First Law of Thermodynamics for an Open System

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This is a topic from Thermodynamics:

Contents

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 ṁin = Σ ṁout

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:

min - mout = ΔmSystem

Considering rate of flow:

in - ṁout = dmSystem/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:

Screen Shot 2013-03-28 at 9.51.13 AM.png

Statement of First Law for Open Systems

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

Ėin - Ėout = ΔĖSystem

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, Ein = Eout, therefore:
ĖSystem = 0
  • The left hand side:
  • Taking into account heat transfer only:
Ėin - Ėout = Q̇
  • Taking into account work transfer only:
Ėin - Ėout = - Ẇ
  • There is also the energy contained by mass flowing in and out. Considering this alone:
Ėin - Ėout = Ėmass in - Ėmass out
  • Thus considering all forms of energy transfer:
Ėin - Ėout = Q̇ - Ẇ + [Ėmass in - Ėmass 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:
flow = F dr/dt
=FV
flow = PAV
  • Also, rearranging ṁ = ρAV:
AV = ṁ/ρ = ṁv
  • Substituting into the equation for flow work:
flow = ṁPv

Considering All Work

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

Ẇ = ẆS + Ẇflow
= ẆS + ṁPv
= ẆS + (ṁPv)in - (ṁPv)out


At this point the law can be rewritten as:

ΔĖSystem = Q̇ - ẆS + (ṁPv)in - (ṁPv)out + [Ėmass in - Ėmass 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:

Ėmass transfer = ĖInternal + ĖKinetic + ĖGrav. 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:

ΔĖSystem = Q̇ - ẆS + (ṁPv)in - (ṁPv)out + ṁin(u + V2/2 + gz)in - ṁout(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:

ΔĖSystem = Q̇ - ẆS + ṁin(u + Pv + V2/2 + gz)in - ṁout(u + Pv + V2/2 + gz)out

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

ΔĖSystem = Q̇ - ẆS + ṁin(h + V2/2 + gz)in - ṁout(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 ṁin ≠ ṁout. In this case ΔĖSystem isn't zero as it is in the SSSF condition, but rather there exists a ΔĖSystem = ṁ2u2 - ṁ1u1. This means the first law must be rewritten:

Q̇ - ẆS = ṁ2u2 - ṁ1u1 + ṁout(h + V2/2 + gz)out - ṁin(h + V2/2 + gz)in

Tips for Problem Solving

  • Assume Vin = Vout = 0 unless the device is a nozzle or a diffuser which are designed to drastically change the velocity (increase and decrease respectively).
  • Assume WS ≠ 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.
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