Parallel Flow Heat Exchanger: Calculate Tube Diameter for Oil Cooling

To\u0020find\u0020the\u0020diameter\u0020of\u0020the\u0020tube\u0020through\u0020which\u0020the\u0020oil\u0020flows,\u0020we\u0020can\u0020use\u0020the\u0020equation\u0020for\u0020the\u0020heat\u0020transfer\u0020rate\u0020in\u0020a\u0020parallel\u0020flow\u0020heat\u0020exchanger:\u000A\u000AQ\u0020\u003D\u0020U\u0020*\u0020A\u0020*\u0020\u0394Tlm\u000A\u000AWhere:\u000AQ\u0020\u003D\u0020Heat\u0020transfer\u0020rate\u0020(W)\u000AU\u0020\u003D\u0020Overall\u0020heat\u0020transfer\u0020coefficient\u0020(W/m^2-K)\u000AA\u0020\u003D\u0020Surface\u0020area\u0020of\u0020the\u0020tube\u0020(m^2)\u000A\u0394Tlm\u0020\u003D\u0020Logarithmic\u0020mean\u0020temperature\u0020difference\u0020(K)\u000A\u000AFirst,\u0020let's\u0020calculate\u0020the\u0020logarithmic\u0020mean\u0020temperature\u0020difference:\u000A\u000A\u0394T1\u0020\u003D\u0020Th,i\u0020-\u0020Tc,i\u0020\u003D\u0020150\u00B0C\u0020-\u002015\u00B0C\u0020\u003D\u0020135\u00B0C\u0020\u003D\u0020135\u0020K\u000A\u0394T2\u0020\u003D\u0020Th,i\u0020-\u0020Th,o\u0020\u003D\u0020150\u00B0C\u0020-\u002075\u00B0C\u0020\u003D\u002075\u00B0C\u0020\u003D\u002075\u0020K\u000A\u000A\u0394Tlm\u0020\u003D\u0020(\u0394T1\u0020-\u0020\u0394T2)\u0020/\u0020ln(\u0394T1\u0020/\u0020\u0394T2)\u000A\u0394Tlm\u0020\u003D\u0020(135\u0020K\u0020-\u002075\u0020K)\u0020/\u0020ln(135\u0020K\u0020/\u002075\u0020K)\u000A\u0394Tlm\u0020\u003D\u002060\u0020K\u0020/\u0020ln(1.8)\u000A\u000ANext,\u0020let's\u0020calculate\u0020the\u0020surface\u0020area\u0020of\u0020the\u0020tube:\u000A\u000AA\u0020\u003D\u00202\u03c0rL\u000A\u000AWhere:\u000AA\u0020\u003D\u0020Surface\u0020area\u0020of\u0020the\u0020tube\u0020(m^2)\u000Ar\u0020\u003D\u0020Radius\u0020of\u0020the\u0020tube\u0020(m)\u000AL\u0020\u003D\u0020Length\u0020of\u0020the\u0020tube\u0020(m)\u000A\u000ASince\u0020the\u0020tube\u0020is\u0020thin-walled,\u0020we\u0020can\u0020assume\u0020that\u0020the\u0020thickness\u0020of\u0020the\u0020tube\u0020is\u0020negligible\u0020compared\u0020to\u0020the\u0020radius.\u0020Therefore,\u0020the\u0020diameter\u0020of\u0020the\u0020tube\u0020is\u0020equal\u0020to\u00202r.\u000A\u000AA\u0020\u003D\u0020\u03c0dL\u000A\u000AWhere:\u000Ad\u0020\u003D\u0020Diameter\u0020of\u0020the\u0020tube\u0020(m)\u000A\u000ANow,\u0020let's\u0020substitute\u0020the\u0020values\u0020into\u0020the\u0020equation\u0020for\u0020the\u0020heat\u0020transfer\u0020rate\u0020and\u0020solve\u0020for\u0020the\u0020diameter\u0020of\u0020the\u0020tube:\u000A\u000AQ\u0020\u003D\u0020U\u0020*\u0020A\u0020*\u0020\u0394Tlm\u000AQ\u0020\u003D\u0020750\u0020W/m^2-K\u0020*\u0020\u03c0dL\u0020*\u002060\u0020K\u0020/\u0020ln(1.8)\u000A\u000AWe\u0020know\u0020that\u0020the\u0020mass\u0020flow\u0020rate\u0020of\u0020the\u0020oil\u0020is\u0020equal\u0020to\u0020the\u0020mass\u0020flow\u0020rate\u0020of\u0020the\u0020water,\u0020so:\u000A\u000Am_oil\u0020*\u0020cp,oil\u0020*\u0020(Th,i\u0020-\u0020Th,o)\u0020\u003D\u0020m_water\u0020*\u0020cp,water\u0020*\u0020(Tc,o\u0020-\u0020Tc,i)\u000A\u000ASubstituting\u0020the\u0020given\u0020values:\u000A\u000A2.4\u0020kg/s\u0020*\u00202613\u0020J/kg-K\u0020*\u0020(150\u00B0C\u0020-\u002075\u00B0C)\u0020\u003D\u00203.0\u0020kg/s\u0020*\u00204180\u0020J/kg-K\u0020*\u0020(Tc,o\u0020-\u002015\u00B0C)\u000A\u000ASolving\u0020for\u0020Tc,o:\u000A\u000ATc,o\u0020\u003D\u0020(2.4\u0020kg/s\u0020*\u00202613\u0020J/kg-K\u0020*\u0020(150\u00B0C\u0020-\u002075\u00B0C)\u0020+\u00203.0\u0020kg/s\u0020*\u00204180\u0020J/kg-K\u0020*\u002015\u00B0C)\u0020/\u0020(3.0\u0020kg/s\u0020*\u00204180\u0020J/kg-K)\u000A\u000ANow,\u0020let's\u0020substitute\u0020the\u0020calculated\u0020values\u0020into\u0020the\u0020equation\u0020for\u0020the\u0020heat\u0020transfer\u0020rate\u0020and\u0020solve\u0020for\u0020the\u0020diameter\u0020of\u0020the\u0020tube:\u000A\u000AQ\u0020\u003D\u0020750\u0020W/m^2-K\u0020*\u0020\u03c0dL\u0020*\u002060\u0020K\u0020/\u0020ln(1.8)\u000AQ\u0020\u003D\u0020(2.4\u0020kg/s\u0020*\u00202613\u0020J/kg-K\u0020*\u0020(150\u00B0C\u0020-\u002075\u00B0C)\u0020+\u00203.0\u0020kg/s\u0020*\u00204180\u0020J/kg-K\u0020*\u002015\u00B0C)\u0020/\u0020(3.0\u0020kg/s\u0020*\u00204180\u0020J/kg-K)\u0020*\u0020\u03c0dL\u0020*\u002060\u0020K\u0020/\u0020ln(1.8)\u000A\u000ASolving\u0020for\u0020d:\u000A\u000Ad\u0020\u003D\u0020Q\u0020/\u0020(750\u0020W/m^2-K\u0020*\u0020\u03c0L\u0020*\u002060\u0020K\u0020/\u0020ln(1.8))