The energy balance for a system, which has been previously introduced, is:

      (Q_in - Q_out ) + (W_in - W_out) +
      (E_mass,in - E_mass,out)
            = ΔE_system = (ΔU + ΔKE + ΔPE)_system

For a closed system, the only forms of energy that can be supplied or removed from a system are heat and work.

(Q_in - Q_out ) + (W_in - W_out) = (ΔU + ΔKE + ΔPE)_system

Sign Convention for Heat Transfer

Sign Convention for Work

If the adopted sign convention is such that the heat entering the system is positive, and the work done by the system is positive for a process from state 1 to state 2, then the energy balance for a closed system becomes:

      Q - W = E₂ - E₁ = ΔE_system
                = (ΔU + ΔKE + ΔPE)_system

For a stationary system, in which no velocity and elevation changes during a process, the change of the total energy of the system is due to the change of the internal energy only. That is,

      Q- W = U₂ - U₁

The definitions of constant volume and constant pressure specific heats have been introduced previously. They are

For most solids and liquids, they can be approximated as incompressible substances, hence the constant volume and constant pressure specific heats are the same. That is,

      c_P = c_v = c

The specific heats of incompressible substances depend on the temperature only. Hence the specific heats are simplified as:

      c_v = du/dT            c_P = dh/dT

From the definition of specific heat, the change of internal energy becomes

      du = c_vdT = c(T) dT

For a process from state 1 to state 2, the change of internal energy is obtained by integrating the above equation from state 1 to state 2.

For small temperature intervals, an average specific heat (c) at the average temperature is used and treated as a constant, yielding

Enthalpy is another temperature dependent variable. The definition of enthalpy is:

      H = U + PV

It can be rewritten in terms of per unit mass as follows:

      h = u+ Pv

Note that v is a constant, so the differential form of the above equation is:

Integrating from state 1 to state 2 yields

      Δh = Δu + v ΔP + vΔP

For solids, the term vΔP is insignificant.

      Δh = Δu

For liquids, two cases are encountered. They are:

• Constant pressure process, ΔP = 0, Δh = Δu
• Constant temperature process, ΔT = 0, Δh = vΔP