4.12. HEAT TRANSFER OF NON-NEWTONIAN FLUIDS

4.12A. Introduction

Most of the studies on heat transfer with fluids have been done with Newtonian fluids. However, a wide variety of non-Newtonian fluids are encountered in the industrial chemical, biological, and food processing industries. To design equipment to handle these fluids, the flow-property constants (rheological constants) must be available or must be measured experimentally. Section 3.5 gave a detailed discussion of rheological constants for non-Newtonian fluids. Since many non-Newtonian fluids have high effective viscosities, they are often in laminar flow. Since the majority of non-Newtonian fluids are pseudoplastic fluids, which can usually be represented by the power law, Eq. (3.5-2), the discussion will be concerned with such fluids. For other fluids, the reader is referred to Skelland (S3).

4.12B. Heat Transfer Inside Tubes

1. Laminar flow in tubes

A large portion of the experimental investigations have been concerned with heat transfer of non-Newtonian fluids in laminar flow through cylindrical tubes. The physical properties that are needed for heat-transfer coefficients are density, heat capacity, thermal conductivity, and the rheological constants K' and n' or K and n.

In heat transfer in a fluid in laminar flow, the mechanism is primarily one of conduction. However, for low flow rates and low viscosities, natural convection effects can be present. Since many non-Newtonian fluids are quite "viscous," natural convection effects are reduced substantially. For laminar flow inside circular tubes of power-law fluids, the equation of Metzner and Gluck (M2) can be used with highly "viscous" non-Newtonian fluids with negligible natural convection for horizontal or vertical tubes for the Graetz number NGz > 20 and n' > 0.10:

Equation 4.12-1


where

Equation 4.12-2


Equation 4.12-3


The viscosity coefficients γb at temperature Tb and γw at Tw are defined as

Equation 4.12-4


The nomenclature is as follows: k in W/m · K, cp in J/kg · K, ρ in kg/m3, flow rate m in kg/s, length of heated section of tube L in m, inside diameter D in m, the mean coefficient ha in W/m2 · K, and K and n' rheological constants (see Section 3.5). The physical properties and Kb are all evaluated at the mean bulk temperature Tb and Kw at the average wall temperature Tw.

The value of the rheological constant n' or n has been found not to vary appreciably over wide temperature ranges (S3). However, the rheological constant K' or K has been found to vary appreciably. A plot of log K' versus 1/Tabs (C1) or versus T°C (S3) can often be approximated by a straight line. Often data for the temperature effect on K are not available. Since the ratio Kb/Kw is taken to the 0.14 power, this factor can sometimes be neglected without causing large errors. For a value of the ratio of 2:1, the error is only about 10%. A plot of log viscosity versus 1/T for Newtonian fluids is also often a straight line. The value of ha obtained from Eq. (4.12-1) is the mean value to use over the tube length L with the arithmetic temperature difference ΔTa:

Equation 4.12-5


when Tw is the average wall temperature for the whole tube and Tbi is the inlet bulk temperature and Tbo the outlet bulk temperature. The heat flux q is

Equation 4.12-6


EXAMPLE 4.12-1. Heating a Non-Newtonian Fluid in Laminar Flow

A non-Newtonian fluid flowing at a rate of 7.56 × 10-2 kg/s inside a 25.4-mm-ID tube is being heated by steam condensing outside the tube. The fluid enters the heating section of the tube, which is 1.524 m long, at a temperature of 37.8°C. The inside wall temperature Tw is constant at 93.3°C. The mean physical properties of the fluid are ρ = 1041 kg/m3, cpm = 2.093 kJ/kg · K, and k = 1.212 W/m · K. The fluid is a power-law fluid having the following flow-property (rheological) constants: n = n' = 0.40, which is approximately constant over the temperature range encountered, and K = 139.9 N · sn'/m2 at 37.8°C and 62.5 at 93.3°C. For this fluid a plot of log K versus T°C is approximately a straight line. Calculate the outlet bulk temperature of the fluid if it is in laminar flow.

Solution: The solution is trial and error, since the outlet bulk temperature Tbo of the fluid must be known in order to calculate ha from Eq. (4.12-2). Assuming Tbo = 54.4°C for the first trial, the mean bulk temperature Tb is (54.4 + 37.8)/2, or 46.1°C.

Plotting the two values of K given at 37.8 and 93.3°C as log K versus T°C and drawing a straight line through these two points, a value for Kb of 123.5 at Tb = 46.1°C is read from the plot. At Tw = 93.3°C, Kw = 62.5.

Next, δ is calculated using Eq. (4.12-2):


Substituting into Eq. (4.12-3),


From Eq. (4.12-4),


Then substituting into Eq. (4.12-1),

Equation 4.12-1


Solving, ha = 448.3 W/m2 · K.

By a heat balance, the value of q in W is as follows:

Equation 4.12-7


This is equated to Eq. (4.12-6) to obtain

Equation 4.12-8


The arithmetic mean temperature difference ΔTa by Eq. (4.12-5) is


Substituting the known values in Eq. (4.12-8) and solving for Tbo,


This value of 54.1°C is close enough to the assumed value of 54.5°C that a second trial is not needed. Only the value of Kb would be affected. Known values can be substituted into Eq. (3.5-11) for the Reynolds number to show that it is less than 2100 and that the flow is laminar.


For less "viscous" non-Newtonian power-law fluids in laminar flow, natural convection may affect the heat-transfer rates. Metzner and Gluck (M2) recommend use of an empirical correction to Eq. (4.12-1) for horizontal tubes.

2. Turbulent flow in tubes

For turbulent flow of power-law fluids through tubes. Clapp (C4) presents the following empirical equation for heat transfer:

Equation 4.12-9


where NRe,gen is defined by Eq. (3.5-11) and hL is the heat-transfer coefficient based on the log mean temperature driving force. The fluid properties are evaluated at the bulk mean temperature. Metzner and Friend (M3) also give equations for turbulent heat transfer.

4.12C. Natural Convection

Acrivos (A1, S3) gives relationships for natural convection heat transfer to power-law fluids from various geometries of surfaces such as spheres, cylinders, and plates.

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