# Optimization of Replaced Grinding Wheel Diameter for Minimum Grinding Cost in Internal Grinding
## 2 Cost Analysis  
The internal grinding cost per part \(C_{ig}\) is determined by [16]:  
\[C_{ig}=C_{m,h}\cdot t_{s}+C_{gu,p}, \tag{1}\]  
where, \(C_{m,h}\) is the cost of the grinding machine (USD/h), including the wages, the overhead cost, the labor cost etc. ; \(t_{s}\) is the grinding time which is discussed later in this section; \(C_{gu,p}\) is the wheel cost per part (USD/p) which is calculated by:  
\[C_{gu,p}=C_{gu}/n_{p,w}. \tag{2}\]  
In which, \(C_{gu}\) is the cost of a piece of wheel (USD/p); \(n_{p,w}\) is the entire parts ground per wheel; \(n_{p,w}\) can be determined as follows [17]:  
\[n_{p,w}=(D_{0}-D_{e})\cdot n_{p,d}/\left[2\left(W_{pd}+a_{ed}\right)\right], \tag{3}\]  
where, \(D_{0}\) is the initial diameter of wheel (mm); \(D_{e}\) is the replaced diameter of the wheel (mm); \(W_{pd}\) is the radial grinding wheel wear per dress (mm/dress); \(a_{ed}\) is the dressing depth of cut (mm); \(n_{p,d}\) is the number of parts per dress which is found by:  
\[n_{p,d}=T_{w}/t_{c}, \tag{4}\]  
wherein, \(T_{w}\) is the wheel life (h); \(t_{c}\) is the grinding time (h) which is calculated as follows:  
\[t_{c}=l_{w}\cdot a_{e,tot}/\left(v_{fa}\cdot f_{r}\right), \tag{5}\]  
\[f_{r}=f_{r,tab}\cdot c_{1}\cdot c_{2}\cdot c_{3}\cdot c_{4}. \tag{6}\]  
In Equations (5) and (6), \(v_{fa}\) and \(f_{r,tab}\) can be determined by the following Equations [18]:  
\[v_{fa}=22.88\cdot W_{gu}^{0.9865}\cdot d_{w}^{0.0821}\cdot S_{rg}^{-2.9833} \cdot n_{w}^{1.2471}, \tag{7}\]  
and  
\[f_{r,tab}=30.2944\cdot a_{e,tot}^{0.567}\cdot v_{fa}^{-0.9693}\cdot d_{w}^{-0. 1269}. \tag{8}\]  
In the above equations, \(l_{w}\) is the length of part (mm); \(a_{e,tot}\) is the total depth of cut (mm); \(v_{fa}\) is the axial feed speed (mm/min); \(f_{r,tab}\) is the tabled radial wheel feed (mm/stroke); \(f_{r}\) is the radial wheel feed (mm/stroke); \(c_{1}\), \(c_{2}\), \(c_{3}\) and \(c_{4}\) are the coefficients for determining the radial wheel feed (Table 1); \(W_{gu}\) is the grinding wheel width; \(d_{w}\) is the work-piece diameter; \(S_{rg}\) is the surface roughness grade; \(n_{w}\) is the work-piece speed; As it is grinding stainless steel, \(n_{w}\) can be determined by [18]:  
\[n_{w}=1255.8\cdot d_{w}^{-0.3491}. \tag{9}\]\(t_{s}\) is the grinding time, including auxiliary time (h), which is determined by:  
\[t_{s}=t_{c}+t_{sp}+t_{d,p}+t_{wr,p}+t_{lu}. \tag{10}\]  
In which, \(t_{c}\) is the grinding time (h); \(t_{sp}\) is the spark-out time (h); \(t_{d,p}\) is the dressing time per piece (h); \(t_{cw}\) is the wheel replacing time (h) and \(t_{lu}\) is the loading and unloading work-piece time (h). These time components can be determined by the equations presented in Table 2 in which \(t_{d}\) is the dressing time (h). To investigate the influence of input factors on the grinding cost, a program was conducted based on the above cost analysis. From the results of the program, the effects of several input factors on the grinding cost are exhibited in \[C_{gu,p}=C_{gu}/n_{p,w}. \tag{2}\]0 Moreover, the relation between the grinding cost and the replaced wheel diameter is described in \[C_{gu,p}=C_{gu}/n_{p,w}. \tag{2}\]1 This relation was calculated by Equation (1), in which D\({}_{0}\) = 20 (mm); W\({}_{\text{gw}}\) = 25 (mm); a\({}_{\text{ed}}\) = 0.12 (mm); C\({}_{\text{mh}}\) = 5 (USD/h); C\({}_{\text{gw}}\) = 3 (USD); T\({}_{\text{w}}\) = 20 (min. ); W\({}_{\text{pd}}\) = 0.02 (mm/dress); S\({}_{\text{Tg}}\) = 7; t\({}_{\text{g}}\) = 7; R\({}_{\text{ld}}\) = 2. As it was reported in [16], the grinding cost is powerfully affected by the replaced grinding wheel diameter. In addition, this cost is minimum when the replaced wheel diameter reaches an optimum value (in this case D\({}_{\text{e,op}}\) = 17.2 (mm)). Besides, the optimum diameter is considerably bigger than the conventional replaced grinding wheel diameter (about 13 to 14 mm [20]).