Physics">
9 - Análisis Dimensional
9 - Análisis Dimensional
9 - Análisis Dimensional
EJERCICIOS
𝑀 = 𝑓( Ω, 𝑅, µ, ø)
Unidades L T M
M Kgmˆ2/sˆ2 2 -2 1
Ω 1/s 0 -1 0
µ Kg/ms -1 -1 1
ø 1 0 0 0
R m 1 0 0
𝜋! = 5 − 3 = 2
𝜋" :
[ Ω 𝑅 µ ][ø]
# # #
𝐿 𝑇 𝑀
= 𝐿$%& 𝑇 $%& 𝑀"& 𝐿"' 𝑇 #' 𝑀#' 𝐿$"( 𝑇 $"( 𝑀"( 𝐿# 𝑇 # 𝑀#
𝐿: 0 = −2𝑎 + 𝑏 − 𝑐 + 0
𝑇: 0 = −2𝑎 − 0 − 𝑐 + 0
𝑀: 0 = 𝑎 + 0 + 𝑐 + 0
𝑏=0
𝑐=0
𝑎=0
𝜋" = Ω# 𝑅# µ# ø
𝜋" = ø
𝜋% :
[ Ω 𝑅 µ ][𝑀]
𝐿# 𝑇 # 𝑀#
= 𝐿$%& 𝑇 $%& 𝑀"& 𝐿"' 𝑇 #' 𝑀#' 𝐿$"( 𝑇 $"( 𝑀"( 𝐿% 𝑇 $% 𝑀"
𝐿: 0 = −2𝑎 + 𝑏 − 𝑐 + 2
𝑇: 0 = −2𝑎 − 0 − 𝑐 − 2
𝑀: 0 = 𝑎 + 0 + 𝑐 + 1
𝑏 = −3
𝑐 = −1
𝑎 = −1
π% = Ω$" R$) µ$" M
𝑀
𝜋% = )
ΩR µ
𝑀
=∝
Ω R) µø
Unidades L T M
Diámetro m 1 0 0
Velocidad m/s 1 -1 0
Densidad kg/m³ -3 0 1
Viscosidad kg/ms -1 -1 1
Gravedad m/s² 1 -2 0
𝜋! = 5 − 3 = 2
Comprobando la independencia de 𝐷 𝜌 µ:
1 0 0
C−3 0 1C Det ≠ 0
−1 −1 1
𝜋" :
[𝐷 𝜌 µ][U]
𝐿! 𝑇 ! 𝑀! = 𝐿" 𝑇 !" 𝑀!" 𝐿#$% 𝑇 !% 𝑀&% 𝐿#&' 𝑇 #&' 𝑀' 𝐿& 𝑇 #& 𝑀!
𝐿: 0 = 𝑎 − 3𝑏 + 𝑐 − 1
𝑇: 0 = 0 − 0 − 𝑐 − 1
𝑀: 0 = 0 + 𝑏 + 𝑐 + 0
𝑎=1
𝑏=1
𝑐 = −1
𝐷𝜌
𝜋" = U = 𝑅𝑒
µ
𝜋% :
[𝐷 𝜌 µ][𝛾]
𝑎=3
𝑏=2
𝑐 = −2
𝐷³𝜌² 𝐷³
𝜋% = 𝑔= 𝑔
µ² 𝜐²
𝑈. 𝐷. 𝑈/ 𝐷/
= ; 15𝐷. = 𝐷/
𝜈. 𝜈/
Presión kg/ms² -1 -2 1
Rugosidad m 1 0 0
𝜋! = 7 − 3 = 4
Comprobando la independencia de 𝐷 𝜌 𝑉:
1 0 0
C−3 0 1C Det ≠ 0
1 −1 0
𝜋" :
[𝐷 𝜌 𝑉][P]
𝐿! 𝑇 ! 𝑀! = 𝐿" 𝑇 !" 𝑀!" 𝐿#$% 𝑇 !% 𝑀&% 𝐿&' 𝑇 #&' 𝑀!' 𝐿#& 𝑇 #( 𝑀&
𝐿: 0 = 𝑎 − 3𝑏 + 𝑐 − 1
𝑇: 0 = 0 − 0 − 𝑐 − 2
𝑀: 0 = 0 + 𝑏 + 0 + 1
𝑎=0
𝑏 = −1
𝑐 = −2
𝑃
𝜋" =
𝜌𝑉 %
𝜋% :
[𝐷 𝜌 𝑉][L]
𝑎 = −1
𝑏=0
𝑐=0
𝐿
𝜋% =
𝐷
𝜋) :
[𝐷 𝜌 𝑉][µ]
𝐿! 𝑇 ! 𝑀! = 𝐿" 𝑇 !" 𝑀!" 𝐿#$% 𝑇 !% 𝑀&% 𝐿&' 𝑇 #' 𝑀!' 𝐿#& 𝑇 #& 𝑀&
𝐿: 0 = 𝑎 − 3𝑏 + 𝑐 − 1
𝑇: 0 = 0 − 0 − 𝑐 − 1
𝑀: 0 = 0 + 𝑏 + 0 + 1
𝑎 = −1
𝑏 = −1
𝑐 = −1
µ
𝜋) =
𝐷𝜌𝑉
𝜋, :
[𝐷 𝜌 𝑉][𝜀]
𝑎 = −1
𝑏=0
𝑐=0
𝜀
𝜋, =
𝐷
𝐿 µ 𝜀 𝐿 𝜀
∆𝑃 = 𝑓 \ , , ] = 𝑓 \ , 𝑅𝑒, ]
𝐷 𝐷𝜌𝑉 𝐷 𝐷 𝐷