# Kinetics of Sludge Superheated Steam Drying and Optimization of Process Parameters

Authors: Xukun Zhang; Pu Xing; Shengping Liu; Chengzhen Zhang; Weiliang Fu
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

The purpose of this paper is to study the effects of steam temperature, sludge thickness, and steam flow on the drying characteristics and drying kinetics of sludge superheated steam and optimize the process parameters of sludge superheated steam drying by using the response surface method and taking relative unit energy consumption as the response value. Single-factor test of superheated steam drying of sludge was conducted under different drying conditions of steam temperature (180, 195, 220, and 260°C), sludge thickness (2, 3.5, 6, and 10 mm), and steam flow rate (22, 29, 36, and 40 m3/h). Taking relative unit energy consumption as the response value and steam temperature, sludge mass (corresponding to sludge thickness), and steam flow as the test factors, the three-factor and five-level response surface test was conducted on the sludge. The experimental results show that steam temperature and sludge thickness had a significant influence on the drying characteristics of superheated sludge steam (p<0.05). With the increase of steam temperature, the decrease of sludge thickness, and the increase of steam flow, the drying time of sludge will be shortened, the drying efficiency of sludge will be improved, and the maximum condensation amount and recovery time will be reduced. The quadratic polynomials, linear equation, and logarithmic model can better fit the change law of water variation of the sludge of superheated steam condensation section, recovery section, and drying section. With the increase of steam temperature, sludge thickness, and steam flow rate, the effective diffusion coefficient also increases. The activation energy of sludge superheated steam drying was 13.42 kJ/mol at a sludge thickness of 6 mm, steam flow rate of 22 m3/h, and different steam temperatures. The order of influence degree of three test factors on relative unit energy consumption is as follows: sludge mass (corresponding to sludge thickness) > steam flow rate > steam temperature. According to response surface test analysis, the optimal drying process with relative unit energy consumption is steam temperature of 215°C, sludge mass of 25 g (corresponding sludge thickness of 8 mm), and steam flow of 30 m3/h. Under this drying process parameter, the minimum energy consumption of sludge superheated steam drying is 284.61 kJ/g. Compared with the relative unit energy consumption value obtained in the verification test, the relative error is 1.65%.


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## Body

## 1. Introduction

Sludge is a solid or semisolid sediment with different water content produced during sewage treatment. With the continuous acceleration of China’s urbanization process, sludge production is increasing rapidly. According to statistics, China’s total output of sludge containing 80% water content has exceeded 60 million tons by 2019, and the annual output of sludge is expected to exceed 90 million tons by 2025 [1, 2]. Sludge contains a large number of pathogenic bacteria, heavy metals, and other harmful substances if not timely treatment will cause great pollution to the environment. On the other hand, the sludge contains a large amount of nitrogen, phosphorus, potassium, and other organic matters and a variety of trace elements, which can be used as raw materials for the preparation of garden and road greening granular fertilizer with certain economic recycling value. Therefore, sludge treatment is an urgent problem to be solved properly. At present, sludge treatment is mainly divided into concentration, digestion, dehydration, hot drying, burning, and final disposal of six processes, among which hot drying is an essential intermediate link in sludge treatment [3]. Hot drying treatment of sludge can greatly reduce the moisture content of sludge, effectively reduce the volume of sludge, remove toxic and harmful substances in the sludge, and reduce pollution.Superheated steam drying is a new drying method that uses superheated steam to directly contact materials to remove moisture. Compared with traditional hot air drying, superheated steam drying has the advantages of good material quality, high heat and mass transfer coefficient, high drying efficiency, and low energy consumption after drying, which is considered one of the development directions of drying technology in the future. Superheated steam drying method is especially suitable for drying materials with high water content, large quantity, and low individual value. Liu et al. [4]used superheated steam to dry soybean residue to study the drying characteristics of soybean residue at different vapor temperatures, different masses, and different vapor flows. The results showed that, compared with the soybean residue before drying, the protein of soybean residue after drying after superheated steam was not significantly reduced. Bao et al. [5] compared and studied the hygroscopicity and dimensional stability of wood after superheated steam drying and hot wind drying. The results show that the wood dried by superheated steam has lower hygroscopicity and higher dimensional stability. Beeby et al. [6] used superheated steam and hot wind to dry cracking catalyst and fine sand, respectively. The results show that the heat transfer coefficient of superheated steam is 400–500 W/(m2·K), while the heat transfer coefficient of hot air drying is 200–300 W/(m2·K). The heat transfer coefficient of superheated steam is higher. Tatemoto et al. [7] used convection drying combined with a fluidized bed to dry carrots, and the drying medium used hot wind and superheated steam, respectively. The results show that when the drying medium is superheated steam, the drying rate of superheated steam combined fluidized bed is much higher than that of other conditions. Choicharoen et al. [8] compared the drying energy consumption of superheated steam drying and hot air drying of high water-containing materials. The results show that, under the same drying conditions, superheated steam drying can save up to 46% energy consumption compared with hot air drying.At present, many researchers have studied the drying characteristics of lignite [9], food [10], wood [11], and other materials dried by superheated steam. However, there are few reports on the desiccation of sludge by superheated steam. The effect of sludge superheated steam drying parameters (steam temperature, sludge thickness, and steam flow) on sludge drying characteristics was studied. Taking steam temperature, sludge mass (corresponding to sludge thickness), and steam flow as test indexes and taking relative unit energy consumption as response value, the superheated steam drying process of sludge was optimized in order to provide a reference for the development of equipment for sludge superheated steam drying.


## 2. Materials and Methods

### 2.1. Test Equipment

The test device is shown in Figure1. It mainly includes LDR0.05-0.6 type electric heating steam boiler (rated steam capacity is 0.05 t/h; Jiujiang Xinyuan Boiler Co., Ltd.), Φ159-0.7-II type cylinder (Jiujiang Xinyuan Boiler Co., Ltd.), 3 u-shaped heating tubes, drying oven (homemade), Wss-411 thermometer (Tianjin Wanda Foster Instrument Co., Ltd.), YK250A40 metal tube float flowmeter (Dalian Youke Instrument Development Center), SCR3-12. LA type artificial intelligence industrial power regulator (Nanchang Yulong Instrument Electric Appliance Co., Ltd.), AI-808 AN-24 VDC temperature control instrument (control accuracy:  ± 1°C; Yueqing Meiger Electronic Appliances Co., Ltd.), watt-hour meter (accuracy: 0.5 level; Shanghai Renmin Hi-Tech Instrument Co., Ltd.), and temperature sensor (accuracy class: 0.5 level; Nanchang Yulong Instrument and Electrical Co., Ltd.). Other test instruments include DHG-9075a electric blast drying oven (Shanghai Yiheng Scientific Instrument Co., Ltd.), ML1602 electronic balance (readability: 0.01 g, repeatability: 0.01 g; Mettler Toledo International Trading (Shanghai) Co., Ltd.), DT1000A type electronic balance (resolution: 0.01 g; Changshu Yiou Instrument Co., Ltd.), one computer, and some glass Petri dishes.Figure 1
Schematic diagram of superheated steam drying experimental setup. (1) Thermometer. (2) Drying chamber. (3) Electronic balance. (4) Sample. (5) Metal tube rotameter. (6) Heater. (7) Computer. (8) Stop valve. (9) Pressure gauge. (10) Safety valve. (11) Steam boiler. (12) Subcylinder. (13) Throttle valve. (14) Drain valve. (15) Throttle valve. (16) Temperature sensor. (17) Control cabinet.


### 2.2. Test Materials

The raw materials were extracted from a sewage treatment plant in Nanchang and dehydrated by the mesh belt filter press. In the loose state, the density is about 979 kg/m3, and the wet-base moisture content is about 75% (w.b). The self-made wooden model was placed on the 0.3 mm thin aluminum sheet, and the wet sludge was loaded into the wooden model and gently pressed. After forming, the size of the thin-layer sludge was 50 × 50 mm2, and the thickness was 2, 3.5, 6, 8, and 10 mm (the corresponding sludge mass was 6, 11, 18, 25, and 30 g).


### 2.3. Test Content and Design

#### 2.3.1. Single-Factor Test Design

In order to further study the characteristics and dynamics of sludge superheated steam drying, based on the preliminary experiments of the research group and relevant literature and combined with the characteristics of relevant test equipment, the temperature of superheated steam, sludge thickness, and steam flow were selected as the test factors. In order to study the influence of superheated steam temperature on the drying characteristics of superheated sludge, the fixed sludge thickness of 6 mm, the steam flow rate of 29 m3/h, and different superheated steam temperatures of 180, 195, 220, 245, and 260°C were selected for single-factor drying test. In order to study the influence of sludge thickness on the drying characteristics of sludge superheated steam, the fixed superheated steam temperature was 220°C, the steam flow rate was 29 m3/h, and the sludge thickness was 2, 3.5, 6, 8, and 10 mm (the corresponding mass was 6, 11, 18, 25, and 30 g), respectively. In order to study the influence of steam flow rate on the drying characteristics of sludge superheated steam, the fixed superheated steam temperature was 220°C, the sludge thickness was 6 mm, and different steam flow rates were 18, 22, 29, 36, and 40 m3/h for single-factor drying test.


#### 2.3.2. Response Surface Test

In order to find the optimal combination of sludge superheated steam drying process parameters, three factors, including superheated steam temperatureX1, sludge mass X2 (the mass is 6, 11, 18, 25, and 30 g, resp. (the corresponding sludge thickness is 2, 3.5, 6, 8, and 10 mm)), and steam flow X3, were selected as independent variables for response surface test. The factors and codes are shown in Table 1.Table 1
Experimental factors and levels.
CodeSteam temperatureX1 (°C)Mass of sludgeX2 (g)Steam flowX3 (m3/h)−1.628180618−1195112202201829124525361.6282603040


#### 2.3.3. Response Surface Test Evaluation Index

Sludge drying is an extremely energy-consuming process, and the sludge drying process with low energy consumption is an important goal of all sludge drying processes. In order to find the process of sludge superheated steam drying with the lowest energy consumption, the relative unit energy consumption was selected as the evaluation index of response surface test. The relative unit energy consumption is defined as follows: during the drying process, the heater heats the steam and keeps the superheated steam at the temperature required by the test, and the ratio of the electric energy consumed to the water mass in the material removed is calculated as(1)W=3600×W2−W1M1−M2,where W is relative unit energy consumption, kJ/g; W1 is the watt-hour meter reading before drying, kW·h; W2 is the reading of the watt-hour meter after drying, kW·h; M1 is the total mass of the material before drying, g; M2 is the total mass of the material after drying, g.


### 2.4. Test Method

#### 2.4.1. Test Process

Open the electric steam boiler, and the steam is produced as a drying medium. When the steam is not injected into the drying system, turn on the artificial intelligence temperature controller and set it as the intelligent setting mode to obtain the self-setting parameters. After the steam is injected, turn on the heater to preheat the internal temperature of the system to the preset temperature of the test and set the power regulator as the working mode. After turning off the heater and steam, manually record the reading of the electricity meter and put the prepared sludge into the desiccator. After closing the door of the drying chamber, the saturated steam is quickly admitted, and the heater is turned on. When the online weighing shows that the moisture content of the desiccated sludge is 20% (w.b), the power supply of the power control cabinet is turned off, the steam is stopped, and then the reading of the electricity meter is recorded manually. In order to ensure that the temperature in the drying room at the beginning of drying is consistent with the preset temperature, heat the system to the set temperature each time before adding sludge for drying. The ML1602 electronic balance carries out data transmission with the computer through the RS232 serial port line, and the data collection frequency is 0.2 Hz. The collected data is written to the Excel file and saved automatically. Each test corresponds to 1 superheated steam flow rate, 1 drying temperature, and 1 sludge thickness (corresponding to sludge mass), and each test is repeated 3 times.


#### 2.4.2. Calculation of Drying Parameters

The calculation formula of the moisture ratio of sludge at different time is as follows:(2)MR=Mt−MeM0−Me,where MR is the sludge moisture ratio; M0 is the initial dry base moisture content of sludge, g/g; Me is the balance dry base moisture content of sludge, g/g; Mt is the dry basis moisture content of sludge at time t of drying process, g/g.SinceMe is small relative to M0 and Mt and can be ignored, equation (1) can be simplified as [12](3)MR=MtM0.The effect of sludge drying to remove water is expressed by sludge drying rate (DR) (g/(g·min)), and the calculation formula is as follows [13]:(4)DR=−Mt+Δt−MtΔt,where M(t+Δt) to (t + Δt) is the process of drying time, water quality unit: g.The initial moisture content of the material is determined by the material heating method. The specific method is as follows.The 3 Petri dishes were numbered 1, 2, and 3 as mass. The proper amount of sludge was evenly spread in the Petri dish, and the total mass was weighed. Then, it was put into the electrothermal incubator at 105°C for constant temperature drying for 3 h, and the mass was weighed by electronic balance. Then, it was dried again until the mass of each Petri dish changed to 0.01 g twice before and after drying. At this point, the mass of the three samples remains constant, which is considered to be the absolute dry mass of the sludge sample.The coefficient of determinationR2, chi-square 2, and root-mean-square error (RMSE) was used to determine the fit between the experimental values and the thin-layer drying model. The larger the R2 and the smaller the chi-square 2 and RMSE, the better the fitting degree of test value and drying model. The expressions of the three parameters were as follows [14]:(5)R2=1−∑i=1NMRexp,i−MRpre,i2∑i=1NMRexp,i−2,(6)χ2=∑i=1NMRexp,i−MRpre,i2N−Z,(7)RMSE=1N∑i=1NMRexp,i−MRpre,i2,where MRexpi is the ith test value of water ratio, MRprei is the theoretical value of the ith water ratio, N is the number of observed values, and Z is the number of model coefficients.Thin-layer sludge drying is the process of water migration to the outside. According to Fick’s second law, the diffusion coefficient of water in the drying process of thin-layer sludge can be calculated through the experimental data and empirical formula as follows [15]:(8)MR=8π2∑n=0∞12n+12exp−2n+12πDefft4δ2,where MR is the sludge moisture ratio; Deff is the effective diffusion coefficient, m2/s; t is the test time, s; δ is the thickness of the thin-layer sludge test sample, m; n is the number of test samples.By taking the natural logarithm of both sides of equation (7), the linear relationship between ln MR and time t can be obtained:(9)lnMR=ln8π2−π2Defft4δ2.According to formula (8), the slope expression of the linear relationship between ln MR and time t is(10)Slope=−π2Deff4δ2.The test data were fitted to obtain the slope value of the linear relationship between ln MR and timet, and then the effective diffusion coefficient Deff of water was obtained from (9).The relationship between effective diffusion coefficientDeff and activation energy can be established according to the Arrhenius equation [16]:(11)Deff=D0exp−EaRT,where Deff is the effective diffusion coefficient, m2/s; D0 is the preexponential factor of the Arrhenius equation, m2/s; Ea is the activation energy, kJ/mol; R is the gas constant, 8.314 J/(mol·K); T is the control temperature of the drying test, K.By taking the natural logarithm of both sides of equation (10), the linear relation expression between ln Deff and 1/T can be obtained:(12)lnDeff=lnD0−EaR1T.By linear fitting of the experimental data, the slope value of the linear relationship between lnDeff and 1/T can be obtained from equation (11), Ea/R, so that the activation energy Ea can be calculated.


### 2.5. The Data Processing

The test data were processed by software OriginPro 8.5.1, Matlab R2012b, Excel 2010, and Design Expert V8.0.6.1.


## 2.1. Test Equipment

The test device is shown in Figure1. It mainly includes LDR0.05-0.6 type electric heating steam boiler (rated steam capacity is 0.05 t/h; Jiujiang Xinyuan Boiler Co., Ltd.), Φ159-0.7-II type cylinder (Jiujiang Xinyuan Boiler Co., Ltd.), 3 u-shaped heating tubes, drying oven (homemade), Wss-411 thermometer (Tianjin Wanda Foster Instrument Co., Ltd.), YK250A40 metal tube float flowmeter (Dalian Youke Instrument Development Center), SCR3-12. LA type artificial intelligence industrial power regulator (Nanchang Yulong Instrument Electric Appliance Co., Ltd.), AI-808 AN-24 VDC temperature control instrument (control accuracy:  ± 1°C; Yueqing Meiger Electronic Appliances Co., Ltd.), watt-hour meter (accuracy: 0.5 level; Shanghai Renmin Hi-Tech Instrument Co., Ltd.), and temperature sensor (accuracy class: 0.5 level; Nanchang Yulong Instrument and Electrical Co., Ltd.). Other test instruments include DHG-9075a electric blast drying oven (Shanghai Yiheng Scientific Instrument Co., Ltd.), ML1602 electronic balance (readability: 0.01 g, repeatability: 0.01 g; Mettler Toledo International Trading (Shanghai) Co., Ltd.), DT1000A type electronic balance (resolution: 0.01 g; Changshu Yiou Instrument Co., Ltd.), one computer, and some glass Petri dishes.Figure 1
Schematic diagram of superheated steam drying experimental setup. (1) Thermometer. (2) Drying chamber. (3) Electronic balance. (4) Sample. (5) Metal tube rotameter. (6) Heater. (7) Computer. (8) Stop valve. (9) Pressure gauge. (10) Safety valve. (11) Steam boiler. (12) Subcylinder. (13) Throttle valve. (14) Drain valve. (15) Throttle valve. (16) Temperature sensor. (17) Control cabinet.


## 2.2. Test Materials

The raw materials were extracted from a sewage treatment plant in Nanchang and dehydrated by the mesh belt filter press. In the loose state, the density is about 979 kg/m3, and the wet-base moisture content is about 75% (w.b). The self-made wooden model was placed on the 0.3 mm thin aluminum sheet, and the wet sludge was loaded into the wooden model and gently pressed. After forming, the size of the thin-layer sludge was 50 × 50 mm2, and the thickness was 2, 3.5, 6, 8, and 10 mm (the corresponding sludge mass was 6, 11, 18, 25, and 30 g).


## 2.3. Test Content and Design

### 2.3.1. Single-Factor Test Design

In order to further study the characteristics and dynamics of sludge superheated steam drying, based on the preliminary experiments of the research group and relevant literature and combined with the characteristics of relevant test equipment, the temperature of superheated steam, sludge thickness, and steam flow were selected as the test factors. In order to study the influence of superheated steam temperature on the drying characteristics of superheated sludge, the fixed sludge thickness of 6 mm, the steam flow rate of 29 m3/h, and different superheated steam temperatures of 180, 195, 220, 245, and 260°C were selected for single-factor drying test. In order to study the influence of sludge thickness on the drying characteristics of sludge superheated steam, the fixed superheated steam temperature was 220°C, the steam flow rate was 29 m3/h, and the sludge thickness was 2, 3.5, 6, 8, and 10 mm (the corresponding mass was 6, 11, 18, 25, and 30 g), respectively. In order to study the influence of steam flow rate on the drying characteristics of sludge superheated steam, the fixed superheated steam temperature was 220°C, the sludge thickness was 6 mm, and different steam flow rates were 18, 22, 29, 36, and 40 m3/h for single-factor drying test.


### 2.3.2. Response Surface Test

In order to find the optimal combination of sludge superheated steam drying process parameters, three factors, including superheated steam temperatureX1, sludge mass X2 (the mass is 6, 11, 18, 25, and 30 g, resp. (the corresponding sludge thickness is 2, 3.5, 6, 8, and 10 mm)), and steam flow X3, were selected as independent variables for response surface test. The factors and codes are shown in Table 1.Table 1
Experimental factors and levels.
CodeSteam temperatureX1 (°C)Mass of sludgeX2 (g)Steam flowX3 (m3/h)−1.628180618−1195112202201829124525361.6282603040


### 2.3.3. Response Surface Test Evaluation Index

Sludge drying is an extremely energy-consuming process, and the sludge drying process with low energy consumption is an important goal of all sludge drying processes. In order to find the process of sludge superheated steam drying with the lowest energy consumption, the relative unit energy consumption was selected as the evaluation index of response surface test. The relative unit energy consumption is defined as follows: during the drying process, the heater heats the steam and keeps the superheated steam at the temperature required by the test, and the ratio of the electric energy consumed to the water mass in the material removed is calculated as(1)W=3600×W2−W1M1−M2,where W is relative unit energy consumption, kJ/g; W1 is the watt-hour meter reading before drying, kW·h; W2 is the reading of the watt-hour meter after drying, kW·h; M1 is the total mass of the material before drying, g; M2 is the total mass of the material after drying, g.


## 2.3.1. Single-Factor Test Design

In order to further study the characteristics and dynamics of sludge superheated steam drying, based on the preliminary experiments of the research group and relevant literature and combined with the characteristics of relevant test equipment, the temperature of superheated steam, sludge thickness, and steam flow were selected as the test factors. In order to study the influence of superheated steam temperature on the drying characteristics of superheated sludge, the fixed sludge thickness of 6 mm, the steam flow rate of 29 m3/h, and different superheated steam temperatures of 180, 195, 220, 245, and 260°C were selected for single-factor drying test. In order to study the influence of sludge thickness on the drying characteristics of sludge superheated steam, the fixed superheated steam temperature was 220°C, the steam flow rate was 29 m3/h, and the sludge thickness was 2, 3.5, 6, 8, and 10 mm (the corresponding mass was 6, 11, 18, 25, and 30 g), respectively. In order to study the influence of steam flow rate on the drying characteristics of sludge superheated steam, the fixed superheated steam temperature was 220°C, the sludge thickness was 6 mm, and different steam flow rates were 18, 22, 29, 36, and 40 m3/h for single-factor drying test.


## 2.3.2. Response Surface Test

In order to find the optimal combination of sludge superheated steam drying process parameters, three factors, including superheated steam temperatureX1, sludge mass X2 (the mass is 6, 11, 18, 25, and 30 g, resp. (the corresponding sludge thickness is 2, 3.5, 6, 8, and 10 mm)), and steam flow X3, were selected as independent variables for response surface test. The factors and codes are shown in Table 1.Table 1
Experimental factors and levels.
CodeSteam temperatureX1 (°C)Mass of sludgeX2 (g)Steam flowX3 (m3/h)−1.628180618−1195112202201829124525361.6282603040


## 2.3.3. Response Surface Test Evaluation Index

Sludge drying is an extremely energy-consuming process, and the sludge drying process with low energy consumption is an important goal of all sludge drying processes. In order to find the process of sludge superheated steam drying with the lowest energy consumption, the relative unit energy consumption was selected as the evaluation index of response surface test. The relative unit energy consumption is defined as follows: during the drying process, the heater heats the steam and keeps the superheated steam at the temperature required by the test, and the ratio of the electric energy consumed to the water mass in the material removed is calculated as(1)W=3600×W2−W1M1−M2,where W is relative unit energy consumption, kJ/g; W1 is the watt-hour meter reading before drying, kW·h; W2 is the reading of the watt-hour meter after drying, kW·h; M1 is the total mass of the material before drying, g; M2 is the total mass of the material after drying, g.


## 2.4. Test Method

### 2.4.1. Test Process

Open the electric steam boiler, and the steam is produced as a drying medium. When the steam is not injected into the drying system, turn on the artificial intelligence temperature controller and set it as the intelligent setting mode to obtain the self-setting parameters. After the steam is injected, turn on the heater to preheat the internal temperature of the system to the preset temperature of the test and set the power regulator as the working mode. After turning off the heater and steam, manually record the reading of the electricity meter and put the prepared sludge into the desiccator. After closing the door of the drying chamber, the saturated steam is quickly admitted, and the heater is turned on. When the online weighing shows that the moisture content of the desiccated sludge is 20% (w.b), the power supply of the power control cabinet is turned off, the steam is stopped, and then the reading of the electricity meter is recorded manually. In order to ensure that the temperature in the drying room at the beginning of drying is consistent with the preset temperature, heat the system to the set temperature each time before adding sludge for drying. The ML1602 electronic balance carries out data transmission with the computer through the RS232 serial port line, and the data collection frequency is 0.2 Hz. The collected data is written to the Excel file and saved automatically. Each test corresponds to 1 superheated steam flow rate, 1 drying temperature, and 1 sludge thickness (corresponding to sludge mass), and each test is repeated 3 times.


### 2.4.2. Calculation of Drying Parameters

The calculation formula of the moisture ratio of sludge at different time is as follows:(2)MR=Mt−MeM0−Me,where MR is the sludge moisture ratio; M0 is the initial dry base moisture content of sludge, g/g; Me is the balance dry base moisture content of sludge, g/g; Mt is the dry basis moisture content of sludge at time t of drying process, g/g.SinceMe is small relative to M0 and Mt and can be ignored, equation (1) can be simplified as [12](3)MR=MtM0.The effect of sludge drying to remove water is expressed by sludge drying rate (DR) (g/(g·min)), and the calculation formula is as follows [13]:(4)DR=−Mt+Δt−MtΔt,where M(t+Δt) to (t + Δt) is the process of drying time, water quality unit: g.The initial moisture content of the material is determined by the material heating method. The specific method is as follows.The 3 Petri dishes were numbered 1, 2, and 3 as mass. The proper amount of sludge was evenly spread in the Petri dish, and the total mass was weighed. Then, it was put into the electrothermal incubator at 105°C for constant temperature drying for 3 h, and the mass was weighed by electronic balance. Then, it was dried again until the mass of each Petri dish changed to 0.01 g twice before and after drying. At this point, the mass of the three samples remains constant, which is considered to be the absolute dry mass of the sludge sample.The coefficient of determinationR2, chi-square 2, and root-mean-square error (RMSE) was used to determine the fit between the experimental values and the thin-layer drying model. The larger the R2 and the smaller the chi-square 2 and RMSE, the better the fitting degree of test value and drying model. The expressions of the three parameters were as follows [14]:(5)R2=1−∑i=1NMRexp,i−MRpre,i2∑i=1NMRexp,i−2,(6)χ2=∑i=1NMRexp,i−MRpre,i2N−Z,(7)RMSE=1N∑i=1NMRexp,i−MRpre,i2,where MRexpi is the ith test value of water ratio, MRprei is the theoretical value of the ith water ratio, N is the number of observed values, and Z is the number of model coefficients.Thin-layer sludge drying is the process of water migration to the outside. According to Fick’s second law, the diffusion coefficient of water in the drying process of thin-layer sludge can be calculated through the experimental data and empirical formula as follows [15]:(8)MR=8π2∑n=0∞12n+12exp−2n+12πDefft4δ2,where MR is the sludge moisture ratio; Deff is the effective diffusion coefficient, m2/s; t is the test time, s; δ is the thickness of the thin-layer sludge test sample, m; n is the number of test samples.By taking the natural logarithm of both sides of equation (7), the linear relationship between ln MR and time t can be obtained:(9)lnMR=ln8π2−π2Defft4δ2.According to formula (8), the slope expression of the linear relationship between ln MR and time t is(10)Slope=−π2Deff4δ2.The test data were fitted to obtain the slope value of the linear relationship between ln MR and timet, and then the effective diffusion coefficient Deff of water was obtained from (9).The relationship between effective diffusion coefficientDeff and activation energy can be established according to the Arrhenius equation [16]:(11)Deff=D0exp−EaRT,where Deff is the effective diffusion coefficient, m2/s; D0 is the preexponential factor of the Arrhenius equation, m2/s; Ea is the activation energy, kJ/mol; R is the gas constant, 8.314 J/(mol·K); T is the control temperature of the drying test, K.By taking the natural logarithm of both sides of equation (10), the linear relation expression between ln Deff and 1/T can be obtained:(12)lnDeff=lnD0−EaR1T.By linear fitting of the experimental data, the slope value of the linear relationship between lnDeff and 1/T can be obtained from equation (11), Ea/R, so that the activation energy Ea can be calculated.


## 2.4.1. Test Process

Open the electric steam boiler, and the steam is produced as a drying medium. When the steam is not injected into the drying system, turn on the artificial intelligence temperature controller and set it as the intelligent setting mode to obtain the self-setting parameters. After the steam is injected, turn on the heater to preheat the internal temperature of the system to the preset temperature of the test and set the power regulator as the working mode. After turning off the heater and steam, manually record the reading of the electricity meter and put the prepared sludge into the desiccator. After closing the door of the drying chamber, the saturated steam is quickly admitted, and the heater is turned on. When the online weighing shows that the moisture content of the desiccated sludge is 20% (w.b), the power supply of the power control cabinet is turned off, the steam is stopped, and then the reading of the electricity meter is recorded manually. In order to ensure that the temperature in the drying room at the beginning of drying is consistent with the preset temperature, heat the system to the set temperature each time before adding sludge for drying. The ML1602 electronic balance carries out data transmission with the computer through the RS232 serial port line, and the data collection frequency is 0.2 Hz. The collected data is written to the Excel file and saved automatically. Each test corresponds to 1 superheated steam flow rate, 1 drying temperature, and 1 sludge thickness (corresponding to sludge mass), and each test is repeated 3 times.


## 2.4.2. Calculation of Drying Parameters

The calculation formula of the moisture ratio of sludge at different time is as follows:(2)MR=Mt−MeM0−Me,where MR is the sludge moisture ratio; M0 is the initial dry base moisture content of sludge, g/g; Me is the balance dry base moisture content of sludge, g/g; Mt is the dry basis moisture content of sludge at time t of drying process, g/g.SinceMe is small relative to M0 and Mt and can be ignored, equation (1) can be simplified as [12](3)MR=MtM0.The effect of sludge drying to remove water is expressed by sludge drying rate (DR) (g/(g·min)), and the calculation formula is as follows [13]:(4)DR=−Mt+Δt−MtΔt,where M(t+Δt) to (t + Δt) is the process of drying time, water quality unit: g.The initial moisture content of the material is determined by the material heating method. The specific method is as follows.The 3 Petri dishes were numbered 1, 2, and 3 as mass. The proper amount of sludge was evenly spread in the Petri dish, and the total mass was weighed. Then, it was put into the electrothermal incubator at 105°C for constant temperature drying for 3 h, and the mass was weighed by electronic balance. Then, it was dried again until the mass of each Petri dish changed to 0.01 g twice before and after drying. At this point, the mass of the three samples remains constant, which is considered to be the absolute dry mass of the sludge sample.The coefficient of determinationR2, chi-square 2, and root-mean-square error (RMSE) was used to determine the fit between the experimental values and the thin-layer drying model. The larger the R2 and the smaller the chi-square 2 and RMSE, the better the fitting degree of test value and drying model. The expressions of the three parameters were as follows [14]:(5)R2=1−∑i=1NMRexp,i−MRpre,i2∑i=1NMRexp,i−2,(6)χ2=∑i=1NMRexp,i−MRpre,i2N−Z,(7)RMSE=1N∑i=1NMRexp,i−MRpre,i2,where MRexpi is the ith test value of water ratio, MRprei is the theoretical value of the ith water ratio, N is the number of observed values, and Z is the number of model coefficients.Thin-layer sludge drying is the process of water migration to the outside. According to Fick’s second law, the diffusion coefficient of water in the drying process of thin-layer sludge can be calculated through the experimental data and empirical formula as follows [15]:(8)MR=8π2∑n=0∞12n+12exp−2n+12πDefft4δ2,where MR is the sludge moisture ratio; Deff is the effective diffusion coefficient, m2/s; t is the test time, s; δ is the thickness of the thin-layer sludge test sample, m; n is the number of test samples.By taking the natural logarithm of both sides of equation (7), the linear relationship between ln MR and time t can be obtained:(9)lnMR=ln8π2−π2Defft4δ2.According to formula (8), the slope expression of the linear relationship between ln MR and time t is(10)Slope=−π2Deff4δ2.The test data were fitted to obtain the slope value of the linear relationship between ln MR and timet, and then the effective diffusion coefficient Deff of water was obtained from (9).The relationship between effective diffusion coefficientDeff and activation energy can be established according to the Arrhenius equation [16]:(11)Deff=D0exp−EaRT,where Deff is the effective diffusion coefficient, m2/s; D0 is the preexponential factor of the Arrhenius equation, m2/s; Ea is the activation energy, kJ/mol; R is the gas constant, 8.314 J/(mol·K); T is the control temperature of the drying test, K.By taking the natural logarithm of both sides of equation (10), the linear relation expression between ln Deff and 1/T can be obtained:(12)lnDeff=lnD0−EaR1T.By linear fitting of the experimental data, the slope value of the linear relationship between lnDeff and 1/T can be obtained from equation (11), Ea/R, so that the activation energy Ea can be calculated.


## 2.5. The Data Processing

The test data were processed by software OriginPro 8.5.1, Matlab R2012b, Excel 2010, and Design Expert V8.0.6.1.


## 3. Results and Analysis

### 3.1. Superheated Steam Drying Characteristics of Sludge

#### 3.1.1. Influence of Steam Temperature on the Drying Characteristics of Superheated Steam from Sludge

Figures2(a) and 2(b) show the drying curves and drying rate curves of 6 mm sludge (corresponding to the sludge mass of 18 g) from 75.29% to 20% (w.b.) under the drying conditions of steam flow rate of 22 m3/h at different steam temperatures (180, 195, 220, and 260°C). In the figure, values of data points are taken every 1 min. As can be seen from Figure 2(a), the drying time required for 6 mm sludge to reach 20% under the drying conditions of steam flow rate of 22 m3/h and different steam temperatures (180, 195, 220, and 260°C) is 25, 22, 19, and 15 min, respectively. Increasing the steam temperature can significantly reduce the drying time. As can be seen from Figure 2(a), during the initial drying period, the moisture ratio of the sludge increases gradually. This is because the high-temperature steam meets the low-temperature material in the initial drying period, and the condensation of the steam leads to the gradual increase of water on the surface of the material. It can also be seen from Figure 2(b) that the initial drying rate of sludge is negative, which is also caused by steam condensation. It can also be seen from Figure 2(b) that the drying rate curves at different steam temperatures all increase first and then gradually decrease, which is the same as the conclusion of Qiao et al. on the drying characteristics of superheated steam in corm [17]. For the whole drying test at different steam temperatures, the drying rate of sludge was in the following order from high to low: 260°C > 220°C > 195°C > 180°C. The higher the steam temperature, the more intense the movement of water molecules in the sludge, the faster the water migration rate, and the higher the drying rate, so the sludge drying efficiency increases with the increase of steam temperature.Figure 2
Drying curves and drying rate curves of 6 mm sludge at different steam temperature under steam flow of 22 m3/h. (a) Drying curves. (b) Drying rate curves.
(a)(b)


#### 3.1.2. Effect of Sludge Thickness on the Drying Characteristics of Superheated Steam of Sludge

Figures3(a) and 3(b) show the drying curve and drying rate curve when the sludge is dried from 75.29% to 20% (w.b.) under the drying conditions of steam temperature 220°C, steam flow rate 22 m3/h, and different sludge thickness of 2, 3.5, 6, and 10 mm (corresponding to sludge mass of 6, 11, 18, and 30 g). In the figure, values of data points are taken every 1 min. The time to reduce the initial moisture content of sludge from 75.29% to 20% under different sludge thicknesses was 8 min for 2 mm, 14 min for 3.5 mm, 20 min for 6 mm, and 27 min for 10 mm. Reducing sludge thickness can shorten drying time and improve drying efficiency, and sludge thickness has a significant impact on drying time (p<0.05). It can be seen from Figure 3(b) that, under different sludge thickening conditions, the drying rate of sludge increases first and then decreases gradually. Steam condensation in the initial stage of sludge superheated steam drying results in a negative initial drying rate of sludge. For the whole drying test with different sludge thickness, the drying rate of the sludge was 2 mm > 3.5 mm > 6 mm > 10 mm from high to low. In the desiccation stage of sludge superheated steam drying, the desiccation rate mainly depends on the heat and mass transfer characteristics inside the material [18]. Under the same steam temperature and steam flow rate and different sludge thickness drying conditions, the thicker the sludge, the longer the water migration path and the slower the water and heat migration efficiency. In addition, the thicker the sludge is, the more the water inside the sludge needs to be transferred from the interior to the surface of the sludge and the longer it takes to reach the target moisture content. Therefore, the drying efficiency of the sludge decreases with the increase of the sludge thickness [19, 20].Figure 3
Drying curves and drying rate curves of sludge at different sludge thicknesses under steam temperature of 220°C and steam flow of 22 m3/h. (a) Drying curves. (b) Drying rate curves.
(a)(b)


#### 3.1.3. Influence of Steam Flow Rate on the Drying Characteristics of Superheated Steam of Sludge

Figures4(a) and 4(b), respectively, show the drying curve and drying rate curve of the sludge from 75.29% to 20% under the drying conditions of steam temperature 220°C, 6 mm thickness, and different steam flow rates. In the figure, values of data points are taken every 1 min. As can be seen from Figure 4(a), under the drying conditions with steam flow rates of 22, 29, 36, and 40 m3/h, respectively, it took 23, 20, 17, and 15 min for the sludge to go from 75.29% to 20% wet basis moisture content. As can be seen from Figure 4(b), under different steam flow rates, the overall drying rate of sludge increases first and then decreases gradually. In the initial stage of sludge superheated steam drying, high-temperature steam meets low-temperature material and causes steam condensation, thus making the initial drying rate of sludge negative. For the whole drying test with different steam flow rates, the overall drying rate of sludge is 40 m3/h > 36 m3/h > 29 m3/h > 22 m3/h from high to low. The greater the steam flow, the more sensible the heat of the steam and the more the heat provided to the material per unit time. The faster the material heats up, the higher the convective heat transfer and mass transfer efficiency in the material are. Therefore, the higher the steam flow, the higher the drying rate and the shorter the drying time. This was consistent with Pronky’s study of using superheated steam to dry beet residues [21].Figure 4
Drying curves and drying rate curves of 6 mm sludge at different sludge flow under steam temperature of 220°C. (a) Drying curves. (b) Drying rate curves.
(a)(b)


### 3.2. Characteristics of Initial Condensation of Superheated Steam Drying of Sludge

In the initial stage of superheated steam drying, when the high-temperature steam meets the low-temperature material, the high-temperature steam tends to condense. The condensed water adheres to the surface of the material, causing the material water to increase and the material water ratio to increase gradually. When the moisture ratio of the material increases to the maximum moisture ratio, the relative drying time at this time is called condensation time. The stage from the initial water ratio to the maximum water ratio is called the condensation stage. With the drying process, the evaporation rate of water exceeds the condensation rate. When the water ratio of the material gradually decreases from the maximum water ratio to the initial water ratio, the corresponding drying time is the recovery time, which is called the recovery stage. After the recovery time, the material moisture ratio gradually decreases from the initial moisture ratio to the sludge moisture content of 20% of the wet basis moisture content; so far, the whole drying process ends, and this stage is called the drying section. The condensation stage increases the drying time of the whole sludge, which has a negative effect on the whole drying process [22].Figures5(a)–5(c), respectively, represent the relationship between different steam temperature, sludge thickness, and steam flow and maximum condensation amount and recovery time of sludge. As can be seen from Figure 5(a), under the drying conditions with sludge thickness of 6 mm, steam flow rate of 22 m3/h, and different steam temperatures, the maximum condensation amount and recovery time of the sludge are when the steam temperature is 180°C, the corresponding maximum condensation amount is 1.06 g, and the maximum recovery time is 105 s. It can also be seen from Figure 5(a) that, with the increase of steam temperature, the maximum condensation amount and recovery time of sludge decrease. Kozanoglu et al. [23] also found that high temperature led to less vapor condensation of grains when they were dried in a low-pressure superheated steam fluidized bed. As can be seen from Figure 5(b), with the increase of sludge thickness, the maximum condensation amount and the recovery time of the sludge increase continuously. When the steam temperature is 220°C, the steam flow is 22 m3/h, and the sludge thickness is 2, 3.5, 6, and 10 mm, the corresponding maximum condensate volume is 0.26, 0.47, 0.74, and 1.03 g, respectively. The corresponding recovery time is 25, 40, 60, and 90 s, respectively. As can be seen from Figure 5(c), with the increase of steam flow rate, the maximum condensation amount and recovery time of sludge decrease. Under the drying conditions of 220°C steam temperature, 6 mm sludge thickness, and different steam flow rates (22, 29, 36, and 40 m3/h), the corresponding maximum amount of condensate was 0.98, 0.74, 0.7, and 0.47 g, respectively, and the corresponding recovery time was 90, 65, 60, and 30 s. Tang et al. [24] also found that, under the lower steam flow rate, the surface of materials would adhere to more water when they were drying thin-layer distillers’ grains with superheated steam. The maximum amount of condensation is also greater.Figure 5
Relationship between the amount of maximum condensation and restoration times and different drying conditions. (a) Relationship between the amount of maximum condensation and restoration times and steam temperature. Note: sludge thickness was 6 mm, and steam flow was 22 m3/h. (b) Relationship between the amount of maximum condensation and restoration times and sludge thickness. Note: steam temperature was 220°C, and steam flow was 22 m3/h. (c) Relationship between the amount of maximum condensation and restoration times and steam flow. Note: steam temperature was 220°C, and sludge thickness was 6 mm.
(a)(b)(c)


### 3.3. Mathematical Model of Sludge Superheated Steam Drying

#### 3.3.1. Establishment of Sludge Superheated Steam Drying Model

In the process of superheated steam drying, sludge will show steam condensation, a condensation phenomenon. The whole process of superheated steam drying sludge can be divided into three stages: condensation stage, recovery stage, and drying stage. Different mathematical models are used to fit and analyze the process.


#### 3.3.2. Drying Model of Superheated Steam Condensation Section of Sludge

The condensation stage refers to the process in which the sludge moisture ratio increases from the initial value to the maximum value during the process of sludge superheated steam drying. The law of moisture change in the condensation stage can be described by quadratic polynomials.(13)MR=1+At+Bt2.Formula (13) is the expression of a quadratic polynomial. MR is the drying water ratio of superheated steam. t is time, s. At the beginning of the drying process, Mt was equal to M0, so the value of the constant term of the formula (13) was 1. A and B are the coefficients.Table2lists the fitting results and fitting parameter values of the sludge superheated steam drying and condensation stage under different drying conditions.Table 2
Fitting results of condensation stage of sludge superheated steam drying under different drying conditions.
Fixed factor conditionVariable factor conditionModel coefficientsR2χ2 ×10−6RMSE ×10−36 mm, 22 m3/h180°CA = 6.90623 × 10−4; B = 2.50777 × 10−60.988199.127143.02195°CA = 3.09735 × 10−4; B = 1.65517 × 10−50.9491940.57936.37220°CA = 0.00196; B = −1.59104 × 10−50.9572916.43214.05260°CA = 8.79372 × 10−4; B = −2.223 × 10−60.956669.29863.05220°C, 22 m3/h2 mmA = 0.00176; B = −1.19441 × 10−50.992443.340661.833.5 mmA = 1.09123 × 10−4; B = 1.1279 × 10−50.970849.867993.146 mmA = 0.00115; B = −5.69471 × 10−60.994461.7981.3410 mmA = 1.73354 × 10−4; B = 8.0251 × 10−60.964028.53692.92220°C, 6 mm22 m3/hA = 0.00196; B = −8.86327 × 10−60.9824513.35073.6529 m3/hA = 1.82449 × 10−4; B = 9.2205 × 10−60.9710111.28973.3636 m3/hA = 6.4653 × 10−4; B = 2.38935 × 10−50.982037.514662.7440 m3/hA = 1.61425 × 10−4; B = 7.2576 × 10−60.995310.733440.8564It can be seen from Table1 that quadratic polynomials are adopted to fit the moisture change law of the superheated steam condensation stage of sludge under different drying conditions, and the fitting degree is relatively high. It can be used to describe the law of moisture change in the superheated steam condensation stage of sludge. The parameters A and B of the quadratic polynomial are considered as functions of steam temperature, sludge thickness, and steam flow. First-order function polynomials can be used to express the relationship between parameters A and B and steam temperature, sludge thickness, and steam flow. Let(14)A=B0+B1T+B2δ+B3Q+B4Tδ+B5TQ+B6δQ,(15)B=C0+C1T+C2δ+C3Q+C4Tδ+C5TQ+C6δQ.The values of parameters A and B obtained through fitting in Table1 and the corresponding drying conditions are substituted into equations (14) and (15). By using the least square method in Matlab to solve the overdetermined equations, it can be concluded that(16)B]=B0B1B2B3B4B5B6=7.77786839715×10−71.1601523864072×10−5−2.2261242076987×10−61.8271958357701×10−54.96376969165552×10−41.09009466012×10−74.975428958102×10−5T,C=C0C1C2C3C4C5C6=7.105650084297×10−8−1.328649949967×10−72.76505892957×10−81.57651654081×10−65.988537873284×10−6−2.85498486235×10−96.083926557042×10−7T.The seven elements in the matrix [B] and [C] are the coefficients of quadratic polynomials A and B.


#### 3.3.3. Drying Model of Sludge Superheated Steam Recovery Section

The recovery stage of sludge superheated steam drying is a process in which the sludge moisture ratio gradually decreases from the maximum value to the initial value. The superheated steam recovery section of sludge is mainly to remove condensed water condensed by the condensation section. The condensed water adheres to the surface of the material and can be considered constant speed drying. Therefore, the law of moisture change in the recovery phase can be described by the linear relationship shown in formula (17).(17)MR=k0t−tcon+MRmax,where tcon is the condensation time, s; MRmax is the maximum water ratio; MR is the water ratio in the recovery stage.The fitting results and fitting parameters of the sludge superheated steam recovery stage are listed in Table3.Table 3
Fitting results of restoration stage of sludge superheated steam drying under different drying conditions.
Fixed factor conditionVariable factor conditionModel coefficientsk0R2χ2 ×10−6RMSE ×10−36 mm, 22 m3/h180°C−7.02597 × 10−40.99055.705182.39195°C−8.00787 × 10−40.984377.707332.78220°C−8.32655 × 10−40.975411.88853.45260°C−9.16322 × 10−40.980185.186842.28220°C, 22 m3/h2 mm−0.003010.9828317.31444.163.5 mm−0.001310.9445526.02875.16 mm−9.32495 × 10−40.988954.01967210 mm−5.40755 × 10−40.9547911.60663.41220°C, 6 mm22 m3/h−7.95143 × 10−40.987697.2812.729 m3/h−8.51206 × 10−40.9647214.26723.7836 m3/h−8.0649 × 10−40.9167531.87075.6540 m3/h−0.001110.961078.793752.97It can be seen from Table3 that the linear equation used to fit the superheated steam recovery stage of sludge has a higher fitting degree, which can be used to describe the water change law in the recovery stage. The slope of the linear equation is determined by steam temperature, sludge thickness, and steam flow. Let(18)k0=D0+D1T+D2δ+D3Q+D4Tδ+D5TQ+D6δQ+D7T2+D8δ2+D9Q2.The slope value in Table3 and the steam temperature, sludge thickness, and steam flow value are substituted into formula (18), the least square method is adopted by Matlab to solve the overdetermined equations, and the 10 coefficients of the slope k0 can be obtained.(19)D=D0D1D2D3D4D5D6D7D8D9=1.801451177×10−6−1.7314443107×10−51.956168515×10−54.138470927×10−54.271908539226×10−31.23440953×10−74.27449539855×10−4−3.0030188×10−8−56.590453167280614−1.402830725×10−6T.The 10 elements in the matrix [D] are the 10 coefficients of the slope k of the linear equation.


#### 3.3.4. Drying Model of Sludge Superheated Steam Drying Section

The sludge superheated steam drying stage refers to the stage where the sludge moisture content is lower than the initial moisture content after the sludge superheated steam drying and recovery stage. In the superheated steam drying section of sludge, the moisture ratio of sludge can be described by the thin-layer drying model. Common thin-layer drying models are listed in Table4 [25].Table 4
Mathematical thin-layer drying models.
No.Model nameModel equation1LewisMR = exp(−kt)2PageMR = exp(−ktn)3Henderson and PabisMR = aexp(−kt)4OverhultsMR = exp[− (kt)n]5LogarithmicMR = aexp(−kt) + b6Two-term modeMR = aexp(−kt) + (1 − a)exp(−kat)Tables5–7 are the fitting results of the thin-layer drying model in the superheated steam drying stage of sludge under different steam temperature, sludge thickness, and steam flow. From Tables 5 to 7, it can be seen that the determination coefficient of the logarithmic model is above 0.998, which is better than other thin-layer drying models. The chi-square and root-mean-square error values of the logarithmic model are also smaller than other thin-layer drying models. Therefore, the logarithmic model is the most suitable thin-layer drying model to describe the moisture change rule during the superheated steam drying stage of sludge.Table 5
Fitting results of drying stage of sludge superheated steam drying under different steam temperature.
Model nameSteam temperature (°C)180195220260R2χ2 ×10−4RMSER2χ2 ×10−4RMSER2χ2 ×10−4RMSER2χ2 ×10−4RMSEPage0.997471.887880.013740.997411.95740.013990.99632.793740.016710.997891.613450.0127Henderson and Pabis0.97997150.038680.983112.80.035710.9850211.30.033650.9834812.60.03554Overhults0.997481.886010.013730.997411.957190.013990.996312.790640.016710.99791.606640.01268Logarithmic0.999570.3260.005710.99970.228270.004780.999730.207050.004550.999240.582680.00455Wis0.85204110.10.104920.88387.90.093760.85785106.80.103360.86979.90.09949Two-term model0.994354.221130.020550.995823.154410.017760.996422.706110.016450.996832.426280.01558Table 6
Fitting results of drying stage of sludge superheated steam drying under different thickness of sludge.
Model nameThickness of sludge (mm)2 mm3.5 mm6 mm10 mmR2χ2 ×10−4RMSER2χ2 ×10−4RMSER2χ2 ×10−4RMSER2χ2 ×10−4RMSEPage0.996392.663440.016320.998091.537110.01240.997921.64820.012840.997322.040150.01428Henderson and Pabis0.9760717.70.042030.9758519.40.044070.9837912.90.035860.988748.565590.02927Overhults0.996392.663330.016320.998091.536130.012390.997931.639360.01280.997332.031510.01425Logarithmic0.999770.173040.004160.998741.02110.01010.998521.176490.010850.999440.424220.00651Lewis0.8292124.40.111530.84284125.50.112040.87315100.10.100060.8994276.20.08732Two-term model0.990517.00520.026470.992665.906620.02430.997352.10470.014510.998591.071030.01305Table 7
Fitting results of drying stage of sludge superheated steam drying under different steam flow.
Model nameSteam flow (m3/h)22 m3/h29 m3/h36 m3/h40 m3/hR2χ2 ×10−4RMSER2χ2 ×10−4RMSER2χ2 ×10−4RMSER2χ2 ×10−4RMSEPage0.998191.35310.011630.997811.649660.012840.997971.564970.012510.997561.936580.01392Henderson and Pabis0.9810614.20.037640.98407120.034660.9842912.10.034760.985311.70.03417Overhults0.998191.353050.011630.997811.648280.012840.997971.563620.01250.997581.923970.01387Logarithmic0.999650.265170.005150.999480.392330.006260.99950.389690.006240.998850.918010.00958Lewis0.8878283.60.091420.8745694.10.0970.8937381.30.090150.8843891.30.09555Two-term model0.995283.530780.018790.996742.455650.015670.997032.286260.015120.998131.14880.0122Table8 shows different drying conditions, the logarithmic model fitting parameters, and the fitting results. It can be seen from Table 8 that with the increase of steam temperature, sludge thickness and steam flow, the model coefficient a of the logarithmic equation has an obvious decreasing trend, and with the increase of steam pressure, sludge thickness, and steam flow, the model coefficient b of the logarithmic equation also has an obvious increasing trend. With the increase of steam temperature and steam flow, the sludge thickness decreases, and the model coefficient K of the logarithmic equation has an increasing trend. The three coefficients a, b, and k of the logarithmic model equation are determined by the steam temperature, sludge thickness, and steam flow. The relationship between the model parameters and the steam temperature, sludge thickness, and steam flow rate can be expressed by quadratic polynomial. Let(20)a=E0+E1T+E2δ+E3Q+E4Tδ+E5TQ+E6δQ+E7T2+E8δ2+E9Q2,(21)k=F0+F1T+F2δ+F3Q+F4Tδ+F5TQ+F6δQ+F7T2+F8δ2+F9Q2,(22)b=G0+G1T+G2δ+G3Q+G4Tδ+G5TQ+G6δQ+G7T2+G8δ2+G9Q2,where a, k, and b are the parameters of the model of the logarithmic equation; T, δ, and Q represent steam temperature, sludge thickness, and steam flow, respectively.Table 8
Fitting parameter and results from the logarithmic model under different drying conditions.
Fixed factor conditionVariable factor conditionModel coefficientsR2χ2 ×10−4RMSE ×10−36 mm, 22 m3/h180°Ca = 1.90262, k = 6.56429 × 10−4, and b = −0.658080.99570.3265.71195°Ca = 1.72213, k = 8.21507 × 10−4, and b = −0.514010.99970.2282654.78220°Ca = 1.71767, k = 0.00111, and b = −0.440290.999730.2070484.55260°Ca = 1.70801, k = 0.00133, and b = −0.462580.999240.5826827.63220°C, 22 m3/h2 mma = 2.22072, k = 0.00172, and b = −0.97250.999770.1730394.163.5 mma = 1.93788, k = 0.00117, and b = −0.68020.9998741.021110.16 mma = 1.64436, k = 0.00111, and b = −0.388030.998521.1764910.8510 mma = 1.52454, k = 9.04004 × 10−4, and b = −0.299540.999440.4242166.51220°C, 6 mm22 m3/ha = 1.77499, k = 7.2396 × 10−4, and b = −0.58980.999650.2651675.1529 m3/ha = 1.69364, k = 0.00101, and b = −0.4570.999480.3923256.2636 m3/ha = 1.63835, k = 0.0012, and b = −0.437390.99950.3896926.2440 m3/ha = 1.6045, k = 0.00148, and b = −0.366320.998850.9180129.58Note. a, k, and b are fit parameters.The parameter values in Table8 are substituted into equations (18)–(20), and the least square method is used to solve the overdetermined equations with Matlab software. It can be calculated as(23)G=G0G1G2G3G4G5G6G7G8G9=−0.0077855990.0093344690.004262615−0.1722097790.9425756980.0006415110.093218708−6.11992×10−5−10729.754880.000566074T.In matrix [E], [F], [G] the 10 elements correspond to the 10 coefficients of the three model parameters a, b, and k in the logarithmic equation.


#### 3.3.5. Model Validation

In order to verify the drying model of the above three stages, the sludge was tested under the drying conditions of steam temperature of 215°C, sludge thickness of 8 mm, and steam flow of 30 m3/h. The experimental values are compared with the predicted values of the drying model, and the results are shown in Figure 6. The experimental value and the predicted value were fitted, and the results showed that the fitting degree of the two was very high, with R2 = 0.99951, RMSE = 0.00664, and χ2 = 4.40285 × 10−5. The above three drying models can be used to describe the moisture variation of the whole sludge superheated steam drying stage.Figure 6
Verification of sludge superheated steam drying model.


### 3.4. Effective Diffusion Coefficient and Activation Energy

Table9 shows the calculation results of effective diffusion coefficient of sludge superheated steam drying under different drying conditions. It can be seen from Table 9 that the effective diffusion coefficient of sludge superheated steam drying is 2.684606 × 10−8∼4.5229776 × 10−8 under the drying conditions of sludge thickness of 6 mm, steam flow rate of 22 m3/h, and different steam temperatures. Under the drying conditions of steam temperature of 220°C, steam flow rate of 22 m3/h, and different sludge thickness, the effective diffusion coefficient of sludge superheated steam drying is 1.00024 × 10−8∼6.9709 × 10−8. Under the drying conditions of 220°C steam temperature, 6 mm sludge thickness, and different steam flow rates, the effective diffusion coefficient of sludge superheated steam drying was 2.80133 × 10−8∼4.50839 × 10−8. It can also be seen from Table 8 that, under the drying conditions of the same sludge thickness, steam flow rate, and different steam temperature, the effective diffusion coefficient of sludge increases with the increase of steam temperature. This is because the higher the steam temperature, the higher the heat and mass transfer efficiency and the faster the water diffusion and evaporation rate. Under the same steam temperature, steam flow rate, and different sludge thickness drying conditions, the effective diffusion coefficient of the sludge increases with the increase of the sludge thickness. It can be seen from formula (10) that the water effective diffusion coefficient Deff is proportional to 4δ2, the thicker the sludge, the greater the effective diffusion coefficient of water. The effective diffusion coefficient of sludge increases with the increase of steam flow. This is because the greater the steam flow, the more the heat transferred to the material through the drying medium and the faster the water diffusion [26].Table 9
Moisture effective diffusion coefficients of sludge under different superheated steam drying condition.
Fixed factor conditionVariable factor conditionSlopekR2Deff/(m2·s−1)6 mm, 22 m3/h180°C−0.001840.963412.6846061 × 10−8195°C−0.002020.968282.9472306 × 10−8220°C−0.002520.971063.6767431 × 10−8260°C−0.003010.974554.5229776 × 10−8220°C, 22 m3/h2 mm−0.006170.950011.00024 × 10−83.5 mm−0.003190.964561.68305 × 10−86 mm−0.00240.982073.50166 × 10−810 mm−0.001720.981656.9709 × 10−8220°C, 6 mm22 m3/h−0.001920.966922.80133 × 10−829 m3/h−0.002340.973963.41412 × 10−836 m3/h−0.002730.972923.98314 × 10−840 m3/h−0.003090.979594.50839 × 10−8Note. Deff is the effective diffusion coefficient of sludge.According to the effective diffusion coefficient of sludge calculated in Table9 under the drying conditions of 6 mm sludge thickness, 22 m3/h steam flow, and different steam temperatures, the relationship between effective diffusion coefficient of sludge and steam temperature was made from formula (11), as shown in Figure 7.Figure 7
Fitting results between lnDeff and 1/T under different steam temperatures with sludge thickness of 6 mm and steam flow of 22 m3/h.It can be seen from Figure7 that the logarithm of the effective diffusion coefficient has a linear relationship with the reciprocal of the vapor temperature. The linear equation is ln Deff = −13.874–1614.2 (1/T) (R2 = 0.9924). According to formula (12), the activation energy of sludge superheated steam drying under the drying conditions of sludge thickness of 6 mm, steam flow rate of 22 m3/h, and different steam temperatures can be calculated as 13.42 kJ/mol.


## 3.1. Superheated Steam Drying Characteristics of Sludge

### 3.1.1. Influence of Steam Temperature on the Drying Characteristics of Superheated Steam from Sludge

Figures2(a) and 2(b) show the drying curves and drying rate curves of 6 mm sludge (corresponding to the sludge mass of 18 g) from 75.29% to 20% (w.b.) under the drying conditions of steam flow rate of 22 m3/h at different steam temperatures (180, 195, 220, and 260°C). In the figure, values of data points are taken every 1 min. As can be seen from Figure 2(a), the drying time required for 6 mm sludge to reach 20% under the drying conditions of steam flow rate of 22 m3/h and different steam temperatures (180, 195, 220, and 260°C) is 25, 22, 19, and 15 min, respectively. Increasing the steam temperature can significantly reduce the drying time. As can be seen from Figure 2(a), during the initial drying period, the moisture ratio of the sludge increases gradually. This is because the high-temperature steam meets the low-temperature material in the initial drying period, and the condensation of the steam leads to the gradual increase of water on the surface of the material. It can also be seen from Figure 2(b) that the initial drying rate of sludge is negative, which is also caused by steam condensation. It can also be seen from Figure 2(b) that the drying rate curves at different steam temperatures all increase first and then gradually decrease, which is the same as the conclusion of Qiao et al. on the drying characteristics of superheated steam in corm [17]. For the whole drying test at different steam temperatures, the drying rate of sludge was in the following order from high to low: 260°C > 220°C > 195°C > 180°C. The higher the steam temperature, the more intense the movement of water molecules in the sludge, the faster the water migration rate, and the higher the drying rate, so the sludge drying efficiency increases with the increase of steam temperature.Figure 2
Drying curves and drying rate curves of 6 mm sludge at different steam temperature under steam flow of 22 m3/h. (a) Drying curves. (b) Drying rate curves.
(a)(b)


### 3.1.2. Effect of Sludge Thickness on the Drying Characteristics of Superheated Steam of Sludge

Figures3(a) and 3(b) show the drying curve and drying rate curve when the sludge is dried from 75.29% to 20% (w.b.) under the drying conditions of steam temperature 220°C, steam flow rate 22 m3/h, and different sludge thickness of 2, 3.5, 6, and 10 mm (corresponding to sludge mass of 6, 11, 18, and 30 g). In the figure, values of data points are taken every 1 min. The time to reduce the initial moisture content of sludge from 75.29% to 20% under different sludge thicknesses was 8 min for 2 mm, 14 min for 3.5 mm, 20 min for 6 mm, and 27 min for 10 mm. Reducing sludge thickness can shorten drying time and improve drying efficiency, and sludge thickness has a significant impact on drying time (p<0.05). It can be seen from Figure 3(b) that, under different sludge thickening conditions, the drying rate of sludge increases first and then decreases gradually. Steam condensation in the initial stage of sludge superheated steam drying results in a negative initial drying rate of sludge. For the whole drying test with different sludge thickness, the drying rate of the sludge was 2 mm > 3.5 mm > 6 mm > 10 mm from high to low. In the desiccation stage of sludge superheated steam drying, the desiccation rate mainly depends on the heat and mass transfer characteristics inside the material [18]. Under the same steam temperature and steam flow rate and different sludge thickness drying conditions, the thicker the sludge, the longer the water migration path and the slower the water and heat migration efficiency. In addition, the thicker the sludge is, the more the water inside the sludge needs to be transferred from the interior to the surface of the sludge and the longer it takes to reach the target moisture content. Therefore, the drying efficiency of the sludge decreases with the increase of the sludge thickness [19, 20].Figure 3
Drying curves and drying rate curves of sludge at different sludge thicknesses under steam temperature of 220°C and steam flow of 22 m3/h. (a) Drying curves. (b) Drying rate curves.
(a)(b)


### 3.1.3. Influence of Steam Flow Rate on the Drying Characteristics of Superheated Steam of Sludge

Figures4(a) and 4(b), respectively, show the drying curve and drying rate curve of the sludge from 75.29% to 20% under the drying conditions of steam temperature 220°C, 6 mm thickness, and different steam flow rates. In the figure, values of data points are taken every 1 min. As can be seen from Figure 4(a), under the drying conditions with steam flow rates of 22, 29, 36, and 40 m3/h, respectively, it took 23, 20, 17, and 15 min for the sludge to go from 75.29% to 20% wet basis moisture content. As can be seen from Figure 4(b), under different steam flow rates, the overall drying rate of sludge increases first and then decreases gradually. In the initial stage of sludge superheated steam drying, high-temperature steam meets low-temperature material and causes steam condensation, thus making the initial drying rate of sludge negative. For the whole drying test with different steam flow rates, the overall drying rate of sludge is 40 m3/h > 36 m3/h > 29 m3/h > 22 m3/h from high to low. The greater the steam flow, the more sensible the heat of the steam and the more the heat provided to the material per unit time. The faster the material heats up, the higher the convective heat transfer and mass transfer efficiency in the material are. Therefore, the higher the steam flow, the higher the drying rate and the shorter the drying time. This was consistent with Pronky’s study of using superheated steam to dry beet residues [21].Figure 4
Drying curves and drying rate curves of 6 mm sludge at different sludge flow under steam temperature of 220°C. (a) Drying curves. (b) Drying rate curves.
(a)(b)


## 3.1.1. Influence of Steam Temperature on the Drying Characteristics of Superheated Steam from Sludge

Figures2(a) and 2(b) show the drying curves and drying rate curves of 6 mm sludge (corresponding to the sludge mass of 18 g) from 75.29% to 20% (w.b.) under the drying conditions of steam flow rate of 22 m3/h at different steam temperatures (180, 195, 220, and 260°C). In the figure, values of data points are taken every 1 min. As can be seen from Figure 2(a), the drying time required for 6 mm sludge to reach 20% under the drying conditions of steam flow rate of 22 m3/h and different steam temperatures (180, 195, 220, and 260°C) is 25, 22, 19, and 15 min, respectively. Increasing the steam temperature can significantly reduce the drying time. As can be seen from Figure 2(a), during the initial drying period, the moisture ratio of the sludge increases gradually. This is because the high-temperature steam meets the low-temperature material in the initial drying period, and the condensation of the steam leads to the gradual increase of water on the surface of the material. It can also be seen from Figure 2(b) that the initial drying rate of sludge is negative, which is also caused by steam condensation. It can also be seen from Figure 2(b) that the drying rate curves at different steam temperatures all increase first and then gradually decrease, which is the same as the conclusion of Qiao et al. on the drying characteristics of superheated steam in corm [17]. For the whole drying test at different steam temperatures, the drying rate of sludge was in the following order from high to low: 260°C > 220°C > 195°C > 180°C. The higher the steam temperature, the more intense the movement of water molecules in the sludge, the faster the water migration rate, and the higher the drying rate, so the sludge drying efficiency increases with the increase of steam temperature.Figure 2
Drying curves and drying rate curves of 6 mm sludge at different steam temperature under steam flow of 22 m3/h. (a) Drying curves. (b) Drying rate curves.
(a)(b)


## 3.1.2. Effect of Sludge Thickness on the Drying Characteristics of Superheated Steam of Sludge

Figures3(a) and 3(b) show the drying curve and drying rate curve when the sludge is dried from 75.29% to 20% (w.b.) under the drying conditions of steam temperature 220°C, steam flow rate 22 m3/h, and different sludge thickness of 2, 3.5, 6, and 10 mm (corresponding to sludge mass of 6, 11, 18, and 30 g). In the figure, values of data points are taken every 1 min. The time to reduce the initial moisture content of sludge from 75.29% to 20% under different sludge thicknesses was 8 min for 2 mm, 14 min for 3.5 mm, 20 min for 6 mm, and 27 min for 10 mm. Reducing sludge thickness can shorten drying time and improve drying efficiency, and sludge thickness has a significant impact on drying time (p<0.05). It can be seen from Figure 3(b) that, under different sludge thickening conditions, the drying rate of sludge increases first and then decreases gradually. Steam condensation in the initial stage of sludge superheated steam drying results in a negative initial drying rate of sludge. For the whole drying test with different sludge thickness, the drying rate of the sludge was 2 mm > 3.5 mm > 6 mm > 10 mm from high to low. In the desiccation stage of sludge superheated steam drying, the desiccation rate mainly depends on the heat and mass transfer characteristics inside the material [18]. Under the same steam temperature and steam flow rate and different sludge thickness drying conditions, the thicker the sludge, the longer the water migration path and the slower the water and heat migration efficiency. In addition, the thicker the sludge is, the more the water inside the sludge needs to be transferred from the interior to the surface of the sludge and the longer it takes to reach the target moisture content. Therefore, the drying efficiency of the sludge decreases with the increase of the sludge thickness [19, 20].Figure 3
Drying curves and drying rate curves of sludge at different sludge thicknesses under steam temperature of 220°C and steam flow of 22 m3/h. (a) Drying curves. (b) Drying rate curves.
(a)(b)


## 3.1.3. Influence of Steam Flow Rate on the Drying Characteristics of Superheated Steam of Sludge

Figures4(a) and 4(b), respectively, show the drying curve and drying rate curve of the sludge from 75.29% to 20% under the drying conditions of steam temperature 220°C, 6 mm thickness, and different steam flow rates. In the figure, values of data points are taken every 1 min. As can be seen from Figure 4(a), under the drying conditions with steam flow rates of 22, 29, 36, and 40 m3/h, respectively, it took 23, 20, 17, and 15 min for the sludge to go from 75.29% to 20% wet basis moisture content. As can be seen from Figure 4(b), under different steam flow rates, the overall drying rate of sludge increases first and then decreases gradually. In the initial stage of sludge superheated steam drying, high-temperature steam meets low-temperature material and causes steam condensation, thus making the initial drying rate of sludge negative. For the whole drying test with different steam flow rates, the overall drying rate of sludge is 40 m3/h > 36 m3/h > 29 m3/h > 22 m3/h from high to low. The greater the steam flow, the more sensible the heat of the steam and the more the heat provided to the material per unit time. The faster the material heats up, the higher the convective heat transfer and mass transfer efficiency in the material are. Therefore, the higher the steam flow, the higher the drying rate and the shorter the drying time. This was consistent with Pronky’s study of using superheated steam to dry beet residues [21].Figure 4
Drying curves and drying rate curves of 6 mm sludge at different sludge flow under steam temperature of 220°C. (a) Drying curves. (b) Drying rate curves.
(a)(b)


## 3.2. Characteristics of Initial Condensation of Superheated Steam Drying of Sludge

In the initial stage of superheated steam drying, when the high-temperature steam meets the low-temperature material, the high-temperature steam tends to condense. The condensed water adheres to the surface of the material, causing the material water to increase and the material water ratio to increase gradually. When the moisture ratio of the material increases to the maximum moisture ratio, the relative drying time at this time is called condensation time. The stage from the initial water ratio to the maximum water ratio is called the condensation stage. With the drying process, the evaporation rate of water exceeds the condensation rate. When the water ratio of the material gradually decreases from the maximum water ratio to the initial water ratio, the corresponding drying time is the recovery time, which is called the recovery stage. After the recovery time, the material moisture ratio gradually decreases from the initial moisture ratio to the sludge moisture content of 20% of the wet basis moisture content; so far, the whole drying process ends, and this stage is called the drying section. The condensation stage increases the drying time of the whole sludge, which has a negative effect on the whole drying process [22].Figures5(a)–5(c), respectively, represent the relationship between different steam temperature, sludge thickness, and steam flow and maximum condensation amount and recovery time of sludge. As can be seen from Figure 5(a), under the drying conditions with sludge thickness of 6 mm, steam flow rate of 22 m3/h, and different steam temperatures, the maximum condensation amount and recovery time of the sludge are when the steam temperature is 180°C, the corresponding maximum condensation amount is 1.06 g, and the maximum recovery time is 105 s. It can also be seen from Figure 5(a) that, with the increase of steam temperature, the maximum condensation amount and recovery time of sludge decrease. Kozanoglu et al. [23] also found that high temperature led to less vapor condensation of grains when they were dried in a low-pressure superheated steam fluidized bed. As can be seen from Figure 5(b), with the increase of sludge thickness, the maximum condensation amount and the recovery time of the sludge increase continuously. When the steam temperature is 220°C, the steam flow is 22 m3/h, and the sludge thickness is 2, 3.5, 6, and 10 mm, the corresponding maximum condensate volume is 0.26, 0.47, 0.74, and 1.03 g, respectively. The corresponding recovery time is 25, 40, 60, and 90 s, respectively. As can be seen from Figure 5(c), with the increase of steam flow rate, the maximum condensation amount and recovery time of sludge decrease. Under the drying conditions of 220°C steam temperature, 6 mm sludge thickness, and different steam flow rates (22, 29, 36, and 40 m3/h), the corresponding maximum amount of condensate was 0.98, 0.74, 0.7, and 0.47 g, respectively, and the corresponding recovery time was 90, 65, 60, and 30 s. Tang et al. [24] also found that, under the lower steam flow rate, the surface of materials would adhere to more water when they were drying thin-layer distillers’ grains with superheated steam. The maximum amount of condensation is also greater.Figure 5
Relationship between the amount of maximum condensation and restoration times and different drying conditions. (a) Relationship between the amount of maximum condensation and restoration times and steam temperature. Note: sludge thickness was 6 mm, and steam flow was 22 m3/h. (b) Relationship between the amount of maximum condensation and restoration times and sludge thickness. Note: steam temperature was 220°C, and steam flow was 22 m3/h. (c) Relationship between the amount of maximum condensation and restoration times and steam flow. Note: steam temperature was 220°C, and sludge thickness was 6 mm.
(a)(b)(c)


## 3.3. Mathematical Model of Sludge Superheated Steam Drying

### 3.3.1. Establishment of Sludge Superheated Steam Drying Model

In the process of superheated steam drying, sludge will show steam condensation, a condensation phenomenon. The whole process of superheated steam drying sludge can be divided into three stages: condensation stage, recovery stage, and drying stage. Different mathematical models are used to fit and analyze the process.


### 3.3.2. Drying Model of Superheated Steam Condensation Section of Sludge

The condensation stage refers to the process in which the sludge moisture ratio increases from the initial value to the maximum value during the process of sludge superheated steam drying. The law of moisture change in the condensation stage can be described by quadratic polynomials.(13)MR=1+At+Bt2.Formula (13) is the expression of a quadratic polynomial. MR is the drying water ratio of superheated steam. t is time, s. At the beginning of the drying process, Mt was equal to M0, so the value of the constant term of the formula (13) was 1. A and B are the coefficients.Table2lists the fitting results and fitting parameter values of the sludge superheated steam drying and condensation stage under different drying conditions.Table 2
Fitting results of condensation stage of sludge superheated steam drying under different drying conditions.
Fixed factor conditionVariable factor conditionModel coefficientsR2χ2 ×10−6RMSE ×10−36 mm, 22 m3/h180°CA = 6.90623 × 10−4; B = 2.50777 × 10−60.988199.127143.02195°CA = 3.09735 × 10−4; B = 1.65517 × 10−50.9491940.57936.37220°CA = 0.00196; B = −1.59104 × 10−50.9572916.43214.05260°CA = 8.79372 × 10−4; B = −2.223 × 10−60.956669.29863.05220°C, 22 m3/h2 mmA = 0.00176; B = −1.19441 × 10−50.992443.340661.833.5 mmA = 1.09123 × 10−4; B = 1.1279 × 10−50.970849.867993.146 mmA = 0.00115; B = −5.69471 × 10−60.994461.7981.3410 mmA = 1.73354 × 10−4; B = 8.0251 × 10−60.964028.53692.92220°C, 6 mm22 m3/hA = 0.00196; B = −8.86327 × 10−60.9824513.35073.6529 m3/hA = 1.82449 × 10−4; B = 9.2205 × 10−60.9710111.28973.3636 m3/hA = 6.4653 × 10−4; B = 2.38935 × 10−50.982037.514662.7440 m3/hA = 1.61425 × 10−4; B = 7.2576 × 10−60.995310.733440.8564It can be seen from Table1 that quadratic polynomials are adopted to fit the moisture change law of the superheated steam condensation stage of sludge under different drying conditions, and the fitting degree is relatively high. It can be used to describe the law of moisture change in the superheated steam condensation stage of sludge. The parameters A and B of the quadratic polynomial are considered as functions of steam temperature, sludge thickness, and steam flow. First-order function polynomials can be used to express the relationship between parameters A and B and steam temperature, sludge thickness, and steam flow. Let(14)A=B0+B1T+B2δ+B3Q+B4Tδ+B5TQ+B6δQ,(15)B=C0+C1T+C2δ+C3Q+C4Tδ+C5TQ+C6δQ.The values of parameters A and B obtained through fitting in Table1 and the corresponding drying conditions are substituted into equations (14) and (15). By using the least square method in Matlab to solve the overdetermined equations, it can be concluded that(16)B]=B0B1B2B3B4B5B6=7.77786839715×10−71.1601523864072×10−5−2.2261242076987×10−61.8271958357701×10−54.96376969165552×10−41.09009466012×10−74.975428958102×10−5T,C=C0C1C2C3C4C5C6=7.105650084297×10−8−1.328649949967×10−72.76505892957×10−81.57651654081×10−65.988537873284×10−6−2.85498486235×10−96.083926557042×10−7T.The seven elements in the matrix [B] and [C] are the coefficients of quadratic polynomials A and B.


### 3.3.3. Drying Model of Sludge Superheated Steam Recovery Section

The recovery stage of sludge superheated steam drying is a process in which the sludge moisture ratio gradually decreases from the maximum value to the initial value. The superheated steam recovery section of sludge is mainly to remove condensed water condensed by the condensation section. The condensed water adheres to the surface of the material and can be considered constant speed drying. Therefore, the law of moisture change in the recovery phase can be described by the linear relationship shown in formula (17).(17)MR=k0t−tcon+MRmax,where tcon is the condensation time, s; MRmax is the maximum water ratio; MR is the water ratio in the recovery stage.The fitting results and fitting parameters of the sludge superheated steam recovery stage are listed in Table3.Table 3
Fitting results of restoration stage of sludge superheated steam drying under different drying conditions.
Fixed factor conditionVariable factor conditionModel coefficientsk0R2χ2 ×10−6RMSE ×10−36 mm, 22 m3/h180°C−7.02597 × 10−40.99055.705182.39195°C−8.00787 × 10−40.984377.707332.78220°C−8.32655 × 10−40.975411.88853.45260°C−9.16322 × 10−40.980185.186842.28220°C, 22 m3/h2 mm−0.003010.9828317.31444.163.5 mm−0.001310.9445526.02875.16 mm−9.32495 × 10−40.988954.01967210 mm−5.40755 × 10−40.9547911.60663.41220°C, 6 mm22 m3/h−7.95143 × 10−40.987697.2812.729 m3/h−8.51206 × 10−40.9647214.26723.7836 m3/h−8.0649 × 10−40.9167531.87075.6540 m3/h−0.001110.961078.793752.97It can be seen from Table3 that the linear equation used to fit the superheated steam recovery stage of sludge has a higher fitting degree, which can be used to describe the water change law in the recovery stage. The slope of the linear equation is determined by steam temperature, sludge thickness, and steam flow. Let(18)k0=D0+D1T+D2δ+D3Q+D4Tδ+D5TQ+D6δQ+D7T2+D8δ2+D9Q2.The slope value in Table3 and the steam temperature, sludge thickness, and steam flow value are substituted into formula (18), the least square method is adopted by Matlab to solve the overdetermined equations, and the 10 coefficients of the slope k0 can be obtained.(19)D=D0D1D2D3D4D5D6D7D8D9=1.801451177×10−6−1.7314443107×10−51.956168515×10−54.138470927×10−54.271908539226×10−31.23440953×10−74.27449539855×10−4−3.0030188×10−8−56.590453167280614−1.402830725×10−6T.The 10 elements in the matrix [D] are the 10 coefficients of the slope k of the linear equation.


### 3.3.4. Drying Model of Sludge Superheated Steam Drying Section

The sludge superheated steam drying stage refers to the stage where the sludge moisture content is lower than the initial moisture content after the sludge superheated steam drying and recovery stage. In the superheated steam drying section of sludge, the moisture ratio of sludge can be described by the thin-layer drying model. Common thin-layer drying models are listed in Table4 [25].Table 4
Mathematical thin-layer drying models.
No.Model nameModel equation1LewisMR = exp(−kt)2PageMR = exp(−ktn)3Henderson and PabisMR = aexp(−kt)4OverhultsMR = exp[− (kt)n]5LogarithmicMR = aexp(−kt) + b6Two-term modeMR = aexp(−kt) + (1 − a)exp(−kat)Tables5–7 are the fitting results of the thin-layer drying model in the superheated steam drying stage of sludge under different steam temperature, sludge thickness, and steam flow. From Tables 5 to 7, it can be seen that the determination coefficient of the logarithmic model is above 0.998, which is better than other thin-layer drying models. The chi-square and root-mean-square error values of the logarithmic model are also smaller than other thin-layer drying models. Therefore, the logarithmic model is the most suitable thin-layer drying model to describe the moisture change rule during the superheated steam drying stage of sludge.Table 5
Fitting results of drying stage of sludge superheated steam drying under different steam temperature.
Model nameSteam temperature (°C)180195220260R2χ2 ×10−4RMSER2χ2 ×10−4RMSER2χ2 ×10−4RMSER2χ2 ×10−4RMSEPage0.997471.887880.013740.997411.95740.013990.99632.793740.016710.997891.613450.0127Henderson and Pabis0.97997150.038680.983112.80.035710.9850211.30.033650.9834812.60.03554Overhults0.997481.886010.013730.997411.957190.013990.996312.790640.016710.99791.606640.01268Logarithmic0.999570.3260.005710.99970.228270.004780.999730.207050.004550.999240.582680.00455Wis0.85204110.10.104920.88387.90.093760.85785106.80.103360.86979.90.09949Two-term model0.994354.221130.020550.995823.154410.017760.996422.706110.016450.996832.426280.01558Table 6
Fitting results of drying stage of sludge superheated steam drying under different thickness of sludge.
Model nameThickness of sludge (mm)2 mm3.5 mm6 mm10 mmR2χ2 ×10−4RMSER2χ2 ×10−4RMSER2χ2 ×10−4RMSER2χ2 ×10−4RMSEPage0.996392.663440.016320.998091.537110.01240.997921.64820.012840.997322.040150.01428Henderson and Pabis0.9760717.70.042030.9758519.40.044070.9837912.90.035860.988748.565590.02927Overhults0.996392.663330.016320.998091.536130.012390.997931.639360.01280.997332.031510.01425Logarithmic0.999770.173040.004160.998741.02110.01010.998521.176490.010850.999440.424220.00651Lewis0.8292124.40.111530.84284125.50.112040.87315100.10.100060.8994276.20.08732Two-term model0.990517.00520.026470.992665.906620.02430.997352.10470.014510.998591.071030.01305Table 7
Fitting results of drying stage of sludge superheated steam drying under different steam flow.
Model nameSteam flow (m3/h)22 m3/h29 m3/h36 m3/h40 m3/hR2χ2 ×10−4RMSER2χ2 ×10−4RMSER2χ2 ×10−4RMSER2χ2 ×10−4RMSEPage0.998191.35310.011630.997811.649660.012840.997971.564970.012510.997561.936580.01392Henderson and Pabis0.9810614.20.037640.98407120.034660.9842912.10.034760.985311.70.03417Overhults0.998191.353050.011630.997811.648280.012840.997971.563620.01250.997581.923970.01387Logarithmic0.999650.265170.005150.999480.392330.006260.99950.389690.006240.998850.918010.00958Lewis0.8878283.60.091420.8745694.10.0970.8937381.30.090150.8843891.30.09555Two-term model0.995283.530780.018790.996742.455650.015670.997032.286260.015120.998131.14880.0122Table8 shows different drying conditions, the logarithmic model fitting parameters, and the fitting results. It can be seen from Table 8 that with the increase of steam temperature, sludge thickness and steam flow, the model coefficient a of the logarithmic equation has an obvious decreasing trend, and with the increase of steam pressure, sludge thickness, and steam flow, the model coefficient b of the logarithmic equation also has an obvious increasing trend. With the increase of steam temperature and steam flow, the sludge thickness decreases, and the model coefficient K of the logarithmic equation has an increasing trend. The three coefficients a, b, and k of the logarithmic model equation are determined by the steam temperature, sludge thickness, and steam flow. The relationship between the model parameters and the steam temperature, sludge thickness, and steam flow rate can be expressed by quadratic polynomial. Let(20)a=E0+E1T+E2δ+E3Q+E4Tδ+E5TQ+E6δQ+E7T2+E8δ2+E9Q2,(21)k=F0+F1T+F2δ+F3Q+F4Tδ+F5TQ+F6δQ+F7T2+F8δ2+F9Q2,(22)b=G0+G1T+G2δ+G3Q+G4Tδ+G5TQ+G6δQ+G7T2+G8δ2+G9Q2,where a, k, and b are the parameters of the model of the logarithmic equation; T, δ, and Q represent steam temperature, sludge thickness, and steam flow, respectively.Table 8
Fitting parameter and results from the logarithmic model under different drying conditions.
Fixed factor conditionVariable factor conditionModel coefficientsR2χ2 ×10−4RMSE ×10−36 mm, 22 m3/h180°Ca = 1.90262, k = 6.56429 × 10−4, and b = −0.658080.99570.3265.71195°Ca = 1.72213, k = 8.21507 × 10−4, and b = −0.514010.99970.2282654.78220°Ca = 1.71767, k = 0.00111, and b = −0.440290.999730.2070484.55260°Ca = 1.70801, k = 0.00133, and b = −0.462580.999240.5826827.63220°C, 22 m3/h2 mma = 2.22072, k = 0.00172, and b = −0.97250.999770.1730394.163.5 mma = 1.93788, k = 0.00117, and b = −0.68020.9998741.021110.16 mma = 1.64436, k = 0.00111, and b = −0.388030.998521.1764910.8510 mma = 1.52454, k = 9.04004 × 10−4, and b = −0.299540.999440.4242166.51220°C, 6 mm22 m3/ha = 1.77499, k = 7.2396 × 10−4, and b = −0.58980.999650.2651675.1529 m3/ha = 1.69364, k = 0.00101, and b = −0.4570.999480.3923256.2636 m3/ha = 1.63835, k = 0.0012, and b = −0.437390.99950.3896926.2440 m3/ha = 1.6045, k = 0.00148, and b = −0.366320.998850.9180129.58Note. a, k, and b are fit parameters.The parameter values in Table8 are substituted into equations (18)–(20), and the least square method is used to solve the overdetermined equations with Matlab software. It can be calculated as(23)G=G0G1G2G3G4G5G6G7G8G9=−0.0077855990.0093344690.004262615−0.1722097790.9425756980.0006415110.093218708−6.11992×10−5−10729.754880.000566074T.In matrix [E], [F], [G] the 10 elements correspond to the 10 coefficients of the three model parameters a, b, and k in the logarithmic equation.


### 3.3.5. Model Validation

In order to verify the drying model of the above three stages, the sludge was tested under the drying conditions of steam temperature of 215°C, sludge thickness of 8 mm, and steam flow of 30 m3/h. The experimental values are compared with the predicted values of the drying model, and the results are shown in Figure 6. The experimental value and the predicted value were fitted, and the results showed that the fitting degree of the two was very high, with R2 = 0.99951, RMSE = 0.00664, and χ2 = 4.40285 × 10−5. The above three drying models can be used to describe the moisture variation of the whole sludge superheated steam drying stage.Figure 6
Verification of sludge superheated steam drying model.


## 3.3.1. Establishment of Sludge Superheated Steam Drying Model

In the process of superheated steam drying, sludge will show steam condensation, a condensation phenomenon. The whole process of superheated steam drying sludge can be divided into three stages: condensation stage, recovery stage, and drying stage. Different mathematical models are used to fit and analyze the process.


## 3.3.2. Drying Model of Superheated Steam Condensation Section of Sludge

The condensation stage refers to the process in which the sludge moisture ratio increases from the initial value to the maximum value during the process of sludge superheated steam drying. The law of moisture change in the condensation stage can be described by quadratic polynomials.(13)MR=1+At+Bt2.Formula (13) is the expression of a quadratic polynomial. MR is the drying water ratio of superheated steam. t is time, s. At the beginning of the drying process, Mt was equal to M0, so the value of the constant term of the formula (13) was 1. A and B are the coefficients.Table2lists the fitting results and fitting parameter values of the sludge superheated steam drying and condensation stage under different drying conditions.Table 2
Fitting results of condensation stage of sludge superheated steam drying under different drying conditions.
Fixed factor conditionVariable factor conditionModel coefficientsR2χ2 ×10−6RMSE ×10−36 mm, 22 m3/h180°CA = 6.90623 × 10−4; B = 2.50777 × 10−60.988199.127143.02195°CA = 3.09735 × 10−4; B = 1.65517 × 10−50.9491940.57936.37220°CA = 0.00196; B = −1.59104 × 10−50.9572916.43214.05260°CA = 8.79372 × 10−4; B = −2.223 × 10−60.956669.29863.05220°C, 22 m3/h2 mmA = 0.00176; B = −1.19441 × 10−50.992443.340661.833.5 mmA = 1.09123 × 10−4; B = 1.1279 × 10−50.970849.867993.146 mmA = 0.00115; B = −5.69471 × 10−60.994461.7981.3410 mmA = 1.73354 × 10−4; B = 8.0251 × 10−60.964028.53692.92220°C, 6 mm22 m3/hA = 0.00196; B = −8.86327 × 10−60.9824513.35073.6529 m3/hA = 1.82449 × 10−4; B = 9.2205 × 10−60.9710111.28973.3636 m3/hA = 6.4653 × 10−4; B = 2.38935 × 10−50.982037.514662.7440 m3/hA = 1.61425 × 10−4; B = 7.2576 × 10−60.995310.733440.8564It can be seen from Table1 that quadratic polynomials are adopted to fit the moisture change law of the superheated steam condensation stage of sludge under different drying conditions, and the fitting degree is relatively high. It can be used to describe the law of moisture change in the superheated steam condensation stage of sludge. The parameters A and B of the quadratic polynomial are considered as functions of steam temperature, sludge thickness, and steam flow. First-order function polynomials can be used to express the relationship between parameters A and B and steam temperature, sludge thickness, and steam flow. Let(14)A=B0+B1T+B2δ+B3Q+B4Tδ+B5TQ+B6δQ,(15)B=C0+C1T+C2δ+C3Q+C4Tδ+C5TQ+C6δQ.The values of parameters A and B obtained through fitting in Table1 and the corresponding drying conditions are substituted into equations (14) and (15). By using the least square method in Matlab to solve the overdetermined equations, it can be concluded that(16)B]=B0B1B2B3B4B5B6=7.77786839715×10−71.1601523864072×10−5−2.2261242076987×10−61.8271958357701×10−54.96376969165552×10−41.09009466012×10−74.975428958102×10−5T,C=C0C1C2C3C4C5C6=7.105650084297×10−8−1.328649949967×10−72.76505892957×10−81.57651654081×10−65.988537873284×10−6−2.85498486235×10−96.083926557042×10−7T.The seven elements in the matrix [B] and [C] are the coefficients of quadratic polynomials A and B.


## 3.3.3. Drying Model of Sludge Superheated Steam Recovery Section

The recovery stage of sludge superheated steam drying is a process in which the sludge moisture ratio gradually decreases from the maximum value to the initial value. The superheated steam recovery section of sludge is mainly to remove condensed water condensed by the condensation section. The condensed water adheres to the surface of the material and can be considered constant speed drying. Therefore, the law of moisture change in the recovery phase can be described by the linear relationship shown in formula (17).(17)MR=k0t−tcon+MRmax,where tcon is the condensation time, s; MRmax is the maximum water ratio; MR is the water ratio in the recovery stage.The fitting results and fitting parameters of the sludge superheated steam recovery stage are listed in Table3.Table 3
Fitting results of restoration stage of sludge superheated steam drying under different drying conditions.
Fixed factor conditionVariable factor conditionModel coefficientsk0R2χ2 ×10−6RMSE ×10−36 mm, 22 m3/h180°C−7.02597 × 10−40.99055.705182.39195°C−8.00787 × 10−40.984377.707332.78220°C−8.32655 × 10−40.975411.88853.45260°C−9.16322 × 10−40.980185.186842.28220°C, 22 m3/h2 mm−0.003010.9828317.31444.163.5 mm−0.001310.9445526.02875.16 mm−9.32495 × 10−40.988954.01967210 mm−5.40755 × 10−40.9547911.60663.41220°C, 6 mm22 m3/h−7.95143 × 10−40.987697.2812.729 m3/h−8.51206 × 10−40.9647214.26723.7836 m3/h−8.0649 × 10−40.9167531.87075.6540 m3/h−0.001110.961078.793752.97It can be seen from Table3 that the linear equation used to fit the superheated steam recovery stage of sludge has a higher fitting degree, which can be used to describe the water change law in the recovery stage. The slope of the linear equation is determined by steam temperature, sludge thickness, and steam flow. Let(18)k0=D0+D1T+D2δ+D3Q+D4Tδ+D5TQ+D6δQ+D7T2+D8δ2+D9Q2.The slope value in Table3 and the steam temperature, sludge thickness, and steam flow value are substituted into formula (18), the least square method is adopted by Matlab to solve the overdetermined equations, and the 10 coefficients of the slope k0 can be obtained.(19)D=D0D1D2D3D4D5D6D7D8D9=1.801451177×10−6−1.7314443107×10−51.956168515×10−54.138470927×10−54.271908539226×10−31.23440953×10−74.27449539855×10−4−3.0030188×10−8−56.590453167280614−1.402830725×10−6T.The 10 elements in the matrix [D] are the 10 coefficients of the slope k of the linear equation.


## 3.3.4. Drying Model of Sludge Superheated Steam Drying Section

The sludge superheated steam drying stage refers to the stage where the sludge moisture content is lower than the initial moisture content after the sludge superheated steam drying and recovery stage. In the superheated steam drying section of sludge, the moisture ratio of sludge can be described by the thin-layer drying model. Common thin-layer drying models are listed in Table4 [25].Table 4
Mathematical thin-layer drying models.
No.Model nameModel equation1LewisMR = exp(−kt)2PageMR = exp(−ktn)3Henderson and PabisMR = aexp(−kt)4OverhultsMR = exp[− (kt)n]5LogarithmicMR = aexp(−kt) + b6Two-term modeMR = aexp(−kt) + (1 − a)exp(−kat)Tables5–7 are the fitting results of the thin-layer drying model in the superheated steam drying stage of sludge under different steam temperature, sludge thickness, and steam flow. From Tables 5 to 7, it can be seen that the determination coefficient of the logarithmic model is above 0.998, which is better than other thin-layer drying models. The chi-square and root-mean-square error values of the logarithmic model are also smaller than other thin-layer drying models. Therefore, the logarithmic model is the most suitable thin-layer drying model to describe the moisture change rule during the superheated steam drying stage of sludge.Table 5
Fitting results of drying stage of sludge superheated steam drying under different steam temperature.
Model nameSteam temperature (°C)180195220260R2χ2 ×10−4RMSER2χ2 ×10−4RMSER2χ2 ×10−4RMSER2χ2 ×10−4RMSEPage0.997471.887880.013740.997411.95740.013990.99632.793740.016710.997891.613450.0127Henderson and Pabis0.97997150.038680.983112.80.035710.9850211.30.033650.9834812.60.03554Overhults0.997481.886010.013730.997411.957190.013990.996312.790640.016710.99791.606640.01268Logarithmic0.999570.3260.005710.99970.228270.004780.999730.207050.004550.999240.582680.00455Wis0.85204110.10.104920.88387.90.093760.85785106.80.103360.86979.90.09949Two-term model0.994354.221130.020550.995823.154410.017760.996422.706110.016450.996832.426280.01558Table 6
Fitting results of drying stage of sludge superheated steam drying under different thickness of sludge.
Model nameThickness of sludge (mm)2 mm3.5 mm6 mm10 mmR2χ2 ×10−4RMSER2χ2 ×10−4RMSER2χ2 ×10−4RMSER2χ2 ×10−4RMSEPage0.996392.663440.016320.998091.537110.01240.997921.64820.012840.997322.040150.01428Henderson and Pabis0.9760717.70.042030.9758519.40.044070.9837912.90.035860.988748.565590.02927Overhults0.996392.663330.016320.998091.536130.012390.997931.639360.01280.997332.031510.01425Logarithmic0.999770.173040.004160.998741.02110.01010.998521.176490.010850.999440.424220.00651Lewis0.8292124.40.111530.84284125.50.112040.87315100.10.100060.8994276.20.08732Two-term model0.990517.00520.026470.992665.906620.02430.997352.10470.014510.998591.071030.01305Table 7
Fitting results of drying stage of sludge superheated steam drying under different steam flow.
Model nameSteam flow (m3/h)22 m3/h29 m3/h36 m3/h40 m3/hR2χ2 ×10−4RMSER2χ2 ×10−4RMSER2χ2 ×10−4RMSER2χ2 ×10−4RMSEPage0.998191.35310.011630.997811.649660.012840.997971.564970.012510.997561.936580.01392Henderson and Pabis0.9810614.20.037640.98407120.034660.9842912.10.034760.985311.70.03417Overhults0.998191.353050.011630.997811.648280.012840.997971.563620.01250.997581.923970.01387Logarithmic0.999650.265170.005150.999480.392330.006260.99950.389690.006240.998850.918010.00958Lewis0.8878283.60.091420.8745694.10.0970.8937381.30.090150.8843891.30.09555Two-term model0.995283.530780.018790.996742.455650.015670.997032.286260.015120.998131.14880.0122Table8 shows different drying conditions, the logarithmic model fitting parameters, and the fitting results. It can be seen from Table 8 that with the increase of steam temperature, sludge thickness and steam flow, the model coefficient a of the logarithmic equation has an obvious decreasing trend, and with the increase of steam pressure, sludge thickness, and steam flow, the model coefficient b of the logarithmic equation also has an obvious increasing trend. With the increase of steam temperature and steam flow, the sludge thickness decreases, and the model coefficient K of the logarithmic equation has an increasing trend. The three coefficients a, b, and k of the logarithmic model equation are determined by the steam temperature, sludge thickness, and steam flow. The relationship between the model parameters and the steam temperature, sludge thickness, and steam flow rate can be expressed by quadratic polynomial. Let(20)a=E0+E1T+E2δ+E3Q+E4Tδ+E5TQ+E6δQ+E7T2+E8δ2+E9Q2,(21)k=F0+F1T+F2δ+F3Q+F4Tδ+F5TQ+F6δQ+F7T2+F8δ2+F9Q2,(22)b=G0+G1T+G2δ+G3Q+G4Tδ+G5TQ+G6δQ+G7T2+G8δ2+G9Q2,where a, k, and b are the parameters of the model of the logarithmic equation; T, δ, and Q represent steam temperature, sludge thickness, and steam flow, respectively.Table 8
Fitting parameter and results from the logarithmic model under different drying conditions.
Fixed factor conditionVariable factor conditionModel coefficientsR2χ2 ×10−4RMSE ×10−36 mm, 22 m3/h180°Ca = 1.90262, k = 6.56429 × 10−4, and b = −0.658080.99570.3265.71195°Ca = 1.72213, k = 8.21507 × 10−4, and b = −0.514010.99970.2282654.78220°Ca = 1.71767, k = 0.00111, and b = −0.440290.999730.2070484.55260°Ca = 1.70801, k = 0.00133, and b = −0.462580.999240.5826827.63220°C, 22 m3/h2 mma = 2.22072, k = 0.00172, and b = −0.97250.999770.1730394.163.5 mma = 1.93788, k = 0.00117, and b = −0.68020.9998741.021110.16 mma = 1.64436, k = 0.00111, and b = −0.388030.998521.1764910.8510 mma = 1.52454, k = 9.04004 × 10−4, and b = −0.299540.999440.4242166.51220°C, 6 mm22 m3/ha = 1.77499, k = 7.2396 × 10−4, and b = −0.58980.999650.2651675.1529 m3/ha = 1.69364, k = 0.00101, and b = −0.4570.999480.3923256.2636 m3/ha = 1.63835, k = 0.0012, and b = −0.437390.99950.3896926.2440 m3/ha = 1.6045, k = 0.00148, and b = −0.366320.998850.9180129.58Note. a, k, and b are fit parameters.The parameter values in Table8 are substituted into equations (18)–(20), and the least square method is used to solve the overdetermined equations with Matlab software. It can be calculated as(23)G=G0G1G2G3G4G5G6G7G8G9=−0.0077855990.0093344690.004262615−0.1722097790.9425756980.0006415110.093218708−6.11992×10−5−10729.754880.000566074T.In matrix [E], [F], [G] the 10 elements correspond to the 10 coefficients of the three model parameters a, b, and k in the logarithmic equation.


## 3.3.5. Model Validation

In order to verify the drying model of the above three stages, the sludge was tested under the drying conditions of steam temperature of 215°C, sludge thickness of 8 mm, and steam flow of 30 m3/h. The experimental values are compared with the predicted values of the drying model, and the results are shown in Figure 6. The experimental value and the predicted value were fitted, and the results showed that the fitting degree of the two was very high, with R2 = 0.99951, RMSE = 0.00664, and χ2 = 4.40285 × 10−5. The above three drying models can be used to describe the moisture variation of the whole sludge superheated steam drying stage.Figure 6
Verification of sludge superheated steam drying model.


## 3.4. Effective Diffusion Coefficient and Activation Energy

Table9 shows the calculation results of effective diffusion coefficient of sludge superheated steam drying under different drying conditions. It can be seen from Table 9 that the effective diffusion coefficient of sludge superheated steam drying is 2.684606 × 10−8∼4.5229776 × 10−8 under the drying conditions of sludge thickness of 6 mm, steam flow rate of 22 m3/h, and different steam temperatures. Under the drying conditions of steam temperature of 220°C, steam flow rate of 22 m3/h, and different sludge thickness, the effective diffusion coefficient of sludge superheated steam drying is 1.00024 × 10−8∼6.9709 × 10−8. Under the drying conditions of 220°C steam temperature, 6 mm sludge thickness, and different steam flow rates, the effective diffusion coefficient of sludge superheated steam drying was 2.80133 × 10−8∼4.50839 × 10−8. It can also be seen from Table 8 that, under the drying conditions of the same sludge thickness, steam flow rate, and different steam temperature, the effective diffusion coefficient of sludge increases with the increase of steam temperature. This is because the higher the steam temperature, the higher the heat and mass transfer efficiency and the faster the water diffusion and evaporation rate. Under the same steam temperature, steam flow rate, and different sludge thickness drying conditions, the effective diffusion coefficient of the sludge increases with the increase of the sludge thickness. It can be seen from formula (10) that the water effective diffusion coefficient Deff is proportional to 4δ2, the thicker the sludge, the greater the effective diffusion coefficient of water. The effective diffusion coefficient of sludge increases with the increase of steam flow. This is because the greater the steam flow, the more the heat transferred to the material through the drying medium and the faster the water diffusion [26].Table 9
Moisture effective diffusion coefficients of sludge under different superheated steam drying condition.
Fixed factor conditionVariable factor conditionSlopekR2Deff/(m2·s−1)6 mm, 22 m3/h180°C−0.001840.963412.6846061 × 10−8195°C−0.002020.968282.9472306 × 10−8220°C−0.002520.971063.6767431 × 10−8260°C−0.003010.974554.5229776 × 10−8220°C, 22 m3/h2 mm−0.006170.950011.00024 × 10−83.5 mm−0.003190.964561.68305 × 10−86 mm−0.00240.982073.50166 × 10−810 mm−0.001720.981656.9709 × 10−8220°C, 6 mm22 m3/h−0.001920.966922.80133 × 10−829 m3/h−0.002340.973963.41412 × 10−836 m3/h−0.002730.972923.98314 × 10−840 m3/h−0.003090.979594.50839 × 10−8Note. Deff is the effective diffusion coefficient of sludge.According to the effective diffusion coefficient of sludge calculated in Table9 under the drying conditions of 6 mm sludge thickness, 22 m3/h steam flow, and different steam temperatures, the relationship between effective diffusion coefficient of sludge and steam temperature was made from formula (11), as shown in Figure 7.Figure 7
Fitting results between lnDeff and 1/T under different steam temperatures with sludge thickness of 6 mm and steam flow of 22 m3/h.It can be seen from Figure7 that the logarithm of the effective diffusion coefficient has a linear relationship with the reciprocal of the vapor temperature. The linear equation is ln Deff = −13.874–1614.2 (1/T) (R2 = 0.9924). According to formula (12), the activation energy of sludge superheated steam drying under the drying conditions of sludge thickness of 6 mm, steam flow rate of 22 m3/h, and different steam temperatures can be calculated as 13.42 kJ/mol.


## 4. Response Surface Tested Results and Analysis

### 4.1. Tested Scheme and Results

An analytical experiment on the level of 3 factors 5 is designed according to the design principle of Central Composite Design [27]. Take the relative unit energy consumption as the response value, and with steam temperature, sludge mass, and steam flow as the tested factors, the relationship between the three tested factors and the response value is studied by the response surface method so as to optimize the parameters of the sludge superheated steam process. Table 10 shows the tested design and results.Table 10
Experimental designs and results.
Serial numberSteam temperatureX1The mass of sludgeX2Steam flowX3The relative unit consumption of energyY (kJ∙g−1)1111322.9166667211−1312.531−11402.631578941−1−1473.68421055−111281.256−11−1322.91666677−1−11416.45244228−1−1−14509−1.68200358.8424437101.68200352.7710843110−1.6820520.48192771201.6820293.80700911300−1.682347.26688114001.682334.624322215000289.389067516000318.327974317000260.450160818000318.072289219000346.987951820000347.26688121000376.205787822000318.327974323000289.3890675


### 4.2. Regression Equation and Parameter Analysis

Through the fitting analysis of the tested data in Table10 by Design Expert v8.0.6.1 software, the multiple regression equation between the relative unit energy consumption (y) and steam temperature (X1), sludge mass (X2), and steam flow (X3) is obtained, as shown in formula (21):(24)Y=2148.31461−10.09451X1−41.40060X2−17.23776X3+0.015276X1X2+0.01413X1X3+0.18712X2X3+0.021837X12+0.64895X22+0.17310X32,where Y is the relative unit energy consumption of sludge superheated steam drying, kJ/g.The variance analysis of multiple regression equation is shown in Table11. It can be seen from Table 11 that the p value of the regression equation model is less than 0.05 (significant). The p value of the missing item is much greater than 0.05 (not significant). The determination coefficient R2 is 0.8708, which indicates that the regression equation model can be used for prediction and analysis [28].Table 11
Experimental analysis of variance table on relative unit consumption energy.
ItemQuadratic sumDOFRMSEF valuep valueModel79876.3498875.159.730.0002X169.93169.930.00770.7862X257273.25157273.2562.81<0.0001X31807.4711807.471.980.1826X122959.6112959.613.250.0948X2216066.41116066.4117.620.0010X321119.4511119.451.230.2879X1X257.17157.170.0630.8062X1X326.57126.570.0290.8671X2X3672.531672.530.740.4060Remainder11853.3013911.79Lack of fit1819.815363.960.290.9055Error10033.4981254.19Sum91729.6522R2 = 0.8708The influence degree of influencing factors can be judged by the value of partial regression coefficientF [29]. The greater the value of F, the higher the degree of influence. It can be seen from Table 11 that the F values of steam temperature (X1), sludge mass (X2), and steam flow (X3) are 0.0077, 62.81, and 1.98, respectively, which shows that the order of the influence of the three factors on the relative unit energy consumption is sludge mass > steam flow > steam temperature.


### 4.3. The Influence of Each Tested Factor on the Test Evaluation Index

Figure8 shows the influence of various tested factors on relative unit energy consumption. Figure 8(a) shows a three-dimensional response surface and contour map of the effect of steam temperature and sludge mass on relative unit energy consumption. It can be seen from Figure 8(a) that when the sludge mass is fixed, the relative unit energy consumption will decrease first and then increase with the increase of steam temperature. This is because when the steam temperature is low, with the increase of steam temperature, the drying rate increases, the drying time shortens, and the relative unit energy consumption decreases. When the steam temperature is high, with the increase of steam temperature, the energy consumption for maintaining the system to dry at high temperature is high, which makes the relative unit energy consumption increase. Therefore, choosing the appropriate steam temperature value can make the relative unit energy consumption at a low level. It can also be seen from Figure 8(a) that, at a certain steam temperature, with the continuous increase of sludge mass (from the level of −1.682 to about 0.841), the relative unit energy consumption presents a decreasing trend, while with the further increase of sludge mass (from about 0.841 to 1.682), the relative unit energy consumption presents an increasing trend. This is because when the sludge mass is small, with the increase of the sludge mass, the water removed from the sludge increases, and the relative unit energy consumption decreases. When the sludge mass is large, the longer drying time leads to an increase in the energy consumption of the maintenance system under this drying condition and the test parameters. So that when the sludge mass is large, the relative unit energy consumption presents an increasing trend. It can also be seen from Figure 8(a) that the effect of sludge mass on relative unit energy consumption is higher than that of steam temperature, which is because the main determinant of relative unit energy consumption is the quality of water removed from the sludge.Figure 8
Effects of relative unit energy consumption on mass of sludge temperature, steam flow, and mass of sludge. (a) Response surface plot and contour map of relative energy consumption on steam temperature and mass of sludge. Note: steam flow is fixed at 0 level. (b) Response surface plot and contour map of relative energy consumption on steam temperature and steam flow. Note: mass of sludge is fixed at 0 level. (c) Response surface plot and contour map of relative energy consumption on mass of sludge and steam flow. Note: steam temperature is fixed at 0 level.
(a)(b)(c)(d)(e)(f)The three-dimensional response surface and contour diagram of steam temperature and steam flow are shown in Figure8(b). As can be seen from Figure 8(b), with the increase of steam temperature and steam flow, the relative unit energy consumption first decreases and then increases. This is because when the steam temperature or steam flow is small, with the increase of steam temperature and steam flow, the heat and mass transfer efficiency increases, and the drying time is shortened. Therefore, the relative unit energy consumption decreases. When the steam temperature or steam flow is large, with the increase of the steam temperature or steam flow, more energy is needed to heat the steam of the same volume to a higher temperature or to heat more steam of the same volume to the same temperature, which results in an increasing trend of relative unit energy consumption.When the steam temperature is at the level of −0.841∼0 and the steam flow is at the level of 0∼0.841, the minimum relative unit energy consumption can be obtained.The three-dimensional response surface and contour diagram of sludge mass and steam flow are shown in Figure8(c). As can be seen from Figure 8(c), with the increase of sludge mass and steam flow, the relative unit energy consumption first decreases and then increases. This is because when the steam flow is constant and the sludge mass is small, with the increase of sludge mass, the water content of sludge removal increases, resulting in the trend of decreasing relative unit energy consumption. When the sludge mass is large, the drying time increases, and the energy consumption required to maintain the system under the drying condition increases, resulting in the trend of increasing relative unit energy consumption. When the mass of sludge is fixed and the steam flow is small, with the increase of steam flow, the efficiency of heat and mass transfer increases, more heat exchange can be carried out with sludge, drying time is shortened, and the relative unit energy consumption is reduced; when the steam flow is large, with the increase of steam flow, more energy is needed to heat more volume of steam. Therefore, the relative unit energy consumption is increasing. It can also be seen from Figure 8(c) that the effect of sludge mass on relative unit energy consumption is higher than that of steam flow because the main determinant of relative unit energy consumption is the mass of water removed from sludge.


### 4.4. Determination of Optimum Drying Conditions and Model Validation

The drying process with low energy consumption is the key to reduce the drying cost. The multiple regression equation is optimized by using Design Expert V8.0.6.1, and the optimized result is as follows:X1 = −0.192; X2 = 1.002; X3 = 0.160; Ymin = 284.609 kJ/g. That is, the steam temperature is 215.19°C, the sludge mass is 25.02 g (sludge thickness 8 mm), and the steam flow is 30.12 m3/h. In actual engineering, the modified drying process parameters were steam temperature 215°C, sludge mass 25 g (sludge thickness 8 mm), and steam flow 30 m3/h. Three tests of sludge superheated steam drying were carried out under the modified drying process. Take the average of its relative unit energy consumption. The results are compared with the predicted values to verify the reliability of the model optimized drying process. The average value of sludge energy consumption per unit is 284.61 kJ/g; compared with the predicted value, the relative error is 1.65%, which shows that the drying process parameters optimized by the model are more reliable. The value of 284.61 kJ/g is higher than that in the published literature, and the reason is that the quality of the test sludge is small, so the test results were compared and analyzed with the relative unit energy consumption.


## 4.1. Tested Scheme and Results

An analytical experiment on the level of 3 factors 5 is designed according to the design principle of Central Composite Design [27]. Take the relative unit energy consumption as the response value, and with steam temperature, sludge mass, and steam flow as the tested factors, the relationship between the three tested factors and the response value is studied by the response surface method so as to optimize the parameters of the sludge superheated steam process. Table 10 shows the tested design and results.Table 10
Experimental designs and results.
Serial numberSteam temperatureX1The mass of sludgeX2Steam flowX3The relative unit consumption of energyY (kJ∙g−1)1111322.9166667211−1312.531−11402.631578941−1−1473.68421055−111281.256−11−1322.91666677−1−11416.45244228−1−1−14509−1.68200358.8424437101.68200352.7710843110−1.6820520.48192771201.6820293.80700911300−1.682347.26688114001.682334.624322215000289.389067516000318.327974317000260.450160818000318.072289219000346.987951820000347.26688121000376.205787822000318.327974323000289.3890675


## 4.2. Regression Equation and Parameter Analysis

Through the fitting analysis of the tested data in Table10 by Design Expert v8.0.6.1 software, the multiple regression equation between the relative unit energy consumption (y) and steam temperature (X1), sludge mass (X2), and steam flow (X3) is obtained, as shown in formula (21):(24)Y=2148.31461−10.09451X1−41.40060X2−17.23776X3+0.015276X1X2+0.01413X1X3+0.18712X2X3+0.021837X12+0.64895X22+0.17310X32,where Y is the relative unit energy consumption of sludge superheated steam drying, kJ/g.The variance analysis of multiple regression equation is shown in Table11. It can be seen from Table 11 that the p value of the regression equation model is less than 0.05 (significant). The p value of the missing item is much greater than 0.05 (not significant). The determination coefficient R2 is 0.8708, which indicates that the regression equation model can be used for prediction and analysis [28].Table 11
Experimental analysis of variance table on relative unit consumption energy.
ItemQuadratic sumDOFRMSEF valuep valueModel79876.3498875.159.730.0002X169.93169.930.00770.7862X257273.25157273.2562.81<0.0001X31807.4711807.471.980.1826X122959.6112959.613.250.0948X2216066.41116066.4117.620.0010X321119.4511119.451.230.2879X1X257.17157.170.0630.8062X1X326.57126.570.0290.8671X2X3672.531672.530.740.4060Remainder11853.3013911.79Lack of fit1819.815363.960.290.9055Error10033.4981254.19Sum91729.6522R2 = 0.8708The influence degree of influencing factors can be judged by the value of partial regression coefficientF [29]. The greater the value of F, the higher the degree of influence. It can be seen from Table 11 that the F values of steam temperature (X1), sludge mass (X2), and steam flow (X3) are 0.0077, 62.81, and 1.98, respectively, which shows that the order of the influence of the three factors on the relative unit energy consumption is sludge mass > steam flow > steam temperature.


## 4.3. The Influence of Each Tested Factor on the Test Evaluation Index

Figure8 shows the influence of various tested factors on relative unit energy consumption. Figure 8(a) shows a three-dimensional response surface and contour map of the effect of steam temperature and sludge mass on relative unit energy consumption. It can be seen from Figure 8(a) that when the sludge mass is fixed, the relative unit energy consumption will decrease first and then increase with the increase of steam temperature. This is because when the steam temperature is low, with the increase of steam temperature, the drying rate increases, the drying time shortens, and the relative unit energy consumption decreases. When the steam temperature is high, with the increase of steam temperature, the energy consumption for maintaining the system to dry at high temperature is high, which makes the relative unit energy consumption increase. Therefore, choosing the appropriate steam temperature value can make the relative unit energy consumption at a low level. It can also be seen from Figure 8(a) that, at a certain steam temperature, with the continuous increase of sludge mass (from the level of −1.682 to about 0.841), the relative unit energy consumption presents a decreasing trend, while with the further increase of sludge mass (from about 0.841 to 1.682), the relative unit energy consumption presents an increasing trend. This is because when the sludge mass is small, with the increase of the sludge mass, the water removed from the sludge increases, and the relative unit energy consumption decreases. When the sludge mass is large, the longer drying time leads to an increase in the energy consumption of the maintenance system under this drying condition and the test parameters. So that when the sludge mass is large, the relative unit energy consumption presents an increasing trend. It can also be seen from Figure 8(a) that the effect of sludge mass on relative unit energy consumption is higher than that of steam temperature, which is because the main determinant of relative unit energy consumption is the quality of water removed from the sludge.Figure 8
Effects of relative unit energy consumption on mass of sludge temperature, steam flow, and mass of sludge. (a) Response surface plot and contour map of relative energy consumption on steam temperature and mass of sludge. Note: steam flow is fixed at 0 level. (b) Response surface plot and contour map of relative energy consumption on steam temperature and steam flow. Note: mass of sludge is fixed at 0 level. (c) Response surface plot and contour map of relative energy consumption on mass of sludge and steam flow. Note: steam temperature is fixed at 0 level.
(a)(b)(c)(d)(e)(f)The three-dimensional response surface and contour diagram of steam temperature and steam flow are shown in Figure8(b). As can be seen from Figure 8(b), with the increase of steam temperature and steam flow, the relative unit energy consumption first decreases and then increases. This is because when the steam temperature or steam flow is small, with the increase of steam temperature and steam flow, the heat and mass transfer efficiency increases, and the drying time is shortened. Therefore, the relative unit energy consumption decreases. When the steam temperature or steam flow is large, with the increase of the steam temperature or steam flow, more energy is needed to heat the steam of the same volume to a higher temperature or to heat more steam of the same volume to the same temperature, which results in an increasing trend of relative unit energy consumption.When the steam temperature is at the level of −0.841∼0 and the steam flow is at the level of 0∼0.841, the minimum relative unit energy consumption can be obtained.The three-dimensional response surface and contour diagram of sludge mass and steam flow are shown in Figure8(c). As can be seen from Figure 8(c), with the increase of sludge mass and steam flow, the relative unit energy consumption first decreases and then increases. This is because when the steam flow is constant and the sludge mass is small, with the increase of sludge mass, the water content of sludge removal increases, resulting in the trend of decreasing relative unit energy consumption. When the sludge mass is large, the drying time increases, and the energy consumption required to maintain the system under the drying condition increases, resulting in the trend of increasing relative unit energy consumption. When the mass of sludge is fixed and the steam flow is small, with the increase of steam flow, the efficiency of heat and mass transfer increases, more heat exchange can be carried out with sludge, drying time is shortened, and the relative unit energy consumption is reduced; when the steam flow is large, with the increase of steam flow, more energy is needed to heat more volume of steam. Therefore, the relative unit energy consumption is increasing. It can also be seen from Figure 8(c) that the effect of sludge mass on relative unit energy consumption is higher than that of steam flow because the main determinant of relative unit energy consumption is the mass of water removed from sludge.


## 4.4. Determination of Optimum Drying Conditions and Model Validation

The drying process with low energy consumption is the key to reduce the drying cost. The multiple regression equation is optimized by using Design Expert V8.0.6.1, and the optimized result is as follows:X1 = −0.192; X2 = 1.002; X3 = 0.160; Ymin = 284.609 kJ/g. That is, the steam temperature is 215.19°C, the sludge mass is 25.02 g (sludge thickness 8 mm), and the steam flow is 30.12 m3/h. In actual engineering, the modified drying process parameters were steam temperature 215°C, sludge mass 25 g (sludge thickness 8 mm), and steam flow 30 m3/h. Three tests of sludge superheated steam drying were carried out under the modified drying process. Take the average of its relative unit energy consumption. The results are compared with the predicted values to verify the reliability of the model optimized drying process. The average value of sludge energy consumption per unit is 284.61 kJ/g; compared with the predicted value, the relative error is 1.65%, which shows that the drying process parameters optimized by the model are more reliable. The value of 284.61 kJ/g is higher than that in the published literature, and the reason is that the quality of the test sludge is small, so the test results were compared and analyzed with the relative unit energy consumption.


## 5. Conclusions

In the process of sludge superheated steam drying, steam temperature and sludge thickness have a significant influence on drying kinetics. When the steam temperature is higher, the sludge thickness is thinner, and the steam flow is larger, the drying time required for the superheated steam to dry the sludge to the target moisture content is shorter, and the drying efficiency is higher. The superheated steam drying process of sludge can be divided into three stages: condensation stage, recovery stage, and drying stage. When the steam temperature is lower, the sludge thickness is thicker, and the steam flow is smaller, the maximum condensation amount of the sludge is larger, and the recovery time is longer. The thin-layer drying model Wang and Singh can well predict the moisture change rule in the drying and condensation stage of superheated sludge steam. The linear equation can fit the recovery stage well. The logarithmic model can make a good prediction of the moisture change in the drying stage. Under the drying conditions of different steam temperature, sludge thickness, and steam flow, the effective diffusion coefficients of the sludge were 2.684606 × 10–8～4.5229776 × 10–8, 1.00024 × 10–8～6.9709 × 10–8, and 2.80133 × 10–8～4.50839 × 10–8, respectively. Under the drying conditions of sludge thickness of 6 mm, steam flow rate of 22 m3/h, and different steam temperature, the activation energy of sludge superheated steam drying is 13.42 kJ/mol. The response surface method was adopted to optimize the sludge superheated steam drying process by taking relative unit energy consumption as the response value and steam temperature, sludge mass, and steam flow as the test factors. The optimized drying process is as follows: steam temperature 215°C, sludge mass 25 g, and steam flow 30 m3/h. Under this process, the relative unit energy consumption is the smallest, which is 284.61 kJ/g, and the relative error is 1.65% through test verification. The results show that the drying process parameters obtained by the response surface optimization method are reliable.



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*Source: 3103710-2022-09-05.xml*