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Thermodynamics and Propulsion

16.3 Steady-State One-Dimensional Conduction

Figure 16.3: One-dimensional heat conduction
Image fig8OneDimHeatConduction_web

For one-dimensional heat conduction (temperature depending on one variable only), we can devise a basic description of the process. The first law in control volume form (steady flow energy equation) with no shaft work and no mass flow reduces to the statement that $ \sum \dot{Q}$ for all surfaces $ = 0$ (no heat transfer on top or bottom of Figure 16.3). From Equation (16.6), the heat transfer rate in at the left (at $ x$ ) is

$\displaystyle \dot{Q}(x) = -k\left(A\frac{dT}{dx}\right)_x.$ (16..9)

The heat transfer rate on the right is

$\displaystyle \dot{Q}(x +dx) = \dot{Q}(x) + \frac{d\dot{Q}}{dx}\biggr\vert _x dx + \dots.$ (16..10)

Using the conditions on the overall heat flow and the expressions in (16.9) and (16.10)

$\displaystyle \dot{Q}(x) - \left(\dot{Q}(x) + \frac{d\dot{Q}}{dx}(x) dx + \dots\right) =0.$ (16..11)

Taking the limit as $ dx$ approaches zero we obtain

$\displaystyle \frac{d\dot{Q}(x)}{dx} =0,$ (16..12)


$\displaystyle \frac{d}{dx}\left(kA\frac{dT}{dx}\right)=0.$ (16..13)

If $ k$ is constant (i.e. if the properties of the bar are independent of temperature), this reduces to

$\displaystyle \frac{d}{dx}\left(A\frac{dT}{dx}\right)=0,$ (16..14)

or (using the chain rule)

$\displaystyle \frac{d^2T}{dx^2} +\left(\frac{1}{A}\frac{dA}{dx}\right)\frac{dT}{dx}=0.$ (16..15)

Equation (16.14) or (16.15) describes the temperature field for quasi-one-dimensional steady state (no time dependence) heat transfer. We now apply this to an example.

16.3.1 Example: Heat transfer through a plane slab

Figure 16.4: Temperature boundary conditions for a slab
Image fig8OneDimPlaneSlab_web

For this configuration (Figure 16.4), the area is not a function of $ x$ , i.e. $ A =\textrm{ constant}$ . Equation (16.15) thus becomes

$\displaystyle \frac{d^2T}{dx^2} = 0.$ (16..16)

Equation (16.16) can be integrated immediately to yield

$\displaystyle \frac{dT}{dx} = a$ (16..17)


$\displaystyle T = ax + b.$ (16..18)

Equation (16.18) is an expression for the temperature field where $ a$ and $ b$ are constants of integration. For a second order equation, such as (16.16), we need two boundary conditions to determine $ a$ and $ b$ . One such set of boundary conditions can be the specification of the temperatures at both sides of the slab as shown in Figure 16.4, say $ T(0) = T_1$ ; $ T(L)
= T_2$ .

The condition $ T(0) = T_1$ implies that $ b = T_1$ . The condition $ T_2 = T(L)$ implies that $ T_2 = aL + T_1$ , or $ a =(T_2 - T_1)/L$ . With these expressions for $ a$ and $ b$ the temperature distribution can be written as

$\displaystyle T(x)= T_1 + \left(\frac{T_2-T_1}{L}\right)x.$ (16..19)

This linear variation in temperature is shown in Figure 16.5 for a situation in which $ T_1
> T_2$ .

Figure 16.5: Temperature distribution through a slab
Image fig8SlabTempDistribution_web

The heat flux $ \dot{q}$ is also of interest. This is given by

$\displaystyle \dot{q} = -k \frac{dT}{dx} = -k\frac{(T_2 - T_1)}{L} =\textrm{ constant}.$ (16..20)

Muddy Points

How specific do we need to be about when the one-dimensional assumption is valid? Is it enough to say that $ dA/dx$ is small? (MP 16.2)

Why is the thermal conductivity of light gases such as helium (monoatomic) or hydrogen (diatomic) much higher than heavier gases such as argon (monoatomic) or nitrogen (diatomic)? (MP 16.3)