Chemical Engineering - Heat Transfer Theory

Walls or heat exchangers - calculate overall heat transfer coefficients.

Heat transfer through a surface like a wall can be calculated as

q = U A dT                        (1)

where

q = heat transfer (W (J/s))

U = overall heat transfer coefficient (W/(m²K))

A = wall area (m²)

dT = (t₁ - t₂) = temperature difference over wall (^oC)

The overall heat transfer coefficient for a multi-layered wall, pipe or heat exchanger - with fluid flow on each side of the wall - can be calculated as

1 / U A = 1 / h_ci A_i + Σ (s_n / k_n A_n) + 1 / h_co A_o                                      (2)

where

U = the overall heat transfer coefficient (W/(m² K))

k_n = thermal conductivity of material in layer n  (W/(m K))

h_{c i,o} = inside or outside wall individual fluid convection heat transfer coefficient (W/(m² K))

s_n = thickness of layer n (m)

A plane wall with equal area in all layers - can be simplified to

1 / U = 1 / h_ci + Σ (s_n / k_n) + 1 / h_co                               (3)

Thermal conductivity - k - for some typical materials (not that conductivity is a property that may vary with temperature)

• Polypropylene PP : 0.1 - 0.22 W/(m K)
• Stainless steel : 16 - 24 W/(m K)
• Aluminium : 205 - 250 W/(m K)

The convection heat transfer coefficient - h - depends on

• type of fluid - if its gas or liquid
• flow properties like velocity
• other flow and temperature dependent properties

Convective heat transfer coefficient for some common fluids:

• Air - 10 to 100 W/m²K
• Water - 500 to 10 000 W/m²K

Multi-layered Walls - Heat Transfer Calculator

This calculator can be use to calculate the overall heat transfer coefficient and the heat transfer through a multi-layered wall. The calculator is generic and can be used for metric or imperial units as long as the use of units is consistent.

A - area (m²)

t₁ - temperature 1 (^oC)

t₂ - temperature 2 (^oC)

h_ci - convective heat transfer coefficient inside wall (W/(m² K))

s₁ - thickness 1 (m)

k₁ - thermal conductivity 1 (W/(m K))

s₂ - thickness 2 (m)

k₂ - thermal conductivity 2 (W/(m K))

s₃ - thickness 3 (m)

k₃ - thermal conductivity 3 (W/(m K))

h_co - convective heat transfer coefficient outside wall (W/(m² K))

Heat Transfer Thermal Resistance

Heat transfer resistance can be expressed as

R = 1 / U                               (4)

where

R = heat transfer resistance (m²K/W)

The wall is split in sections of thermal resistance where

• the heat transfer between the fluid and the wall is one resistance
• the wall it self is one resistance
• the transfer between the wall and the second fluid is a thermal resistance

Surface coatings or layers of "burned" product adds extra thermal resistance to the wall decreasing the overall heat transfer coefficient.

Some typical heat transfer resistances

• static layer of air, 40 mm (1.57 in)  : R = 0.18 m²K/W
• inside heat transfer resistance, horizontal current : R = 0.13 m²K/W
• outside heat transfer resistance, horizontal current : R = 0.04 m²K/W
• inside heat transfer resistance, heat current from down upwards : R = 0.10 m²K/W
• outside heat transfer resistance, heat current from above downwards : R = 0.17 m²K/W

Example - Heat Transfer in Air to Air Heat Exchanger

An air to air plate exchanger with area 2 m² and wall thickness 0.1 mm can be made in polypropylene PP, aluminium or stainless steel.

The heat transfer convection coefficient for air is 50 W/m²K. Inside temperature in the exchanger is 100 ^oC and outside temperature is 20 ^oC.

The overall heat transfer coefficient U per unit area can be calculated by modifying (3) to

U = 1 / (1 / h_ci + s / k + 1 / h_co)                                               (3b)

The overall heat transfer coefficient for heat exchanger in

• polypropylene with thermal conductivity 0.1 W/mK is

U_PP = 1 / (1 / (50 W/m²K) + (0.1 mm) (10^-3 m/mm)/ (0.1 W/mK) + 1 / (50 W/m²K))

  = 24.4 W /m²K

The heat transfer is

q = (24.4 W /m²K) (2 m²) ((100 ^oC) - (20 ^oC))

  = 3904 W

  = 3.9 kW 

• stainless steel with thermal conductivity 16 W/mK:

U_SS = 1 / (1 / (50 W/m²K) + (0.1 mm) (10^-3 m/mm)/ (16 W/mK) + 1 / (50 W/m²K))

  = 25 W /m²K

The heat transfer is

q = (25 W /m²K) (2 m²) ((100 ^oC) - (20 ^oC))

  = 4000 W

  = 4 kW 

• aluminum with thermal conductivity 205 W/mK:

U_Al = 1 / (1 / (50 W/m²K) + (0.1 mm) (10^-3 m/mm)/ (205 W/mK) + 1 / (50 W/m²K))

  = 25 W /m²K

The heat transfer is

q = (25 W /m²K) (2 m²) ((100 ^oC) - (20 ^oC))

  = 4000 W

  = 4 kW 

Typical Overall Heat-Transfer Coefficients

• Free Convection Gas - Free Convection Gas : U = 1 - 2 W/m²K  (typical window, room to outside air through glass)
• Free Convection Gas - Forced liquid (flowing) water : U = 5 - 15 W/m²K  (typical radiator central heating)
• Free Convection Gas - Condensing Vapor Water : U = 5 - 20 W/m²K  (typical steam radiators)
• Forced Convection (flowing) Gas - Free Convection Gas : U = 3 - 10 W/m²K (superheaters)
• Forced Convection (flowing) Gas - Forced Convection Gas : U = 10 - 30 W/m²K (heat exchanger gases)
• Forced Convection (flowing) Gas - Forced liquid (flowing) water  : U = 10 - 50 W/m²K (gas coolers)
• Forced Convection (flowing) Gas - Condensing Vapor Water  : U = 10 - 50 W/m²K (air heaters)
• Liquid Free Convection - Forced Convection Gas : U = 10 - 50 W/m²K (gas boiler)
• Liquid Free Convection - Free Convection Liquid : U = 25 - 500 W/m²K (oil bath for heating)
• Liquid Free Convection - Forced Liquid flowing (Water) : U = 50 - 100 W/m²K (heating coil in vessel water, water without steering), 500 - 2000 W/m²K (heating coil in vessel water, water with steering)
• Liquid Free Convection - Condensing vapor water : U = 300 - 1000 W/m²K (steam jackets around vessels with stirrers, water), 150 - 500 W/m²K (other liquids)
• Forced liquid (flowing) water - Free Convection Gas : U = 10 - 40 W/m²K (combustion chamber + radiation)
• Forced liquid (flowing) water - Free Convection Liquid : U = 500 - 1500 W/m²K (cooling coil - stirred)
• Forced liquid (flowing) water - Forced liquid (flowing) water : U = 900 - 2500 W/m²K (heat exchanger water/water)
• Forced liquid (flowing) water - Condensing vapor water : U = 1000 - 4000 W/m²K (condensers steam water)
• Boiling liquid water - Free Convection Gas : U = 10 - 40 W/m²K (steam boiler + radiation)
• Boiling liquid water - Forced Liquid flowing (Water) : U = 300 - 1000 W/m²K (evaporation of refrigerators or brine coolers)
• Boiling liquid water - Condensing vapor water : U = 1500 - 6000 W/m²K (evaporators steam/water)