Jacketed Vessel Heat Transfer (Half Pipe Coil)
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
Process Side, hi
Process side film coefficient, hi depends upon type of impeller, 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 nonstandard 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.
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.
Resources
 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