.96 10 Tfs  2.95Tfs  323.36

Without stirring, the heat transfer between the coiled tube and the slurry occurred as natural convection and the slurry flowed caused by buoyancy. Owing to the small temperature difference between the tube wall and the slurry, liquidity of the slurry was weak and the shear rate was small. Therefore, ZSV should be used as the apparent viscosity of the slurry under natural convection conditions for the numerical simulation. The applicable range for equations (29) and (30) was 20 s-1< γ <200 s-1, which is suitable for simulating the flow or heat transfer performance of the slurry under stirring conditions or forced convection. 2.3 Numerical method 2.3.1 Mesh of the model The geometrical physical model was established by CATIA V5, and the polyhedral grids were generated by Fluent 2019 R2 (Fig. 2). Compared with tetrahedral grids, polyhedral grids have more adjacent elements which can more accurately calculate the gradient of variables and predict the flow situation in local areas. The boundary layer grids were generated near the tube wall and the digester wall.

Outlet

Inlet

Heating coil Digester

Soil column

Fig. 2. Mesh of the model. 2.3.2 Settings in CFD software Table 1 respectively shows the density, thermal conductivity and specific heat capacity of the soil, insulation material, coiled tube, water and biogas slurry. Table 1 Thermophysical properties of the materials used in this study. Material

Density (kgm-3)

Thermal conductivity coefficient (Wm-2K-1)

Soil Rubber plastic PEX Copper Stainless steel Water Slurry(TS=7.5%) Slurry(TS=9.1%) Slurry(TS=9.5%) Slurry(TS=12.1%)

1886.6 78 960 8954 7817 998.2 Eq. (22) Eq. (23) Eq. (24) Eq. (25)

0.92 0.035 0.25 398 16.3 0.60 0.57 0.55 0.55 0.53

Specific heat capacity (Jkg-1·K-1) 2639.3 2260 2260 384 460 4183.0 3983.8 3939.8 3926.3 3857.3

The conjugate heat transfer model was numerically calculated by Fluent 2019 R2. The relevant parameter settings used in Fluent are shown in Table 2. Table 2 The parameter settings in Fluent. Items General

Setting details Solver: Pressure based

Models

Solution methods

Materials

Boundary conditions

Solution Initialization

Geometry dimension: 3D Velocity formulation: Absolute Time: Unsteady Gravitational acceleration: Z-axis direction, g=9.8ms-2 Energy mode: ON Viscous model: Standard k-ε, RNG k-ε, Realizable k-ε Full buoyancy effects: ON Scheme: SIMPLE Gradient: green-gauss node based Pressure: PRESTO Momentum: QUICK Turbulent kinetic energy: Second order upwind Turbulent dissipation rate: Second order upwind Energy: Second order upwind Time dispersion: Second order implicit The thermal property parameters of soil, insulation, coil pipe, slurry and water were set according to Table 1. The thermophysical parameters of the FS with different TS were calculated by equations (22-27). The apparent viscosity and the density of the FS was imported into the Fluent through the user defined function (UDF). The boundary conditions were set according to the equations (16-19). The outdoor air temperature fitting function (17) was imported into the boundary conditions through UDF. The soil initial temperature field function (Eq. (15)) was imported into Fluent through UDF. The initial temperature of slurry and heat medium were patched according to equation (14).

The time step size can affect the convergence of the calculation and the accuracy of the results. Gómez et al. studied the thermal performance of a coiled tube heat exchanger in a hot water storage tank [21]. The conjugate heat transfer between the heat medium in coiled tube and the water in tank is similar to the heat transfer process of FS heated by the HCT in this study. After comparing the tank temperatures, net heat transferred and velocity of the buoyancy flow, Gómez et al. concluded that there were no significant differences in the simulation results when time steps were lower than 20 s were used in the Boussinesq model. Consequently, the time step used for the transient simulation in this study was 5 s. 2.3.3 Grid independence study

In order to analyse the influence of the number of grids on the numerical solution, the models with different grid numbers (548,000, 873,000, 1,360,000, and 3,035,000) were established. As shown in Fig. 3, the variation of water temperature at the HCT outlet over time was calculated by using the models with different grid numbers. The outlet temperature of the coiled tube increased with the increasing mesh number. When the mesh numbers increased from 548,000 to 1,360,000, the average temperature difference at the outlet was 0.3 °C. When the mesh numbers increased from 1,360,000 to 3,035,000, the average temperature difference at the outlet was only 0.04 °C. Considering the computational accuracy and cost, the model with 1,360,000 cells is selected for the following simulation. 59.5 59.0

Outlet temperature (°C)

58.5 58.0 57.5 57.0 56.5

548,000 cells 873,000 cells 1,360,000 cells 3,035,000 cells

48 47 46 45

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

Time (s)

Fig. 3. Comparison of water temperature at the outlet of HCTs under different mesh numbers 3. Results and discussion 3.1 Model Validation To validate the numerical model, the simulation results are compared to the experimental results and root mean square deviation (RMSD) has been evaluated by following equation [48]:

RMSD =

1 n [(X sim,i  X exp,i )/X exp,i ]2  i 1 n

(35)

Fig. 4 shows the measured and simulated values of the inlet and outlet temperatures of the HCT. In the experiment, an electric hot water boiler was used as the heating source for the BD heating system, and the water supply temperature was set to 60 ± 2 °C. Although the measured inlet temperature fluctuated continuously, the average value was approximately 60 °C. Accordingly, it was reasonable to set the inlet temperature of the coiled tube to 60 °C in the CFD simulation. In addition, the measured outlet temperature also fluctuated with the inlet temperature and the average values agreed well with the simulated values. Reynolds-Averaged Navier-Stokes (RANS) turbulence models have been widely used in general engineering problems and research. RANS turbulence models mainly include the Reynolds stress model (RSM), k-epsilon and k-omega two-equation turbulence model [49]. With respect to the numerical study of the HCT heat exchanger thermal performance, the k-epsilon two-equation turbulence model is more widely used than the other models [50-52]. The simulated values using the k-epsilon turbulence model (Realizable, Standard and RNG k-ε model) were compared with the experimental values (Fig. 4). When the water was heated by a coiled tube, the RMSDs of the three models were 3.1%, 3.4% and 4.0% respectively. When the high-solids FS (TS = 9.5%) is heated, the RMSDs were 5.9%, 6.1% and 6.2% respectively. Accordingly, the Realizable k-ε model is superior to predict the conjugate heat transfer of HCT. The RMSD of the simulation result was 27.4% while the physical parameters of FS were replaced by that of water. Therefore, the accuracy of the numerical model proposed in this study, whose physical parameters were based on the rheological and thermal properties of FS, was significantly higher than the existing models based on the corresponding parameters of water.

Temperature (°C)

Set-point temperature of inlet in CFD Outlet water temperature (CFD) Inlet water temperature (EXP) Outlet water temperature (EXP)

50 45

Water temperature (°C)

75

64 60

70

56 52

65

48

40

Inlet

Outlet

35

60

30

55

25 50 20

Experiment date Realizable k-ε RNG k-ε Standard k-ε

15 10

0

2000

4000

6000

8000

10000

12000

14000

16000

45

Inlet and outlet temperature (°C)

55

40 18000

Time (s)

(a) 40 Temperature (°C)

75 64

Slurry temperature (°C)

60

70

56

65

52

30

Outlet

Inlet 60

25 55 20 50 Experiment date Realizable k-ε Standard k-ε RNG k-ε

15

10

0

2000

4000

6000

8000

10000

12000

14000

16000

45

Inlet and outlet temperature (°C)

Set-point temperature of inlet in CFD Outlet water temperature (CFD) Inlet water temperature (EXP) Outlet water temperature (EXP)

35

40 18000

Time (s)

(b) Fig. 4. Comparison of simulation results with experimental results: (a) water and (b) FS (TS=9.5%). 3.2 Comparison of HTP of the coiled tube in water and high-solids FS Fig. 5 shows the variation of temperature and HF of coiled tube in water and FS with different TS. The temperature rise (TR) of water is significantly higher than that of high-solids FS. Moreover, increasing the TS resulted in a decreasing of TR and HF of the coiled tube. After heating for 5 h, the water temperature increased by approximately 15.7°C, while the slurry temperature with different TS (7.5%, 9.1% and

12.1%) increased by 11.50 °C, 10.67 °C, and 8.17 °C, respectively. These results indicate that the solid organic matter of the slurry inhibit the heat transfer of the HCT. The HF of the tube wall varied greatly between the water and high-solids FS. The HF in water increased rapidly during the initial stage and then decreases gradually, whereas that in the high-solids FS decreases rapidly during the initial stage, then increased slightly and finally decreased slowly until the end. The mean HF of the coiled tube in the water is 3.8 kWm-2, which was approximately double that in the high-solids FS with TS = 12.1%. It can be seen that the HTP of the coiled tube varied greatly between the water and high-solids FS, which is in accordance with previously published experimental results [36]. Consequently, the rheological and thermal parameters of FS should be used to simulate the heat transfer of the coiled tube in high-solids HBD. 35

6000 Temperature Heat flux

Temperature (°C)

5000

25

4000

20

3000

15

2000

10

0

2000

4000

6000

8000

Heat flux (Wm-2)

Water Slurry(7.5%) Slurry(9.1%) Slurry(12.1%)

30

1000 10000 12000 14000 16000 18000

Time (s)

Fig. 5. The variation of temperature and heat flux of coiled tube in water and FS with different TS with time. The effect of TS on the HTP of the coiled tube can be regarded as a comprehensive influence of the rheological and thermal properties of the slurry. The thermophysical properties and mean strain rate of water and FS of different TS (7.5%, 9.1% and 12.1%) were calculated (Fig. 6). Fig 6 (a) shows that as the TS increased, the thermal conductivity and specific heat capacity of the fluid gradually decreased, while the

density and viscosity gradually increased. The thermal diffusivity of the liquid decreases as the TS increases. Specially, at TS of 0.0%, 7.5%, 9.1%, and 12.1% the corresponding thermal diffusivity was 1.48×10-7 m2s-1, 1.40×10-7 m2s-1, 1.36×10-7 m2s-1, and 1.33×10-7 m2s-1, respectively, which is one of the reasons that the temperature rise rate of water was higher than that of high-solids FS. In addition, the apparent viscosity of the high-solids FS is much larger than that of the water, which caused the convection performance of the high-solids FS after heating to be much smaller than that of water. Fig. 6 (b) shows that the average strain rate of water is 0.155 s-1 when heated for 5 hours, while the mean shear rate of the slurry is small with a range of 0.002~0.006 s-1. Thus, increasing the TS of the slurry cause the viscosity increase, which resulted in a decreased fluidity and seriously affected the natural convective heat transfer between the coiled tube and the slurry. 0.7

1200

5000

Thermal conductivity

Specific heat capacity

Density

20

Viscosity

18

0.6

1000

4000

16 14

0.5

600 2000

1000

200

0.1

0 0.

7. 5

0

10 8 6 4

400

0.2

0.0

ρ (kgm-3)

0.3

c (Jkg-1K-1)

λ (Wm-1K-1)

0.4

3000

1 1 0 1 1 0 9. 12. 0. 7.5 9. 12. 0. 7.5

1 1 0 9. 12. 0. 7.5

Total solid concentration (%)

(a)

1 1 9. 12.

0

2

0.004 0.003 0.002 0.001 0.000

μ (Nsm-2)

12

800

0.20

Mean Strain Rate (s-1)

0.15

0.10

0.05

0.006 0.004 0.002 0.000

TS=0.0%

TS=7.5%

TS=9.1%

TS=12.1%

(b) Fig. 6. Thermophysical properties and mean strain rate of the water and FS under different TS: (a) thermophysical properties and (b) mean strain rate. Fig. 7 shows the temperature field of the vertical cross-section in the BDs for water and slurry with different TS (7.5%, 9.1%, 12.1%). Significantly different temperature fields heated with the same HCT were observed for the water and high-solids FS. During the initial 10 min of heating, the temperature stratification was formed in the water, but the coiled tube in the high-solids FS is surrounded by a high-temperature liquid region. Moreover, the shape of the high temperature region varied under different slurry TS. The regions were shaped as a diamond, dumbbell, and circle under a TS of 7.5%, 9.1% and 12.1%, respectively. The main reason for this is that the fluidity of the high-solids FS was poor, which affected the diffusion and mixing of the high temperature slurry around the tube to the surrounding low temperature area. The high temperature region caused the temperature difference between the inside and outside of the tube to decrease, which in turn reduced the heat transfer of the coiled tube. As the slurry flowed and mixed under buoyancy, the area of the high temperature region continuously changes. After heating for 2 h, the area and the temperature of the high temperature region in the slurry decreased. After heating for 5 h, there was a significant vertical temperature gradient in the water and slurry. As the TS of the slurry increased,

the volume fraction of the slurry that met the mesophilic fermentation conditions (FS

Temperature (K)

temperature >25 °C) decreases.

10 min

30 min

2 hour

5 hour

2 hour

5 hour

2 hour

5 hour

2 hour

5 hour

Temperature (K)

(a)

10 min

30 min

Temperature (K)

(b)

10 min

30 min

Temperature (K)

(c)

10 min

30 min

(d) Fig. 7. Temperature contour in BD during heating process: (a) water, (b) FS (TS=7.5%), (c) FS (TS=9.1%), (d) FS (TS=12.1%). Fig. 8 (a) and (b) show the velocity vectors of FS and water in biogas digester, respectively. It can be seen that the natural convection exists in both water and FS under the action of buoyancy. When the FS was heated, the flow of fluid near the tube wall was stronger than other regions. The warmed slurry continuously flowed upward and accumulated in the upper part of the BD to form a high temperature region. The temperature stratification in the biogas digester was obvious. Because of the small

viscosity of water, strong natural convection was occurred near the tube wall. The warmed fluid not only flowed upward, but also circulated in the horizontal direction under the thermal action of the adjacent tube wall, thereby effecting mixing of fluids of different temperatures. Therefore, it can be seen from the temperature contours (Fig. 7) that the temperature distribution in water is relatively uniform, especially in the horizontal direction. In the FS, a high temperature region was generated near the tube wall, which was disadvantageous for the heat transfer between the helical coiled tube and the slurry.

(a)

(b) Fig. 8. Velocity vector of fluid in biogas digester: (a) FS (TS=9.1%) and (b) water.

3.3 Effect of pitch and helical diameter Pitch and helical diameter are two vital structural parameters for the HCT [41]. In this study, the effects of pitch and helical diameter on the HTP of the coiled tube in water and high-solids FS were compared and analysed. In the CFD simulation, the pitches of the coiled tubes were set to 25 mm, 50 mm, 75 mm, 100 mm and 125 mm (tube length: 8.8 m, helical diameter: 50 cm) while the helical diameters of the coiled tubes were set to 25 cm, 40 cm, 50 cm, 75 cm and 95 cm (tube length: 10.3 m, pitch: 50 mm). Table 3 shows the TR and average heat transfer coefficient (AHTC) for the different helical diameters and pitches in the water and slurry with TS of 9.1%. In the water and high-solids FS, the TR and AHTC increased with the increasing pitch and decrease with the increasing helical diameter. Furthermore, the influence of pitch and helical diameter on the HTP of the coiled tube in the high-solids FS was greater than that in the water; in particular, the effect of pitch was much more significant. Although the ratio of pitch to tube diameter (P/dc) increased from 1.56 to 7.81, the AHTC of the coiled tube in the water and high-solids FS (TS=9.1%) increased by 3.5% and 14.9%, respectively. In comparison, the ratio of helical diameter to digester diameter (Dc/Db) increased from 0.25 to 0.95, and the AHTC of the coiled tube in water and high-solids FS (TS=9.1%) decreased by 6.0% and 8.6%, respectively. Table 3 Temperature rise and average heat transfer coefficient under different pitches and helical diameters. Type

P/dc

Dc/Db

Date 1.56 3.13 4.69 6.25 7.81 0.25 0.40 0.50

Water 9.80 10.11 10.45 10.73 11.24 11.95 11.55 11.38

TR θ(°C) Slurry (9.1%) 6.15 6.76 7.08 7.28 7.57 8.35 8.03 7.92

AHTC k (Wm-2K-1) Water Slurry (9.1%) 93.32 48.40 94.41 52.35 95.79 53.96 96.17 55.37 96.72 56.85 97.63 56.32 94.70 54.98 94.79 52.51

0.75 0.95

11.11 10.79

7.73 7.48

90.66 91.75

50.97 51.47

Fig. 9 (a) and (b) show the temperature field in the BD under different HCT pitches in the water and FS. With the increase of pitch, the proportion of high temperature water in the digester gradually increased. The main reason for this was that the height of the coiled tube increased with increasing pitch; thus, the heat-exchange between the coiled tube and liquids of different depths was more sufficient, and the effect of temperature stratification on the HTP of the coiled tube was weakened. Furthermore, the water temperature near the coiled tube was low. As a result, the influence of the heat transfer between the adjacent loops could be ignored. However, the temperature of the fluid near the coiled tube was high in the high-solids FS, which resulted in heat transfer interactions of adjacent coiled tubes. In particular, when the P/dc was relatively small, the thermal interaction between adjacent loops substantially affected the overall HTP

(a)

25 mm 50 mm 75 mm 100 mm 125 mm

Temperature (K)

Temperature (K)

of the coil.

(b)

25 mm 50 mm 75 mm 100 mm 125 mm

Fig. 9. Temperature contour in the BD under different pitches (heating for 5 h): (a) water and (b) FS (TS=9.1%). Fig. 10 (a) and (b) show the temperature field in the BD under different helical diameters of the HCT in water and slurry. Since the length and the pitch of the coiled tube remained constant, the coil number decreased with increasing helical diameter and the height of the coiled tube decreased accordingly. Consequently, as the helical diameter increased, the proportion of the high temperature liquid decreased. In conclusion, when designing the heat exchanger of the HCT in a BD, the P/dc should not be less than 3.13, and the Dc/Db should be appropriately reduced so that the length

(a)

25 cm 40 cm 50 cm 75 cm 95 cm

Temperature (K)

Temperature (K)

of the tube and the height of the digester are sufficient.

25 cm 40 cm 50 cm 75 cm 95 cm

(b)

Fig. 10. Temperature contour in the BD under different helical diameters (heating for 5 h): (a) water and (b) fermentation slurry (TS=9.1%). 3.4 Comparison of HTP of different tube materials in water and high-solids FS

Copper, stainless steel and PEX coiled tubes have been used in studies of heat exchange coiled tubes in BDs. The heat transfer effects of these HCT materials in water and high-solids FS are compared and analysed. Fig. 11 shows the variation of temperature and HF of the coiled tubes in water and high-solids FS. In the water, the TR and HF of the copper coiled tube were the largest, followed by stainless steel. In the high-solids FS (TS=9.1%), the TR and HF of the copper are slightly larger than those of the stainless steel coiled tube. After heating for 5 h, the temperature differences between the metal and PEX coiled tubes in the water and high-solids FS were approximately 20 °C and 10 °C, respectively. 70 60

PEX Stainless steel Copper

Heat flux

50

Temperature (°C)

28000

Water

24000 20000

40

16000

30

12000

20

8000

10

4000

0

0

2000

4000

6000

8000

Heat flux (Wm-2)

Slurry (9.1%) Temperature

0 10000 12000 14000 16000 18000

Time (s)

Fig. 11. The variation of temperature and heat flux of coiled tube in water and FS (TS=9.1%) with different pipe material with time. The thermal resistances of the HCTs of different materials in water and high-solids FS (TS=9.1%) are presented in Fig. 12. The total thermal resistance of coiled tubes is a summation of the convection thermal resistance inside the tube, the conduction thermal resistance of the tube wall and the convection thermal resistance outside the tube. The total thermal resistance of the PEX coiled tube was about 6 times larger than that of the metal tube when the water was heated. Furthermore, the convective thermal resistance outside the tube was the largest among all the thermal resistance values of the metal

coiled tubes, while the wall thermal resistance of the PEX coiled tube was the largest, accounting for 81.2%. When the high-solids FS was heated, the convective thermal resistance outside the tube increased significantly, which resulted in the total thermal resistance of the metal coiled tube increasing by about 3.5 times, while that of the PEX coiled tube increased by only 1.7 times. At the same time, the difference between the thermal resistance of the PEX and the metal coiled tube decreased from 6 times to 2.7 times. Thus, it can be concluded that the heat transfer performance of the metal coiled tube is better than that of PEX tube in water, while the enhanced heat transfer advantage of the metal tube is greatly reduced in high-solids FS. The above analysis provides a basis for the design and selection of an HCT heat exchanger of a BD heating system. For example, during anaerobic fermentation with low FS and high load, metal coiled tubes should be used in order to improve the heat exchange efficiency and reduce the digester volume occupied by heat exchanger. For the rural HBDs, the heat transfer efficiency between the metal and PEX coiled tubes is relatively small due to the high-solids FS. However, the cost of a PEX tube is much lower than that of a metal tube (the approximate price of a coiled tube: copper, 60 RMBm-1; stainless steel, 30 RMBm-1; and PEX, 10 RMBm-1) [53]; therefore, it is suitable to use a PEX HCT heat exchanger. 0.30 Outside the coil

Thermal resistance (mKW-1)

0.25 Coil

Water t1 ti ti Rii Rw

0.20

Coil wall

Inside the coil

Water t2

to

1.4%

Rt=0.225

Ri=0.003

Ro Rt=Ri+Rw+Ro

0.15 81.2% Rw=0.183

0.10

0.05

8.3% 7.1% 84.6%

0.00

Rt=0.039

Ri=0.003 9.2% Rt=0.035 Ri=0.003 Rw=0.003 0.3% Rw=1×10-4 Ro=0.033 90.5% Ro=0.032 17.4%

Stainless steel

Copper

(a)

Ro=0.039

PEX

0.50 Coil wall

Outside the coil

Inside the coil

Thermal resistance (mKW-1)

0.45 0.40 Coil

Water t1 ti ti Rii Rw

0.35

Slurry t2

to

0.9%

Rt=0.372

Ro Rt=Ri+Rw+Ro

0.30

Ri=0.003

49.2%

Rw=0.183

49.9%

Ro=0.186

0.25 0.20 0.15

2.4% Rt=0.138 Ri=0.003 2.5% Rt=0.133 Ri=0.003 2.0% Rw=0.003 0.1% Rw=1×10-4

0.10 0.05 0.00

95.6%

Ro=0.132

Stainless steel

Ro=0.130

97.4%

Copper

PEX

(b) Fig. 12. Thermal resistances of helical coiled tubes: (a) water and (b) FS (TS=9.1%). 4. Conclusions A CFD model for the conjugate heat transfer of a HCT in a high-solids HBD was developed based on the rheological and thermal properties of FS. The model was subsequently validated by experimental results. The following conclusions were drawn from the results: 1) Compared with the existing models that use the physical parameters of water, the accuracy of the numerical model proposed in this study whose physical parameters were based on the rheological and thermal properties of high-solids FS is significantly improved. 2) The ZSV of slurry varied more greatly with the TS compared to density, specific heat capacity and thermal conductivity. As the TS increased, the ZSV of the FS significantly increased while the HTP of the HCT was notably weakened. 3) The AHTC of the coiled tube increased with increasing pitch and decreased with increasing of helical diameter. In the high-solids FS, the influence of pitch on the AHTC was greater than that of helical diameter, and the AHTC decreased by 14.9% as the P/dc was decreased from 7.81 to 1.56.

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Highlights: 

CFD models were developed for helical coiled tube in high-solids slurry.



The models were validated by field experimental data.



The thermal performance of coils in water and high-solids slurry was compared.



The effect of pitch, diameter, and material of helical coil pipe were analysed.

Conflict of Interest

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