Energy Balance for Closed Systems

Energy Balance for Closed System

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

      (Qin - Qout ) + (Win - Wout) +
      (Emass,in - Emass,out)
            = ΔEsystem = (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.

(Qin - Qout ) + (Win - Wout) = (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 = ΔEsystem
                = (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

  Specific Heats of Solids and Liquids


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,

      cP = cv = c

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

      cv = du/dT            cP = dh/dT

    Internal Energy and Enthalpy Difference of Solids and Liquids

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

      du = cvdT = 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.



Internal Energy and Enthalpy of
Solids and Liquids

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