Biochemical Engineering Fundamentals, Part 1 - Bailey J., Ollis D. 1989

Transport phenomena in biotechnological systems
Heat transfer
Heat transfer equations

From equation (8.118) and the overall heat balance [equation (8.119)] of the heating, cooling, and sterilization processes, the following general equation for calculating heat transfer is derived:

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To use this equation in practical calculations, an expression for the overall heat transfer coefficient h must be found, analogous to the previously discussed expressions for the overall mass transfer coefficient in a gas–liquid system. The continuity of heat flux requires that, under steady-state conditions, heat transfer through a flat wall of thickness L_w separating the culture broth at Temperature T_liq,1 and the heating or cooling fluid at temperature T_liq,2 must satisfy the equation

Here, k_s is the thermal conductivity of the wall, expressed in kcal/(cm·s·deg). If the overall heat transfer coefficient h, defined as

then equation (8.124) can be transformed into

By analogy with the mass transfer we have already studied, the overall resistance to heat transfer is equal to the sum of three resistances. For heat transfer through the cylindrical tubular surface of heating or cooling coils, the surface area does not remain constant but increases proportionally to the distance from the inner surface of the tube. In this case, the equation for must be modified as follows:

Here, d_i and d_o are the inside and outside diameters of the tube, respectively. Note that the subscript o in the coefficient indicates that the heat transfer area calculation is based on the outer diameter of the tube. The thermal conductivity k_s of solids depends on their nature. For instance, at 100 °C, k_s is 0.908 cal/(s·cm·K) for copper and 0.107 cal/(s·cm·K) for steel; as the temperature decreases, k_s slowly increases. Values of k_s for various Materials used in heat exchangers can be found in any engineering reference book.

Studying momentum and heat transfer at any fluid–solid interface allows for the Determination of the corresponding individual heat transfer coefficients at the phase boundaries, h_w1, h_w2 or h_o, h_i in equations (8.126) and (8.127), respectively. If these individual coefficients vary depending on the position on the heat transfer surface, the overall local heat transfer coefficient is determined by equations of the type (8.126) and (8.127), and calculating the total heat transferred requires integration over the entire heat transfer surface.

In studying heat transfer in a liquid–solid system, the following dimensionless groups are used:

Here, k_f is the thermal conductivity of the fluid, cal/(s·cm·deg); C_p is the specific heat capacity, cal/(g·deg); d is the distance (or tube diameter), cm; μ is the viscosity, P; u is the velocity, cm/s; g is the gravitational acceleration, cm/s².

As indicated in reference [5], the Brinkman number represents The ratio of heat produced by viscous dissipation to heat transferred by conduction; under our conditions of heat transfer through the heat exchanger surface, this value can be neglected. (The Brinkman number can be large near the impeller blade.) Similarly, in baffled reactors or those with an off-center agitator, the Froude number is usually negligible.

The heat transfer coefficient h, expressed as the dimensionless Nusselt number (Nu), is a function of Pr and Re:

Nu = f(Pr, Re)      (8.129a)

For laminar flow in straight tubes, heat transfer also depends on the ratio of the tube length L to its diameter d (L/d); therefore, equations of the following type have been proposed:

Temperature changes cause variations in various fluid properties near the heat transfer surface. Of these properties, viscosity is the most important; therefore, the ratio μ_b/μ₀, where μ_b and μ₀ are the fluid viscosities at the bulk fluid temperature and the wall temperature, respectively, should also be taken into account in the equation:

Since h (and consequently Nu) can be defined either as a local transfer coefficient or as a coefficient averaged over the heat transfer surface in one of several ways, when using literature values of Nu_loc or Nu_av, ∆T_loc or ∆T_av should be applied accordingly.

For heat transfer in the turbulent flow of fluids with viscosities close to that of Water, the following equation has been proposed, which is useful for both heating and cooling of the reaction mixture*:

This equation was found to be valid for

10⁴≤ Re ≤ 1.2⋅10⁵ (turbulent flow)

0.7≤Рr≤120 (for all liquids except molten metals)

L/d≥60 (long tubes)

As modified by Sieder and Tate**, the latter equation is also applicable at large temperature differences; it has proven useful for determining heat transfer in viscous liquids, such as oils:

If the liquid phase is non-uniform in density, natural convection also contributes significantly to heat transfer; in these cases, the Grashof number is introduced into the corresponding equations:

* McAdams W. H., Heat Transmission, 3rd ed., p. 152, McGraw-Hill Book Company, New York, 1954.

** Sieder E. N., Tate G. E., Heat Transfer and Pressure Drop of Liquids in Tubes, Ind. Eng. Chem., 28, 1429 (1936).

Then the equation for Nu for fluid flow in horizontal tubes takes the form*

(In the case of vertical tubes, a coefficient of 1.0 is used instead of the viscosity ratio, and the constant 0.04 is replaced by 0.0722.)

Heat transfer in Non-Newtonian fluids is described by other equations. Thus, for pseudoplastic fluids (Section 8.8), two equations have been used**:

or

Here η_v is the apparent viscosity [Equation (8.92)], determined at the temperature of the liquid phase or the wall. Note that for a Newtonian fluid (n = 1) with a small temperature difference between the liquid phase and the wall, Equation (8.134b) reduces to Equation (8.133). In Equation (8.134a), the temperature dependence of viscosity is taken into account to the same extent as in Equation (8.133).

In principle, heat exchangers can have A wide variety of surfaces and flow regimes. Some equations for several specific situations are given in Example 8.3; others can be found in standard heat transfer handbooks and in Chapter 3 of Charm's monograph (reference [30] in Chapter 9). A number of Examples of calculating heat transfer coefficients and heat exchanger duties are covered in the exercises. Sometimes the culture broth forms a deposit on the heater surface; this leads to system fouling, A change in the wall heat transfer coefficient over time, and an increase in h on the liquid side.

Example 8.3. Heat Transfer Equations

Natural convection heat transfer from a vertical plane or from a cylinder surface***:

Here L is the wall length or cylinder diameter. All parameters are evaluated at (T_liq+T_wall)/2, as well as at

* Martinelli R. C., Boelter L. M., AIChE Mtg., 1942 (cited in McAdams W. H., see above).

** Charm S. Е., Merill Е. W., Heat Transfer Coefficients in Straight Tubes for Pseudoplastic Food Materials in Streamline Flow, Food Res., 24, 319 (1959).

*** King W. J., Free Convection, Mech. Eng., 54, 347 (1932).

Heat transfer in the space between coaxial cylinders* Laminar regime:

where d₀, d_і are the outer and inner diameters. Turbulent regime:

Gravity flow over horizontal tube surfaces

Here w is the fluid velocity; L is the tube length; d₀ is the outer diameter.

Turbulent flow in tubes***:

Flow normal to an isolated cylinder***:

Flow normal to a row of tubes spaced two diameters apart***:

* Monrad С. С., Pelton J. F., in McAdams W. H., Heat Transmission, McGraw-Hill Book Company, New York, 1954.

** Dittus F. W., Boelter С. M. K., Univ. Calif. Publ. Eng., 2, 443 (1930); (cited in McAdams W. H., Heat Transmission, see above).

*** Charm S. E., Fundamentals of Food Engineering, 2d ed., Chap. 4, Avi Pwblishing, Westport, Conn., 1971.

Here Re_m is the Re at u_max, and the distance 2d is defined as follows:

Staggered successive rows of tubes of the same type*:

Equation (8П3.7) is used, with the coefficient 0.21 replaced by 0.27 for 3 rows, 0.30 for 5 rows, and 0.33 for 10 or more rows of tubes.

A vast number of studies have been dedicated to mass and heat transfer. In this chapter, we have considered only the Basic principles of determining parameters of greatest interest for biochemical technology. As pointed out in some of the general References listed at the end of this chapter, a very large number of empirical and semi-empirical equations have been published to date for determining heat and mass transfer under a wide variety of conditions. In practice, one should always use the equations that best match the process under study; furthermore, the limits of applicability of these equations must always be taken into account.

* Charm S. Е., Fundamentals of Food Engineering, 2d ed., Chap. 4, Avi Publishing Westport, Conn., 1971.