Determining the thickness of sludge on the heat exchanger tube inside an anaerobic digester

The paper presents a simplified method of determining the thickness of sludge on the walls of the heat exchanger piping in a biogas plant digester. The evaluation of the thickness of a sludge layer is based on the biogas plant operation parameters, including the inlet and outlet temperature of the heat exchanger, mass flow and the geometric characteristics of the heat exchanger and physical parameters of the substrate working inside a fermentation chamber. Measurement of the thickness of the sludge layer on the walls of a heat exchanger is only possible at the time of general cleaning of a digester, which necessitates switching off the biogas plant operation. The paper compares the results of predictions with experimental data of the work of a biogas plant digester located in north-eastern Poland, in Ryboły. The presentation of the obtained numerical results is supplemented by the uncertainty analysis. The significance of undertaking such research lies in its applicational aspects, as during the operation of a biogas plant sludge accumulates on the walls of a digester, which provides additional thermal resistance and reduces the thermal efficiency of the heat exchanger.


Introduction
Biogas plants are among the most cost-effective alternative energy sources [1,2]. Biogas is produced from organic substances that are degraded by microorganisms in the methane fermentation process [3]. The products of the process are methane and carbon dioxide, as well as small amounts of hydrogen sulphide, nitrogen and hydrogen. The growing interest in biogas production means that more and more substrates, both waste and deliberately produced, coming from industry, agriculture or urban areas are used in its production. The most frequently used raw materials include: waste from food production, liquid or solid animal waste (manure, slurry), food leftovers, organic municipal waste, post-slaughter waste, waste from crop production, energy crops (maize, alfalfa) and biomass forest [4][5][6][7]. It should be noted that some organic waste, e.g. medical waste, due to the risk of hygienic contamination and the possibility of pathogens, cannot be used as a substrate in fermentation chambers [8,9]. The residues from the fermentation process can be used as a fertiliser provided they are free from pathogens [10,11]. The work of a biogas plant also involves many problems related to heat exchange. The main problem is maintaining a constant temperature in the fermentation chambers, especially during the winter. The main causes of the problem related to maintaining a constant temperature inside the fermentation chamber are too thin a layer of thermal insulation of the digester, and inefficient operation of the heat exchanger. The issue of thermal insulation of building partitions has been described in the literature [12][13][14], while the work of the heat exchanger is presented in the publication [15]. The work of the heat exchanger is mainly influenced by the sludge that appears on the walls of the heat exchanger tubing during the operation of the biogas plant. In the work [15], a simplified model of the simulation of thermal efficiency of the heat exchanger is described, depending on the thickness of the sludge layer on the exchanger walls. The increase in the thickness of the sludge layer on the walls of the exchanger is an additional thermal resistance and causes a decrease in the thermal efficiency of the heat exchanger. An increase in the efficiency of the exchanger can be achieved by increasing the flow of the heating medium through the exchanger, increasing the heat exchange surface by adding additional loops in the heat exchanger tubing or raising the flow temperature. The first method is limited by the circulation pump operating parameters. The second method involves additional costs of heat exchanger expansion, while the third method can lead to the death of microorganisms in the fermentation chamber due to the temperature increase within the heat exchanger if the temperature of the heat exchanger surface is too high. Examining the thickness of the insulation on the walls of the exchanger is extremely difficult, because it requires switching off the biogas plant operation, emptying the fermentation chamber from the substrate, and then measuring the thickness of the sludge layer on the walls of the exchanger tubing. Due to the inability to monitor the thickness of the sludge on the walls of the heat exchanger, it seems appropriate to create a model and procedure to determine the heat exchanger layer based on measurement data such as the thermal efficiency of the heat exchanger in the digester. The heat exchanger research was performed based on the example of a biogas plant located in the village of Ryboły in the north-eastern part of Poland. The photograph in Fig. 1 shows a general view of fermentation chambers in the biogas plant in Ryboły. A detailed description of the biogas plant can be found in the publications [14,15]. The present work proposes a method that allows the determination of the thickness of the sludge layer based on the heat exchanger's thermal efficiency measurement. More precisely, the thickness of the sludge layer is inferred from the measured outlet temperature from the exchanger. The obtained estimates turn out to be in good agreement with their true values (known from measurement), which suggests that the method can find a practical application. Moreover, the present study contains uncertainty analysis referring to the parameter being determined (thickness of the sludge layer), in order to control the error of the method.

Mathematical formulation
Heat exchangers in the biogas fermentation chambers are made of circular tubes attached to the walls of the loopshaped fermentation chamber [15][16][17]. A physical scheme of the piping fragment is shown in Fig. 2. Due to the large ratio of the loop diameter to the heat exchanger's piping diameter, a straight duct was assumed for further consideration.

Adapting a model for heat exchange
A simplified model, described in [15], was used for calculations, in which the flow inside the heat exchanger pipeline and external flow of the substrate around the heat exchanger tubing are considered (Fig.2). The return temperature T m,o from the heat exchanger is described by the known relationship [18]: where T a is the temperature of the substrate, T m,i is the temperature at the inlet to the heat exchanger, c pm is the specific heat capacity of the working medium in the exchanger, L is the length of a single heat exchanger loop in the digester, d 3 is the outside diameter of the heat exchanger pipe, taking into account the thickness of the sludge, m is the mass flow of the heat exchanger medium, and U is the overall heat transfer coefficient, which is defined by the formula: where d 1 is the inner diameter of the heat exchanger, d 2 is the outer diameter of the heat exchanger, k w is the thermal conductivity of the steel wall of the heat exchanger, k sl is the thermal conductivity of the sludge on the heat exchanger, and h i and h o are convective heat transfer coefficients for internal (3) and external (4) flows, respectively: where k m is the thermal conductivity of the working medium in the exchanger, k s is the thermal conductivity of the substrate, Nu d1 and Nu d3 are Nusselt numbers for internal and external flows, respectively. In the case of the internal flow, the Nusselt number Nu d1 can be determined from the Dittus-Boelter correlation: where Re d3 is the Reynolds number for the substrate flow through the heat exchanger tubes, v s is the substrate velocity in the fermentation chamber,  s is the substrate density,  s is the dynamic viscosity of the substrate, and c ps is the specific heat capacity of the substrate.

Method for determining the thickness of the sludge layer
Equation (1), crucial in the following considerations, allows us to directly calculate the temperature T m,o of the heat exchanger outlet, provided the other model variables and parameters, and especially d 3 , are given.
Here we invert this reasoning, assuming that T m,o is given while the outer diameter of the sludge layer has to be determined. This is a so-called inverse geometric problem (because the problem consists in searching for some missing geometric characteristics of the model).
It can be solved for d 3 if all the remaining variables are known from measurement or previous calculation and the solution will be unique. Having determined d 3 , we easily find the thickness of the sludge layer, s, from the dependence s=(d 3 -d 2 )/2. Since (7) is a non-algebraic equation, only an approximate solution can be found. For the purpose of calculating d 3 from (7), one has to use a numerical procedure like MathCad's built-in function based on the Levenberg-Marquardt method and dedicated to root finding, which was successfully applied by the authors. Typically, such procedures require an initial guess for the solution. Since the sludge layer is by its nature relatively thin, d 2 must be close to the exact solution so it can stand for the first guess in the problem considered. There appears another possibility to infer the sludge thickness from the experimental data available. Namely, one can make use of the total heat transfer rate defined by Inserting (1) into (8) and after appropriate rearrangement, equation (8) becomes (9) with U defined by (2). The solution procedure for determining d 3 , similar to that based on equation (7), requires q (instead of T m,i ) to be known.

Results and discussion
This section compares the results of calculations with the actual parameters of the biogas plant operation and presents simulation results. Below is a comparison of the results of sludge thickness calculations on the exchanger walls and outlet temperature from the heat exchanger with the results measured during the biogas plant operation. Two variants of biogas plant operation were adopted: (i) after three years of biogas plant operation just before the general biogas plant maintenance and (ii) after general biogas plant maintenance and purification of the heat exchanger tubing from the accumulated sludge. The general maintenance of the biogas plant consists in discontinuing the operation of the biogas plant, then dismantling the pneumatic roof from the fermentation chamber and emptying the chamber of the substrate, and then on cleaning the biogas plant walls and the heat exchanger. An example picture from the cleaning of the digester is in the publication [19]. While cleaning the biogas plant, measurements of the sludge thickness were made on the heat exchanger tubing for the first variant. In the case of the second variant, the thickness of the sludge on the walls of the heat exchanger should be close to zero, due to the short period of operation of the biogas plant after general maintenance, which was three months. It should be noted here that the sludge thickness measurement on the walls of the heat exchanger is possible only when the digester is completely emptied of the sludge, which means that the biogas plant is shut down. Sample photos of the tested heat exchanger and sludge can be found in the publication [15].

Experimental data and calculation results
The calculations were performed for the following values of parameters:  In the first variant, the absolute error of the sludge thickness on the heat exchanger was 1mm, while in the case of the second variant the thickness of the sludge is close to zero, which proves the high accuracy of the model. It should be noted here that the thickness of the sludge and the conduction coefficient of the sludge depend mainly on the type of substrate working in the digester. In this case, the presented biogas plant consists of maize silage, chicken manure and potato pulp. At the temperature of the outlet from the heat exchanger, the absolute error was about 1 o C. Based on the reading of the temperature rise on the return from the heat exchanger, the thickness of the sludge layer on the walls of the heat exchanger can be read. Figure 3 presents the simulations of the sludge thickness, s, as a function of the return temperature from the heat exchanger at a constant supply temperature and for different values of the sludge heat transfer coefficient. The predictions were obtained for k sl =0.3, 0.6 and 1.0 W/m/K, while the other parameters were the same as listed in this section, except for T m,i =57 o C and T a =40 o C. The uncertainty estimates corresponding to this case remain at an acceptable level.

Conclusions
The paper presents a numerical method of determining the thickness of sludge on the surface of a heat exchanger as a function of the fermentation chamber operation parameters. The sludge accumulated on the surface of the heat exchanger tubes significantly affects the thermal efficiency of the heat exchanger. The mathematical model of the problem is based on the equation relating the following variables to one another: inlet and outlet temperature (of the working medium), substrate temperature and the average overall heat transfer coefficient, the latter containing the thickness of sludge which we need to determine. In terms of computation, the problem is reduced to solving a non-algebraic equation with one unknown, which we perform with the use of a commercial computational program. The obtained numerical results were verified by comparison to the real experimental values and they turned out to be sufficiently accurate. Besides, based on the conducted numerical experiment, we present theoretical predictions of the thickness of the sludge layer, and these results agree with the physics of the considered problem. Moreover, the calculated uncertainties of the obtained results were at an acceptable level and therefore could be considered credible. The presented study shows an alternative proposal for determining the thickness of sludge making use of the total heat transfer rate of the heat exchanger. Finally, the developed method has some potential for supporting the work of a biogas plant.