Flow Processes

Processes in Open Systems:

Open systems involve mass flow across the boundary . The closed system can be considered as a special case of an open system with no mass flow across the boundary. Open systems can have work and heat interactions across the boundary and in addition have matter flowing across the boundary. They can be subjected to all the processes that closed systems can be subjected to. Process carried out on matter in flow are called flow processes. It should be noted that it is not possible to talk about an equilibrium state in an open system with mass flow since the matter inside the system is always changing.

The Equation of Continuity:

Consider the open system within the control volume CV in figure2 below ;

Matter is crossing the control surface( the system boundary) at n locations. Let us call each of these flows as a stream and use the suffix i to identify the ith stream. In keeping with our sign convention for boundary interactions , we will attach a plus(+) sign to the mass of matter entering and a minus(-) sign to the mass of matter leaving. By conservation of matter , we can say that the change in the mass(M) of the system over a small interval of time δt is given by ;

 δM = ∑ δm_i

Where δm_i is the mass entering the control volume along stream i. For a finite interval of time from t₁ to t₂ , we may write ;

M₂ - M₁ = ∑ m_i

Where m_i is the mass entering the control volume during the time interval  t₁ to  t₂ along stream i. The above equations are two forms of the Equation of Mass Continuity or simply the Equation of Continuity.

Application of the first law to Open systems:

By considering the matter crossing the boundary of the open system over a small interval of time δt_i we can define a closed system whose boundary changes from CS1 to CS2 (see figure 3).

Let us consider the matter of mass δm_i crossing the boundary of the open system along stream i. If the specific volume of the matter is v_i the boundary expansion during time δt is (v_iδm_i). If the local pressure at the stream is p_i , the boundary expansion work done by the surroundings on the system is p_iv_iδm_i. The product pv is commonly referred to as ‘work of introduction’ or ‘flow work’ in flow processes and represents the work done in transferring a unit mass across a control surface. The total flow work is given by ;

∑ p_iv_iδm_i

If there are heat and work interactions of δW and δQ across the system boundary and each stream has in addition to internal energy , kinetic and potential energy , using the First law we may write ;

E + δE = E + δQ + δW + ∑ p_iv_iδm_i + ∑(u_i + c_i²/2 + gz_i) δm_i

Where E and (E + dE) refer to the total energy within the control volume at times t and (t + dt) , respectively , where ;

E = U + E_k + E_p

Above equation may be rewritten as ;

δE = δQ + δW + ∑ p_iv_iδm_i + ∑(u_i + c_i²/2 + gz_i) δm_i

where u_i is the specific internal energy and c_i the velocity of the fluid in stream i , z_i is the elevation of stream i and g is the local acceleration due to gravity. The sum (p_iv_i + u_i) is a function of , the state of the fluid in stream i and it is therefore a property of the fluid. We use the symbol h to denote it and call it specific enthalpy.

h = u + pv 

For given mass m , the enthalpy is ;

H = mh

And we may write ;

H = U + pV

Above equation can be rewritten as ;

δE = δQ + δW + ∑(h_i + c_i²/2 + gz_i) δm_i

Where the changes in the values of the kinetic and potential energies of the open system are negligible, we may write above equation as ;

δU = δQ + δW + ∑(h_i + c_i²/2 + gz_i) δm_i

The Steady Flow Energy Equation:

An open system can be said to be under steady conditions when the state at any point within it does not change with the time. This means that the mass and the total energy of the open system remain constant.

Equations   δM = ∑ δm_i   and   H = U + pV   , respectively , become ;

 ∑ δm_i = 0

And ;

δQ + δW + ∑(h_i + c_i²/2 + gz_i) δm_i = 0

Above equation is known as the Steady Flow Energy Equation (SFEE). The Continuity Equation (CE) and the SFEE for a steady flow process can be written as ;

 ∑ m_i = 0

Q + W + ∑(h_i + c_i²/2 + gz_i) m_i = 0

For a finite interval of time.

They may be also written as ;

 Q̇ + Ẇ + ∑(h_i + c_i²/2 + gz_i) δṁ_i = 0

Where the dot placed above each of the symbols m , Q ,W refer to differentiation with respect to time. 

( ṁ = dm/dt = mass flow rate ; Q̇ = dQ/dt =heat flow rate ; Ẇ = dW/dt = power supplied )