Review of thermodynamics applied to flow — first law for open systems

3.1.1 · Physics › Compressible Flow & Aerodynamics

HUM KYA KAR RAHE HAIN

KAISE: Steady-flow energy equation scratch se derive karna

Hum ek chhota sa mass $δ m$ follow karte hain jab woh inlet 1 ke thoda bahar se outlet 2 ke thoda bahar tak move karta hai, aur us chalte hue lump par closed-system first law apply karte hain (kyunki yeh same matter hai throughout, isliye closed-system law legal hai).

Step 1 — Lump ke liye Closed-system first law. $Δ E = Q - W_{total}$ Yeh step kyun? Fixed chunk of matter ke liye energy conserved hoti hai; yahi woh ek law hai jis par hum truly sure hain, isliye hum yahan se shuru karte hain.

Step 2 — Lump mein kaunsi energy hai? Internal + kinetic + potential, per unit mass: $e = u + \frac{1}{2} V^{2} + g z$ Kyun? Ye woh storable energies hain jo fluid carry karta hai; humein 1 aur 2 ke beech inka change chahiye.

Step 3 — Work ko flow work + shaft work mein split karo. Total work ke do parts hain:

Flow (displacement) work: surroundings lump ko 1 par push karke andar daalta hai aur woh fluid ko 2 par bahar push karta hai. Pressure $p$ par ek face ke paas volume $V_{v o l} = v δ m$ push karne mein work $p V_{v o l} = p v δ m$ lagta hai.
Shaft work $W_{s}$ : shaft/blade ke zariye useful work (turbine out, compressor in).

Toh per unit mass: $W_{total} = w_{s} + (p_{2} v_{2} - p_{1} v_{1})$ Sign kyun? Outlet par system downstream fluid par $p_{2} v_{2}$ work karta hai (work out, +). Inlet par upstream fluid hamare system par $p_{1} v_{1}$ work karta hai (isliye subtract karo). Yahi derivation ka core hai.

Step 4 — Assemble karo. Per unit mass ( $q = Q / δ m$ , etc.): $(u_{2} + \frac{1}{2} V_{2}^{2} + g z_{2}) - (u_{1} + \frac{1}{2} V_{1}^{2} + g z_{1}) = q - w_{s} - (p_{2} v_{2} - p_{1} v_{1})$

Step 5 — $u + p v$ ko enthalpy mein group karo. Flow-work terms ko left side le jao: $(u_{2} + p_{2} v_{2}) + \frac{1}{2} V_{2}^{2} + g z_{2} = (u_{1} + p_{1} v_{1}) + \frac{1}{2} V_{1}^{2} + g z_{1} + q - w_{s}$

Aerodynamics specialisation: adiabatic, no work

Nozzle/diffuser/free stream mein: $w_{s} = 0$ (koi shaft nahi), $q = 0$ (adiabatic, fast flow), $g z$ negligible (gas). SFEE simplify hokar:

Worked examples

Common mistakes

Active recall

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ek bheed wali train (pipe). Kisi ko train mein andar dalne ke liye tumhe bheed ko push karke jagah banana padega — yahi pushing "flow work" hai. Kisi ko bahar nikalne ke liye andar ki bheed use bahar dhakelta hai — woh bhi work hai. Toh har aadmi apna backpack of energy ( $u$ ) carry karta hai plus darwaze se squeeze hone ke liye jo push chahiye ( $p v$ ). Hum in dono ko chipka dete hain aur bundle ko "enthalpy" kehte hain. Ab energy in = energy out: jo heat tum add karo + jo work fan kare = sabke bundles mein change + unka bhagne ki energy. Jab train ek tangi tunnel se speed mein guzarti hai aur koi heat add nahi karta, log thoda thande pad jaate hain kyunki unka bundle energy, tez daudne ki energy mein badal gayi.

Connections

Stagnation properties & isentropic relations
Speed of sound and Mach number
Conservation of mass — continuity equation
Closed-system first law of thermodynamics
Bernoulli equation as low-speed limit of SFEE
Nozzles and diffusers

Open-system energy balances mein enthalpy internal energy ki jagah kyun leta hai?

Kyunki boundary cross karne wale mass ko flow work

p v

ki zaroorat padti hai, jo hamesha

u

ke saath pair hota hai;

h = u + p v

define karne se yeh cleanly absorb ho jaata hai.

Steady-flow energy equation batao.

h_{1} + \frac{1}{2} V_{1}^{2} + g z_{1} + q = h_{2} + \frac{1}{2} V_{2}^{2} + g z_{2} + w_{s}

Pressure

p

ke port par flow (displacement) work per unit mass kya hota hai?

p v

— specific volume

v

wale unit mass ko boundary ke paas push karne ka work.

Stagnation (total) enthalpy define karo.

h_{0} = h + \frac{1}{2} V^{2}

; woh enthalpy jo milti agar flow ko adiabatically rest par laya jaata. Adiabatic, no-shaft flow mein constant hota hai.

Calorically perfect gas ke liye

T_{0}

aur

T

ka relation batao.

T_{0} = T + \frac{V ^{2}}{2 c _{p}}

, kyunki

h = c_{p} T

Adiabatic nozzle mein velocity badhne par temperature ka kya hota hai?

Temperature girta hai, kyunki

h + \frac{1}{2} V^{2} =

const hota hai isliye KE badhti hai enthalpy ki keemat par.

Turbine vs compressor ke liye shaft work

w_{s}

ka sign (work-out convention)?

Turbine:

w_{s} > 0

(work out). Compressor:

w_{s} < 0

(work in).

Flow work aur shaft work mein kya fark hai?

Flow work

p v

fluid ko ports se push karta hai (

h

ke andar chhupa hua); shaft work ek moving boundary/blade ke zariye extra useful work hai — kabhi double-count mat karo.