F5. Heat Transfer Formulas

Conduction: One of the heat-transfer mechanisms is conduction. At steady state, heat transfer rate, Q is given as

Where ΔT is the driving force and R is the resistance to heat transfer. If Q is given in Btu/hr, and ΔT in °F, then units of resistance are °F·hr/Btu. The heat transfer resistance due to conduction is given as

where L is the length of the medium, k is thermal conductivity, and A is the area of heat transfer normal to the flow. The heat transfer resistance due to convection is given as

Conduction Through A Cylindrical Body: For objects with cylindrical dimensions, the relevant equations are provided in Table 5.1.

Critical radius of insulation: Heat loss from an insulated pipe varies as radius of insulation. Heat loss is minimum at critical radius. The thickness of insulation corresponding to critical radius of insulation is known as critical insulation thickness. If we insulate beyond this point, heat loss rate increases. This radius at critical heat loss is given as

Free Convection: Another mode of heat transfer is natural (free) convection. A number of correlations have been developed which are functions of Prandtl number (Pr), Grashof number (Gr), geometry and orientation of the heat transfer surface. See Table 5.2, where Nu is the Nusselt number, L is the characteristic length. The product Gr·Pr is termed as Raleigh number (Ra). Values of a and m are reported in Table 5.3.

For ideal gases β is equal to 1/T. For liquids, this value is calculated from density data. For water, this value is obtained from steam tables. Air properties are calculated at the film temperature that is the average of surface and bulk temperatures.

Forced Convection: The most common mode of heat transfer is forced convection. A number of correlations have been developed for situations where fluid is flowing through pipes and annuli. These correlations are functions of Reynolds number (Re), Prandtl number (Pr), and Graetz number (Gz) (see Tables 5.4 and 5.5). During computation of these numbers, the properties of the fluid are taken at bulk temperatures.

Flow through annuli: If the fluid is flowing through an annulus, then heat transfer correlations are given in Table 5.5.

Tube Wall Temperature, t_w: It should be noted that film coefficients are calculated at bulk temperatures whereas heat transfer occurs at the wall. A viscosity correction factor is employed to find the correct heat transfer coefficient. Tube wall temperature relationships depend upon whether cold or hot fluid is flowing in the pipe. See Table 5.6.

Clean Coefficient, U_C and Fouling Resistance, R_d: Once film coefficients are known, clean coefficient for heat exchangers can be found. It is important to allow for fouling of exchangers. The relationships involved are listed in Table 5.7.

Driving Force for Heat Transfer, ΔT: Heat transfer is a strong function of the temperature driving force. This depends on the type and arrangement of heat exchangers. These equations are provided in Table 5.8.

Shell and Tube Exchangers: The tube side heat transfer is handled in a manner similar to that of flow through a pipe. For shell-side of the exchanger, an equivalent diameter is computed. This equivalent diameter depends on the arrangement of tubes (square, triangular pitch). For a square pitch arrangement, this is given as

Where P_t = tube pitch; d_o = outer diameter of the tube. Cross-sectional area a is given as

Where ID = inner diameter of shell; C = clearance; and B = baffle spacing The heat transfer through a shell can be calculated as

Condensation of Vapors: Heat exchanger-coefficients for situations where phase change occurs (condensation) depend on the orientation of heat transfer surface (vertical, horizontal). Simplified correlations are developed for film type condensation. If a phase is condensing (steam), then heat transfer coefficient for a steam inside a circular tube assuming film-type condensation is given as a group, f_D, a function of both Reynolds number and Prandtl number. Figure 5.3 (Figure 7.17 Chopey) presents Dukler's results. In this Figure f_D given by

and is plotted against Reynolds number with Prandtl number as a parameter. In this equation, h_o is heat transfer coefficient without any shear. In order to account for interfacial shear, Dukler demonstrated that a ratio h/h_o is a function of Re_T and A_D, where Re_T is terminal Reynolds number (Chopey, Table 7.3) The parameter A_D is given as

Where subscripts L and G stand for liquid and vapor.

Condensation in a horizontal tube: If a vapor is condensing in a horizontal tube, then assuming the flow is stratified with laminar film condensation, the heat transfer coefficient can be found as

Where L = length of the tube; W = steam condensation rate; μ = viscosity of condensate;

ρ = density of condensate; n = number of segments

It is suggested that this value should be corrected by a factor h/h_c that depends on condensate loading. This factor is reported as a function of two parameters, Reynolds number, and A_s where

Batch Processes: Some of the industrial applications include (1) cooling/heating of the tank contents by an isothermal medium, (2) heating of the tank contents by a non-isothermal medium (coil-in-tank, jacketed vessel), and (3) cooling of the tank contents by a non-isothermal medium. These relations are provided in Table 5.10. In these equations M is the size of the batch, and U is the heat transfer coefficient.

Non-isothermal batch cooling: Another useful situation arises when cooling is provided in a tank by a fluid, which enters at leaves at different temperatures. (Table 5.10, Equations 3 and 4).

Heating of tank contents by an external heat exchanger: A more complicated application is applicable when tank contents are being heated by an external heat exchanger (see Table 5.10 Equation 5).

Agitated Vessel: This is another application of heat transfer from a coil that is immersed in an agitated vessel. The governing equation is given in Table 5.10 as Equation 6.

Radiation: When heat is transferred through radiation, heat transfer rate, Q, is give as

Where F = factor depending upon the view emissivity factors.

σ = Stefan-Boltzmann constant, 0.1713´10^-8 Btu/(h·°F⁴).

If the surfaces are normal to each other then the view factor is 1. The emissivity factor, F_ε, can be calculated as

Where ε_i is an emissivity of body i having surface temperature T_i.

Radiation heat loss with a shield can be calculated once we know the temperature of the shield. The radiative flux from the first body to the shield is equal to the radiative flux from the shield to the second body. This is true because no heat is stored in the shield.

Upon rearrangement, T₃ can be found to be

This value of temperature can now be used to compute heat transfer, Q.

Unsteady State Heat Transfer (Conduction): Another area of heat transfer is unsteady state transfer. Gurnie-Lurie has developed a set of charts for different geometries of the solids. The cases considered are wall of infinite thickness heated on one side, wall of finite thickness heated on both sides, wall of finite thickness heated by a fluid with a contact resistance.

Wall of infinite thickness is heated on one side: In this situation, the temperature of the material at a distance x from the surface at θ hours is given by Table 5.11 Equation 1, where T_o is initial temperature, T_s is surface temperature, χ is a dimensionless length, α is thermal diffusivity, and q is flux passing the surface at any time θ.

Wall of finite thickness is heated on both sides: In this situation, the temperature of the material at a distance x from the center is given as Table 5.11 Equation 3.

Finite body is heated by a fluid with contact resistance: In this case, the unsteady state solution to the problem involves three parameters

The dimensionless temperature Y is given as a function of these parameters for different types of geometries by Gurnie-Lurie.