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![]() The main water line into a tall building has a pressure of 600 kPa at 5 m below ground level, as shown in<image 1>. A pump brings the pressure up so thatnthe water can be delivered at 200 kPa at the top floor 100 m above ground level. Assume a flow rate of 10 kg/s liquid water at 10°C and neglect any difference in kinetic energy and internal energy u. Find the pump work. | Continuity Eq.:
Bernoulli Eq.:
Assuming the water main is large so that the flow velocity there is negligible, and using subscripts 1 for the main and 2 for the top floor, we get:
Substituting for V2 and rearranging:
From the table,
Expected Answer: A Difficulty: Hard Subfield: Thermodynamics |
{ "id": "gen-veskO04wr8Z8Hee0mBtKhGFvEVrv", "model": "google/gemini-pro-vision", "choices": [ { "message": { "role": "assistant", "content": " Continuity Eq.:\n\n$$\\rho AV=\\text{const}\\rightarrow \\rho_1A_1V_1=\\rho_2A_2V_2$$\n\n$$\\rightarrow V_1=V_2\\frac{\\rho_2}{\\rho_1}\\frac{A_2}{A_1}$$\n\nBernoulli Eq.:\n\n$$p_1+\\frac{1}{2}\\rho_1V_1^2+\\rho_1gy_1=p_2+\\frac{1}{2}\\rho_2V_2^2+\\rho_2gy_2+W_{pump}$$\n\nAssuming the water main is large so that the flow velocity there is negligible, and using subscripts 1 for the main and 2 for the top floor, we get:\n\n$$p_1+\\rho_1gy_1=p_2+\\rho_2gy_2+W_{pump}$$\n\nSubstituting for V2 and rearranging:\n\n$$W_{pump}=(p_2-p_1)+(\\rho_2-\\rho_1)gy_2-\\frac{1}{2}\\rho_2V_1^2\\left(\\frac{\\rho_1}{\\rho_2}\\frac{A_2}{A_1}\\right)^2$$\n\nFrom the table,\n\n$$\\rho_1=999\\text{ kg/m}^3, \\rho_2=997\\text{ kg/m}^3$$\n\n>>>>>>We don't know A1 and A2, but their ratio will cancel out:\n\n$$W_{pump}=600\\times10^3\\text{ Pa}+997\\times9.81\\text{ m/s}^2\\times105\\text{ m}$$\n\n$$-999\\times9.81\\text{ m/s}^2\\times5\\text{ m}-\\frac{1}{2}\\times997\\times\\left(\\frac{10}{999}\\frac{A_2}{A_1}\\right)^2\\text{ m}^2/\\text{s}^2$$\n\n$$=5.94\\times10^6\\text{ W}=\\boxed{5.94\\text{ MW}}$$" }, "finish_reason": "STOP" } ], "object": "chat.completion", "created": 1708556766, "usage": { "prompt_tokens": 1440, "completion_tokens": 1932, "total_tokens": 3372 } }