(Solved): Explain clearly Design a cylindrical riser, with a height equal to twice its diameter, that will com ...

Design a cylindrical riser, with a height equal to twice its diameter, that will compensate for shrinkage in a \\( 2 \\mathrm{~cm} \\times 8 \\) cm \\( 16 \\mathrm{~cm} \\) casting. Solution \\[ \\mathrm{H}=2 \\mathrm{D} \\] TST\\( _{\\mathrm{r}}=1.25 \\mathrm{TST}_{\\mathrm{c}} \\) \\[ C_{m}\\left(V_{r} / A_{r}\\right)^{2}=1.25 C_{m}\\left(V_{c} / A_{c}\\right)^{2} \\] \\[ \\begin{array}{l} \\left(\\mathrm{V}_{\\mathrm{r}} / \\mathrm{A}_{\\mathrm{r}}\\right)^{2}=1.25\\left(\\mathrm{~V}_{\\mathrm{c}} / \\mathrm{A}_{\\mathrm{c}}\\right)^{2} \\\\ \\left(0.25 \\pi \\mathrm{D}^{2} \\mathrm{H} / 2 \\times 0.25 \\pi \\mathrm{D}^{2}+\\pi \\mathrm{DH}\\right)^{2}=1.25\\left(8 \\times 16 \\times 2 / 2(8 \\times 16+2 \\times 8+16 \\times 2)^{2}\\right. \\\\ \\left.0.25 \\pi \\mathrm{D}^{2} \\mathrm{H} / 2 \\times 0.25 \\pi \\mathrm{D}^{2}+\\pi \\mathrm{DH}\\right)^{2}=1.25(256 / 352)^{2} \\\\ \\left.0.25 \\pi \\mathrm{D}^{2} \\mathrm{H} / 2 \\times 0.25 \\pi \\mathrm{D}^{2}+\\pi \\mathrm{DH}\\right)^{2}=0.661 \\\\ (0.25 \\mathrm{D} 2 \\mathrm{D} / 0.5 \\mathrm{D}+2 \\mathrm{D})^{2}=0.661 \\\\ (0.5 \\mathrm{D} / 2.5 \\mathrm{D})^{2}=0.661 \\\\ (0.5 \\mathrm{D} / 2.5)^{2}=0.661 \\\\ (0.5 \\mathrm{D} / 2.5)=(0.661)^{0.5} \\\\ (0.2 \\mathrm{D})=0.813 \\\\ \\mathrm{D}=4.1 \\mathrm{~cm} \\quad \\mathrm{H}=8.2 \\mathrm{~cm} \\end{array} \\] The geometry of the casting and riser