How do we calculate Loss of Head due to Friction in Pipe? What is the major cause of loss of head or energy in a pipe?
According to Darcy's formula, the loss of head due to friction in the pipe (hf) is given by
![](data:image/png;base64,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)
where
f = Darcy's coefficient,
l = Length of pipe,
v = Mean velocity of liquid in pipe, and
d = Diameter of pipe.
The major loss of head or energy is due to friction. The minor loss of head includes the following cases:
(a) Loss of head due to sudden enlargement,
![](data:image/png;base64,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)
(b) Loss of head due to sudden contraction,
![](data:image/png;base64,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)
where
Cc= Coefficient of contraction.
(c) Loss of head at the inlet of a pipe,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGYAAAAxCAIAAACnNkt6AAAZBElEQVRogaV7+XMc15Fm/48TsbGzO7ZnVrJli7K0q4Ne3RJPU6Ql8xBtixQp8b4vAAQpUjx1gKJI8AYJEgRxAw10d9U7MvMdVb0/ZHX2A6jd2JmJeIEoVFdX1fvel/m+PLpWtGMsQyhcKJwL6AL6SL7tfduHdnBtkkElQrTpoBKpRCxgyXksgAp0Jbk2YQF82ZLh2uTbHkJ1ffWt5IbdK0uywVCBVCAW8IsDorXB8LfSfyFY7ZQNBqP1EXyEUGAsKRToI7hgyRsXqo9csDL4Mh58AR9TgTWBrGhHHwmdBWdcm0I7uJJ82y9BTWaFcdEMecKu7CLLuDwP2ZIzxmvtFM9cLuD7+LbnB2EEGwwWYIPhkWKaLht/JKjxzSFaF4G8EdQYMvIGSKHTjAifl0/TwaiFArGAmi9daAceRTsWZQyFc22iAiFY5gIPV1IXmrZ3bU8l8WBobDDaKe2U8bpz2S9DhgVggTwl47UNVpglrIFo+aFUVJR0JaXMev6eEK3xmsGVheRbMWQu8Px5WPIdlhXgIrhoXQTHGKWQFUjBUrA+AkRbwwJkYgwNT7VrYvJmBVLhsSBXBio9FkSFd2VwbS9mqzDLoQXeMC/4DswaISlEUKi0M9oZ4y0WRKXXzijMMYJ2CiNgAb50zDVZUd/2VGDX6DpcXkQxb3jljNcpZOgtkGG3A2QsaiCDzlqn+UpeP7m54C7Q84JRiTUGiOfTPegQasnCYqRqkoXnqWJBru1lqY3XCjM2NF7z1EgZAuOtdkaTtgGwIIhIpdekNeUMFjsv4/USt1BxP7G+akWL6tHMcU05D346n7dOW9DorOBVHbiuB2Tj4AM2/+q7wVCJfIZKrDGhUnqLExGidWwNZYZUeIiIBWFBWKCYg9iaWCg/STydaxNEMN4y0WwAG4AKb7zVTour4ltRgcJ9hi919q4kWVH5YuUcKM9sM7PNatqUW6fRWUAzOzdx9+71+3d/rs9Pa5tbp8WSWqbB/9pgNOUMFsNEBfJ9sIAaO2BBWggpROVL+fvGW6GGDWC8hYjGVw8QiNkqU+wqICKvodGk2SptAE1aodKkc8h4nbsr3GGK0FyshhcDok0XRtbJeJ3ZZss0uteg0jafq0+eP3ty5Yp3Xnnlt+fOHs9Vw7ru/ecaU/wVhZk8mj+VVzJe1/is8ZqR5kv5CoUZX6cw45PifSCi8cA00aTYEsXrQbQMoqZ8uj620JoV5cGQKVQ2ALszhcp4m9msoeo8AZ62diqHVmabPAF5T6EAmxuziS1RYcZugV8+h1Z1H9vMbUvb/PHwnbt3r9fnp7dsXv/1jq2zcxPs+/j1Uug781UdruiWafAr1dINGwvIsbvUjBTPn08abzVpNkkmmg1gvGGKLZEOCrOWadwb+nl8aoT5xQxnZvF3tTM55Ew6nnzFTcpT4rBFpB6nOyXKM9vkF+Ap8YRTyBRmuWlpm2ubG1AG1KEDX509c2y+Pm1ILaEtL4lYmDyoekqwNaEcBEslVs47gtiFvLQNho3IeMvOiP2R8ZUJs+sRA9eUN/X85OxoI6/LvoMFaNIt02KGZjZjxkFETSqHFj+UX0NTztNmvvPiKcwYHeYa2yATLYcWL3MFE7SqA9vKTSvXTQOKPAw/vP3tN6fGxoaRLPoOXqmBdxwRz11w5Jep8UuwN6US+SWoQOuUodw6BV5DMOA1BKtJaWKb0i3TVJhrp7XTvPKuJN8mKkmcjqZ8dn6yoeopB403OWTGG4ioMNekbLC8h/CEK6USqhfFCIYUeAPBWqfBGxeRxQEPQadlGplt5rxjdt7fYG4g0zZTqqFtNjM7dvzYnmsDF2dnx5XJ0HeJ3PU/nQ03dWr8r8KsxlhW+0KJ/JkLFp1GVx0AKfKG/CIHIV5PDErEBBVog7FOK8wa2VxmmhCtqAR+OX5oJa+CrZwur22HSszcUDjywCEdkOFjS1p2FeGXwizHlmICYq5ty0BmIbOQadvK8vpcffzoka83bVx7cP+Xp3sPTk4+sU61TENhltnmQj7HqAlt5YBP8sLUJMRJ9ZdEDHzQCbUg3f4W7VMdZyfBI7szJsWS+DFVEhIPLd37OqZBBfpIPlIoXCwDeUBnXUBwhn0FR5EMmXUqx0x7pZ3SmE1Pjw78cO7xo0EkNfxo8MaNK8OPBo8f2/3pX1auXPG/T57YO1cf15jxeufQqjenGb6GqufQ4jny/iMHmvIaT2aJ0JeINJZUlK5oOw7B2MfJ/tKVjk5hrDZEgQyiZTh4cNwjcSg/TqQMX9+VJnzPAiBaH4nDOEkf+EgUgB0IlZ23CpaDB+WV8Rq8Hn40uGbVu/t2/x0w7z25b+vn60dH7xvILObatgCVxVxBi70Vb1ZM2PlsVvZoBpEKZP1RiQwhF9sOFiBhqkSkLlhLXdUj5KqsjPIk/OwozCV4dWImVhsMKAS7JD+RitJqDQIWZYxFYMj4rytI9tDU6iFaW1gswEXI8vqunVs+37wuy+bOnz16/Njuhca0MU2LOTlD3qDTebLbMm3ZZ/EqLnEXlfqXOCkNazj7wbkRIAWkLOYGM+O16cxHvK+AKMGDuKfng2QmhUBWHYdu1oiNlC+AYLEAchDLEKJj85TRjQFDtVlXT+EwIwI6fbbv0NYtn9y7M3C69+DNm1ct5so00GmOxtFpBS1e7zQsZatk6vHeuAiyNHwTT5RmSCzmc/VxAxk6DWlSLP5CukpCS/FKqV6XjZVxSWko9+HwFrxhKKlAIOMjheiYYjx86Z53jjYYKIDaVSaGvDl35uinG1Yc2v/lxXPH5xemtG0p0+CED29oGnMBhd+ZCkykezfs432gYlnFjqKim+uYpLYtbVs//3z5q51bRh7fIW/EiESCsTrtMqgjL5hlz0PGH4E3o+MPL13pu/L92Zn6uDgOGdbpp+NDT0bvPx0fun37GqMmvoxHtTt33r+iSQFYcmRKocDLF3pWfvynnuN7J8aH2VYs5OSNEAJ8lz6SwJC1xzTvRrl2igqsVd4kAgTrCnIluYJCQSFSK6s3W3P7925/9dXfDT+65QInlQAk0m4jlIgR0BsK1kVAbyAYCha9sU4bUtZp9BaCdRExAgVAb4ceDm7a8sm+fV8ePPjVjh1bpmfHwBnrFHpjKKdg5xvTq9a8u37DihWr3v76q63oDDlL3nZSXUDBQjAYLQRDBVAB4LR1ioL1BfoCfYmhwJGRO5cvnJqceOKDZVpJslBUgYvgSvQl8RepABfBl+RK8qVzEV1EngIEcBFrFjX7CM5iy4hFQGdbrfnt2zatXv2+AeU8sMu0pIzXUAKUoAprg0GnkTRLOeY8Om1RcU4KnUWy5IElgrb5po3rPvroT8+ePZyafvbS7//t6uUzSjWBNJAykLlgT/ce3LJ53ckTe/fv3Xb39oCP4LwFUvxXPKw8jp+IpMkZSU+7YA1k2rZcsC6AQCZDkohpwrqz42EoXCwCI0MeXEDOuNXQdXdxH0k+9pGcx3p96oP33/py++aRkfvfnjv1bPSBCxa90V6baExhVGG078IkGXSGDJ0lB0iWJShD9nj4zuuvv3xg3475hRny8NZbf1z58TuzcxMGcos5o/a7l379xT8+vT5w0ULuvOW/zttWNnf+m2Mjj+9YyIGUhXxq+um33xyfmXlGzpAz8iYixX0EH4DT1vJRCp+QTgQ8eRM6Ww0TiF+eRy2WoShj2S64CMCosXpEZ8fGhl9++YXdu77oOXngww+X9/Uc4MfYYEw0prC6BO21huzJ49tn+g/3ntp//Njuwwd3nji+5+iRXbduDWiT85Pk2Rcv9L7xxrIL53uybAHIrF79/ou//fXY2LC2LYs5Op2r+c8+XbXi4z+9/vrvN/51zVx9PFfz/NxczZ//5tiliz03blzRtjU2NvT1V5/v+HLTfH3SeevDopKHCzZEjAVJiURkAI+0gMIX8Kc+gmw1IqRdQP63giyWgSGrXGx0zqO2+ZXL/S+8+Kv+viM//vDt8uV/PHliL5sDBIslQom2jRCtJTU58eT6wMXvvztz+VLv5Yu9P3x39url/ifDdwGN6ANejJMn9i9b9uJ3V88qkwGaz7dseOHFX42PP2GLc8ECqanppw/uXT/bd/i//8s/9fUcsJjzZBjQ4UeDO3ds3rP773/buqH31P5nzx740HVPjNTzdidKM0QUiqHT8l25JpYUS5diwgf8tyZYpto6FM4HWmjMbt70yVtv/XFqavThw1tvvLHsp4EL9frEgwc3NCkbLZZk2w4KoGDnF6aG7v988+bVwcHvhu7/PDJyZ3T0wcLCrPPoAsqdYxkune9dtuzFH74/r22Ozm7auO5f/+2fp6fHWF6GAskbHwBQ+Qiffbpq1cq3nbfsqoAUOcNce+GF//b22681m7NsdzxhH2BqcuTc2aNHj3zd33foTP/hE8f39PceOtN/+NyZo1cu9V262HP5Uu/9+9cHB7/76dqF6wMXrw18+9O1C3P1cb5J0XZF2xXtyINrbzJiGWqiqmVW1Q7gcXLy6R/+8D92ff33Zqvee+rg8uV/nJ0dm5l5NjR0UzttC2tL0KWFwlIw42MPL1/q/ebMkd5T+w8f3Ll7198OHdh5//4NA0ogY7aPjg69887rJ47va7bq6OyaNR8sX/7qfH2aGcHL7rw1kAGpIwe/euutZUCKSUHeaN0YGrqxdu37H7z3xnvvvT78aNAFK3bnI0xNjvT3HTp2dFdfz4Ez/Yf7+w6d7j147Oiukyf2nuk/3HNy34nje7672n+m//DxY7v7+w71nTrQc3Lf+NgjXrAqTOR6WzumYpBHjTcFXn/GzkciB+js7dvXfvOv/3VoaHBq+tn69St/99JvLpw/eX3g4o8/nDNe22htAbYEGy0Fq9TC9MzoxMTw2NjDpyP3Hjz4+eHDwXp9ihyEwknE4yPlurlu3cdr1344Ofl0cvLpsmUv7t71RZ43bty4eufOgIFsfmGKvTuQ2rlj887tm9gl+wDkzePhW5988tHftm64c+uHwwd3bvzrmru3B2JJoajMUNvWwsJ0fX5yYWF6oTHdyuYajZnZubHZ2bG5+vjMzLOp6aczs8+mpkYmJ59MTz2dnHwyMTGc5fXUhFOWpaNoxxoL61gECUd8JNZTj4Zvnzi5r6UWpufGdu/+YvWa965ePn1w/5d9PQfQGxcBu7VxYHHHAs2XRFWMiVSQL0Nox9COvh0wInk78OO3b7/92onje1etfOfNN19++HCQvN24ed0X2zdmeuEf2z5b/5cVNwe/u3K1/3++8YexiUcQKrnXyOe2/n3Dnz/5aPjJHYtqZmbswN4v1679YL4+Rd66iBQQvEUPLmCI5AN6Dz5gLEjK45Kh8RF8RB8RnUFnQ8EujEeQ3AlbnrivmvPoI/lA4tTQWUMKIzSyufls1ni9kM+Njj988vTe8MPBTRvX3rkzgKRdsOCUwRyctklhsQrvq6qwp9K7Mrgy+Hb07UilJ28XGtM3bly58dPlb785fu/eT9q2fIQ79366//CGdfr6jSt7922/eKn32vVLP1670A3FgtWY37o7cPfB9dy0XEAkO1efvHVrIFcNjhBspyIBAcgDOSCyzkGIlfJiJZRARqK8ltpgEYoyipeXv7WUXwwZeWDhzllQQ6qh6k09b0gN3rj60UdvLTSm2VI4i2JwUTEmKYZ6346uXdWJu6gFq23LQt5qzXH44gOg09ZpBRl4Y0jVG9Mz8xOz85NSu+KIXTsF3uTQAmd4+ycPSBbJorPkQaPOUBlv0QM6a6yyJvceQ+wqr3QzFY/kIzFGPBgyxqQoF1loTfyXT8QbeMOJYypQO9XU8009ryC7efPqB++/8Xj4FjvjKmojJdWNKmHNEWhRsYxrK1IqBtIGMs5buWDJmVCgD4AROEgGbzjnpzHXmEv6VyrhGMHFSjz6QEDGWIXOWtQKVA55y7Qmp0b37tn+yrIX/9drvz9xZE+jMStybHHrCjLW5CEWobs5lpGtMkTHJih0q/lIoSAhHn8/TZlyQpJTY1PTTwd+PN9szrIgqLJDXkseTXL8rk2u9EIxridBROMtkEanWeinSpLbLzgS5qXSTinIJIvJlVDtFKfeZKMnD1wHQW81KuNtI5vv6Tn05z9/dOjAV+vWfPjKyy9ePHdStBjrO2a3WOXzhsl9PdVx2T2upGwKGboqoLfBcCWxsg6vLeRZXuedi70pOo2dJKXYZge1LmRptRidEcik06ZoO0lD8a2kxsMZZC7o8HlOWIq25DYLFxA9aNQK9ej446vfn388fKfZnHv0YHDDuhX7dm/L8noVKjnjvO3EdiAxYgoZ63mGiXnHYWXRLiqRIc8mD5a6CfiWaXCNgO0ixKrHCJ3mqZI36LullrQqmkLG/KqK7d6i0yy7ZMGDpJ6SJiLJvnNS27XJdir+zLJYBo6WOVJGbxUoBaqRzdfnZwANeajPTf7j80/3794GqIBUns9futizfdtfV69+99CBHU+f3ufEhA+UQobOZvmCRa1MJltEiK5sF7VUl7N3MNBtBjFec7eAphyC4Wztkg4sYJiKRW1lrk0pZIwXbwIuAsfe7Ik5AosFpZBJkSnNILLhc9ebb3uxHSk7obMGlSGjURvU6KwP+ON35zf/dd39uz/xfvXNmSMrPv7TxQuntn6+/rXXfndt4AIzdIlhuoAWtbZ5rpt8XmhYs8Fw0wpvc+wv0rYU66oaMBXImSkn+W7uunmuFUf8IMRFOVv2VlXWP+mDpE7vge/0XQnLfNu7oiuzqy0siWCqhriAsQghOvIAnTQGkq7XJ04c33Om/7AxGTkA1If27/zs0zUjI/ePHtm1du0Hjx7dTlUF48KeS/IIaehSQVaVSNqeu5S4rNLU81Kvt53WHew0zkknmrBAcpjSidcta3Z6fqoWu87+IHlzIRe/BnX6Pav+siSYS7FL4z62DxeQ82LotDKNhcb06d6Dhw/ubDRnyFt2WHt2b1u18t3TvYdff/3l/Xu/bGXzLC8Er9SpcUPVkofW0jq2dOJxbcp2GjqkMTFVEpJEtov7cGynr1V4J0I3LSMJpokhd4t1XFXh3lJhFr9x2S6kU3WJJuhIcY2kjWleudx37Oiuhca0jzA9NToxMQJkLl/oW7Xy3cMHvh4aGsyyBRb3bJsc0snNy3bBKPOjmXGxDDUpxzHdxJFzSRWSfr60DsKVIeaC5N27RQenRRmIP6p6Vkuq2oWT7kPpcHElSXWa/8UC2LXLZNJUTMoygYy8UWqhv+/QS7//zauv/vbDD99cvvyVDRtWDg0Nkocfv/92+fJXt275y8T4iEVdJQ4dsPgq2hGd5dJMSm2WbwaUj1RVmMQc0t68VDpAp7MyrTam1wjRoNP+KXgJLiJ0seO8+CZilVXJrkQqkSvtNlTdmqkjk4So9EbHMpTtomwXsXQ+wOTEkwP7tq9e/e6HH765auXbn3266uKFnlw3nzy+t+2LjSs+fvtXv/4v77335tOnD0TfSzqPPKTVGYkxRIjUxDSqZtRyceNgSYuCx06LilioKFihGG+1KY9kJ1nSxpDW56WQqjDj81X/JmbSXZCKTAZIsBPShcJx4ihX87leMKaZq/ksr1tUuW4e2LfjdO/hRnPu7t3rr7320pn+o1XKvuO/uO0jyxeYUOnCyCNqS159yb/PD9FNqQLobpedJgRGTa5kE2aDtYt7UAU+Rk2qYbbTWZP2ZEgwyGCJYXbxClAp+wiSmObyhwW9c8fnB/fvnKtP9pw8sH79ygcPbsq+UZl2EZxHNlgJutPNuuvLpC1DtkIZKad+EdnUkUsjtqiEtDYs5vx8xZDZKu0aad+/GIW4lVA4YZlgFzuFC7e4/sZpSyDtAtbnp89/c3LbFxsvXzo9NzcpuKSGKYQSe08hK9tFTXBhjyZigoETecETSDVX6qTkWKyVy90pN1OgU9+XRg6+7TkwEuVBJS4xyXSkkKWk64xOXrBc9OkScSdn0pG6JipRfjji274mykgcSmqY8tEiKj3XUZBufM/7KflWqjbkpwJpvoghw6TpKrTDfwKy8DxA/++Lu+q/TaEdqMTUgTBpaunvR/giCeWWzK3qICsQOu3bz9On+jFIR4WmLJOfGsjypF5Mfi/F1/Be9J9mWfhFWv07IEv7MyiHaF2baukGnzpg3iuXbHwQLZUk8go7v2RKgcPkBxZLWCYU4wZwpnpq9bT4Fy68nP9hyNge/38g+wVutgO7KdvpA81sk0V+LfVBouNFoHIbU7XHdXp8GSkI3V4VceQyVUwSOInEVbbTlCtXUoksXOW3eamjYMOU3GcFRCdrukSjLZG4S+BLd4zu1lHG1Ot3s9Mdy+NfAUnQ3e1ipCRaFBSk3UX6WlPiCL7y9efFsPh13kC5XatqmUryPHIy5ZfAWrYLduESxzBM/DfEDpTPQfZ/M7r0JpW+L6oFkAsqvd3pubTBZLbJG1ot9dnSdAydnx9xuortKIUMouUuaWnzlkSF+ME0t0GdhmuOCtIdmbWr3Dx1lFR1oYZUZ6T1sV9kWer4f3EswreMnLEQK+ZPpXdVGs1apsGW938A78xDkwaGZZMAAAAASUVORK5CYII=)
(d) Loss of head at the outlet of a pipe,
![](data:image/png;base64,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)
According to Darcy's formula, the loss of head due to friction in the pipe (hf) is given by
where
f = Darcy's coefficient,
l = Length of pipe,
v = Mean velocity of liquid in pipe, and
d = Diameter of pipe.
The major loss of head or energy is due to friction. The minor loss of head includes the following cases:
(a) Loss of head due to sudden enlargement,
(b) Loss of head due to sudden contraction,
where
Cc= Coefficient of contraction.
(c) Loss of head at the inlet of a pipe,
(d) Loss of head at the outlet of a pipe,
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