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PLE 5–10 Bernoulli Equation for Compressible Flow the Bernoulli equation when the compressibility effects are not negli- for an ideal gas undergoing (a) an isothermal process and (b) an isen- process. TION The Bernoulli equation for compressible flow is to be obtained ideal gas for isothermal and isentropic processes. ptions 1 The flow is steady and frictional effects are negligible. 2 The s an ideal gas, so the relation P = pRT is applicable. 3 The specific are constant so that Plp* = constant during an isentropic process. is (a) When the compressibility effects are significant and the flow t be assumed to be incompressible, the Bernoulli equation is given by 40 as

PLE 5–10 Bernoulli Equation for Compressible Flow the Bernoulli equation when the compressibility effects are not negli- for an ideal gas undergoing (a) an isothermal process and (b) an isen- process. TION The Bernoulli equation for compressible flow is to be obtained ideal gas for isothermal and isentropic processes. ptions 1 The flow is steady and frictional effects are negligible. 2 The s an ideal gas, so the relation P = pRT is applicable. 3 The specific are constant so that Plp* = constant during an isentropic process. is (a) When the compressibility effects are significant and the flow t be assumed to be incompressible, the Bernoulli equation is given by 40 as

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
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EXAMPLE 5–10 Bernoulli Equation for Compressible Flow Derive the Bernoulli equation when the compressibility effects are not negli- gible for an ideal gas undergoing (a) an isothermal process and (b) an isen- tropic process. SOLUTION The Bernoulli equation for compressible flow is to be obtained for an ideal gas for isothermal and isentropic processes. Assumptions 1 The flow is steady and frictional effects are negligible. 2 The fluid is an ideal gas, so the relation P = pRT is applicable. 3 The specific heats are constant so that Plp* = constant during an isentropic process. Analysis (a) When the compressibility effects are significant and the flow cannot be assumed to be incompressible, the Bernoulli equation is given by Eq. 5-40 as dP 4 V². v? + gz = constant (along a streamline) (1) The compressibility effects can be properly accounted for by expressing p in terms of pressure, and then performing the integration S dPlp in Eq. 1. But this requires a relation between P and p for the process. For the isothermal expansion or compression of an ideal gas, the integral in Eq. 1 can be per- formed easily by noting that T = constant and substituting p = PIRT. It gives dP = RT In P PIRT Substituting into Eq. 1 gives the desired relation, Isothermal process: RT In P + + gz = constant (2) (b) A more practical case of compressible flow is the isentropic flow of ideal gases through equipment that involves high-speed fluid flow such as nozzles, diffusers, and the passages between turbine blades. Isentropic (i.e., reversible and adiabatic) flow is closely approximated by these devices, and it is characterized by the relation Plp* = C = constant, where k is the spe- cific heat ratio of the gas. Solving for p from Plp* = C gives p = C-/k pl./k. Performing the integration, P- +1 - 1/k + 1 dP = C (3) P - 1/k + 1 Substituting, the Bernoulli equation for steady, isentropic, compressible flow of an ideal gas becomes k P Isentropic flow: + gz = constant (4a) - 1 + gz, = + gz (4b) A common practical situation involves the acceleration of a gas from rest (stagnation conditions at state 1) with negligible change in elevation. In that case we have z = z, and V, = 0. Noting that p = PIRT for ideal .......
Transcribed Image Text:EXAMPLE 5–10 Bernoulli Equation for Compressible Flow Derive the Bernoulli equation when the compressibility effects are not negli- gible for an ideal gas undergoing (a) an isothermal process and (b) an isen- tropic process. SOLUTION The Bernoulli equation for compressible flow is to be obtained for an ideal gas for isothermal and isentropic processes. Assumptions 1 The flow is steady and frictional effects are negligible. 2 The fluid is an ideal gas, so the relation P = pRT is applicable. 3 The specific heats are constant so that Plp* = constant during an isentropic process. Analysis (a) When the compressibility effects are significant and the flow cannot be assumed to be incompressible, the Bernoulli equation is given by Eq. 5-40 as dP 4 V². v? + gz = constant (along a streamline) (1) The compressibility effects can be properly accounted for by expressing p in terms of pressure, and then performing the integration S dPlp in Eq. 1. But this requires a relation between P and p for the process. For the isothermal expansion or compression of an ideal gas, the integral in Eq. 1 can be per- formed easily by noting that T = constant and substituting p = PIRT. It gives dP = RT In P PIRT Substituting into Eq. 1 gives the desired relation, Isothermal process: RT In P + + gz = constant (2) (b) A more practical case of compressible flow is the isentropic flow of ideal gases through equipment that involves high-speed fluid flow such as nozzles, diffusers, and the passages between turbine blades. Isentropic (i.e., reversible and adiabatic) flow is closely approximated by these devices, and it is characterized by the relation Plp* = C = constant, where k is the spe- cific heat ratio of the gas. Solving for p from Plp* = C gives p = C-/k pl./k. Performing the integration, P- +1 - 1/k + 1 dP = C (3) P - 1/k + 1 Substituting, the Bernoulli equation for steady, isentropic, compressible flow of an ideal gas becomes k P Isentropic flow: + gz = constant (4a) - 1 + gz, = + gz (4b) A common practical situation involves the acceleration of a gas from rest (stagnation conditions at state 1) with negligible change in elevation. In that case we have z = z, and V, = 0. Noting that p = PIRT for ideal .......
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