### 1. Introduction

C/N ratio correction

Compensation of the rare nutrients for microorganisms

Reducing the potential of toxic substances

Stimulation of synergistic effects on microorganisms

*Chlorella Vulgaris*microalgae (25% COD), increased the efficiency of microalgae methane production by 17% compared to the theoretical value.

*Chlorella sp.*and activated sludge residue mixture modified the reduction of volatile solids, which increased the efficiency of hydrolysis and microalgae biogas efficiency by 10% [16]. Similarly, anaerobic co-digestion of the

*Chlorella sp.*and sewage sludge (63% VS) mixture with

*Scenedesmus sp.*(37% VS) increased methane production by 23% compared to the mono-digestion of sewage sludge [8].

_{CH4}/g VS. The maximum methane production by algae is reported between 140–270 ml

_{CH4}/g VS, which is similar to the amount of methane production in the case of sludge [20]. The amount of methane produced depends on the type of the algae for digestion and the operational conditions of the digester.

*Cladophora*green algae, analysis of the synergy effects, and optimization of the response variable using a mixture design.

### 2. Materials and Methods

### 2.1. Preparation of Substrate

#### 2.1.1. Primary and secondary sludge

### 2.2. Reactors Start-Up Procedure

### 2.3. Biogas Production and Measurement System

### 2.4. Methods of Analysis

### 2.5. Design of Experiments By the Optimal Mixture Design Method

_{i}” indicates the “i” ratio of ingredients in the mixture, then [25]:

##### (1)

$$\sum _{i=1}^{q}{x}_{i}={x}_{1}+{x}_{2}+\dots +{x}_{q}=1.0;\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{x}_{i>0;\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{i}=1,2,3,\dots \text{q}}$$_{j}) and an upper limit (U

_{j}). The general form of the constrained mixture problems is defined as Eq. (2):

##### (2)

$${\sum}_{f}{X}_{f}=1\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{and\hspace{0.28em}}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{L}_{f}\le {X}_{f}\le {U}_{f}$$_{j}) as shown by Eq. (3):

*Cladophora*microalgae.

Choosing the right mixture design: several mixture design techniques such as Optimal Mixture Designs, Simplex-Centroid Design, and Simplex Lattice Design, where a suitable technique must be selected based on the range of independent variables or boundary constraints. When the scope of the independent variables is the same, a simple mixture design is used. On the other hand, an optimal strategy is a good choice when the boundary constraints are non-simple and have non-uniform sizes. Three types of optimal strategies, which are called D-optimal, A-optimal, and I-optimal, are available in Design-Expert software. Optimization of D is favorable for factorial and screening designs to recognize the essential variables. The algorithm selects points that minimize the size of the confidence interval of the coefficients. The A-optimal strategy reduces the mean of polynomial coefficients variance. The I-optimal strategy uses an integrated variance criterion which leads to minimizing the mean-variance of responses in a particular interest area.

Selecting the name, unit, and boundary constraints of the components

Selecting the name and unit of answers

Suggest a suitable plan for finding the relationship between the answers and the components of the mixture

Performing all experiments proposed by the plan one by one similar to the numbers

Recording the obtained answers from the experimental results in the answers column of the Office-Word software

### 2.6. Statistical Analysis

### 3. Results and Discussion

### 3.1. Design of Experiments

##### (4)

$$\text{Biogas}=162.20\text{A}+79.23\text{B}+94.43\text{C}+376.81\text{AC}+798.79\text{AC}(\text{A}-\text{C})$$^{−4}confirms that the model is significant. Also, the ANOVA chart shows that the mixture components, including A, B, C, AC, and AC (A-C), are substantial terms of the model because their P-value is less than 0.05.

### 3.2. Diagnostics

### 3.3. Effects of Mixture Components and Optimization

*Cladophora*algae (C) in Fig. [4a]. The sum of fractions of A, B, and C is 100, and there is the highest and lowest amount of biogas in the study area. Also, Fig. [4b] Shows the contour diagram a 2-D representation of the response. This 2-D diagram is shown in the form of a triangular graph, in which the contour lines show the amount of biogas in each composition. When the other two factors remain constant, it provides information about the impact of all 3 independent parameters on biogas production. Biogas production in experiments of 3, 5, 14, 15, and 21 was more than 250 ml/g VS. In the experiments of 3, 5, and 15, the amount of secondary sludge in the composition is zero and in the investigations of 14 and 21 is less than 20% of the total substrate composition. Also, in these experiments, the amount of primary sludge is almost more than 50% of the substrate composition. Therefore, it can be concluded that the presence of primary sludge in the substrate composition improves the performance of biogas production, and secondary sludge reduces its performance. On the other hand, in all these experiments except the experiment of 15, the amount of algae was less than 40% of the composition. This reflects that increasing the ratio of algae in the composition to above 40% reduces biogas production. Meanwhile, in the experiments of 17 and 22, biogas production has reached less than 70 ml/g VS, which is a minimal amount. In these experiments, the proportion of algae in the composition is 70 and 100% of the total composition, respectively. These results are consistent with the results of other scholars [13, 14]. In one study, Sole-Bundo, Cucina [17] showed that anaerobic co-digestion of sewage sludge and microalgae (75%-25% VS) reduces the potential of biotoxicity in the digested feed. Anaerobic co-digestion dilutes the effect of inhibitory compounds. Dilution of potentially toxic compounds, modification in nutrient balance, microorganisms synergistic effects, increased load of biodegradable organic matter, and more stable digestion are the reasons for better biogas efficiency [12, 30, 31].

### 3.4. Response Prediction and Confirmation Experiments

### 4. Conclusion

*Cladophora*algae (C) 18–30%. The model was numerically optimized, and the optimal composition for the highest amount of biogas production (290.86 mg/L) was determined as 74.34% of feed A and 25.66% of feed B without the presence of feed C. The amount of produced biogas under the optimal laboratory conditions was 296.03mg/L, which is very close to the predicted value by the model. Therefore, the optimal mixture design can be utilized as an alternative optimization tool and method in optimizing the composition of substrates in an anaerobic digestion system.