# One-Dimensional Steady-State Convection and Diffusion

## Exact Solution

 $\frac{d(\rho u\varphi )}{dx}=\frac{d}{dx}\left( \Gamma \frac{d\varphi }{dx} \right)$ (1)

where the velocity, u, is assumed to be known. Both density, ρ, and diffusivity, Г, are assumed to be constants. The continuity equation for this one-dimensional problem is

 $\frac{d(\rho u)}{dx}=0$ (2)

Equation (1) is subject to the following boundary conditions:

 $\varphi =\varphi _{0},\text{ }x=0$ (3)
 $\varphi =\varphi _{L},\text{ }x=L$ (4)

By introducing the following dimensionless variables

 $\Phi =\frac{\varphi -\varphi _{0}}{\varphi _{L}-\varphi _{0}},\text{ }X=\frac{x}{L}$ (5)

the one-dimensional steady-state convection and diffusion problem can be nondimensionalized as

 $\text{Pe}\frac{d\Phi }{dX}=\frac{d^{2}\Phi }{dX^{2}}$ (6)
 $\begin{matrix}{}\\\end{matrix}\Phi =0,\text{ X}=0$ (7)
 $\begin{matrix}{}\\\end{matrix}\Phi =1,\text{ }X=1$ (8)

where

 $\text{Pe}=\frac{\rho uL}{\Gamma }$ (9)

is the Peclet number that reflects the relative level of convection and diffusion. Pe becomes zero for the case of pure diffusion and becomes infinite for the case of pure advection. The exact solution of eqs. (6) - (8) can be obtained as

 $\Phi =\frac{\varphi -\varphi _{0}}{\varphi _{L}-\varphi _{0}}=\frac{\exp (\text{Pe}X)-1}{\exp (\text{Pe})-1}$ (10)

which will be used as a criterion to check the accuracy of various discretization schemes.

## Central Difference Scheme

Integrating the governing equation over the control volume P (shaded area in the figure to the right), one obtains

Control volume for one-dimensional problem.
 $(\rho u\varphi )_{e}-(\rho u\varphi )_{w}=\left( \Gamma \frac{d\varphi }{dx} \right)_{e}-\left( \Gamma \frac{d\varphi }{dx} \right)_{w}$ (11)

The right-hand side of eq. (1) can be obtained by assuming the distribution of $\varphi$ between any two neighboring grid points is piecewise linear, i.e.,

$\left( \Gamma \frac{d\varphi }{dx} \right)_{e}=\Gamma _{e}\frac{\varphi _{E}-\varphi _{P}}{(\delta x)_{e}}$

$\left( \Gamma \frac{d\varphi }{dx} \right)_{w}=\Gamma _{w}\frac{\varphi _{P}-\varphi _{W}}{(\delta x)_{w}}$

where Γe and Γw are the diffusivities at the faces of the control volume. To ensure that the flux of $\varphi$ across the faces of the control volume is continuous, the harmonic mean diffusivity at the faces should be used. To evaluate the left hand side of eq. (1), it is necessary to know the values of $\varphi$ at the faces of the control volume. If the piecewise linear profile of $\varphi$ is chosen, it follows that

$\varphi _{e}=\frac{\varphi _{E}+\varphi _{P}}{2}$

$\varphi _{w}=\frac{\varphi _{P}+\varphi _{W}}{2}$

Therefore, eq. (1) becomes

 $(\rho u)_{e}\frac{\varphi _{E}+\varphi _{P}}{2}-(\rho u)_{w}\frac{\varphi _{P}+\varphi _{W}}{2}=\Gamma _{e}\frac{\varphi _{E}-\varphi _{P}}{(\delta x)_{e}}-\Gamma _{w}\frac{\varphi _{P}-\varphi _{W}}{(\delta x)_{w}}$ (12)

Defining the mass flux and diffusive conductance

 $F=\rho u,\text{ }D=\frac{\Gamma }{\delta x}$ (13)

eq. (2) can be rearranged as

 $a_{P}\varphi _{P}=a_{E}\varphi _{E}+a_{W}\varphi _{W}$ (14)

where

 $a_{E}=D_{e}-\frac{1}{2}F_{e}$ (15)
 $a_{W}=D_{w}+\frac{1}{2}F_{w}$ (16)
 $\begin{matrix}{}\\\end{matrix}a_{P}=a_{W}+a_{E}+(F_{e}-F_{w})$ (17)

This scheme is termed the central difference scheme because the values of $\varphi$ at the faces of the control volume are taken as the averaged value between two grid points. The continuity equation requires that Fe = Fw and therefore, eq. (7) reduces to

aP = aW + aE

To evaluate the performance of the central difference scheme, let us consider the case of a uniform grid, i.e., x)e = (δx)w = δx, for which case eq. (2) can be rearranged as

 $\varphi _{P}=\frac{1}{2}\left[ \left( 1-\frac{\text{Pe}_{\Delta }}{2} \right)\varphi _{E}+\left( 1+\frac{\text{Pe}_{\Delta }}{2} \right)\varphi _{W} \right]$ (18)

where

 $\text{Pe}_{\Delta }=\frac{\rho u\delta x}{\Gamma }=\frac{F}{D}$ (19)

is the Peclet number using grid size as the characteristic length, which is referred to as the grid Peclet number. The grid Pe is a ratio of the strength of convection over diffusion. To ensure stability of the discretization scheme, the value of $\varphi _{P}$ should always fall between $\varphi _{E}$ and $\varphi _{W}$, which requires that the coefficients, $\varphi _{E}$ and $\varphi _{W}$, are positive, i.e.,

 $\left| \text{Pe}_{\Delta } \right|\le 2$ (20)

This is the criterion for stability of the central difference scheme. It can be demonstrated that the central difference becomes unstable if eq. (10) is violated. The fact that the central difference scheme is stable under small grid Peclet number indicates that the central difference scheme is accurate only if the convection is not very significant.

## Upwind Scheme

The central difference scheme assumes that the effects of the values of $\varphi$ at two neighboring grid points on the value of $\varphi$ at the face of the control volume are equal. This assumption is valid only if the effect of diffusion is dominant. If, on the other hand, the convection is dominant, one can expect that the effect of the grid point upwind is more significant than that of the point downwind. If we can assume that the value of $\varphi$ at the face of the control volume is dominated by the value of $\varphi$ at the grid point at the upwind side and that the effect of the value of $\varphi$ at the downwind side can be neglected, the two terms on the left hand side of eq. (4.211) can be expressed as

$(\rho u\varphi )_{e}=\left\{ \begin{matrix} F_{e}\varphi _{P},\text{ }F_{e}>0 \\ F_{e}\varphi _{E},\text{ }F_{e}<0 \\ \end{matrix} \right.$

$(\rho u\varphi )_{w}=\left\{ \begin{matrix} F_{w}\varphi _{W},\text{ }F_{w}>0 \\ F_{w}\varphi _{P},\text{ }F_{w}<0 \\ \end{matrix} \right.$

The above two equations can be expressed in the following compact form:

$(\rho u\varphi )_{e}=\varphi _{P}\left[\!\left[ F_{e},0 \right]\!\right]-\varphi _{E}\left[\!\left[ -F_{e},0 \right]\!\right]$

$(\rho u\varphi )_{w}=\varphi _{W}\left[\!\left[ F_{w},0 \right]\!\right]-\varphi _{P}\left[\!\left[ -F_{w},0 \right]\!\right]$

where the operator $\left[\!\left[ A,B \right]\!\right]$ denotes the greater of A and B (Patankar, 1980). Substituting the above expression into the left hand side of eq. (4.211) and using central difference for the right hand side of eq. (4.211), the discretized equation becomes

 $a_{P}\varphi _{P}=a_{E}\varphi _{E}+a_{W}\varphi _{W}$ (21)

where

 $a_{E}=D_{e}+\left[\!\left[ -F_{e},0 \right]\!\right]$ (22)
 $a_{W}=D_{w}+\left[\!\left[ F_{w},0 \right]\!\right]$ (23)
 $\begin{matrix}{}\\\end{matrix}a_{P}=a_{W}+a_{E}+(F_{e}-F_{w})$ (24)

The above scheme is referred to as the upwind scheme because the value of $\varphi$ at the grid point on the upwind side was used as the value of $\varphi$ at the face of the control volume to discretize the convection term. The upwind scheme ensures that the coefficients in eq. (4.221) are always positive so that a physically unrealistic solution can be avoided.

## Hybrid Scheme

The upwind scheme uses the value of $\varphi$ from the grid point at the upwind side as the value of $\varphi$ at the face of the control volume regardless of the grid Peclet number. While this treatment can yield accurate results for cases with high Peclet number, the result will not be accurate for cases where the grid Peclet number is near zero; for which cases the central difference scheme can produce better results. Spalding (1972) proposed a hybrid scheme that uses the central difference scheme when $\left| \text{Pe}_{\Delta } \right|\le 2$ and the upwind scheme when $\left| \text{Pe}_{\Delta } \right|>2$.

To observe the difference between the central difference and upwind schemes, the coefficient for the east neighboring grid point, eqs. (4.215) and (4.222), can be rewritten as

 $a_{E}/D_{e}=1-\frac{1}{2}\text{Pe}_{\Delta e},\text{ Central difference scheme}$ (25)
 $a_{E}/D_{e}=1+\left[\!\left[ -\text{Pe}_{\Delta e},0 \right]\!\right],\text{ Upwind scheme}$ (26)

The hybrid scheme can then be expressed as

$a_{E}/D_{e}=\left\{ \begin{matrix} -\text{Pe}_{\Delta e}\text{ Pe}_{\Delta e}<-2 \\ 1-\frac{1}{2}\text{Pe}_{\Delta e}\text{ }-\text{2}\le \text{ Pe}_{\Delta e}\le 2 \\ 0\text{ Pe}_{\Delta e}>2 \\ \end{matrix} \right.$

which can be rewritten in the following compact form

 $a_{E}/D_{e}=\left[\!\left[ -\text{Pe}_{\Delta e},1-\frac{1}{2}\text{Pe}_{\Delta e},0 \right]\!\right]$ (27)

The coefficient for the west neighbor grid point can be obtained using a similar approach.

 $a_{W}/D_{w}=\left[\!\left[ \text{Pe}_{\Delta w},1+\frac{1}{2}\text{Pe}_{\Delta w},0 \right]\!\right]$ (28)

The above hybrid scheme combines the advantages of the central difference and upwind schemes to yield better results for cases where $\left| \text{Pe}_{\Delta } \right|\to \infty$ or $\left| \text{Pe}_{\Delta } \right|\sim 0$. However, there is still room for improvement of the solution when $\left| \text{Pe}_{\Delta } \right|$ is near 2 (see Problem 4.23).

## Exponential and Power Law Schemes

Since the exact solution of eq. (4.201) exists, one can reasonably expect that an accurate scheme can be derived if the result of the exact solution, eq. (4.210), is utilized. Equation (4.201) can be rewritten as

 $\frac{d}{dx}\left( \rho u\varphi -\Gamma \frac{d\varphi }{dx} \right)=0$ (29)

Defining the total flux of $\varphi$ due to convection and diffusion

 $J=\rho u\varphi -\Gamma \frac{d\varphi }{dx}$ (30)

eq. (4.229) becomes

 $\frac{dJ}{dx}=0$ (31)

Integrating eq. (4.231) over the control volume P (shaded area in Fig. 4.17), yields

 $\begin{matrix}{}\\\end{matrix}J_{e}=J_{w}$ (32)

Instead of assuming piecewise linear distribution of $\varphi$ as with central difference scheme or assuming $\varphi$ at the face of the control volume is equal to the value of $\varphi$ at the grid point on the upwind side in the upwind scheme, the distribution of $\varphi$ between grid points can be taken as that obtained from the exact solution, eq. (4.210). Applying eq. (4.210) between grid points E and P, we have

 $\frac{\varphi (x)-\varphi _{P}}{\varphi _{E}-\varphi _{P}}=\frac{\exp [\text{Pe}_{\Delta \text{e}}(x-x_{P})/(\delta x)_{e}]-1}{\exp (\text{Pe}_{\Delta \text{e}})-1}$ (33)

Substituting eq. (4.233) into eq. (4.230) and evaluating the result at x = xe, the total flux of $\varphi$ at the face of control volume becomes

 $J_{e}=F_{e}\left[ \varphi _{P}+\frac{\varphi _{P}-\varphi _{E}}{\exp (\text{Pe}_{\Delta e})-1} \right]$ (34)

Similarly, the total flux at the west face of the control volume is

 $J_{w}=F_{w}\left[ \varphi _{W}+\frac{\varphi _{W}-\varphi _{P}}{\exp (\text{Pe}_{\Delta w})-1} \right]$ (35)

Substituting eqs. (4.234) and (4.235) into eq. (4.232) and rearranging the resulting equation yields

 $a_{P}\varphi _{P}=a_{E}\varphi _{E}+a_{W}\varphi _{W}$ (36)

where

 $a_{E}=\frac{F_{e}}{\exp (\text{Pe}_{\Delta e})-1}$ (37)
 $a_{W}=\frac{F_{w}\exp (\text{Pe}_{\Delta w})}{\exp (\text{Pe}_{\Delta w})-1}$ (38)
 $\begin{matrix}{}\\\end{matrix}a_{P}=a_{W}+a_{E}+(F_{e}-F_{w})$ (39)

Equations (4.237) and (4.238) can be rewritten in a format similar to that of eqs. (4.225) – (4.228), i.e.,

 $a_{E}/D_{e}=\frac{\text{Pe}_{\Delta e}}{\exp (\text{Pe}_{\Delta e})-1}$ (40)
 $a_{W}/D_{w}=\frac{\text{Pe}_{\Delta w}\exp (\text{Pe}_{\Delta w})}{\exp (\text{Pe}_{\Delta w})-1}$ (41)

The comparison of aE / De for different schemes is shown in Fig. 1. It can be seen that the hybrid scheme can be viewed as an envelope of the exponential scheme. The hybrid scheme is a good approximation if the absolute value of the grid Peclet number is either very large or near zero.

Figure 1: Comparison among different schemes.

While the exponential scheme is accurate, the computational time is much longer than for the central difference, upwind or hybrid schemes. Patankar (1981) proposed a power law scheme that has almost the same accuracy as the exponential scheme but a substantially shorter computational time. The coefficient of the neighbor grid point on the east side can be obtained by

$a_{E}/D_{e}=\left\{ \begin{matrix} -\text{Pe}_{\Delta e}\text{ Pe}_{\Delta e}<-10 \\ (1+0.1\text{Pe}_{\Delta e})^{5}-\text{Pe}_{\Delta e}\text{ }-1\text{0}\le \text{Pe}_{\Delta e}<0 \\ (1-0.1\text{Pe}_{\Delta e})^{5}\text{ 0}\le \text{Pe}_{\Delta e}\le 10 \\ 0\text{ Pe}_{\Delta e}>10 \\ \end{matrix} \right.$

which can be rewritten in the following compact form

 $a_{E}/D_{e}=\left[\!\left[ 0,\left( 1-0.1\left| \text{Pe}_{\Delta e} \right| \right)^{5} \right]\!\right]+\left[\!\left[ 0,-\text{Pe}_{\Delta e} \right]\!\right]$ (42)

## A Generalized Expression of Discretization Schemes

The above discretization schemes can be expressed in a single generalized form. The total flux J at the interface between two grid points that were defined in eq. (4.230) can be used to define:

 $J^{*}=\frac{J}{\Gamma /\delta x}=\text{Pe}_{\Delta }\varphi -\frac{d\varphi }{d(x/\delta x)}$ (43)

which relates to the values of $\varphi$ at grid points i and i+1 (see Fig. 4.19). The first term on the right side of eq. (4.243) will be related to some weighted average of $\varphi _{i}$ and $\varphi _{i+1}$, and the second term will be related to the difference between $\varphi _{i}$ and $\varphi _{i+1}$. Thus, one can express J * the total flux as (Patankar, 1980)

 $J^{*}=B(\text{Pe}_{\Delta })\varphi _{i}-A(\text{Pe}_{\Delta })\varphi _{i+1}$ (44)

where A and B are dimensionless coefficients that are functions of the grid Peclet number. If the field of $\varphi$ is uniform, we will have $d\varphi /dx=0$ and eq. (4.243) becomes

 $J^{*}=\text{Pe}_{\Delta }\varphi _{i}=\text{Pe}_{\Delta }\varphi _{i+1}$ (45)

Comparing eqs. (4.244) and (4.245) yields

 $\begin{matrix}{}\\\end{matrix}B(\text{Pe}_{\Delta })-A(\text{Pe}_{\Delta })=\text{Pe}_{\Delta }$ (46)
Figure 1: Total flux at the interface between grid points i and i+1 .

For the grid system shown in Fig. 1, if we reconsider the problem in a reversed coordinate system x' (x' = − x), the grid Peclet number will become − PeΔ and J * becomes

 $J^{*}=B(-\text{Pe}_{\Delta })\varphi _{i+1}-A(-\text{Pe}_{\Delta })\varphi _{i}$ (47)

The symmetric properties of A and B can be obtained by comparing eqs. (4.244) and (4.247), i.e.,

 $\begin{matrix}{}\\\end{matrix}A(-\text{Pe}_{\Delta })=B(\text{Pe}_{\Delta })$ (48)
 $\begin{matrix}{}\\\end{matrix}B(-\text{Pe}_{\Delta })=A(\text{Pe}_{\Delta })$ (49)

For the exponential schemes discussed above, one can obtain J * from eq(4.234)or (4.235), i.e.,

\begin{align} & J^{*}=\text{Pe}_{\Delta }\left[ \varphi _{i}+\frac{\varphi _{i}-\varphi _{i+1}}{\exp (\text{Pe}_{\Delta })-1} \right] \\ & =\frac{\exp (\text{Pe}_{\Delta })\text{Pe}_{\Delta }}{\exp (\text{Pe}_{\Delta })-1}\varphi _{i}-\frac{\text{Pe}_{\Delta }}{\exp (\text{Pe}_{\Delta })-1}\varphi _{i+1} \\ \end{align}

Comparing the above expression with eq. (4.244), one obtains

$B=\frac{\exp (\text{Pe}_{\Delta })\text{Pe}_{\Delta }}{\exp (\text{Pe}_{\Delta })-1},\text{ }A=\frac{\text{Pe}_{\Delta }}{\exp (\text{Pe}_{\Delta })-1}$

It can be verified that the above A and B satisfy eqs. (4.246), and (4.248) – (4.249). The implication of the above properties of A and B is that if the function A(PeΔ) for the case that PeΔ > 0 is known, the expressions of A and B for all PeΔ can be obtained. For example, if PeΔ < 0, eq. (4.246) can be used to obtain

A(PeΔ) = B(PeΔ) − PeΔ

Substituting eq. (4.248) into the above equation yields

A(PeΔ) = A( − PeΔ) − PeΔ

Considering $-\text{Pe}_{\Delta }=\left| \text{Pe}_{\Delta } \right|$ for the case that PeΔ < 0, the above expression can be rewritten as

$A(\text{Pe}_{\Delta })=A(\left| \text{Pe}_{\Delta } \right|)+\left| \text{Pe}_{\Delta } \right|\text{ for Pe}_{\Delta }<0$

Since $A(\text{Pe}_{\Delta })=A(\left| \text{Pe}_{\Delta } \right|)\text{ for Pe}_{\Delta }>0$ the following expression for A under any grid Peclet number can be expressed as

 $A(\text{Pe}_{\Delta })=A(\left| \text{Pe}_{\Delta } \right|)+\left[\!\left[ -\text{Pe}_{\Delta },0 \right]\!\right]$ (50)

Similarly, the expression of B for any grid Peclet number can be expressed as (see Problem 4.24).

 $B(\text{Pe}_{\Delta })=A(\left| \text{Pe}_{\Delta } \right|)+\left[\!\left[ \text{Pe}_{\Delta },0 \right]\!\right]$ (51)

Therefore, different discretization schemes for the convection-diffusion terms can be characterized by different $A(\left| \text{Pe}_{\Delta } \right|)$ . To derive the generalized formula for different discretization schemes, let us begin from eq. (4.232), i.e.,

 $J_{e}^{*}\text{D}_{e}=J_{w}^{*}\text{D}_{w}$ (52)

The total fluxes at the faces of the control volumes can be obtained from eq. (4.244), i.e.,

 $J_{e}^{*}=B(\text{Pe}_{\Delta e})\varphi _{P}-A(\text{Pe}_{\Delta e})\varphi _{E}$ (53)
 $J_{w}^{*}=B(\text{Pe}_{\Delta w})\varphi _{W}-A(\text{Pe}_{\Delta w})\varphi _{P}$ (54)

Substituting the above expressions into eq. (4.252) and rearranging the resulting equation yields

$\left[ D_{e}B(\text{Pe}_{\Delta e})+D_{w}A(\text{Pe}_{\Delta w}) \right]\varphi _{P}=D_{e}A(\text{Pe}_{\Delta e})\varphi _{E}+D_{w}B(\text{Pe}_{\Delta w})\varphi _{W}$

which can be rearranged as

 $a_{P}\varphi _{P}=a_{E}\varphi _{E}+a_{W}\varphi _{W}$ (55)

where

 $a_{E}=D_{e}A(\text{Pe}_{\Delta e})=D_{e}\left\{ A(\left| \text{Pe}_{\Delta e} \right|)+\left[\!\left[ -\text{Pe}_{\Delta e},0 \right]\!\right] \right\}$ (56)
 $a_{W}=D_{w}B(\text{Pe}_{\Delta w})=D_{w}\left\{ A(\left| \text{Pe}_{\Delta w} \right|)+\left[\!\left[ \text{Pe}_{\Delta w},0 \right]\!\right] \right\}$ (57)

 aP = aE + aW + (Fe − Fw) (58)
) for different schemes.

In arriving at eqs. (4.256) and (4.257), A and B were obtained from eqs. (4.250) and (4.251). At this point, it is apparent that different discretization schemes can be characterized by different expressions for A(|PeΔ|). By comparing eqs. (4.256) and (4.257) with different expressions of aE and aW for different schemes, the corresponding A(|PeΔ|) for different schemes can be summarized in Table 1 and plotted in Fig. 2. It should be noted that the difference between the power law and exponential scheme is exaggerated for clear presentation. The generalized formula represented by eqs. (4.255) – (4.258) will be very helpful to develop a generalized computer code for all schemes. A special module or subroutine can be written for different schemes.

Table 1 Summary of A(|PeΔ|) for different schemes

 Scheme A(|PeΔ|) Central difference $1-0.5\left| \text{Pe}_{\Delta } \right|$ Upwind 1 Hybrid $\left[\!\left[ 0,1-0.5\left| \text{Pe}_{\Delta } \right| \right]\!\right]$ Exponential $\left| \text{Pe}_{\Delta } \right|/[\exp (\left| \text{Pe}_{\Delta } \right|)-1]$ Power Law $\left[\!\left[ 0,(1-0.1\left| \text{Pe}_{\Delta } \right|)^{5} \right]\!\right]$