# Small Heat Source Used for Combustion of Wheat-Straw Pellets
## 2 Design and Method
### Design of the Heat Exchanger
#### 2.3.1 Numerical Design of Heat Exchanger  
To calculate the heat exchanger properties, we need to know the input parameters, such as the fuel to be used, and the maximal size of the heat exchanger. As was mentioned before, our boiler should not be wider than 600 mm, and the minimal depth of insulation is 100 mm. The maximal width and depth of the exchanger was set to 400 mm. The heat exchanger is made of steel. Our boiler will be used for the combustion of wheat-straw pellets. The target heat power of the heat exchanger was set to 20 kilowatts. A model based on criterial equations was designed to simply calculate the optimal dimensions of the exchanger with concrete type of used fuel. The calculation scheme is described in Figure 4.  
Figure 4: Scheme of calculations of heat exchanger’s dimensions. From Figure 4 it is obvious that, firstly, we have to calculate input parameters of burning fuel from Table 1. Chemical composition of combusted fuel (wheat straw). [21]  
The calorific value of used fuel from the chemical composition of the fuel given in Table 1 is calculated by using Equation (1) [22; 23; 24]. \[Q_{n}=\left[339\times C^{daf}+1029.7\times H^{daf}-108.8\times O^{daf}\right] \times\left[\frac{100-A^{r}-W^{r}}{100}\right]-W^{r}\times 25.1. \tag{1}\]  
Next step is calculation of proposed sizes of heat exchanger by using parameters from Table 2.  
By Equations (2) and (3) the temperature of inlet flue gas, and the amount of inlet flue gas to heat exchanger were calculated. \[t_{11}=\frac{m_{p}\times Q_{n}\times\eta_{sp}\times(1-\varepsilon)+\lambda \times m_{sp-min}\times\left[c_{svp}\right]\times t_{sv}}{m_{sp}\times\left[c_{ sp}\right]}. \tag{2}\]  
Heat balance equation:  
\[Q=m_{1}\times c_{1}\times(t_{11}-t_{12})=m_{2}\times c_{1}\times(t_{22}-t_{21}). \tag{3}\]  
In Equation (3), \(Q\) is demanded power of heat exchanger (W), \(m_{1}\) is flow of flue gas (kg\(\cdot\)s\({}^{-1}\)), \(c_{1}\) is specific heat capacity of flue gas (J\(\cdot\)kg\({}^{-1}\)K\({}^{-1}\)), \(t_{11}\) is inlet temperature of flue gas (\({}^{\circ}\)C), \(t_{12}\) is outlet temperature of flue gas (\({}^{\circ}\)C), \(m_{2}\) is flow of water (kg\(\cdot\)s\({}^{-1}\)), \(c_{2}\) is specific heat capacity of water (J\(\cdot\)kg\({}^{-1}\)K\({}^{-1}\)), \(t_{21}\) is inlet temperature of water (\({}^{\circ}\)C), \(t_{22}\) is outlet temperature of water (\({}^{\circ}\)C). Specific heat capacity of flue gas [25]:  
\[c_{1}=\left(\frac{(t_{11}-t_{12})}{2}\right)^{2}\times 9.991093\times 10^{-5}+ \frac{(t_{11}-t_{12})}{2}\times 1.27767\times 10^{-1}+1007.57. \tag{4}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Inlet temperature of flue gas (\(t_{11}\)) & 1040 & \({}^{\circ}\)C \\ Outlet temperature flue gas (\(t_{12}\)) & 150 & \({}^{\circ}\)C \\ Inlet temperature of water (\(t_{21}\)) & 50 & \({}^{\circ}\)C \\ Outlet temperature of water (\(t_{22}\)) & 80 & \({}^{\circ}\)C \\ Demanded heat power (\(Q\)) & 20 & kW \\ \hline
**Calculated Parameter** & **Value** & \\ \hline Specific heat capacity of flue gas (\(c_{1}\)) & 1118.68 & J\(\cdot\)kg\({}^{-1}\)K\({}^{-1}\) \\ Specific heat capacity of water (\(c_{2}\)) & 4180.56 & \({}^{\circ}\)C \\ Flow of flue gas (\(m_{1}\)) & 2.0128 \(\times\) 10\({}^{-2}\) & kg\(\cdot\)s\({}^{-1}\) \\ Flow of water (\(m_{2}\)) & 1.5968 \(\times\) 10\({}^{-1}\) & kg\(\cdot\)s\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Heat exchanger design inputs. Specific heat capacity of water [25]:  
\[c_{2}=4210-1.363\times\frac{(t_{11}-t_{12})}{2}+1.4\times 10^{-3}\times\left(\frac{(t _{11}-t_{12})}{2}\right)^{2} \tag{5}\]  
The overall heat transfer rate is calculated by the logarithmic mean temperature design method (LMTD) [25,26]:  
\[Q_{r}=h\times S\times\Delta t_{LN} \tag{6}\]  
where \(Q\) represents heat transfer rate (W), k is overall transfer coefficient (W\(\cdot\)m\({}^{-1}\).K\({}^{-1}\)) and \(\Delta t_{LN}\) is logarithmic mean temperature gradient for counter flow (\({}^{\circ}\)C).