Work and Heat

This is a topic from Thermodynamics:

Introduction

Heat is a process, not a physical substance or a property. Heat refers to the process whereby energy is transferred down a temperature gradient.
Sign convention:

• Heat in is positive
• Heat out is negative

Work is the energy transferred when motion is applied against an opposing or resistive force.
Sign convention:

• The opposite of heat
• Work in is negative
• Work out is positive

Mathematical Definition for Work

Recall from the first year physics courses:

W = Fx

For an infinitesimal amount of work:

dW = Fdx

= PA dx

= P dV

Therefore, for a process moving from state 1 to state 2:

Consider the following derivations using this defintion

Work for an Isobaric Process

• Consider an isobaric process such that pressure is constant: P = c
  
• Performing the integral:

• The work done is the area under the graph of the PV diagram:
  

Work for an Polytropic Process

• Consider an polytropic process such that PVⁿ = c
  
• Performing the integral:

Work for an Isothermal Process

• Consider an isothermal process such that PV = c
  
• Performing the integral:

• The work done is the area under the graph of the PV diagram:
  

Types of Work

When a system is open it can be susceptible to more than one form of work.
Consider the following diagram:

1. The work done on/by the piston at the top of the diagram is the type of work with which we are mostly concerned with closed systems. It is known as Boundary Work and is denoted W_b. Boundary work, in which the system boundary undergoes a net motion, is compatible with both closed systems and open systems.
2. The work done by the battery on the left of the diagram is called Electrical Work and is denoted W_e. It refers to the work done in the form of electrical power on the heating element embedded in the system.
3. The work done by the rotor on the right of the diagram is called Shaft Work and is denoted W_s.
4. The work done by the flow of mass in and out of the system seen in the bottom left and top right of the diagram is known as Flow Work and is denoted W_flow. Flow work is the work done when a volume of fluid flows against an opposing pressure. The points of exit and entry into the system is known as ports. This type of work is not possible in a closed system because it involved a transfer of mass across the system boundary.
   
   

The boundary work, electrical work and shaft work are generally grouped as W_sys such that:

W = W_b + W_e + W_s + W_flow becomes

W = W_sys + W_flow

Pump Work

Note: This type of work requires an understanding of concepts covered later in the notes and is not needed until the second law of thermodynamics has been covered.

• For an open system there can also be pump work, which, upon application of the first law for an open system is given as follows:

w = (P_inv_in + V_in²/2 + gz_in) - (P_outv_out + V_out²/2 + gz_out) - (u_out - u_in - q)

• However it can also be shown that q_rev = u_out - u_in for a reversible process, or u_out - u_in > q for an irreversible process.
• Therefore pump work can be rewritten as:

w_rev = (P_inv_in + V_in²/2 + gz_in) - (P_outv_out + V_out²/2 + gz_out)

• Ignoring kinetic and potential energies:

w_rev = P_inv_in - P_outv_out

• The flow is incompressible, and so v_in = v_out = v
• Therefore we have:

w_rev = v(P_in - P_out)

If work is equal to zero, we get Bernoulli's equation:

P_in/ρ + V_in²/2 + gz_in = P_out/ρ + V_out²/2 + gz_out