## Correlations for heat transfer and pressure drop

Nomenclature:

```D       - Hydraulic diameter
L       - Flow length
vel     - Fluid velocity
Dens    - Density
CP      - Specific heat capacity
DV      - Dynamic viscosity
KV      - Kinematic viscosity = DV/Dens
TC      - Thermal conductivity
Beta    - Thermal expansion coefficient
T,Surf  - Surface temperature
T,Amb   - Ambient temperature
dT      = T,Surf - T,Amb
Grav    - Gravity acceleration
Re      - Reynolds number = vel*D/KV
Gr      - Grashof number = Grav*Beta*abs(dT)*D^3/KV^2
Pr      - Prandtl number = KV*CP/TC
Ra      - Rayleigh number = Gr*Pr
Nu      - Nusselt number = HTC*D/TC
HTC     - Heat transfer coefficient
rlRough - Relative coarseness
rFrict  - Friction coefficient for fluid flow pressure drop
Lam     - Laminar
Tr      - Transient
Turb    - Turbulent
```

General forced convection heat transfer:

```Re > 10000:        HTC = HTCTurb = 0.027*(TC/D)*Re^0.8*Pr^(1/3)
Re < 2300:         HTC = HTCLam  = 1.86*(TC/D)*Re*Pr*D/L)^(1/3)
2300 < Re < 10000: HTC = HTCTurb*(1-(1-1.86*(L/D)^(-1/3)*(2300/Re)^(3/2))
```

Laminar forced convection (Kreith & Black):

```Nu = 0.664*sqrt(Re)*Pr^(1/3);
```

Turbulent forced convection (Kreith & Black):

```Nu = 0.036*(Re^0.8-23200)*Pr^(1/3);
```

Forced convection heat transfer for external cross flow over single pipe (Churchill & Bernstein, 1977):

```Nu = 0.3 + 0.62*Re^(1/2)*Pr^(1/3)/(1+(0.4/Pr)^(2/3))^0.25*(1+(Re/282000)^(5/8))^(4/5)
```

Turbulent forced convection heat transfer inside smooth pipes (Sieder & Tate, 1936):

```Nu = 0.027*Re^0.8*Pr^(1/3)*(DV/DV,wall)^0.14, Re > 10,000, 0.5 < Pr < 1E6
```

Turbulent forced convection heat transfer inside smooth pipes (Dittus & Boelter, 1930):

```Nu = 0.023*Re^0.8*Pr^0.4, dT > 0, 2500 < Re < 1.24E5, 0.7 < Pr < 120, L/D > 60
```

Turbulent forced convection heat transfer inside smooth pipes (Dittus & Boelter, 1930):

```Nu = 0.023*Re^0.8*Pr^0.3, dT < 0, 2500 < Re < 1.24E5, 0.7 < Pr < 120, L/D > 60
```

General vertical plate free convection (Churchill & Chu, 1975):

```Nu = [0.825 + 0.387*Ra^(1/6)/[1 + (0.492/Pr)^(9/16)]^(8/27)]^2, 0.1 < Ra < 1E12
```

Vertical plate laminar free convection (Kreith & Black):

```Nu = 0.59*Ra^(1/4)
```

Vertical plate turbulent free convection (Kreith & Black):

```Nu = 0.10*Ra^(1/3)
```

Vertical, short pipe external free convection heat transfer (LeFevre & Ede, 1956):

```Nu = 4/3*[7*Ra*Pr/[5*(20 + 21*Pr)]]^(1/4) + 4*(272 + 315*Pr)*L/[35*(64 + 63*Pr)*D]
```

Vertical long pipe internal free convection heat transfer (A. Bejan, 1984):

```Nu = Ra/128, L/D > Ra
```

Horizontal plate stable free convection (Incropera & DeWitt, 1990):

```Nu = 0.27*Ra^(1/4), 1E5 < Ra < 1E10
```

Horizontal plate unstable laminar free convection (Lloyd & Moran, 1974):

```Nu = 0.54*Ra^(1/4), 1E4 < Ra < 1E7
```

Horizontal plate unstable turbulent free convection (Lloyd & Moran, 1974):

```Nu = 0.15*Ra^(1/3), 1E7 < Ra < 1E9
```

Horizontal pipe external free convection heat transfer (Churchill & Chu, 1975):

```Nu = [0.6 + 0.387*Ra^(1/6)/[1 + (0.559/Pr)^(9/16)]^(8/27)]^2, 1E-5 < Ra < 1E12
```

Correlations for pressure drop:
rlRoughMin interpolated in the following table with respect to the Reynolds number Re:

```Re:         0 20,000 20,000 100,000 1,000,000 10,000,000 100,000,000
rlRoughMin: 1      1  0.067   0.014    0.0017    0.00019    0.000025
```

Smooth pipes: (rlRough < rlRoughMin)

```Re < 2,000:           rFrict = 64/Re,
2,000 < Re < 100,000: rFrict = 0.3164*Re^(-0.25),
Re > 100,000:         rFrict = 0.0032+0.221*Re^(-0.237)
```

Coarse pipes: (rlRough > rlRoughMin)

```Re < 2,000:           rFrict = 64/Re = rFrict,Lam
2,000 < Re < 3,000:   rFrict,Tr = rFrict,Lam+(rFrict,Turb-rFrict,Lam)*(0.001*Re-2)
3,000 < Re < 20,000:  rFrict,0 = 0.3164*Re^(-0.25),
rLam,0 = sqrt(rFrict,0)
rLam,n = 1/(1.14-0.868589*ln(rlRough+9.3/(Re*rLam,n-1)))
rFrict = rLam,n*rLam,n = rFrict,Turb
Re > 20,000:          rLam = 1/(1.14+0.868589*ln(1/rlRough))
rFrict = rLam*rLam
```