Wikipedia

Vloeimeetskyf: Verskil tussen weergawes

Blaai deur geskiedenis
← Ouer wysigingNuwer wysiging →
Content deleted Content added
VisueelWikiteks
Lyn 107:
<math>P_1 + gh_1 + \frac{\rho v_1^2}{2} \quad = \quad P_o + gh_o + \frac{\rho v_o^2}{2}</math>
 
As:
As aanvaar word dat <math>h_1 = h_o</math> dan:
 
<math>v_1 = \frac{Q}{A_1}</math> {{spaces|5}} en {{spaces|5}} <math>v_o = \frac{Q}{A_o}</math> {{spaces|5}} en {{spaces|5}} <math>\rho = \rho_1 = \rho_o</math> {{spaces|5}} en {{spaces|5}} <math>\Delta P = P_1 - P_o</math> {{spaces|5}} en {{spaces|5}} <math>\Delta h = h_1 - h_o</math> {{spaces|5}} en {{spaces|5}} <math>A = \frac{\pi}{4}d^2</math> {{spaces|5}} en {{spaces|5}} <math>\beta = \frac{d_o}{d_1}</math> {{spaces|5}} en {{spaces|5}} <math>\beta^2 = \left(\frac{d_o}{d_1}\right)^2 = \frac{A_o}{A_1}</math>
<math>P_1 + \frac{\rho v_1^2}{2} \quad = \quad P_o + \frac{\rho v_o^2}{2}</math>
 
dan is:
<math>P_1 - P_2 = \Delta P = \frac{\rho v_o^2}{2} - \frac{\rho v_1^2}{2} </math>
 
:<math>\frac{1}{2}\left({Q \over A_1}\right)^2 + gh_1 + \frac{P_1}{\rho} = \frac{1}{2}\left({Q \over A_o}\right)^2 + gh_o + \frac{P_o}{\rho}</math>
::Die verwantskap tussen volumevloei, deursnitarea en snelheid is:
 
:<math>g \left( h_1-h_o \right) + \frac{P_1-P_o}{\rho} = \frac{Q^2}{2} \times \left(\frac{1}{A_o^2} - \frac{1}{A_1^2} \right)</math>
::<math>Q = A\cdot v</math>
 
:<math>\frac{Q^2}{2} \times \frac{1}{A_o^2} \times \left(1 - \frac{A_o^2}{A_1^2} \right) = \frac{\Delta P}{\rho} + g \Delta h</math>
::Die volumevloeitempo is orals dieselfde, dus is:
 
:<math>\Deltafrac{Q^2}{2} P =\times \frac{1}{A_o^2} \rhotimes \left(1 - \betafrac{d_o^4}{d_1^4} \right) = \frac{1}{\rho} \left({Q \overDelta C_dP + \rho g \Delta h A_o}\right)^2</math>▼
::<math>Q = Q_o = Q_1</math>
 
:<math>Q^2 = A_o^2\;\sqrt{\frac{1}{1-2 \beta^4}}\;\sqrt{2left( \Delta P \over+ \rho} \quad = \quad A_o\;\sqrt{2g \Delta Ph \over right)}{\rho \left(1 - \beta^4 \right)}</math>▼
::<math>v_1 = Q/A_1</math> en <math>v_o = Q/A_o</math> :
 
:<math>Q = A_o\;\sqrt{ \frac{2 \;left( \Delta P /+ \rho g \Delta h \right)}{\rho \left(1 -(A_o/A_1) \beta^24 \right)}}</math>▼
Dus:
 
<math>\Delta P = \frac{\rho}{2} \bigg(\frac{Q}{A_o}\bigg)^2 - \frac{\rho}{2} \bigg(\frac{Q}{A_1}\bigg)^2 = \frac{1}{2}\rho Q^2 \bigg(\frac{1}{A_o^2} - \frac{1}{A_1^2}\bigg) = \frac{1}{2}\frac{\rho Q^2}{A_o^2} \bigg(1 - \frac{A_o^2}{A_1^2}\bigg)</math>
 
Los op vir <math>Q</math>:
 
▲<math>Q = A_o\;\sqrt{\frac{2\;\Delta P / \rho}{1-(A_o/A_1)^2}}</math>
 
As <math>A = \pi \frac{d^2}{4}</math>, dan is:
 
<math>Q = A_o\;\sqrt{\frac{1}{1-(d_o/d_1)^4}}\;\sqrt{2 \Delta P \over \rho}</math>
 
Indien die betafaktor gedefinieer word as <math>\beta = d_o/d_1</math>, dan:
 
▲<math>Q = A_o\;\sqrt{\frac{1}{1-\beta^4}}\;\sqrt{2 \Delta P \over \rho} \quad = \quad A_o\;\sqrt{2 \Delta P \over \rho \left(1-\beta^4\right)}</math>
 
Hierdie gee die drukval tussen die punt voor die meetskyf (1) en na die meetskyf (2). Wanneer ideale toestande geld, sal <math>P_1 = P_2</math> wees volgens die [[Bernoulli-beginsel]]. Maar as gevolg van energieverliese in die vorm van hitte en klank, herstel die druk nooit weer ten volle nie. Daarom is dit nodig om die dimensielose [[uitlaatvloeikoëffisiënt]] <math>C_d</math> (<math>C_d > 1</math>) by te voeg en daarom word die vergelyking:
 
<math>Q = C_d A_o\;\sqrt{2 \Delta P \over \rho \left(1-\beta^4\right)}</math> {{spaces|10}} of {{spaces|10}}
▲<math>\Delta P = \frac{1}{2} \rho \left(1-\beta^4\right) \left({Q \over C_d A_o}\right)^2</math>
 
::As <math>C = \frac{C_d}{\sqrt{1-\beta^4}}</math>, dan is:
 
{| class="wikitable"
|-align=center
|
<math>Q = CC_d A_o \;\sqrt{ \frac{2 \left( \Delta P + \overrho g \Delta h \right)}{\rho \left(1 - \beta^4 \right)}}</math> {{spaces|10}} of {{spaces|10}} <math>\Delta P = \frac{\rho}{2}\left({Q \over C A_o}\right)^2</math>
<math>\Delta P = \frac{1}{2} \rho \left(1-\beta^4\right) \left({Q \over C_d A_o}\right)^2 - \rho g \Delta h</math>
 
|}
 
Let wel, die eenhede moet so gekies word sodat al die eenhede uit kanselleer.
 
Gewoonlik is die term <math>\rho g \Delta h</math> weglaatbaar klein.
 
=== Bylaag B: Balansering van eenhede ===
Ontsluit van "https://af.wikipedia.org/wiki/Vloeimeetskyf"