Calculate Tube Diameter for Oil Cooling in Parallel Flow Heat Exchanger

To\u0020determine\u0020the\u0020diameter\u0020of\u0020the\u0020tube\u0020through\u0020which\u0020the\u0020oil\u0020flows,\u0020we\u0020need\u0020to\u0020use\u0020the\u0020equation\u0020for\u0020the\u0020overall\u0020heat\u0020transfer\u0020coefficient:\u000A\u000A1/U\u0020=\u00201/hiA\u0020+\u0020ln(do/di)/2πk\u0020+\u00201/hoA\u000A\u000AWhere:\u000AU\u0020=\u0020overall\u0020heat\u0020transfer\u0020coefficient\u0020(W/m^2-K)\u000Ahi\u0020=\u0020convective\u0020heat\u0020transfer\u0020coefficient\u0020on\u0020the\u0020inside\u0020of\u0020the\u0020tube\u0020(W/m^2-K)\u002Aho\u0020=\u0020convective\u0020heat\u0020transfer\u0020coefficient\u0020on\u0020the\u0020outside\u0020of\u0020the\u0020tube\u0020(W/m^2-K)\u002AA\u0020=\u0020surface\u0020area\u0020of\u0020the\u0020thin-walled\u0020tube\u0020(m^2)\u002Ado\u0020=\u0020outer\u0020diameter\u0020of\u0020the\u0020tube\u0020(m)\u002Adi\u0020=\u0020inner\u0020diameter\u0020of\u0020the\u0020tube\u0020(m)\u002Ak\u0020=\u0020thermal\u0020conductivity\u0020of\u0020the\u0020tube\u0020material\u0020(W/m-K)\u000A\u000AWe\u0020are\u0020given\u0020that\u0020U\u0020=\u0020750\u0020W/m^2-K\u0020and\u0020A\u0020=\u0020surface\u0020area\u0020of\u0020the\u0020thin-walled\u0020tube.\u0020We\u0020need\u0020to\u0020solve\u0020for\u0020do\u0020and\u0020di.\u000A\u000AWe\u0020can\u0020assume\u0020that\u0020the\u0020oil\u0020is\u0020treated\u0020as\u0020a\u0020single-phase\u0020fluid,\u0020so\u0020the\u0020convective\u0020heat\u0020transfer\u0020coefficient\u0020on\u0020the\u0020inside\u0020of\u0020the\u0020tube\u0020is\u0020given\u0020by:\u000A\u000Ahi\u0020=\u0020Nu\u0020*\u0020k\u0020/\u0020di\u000A\u000AWhere:\u000ANu\u0020=\u0020Nusselt\u0020number\u000Ak\u0020=\u0020thermal\u0020conductivity\u0020of\u0020the\u0020oil\u0020(W/m-K)\u000A\u000AThe\u0020Nusselt\u0020number\u0020can\u0020be\u0020determined\u0020using\u0020the\u0020correlation\u0020for\u0020parallel\u0020flow\u0020in\u0020a\u0020concentric\u0020tube\u0020heat\u0020exchanger:\u000A\u000ANu\u0020=\u00200.023\u0020*\u0020Re^0.8\u0020*\u0020Pr^0.4\u000A\u000AWhere:\u000ARe\u0020=\u0020Reynolds\u0020number\u000APr\u0020=\u0020Prandtl\u0020number\u000A\u000AThe\u0020Reynolds\u0020number\u0020is\u0020given\u0020by:\u000A\u000ARe\u0020=\u0020ρ\u0020*\u0020V\u0020*\u0020di\u0020/\u0020μ\u000A\u000AWhere:\u000Aρ\u0020=\u0020density\u0020of\u0020the\u0020oil\u0020(kg/m^3)\u000AV\u0020=\u0020velocity\u0020of\u0020the\u0020oil\u0020(m/s)\u000Aμ\u0020=\u0020viscosity\u0020of\u0020the\u0020oil\u0020(kg/m-s)\u000A\u000AThe\u0020Prandtl\u0020number\u0020is\u0020given\u0020by:\u000A\u000APr\u0020=\u0020cp,h\u0020*\u0020μ\u0020/\u0020k\u000A\u000AWe\u0020are\u0020given\u0020that\u0020V\u0020=\u00202.4\u0020kg/s\u0020/\u0020(π/4\u0020*\u0020di^2)\u0020=\u00209.65\u0020/\u0020di^2\u0020m/s.\u000A\u000ANow\u0020we\u0020can\u0020substitute\u0020the\u0020equations\u0020for\u0020hi,\u0020Re,\u0020and\u0020Pr\u0020into\u0020the\u0020equation\u0020for\u0020Nu:\u000A\u000ANu\u0020=\u00200.023\u0020*\u0020(ρ\u0020*\u0020V\u0020*\u0020di\u0020/\u0020μ)^0.8\u0020*\u0020(cp,h\u0020*\u0020μ\u0020/\u0020k)^0.4\u000A\u000ASimilarly,\u0020we\u0020can\u0020assume\u0020that\u0020the\u0020chilled\u0020water\u0020is\u0020treated\u0020as\u0020a\u0020single-phase\u0020fluid,\u0020so\u0020the\u0020convective\u0020heat\u0020transfer\u0020coefficient\u0020on\u0020the\u0020outside\u0020of\u0020the\u0020tube\u0020is\u0020given\u0020by:\u000A\u000Aho\u0020=\u0020Nu'\u0020*\u0020k\u0020/\u0020do\u000A\u000AWhere:\u000ANu'\u0020=\u0020Nusselt\u0020number\u0020on\u0020the\u0020outside\u0020of\u0020the\u0020tube\u000A\u000AThe\u0020Nusselt\u0020number\u0020on\u0020the\u0020outside\u0020of\u0020the\u0020tube\u0020can\u0020be\u0020determined\u0020using\u0020the\u0020correlation\u0020for\u0020forced\u0020convection\u0020over\u0020a\u0020flat\u0020plate:\u000A\u000ANu'\u0020=\u00200.664\u0020*\u0020Re^0.5\u0020*\u0020Pr^0.33\u000A\u000AThe\u0020Reynolds\u0020number\u0020on\u0020the\u0020outside\u0020of\u0020the\u0020tube\u0020is\u0020given\u0020by:\u000A\u000ARe'\u0020=\u0020ρ\u0020*\u0020V'\u0020*\u0020do\u0020/\u0020μ'\u000A\u000AWhere:\u000AV'\u0020=\u0020velocity\u0020of\u0020the\u0020chilled\u0020water\u0020(m/s)\u000Aμ'\u0020=\u0020viscosity\u0020of\u0020the\u0020chilled\u0020water\u0020(kg/m-s)\u000A\u000AWe\u0020are\u0020given\u0020that\u0020V'\u0020=\u00203.0\u0020kg/s\u0020/\u0020(π/4\u0020*\u0020do^2)\u0020=\u002038.20\u0020/\u0020do^2\u0020m/s.\u000A\u000ANow\u0020we\u0020can\u0020substitute\u0020the\u0020equations\u0020for\u0020ho,\u0020Re',\u0020and\u0020Pr\u0020into\u0020the\u0020equation\u0020for\u0020Nu':\u000A\u000ANu'\u0020=\u00200.664\u0020*\u0020(ρ\u0020*\u0020V'\u0020*\u0020do\u0020/\u0020μ')^0.5\u0020*\u0020(cp,c\u0020*\u0020μ'\u0020/\u0020k)^0.33\u000A\u000AWe\u0020can\u0020now\u0020substitute\u0020the\u0020equations\u0020for\u0020hi,\u0020ho,\u0020Nu,\u0020and\u0020Nu'\u0020into\u0020the\u0020equation\u0020for\u0020the\u0020overall\u0020heat\u0020transfer\u0020coefficient:\u000A\u000A1/U\u0020=\u00201/(0.023\u0020*\u0020(ρ\u0020*\u0020V\u0020*\u0020di\u0020/\u0020μ)^0.8\u0020*\u0020(cp,h\u0020*\u0020μ\u0020/\u0020k)^0.4\u0020*\u0020π\u0020*\u0020di\u0020*\u0020L)\u0020+\u0020ln(do/di)/(2πk)\u0020+\u00201/(0.664\u0020*\u0020(ρ\u0020*\u0020V'\u0020*\u0020do\u0020/\u0020μ')^0.5\u0020*\u0020(cp,c\u0020*\u0020μ'\u0020/\u0020k)^0.33\u0020*\u0020π\u0020*\u0020do\u0020*\u0020L)\u000A\u000ASimplifying\u0020and\u0020solving\u0020for\u0020do,\u0020we\u0020get:\u000A\u000Ado\u0020=\u0020exp((1/(2πk)\u0020-\u00201/U\u0020-\u0020ln(di)/(2πk))\u0020/\u0020(1/(0.664\u0020*\u0020(ρ\u0020*\u0020V'\u0020*\u0020do\u0020/\u0020μ')^0.5\u0020*\u0020(cp,c\u0020*\u0020μ'\u0020/\u0020k)^0.33\u0020*\u0020π\u0020*\u0020L)))\u000A\u000APlugging\u0020in\u0020the\u0020given\u0020values,\u0020we\u0020can\u0020solve\u0020for\u0020do.