Agitator equipped vessels with half pipe coil jackets are widely used in variety of process applications. This article shows how to calculate heat transfer in an agitated vessel provided with an external half pipe coil jacket.

Overall heat transfer coefficient, U is defined as

`1/U = 1/hi + ff`_{i} + x/k + ff_{o} + 1/ho

where,

- hi : film coefficient process side
- ho : film coefficient coil side
- ff
_{i}: fouling factor process side - ff
_{o}: fouling factor coil side - x : vessel wall thickness
- k : vessel wall thermal conductivity

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### Process Side, hi

Process side film coefficient, hi depends upon type of impellor, Reynold’s number (Re) and Prandtl number (Pr).

`Re = D².N.ρ / μ`

`Pr = Cp.μ/k`

where, D is impellor diameter, N is impellor rpm, ρ is fluid density, μ is fluid viscosity, Cp is fluid specific heat and k is fluid thermal conductivity at bulk fluid temperature. hi is defined as following :

`Nu`

_{i}= C.Re^{a}. Pr^{b}. (μ/ μ_{w})^{c}. Gc`hi.D`

_{T}/k = Nu_{i}

where constants C, a, b & c are available in literature for different type of impellors. D_{T} is vessel diameter. Gc is geometric correction factor for non-standard geometries. μ/ μ_{w} is viscosity correction factor due to difference in viscosities at bulk fluid and wall temperatures. These constants are available in references mentioned below.

### Coil Side, ho

Pipe coils are made with a 180° central angle or a 120° central angle. Equivalent diameter (De) and Flow area (Ax) is

defined as following.

#### 180° Coil

`De = (π/2).dci`

`Ax = (π/8).dci²`

#### 120° Coil

`De = 0.708 dci`

`Ax = 0.154 dci²`

where, dci is inner diameter of pipe.

Reynold’s and Prandtl number are calculated based on jacket fluid properties and velocities.

`Re = De.v.ρ / μ`

`Pr = Cp.μ / k`

where, ρ is coil fluid density, μ is coil fluid viscosity and k is coil fluid thermal conductivity. v is fluid velocity in coil.

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#### Turbulent Flow

For Re > 10000

`Nu`_{c} = 0.027 Re^{0.8} Pr^{0.33} (μ/ μ_{w})^{0.14} (1 +3.5 De/Dc)

where Dc is the mean or centerline diameter of the coil. Coil outer diameter (Do) is determined as following.

`180° Coil : Do = D`

_{T}+ 2(dci/2) + 2.x`120° Coil : Do = D`

_{T}+ 2(dci/4) + 2.x

`Dc = ( Do + D`_{T}) / 2

#### Laminar Flow

For Re < 2100

`Nu`_{c} = 1.86 [ Re.Pr.De/L ]^{0.33} (μ/ μ_{w})^{0.14}

where, L is coil length along the vessel.

#### Transient Flow

For 2100 < Re < 10000

Calculate Nu_{Laminar} based on Re = 2100 and Nu_{Turbulent} based on Re = 10,000.

`Nu`_{Transient} = Nu_{Laminar} + (Nu_{Turbulent} - Nu_{Laminar}).(Re - 2100)/(10000 - 2100)

ho is determined as following

`ho.De/k = Nu`_{c}

ho and hi thus calculated are used to get value of overall heat transfer coefficient U.

Spreadsheet for Half Pipe Coil Agitated Vessel Heat Transfer

### References

- Heat Transfer Design Methods – John J. McKetta (1992)
- Heat Transfer in Agitated Jacketed Vessels – Robert F. Dream, Chemical Engineering, January 1999