Mach Number

# It is defined as the ratio of local velocity of fluid to the acoustic or sound velocity.

It is a dimensionless number designated by M.

It is expressed as, M = C/a Where, C = Velocity of fluid (air),
   a = Velocity of sound.

# Mach number can also be expressed as the square root of the ratio of inertia force
of flow to the elastic force.

M² = (Inertia force of flow)/(Elastic force of fluid)

Different Regimes of Compressible flow

Subsonic Flow:
If the flow of fluid with a mach number less than 1, (i.e., C<a) then the flow is called
a subsonic flow. It is characterized by smooth streamlines.

Sonic Flow:

It the fluid flows with Mach number is equal to unity, (i.e., C=a) then the flow is
considered as sonic flow.

Supersonic flow:

If the Mach number is greater than 1 (i.e., C>a) then the flow is called
supersonic flow. It is observed with an oblique shock.

Hypersonic flow:

It the mach number of the flowing fluid is very high, (i.e., M>5) then the flow
is a hypersonic flow. This flow is observed with severe shock in the flow field. 

Note :
M > 5 --> Hypersonic flow

M > 1 --> Supersonic flow

M = 1 --> Sonic flow

M < 1  --> Subsonic flow

COMBINED FIRST AND SECOND LAW OF THERMODYNAMICS

Combined 1^st and 2^nd Law of Thermodynamics.

By first law of thermodynamics, ΔQ = dU + ΔW

Since ΔW = pdV

ΔQ = dU + pdV ----------------- 1.

From second law of thermodynamics, (Entropy concept)

ΔQ = T. dS ----------------------- 2.

Put equation 2. in 1.

We get, TdS = dU + pdv ---------------- 3.

We know that, enthalpy h = u + pv

On differentiating, we get

dh = du + pdv + vdp,

From equation 3. , dh = Tds + vdp

Tds = dh - vdp -------------- 4.

The equations 3. and 4. are the thermodynamic equations relating the properties of system.

The following are the relations obtained from the first and second laws.

1. dQ = dE + dW: Holds good for all process, reversible or irreversible and for all systems.

2. dQ = dU + dW: Holds good for any process undergone by a closed system.

3. dQ = dU + pdV: It is good for a closed system , where pdV work is present. This relation true only for quasi-static* process.

4. dQ = TdS: This equation is true only for a reversible processes.

5. TdS = dH –Vdp: This relation hold good for any process, since there is no path function term in the equation. 

6. TdS = dU + pdV: It is good for any reversible or irreversible process, undergone by a closed system. Since the properties in the relation which are independent of the path. 

Note:

* A quasi-static process is a thermodynamic process that happens very slowly  for the system to be in equilibrium. It is reversible.

Nusselt Number (Nu)

Physical Significance of Nusselt Number

# Consider a layer of fluid of thickness L, difference in temperature  ΔT = T₂-T₁. 

Heat Transfer through a fluid layer

# Heat transfer through the fluid layer is convection when the fluid involves some motion.

# Heat transfer for this case is Q_Conv = h . A . ΔT ---------1.

# On the contrary, heat transfer through the fluid layer is conduction when the fluid is not in motion.

# Heat transfer for this case is Q_Cond = k . A.  ΔT / (L ) --------2.

# By taking their ratio of equation 1. & 2.,

Q_Conv / Q_Cond = (h .  ΔT .  L) / (k .  ΔT)

                         = (h .  L) / (k )

                         = Nu, Nusselt number.

# Nusselt number represents the improvement of heat transfer through a fluid layer as a result of convection relative to conduction across the same fluid.

# The larger the Nusselt number, the more efficient the convection.

# Nusselt number, Nu is unity for fluid layer represents heat flux across the layer by pure conduction.

Note;

k - Thermal Conductivity, W/m K
h - Convection heat transfer coefficient , W/m² K