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1 8 Fundamentals of Heat Transfer René Reyes Mazzoco Universidad de las Américas Puebla, Cholula, Mexico 1 HEAT TRANSFER MECHANISMS 1.1 Conduction Conduction heat transfer is explained through the molecular motion in the solid s structure. Heat is transferred from one molecule to the adjacent molecule by means of vibrational motion. This basic description points out that heat transfer through a solid takes place entirely by conduction, and also states that it occurs to a limited extent in liquids and gases because of their molecular mobility. The mathematical formulation of conduction heat transfer was proposed by Joseph Fourier while solving heat transfer problems in metal casting and template. The first step for this formulation is the recognition that the amount of heat transferred, q(w), from one point of a metal piece to another point of the same medium (continuum) is proportional to the temperature difference between those two points. The evaluation of the temperature difference through the derivative in any direction (s) makes the measurement independent of any two specific points and the distance between them: q dt ds (1)

2 For most cases of conduction heat transfer in solids it is possible to change the proportionality relation to an equality defining the proportionality constant called thermal conductivity, k(w/m K). Thus, the calculation of the amount of heat transferred per cross-sectional surface area, A(m 2 ), is obtained from Fourier s first law: q a = k dt ds (2) Measurement of Thermal Conductivity The thermal conductivity is a physical property characteristic of the medium and is defined similarly for the three states of the matter. The expression for the calculation of the amount of heat transferred by conduction is also used to define the thermal conductivity, and one procedure for its measurement, that is, k, is calculated after measuring the linear temperature difference between two points, the amount of heat transferred, and the cross-sectional area associated with the heat flow trajectory: k = q/a dt/ds The procedure for calculating values of k is especially relevant for the evaluation of this property in mixtures of contaminated media for which no information is normally available. For this evaluation it is easier to calculate the thermal diffusivity, α (where α = k/ρc p, m 2 /s) of the mixture while solving Fourier s second law in cylindrical coordinates: T t =α T r T r r Thus, a cylinder with a substance of unknown thermal conductivity is placed in a bath at constant temperature to obtain the initial and boundary conditions associated with the solution of Eq. (4): T(r, 0) = initial sample temperature T(0, t) = measurement at the cylinder s center T(R, t) = constant bath temperature The value of α can be obtained from the solution of Eq. (4) with these boundary conditions. The values of the density, ρ, and the constant-pressure heat capacity, C p, are normally available in the literature, or are measured experimentally. Thermal conductivity prediction models are available for gases and liquids in several references (e.g., Ref. 1). (3) (4)

3 1.1.2 Effect of Shape on Calculation of Conduction Heat Transfer Evaluation of the integral between two specific points in the direction of heat transfer allows for the calculation of the macroscopic amount of heat. For an object with constant cross-sectional area in the direction of heat flow, integration of Eq. (2) gives: q A = k T 1 T 2 s 2 s 1 (5) Instead, for a cylindrical object, heat flow in the direction of the radius finds a constantly changing cross-sectional area. Thus, integration of Eq. (2) with A = 2πrL gives q = T 1 T 2 ln(r 2 /r 1 )/2πkl which involves a logarithmic distance (radius) difference instead of the thickness of the medium involved Combined Resistances A common problem in heat transfer design is the combination of several layers of solid to provide heat insulation, or layers of solid and fluid as in heat exchanger design. For a rectangular geometry, two combined resistances to heat transfer can be expressed as q A = T 1 T 2 (L 1 /k 1 ) + (L 2 /k 2 ) where L and k are the thickness and the thermal conductivity of components (1) and (2), respectively, and T 1 and T 2 are the temperatures at the external surfaces of the combined wall. Each additional layer of material increases by one the resistances to heat transfer added in the denominator of Eq. (7). The effect of geometry on the combined resistances to heat transfer can be obtained by integrating Eq. (2) for a double-layered cylinder: q = T 2 T 1 [ln(r 2 /r 1 )/(2πk 1 L)] + [ln(r 3 /r 2) /(2πk 2 L)] Relative Magnitude of Values of Thermal Conductivity The values of the thermal conductivity depend on the phase of the material considered: (6) (7) (8)

4 k gas < k liquid < k solid Thus, solid materials are good heat conductors, while for heat insulation trapped gases are the best option. Good electric conductor metals are the best selection for heat conductors. The design of heat insulation follows the criteria for air entrapment in fabrics or ceramics that could be resistant to high temperatures. 1.2 Convection Heat convection is described as heat transport in fluid eddies promoted by the flow derived from a mechanical device, a pump or fan (forced convection), or a density difference (natural convection). The mechanism is associated with the definition of the convective heat transfer coefficient, h(w/m 2 C): h = q A(T 2 T 1) (9) As the turbulent flow process carrying the heat cannot be fully described, the temperature difference is considered at two points (1) and (2) in the direction of heat transfer. It is not possible to describe this process through a differential equation, and Eq. (9) is a definition for h that is related to the specific geometry associated to the surface area, A, and the flow conditions. The convective heat transfer coefficient can be calculated for design purposes from experimental information gathered in the open literature. Experiments have been carried out under geometry, flow range, and similar thermophysical properties conditions that can be encountered in process applications. The information has been grouped in terms of flow conditions and thermophysical properties involved. Flow conditions are described through the Reynolds number (Re) for forced convection. The Reynolds number relates the momentum convection associated to the flow velocity, v, to the momentum diffusivity associated to ν, the kinematic viscosity (ν = µ/ρ), µ is Newtonian viscosity (kg/ms). At low Reynolds numbers, implying low flow velocity, momentum diffusivity dominates, and the fluid displacement is in the laminar flow condition. When the flow velocity is high relative to the kinematic viscosity, the Reynolds number is high, indicating turbulent flow conditions. Re = Lν v (10) L is the flow characteristic length; for internal flow in circular pipes, L is the internal diameter. The Grashof number (Gr) describes flow conditions for natural convection and is used instead of the Reynolds number.

5 Gr = gβ(t w T infinity )L 3 v 2 (11) Here g is the acceleration of gravity, T w is the solid wall temperature, T infinity is the fluid bulk temperature, L is the heat transfer characteristic length, and β is the volume coefficient of expansion: β= (ρ infinity ρ) ρ(t T infinity ) (12) ρ infinity is the fluid bulk density. In natural and forced convection, the Prandtl number describes the influence of thermophysical properties in the calculation of the convective heat transfer coefficient, normally to the 1 3 power. Pr = v α = C pµ k (13) The Nusselt number, Nu, is the ratio of heat convection to diffusion associated to the heat transfer characteristic length, L: Nu = hl k (14) From the exact analysis of the boundary layer between the fluid and the solid wall transferring heat, the correlation in forced convection among Nusselt, Reynolds, and Prandtl numbers is Nu = Re 1/2 Pr 1/3 (15) This theoretical correlation has very limited application, and the dependence of these dimensionless numbers on the geometry makes experimentation necessary to calculate correlations for each geometry. The correlation results are normally reported with the same mathematical formulation: Nu = c 0 Re n Pr m (16) For natural convection, the analysis of the boundary layer provides the correlation of the important dimensionless numbers: Nu = C(Gr Pr) m (17) 1.3 Radiation For practical conditions, radiation emitted (or received) by surface is calculated from an equation that involves the effect of the area, A 12, the emissivity, ε 1, of the emitting surface involved, and a view factor, F 12, that describes the effect

6 of the relative positions of the two surfaces involved on the amount of radiation exchanged. The formulation of the exchanged radiation is q = σε 1 A 12 F 12 (T 1 4 T 2 4 ) (18) All practical terms in Eq. (18) are measured experimentally and are reported in several references (e.g., Ref. 2) 2 HEAT ACCUMULATION Heat accumulation is described through the heat capacity. The specific property normally used to achieve this calculation is the constant-pressure heat capacity, C p (J/kg C). The total amount of material that stores heat should be expressed in the mass or molar terms used for the C p. The heat stored is then a function of the temperature change in the total mass considered: C p = q m(dt/dt) (19) The temperature variation with time allows the evaluation of the heat flow accumulated. 2.1 Sensible Heat The variation of the temperature in a fluid medium defines sensible heat. The calculation of the amount of sensible heat is obtained from Eq. (19). The heat capacity should correlate the fluid and phase considered. 2.2 Latent Heat The process where a change of phase takes place requires the addition of latent heat. The latent heat is used to change phase in a fluid without a change in the medium temperature. The evaluation of the latent heat is necessary to measure the amount of heat required for phase change. Latent heat values and prediction correlations are available in Ref EXPERIMENTAL MEASUREMENT AND PREDICTION OF HEAT TRANSFER THERMOPHYSICAL PROPERTIES 3.1 Constant-Pressure Heat Capacity, C p Measurement of the C p requires the evaluation of temperature change in a fixed mass of material due to a heat flow from the surroundings according to Eq. (19).

7 3.2 Thermal Conductivity, k For the measurement of k, Fourier s first law is normally used to define the parameters involved in the evaluation. The heat flux in Eq. (2) is determined from the heat flow and the body geometry while the temperature gradient is measured directly. 3.3 Convective Heat Transfer, h The convective heat transfer coefficient is experimental measured of forced- and natural-convection conditions. h is part of Nu, while flow conditions are represented by Re or Gr, and the thermophysical properties form Pr. Normally, the values of h are obtained from reported correlations. If it is necessary to evaluate h for conditions not previously studied, the information is gathered and analyzed according to Eq. (16) or (17). 3.4 Thermophysical Properties of Mixtures in Pollution Control Mixtures of contaminated media normally require the experimental evaluation of the thermophysical properties. In some cases, due to nonavailability of the experimental data, correlations for calculating the thermophysical properties are limited. 4 HEAT TRANSFER DESIGN Process efficiency is defined at the design stage. Design algorithms for heat transfer equipment can be found in several classic references (e.g., Ref. 3) and are still used for designing heat transfer equipment. Several software options are also available for efficient heat transfer equipment design; software the description can be obtained from demos downloaded from an Internet search (any search engine) on Heat Exchangers. The basic equation for heat exchange design is q = U o A o T LM F (20) where U o is the overall heat transfer coefficient and includes all the heat transfer conductances around the solid wall transferring heat. For a flat wall transferring heat, U o = 1 1/h inside + G/k + 1/h outside (21) where h inside is the inside convective heat transfer coefficient, G is the wall thickness, and h outside is the outside heat transfer coefficient.

8 The driving force for heat transfer in a heat exchanger is the logarithmic mean temperature difference: T LM F = [T outlet t inlet ) (T inlet t outlet )]F ln[(t outlet t inlet )/(T inlet t outlet )] (22) T is the hot fluid temperature, t is the cold fluid temperature, and F is the efficiency factor adapted for each configuration of shell and tube, plate exchangers, and direct-contact heat exchangers (4). From the calculation of the amount of heat transferred, including the temperature changes involved and the overall heat transfer conductance, the area for heat exchange is determined. Several heat transfer equipment can be used to accomplish the heat exchange between the media in a given process condition. 4.1 Heat Transfer Design and Good Engineering Practices Design defines the efficiency of the operation of a process. Once the optimized design is utilized, it is necessary to maintain good engineering practices. These practices should include pollution control and waste minimization. Heat transfer equipment is subjected to fouling and corrosion, which are among the major hurdles for the operation. Fouling increases heat transfer resistance and waste of energy. Good engineering practices include the use of fouling suppresants in heat transfer fluids and periodic cleaning of the exchanger walls. For water as the cooling or heating medium in industrial operations there are several standard techniques for keeping fouling low. Water in cooling-water circuits has to be treated to keep salts and dirt content low. Common treatments include the addition of coagulants for sedimentation of some salts and particles; addition of biocides, to prevent microbial growth that is another source of fouling; and the addition hardness suppresants such as polyphosphates; among others. Although the materials used in heat transfer fluids treatment are a source of solid waste, handling its final deposition should follow normal procedures. Fouling prevention is not considered a polluting operation. Fouling prevention by-products can be integrated to cement kiln operations when feasible, in order to eliminate waste generation. Corrosion protection of heat transfer surfaces is a suggested practice for pollution control and waste minimization. In order to prevent corrosion, begin with the analysis of the appropriate combination of materials and fluids. For the operating equipment, passive and active cathodic protection are recommended.

9 4.2 Innovations for Efficient Heat Use Efficient energy use is a direct way to reduce pollution and minimize wastes from industrial sources. The ongoing research in energy efficiency and resulting innovations highlight the intensity of scientific activity in this field. New approaches to increase heat transfer efficiency include the following. 1. Fluidized bed combustion is the choice for eliminating solids in solidwaste management schemes. In general, direct contact between the materials increases heat transfer efficiency. Direct contact reduces the heat transfer resistances due to the wall in conventional equipment, and increases the convective heat transfer coefficients due to the higher contact velocities between the materials and fluids. 2. To increase the efficiency in steam generation, direct-contact heat exchangers make use of residual heat from combustion gases to preheat the feed streams to the boiler. Thermal recovery is a possibility from direct-contact heat exchangers and heat pumps. Rotary drums recover heat from a residual discharge in an steam generator and transport it to preheat the inlet streams to the generator. 3. Heat pipes are a promising technology for increasing residual heat usage as heat pumps. Heat pipes use capillary pressure as the driving force for condensing and evaporating the working fluid, thus eliminating the necessity for pump and compressor in the power cycle. The understanding of heat pipe operation is related to the evaluation of convective heat transfer coefficients for change-of-phase heat transfer. 4. Plate heat exchangers are now available for almost any process condition, including high-pressure and corrosivity conditions. Enhanced heat transfer surfaces improve energy management, reducing wastes. Improved surfaces increase the convective heat transfer coefficients for heating cooling operations, and change-of-phase heat transfer. 5. Co-generation in chemical and petrochemical processes makes use of the process integration gained from the use of simulation and pinchpoint techniques to increase energy usage. 5 CONCLUSIONS The understanding of heat transfer fundamentals is a basic step toward the proposition of improved industrial solutions in terms of energy wastes minimization. Clear fundamental concepts make the use of design software straightforward. This is the approach to equipment design that produces the best results for waste minimization.

10 Heat transfer innovations are improving energy handling in industrial processes, reducing pollution and wastes. This research field is active in fundamentals such as enhanced heat transfer or heat pipe development. REFERENCES 1. R. C. Reid, J. M. Prausnitz and B. E. Poling, The Properties of Gases and Liquids, 4th ed. New York: McGraw-Hill, J. P. Holman, Heat Transfer, 8th ed. New York: McGraw-Hill, D. Q. Kern, Process Heat Transfer. New York: McGraw-Hill, O. Levenspiel, Engineering Flow and Heat Exchange, 2nd ed. New York: Plenum Press, 1992.