Comparison between Approximate and Exact Analytical Heat Conduction Rates in Struts of Rectangular Profile

Struts (or extended straight fins of rectangular profile) are extensively used in a multitude of heat exchange devices for the purpose of increasing the heat transfer rate between two opposed planar walls that are heated and an incoming fluid flowing at a low temperature between the planar walls [1].

From heat conduction theory, a strut transfers heat by longitudinal conduction within its defining boundaries, whereas the two exposed surfaces transfer heat by transverse convection to the fluid in which it is immersed. The heat conduction analysis in struts has been traditionally carried out through fin theory using the quasi one-dimensional heat conduction equation [1]. In general, the beneficial features of struts revolve around low mean convection coefficients, as is often the case for natural convection with air and gases or even in situations of forced convection with liquids moving at low velocities.

The formal mathematical formulation of a strut (extended straight fin) with rectangular profile must begin with the two-dimensional heat conduction equation along with the proper boundary conductions [2]. The heat conduction problem has two figures-of-merit: one is the thickness t) and the other is the slenderness ratio $S=\frac{L}{t}$.

The dimensionality of straight fins of rectangular profile has been investigated by Lau and Tan [3], who performed a comparative numerical study of the heat transfer rates utilizing the formal two-dimensional heat conduction equation and the simplified quasi one-dimensional heat conduction equation. The authors used a combination of  $Bi_t=\frac{\bar{h}t}{k}$  (based on the half-thickness t) ranging between 0.01 and 10 and $S=\frac{L}{t}$ taking values from 1 up to 100 (resembling an infinite straight fin). In the publication, the relative errors between the two-dimensional and quasi one-dimensional heat transfer rates were reported in graphical form. Setting a relative error at ε=5 %, it was found that the simplified quasi one-dimensional model rests on two components: 1) the transverse Biot number $Bi_t$≤0.5 and 2) the slenderness ratio S must be of order of 1 or greater. Subsequently, the relative error ascends to ε=10 % when $Bi_t$=1, irrespective of the value of S. Overall, the pattern displayed by the relative errors shows that ε raises gradually with increments in $Bi_t$, whereas ε remains invariable to enlargements in S.

The primary objective of the present study is to utilize the physics-based two-dimensional heat conduction equation. The secondary objective is to develop a systematic procedure to establish a convenient criterion to assess the validity of the approximate, one-term series solution derived from the exact, infinite series solution of the two-dimensional heat conduction equation. The suited criterion shall be able to articulate the geometric, hydrodynamic and thermal quantities affecting the strut of rectangular profile. At the end, the transverse Biot number $Bi_t$  united with the slenderness ratio S establish the definitive borderline between the two-dimensional and the quasi one-dimensional heat transfer rates, along with the intrinsic relative errors for suitable pairs of $Bi_t$ and S.

A literature review brings forth publications that gravitate around the exact two-dimensional heat conduction formulation for the straight fin of rectangular profile. Hence, the relevant publications are those by Sparrow and Hennecke [4], Look [5], Huang and Shah [6], Juca and Prata [7] and Singh et al. [8], which surprisingly did not touch upon the expedient approximate heat conduction formulations.

The method of separation of variables engages the product of two functions

$\theta \,(x,\,y)=X\ (x)\,Y\ (y)$ (3)

and transforms the elliptic partial differential Eq. (1) into a system of two ordinary differential equations of second order [2]. One ordinary differential equation of second order associated with the y-direction is

$\frac{{{d}^{2}}Y}{d{{y}^{2}}}+{{\lambda }^{2}}\,Y=0$  (4)

subject to two homogeneous boundary conditions

$\frac{d\,Y(0)}{dy}=0$  (5a)

$-\,k\frac{\partial Y(t)}{\partial y}\ =\overline{h}\,Y(t)\ $  (5b)

coming from Eqns. (2c) and (2d). Correspondingly, the set of Eqns. (4), (5a) and (5b) defines an eigenvalue problem in the y-direction. Another ordinary differential equation of second order connected to the x-direction is

$\frac{{{d}^{2}}X}{d{{x}^{2}}}-{{\lambda }^{2}}\,X=0$ (6)

subject to a homogeneous boundary condition

$\frac{d\,X\,(L)}{dx}=0$ (7a)

and a non-homogeneous boundary condition

$X(0)\ =\ {{\theta }_{b}}$ (7b)

coming from Eqns. (2a) and (2b).

Conceptually, the set of Eqns. (6) and (7a) and (7b) defines a boundary value problem (BVP) in the x-direction.

On one hand, the general solution of eq. (6) is

${{Y}_{n}}(y)={{C}_{n}}\cos \ {{\lambda }_{n}}y$  (8)

where C_n is an arbitrary constant and l_n are the roots of the transcendental equation

$k\,{{\lambda }_{n}}\left( \sin \,{{\lambda }_{n}}t \right)=\overline{h\,}(\cos \,\,{{\lambda }_{n}}t)$ (9)

On the other hand, the general solution of Eq. (4) is expressed in terms of exponential functions

${{X}_{n}}(x)={{B}_{n}}\ \left[ \,{{e}^{\,{{\lambda }_{n}}x}}+{{e}^{\,(2L\,-\,\,x)\,{{\lambda }_{n}}}} \right]$ (10)

where B_n is an arbitrary constant.

In compliance with the trio of Eqns. (3), (8) and (10), the product solution $\theta$(x, y) is next expressible by the infinite series

$\theta \,(x,\,y)=\sum\limits_{n=1}^{\infty }{{{a}_{n}}\,}\left[ \,e{{\,}^{{{\lambda }_{n}}x}}+e{{\,}^{(2L\,-\,\,x)\,{{\lambda }_{n}}}} \right]\,\ \cos \,{{\lambda }_{n}}y$ (11)

where the new coefficients are

$a_n=B_nC_n$. (12)

Substituting the boundary condition for the base temperature excess $\theta _b$ stated in Eq. (7b) into Eq. (11), delivers the infinite series

$θ_b=\sum_{n=1}^∞a_n(1+e^{2Lλ_n})cosλ_ny$ (13)

The mathematical meaning of this equation signifies that the function f (y) =$\theta_b$  needs to be represented in terms of a Fourier series, where the unknown coefficients  are Fourier series coefficients $a_n(1+e^{2Lλ_n})$. These Fourier series coefficients $a_n(1+e^{2Lλ_n})$ are obtained from the expansion of the cosine Fourier series in Eq. (13) in a series of orthogonal functions. On account of this, the appropriate formula found in Arpaci [2] is

${{a}_{n}}\,\left( \,1+e{{\,}^{2L{{\lambda }_{n}}}} \right)=\frac{\int_{0}^{1}{{{\theta }_{b}}\ \cos \,{{\lambda }_{n}}y\,dy}}{\int_{0}^{1}{\cos {{\,}^{2}}{{\lambda }_{n}}y\,dy}}$  (14)

Performing the integrations while skipping the algebra, the coefficients $a_n(1+e^{2Lλ_n})$ are given at the end by the equation

${{a}_{n}}\,\left( \,1+{{e}^{\,2L{{\lambda }_{n}}}} \right)=2\,{{\theta }_{b}}\left( \frac{\ \sin \,{{\lambda }_{n}}t}{{{\lambda }_{n}}t+\sin \,{{\lambda }_{n}}t\,\ \cos \,{{\lambda }_{n}}t} \right)$ (15)

Introducing Eq. (15) into Eq. (12), delivers the infinite series 

$\theta \,(x,\,y)=2\theta_b\sum\limits_{n=1}^{\infty }{\frac{1}{(1+e^{2Lλ_n})}}(\frac{sin\lambda _nt}{\lambda _nt+sin\lambda _ntcos\lambda _nt})[e^{\lambda _nt}+e^{(2L-x)\lambda _t}]cos\lambda _ny$ (16)

which is the exact, analytical solution of Eq. (1) subject to Eqns. (2a)-(2d).

Returning to Eq. (9), the simultaneous effect of the mean convection coefficient $bar{h}$ and the thermal conductivity k on the roots $\lambda_n$ are adequately encapsulated into a thermo-geometric parameter. That is, knowing that the internal conductive resistance is R_K =$\frac{t}{kA}$  and the external convective resistance is R_C =$\frac{1}{\bar{h}A}$, their ratio sets up the transverse Biot number Bi_t:

$\frac{R_K}{R_C}=\frac{t/kA}{1/\bar{h}A}=\frac{\bar{h}t}{k}=Bi_t$ (17)

where the half-thickness of the strut t acts as the characteristic length.

With the previous background, and resorting to the eigenvalues ${\mu }_{n}$=$\lambda$_nt for compactness, Eq. (9) is reformulated into the dimensionless transcendental equation

${{\mu }_{n}}\tan \,{{\mu }_{n}}=B{{i}_{t}}$ (18)

For a given Bi_t inside the range 0<Bi_t<∞, there is an infinite number of positive eigenvalues ${\mu }_{n}$ that satisfy eq. (18). The first six eigenvalues ${\mu }_{n}$ (n = 1,…, 6) are available in the heat conduction textbooks by Schneider [10], Luikov [11] and Özişik [12]. Advantageously, the entire set of eigenvalues ${\mu }_{n}$= f (Bi_t) can be computed nowadays with explicit equations developed by Milkhailov and Vulchanov [13] and Haji-Sheik and Beck [14].

In view of the foregoing, Eq. (16) can be rewritten in terms of the eigenvalues ${\mu }_{n}$ as follows

$\theta \,(x,\,y)=2\theta_b\sum\limits_{n=1}^{\infty }\frac{[e^{{\mu }_{n}\frac{x}{t}+e^{\frac{1}{t}(2L-x){\mu }_{n}}}]}{(1+e^{2\frac{L}{t}{\mu }_{n}})}(\frac{sin{\mu }_{n}}{{\mu }_{n}+sin{\mu }_{n}cos{\mu }_{n}})cos{\mu }_{n}\frac{y}{t}$ (19)

Actually, Eq. (19) embodies the exact, analytical two-dimensional temperature distribution in the strut of rectangular profile under study. Certainly, this equation is useful in gaining physical insight into the heat conduction features in the strut.

The exact heat transfer rate from the strut of rectangular profile to the nearby fluid is determined from Fourier’s law applied at the left base of the strut x = 0:

$Q_{2-D}=-2kw\int^t_0\frac{\partial\theta(0,y)}{\partial x}dy$ (20)

Next, after introducing the temperature distribution  from Eq. (19), Q is represented by the infinite series

$Q_{2-D}=4kw\theta_b\sum\limits_{n=1}^{\infty }(\frac{sin^2{\mu }_{n}}{{\mu }_{n}+sin{\mu }_{n}cos{\mu }_{n}})tanhS{\mu }_{n}$(21)

where $S=\frac{L}{t}$ stands for the fin slenderness ratio.

In the two Eqns. (19) and (21) for the respective $\theta(x,y)$ and $Q_{2-D}$, the ensuing eigenvalues ${\mu }_{n}$=$f(Bi_t)$ are determined from Eq. (18). 

The approximate mathematical model begins with the mean temperature excess:

$\bar{\\theta}(x)=\frac{1}{t}\int_0^t\theta(x,y)dy$   (27)

where θ(x,y) is the two-dimensional temperature excess in Eq. (19).

A thermodynamic energy balance in a control volume of thickness Δx placed in the strut (extended straight fin) of rectangular profile provides the quasi one-dimensional heat conduction equation

$\frac{d^2\bar{\theta}}{dx^2}-m^2\bar{\theta}=0$ (28)

where $m^2=\frac{\bar{h}}{kt}$  denotes the thermo-geometric parameter with units 1/m² [16]. The boundary conditions for Eq. (28) are Eqns. (2a) and (2b) replacing $\theta$ with $\bar{\theta}$   

$\bar{\theta}=\theta_b$, $x=0$  (29a)

$\frac{d\bar{\theta}}{dx}=0$, $x=L$ (29b)

These boundary conditions are of Dirichlet and null Neumann type.

The general solution of Eq. (28) subject to Eqns. (29a) and (29b) gives the particular solution, which is expressed compactly as

$\frac{\bar{\theta}(x)}{\theta_b}=\frac{cosh[m(L-x)]}{coshmL}$  (30) 

This equation entails to the quasi one-dimensional mean temperature distribution in the strut of rectangular profile. 

The heat transfer rate is calculated with Fourier’s law applied at the base x = 0 of the left part,

${{Q}_{quasi\ \,1-D}}=-\,k(2\,tw)\,\frac{d\overline{\theta \,}(0)}{dx}$  (31)

where the cross-sectional area is A_c=2tw. After introducing Eq. (30) into Eq. (31), the final expression for the heat transfer rate is        

$Q_{quasi 1-D}=2kwθ_b\sqrt{Bi_t}tanh(S\sqrt{Bi_t})$  (32)

Herein, the thermo-geometric parameter m and the transverse Biot number Bit are related by the equality $m^2=\frac{Bi_t}{t^2}$.

The main conclusions that may be drawn from the present study on the determination of heat transfer rates in two-dimensional struts of rectangular profile are listed next:

1) An exact, analytical solution of the two-dimensional heat conduction equation subject to the proper boundary conditions provides the exact benchmark heat transfer rates represented by an infinite series in terms of two controlling parameters: the transversal Biot number Bi_t and the slenderness ratio S.

2) The approximate solution of the quasi one-dimensional heat conduction equation for the straight fin of rectangular profile works well for two-dimensional struts of rectangular profile when the transversal Biot numbers are small of the order of Bi_t=0.01 and 0.1 in conjunction with the slenderness ratios S=1, 5 and 10; all cases manifest relative errors ε of less than –2 %. For Bi_t>0.5 regardless of S, the relative errors increase significantly.

3) Numerical evaluation of the exact infinite series solution of the two-dimensional heat conduction equation is not necessary. It was demonstrated in the present study that the approximate one term series solution display relative errors ε that are less than –5% whenever the transversal Biot number Bi_t≤0.60 is combined with the slenderness ratio S=1. Additionally, the same relative error criterion of –5% applies to situations when the transversal Biot number Bi_t≤0.92 is combined with the slenderness ratios S=5 and 10.