Numerical and Experimental Study of Pump Sump Flows

<table>Diagram of how to obtain circulation on each selected disc. (a) A horizontally sliced disc (black area) at a given elevation. (b) A disc is divided into several small parts, where each dot indicates vorticity <svg height="19.4125" id="M241" style="vertical-align:-4.74141pt" version="1.1" viewbox="0 0 167.95 19.4125" width="167.95" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path d="M615 290q0 -148 -114 -245q-65 -57 -125 -57q-39 0 -62 27t-29 70q-31 -41 -72 -69t-74 -28q-49 0 -82.5 43t-33.5 124q0 97 55 174.5t148 124.5l15 -24q-59 -45 -94 -113t-35 -146q0 -58 19.5 -88.5t48.5 -30.5q33 0 62 34.5t45 101.5q16 73 20 135l80 17l5 -6
q-11 -37 -23.5 -100t-12.5 -92q0 -47 16.5 -71.5t44.5 -24.5q62 0 95.5 67t33.5 160q0 107 -63 152l1 9q14 14 45 14q29 0 57.5 -44t28.5 -114z" id="x1D714"></path></g>
<g transform="matrix(.012,-0,0,-.012,12.15,16.25)"><path d="M480 416q0 -21 -18 -41q-9 -11 -17 -7q-20 9 -42 9q-62 0 -140 -78q23 -69 88 -192q17 -31 27 -42t20 -11q16 0 62 46l17 -20q-64 -92 -119 -92q-35 0 -70 66q-41 73 -84 187q-36 -30 -62 -61q-27 -115 -35 -172q-41 -8 -78 -20l-6 6l140 612q7 28 0.5 34t-37.5 7l-34 1
l5 26q38 4 74 13.5t57 17t25 7.5q12 0 4 -32l-104 -443h2q35 38 97 93q39 35 65.5 56t62 41.5t58.5 20.5q19 0 30.5 -10t11.5 -22z" id="x1D458"></path></g><g transform="matrix(.012,-0,0,-.012,20.003,16.25)"><path d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30
q16 0 76 48z" id="x1D456"></path></g>
<g transform="matrix(.017,-0,0,-.017,23.875,12.162)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,-0,0,-.017,29.757,12.162)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,44.443,12.162)"><path d="M471 379q0 -56 -17 -126t-47 -124q-80 -141 -227 -141q-81 0 -119 46t-38 114q0 92 45 161.5t108 101.5q73 37 137 37q45 0 80 -33v27q0 83 -21 133q-38 90 -138 90q-68 0 -123 -31q-12 0 -26.5 18.5t-14.5 31.5q52 28 138 28q139 0 219 -134q44 -74 44 -199zM382 335
q0 22 -4.5 33t-18.5 20q-35 21 -79 21q-23 0 -57 -14q-36 -15 -74.5 -75t-38.5 -156q0 -58 23 -97t67 -39q17 0 38 7q50 15 97 93t47 207z" id="x1D715"></path></g><g transform="matrix(.017,-0,0,-.017,52.841,12.162)"><path d="M515 96q-37 -44 -84.5 -76t-71.5 -32q-12 0 -18 7t-6 34t10 75l19 89h-2q-87 -113 -181 -176q-43 -29 -83 -29q-46 0 -46 63q0 31 9 67l52 222q7 36 -1 36q-10 0 -76 -50l-13 24q47 45 92 71.5t67 26.5q18 0 21 -16.5t-8 -65.5l-56 -242q-16 -67 13 -67q41 0 114 74
t114 146l32 145l74 26h11q-52 -199 -80 -347q-8 -39 6 -39q7 0 32 17.5t47 39.5z" id="x1D462"></path></g>
<g transform="matrix(.012,-0,0,-.012,62,16.25)"><path d="M556 393q0 -39 -36 -106q-42 -78 -185 -279q-47 -66 -81 -108t-117 -135l-112 -26l-8 22q150 90 251 219q-6 136 -39 340q-8 53 -21 53q-6 0 -27 -19.5t-38 -42.5l-16 26q80 111 127 111q23 0 35 -28t20 -90q18 -137 27 -263h2q142 200 142 279q0 24 -14 48q-4 7 5 26
q13 28 43 28q18 0 30 -15.5t12 -39.5z" id="x1D466"></path></g>
<g transform="matrix(.017,-0,0,-.017,69.512,12.162)"><path d="M368 703l-264 -866h-60l265 866h59z" id="x2F"></path></g><g transform="matrix(.017,-0,0,-.017,76.516,12.162)"><path d="M471 379q0 -56 -17 -126t-47 -124q-80 -141 -227 -141q-81 0 -119 46t-38 114q0 92 45 161.5t108 101.5q73 37 137 37q45 0 80 -33v27q0 83 -21 133q-38 90 -138 90q-68 0 -123 -31q-12 0 -26.5 18.5t-14.5 31.5q52 28 138 28q139 0 219 -134q44 -74 44 -199zM382 335
q0 22 -4.5 33t-18.5 20q-35 21 -79 21q-23 0 -57 -14q-36 -15 -74.5 -75t-38.5 -156q0 -58 23 -97t67 -39q17 0 38 7q50 15 97 93t47 207z" id="x1D715"></path></g><g transform="matrix(.017,-0,0,-.017,84.913,12.162)"><path d="M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37
q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z" id="x1D465"></path></g><g transform="matrix(.017,-0,0,-.017,98.189,12.162)"><path d="M535 230h-483v50h483v-50z" id="x2212"></path></g><g transform="matrix(.017,-0,0,-.017,111.941,12.162)"><path d="M471 379q0 -56 -17 -126t-47 -124q-80 -141 -227 -141q-81 0 -119 46t-38 114q0 92 45 161.5t108 101.5q73 37 137 37q45 0 80 -33v27q0 83 -21 133q-38 90 -138 90q-68 0 -123 -31q-12 0 -26.5 18.5t-14.5 31.5q52 28 138 28q139 0 219 -134q44 -74 44 -199zM382 335
q0 22 -4.5 33t-18.5 20q-35 21 -79 21q-23 0 -57 -14q-36 -15 -74.5 -75t-38.5 -156q0 -58 23 -97t67 -39q17 0 38 7q50 15 97 93t47 207z" id="x1D715"></path></g><g transform="matrix(.017,-0,0,-.017,120.339,12.162)"><path d="M515 96q-37 -44 -84.5 -76t-71.5 -32q-12 0 -18 7t-6 34t10 75l19 89h-2q-87 -113 -181 -176q-43 -29 -83 -29q-46 0 -46 63q0 31 9 67l52 222q7 36 -1 36q-10 0 -76 -50l-13 24q47 45 92 71.5t67 26.5q18 0 21 -16.5t-8 -65.5l-56 -242q-16 -67 13 -67q41 0 114 74
t114 146l32 145l74 26h11q-52 -199 -80 -347q-8 -39 6 -39q7 0 32 17.5t47 39.5z" id="x1D462"></path></g>
<g transform="matrix(.012,-0,0,-.012,129.488,16.25)"><path d="M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37
q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z" id="x1D465"></path></g>
<g transform="matrix(.017,-0,0,-.017,136.762,12.162)"><path d="M368 703l-264 -866h-60l265 866h59z" id="x2F"></path></g><g transform="matrix(.017,-0,0,-.017,143.766,12.162)"><path d="M471 379q0 -56 -17 -126t-47 -124q-80 -141 -227 -141q-81 0 -119 46t-38 114q0 92 45 161.5t108 101.5q73 37 137 37q45 0 80 -33v27q0 83 -21 133q-38 90 -138 90q-68 0 -123 -31q-12 0 -26.5 18.5t-14.5 31.5q52 28 138 28q139 0 219 -134q44 -74 44 -199zM382 335
q0 22 -4.5 33t-18.5 20q-35 21 -79 21q-23 0 -57 -14q-36 -15 -74.5 -75t-38.5 -156q0 -58 23 -97t67 -39q17 0 38 7q50 15 97 93t47 207z" id="x1D715"></path></g><g transform="matrix(.017,-0,0,-.017,152.163,12.162)"><path d="M556 393q0 -39 -36 -106q-42 -78 -185 -279q-47 -66 -81 -108t-117 -135l-112 -26l-8 22q150 90 251 219q-6 136 -39 340q-8 53 -21 53q-6 0 -27 -19.5t-38 -42.5l-16 26q80 111 127 111q23 0 35 -28t20 -90q18 -137 27 -263h2q142 200 142 279q0 24 -14 48q-4 7 5 26
q13 28 43 28q18 0 30 -15.5t12 -39.5z" id="x1D466"></path></g><g transform="matrix(.017,-0,0,-.017,162.006,12.162)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g>
</svg> at its occupied area (<svg height="18.65" id="M242" style="vertical-align:-4.77652pt" version="1.1" viewbox="0 0 18.237499 18.65" width="18.237499" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path d="M716 28l-8 -28h-254l8 28q55 5 66 17.5t5 47.5l-30 145h-232l-74 -138q-21 -43 -11 -54.5t72 -17.5l-6 -28h-235l8 28q53 7 73.5 21t53.5 70l319 540l33 8q6 -43 33 -176l80 -381q10 -49 26.5 -63t72.5 -19zM495 281l-49 264h-2l-149 -264h200z" id="x1D434"></path></g>
<g transform="matrix(.012,-0,0,-.012,12.525,15.475)"><path d="M394 607q0 -25 -15.5 -43t-39.5 -18q-19 0 -32 13t-13 35t16 41.5t40 19.5q20 0 32 -13.5t12 -34.5zM360 349l-57 -283q-38 -188 -122 -263q-72 -64 -114 -64q-16 0 -30 10.5t-14 22.5q0 24 21 45q15 13 26 6q30 -20 54 -20q41 0 65 95q37 147 86 415q6 34 6.5 42.5
t-5.5 11.5q-18 0 -79 -54l-14 25q48 48 96.5 79t71.5 31q29 0 9 -99z" id="x1D457"></path></g>
</svg>).</table>

Mathematical Problems in Engineering

Numerical and Experimental Study of Pump Sump Flows

Figure 18