EQUATIONS _ Thermo

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the relationship between specific volume (v) and pressure (P) in an isentropic process as modeled by the PG model?

The isentropic process relates specific volume and pressure through the equation $T_1 v_1^{k-1} = T_2 v_2^{k-1}$. This shows that specific volume changes with temperature and pressure during an isentropic process.

How is the work done (W) calculated in a general sense during a process?

The work done is calculated using $W = ∫ F ⋅ ds$, where F is the force and ds is the displacement. In terms of pressure-volume work, it can also be expressed as $W = ∫ Pdv$.

What is the expression for total work done in a fluid moving through a boundary according to the work done equations provided?

Total work done is expressed as $W = m ∫ Pdv$, where m is the mass flow rate. This incorporates mass into the work done over volume change.

Define the coefficient of performance (COP) in relation to heat transfer.

The COP is defined as $COP = \frac{q_{in}}{w_{in}}$. It measures the efficiency of a heat pump or refrigerator in converting work input into heat transfer. Signup and view all the answers

In terms of energy properties, how is enthalpy (H) related to internal energy (U) and pressure-volume work?

Enthalpy is given by the equation $H = U + PV$. This defines how internal energy combines with pressure-volume work to assess total energy in a system. Signup and view all the answers

Explain the significance of the equation $du = cv dT$ in thermodynamic processes.

The equation $du = cv dT$ defines the change in internal energy (du) of a system as a function of its heat capacity at constant volume (cv) and the change in temperature (dT). Signup and view all the answers

What do the terms $h_x$, $u_x$, and $v_x$ signify in terms of mixtures of liquid and vapor?

$h_x$, $u_x$, and $v_x$ represent the total enthalpy, internal energy, and specific volume of a mixture, respectively, expressed in terms of mass fractions of liquid and vapor. They are calculated as $h_x = h_f + x(h_g - h_f)$, where x is the quality of the vapor. Signup and view all the answers

What is the relationship between specific heats $c_p$ and $c_v$, and the gas constant (R) according to the information provided?

The relationship is given by $c_p = c_v + R$, indicating that the specific heat at constant pressure is greater than the specific heat at constant volume by the gas constant R. Signup and view all the answers

What is the Ideal Gas Equation and what do the variables represent?

The Ideal Gas Equation is represented as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the universal gas constant, and T is temperature. Signup and view all the answers

What is the specific gas constant for air and how is it derived?

The specific gas constant for air is R_air = 0.287 kJ/(kg·K), derived from the universal gas constant divided by the molecular weight of air. Signup and view all the answers

How is mass balance expressed in an open system?

In an open system, the mass balance is expressed as dmsys/dt = ∑ṁ_in - ∑ṁ_out. Signup and view all the answers

What does the energy balance equation represent in an open system?

The energy balance equation for an open system represents the rate of change of energy in the system being equal to the energy in minus the energy out. Signup and view all the answers

What components are included in the entropy balance for an open system?

The entropy balance for an open system includes contributions from mass flow in and out, along with any entropy generated within the system. Signup and view all the answers

Explain how to calculate the change in entropy for ideal gases.

The change in entropy for ideal gases can be calculated using the formula ∆s = C_p ln(T2/T1) - R ln(P2/P1) or ∆s = C_v ln(T2/T1) + R ln(v2/v1. Signup and view all the answers

What is the significance of the term S_gen in the entropy balance equation?

The term S_gen represents the entropy generation within the system, accounting for irreversible processes. Signup and view all the answers

In terms of energy balance, what does the expression ΔEsys include in closed systems?

In closed systems, the expression ΔEsys includes changes in internal energy, kinetic energy, and potential energy, showing the total energy differences. Signup and view all the answers

Study Notes

Ideal Gas Equation

Pv = RT or PV = mRT, where R = Ru/MW
Rair = 0.287 kJ/kg·K = 0.287 kPa·m³/kg·K (use table A-1 for other gases)

Mass Balance

dm/dt = Σmin - Σmout = ρAV

Energy Balance

Open: dE/dt = E_in - E_out
- E_in = Σ m_in(h_in + V² / 2 + gz_in )
- E_out = Σ m_out(h_out + V² / 2 + gz_out)
Open (Expanded): d(E_sys)/dt = (Qin + Win) − (Qout + Wout) + Σ m_in(h_in + V² / 2 + gz_in) − Σ m_out((h_out + V² / 2 + gz_out))
Closed: ΔE_sys = E_in - E_out

Entropy Balance

Open: ds/dt = ΣS_in - ΣS_out + S_gen /dt
- Where S_in and S_out can be mass, heat transfer, or both
Closed: ΔS_sys = ∑ S_in − ∑ S_out + S_gen

Entropy Changes of Ideal Gases

Δs = C_v ln(T₂/T₁) + R ln(V₂/V₁) or Δs = C_p ln(T₂/T₁) − R ln(P₂/P₁)
NOTE: This assumes constant specific heats.

Isentropic Process

T₁v¹⁻ᵏ = T₂v₂¹⁻ᵏ
Pvᵏ = P₂v₂ᵏ

Different Types of Work

General: W = ∫ F ⋅ ds
Boundary: W = ∫ Pdv (Specific) W = m∫ Pdv (Total)
Electrical: W = V ⋅ I
Shaft: Wshaft = 2πnτ

Reversible Steady Flow Work

W = ∫ v dP (Specific) = m∫ v dP (Total)

Efficiency and COP

η = W_net,out / W_in
COP_R=Q_out / W_in
COP_HP = Q_in / W_in

Properties

h = u + Pv or H = U + PV
du = C_v dT , dh = C_p dT, Cp = Cv + R
For any property, Y = mY (upper case is total, lower case is specific)
- X = (mass_vapor / mass_total), V_x = V_f + X(V_g-V_f)
- U_x = U_f + X(U_g - U_f), h_x = h_f + X(h_g - h_f)

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Test your knowledge on the ideal gas equation, mass, energy, and entropy balances. This quiz covers equations and principles applicable to open and closed systems, including isentropic processes. Ideal for students studying thermodynamics and related fields.