SCHEMAT STANOWISKA POMIAROWEGO
WZORY
Teoretyczny współczynnik ciśnienia
$${\overset{\overline{}}{p}}_{t} = 1 - 4\sin^{2}\alpha$$
Różnica ciśnień Δp
p − p∞ = ρmghrzecz
Ciśnienie nasycenia pary wodnej
$$p_{s} = 9,8065*10^{5}\frac{e^{0,01028 - \frac{7821,541}{T} + 82,86568}}{T^{11,48776}}$$
Gęstość czynnika roboczego (przepływające powietrze)
$$\rho_{p} = \frac{1}{R_{s}}\frac{1 + \frac{0,622\varphi p_{s}}{p - \varphi p_{s}}}{1 + \frac{\varphi p_{s}}{p - \varphi p_{s}}}\frac{p}{T}$$
Prędkość przepływającego powietrza
$$v_{\infty} = \sqrt{\frac{2 \bullet \left( p - p_{\infty} \right)}{\rho_{\text{pow}}}}$$
Doświadczalny współczynnik ciśnienia
$$\overset{\overline{}}{p} = \frac{p - p_{\infty}}{\frac{v_{\infty}^{2} \bullet \rho_{\text{pow}}}{2}}$$
Współczynnik oporu powietrza
$$c_{\text{xp}} = \frac{P_{c}}{\frac{v_{\infty}^{2} \bullet \rho_{\text{pow}}}{2} \bullet A}$$
gdzie:
Wartość siły oporu ciśnieniowego:
$$P_{c} = A\int_{0}^{\pi}{p\cos\alpha\text{dα}} = A\left( \frac{\alpha}{2}p_{1}\cos\alpha_{1} + \alpha\sum_{i = 2}^{60}{p_{i}\cos\alpha_{i}} + \frac{\alpha}{2}p_{61}\cos\alpha_{61} \right)$$
Dane otoczenia:
p0=1013hPa
$\rho_{m} = 827\frac{\text{kg}}{m^{3}}$
ϕ0=31%
TABELA POMIARÓW I WYNIKÓW OBLICZEŃ
| L.p. | α | ∆ h | T | T | ps | ρp | pt | p-p∞ | przecz | alfa |
|---|---|---|---|---|---|---|---|---|---|---|
| - | ◦ | mm | ◦ C | K | Pa | kg/m3 | Pa | Pa | Pa | Rad |
| 1. | 0 | 91 | 17,9 | 291,05 | 1994,700 | 1,2095 | 1,000 | 369,135585 | 1,000 | 0,000 |
| 2. | 3 | 91 | 18,1 | 291,25 | 2020,026 | 1,2086 | 0,989 | 369,135585 | 1,001 | 0,052 |
| 3. | 6 | 90 | 18,2 | 291,35 | 2032,794 | 1,2082 | 0,956 | 365,07915 | 0,990 | 0,105 |
| 4. | 9 | 88 | 18,3 | 291,45 | 2045,632 | 1,2078 | 0,902 | 356,96628 | 0,968 | 0,157 |
| 5. | 12 | 85 | 18,4 | 291,55 | 2058,542 | 1,2073 | 0,827 | 344,796975 | 0,936 | 0,209 |
| 6. | 15 | 80 | 18,5 | 291,65 | 2071,522 | 1,2069 | 0,732 | 324,5148 | 0,881 | 0,262 |
| 7. | 18 | 74 | 18,5 | 291,65 | 2071,522 | 1,2069 | 0,618 | 300,17619 | 0,815 | 0,314 |
| 8. | 21 | 66 | 18,5 | 291,65 | 2071,522 | 1,2069 | 0,486 | 267,72471 | 0,727 | 0,367 |
| 9. | 24 | 59 | 18,6 | 291,75 | 2084,574 | 1,2065 | 0,338 | 239,329665 | 0,650 | 0,419 |
| 10. | 27 | 50 | 18,7 | 291,85 | 2097,698 | 1,2060 | 0,176 | 202,82175 | 0,551 | 0,471 |
| 11. | 30 | 39 | 18,8 | 291,95 | 2110,894 | 1,2056 | 0,000 | 158,200965 | 0,430 | 0,524 |
| 12. | 33 | 30 | 18,9 | 292,05 | 2124,163 | 1,2052 | -0,187 | 121,69305 | 0,331 | 0,576 |
| 13. | 36 | 11 | 18,9 | 292,05 | 2124,163 | 1,2052 | -0,382 | 17,848314 | 0,049 | 0,628 |
| 14. | 39 | 5 | 18,9 | 292,05 | 2124,163 | 1,2052 | -0,584 | 7,4638404 | 0,020 | 0,681 |
| 15. | 42 | -15 | 19,1 | 292,25 | 2150,919 | 1,2043 | -0,791 | -60,846525 | -0,166 | 0,733 |
| 16. | 45 | -46 | 19,2 | 292,35 | 2164,407 | 1,2039 | -1,000 | -186,59601 | -0,508 | 0,785 |
| 17. | 48 | -74 | 19,3 | 292,45 | 2177,969 | 1,2035 | -1,209 | -300,17619 | -0,817 | 0,838 |
| 18. | 51 | -93 | 19,2 | 292,35 | 2164,407 | 1,2039 | -1,416 | -377,248455 | -1,027 | 0,890 |
| 19. | 54 | -113 | 19,4 | 292,55 | 2191,605 | 1,2030 | -1,618 | -458,377155 | -1,248 | 0,942 |
| 20. | 57 | -131 | 19,4 | 292,55 | 2191,605 | 1,2030 | -1,813 | -531,392985 | -1,447 | 0,995 |
| 21. | 60 | -142 | 19,5 | 292,65 | 2205,316 | 1,2026 | -2,000 | -576,01377 | -1,569 | 1,047 |
| 22. | 63 | -142 | 19,4 | 292,55 | 2191,605 | 1,2030 | -2,176 | -576,01377 | -1,569 | 1,100 |
| 23. | 66 | -124 | 19,5 | 292,65 | 2205,316 | 1,2026 | -2,338 | -502,99794 | -1,370 | 1,152 |
| 24. | 69 | -108 | 19,5 | 292,65 | 2205,316 | 1,2026 | -2,486 | -438,09498 | -1,194 | 1,204 |
| 25. | 72 | -98 | 19,5 | 292,65 | 2205,316 | 1,2026 | -2,618 | -397,53063 | -1,083 | 1,257 |
| 26. | 75 | -95 | 19,5 | 292,65 | 2205,316 | 1,2026 | -2,732 | -385,361325 | -1,050 | 1,309 |
| 27. | 78 | -91 | 19,6 | 292,75 | 2219,102 | 1,2022 | -2,827 | -369,135585 | -1,006 | 1,361 |
| 28. | 81 | -90 | 19,7 | 292,85 | 2232,963 | 1,2017 | -2,902 | -365,07915 | -0,995 | 1,414 |
| 29. | 84 | -88 | 19,6 | 292,75 | 2219,102 | 1,2022 | -2,956 | -356,96628 | -0,973 | 1,466 |
| 30. | 87 | -86 | 19,6 | 292,75 | 2219,102 | 1,2022 | -2,989 | -348,85341 | -0,951 | 1,518 |
| 31. | 90 | -88 | 19,7 | 292,85 | 2232,963 | 1,2017 | -3,000 | -356,96628 | -0,973 | 1,571 |
| 32. | 93 | -90 | 19,7 | 292,85 | 2232,963 | 1,2017 | -2,989 | -365,07915 | -0,995 | 1,623 |
| 33. | 96 | -88 | 19,7 | 292,85 | 2232,963 | 1,2017 | -2,956 | -356,96628 | -0,973 | 1,676 |
| 34. | 99 | -88 | 19,8 | 292,95 | 2246,900 | 1,2013 | -2,902 | -356,96628 | -0,974 | 1,728 |
| 35. | 102 | -88 | 19,8 | 292,95 | 2246,900 | 1,2013 | -2,827 | -356,96628 | -0,974 | 1,780 |
| 36. | 105 | -87 | 19,9 | 293,05 | 2260,912 | 1,2009 | -2,732 | -352,909845 | -0,963 | 1,833 |
| 37. | 108 | -87 | 19,8 | 292,95 | 2246,900 | 1,2013 | -2,618 | -352,909845 | -0,963 | 1,885 |
| 38. | 111 | -90 | 19,9 | 293,05 | 2260,912 | 1,2009 | -2,486 | -365,07915 | -0,996 | 1,937 |
| 39. | 114 | -89 | 19,9 | 293,05 | 2260,912 | 1,2009 | -2,338 | -361,022715 | -0,985 | 1,990 |
| 40. | 117 | -89 | 19,9 | 293,05 | 2260,912 | 1,2009 | -2,176 | -361,022715 | -0,985 | 2,042 |
| 41. | 120 | -89 | 19,9 | 293,05 | 2260,912 | 1,2009 | -2,000 | -361,022715 | -0,985 | 2,094 |
| 42. | 123 | -89 | 19,9 | 293,05 | 2260,912 | 1,2009 | -1,813 | -361,022715 | -0,985 | 2,147 |
| 43. | 126 | -91 | 19,9 | 293,05 | 2260,912 | 1,2009 | -1,618 | -369,135585 | -1,007 | 2,199 |
| 44. | 129 | -89 | 19,9 | 293,05 | 2260,912 | 1,2009 | -1,416 | -361,022715 | -0,985 | 2,251 |
| 45. | 132 | -90 | 19,9 | 293,05 | 2260,912 | 1,2009 | -1,209 | -365,07915 | -0,996 | 2,304 |
| 46. | 135 | -90 | 19,9 | 293,05 | 2260,912 | 1,2009 | -1,000 | -365,07915 | -0,996 | 2,356 |
| 47. | 138 | -90 | 19,9 | 293,05 | 2260,912 | 1,2009 | -0,791 | -365,07915 | -0,996 | 2,409 |
| 48. | 141 | -89 | 19,9 | 293,05 | 2260,912 | 1,2009 | -0,584 | -361,022715 | -0,985 | 2,461 |
| 49. | 144 | -89 | 20 | 293,15 | 2275,001 | 1,2004 | -0,382 | -361,022715 | -0,985 | 2,513 |
| 50. | 147 | -89 | 20 | 293,15 | 2275,001 | 1,2004 | -0,187 | -361,022715 | -0,985 | 2,566 |
| 51. | 150 | -89 | 20 | 293,15 | 2275,001 | 1,2004 | 0,000 | -361,022715 | -0,985 | 2,618 |
| 52. | 153 | -85 | 20 | 293,15 | 2275,001 | 1,2004 | 0,176 | -344,796975 | -0,941 | 2,670 |
| 53. | 156 | -86 | 20 | 293,15 | 2275,001 | 1,2004 | 0,338 | -348,85341 | -0,952 | 2,723 |
| 54. | 159 | -86 | 20 | 293,15 | 2275,001 | 1,2004 | 0,486 | -348,85341 | -0,952 | 2,775 |
| 55. | 162 | -85 | 20 | 293,15 | 2275,001 | 1,2004 | 0,618 | -344,796975 | -0,941 | 2,827 |
| 56. | 165 | -84 | 20 | 293,15 | 2275,001 | 1,2004 | 0,732 | -340,74054 | -0,930 | 2,880 |
| 57. | 168 | -82 | 19,9 | 293,05 | 2260,912 | 1,2009 | 0,827 | -332,62767 | -0,908 | 2,932 |
| 58. | 171 | -85 | 20 | 293,15 | 2275,001 | 1,2004 | 0,902 | -344,796975 | -0,941 | 2,985 |
| 59. | 174 | -82 | 20,1 | 293,25 | 2289,166 | 1,2000 | 0,956 | -332,62767 | -0,908 | 3,037 |
| 60. | 177 | -83 | 20,1 | 293,25 | 2289,166 | 1,2000 | 0,989 | -336,684105 | -0,919 | 3,089 |
| 61. | 180 | -86 | 20,1 | 293,25 | 2289,166 | 1,2000 | 1,000 | -348,85341 | -0,953 | 3,142 |
TABELA SUMUJĄCA DLA Pc:
| 1. | 1,497944 | 11. | 1,431609 | 21. | -2,17096 | 31. | -1,7E-16 | 41. | 1,47757 | 51. | 2,560141 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 2. | 2,995889 | 12. | 1,081771 | 22. | -2,13748 | 32. | 0,152813 | 42. | 1,609485 | 52. | 2,633991 |
| 3. | 2,985704 | 13. | 0,802996 | 23. | -1,91431 | 33. | 0,312144 | 43. | 1,736988 | 53. | 2,579244 |
| 4. | 2,933654 | 14. | 0,113133 | 24. | -1,47339 | 34. | 0,456765 | 44. | 1,901522 | 54. | 2,666819 |
| 5. | 2,841771 | 15. | 0,04524 | 25. | -1,10655 | 35. | 0,607288 | 45. | 1,977375 | 55. | 2,71674 |
| 6. | 2,711566 | 16. | -0,35117 | 26. | -0,84099 | 36. | 0,755982 | 46. | 2,113078 | 56. | 2,727131 |
| 7. | 2,513676 | 17. | -1,01946 | 27. | -0,65489 | 37. | 0,892667 | 47. | 2,220772 | 57. | 2,729147 |
| 8. | 2,282424 | 18. | -1,54298 | 28. | -0,47217 | 38. | 1,034859 | 48. | 2,322379 | 58. | 2,689192 |
| 9. | 1,991989 | 19. | -1,81052 | 29. | -0,31214 | 39. | 1,215469 | 49. | 2,390758 | 59. | 2,807867 |
| 10. | 1,737406 | 20. | -2,03985 | 30. | -0,15276 | 40. | 1,341605 | 50. | 2,479275 | 60. | 2,720927 |
| Δα | 61. | 1,497944 | |||||||||
| 3 | Σ | 65,29508 |
PRZYKŁADY OBLICZEŃ (dla pomiaru nr 1)
Teoretyczny współczynnik ciśnienia
$${\overset{\overline{}}{p}}_{t} = 1 - 4\sin^{2}\alpha = 1 - 4\sin^{2}0,157 = 1$$
Różnica ciśnień Δp
$$p - p_{\infty} = \rho_{m}gh_{\text{rzecz}} = 827\frac{\text{kg}}{m^{3}}*9,81\frac{m}{s^{2}}*91*10^{- 3}*0,5 = 369,14\text{\ Pa}\ $$
Ciśnienie nasycenia pary wodnej
$$p_{s} = 9,8065*10^{5}\frac{e^{0,01028 - \frac{7821,541}{T} + 82,86568}}{T^{11,48776}} = 9,8065*10^{5}\frac{e^{0,01028 - \frac{7821,541}{291,05} + 82,86568}}{{291,05}^{11,48776}} = \ 1994,70\ \text{Pa}$$
Gęstość czynnika roboczego
$$\rho_{p} = \frac{1}{R_{s}}\frac{1 + \frac{0,622\varphi p_{s}}{p - \varphi p_{s}}}{1 + \frac{\varphi p_{s}}{p - \varphi p_{s}}}\frac{p}{T} = \ \frac{1}{287,1}*\frac{1 + \frac{0,622*0,31*1994,70}{1,013*10^{5} - 0,31*1994,70}}{1 + \frac{0,31*1994,70}{1,013*10^{5} - 0,31*1994,70}}*\frac{1,013*10^{5}}{291,05} = = \ 1,2095\frac{\text{kg}}{m^{3}}$$
Prędkość przepływającego powietrza
$$v_{\infty} = \sqrt{\frac{2 \bullet \left( 369,14 \right)}{1,2095}} = 24,71\frac{m}{s}$$
Doświadczalny współczynnik ciśnienia
$$\overset{\overline{}}{p} = \frac{p - p_{\infty}}{\frac{v_{\infty}^{2} \bullet \rho_{\text{pow}}}{2}} = \frac{369,14}{\frac{{24,71}^{2} \bullet 1,2095}{2}} = \ 1,000\text{\ Pa}$$
Wartość siły oporu ciśnieniowego:
$$P_{c} = A\int_{0}^{\pi}{p\cos\alpha\text{dα}} = \left( \frac{\alpha}{2}p_{1}\cos\alpha_{1} + \alpha\sum_{i = 2}^{60}{p_{i}\cos\alpha_{i}} + \frac{\alpha}{2}p_{61}\cos\alpha_{61} \right) = 65,3\ $$
Współczynnik oporu powietrza
$$c_{\text{xp}} = \frac{P_{c}}{\frac{v_{\infty}^{2} \bullet \rho_{\text{pow}}}{2}} = \frac{65,3}{\frac{{24,71}^{2} \bullet 1,2095}{2}} = 0,708\ $$
WNIOSKI
Celem ćwiczenia było doświadczalne i teoretyczne wyznaczenie rozkładu ciśnień na powierzchni ciała opływowego (walca) ustaloną równoległą strugą płynu. Jak widać na wykresie rozkład ciśnienia na obwodzie walca podczas przeprowadzania pomiarów ustabilizował się jako opływ z oderwaniem laminarnej warstwy przyściennej. Wykres potwierdził również teorię punktu oderwania , leżącego w przedniej części walca i wznoszącego w przybliżeniu 80°.