Open AccessArticle

Estimating Sludge Deposition on the Heat Exchanger in the Digester of a Biogas Plant

Tomasz Janusz Teleszewski

^1,*

and

Leszek Hożejowski

Department of HVAC Engineering, Bialystok University of Technology, 15-351 Bialystok, Poland

Faculty of Management and Computer Modelling, Kielce University of Technology, al.Tysiąclecia P.P. 7, 25-314 Kielce, Poland

Author to whom correspondence should be addressed.

Sustainability 2024, 16(18), 7981; https://doi.org/10.3390/su16187981

Submission received: 27 June 2024 / Revised: 4 September 2024 / Accepted: 10 September 2024 / Published: 12 September 2024

(This article belongs to the Special Issue Sustainable and Renewable Thermal Energy Systems)

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Browse Figures

Figure 1
General view of the biogas plant fermentation chamber. "> Figure 2
Diagram of the digester with a photo of a fragment of the heat exchanger before and after cleaning the heat exchanger walls. "> Figure 3
Boxplot of the measured sludge thickness on the outer walls of the heat exchanger. "> Figure 4
Diagram of the heat exchanger adopted for the model. "> Figure 5
Calculated thickness of the sludge layer on the heat exchanger walls in the winter months of 2017–2020 with added start and end points. "> Figure 6
Dependence of the overall heat transfer coefficient on the thickness of the deposits on the walls of the heat exchanger (based on the 2D model). "> Figure 7
Dependence of the thermal power of a single heat exchanger loop and the temperature at the outlet of the heat exchanger as a function of the sludge thickness. "> Figure 8
Schematic diagram of sludge monitoring on the heat exchanger in a digester: 1—flow meter, 2—substrate temperature sensor inside the digester, 3—heating medium temperature sensor on the supply pipe of the heat exchanger, 4—heating medium temperature sensor on the return from the heat exchanger, 5—heat exchanger operation monitor, 6—supply pipe, 7—return pipe, 8—fermentation chamber wall. ">

Versions Notes

Abstract

The presented research addresses a problem occurring in a biogas plant, which we know plays an important role in sustainable development. The sludge deposited on the walls of the digester’s heat exchanger impairs heat transfer to the substrate. It leads to a temperature drop inside the biogas plant and threatens its correct operation. The thickness of the sludge layer cannot be directly measured when the plant is operating. Therefore, the aim of this work was to develop and then validate a method for estimating, based on the operating parameters of the exchanger, the thickness of the sludge layer and to give theoretical foundations for designing an automatic sludge monitoring system. Two mathematical models (and methods) were developed: one- and two-dimensional. The former model was solved analytically while the latter by the Trefftz method. The numerical results from these two approaches showed very good agreement with each other and with the actual measurement taken directly after removing the substrate from the fermentation chamber. According to the calculation results, the growth of the sludge layer was linear with time, and its rate was 0.0064 mm per day. Finally, a schematic diagram of an intended sludge monitoring system was proposed. It could optimize biogas plant operation and thus become a step towards more sustainable energy production.

Keywords:

biogas plant; heat exchanger; fermentation chamber; Trefftz method; sludge deposition

1. Introduction

The sustainable development of both rural and urban areas is increasingly related to the production of electricity and heat in biogas plants [1,2,3,4,5]. Biogas plants also constitute an important element of the circular bioeconomy [6] and can be used to stabilize the energy system [7]. A typical agricultural biogas plant in Poland, most often used for production of electricity and heat, consists of the following elements: preparation tank, anaerobic digester, biogas storage, and cogeneration unit [8]. The basis for the proper operation of the fermentation tank is a constant temperature inside the chamber, which is usually around 40 °C [9]. A drop or an excessive increase in temperature in the digester causes a reduction in methane production, which may also affect the stability of the energy system [10].

The most common problems with maintaining the right temperature may occur during winter in temperate climates [11]. The greatest heat losses are generated by the pneumatic roof of the gas chambers and amount to approximately 90% of all heat losses [12]. Insulating a pneumatic roof is problematic. In order to ensure the required temperature inside the digester chamber of the biogas plant, special attention should be paid to the operation of the heat exchanger used to heat the substrate.

The most commonly used heat exchangers in fermentation chambers are pipe-shaped heat exchangers, which are mounted on the walls of the fermentation chambers using clamps. In the process of designing a biogas plant, the size of the heat exchanger is selected to match the size of the fermentation chamber by determining the appropriate surface area of the heat exchanger walls and determining the heat exchanger’s supply temperature. It should be noted here that the supply temperature of the heat exchanger cannot be too high because high temperatures may lead to the death of methane-producing microorganisms. A serious problem in the operation of the heat exchanger is the sludge deposited on its outer walls, which causes additional thermal resistance to heat exchange [12]. In the literature on the subject, we have not found works reporting such problems concerning digesters. There are many publications devoted to problems with biogas plants connected with temperature, not fouling [13,14]. Additionally, unlike in the present paper, researchers have focus their attention mostly on the deposits inside a heat exchanger, e.g., [15,16,17,18,19,20,21]. One paper [22] stands out from these publications as it addresses the problem of the sediment occurring both inside and outside the digester. However, the study assumed a constant (time-independent) thickness of the outer sludge layer and determined its growth inside the exchanger, so comparison to the present research would be problematic.

When the thickness of the sludge deposit on the walls of the heat exchanger is so large that further heating of the substrate will be significantly hampered by the thermal resistance of the sludge, the heat exchanger is cleaned. Cleaning the heat exchanger involves stopping the operation of the biogas plant and replacing the entire substrate inside the fermentation chamber.

Due to the fact that, in a working digester, it is not possible to measure the thickness of the sludge on the walls of the heat exchanger, a reasonable solution is a mathematical calculation based on the operating conditions of the heat exchanger. In terms of applied mathematics, such a problem is referred to as an inverse geometric problem.

To solve the aforementioned problem, the presented research aimed to propose and then validate with actual data a computational method for estimating the thickness of the sludge deposit on the outer walls of the heat exchanger inside a fermentation chamber during the fermentation process. The method is intended for the design of an automatic control system for monitoring the growth of the sludge layer. The installation of such a system is being considered in the future. It would certainly help to optimize the production of biogas and, to a certain extent, make the process of production less energy-consuming, thus contributing to the sustainable development of biogas plants.

The following sections outline the remaining content of this paper. Section 2 describes in detail the fermentation chamber in the biogas plant. Then, two mathematical models for the considered problem are built, each with the appropriate solution method for determining the sludge layer thickness. Additionally, the overall heat transfer coefficient is determined. Section 3 presents the numerical results and their discussion. The last section contains the conclusions of this work.

2. Materials and Methods

2.1. Subject of This Research

The subject of this research was the heat exchanger of the fermentation chamber (Figure 1 and Figure 2) of a 1 MW biogas plant located in north-eastern Poland, which has a temperate climate, where winters can be both frosty and mild, and summers can be hot or rainy. The tested biogas plant normally operated continuously for a three-year period, from the moment the biogas plant was turned on until it was turned off, in order to clean the fermentation chamber and the heat exchanger. Fermentation chambers were filled with substrate consisting of corn silage, poultry manure, and potato pulp. Due to problems with maintaining the proper temperature in the tested digesters during the winter, modernization works were carried out in August 2017 by adding additional heat exchanger loops inside the digester.

The length of a single loop of the tubular heat exchanger was 90 m, with an internal diameter d₁ and external diameter d₂ equal to 56.3 mm and 60.3 mm, respectively. The heat exchanger pipe was made of austenitic corrosion resistant steel with a thermal conductivity k_p of 15 W/(m K). The average values of the thermal conductivity k_w, the dynamic viscosity coefficient μ_w, and the specific heat capacity of the heating medium (water) c_pw in the heat exchanger were assumed to be 0.64 W/(m K), 0.000509 kg/(m s), and 4180 J/(kg K), respectively. Based on the literature [23,24,25,26], the following substrate and sludge parameters were assumed: substrate thermal conductivity k_s equal to 0.62 W/(m K), substrate specific heat c_ps equal to 4184 J/(kg K), substrate velocity v_s equal to 0.005 m/s, substrate dynamic viscosity coefficient μ_s equal to 0.03 kg/(m s), and thermal conductivity of the sludge k_sl equal to 0.8 W/(m K). Due to the use of demineralized water in the heating installation, it was assumed that there was no fouling layer inside the heat exchanger.

The heat exchanger’s operating parameters were measured using a Kamstrup heat meter with the following accuracies: temperature (PT500 temperature sensors manufactured by Kamstrup (Skanderborg, Denmark)) ± 0.3 °C, flow (Ultraflow flowmeter manufactured by Kamstrup) with a maximum error of 5% of the measured value. The average flow rate of the heating medium in winter through the heat exchanger loop was 0.29 kg/s.

Figure 2 shows a diagram of the fermentation chamber, photographs of a fragment of the heat exchanger with sludge after a three-year period of operation, and the same fragment after cleaning.

To assess the thickness of the sludge on the outer wall of the heat exchanger, measurements were made at 30 points, using an Insize 1141 depth gauge. Figure 3 shows a boxplot for the measurement data. Based on this, the average value of the deposits layer on the outer walls of the heat exchanger was 7.1 mm (August 2020). It should be noted that the measurement was made after removing the substrate from the fermentation chamber because measuring the sludge thickness during the operation of the biogas plant was not possible.

2.2. Model of Heat Transfer in the Heat Exchanger

The mathematical model adopted to describe thermal processes in the considered heat exchanger had the following assumptions: steady-state conditions, fully developed flow, incompressible liquid (substrate), and negligible viscous dissipation. Heat exchange between the fermentation chamber and its surroundings was not taken into account.

The heating pipe that runs along the walls of the fermentation chamber creates loops so large in diameter that the pipe can be thought of as straight. The geometric model of the considered heat exchanger was therefore a long hollow cylinder with the outer shell being a layer of sludge, as shown in Figure 4.

2.2.1. Determining the Sludge Layer Thickness: One-Dimensional Approach

In the one-dimensional approach, it was assumed that the temperature varies only in the radial direction. In this case, the following heat equation was adopted:

\frac{d}{d r} (r \frac{d T}{d r}) = 0,

(1)

and it is satisfied by both the steel pipe temperature (T_p) and the sludge temperature (T_sl).

The following boundary conditions were assumed for Equation (1), corresponding to the physics of the problem:

T_{p} = T_{w} at r = r_{1},

(2)

- k_{p} \frac{d T_{p}}{d r} = q at r = r_{1},

(3)

T_{p} = T_{s l} at r = r_{2},

(4)

- k_{p} \frac{d T_{p}}{d r} = - k_{s l} \frac{d T_{s l}}{d r} at r = r_{2},

(5)

where T_w is water temperature, q is heat flux density, and k_p and k_sl are the thermal conductivity of the steel pipe and the sludge, respectively.

It should be noted that strict compliance with the model assumptions requires that the water temperature must be constant. The temperature actually drops as water flows through the pipe. Therefore, to ensure a constant T_w in boundary condition (2), the arithmetic mean of the temperatures at the inlet (T_w,i) and outlet (T_w,o) of the heat exchanger was adopted.

The total heat flux Q is given by the following equation:

Q = {m c}_{p w} Δ T = m c_{p w} (T_{w, o} - T_{w, i}),

(6)

where m and c_pw are the mass flow and specific heat of the heating medium, respectively. For simplicity, it was assumed that the heat flux density q is constant and is described by the following relationship:

q = \frac{Q}{A},

(7)

where A denotes the inner surface of the pipe.

Analytically integrating the governing Equation (1), the following solutions were obtained:

T_{p} = C_{1} \ln r + C_{2},

(8)

T_{s l} = D_{1} \ln r + D_{2},

(9)

The constants C₁, C₂, D_1, and D₂ are specified using boundary conditions (2)–(5).

After determining the temperature T_sl, the sludge thickness s can be determined on the basis of the convective boundary condition at the interface between the sludge layer and the substrate at temperature T_a:

- k_{s l} \frac{d T_{s l}}{d r} = h_{0} (T_{s l} - T_{a}) at r = r_{2} + s,

(10)

The external convective heat transfer coefficient h₀ present in Equation (10) can be determined using the Churchill–Bernstein correlation [27]:

h_{0} = \frac{N u k_{s}}{d_{3}} at d_{3} = 2 r_{2} + 2 s,

(11)

N u = 0.3 + \frac{0.62 {Re}^{1 / 2} \Pr^{1 / 3}}{{[1 + {(0.4 / \Pr)}^{2 / 3}]}^{1 / 4}} {[1 + {(\frac{Re}{282000})}^{5 / 8}]}^{4 / 5}, Re \Pr \geq 0.2,

(12)

h_{0} = \frac{N u k_{s}}{d_{3}} at d_{3} = 2 r_{2} + 2 s,

(13)

Re = \frac{ρ_{s} v_{s} d_{3}}{μ_{s}},

(14)

\Pr = \frac{c_{p s} μ_{s}}{k_{s}},

(15)

where Re denotes the Reynolds number, and Nu is the Nusselt number.

Combining (9) and (10) gives

- k_{s l} \frac{D_{1}}{r_{2} + s} = h_{0} (D_{1} \ln (r_{2} + s) - T_{a})

(16)

Equation (16) should be solved for s, and it can be solved only numerically. The preliminary results concerning the solutions to Equation (16) were presented in [28].

2.2.2. Determining the Sludge Layer Thickness: Two-Dimensional Approach

In the second approach, reference was made to the previous mathematical model in terms of geometric assumptions (Figure 4) but without neglecting temperature changes in the axial direction (z). Assuming an exponential water temperature profile, the following formula for T_w can be written:

T_{w} (z) = T_{w, i} {(\frac{T_{w, o}}{T_{w, i}})}^{z / L}

(17)

Now, considering a small pipe element with length ∆z, it was assumed that Equations (6) and (7) refer locally to this tubular element, with

Δ T = T_{w} (z + Δ z) - T_{w} (z)

and

A = 2 π r_{1} Δ z

. In this case, Equation (7) gives the average q over the length ∆z of the pipe element. Taking the limit ∆z → 0, the heat flux at point z was obtained:

q (z) = \lim_{Δ z \to 0} \frac{m c Δ T}{2 π r_{1} Δ z} = \frac{m c T_{w, i} {(\frac{T_{w, o}}{T_{w, i}})}^{z / L} \ln (\frac{T_{w, o}}{T_{w, i}})}{2 π r_{1} L}

(18)

The proposed two-dimensional mathematical model of heat transfer is described by

\frac{\partial^{2} T}{\partial r^{2}} + \frac{1}{r} \frac{\partial T}{\partial r} + \frac{\partial^{2} T}{\partial z^{2}} = 0

(19)

which is satisfied by both T_p and T_sl so that T has two meanings, depending on r. The boundary conditions for T_p and T_sl are as follows:

T_{p} = T_{w} at r = r_{1}, 0 < z < L,

(20)

- k_{p} \frac{\partial T_{p}}{\partial r} = q at r = r_{1}, 0 < z < L,

(21)

T_{p} = T_{s l} at r = r_{2}, 0 < z < L,

(22)

- k_{p} \frac{\partial T_{p}}{\partial r} = - k_{s l} \frac{\partial T_{s l}}{\partial r} at r = r_{2}, 0 < z < L,

(23)

Similar to the first approach, the sludge thickness is determined on the basis of convective boundary condition (10), with the difference being that temperature T_sl now also depends on z.

It was decided to choose the Trefftz method for solving (19)–(23). The reason for this choice was in the numerous advantages of this method as well as its proven effectiveness on a wide range of heat transfer problems [29,30,31,32]. The Trefftz method, named after the German mathematician Erich Trefftz, applies to linear partial differential equations. It allows for approximating solutions through certain basis functions that satisfy the governing equation of a considered problem but at the cost of approximate fulfillment of the boundary conditions.

Two types of basis functions are used for the governing Equation (19) [31], V_n⁽¹⁾(r, z), and V_n⁽²⁾(r, z), given by the following formulas:

V_{n}^{(1)} (r, z) = \sum_{k = 0}^{[n / 2]} \frac{{(- 1)}^{k}}{4^{k} {(k!)}^{2} (n - 2 k)!} \cdot r^{2 k} \cdot z^{n - 2 k}

(24)

and

\{\begin{array}{l} V_{0}^{(2)} (r, z) = \ln r \\ V_{1}^{(2)} (r, z) = z \cdot \ln r \\ V_{n}^{(2)} (r, z) = \frac{2 n - 1}{n^{2}} \cdot z \cdot V_{n - 1}^{(2)} (r, z) - \frac{r^{2} + z^{2}}{n^{2}} \cdot V_{n - 2}^{(2)} (r, z) + \frac{2 r}{n^{2}} \cdot \frac{\partial V_{n}^{(1)} (r, z)}{\partial r} \end{array}

(25)

In order to find an approximate solution to problem (19)–(23), temperature T_p was expressed as a linear combination of the basis functions as follows:

T_{p} \approx {\tilde{T}}_{p} = \sum_{n = 0}^{N} C_{n} V_{n}^{(1)} + \sum_{n = 0}^{N} D_{n} V_{n}^{(2)}

(26)

The coefficients C_n and D_n were determined by best matching the solution (26) to the assumed conditions (20) and (21) at the boundaries. “Matching” should be understood in the variational sense as minimization of the error functional. The functional to be minimized has the form

Φ ({\tilde{T}}_{p}) = Φ (C_{1}, \dots, C_{N}, D_{1}, \dots, D_{N}) = {‖{\tilde{T}}_{p} (r_{1}, z) - T_{w} (z)‖}^{2} + {‖q (z) - k_{p} \frac{\partial {\tilde{T}}_{p} (r_{1}, z)}{\partial r}‖}^{2}

(27)

After approximating T_p, temperature T_sl was approximated in the same way to best meet boundary conditions (22) and (23).

In the two-dimensional model, condition (10) should be rewritten using partial differentiation:

- k_{s l} \frac{\partial T_{s l}}{\partial r} (r_{2} + s, z) = h_{0} (T_{s l} (r_{2} + s, z) - T_{a})

(28)

Equation (28) needs to be solved with respect to s, assuming z is fixed. The solutions of Equation (28) for different values of z represent the sludge thickness varying throughout the exchanger pipe. In the last step, the obtained s should be averaged to result in a single value, as we can expect from the actual measurement.

2.2.3. Overall Heat Transfer Coefficient

After determining the thickness of the sludge layer, the overall heat transfer coefficient can be determined:

U = {(\frac{1}{h_{i}} + \frac{d_{1}}{2 k_{p}} \ln \frac{d_{2}}{d_{1}} + \frac{d_{1}}{2 k_{s l}} \ln \frac{d_{3}}{d_{2}} + \frac{d_{1}}{d_{3}} \frac{1}{h_{0}})}^{- 1},

(29)

The internal convective heat transfer coefficient h_i is determined from the Dittus–Boelter correlation [27]:

h_{i} = \frac{N u_{d 1} k_{w}}{d_{1}},

(30)

N u_{d_{1}} = 0.023 {Re}_{d_{1}}^{4 / 5} \Pr^{0.3}, {Re}_{d_{1}} \geq 10000, 0.6 \leq \Pr \leq 16000,

(31)

\Pr = \frac{c_{p w} μ_{w}}{k_{w}},

(32)

{Re}_{d_{1}} = \frac{4 m}{π d_{1} μ_{w}},

(33)

Obviously, the formula for U applies to both the one-and two-dimensional approaches.

3. Results and Discussion

Significant heat losses through the walls of the fermentation chamber and especially through the pneumatic roof [11] occur in the winter months. Therefore, the heat exchanger in the fermentation chamber operates at maximum thermal power during this period. This is the reason why the calculations and validation of the proposed mathematical models are based on data from the winter months, namely January, February, and December in 2017–2020.

Table 1 shows the results of the sludge thickness estimation based on the measured average monthly temperatures inside the fermentation chamber and average monthly operating temperatures of the heat exchanger. During the period under study, the circulation pump worked at the maximum constant rotational speed of the pump impeller, and the average flow of water in a single loop of the heat exchanger was 0.29 kg/s.

As can be seen in Table 1, both mathematical models give similar results for the sludge layer thickness, with absolute differences of 0.17 mm at most.

Figure 5 shows the calculated thickness of the sludge on the walls of the digester heat exchanger in the months of January, February, and December in 2017–2020. Also, it contains the point of starting the digester after thoroughly cleaning the heat exchanger (August 2017) and the point of turning off the digester and cleaning the heat exchanger (August 2020). It must be noted that the sludge thickness at the initial time (August 2017) was assumed to be zero (s = 0), while at the point of shutting down the digester for cleaning (August 2020), the measured thickness was 7.1 mm on average. The thickness of 7.1 mm compared to 12 mm reported in the previous study [28] is 40.8% thinner. Two reasons may explain this difference. Firstly, in 2017, two additional loops were added to the heat exchanger in order to lower the supply temperature in the heat exchanger, and, as shown in [28,33], a lower temperature translates into thinner layer of sludge deposit. Secondly, the present study referred to the operation time of the biogas plant, which was shorter than that before 2017.

Based on the calculations performed, one can observe an almost linear increase in the sludge thickness on the heat exchanger walls. This conclusion follows not from an observation alone but primarily from the coefficient of determination R² being greater than 0.96 (a very good fit of the computed values to the regression line). The linear function describing this relationship was obtained by the least squares method, and it reads

s = 0.0064 t,

(34)

where t denotes time measured in days.

The coefficient of t in (34) was estimated as 0.0064, which means that each increase in time by one day caused an increase in the sludge layer thickness by 0.0064 mm. Furthermore, Formula (34) enables the prediction of the sludge thickness. The value of s is equal to 7.03 mm for t = 1096 days, which agrees with the measurement (see Figure 3).

Figure 6 shows the results of calculating the overall heat transfer coefficient of the heat exchanger in the digester for various sludge thicknesses. The calculations assumed a constant temperature of 39.5 °C inside the fermentation chamber and a supply temperature of 55 °C. As the thickness of the sludge on the piping walls increases, the overall heat transfer coefficient decreases. The value of the overall heat transfer coefficient is approximately two times lower for a sludge thickness of 7.1 mm than for the wall without sludge. The relationship between the overall heat transfer coefficient and the thickness of the deposit on the heat exchanger walls can be well approximated with a third-degree polynomial, which has the following form for the 2D model:

U = - 0.0641 s^{3} + 1.3197 s^{2} - 13.321 s + 115.84

(35)

The value of the overall heat transfer coefficient for the case without sludge deposit was 115.93 W/(m² K); after three years of using the digester, it decreased to 67.8 W/(m² K).

A paper [22] refers to a digester located in Italy, and it presents the results of calculating the overall heat transfer coefficient as a function of the fouling thickness inside the pipe, assuming a constant thermal resistance of the fouling outside the pipe of 0.002 m² K/W. As shown by the authors of [22], the general trend in the overall heat transfer coefficient decreases with an increasing amount of fouling, obviously caused by the additional thermal resistance of the deposit. Furthermore, the growth in the outer fouling alone resulted in a drop in the overall heat transfer coefficient by 27%. To make a comparison to the present study, the same percentage decrease in the overall heat transfer coefficient would be caused by a 3.2 mm sludge layer on the outer surface of the pipe.

Figure 7 shows the thermal power of a single heat exchanger loop and the temperature at the heat exchanger outlet versus the thickness of the sludge on the outer walls. As the thickness of the sludge on the heat exchanger walls increases, the temperature at the exchanger’s outlet increases, while the heating power decreases, which is consistent with the trends in the thermal power and outlet temperature shown in [12]. Monitoring the sludge layer on heat exchanger walls would help maintain a relatively stable temperature inside the digester; this should be desirable because too low or too high temperature in the digester causes a decrease in methane production [10,34,35]. Additionally, fluctuating changes in temperature might kill the methane-producing microorganisms [35].

Currently, the intensive development of automatic control systems equipped with microprocessor controllers has allowed the use of various algorithms controlling the operation of heating devices and heat transport pipelines. An exemplary algorithm for controlling deposits inside heat exchangers in a city heating network is presented in [18]. The algorithm presented in this work can also be used to constantly monitor the level of sludge in the heat exchanger of a biogas plant. Figure 8 shows the proposed scheme for controlling the sludge layer thickness on the heat exchanger walls. The main elements of the sludge monitoring system are a flow meter (1), sensors for measuring the substrate temperature (2), the supply temperature (3), and the return temperature (4) from the heat exchanger piping. The parameters of the substrate, sludge, and water flowing through the pipe should also be entered into the controller (5).

4. Conclusions

The presented research concerns the problem of identifying the thickness of the sludge layer accumulated on the outer wall of a heat exchanger—a problem rarely addressed in the literature. It is worth emphasizing that the results obtained and shown in this paper are based on the operating parameters of an actually existing biogas plant, not on simulations alone.

From a computational point of view, the Trefftz method demonstrated its potential applicability to inverse geometric problems. But, more importantly, from an engineer’s perspective, each of the approaches to the considered problem (one- and two-dimensional) proved to be effective in estimating the sludge layer thickness. The results provided by them are consistent. Since the calculations yielded a mathematical (linear) dependence between the heating time and the sludge layer thickness, it can be used to predict, in quantitative terms, the growth in the sludge deposited on the walls of a heat exchanger. This in turn provides a biogas plant operator with the information necessary to make decisions about cleaning the digester.

Monitoring the thickness of the sludge deposited on the walls of the heat exchanger in a fermentation chamber is particularly important in the winter when there is a risk of underheating of the fermentation chamber, which reduces biogas production. The findings of this study allow for developing a framework for a monitoring system. A schematic diagram of such a system is included in this work.

Controlling the degree of contamination of the heat exchanger surface is one of the tools for planning the stable operation of the biogas plant as a renewable energy source, which is an integral element of sustainable development.

Author Contributions

Conceptualization, L.H. and T.J.T.; methodology, L.H.; software, L.H.; validation, L.H. and T.J.T.; formal analysis, L.H. and T.J.T.; investigation, L.H. and T.J.T.; resources, L.H. and T.J.T.; data curation, L.H. and T.J.T.; writing—original draft preparation, L.H. and T.J.T.; writing—review and editing, L.H.; visualization, L.H. and T.J.T.; supervision, L.H. and T.J.T.; project administration, T.J.T.; funding acquisition, T.J.T. All authors have read and agreed to the published version of the manuscript.

Funding

This study was executed with resources of the statutory work financed by the Ministry of Science and Higher Education in Poland (WZ/WB-IIŚ/8/2023).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

Nomenclature

A	inner surface of the pipe
c_ps	specific heat capacity of the substrate (J/(kg K))
c_pw	specific heat capacity of the heating medium (water) (J/kg/K)
C_n	coefficient (real number)
d	diameter (m)
D_n	coefficient (real number)
h_i	internal convective heat transfer coefficient (W/(m² K)
h_o	external convective heat transfer coefficient (W/(m² K)
k_p	thermal conductivity of the steel pipe (W/(m K))
k_sl	thermal conductivity of the sludge (W/(m K))
k_s	thermal conductivity of the substrate (W/(m K))
k_w	thermal conductivity of the heating medium (water) (W/(m K))
L	length of heat exchanger in a single loop (m)
m	mass flow (kg/s)
n	index (nonnegative integer)
N	number of basis functions
Nu	Nusselt number
Pr	Prandtl number
q	heat flux density (W/m²)
Q	total heat flux (W)
r	radius, radial coordinate (m)
Re	Reynolds number
R²	coefficient of determination
s	sludge thickness (m or mm)
t	time (day)
T_a	substrate temperature (°C)
T_p	pipe temperature (°C)
T_sl	sludge temperature (°C)
T_w	water temperature (°C)
T_w,i	average temperature at the inlet to the heat exchanger (°C)
T_w,o	average temperature at the outlet from the heat exchanger (°C)
U	overall heat transfer coefficient (W/(m² K))
V_n⁽¹⁾	1st kind basis function
V_n⁽²⁾	2nd kind basis function
v_s	substrate velocity (m/s)
z	axial coordinate (m)
Greek symbols
μ_s	dynamic viscosity coefficient of the substrate (kg/(m s))
μ_w	dynamic viscosity coefficient of the heating medium (water) (kg/(m s))
ρ_s	substrate density (kg/m³)
Φ	functional

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Figure 1. General view of the biogas plant fermentation chamber.

Figure 2. Diagram of the digester with a photo of a fragment of the heat exchanger before and after cleaning the heat exchanger walls.

Figure 3. Boxplot of the measured sludge thickness on the outer walls of the heat exchanger.

Figure 4. Diagram of the heat exchanger adopted for the model.

Figure 5. Calculated thickness of the sludge layer on the heat exchanger walls in the winter months of 2017–2020 with added start and end points.

Figure 6. Dependence of the overall heat transfer coefficient on the thickness of the deposits on the walls of the heat exchanger (based on the 2D model).

Figure 7. Dependence of the thermal power of a single heat exchanger loop and the temperature at the outlet of the heat exchanger as a function of the sludge thickness.

Figure 8. Schematic diagram of sludge monitoring on the heat exchanger in a digester: 1—flow meter, 2—substrate temperature sensor inside the digester, 3—heating medium temperature sensor on the supply pipe of the heat exchanger, 4—heating medium temperature sensor on the return from the heat exchanger, 5—heat exchanger operation monitor, 6—supply pipe, 7—return pipe, 8—fermentation chamber wall.

Table 1. Heat exchanger operating parameters and results of calculations of the thickness of sludge on the heat exchanger walls in the winter months of 2017–2020.

Date	T_a	T_w,i	T_w,o	S (1D Model)	S (2D Model)
-	°C	°C	°C	mm	mm
December 2017	40.7	55.0	44.1	0.00	0.00
January 2018	41.9	55.2	45.8	0.30	0.14
February 2018	39.1	55.6	44.2	0.54	0.38
December 2018	42.0	55.8	48.0	3.54	3.37
January 2019	41.3	59.1	49.0	3.44	3.27
February 2019	42.0	58.0	49.0	3.60	3.44
December 2019	42.2	56.9	49.2	4.89	4.73
January 2020	42.0	58.1	50.0	5.67	5.50
February 2020	41.5	58.0	50.0	6.43	6.26

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Teleszewski, T.J.; Hożejowski, L. Estimating Sludge Deposition on the Heat Exchanger in the Digester of a Biogas Plant. Sustainability 2024, 16, 7981. https://doi.org/10.3390/su16187981

AMA Style

Teleszewski TJ, Hożejowski L. Estimating Sludge Deposition on the Heat Exchanger in the Digester of a Biogas Plant. Sustainability. 2024; 16(18):7981. https://doi.org/10.3390/su16187981

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Teleszewski, Tomasz Janusz, and Leszek Hożejowski. 2024. "Estimating Sludge Deposition on the Heat Exchanger in the Digester of a Biogas Plant" Sustainability 16, no. 18: 7981. https://doi.org/10.3390/su16187981

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