Temperature Ratio across Expansion Fan Solution

STEP 0: Pre-Calculation Summary
Formula Used
Temperature Ratio Across Expansion Fan = (1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Ahead of Expansion Fan^2)/(1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Behind Expansion Fan^2)
Te,r = (1+0.5*(γe-1)*Me1^2)/(1+0.5*(γe-1)*Me2^2)
This formula uses 4 Variables
Variables Used
Temperature Ratio Across Expansion Fan - Temperature Ratio Across Expansion Fan is the ratio of downstream temperature to upstream temperature across the expansion fan.
Specific Heat Ratio Expansion Wave - The Specific Heat Ratio Expansion Wave is the ratio of the heat capacity at constant pressure to heat capacity at constant volume.
Mach Number Ahead of Expansion Fan - Mach Number Ahead of Expansion Fan is the Mach number of the upstream flow.
Mach Number Behind Expansion Fan - Mach Number Behind Expansion Fan is the Mach number of downstream flow across expansion fan.
STEP 1: Convert Input(s) to Base Unit
Specific Heat Ratio Expansion Wave: 1.41 --> No Conversion Required
Mach Number Ahead of Expansion Fan: 5 --> No Conversion Required
Mach Number Behind Expansion Fan: 6 --> No Conversion Required
STEP 2: Evaluate Formula
Substituting Input Values in Formula
Te,r = (1+0.5*(γe-1)*Me1^2)/(1+0.5*(γe-1)*Me2^2) --> (1+0.5*(1.41-1)*5^2)/(1+0.5*(1.41-1)*6^2)
Evaluating ... ...
Te,r = 0.73090692124105
STEP 3: Convert Result to Output's Unit
0.73090692124105 --> No Conversion Required
FINAL ANSWER
0.73090692124105 0.730907 <-- Temperature Ratio Across Expansion Fan
(Calculation completed in 00.004 seconds)

Credits

Created by Shikha Maurya
Indian Institute of Technology (IIT), Bombay
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10+ Expansion Waves Calculators

Flow Deflection Angle due to Expansion Wave
Go Flow Deflection angle = (sqrt((Specific Heat Ratio Expansion Wave+1)/(Specific Heat Ratio Expansion Wave-1))*atan(sqrt(((Specific Heat Ratio Expansion Wave-1)*(Mach Number Behind Expansion Fan^2-1))/(Specific Heat Ratio Expansion Wave+1)))-atan(sqrt(Mach Number Behind Expansion Fan^2-1)))- (sqrt((Specific Heat Ratio Expansion Wave+1)/(Specific Heat Ratio Expansion Wave-1))*atan(sqrt(((Specific Heat Ratio Expansion Wave-1)*(Mach Number Ahead of Expansion Fan^2-1))/(Specific Heat Ratio Expansion Wave+1)))-atan(sqrt(Mach Number Ahead of Expansion Fan^2-1)))
Prandtl Meyer Function at Upstream Mach Number
Go Prandtl Meyer Function at Upstream Mach no. = sqrt((Specific Heat Ratio Expansion Wave+1)/(Specific Heat Ratio Expansion Wave-1))*atan(sqrt(((Specific Heat Ratio Expansion Wave-1)*(Mach Number Ahead of Expansion Fan^2-1))/(Specific Heat Ratio Expansion Wave+1)))-atan(sqrt(Mach Number Ahead of Expansion Fan^2-1))
Prandtl Meyer Function
Go Prandtl Meyer Function = sqrt((Specific Heat Ratio Expansion Wave+1)/(Specific Heat Ratio Expansion Wave-1))*atan(sqrt(((Specific Heat Ratio Expansion Wave-1)*(Mach Number^2-1))/(Specific Heat Ratio Expansion Wave+1)))-atan(sqrt(Mach Number^2-1))
Pressure behind Expansion Fan
Go Pressure Behind Expansion Fan = Pressure Ahead of Expansion Fan*((1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Ahead of Expansion Fan^2)/(1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Behind Expansion Fan^2))^((Specific Heat Ratio Expansion Wave)/(Specific Heat Ratio Expansion Wave-1))
Pressure Ratio across Expansion Fan
Go Pressure Ratio Across Expansion Fan = ((1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Ahead of Expansion Fan^2)/(1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Behind Expansion Fan^2))^((Specific Heat Ratio Expansion Wave)/(Specific Heat Ratio Expansion Wave-1))
Temperature behind Expansion Fan
Go Temperature Behind Expansion Fan = Temperature Ahead of Expansion Fan*((1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Ahead of Expansion Fan^2)/(1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Behind Expansion Fan^2))
Temperature Ratio across Expansion Fan
Go Temperature Ratio Across Expansion Fan = (1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Ahead of Expansion Fan^2)/(1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Behind Expansion Fan^2)
Flow Deflection Angle using Prandtl Meyer Function
Go Flow Deflection angle = Prandtl Meyer Function at Downstream Mach no.-Prandtl Meyer Function at Upstream Mach no.
Forward Mach Angle of Expansion Fan
Go Forward Mach Angle = arsin(1/Mach Number Ahead of Expansion Fan)
Rearward Mach Angle of Expansion Fan
Go Rearward Mach Angle = arsin(1/Mach Number Behind Expansion Fan)

Temperature Ratio across Expansion Fan Formula

Temperature Ratio Across Expansion Fan = (1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Ahead of Expansion Fan^2)/(1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Behind Expansion Fan^2)
Te,r = (1+0.5*(γe-1)*Me1^2)/(1+0.5*(γe-1)*Me2^2)

Do you know who developed the centered expansion wave theory?

The centered expansion wave theory was developed by Ludwig Prandtl and his student Theodor Meyerin 1907-1908. It is commonly denoted as Prandtl-Meyer expansion waves.

How to Calculate Temperature Ratio across Expansion Fan?

Temperature Ratio across Expansion Fan calculator uses Temperature Ratio Across Expansion Fan = (1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Ahead of Expansion Fan^2)/(1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Behind Expansion Fan^2) to calculate the Temperature Ratio Across Expansion Fan, The Temperature Ratio across Expansion Fan formula is defined as the ratio of static to stagnation temperature downstream of the expansion fan to the static to stagnation temperature upstream of the expansion fan. Temperature Ratio Across Expansion Fan is denoted by Te,r symbol.

How to calculate Temperature Ratio across Expansion Fan using this online calculator? To use this online calculator for Temperature Ratio across Expansion Fan, enter Specific Heat Ratio Expansion Wave e), Mach Number Ahead of Expansion Fan (Me1) & Mach Number Behind Expansion Fan (Me2) and hit the calculate button. Here is how the Temperature Ratio across Expansion Fan calculation can be explained with given input values -> 0.731707 = (1+0.5*(1.41-1)*5^2)/(1+0.5*(1.41-1)*6^2).

FAQ

What is Temperature Ratio across Expansion Fan?
The Temperature Ratio across Expansion Fan formula is defined as the ratio of static to stagnation temperature downstream of the expansion fan to the static to stagnation temperature upstream of the expansion fan and is represented as Te,r = (1+0.5*(γe-1)*Me1^2)/(1+0.5*(γe-1)*Me2^2) or Temperature Ratio Across Expansion Fan = (1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Ahead of Expansion Fan^2)/(1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Behind Expansion Fan^2). The Specific Heat Ratio Expansion Wave is the ratio of the heat capacity at constant pressure to heat capacity at constant volume, Mach Number Ahead of Expansion Fan is the Mach number of the upstream flow & Mach Number Behind Expansion Fan is the Mach number of downstream flow across expansion fan.
How to calculate Temperature Ratio across Expansion Fan?
The Temperature Ratio across Expansion Fan formula is defined as the ratio of static to stagnation temperature downstream of the expansion fan to the static to stagnation temperature upstream of the expansion fan is calculated using Temperature Ratio Across Expansion Fan = (1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Ahead of Expansion Fan^2)/(1+0.5*(Specific Heat Ratio Expansion Wave-1)*Mach Number Behind Expansion Fan^2). To calculate Temperature Ratio across Expansion Fan, you need Specific Heat Ratio Expansion Wave e), Mach Number Ahead of Expansion Fan (Me1) & Mach Number Behind Expansion Fan (Me2). With our tool, you need to enter the respective value for Specific Heat Ratio Expansion Wave, Mach Number Ahead of Expansion Fan & Mach Number Behind Expansion Fan and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.
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