An Explicit Expression for the Break-Even Radius of Insulation on a Cylinder in Cross-Flow

Abstract

An explicit expression was developed for approximating the break-even radius of insulation on a cylinder in cross flow with forced convective heat transfer where the heat transfer rate equals the heat transfer rate from a bare cylinder.  Similar assumptions for the classical form of the critical radius of insulation were invoked.  However, the assumption that the heat transfer coefficient is independent of the radius of insulation was relaxed.  The explicit expression derived here uses an algebraic approximation for the logarithmic-mean radius of insulation to avoid the computational inconvenience of solving the implicit nonlinear function of the break-even radius of insulation that is sensitive to initial guesses for the root.  The results of the approximation agree with the implicit expression within 1% and 5% for practical insulation thicknesses up to 15 and 25 times the radius of the bare cylinder, respectively.  An analytical expression for the critical radius of insulation where the convective heat transfer coefficient varies with radius of insulation is also included.

Introduction

It is well known that wrapping a cylinder in a uniform layer of insulation may enhance the rate of heat transfer between the cylinder and an external fluid by increasing the external surface area [1, 2].  The heat transfer rate to or from an insulated cylinder reaches a maximum at the critical radius of insulation.  At the break-even radius of insulation, the effect of additional surface area balances with the effect of additional conduction resistance to give the same heat transfer rate as a bare cylinder [2].  Increasing the radius of insulation beyond the break-even value should lower the heat transfer rate between the cylinder and a surrounding fluid.

Information about the critical radius of insulation is useful when the goal is to maximize the heat transfer rate.  However, fin type technology for enhancing the heat transfer area may be preferred in this situation.  The critical radius of insulation has limited usefulness for determining the amount of insulation required to reduce the rate of heat transfer.  Conversely, the break-even radius is essential to design and economic analysis of insulating cylindrical systems, such as pipes.  A classical analytical solution for the critical radius of insulation is widely known for the case of constant cylinder surface temperature and heat transfer coefficient that is independent of the radius of insulation.  However, no simple, analytical solution for the break-even radius of insulation is available due to the nonlinear nature of the heat transfer rate as a function of the radius of insulation [2].

Other investigators [3-5] have considered relaxing the classical assumptions on the critical radius of insulation, including the effects of multidimensional heat transfer, variable heat transfer coefficients, and radiation.  This work describes a new explicit approximation for the break-even radius of insulation under conditions of forced, cross-flow convection heat transfer that includes the effect of varying the radius of insulation.  A new analytical expression for the critical radius of insulation was also derived for these conditions.  This work is limited to one dimensional heat transfer in the radial direction of a cylindrical geometry by conduction through a uniform insulation layer and convection between the outer surface and a surrounding fluid.

Theory

The cross section of an insulated cylinder is illustrated in Figure 1.  The steady-state, convective heat transfer rate per unit length from a bare cylinder is described by Newton’s law of cooling:



where q’_i is the convective heat transfer rate per unit length of cylinder, r_i is the radius of the bare cylinder, h_i is the convective heat transfer coefficient for the bare surface without insulation, T_i is the bare cylinder surface temperature, and T_f is the surrounding bulk fluid temperature.



Figure 1.     Cross section of a cylinder with radius r_i, radius of insulation r_o, bare surface temperature, T_i, surrounded by fluid with bulk temperature, T_f.

A layer of insulation on a cylinder has the combined effects of increasing the resistance to heat transfer by conduction through the insulation while increasing the outside surface area that enhances convective heat transfer:

where q’_o is the heat transfer rate per unit length of cylinder, T_i is the temperature at the cylinder-insulation interface, k is the thermal conductivity of insulation, r_o is the outside radius of insulation, and h_o is the outside convective heat transfer coefficient for the insulated cylinder.  Equation (2) assumes constant properties and heat conduction only in the radial direction.

The ratio of the heat transfer rates described by Equations (1) and (2) for the cylinder with and without insulation gives:

where the Biot number is defined in terms of the bare cylinder conditions:

Baehr and Stephan [6] recommend the correlation of Zukauskas and Zingzda [7] for predicting the forced convection heat transfer coefficient of a cylinder with cross flow:

where k_f, Re, and Pr are the fluid thermal conductivity, Reynolds number and Prandtl number, respectively.  The physical properties of the fluid required in Equation (5) are evaluated at the bulk fluid temperature, T_f.  The constant, c, and exponents, m and n, depend on the Reynolds number, which is defined in terms of the cylinder radius:

where u_f and v_f are the bulk fluid velocity and kinematic viscosity, respectively.  The ratio of heat transfer coefficients for the insulated cylinder relative to the bare cylinder using Equations (5) and (6) gives a power law function of the radii:

The Reynolds number exponent m has values listed in Table 1 for conditions ranging from laminar to turbulent flow.

The exponent m may be treated as constant when the Reynolds number for forced convection on a bare cylinder is of the same order of magnitude as the Reynolds number for the larger diameter of insulation.

As an example, Equation (8) is plotted in Figure 2 for a range of exponents, m, with Bi = 0.4 to compare the critical and break-even radii.  The plot shows how the heat transfer rate initially increases to a maximum as the radius of insulation increases.  The critical radius in this example for m > 0.4 is approximately twice the radius of the cylinder.  This is followed by a decrease in the heat transfer rate back to a level even with that of the bare cylinder, marked by the horizontal line.  The break-even radius for this example is at least five times the cylinder radius for m > 0.4.  The benefits of adding a layer of insulation to inhibit heat transfer are only realized at an insulation thickness in excess of the break-even radius.  However, for laminar flow conditions with m = 0.4, the critical and break-even radii of this example practically equal the bare cylinder radius, indicating that any insulation added lowers the heat transfer rate between the cylinder and the surrounding fluid.

Figure 2.     Dimensionless heat transfer rate from an insulated cylinder with Bi = 0.4 for the full range of Reynolds number exponents, m.

Critical Radius of Insulation

where r_c is the critical radius of insulation.

Substitution from Equation (8) into Equation (9) gives a result for the relative critical radius of insulation with convection in cross-flow when the classical assumption of radius independence is relaxed:

Equation (11) reveals limits on the Biot number for the case of cross flow.  In the limit Bi ® 0, r_c ® ¥.  In the limit Bi ® m, r_c ® r_i.  Note that the bare cylinder radius cannot exceed the critical radius of insulation, such that 0 < Bi £ m.

Break-even Radius of Insulation

The break-even radius occurs when the heat transfer with and without insulation are equal.  In this case Equation (8) simplifies to:

where r_b is the break-even radius of insulation.  Equation (12) does not have an analytical solution for r_b.  An iterative, “trial and error” solution method is required to obtain the root.  The nonlinear nature of Equation (12) is such that convergence to the correct root is sensitive to the initial guess needed to initiate the iterative solution method.  This computational inconvenience is overcome for practical applications by an approximation for the term involving the natural logarithm.

Method of Solution

Equation (12) is rearranged using the following substitution:

The left-hand-side of Equation (14) is the logarithmic mean of 1 and R^-1.

Underwood [8] derived an approximation for the logarithmic mean expression with application to the log-mean temperature difference commonly used for countercurrent heat exchanger design.  Underwood’s approximations for the log-mean has found application in simplifying the analysis of gas membrane permeation [9, 10] and staged separation processes [11].  Underwood’s [8] approximation for the log-mean in Equation (14) gives:

Underwood [8] proposed n = 1/3 for the practical range of temperatures in heat exchanger analysis.  Chen [12] recommended n = 0.3275 to extend the range of usefulness in heat exchanger design.  The approximation for the log-mean in Equation (15) transforms Equation (12) into an explicit function for approximating the break-even radius of insulation:

Results and Discussion

Implicit solutions for the break-even radii were determined by the iterative conjugate gradient method using the computational software MathCADÒ.  Other iterative methods available in the computation software, including the Levenberg-Marquardt and Newton’s method were also used.  It was found that each iterative solution method was sensitive to the initial guess for the root.  For higher values of Bi near the upper limit m, an initial guess within +5% of the root was required for solution convergence.

The classical assumption that the heat transfer coefficient is independent of the radius of insulation was shown in Figure 2 to give upper bounds for the critical and break-even insulation radii.  Results for the case where m = 1 corresponding to the classical assumptions are included in the following discussion for purposes of comparison.  The results for the critical and break-even radii of insulation over the range of m listed Table I reveal the affect the Reynolds number exponent m on the solution.  The critical and break-even radii increase as m increases from 0.4 for laminar flow to 0.8 for turbulent flow.

Equation (16) is plotted against the bare cylinder Biot number in Figure 3 in order to compare the explicit approximation to the implicit result obtained from the trial-and-error solution to Equation (12).  Recall that the upper limit on Bi is the Reynolds number exponent m.  The explicit approximation for the break-even radius agrees with the implicit solution within 1% for r_b/r_i <15 and 5% for r_b/r_i < 25 for all values of m.  Chen’s [12] recommendation for n = 0.3275 extends the range for 1% agreement to r_b/r_i <20 and 5% agreement to r_b/r_i < 40 at lower values for Bi.  However, the results using Underwood’s [8] recommendation for n = 1/3 gives slightly better agreement when Bi > 0.5.  In nearly all cases, the explicit approximations are greater than the implicit solutions providing conservative results when designing insulation for a cylinder.  The range of agreement between the explicit approximation and the implicit function is well within most practical applications of insulating a cylinder to reduce the rate of heat transfer.  The trends in Figure 3 also indicate that the break-even radius of insulation increases as the convective heat transfer coefficient for the bare cylinder decreases for constant m and r_i.  Thus, insulating a cylinder is more important for conditions involving larger heat transfer coefficients where the overall resistance to heat transfer between the fluid and cylinder is reduced.

Figure 3.     Comparison of the explicit and implicit break-even radius results with Underwood’s [8] approximation for the full range of Reynolds number exponents, m.

The break even radius of insulation is compared with the critical radius of insulation in Figure 4.  The results of Figure 4 show that the break-even radius of insulation increases exponentially relative to the critical radius and can exceed the critical radius by one or more orders of magnitude.  Thus, the critical radius of insulation has limited value for decision making when insulating cylinders to lower the heat transfer rate involving conduction and convection.

Figure 4.     Break-even vs. critical radius of insulation for the full range of Reynolds number exponents, m.

Conclusions

Simple, explicit expressions were derived for the critical and break-even radii of insulation on a cylinder with cross-flow, forced convection heat transfer.  The explicit function for the break-even radius of insulation was derived using Underwood’s [8] approximation for the logarithmic mean applied to the dimensionless radius of insulation.  The explicit function for the break-even radius of insulation eliminates the computational inconvenience of solving the nonlinear function that requires a good initial guess and an iterative solution method.  The solution provides practical results for typical applications of insulation where the thickness of insulation is less than 25 times the radius of the cylinder.  The explicit approximation for the break-even radius is limited by the assumptions of one-dimensional heat transfer by conduction and convection and constant properties.  Nevertheless, the results presented here are useful as a guide in economic analysis of insulating cylindrical systems, such as pipes.

Nomenclature

Bi = Biot number or dimensionless radius of insulation

c = constant coefficient in correlation for the convective heat transfer coefficient.

h = convective heat transfer coefficient, W/m²·K

k = thermal conductivity of insulation, W/m·K

Pr = Prandtl number for the bulk fluid

q’ = heat transfer rate per unit length, W/m

r = radius, m

R = dimensionless radius of insulation

Re = Reynolds number for the fluid in cross-flow over a cylinder

T = temperature, K

u = velocity, m/s

v = kinematic viscosity, m²/s

Subscripts/Superscripts

b = break-even conditions

c = critical conditions

f = fluid

i = reference to radius of bare surface

m = heat transfer coefficient exponent

n = Underwood exponent

References

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