First Law of Thermodynamics for an Open System

This is a topic from Thermodynamics:

Introduction

Open systems are those which are neither impervious to mass flow or energy flow.

Steady-State Steady-Flow (SSSF)

Note for this topic, that systems operate in a steady-state steady-flow (SSSF) condition unless you are informed otherwise. The SSSF condition is where total mass doesn't change with respect to time. This means that while mass may flow into the system at a given rate, it flows out of the system at the same rate.
Under the circumstances, Σ ṁ_in = Σ ṁ_out

where ṁ is the change in mass with respect to time - the 'flow'.

Conservation of Mass

In an open system mass transfers may occur. The law of conservation of mass applies however, and so:

m_in - m_out = Δm_System

Considering rate of flow:

ṁ_in - ṁ_out = dm_System/dt

Note that by dimensional analysis:

ṁ = ρAV

where ρ is density,

A is the cross sectional area through which fluid flows,

V is average fluid velocity.

Because density and specific volume are reciprocals,

ṁ = AV / v

where v is specific volume

The integral form for mass flow is:

Statement of First Law for Open Systems

The 1st Law of Thermodynamics for an open system states that:

Ė_in - Ė_out = ΔĖ_System

It is a basic implication of the law of conservation of energy, and as such requires no derivation.

• The right hand side, in a SSSF condition, E_in = E_out, therefore:

Ė_System = 0

• The left hand side:

• Taking into account heat transfer only:

Ė_in - Ė_out = Q̇

• Taking into account work transfer only:

Ė_in - Ė_out = - Ẇ

• There is also the energy contained by mass flowing in and out. Considering this alone:

Ė_in - Ė_out = Ė_{mass in} - Ė_{mass out}

• Thus considering all forms of energy transfer:

Ė_in - Ė_out = Q̇ - Ẇ + [Ė_{mass in} - Ė_{mass out}]

Considering Flow Work Only

Work may cross the system boundary as flow work. This is motion of the fluid across the system boundary against an opposing pressure at locations we call ports.
Consider the following:

• Defining flow work:

Ẇ_flow = F dr/dt

=FV

Ẇ_flow = PAV

• Also, rearranging ṁ = ρAV:

AV = ṁ/ρ = ṁv

• Substituting into the equation for flow work:

Ẇ_flow = ṁPv

Considering All Work

It is given that W = W_sys + W_flow and shortening subscript 'sys' to 'S' (as opposed to 's' which denoted shaft work), consider the following:

Ẇ = Ẇ_S + Ẇ_flow

= Ẇ_S + ṁPv

= Ẇ_S + (ṁPv)_in - (ṁPv)_out

At this point the law can be rewritten as:

ΔĖ_System = Q̇ - Ẇ_S + (ṁPv)_in - (ṁPv)_out + [Ė_{mass in} - Ė_{mass out}]

Considering the Energy of Mass in Flow

The matter entering or leaving the control volume contains energy, which may come in the forms of internal energy, kinetic energy and gravitational potential energy:

Ė_{mass transfer} = Ė_Internal + Ė_Kinetic + Ė_{Grav. Potential}

= ṁ(u + V²/2 + gz)

where g is the acceleration due to gravity,

z is the height (because h, the normal symbol for height, refers to enthalpy in thermodynamics)

V is the average fluid velocity

At this point the law can be rewritten as:

ΔĖ_System = Q̇ - Ẇ_S + (ṁPv)_in - (ṁPv)_out + ṁ_in(u + V²/2 + gz)_in - ṁ_out(u + V²/2 + gz)_out

Regrouping to Include Specific Enthalpy

Regrouping the equation, so that flow work is grouped with mass of energy in flow, gives:

ΔĖ_System = Q̇ - Ẇ_S + ṁ_in(u + Pv + V²/2 + gz)_in - ṁ_out(u + Pv + V²/2 + gz)_out

And recalling that h = u + Pv the equation can be rewritten once more, as:

ΔĖ_System = Q̇ - Ẇ_S + ṁ_in(h + V²/2 + gz)_in - ṁ_out(h + V²/2 + gz)_out

Transient Systems

A transient system is one in which the SSSF condition is not met, because one of more properties change as a function of time, for example the mass might change because ṁ_in ≠ ṁ_out. In this case ΔĖ_System isn't zero as it is in the SSSF condition, but rather there exists a ΔĖ_System = ṁ₂u₂ - ṁ₁u₁. This means the first law must be rewritten:

Q̇ - Ẇ_S = ṁ₂u₂ - ṁ₁u₁ + ṁ_out(h + V²/2 + gz)_out - ṁ_in(h + V²/2 + gz)_in

Tips for Problem Solving

• Assume V_in = V_out = 0 unless the device is a nozzle or a diffuser which are designed to drastically change the velocity (increase and decrease respectively).
• Assume W_S ≠ 0 if the device is a compressor as work is done to achieve the compression.
• A throttle restricts flow. For a throttle the change in enthalpy is zero (h₁ = h₂), W = 0, and Q = 0.
• Be aware that you may need to perform linear extrapolation (rather than interpolation) when using the superheat values on steam tables pages 13-15.