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The purpose of this article is to provide a practical, calculation-ready method to predict fluidized bed expansion using the Richardson–Zaki equation, including how to select the exponent, compute voidage, and convert voidage into expanded bed height for engineering design and troubleshooting.
1. What “bed expansion” means in fluidization design
Fluidized bed expansion is the increase in bed height when a fluid flows upward through a bed of particles and increases the bed voidage. The same solids inventory occupies a larger volume because the void spaces between particles increase. In many liquid–solid and some homogeneous gas–solid regimes, the expanded bed can be approximated as a uniform suspension characterized by a single average voidage and a single expanded height. This approximation is most useful when the bed is not strongly bubbling or slugging and when axial solids concentration gradients are limited.
Note : Richardson–Zaki based expansion predictions are intended for homogeneous expansion conditions. If the bed shows strong bubbling, large clusters, slugging, or a clear dilute splash zone, a single voidage model may not represent the full axial structure.
2. Richardson–Zaki equation and the variables you actually need
The Richardson–Zaki relationship links superficial fluid velocity to bed voidage using a power law. In fluidization practice, it is commonly written using the bed voidage as the key unknown to be solved from an operating velocity.
Richardson–Zaki (velocity–voidage form). U = U_t * ε^n. Where. U = superficial fluid velocity through the empty column cross-section [m/s]. U_t = terminal (free) settling velocity of a single particle in the same fluid [m/s]. ε = bed voidage (volume fraction of fluid in the expanded bed) [-]. n = Richardson–Zaki exponent (hindered settling / expansion index) [-].For expansion prediction, the most common workflow is to choose or calculate U_t and n, compute ε from the operating U, and then compute the expanded bed height from solids conservation. This procedure converts a velocity measurement or design velocity directly into bed height.
2.1 Solve voidage from operating superficial velocity
Voidage from Richardson–Zaki. ε = (U / U_t)^(1/n).
Practical bounds.
0 < ε < 1.
U must be ≤ U_t for ε ≤ 1 in this simple form.2.2 Convert voidage into expanded bed height using solids conservation
If the bed contains a fixed mass of solids and the cross-sectional area is constant, the solids volume in the bed is conserved. Using the solids fraction (1 − ε), the expanded height follows directly from a reference state such as the static packed bed at zero flow.
Solids inventory conservation (same solids mass in the bed). A * H0 * (1 - ε0) = A * H * (1 - ε). Expanded height prediction. H = H0 * (1 - ε0) / (1 - ε). Where. H0 = static bed height at reference condition [m]. ε0 = voidage at reference condition (often packed bed voidage) [-]. H = expanded bed height at velocity U [m]. A = column cross-sectional area [m^2] (cancels out if constant).In most liquid–solid systems, ε0 is often taken as the packed bed voidage of the particles used. In many practical cases with reasonably spherical particles, ε0 is commonly around 0.35 to 0.45 depending on size distribution and packing method. If you have measured H0 and know the solids mass and density, you can also compute ε0 from a packing calculation instead of assuming a value.
| Symbol | Meaning | Typical source in practice | Unit |
|---|---|---|---|
| U | Superficial velocity | Flow rate divided by column area | m/s |
| U_t | Single-particle terminal velocity | Correlation or measurement | m/s |
| ε | Expanded bed voidage | Solved from Richardson–Zaki | - |
| n | Richardson–Zaki exponent | Function of terminal Reynolds number or fit to data | - |
| H0, ε0 | Reference bed height and voidage | Static bed measurement or packing calculation | m, - |
3. Step-by-step calculation procedure for bed expansion prediction
3.1 Step 1. Confirm the bed is actually fluidized
Bed expansion correlations are applied after minimum fluidization is reached. In design and troubleshooting, confirm that the operating U is above the minimum fluidization velocity U_mf for the given particles and fluid. If U is below U_mf, the bed behaves as a fixed bed and the Richardson–Zaki expansion model is not the correct physics. If you do not have U_mf measured, an initial estimate can be obtained from widely used fixed-bed pressure-drop correlations combined with force balance at incipient fluidization.
Note : If the distributor causes channeling or if the bed is non-uniform, a measured pressure drop that is lower than the effective particle weight per area is a warning sign that the bed is not uniformly fluidized, even if the average flow rate suggests U > U_mf.
3.2 Step 2. Determine the terminal settling velocity U_t
Terminal settling velocity is the single-particle velocity at which drag balances buoyant weight. For many engineering calculations, U_t is obtained by solving a drag correlation for a sphere using a force balance. One commonly used approach is to iterate using a drag coefficient correlation expressed in terms of the particle Reynolds number.
Force balance for a single spherical particle at terminal velocity. (π/6) * d_p^3 * (ρ_p - ρ_f) * g = (1/2) * C_D * ρ_f * (π/4) * d_p^2 * U_t^2. Reynolds number based on terminal velocity. Re_t = (ρ_f * U_t * d_p) / μ. Example drag coefficient correlation (Schiller–Naumann form). For Re_t < 1000. C_D = (24/Re_t) * (1 + 0.15 * Re_t^0.687). For Re_t ≥ 1000. C_D ≈ 0.44. Procedure. 1) Guess U_t. 2) Compute Re_t. 3) Compute C_D(Re_t). 4) Compute the U_t implied by the force balance. 5) Repeat until U_t converges.If particle shape is far from spherical, use an effective diameter and a shape-aware drag approach, or directly measure U_t in a settling test. In that case, Richardson–Zaki predictions can still be used, but the exponent and effective properties should be based on the same particle population and fluid conditions.
3.3 Step 3. Select the Richardson–Zaki exponent n
The exponent n is most commonly treated as a function of the terminal Reynolds number Re_t. Two practical approaches are widely used in engineering work. One approach uses piecewise ranges that recover the low-Re and high-Re asymptotes. Another approach uses a smooth continuous equation that transitions across Reynolds number without discontinuities.
Continuous form often used for engineering estimation (Rowe-type form). n = 2 * (2.35 + 0.175 * Re_t^0.75) / (1 + 0.175 * Re_t^0.75). Behavior. As Re_t → 0, n → 4.7. As Re_t becomes large, n approaches about 2.0.A piecewise alternative is often used when you want the high-Re asymptote closer to about 2.4 for turbulent conditions. In practice, the best approach is to treat n as a calibrated parameter when you have any bed expansion data available for the same solids and column, because n can be influenced by particle shape, size distribution, wall effects, and flow regime.
| Method | How n is obtained | Strength | Common engineering use |
|---|---|---|---|
| Continuous Reynolds-based n | Compute Re_t from U_t, then compute n from a smooth equation | No discontinuities at regime boundaries | Design screening and automation |
| Piecewise Reynolds-based n | Select n using Re_t ranges with different formulas per range | Can match regime asymptotes more explicitly | Hand calculations and legacy design bases |
| Fitted n from bed data | Fit log(U/U_t) vs log(ε) slope to estimate n | Highest accuracy for the same system | Commissioning and troubleshooting |
Note : If you have one reliable expansion point (U, H) and know H0 and ε0, you can back-calculate ε and then estimate an effective n for your column. This can be more reliable than a generic n correlation when wall effects or particle irregularity are significant.
3.4 Step 4. Compute ε and then compute the expanded height H
Once U_t and n are available, compute ε = (U/U_t)^(1/n), then compute H = H0 * (1 − ε0)/(1 − ε). This produces a direct bed expansion prediction at the chosen operating velocity. If you need the entire expansion curve, repeat the same steps over a range of U values and tabulate H as a function of U.
4. Worked example calculation for fluidized bed expansion prediction
This example shows the mechanics of converting an operating superficial velocity into voidage and expanded height using a Reynolds-based exponent. The example uses water at room temperature and sand-like particles, which is a common liquid–solid fluidization scenario.
Given. Fluid: water, ρ_f = 998 kg/m^3, μ = 1.0e-3 Pa·s. Solids: ρ_p = 2650 kg/m^3, particle diameter d_p = 500e-6 m. Static bed: H0 = 0.50 m, packed bed voidage ε0 = 0.40. Operating superficial velocity: U = 0.040 m/s. g = 9.80665 m/s^2. Step A. Solve terminal velocity U_t by force balance with a drag correlation. A converged terminal velocity for these inputs is approximately. U_t ≈ 0.0785 m/s. Step B. Compute terminal Reynolds number. Re_t = ρ_f * U_t * d_p / μ. Re_t ≈ 39.2. Step C. Compute exponent n from a continuous Reynolds-based form. n = 2 * (2.35 + 0.175 * Re_t^0.75) / (1 + 0.175 * Re_t^0.75). n ≈ 2.72. Step D. Compute expanded bed voidage. ε = (U / U_t)^(1/n). ε = (0.040 / 0.0785)^(1/2.72) ≈ 0.780. Step E. Compute expanded bed height. H = H0 * (1 - ε0) / (1 - ε). H = 0.50 * (0.60) / (0.220) ≈ 1.36 m. Result. At U = 0.040 m/s, the predicted bed voidage is about 0.78 and the bed expands from 0.50 m to about 1.36 m.This example highlights an important practical feature. A moderate increase in voidage near high ε causes a large increase in height because the solids fraction (1 − ε) becomes small. This is why expansion predictions become sensitive at very high voidage conditions.
Note : When ε becomes very high, small errors in ε can create large errors in H because H scales with 1/(1 − ε). In this regime, use measured expansion data to fit an effective n whenever possible.
5. Building an expansion curve and using it for design decisions
5.1 Generate H versus U for operating window checks
To create an expansion curve, compute H at multiple velocities from slightly above U_mf up to the intended maximum velocity. The curve is useful for verifying that the expanded bed height stays below a freeboard limit, that entrainment risk is acceptable, and that downstream separation equipment is not overloaded by fines carryover.
| Design question | Expansion-curve quantity | What to check |
|---|---|---|
| Freeboard sizing | Maximum predicted H | H must be below the start of carryover or overflow region |
| Process stability | Slope dH/dU | Very steep slopes indicate sensitivity to flow control and property drift |
| Operating flexibility | H at turndown velocity | At low U, confirm the bed remains fluidized and does not defluidize |
| Instrumentation interpretation | Predicted ε | Compare with densitometer or pressure-gradient inferred voidage |
5.2 Use log form to fit n from data when you have measurements
If you can measure bed height at several velocities, you can obtain ε from H and then fit n. This is often the most practical way to adapt the model to a specific column and particle population.
From measured bed height to voidage. ε = 1 - (H0 * (1 - ε0) / H). Fit form for n using multiple data points. U/U_t = ε^n. Take logs. ln(U/U_t) = n * ln(ε). Fit n as the slope of ln(U/U_t) versus ln(ε).6. Common pitfalls and how to avoid incorrect expansion predictions
6.1 Using Richardson–Zaki in non-homogeneous gas–solid regimes
Gas–solid beds often become bubbling and heterogeneous shortly after minimum fluidization for many particle types and column sizes. In those cases, a single voidage may not represent the bed, and the measured bed surface can fluctuate. If you still need a practical estimate, treat Richardson–Zaki as an effective correlation for a limited operating window and rely on fitted n from measured expansion data in the same regime.
6.2 Ignoring wall and diameter effects
When the column diameter is small relative to particle diameter, wall effects influence both terminal velocity and hindered settling behavior. In practice, this appears as expansion behavior that differs from large-diameter expectations. If the system is in this regime, use column-specific calibration data, or incorporate a wall correction for settling velocity and interpret n as an effective parameter for the given diameter ratio.
6.3 Applying the model at extremely high voidage
At very high voidage, the expanded bed height becomes highly sensitive to small changes in ε. If you operate near this region, expand the safety margin on freeboard, use stable flow control, and validate predictions with direct height measurements during commissioning.
6.4 Mixing polydisperse solids without recognizing segregation
With broad particle size distributions, smaller particles can remain suspended at lower velocities while larger particles concentrate near the bottom, which creates axial variation in ε and an apparent stratification. A single Richardson–Zaki curve may not represent this behavior. In those cases, either narrow the size distribution, or model each size class with its own effective U_t and n, and interpret the result as a composite rather than a single uniform bed.
7. Practical checklist for implementation in calculations and automation
Inputs checklist. 1) Fluid properties: ρ_f, μ, temperature, and whether properties change with concentration. 2) Particle properties: d_p (or size distribution), ρ_p, and shape considerations. 3) Column geometry: diameter, cross-sectional area, distributor quality, and freeboard. 4) Reference bed: H0 and ε0 (or solids mass to compute ε0). 5) Operating velocities: range of U values to evaluate.
Computation checklist.
Estimate or measure U_mf to confirm fluidization for the target range.
Compute U_t using a drag-based terminal velocity calculation or measurement.
Compute Re_t and choose n (correlation or fitted from data).
Compute ε(U) and then H(U).
Validate against at least one measured expansion point if available.FAQ
Should I use superficial velocity or interstitial velocity in the Richardson–Zaki equation.
The Richardson–Zaki form used for bed expansion is typically written with superficial velocity U and voidage ε as U = U_t ε^n. Interstitial velocity is U/ε, which is not the standard variable in the classic power-law form. Use superficial velocity when applying the expansion prediction workflow shown here.
What should I do if my predicted voidage is greater than 1.
A computed ε greater than 1 indicates that the inputs are inconsistent with the simple homogeneous Richardson–Zaki form. This commonly happens if U is set above U_t, if U_t was underestimated, or if the bed behavior is not homogeneous. Recheck the terminal velocity calculation, verify properties, and confirm that the operating regime is appropriate for a uniform voidage model.
How can I estimate the packed bed voidage ε0 if I do not want to assume a value.
You can compute ε0 from solids mass and particle density if you know the static bed height H0 and cross-sectional area A. The solids volume is m_s/ρ_p, and the bed volume is A H0, so ε0 = 1 − (m_s/ρ_p)/(A H0). This approach uses measurable quantities and avoids assuming a typical random packing value.
Is Richardson–Zaki valid for three-phase fluidized beds.
Three-phase beds often require more specialized voidage and holdup correlations because gas and liquid phases interact and the solids experience different drag conditions than in a simple liquid–solid bed. Richardson–Zaki can sometimes be used as a building block for the solids phase in limited regimes, but for design-quality prediction in three-phase systems, use correlations developed for gas–liquid–solid operation and validate with system data.
How do I calibrate the exponent n quickly from two measurements.
If you have two operating points with measured heights H1 and H2 at velocities U1 and U2, compute ε1 and ε2 from solids conservation, then use n = ln(U1/U2) / ln(ε1/ε2) if U_t cancels in the ratio. If you also have U_t, you can fit ln(U/U_t) versus ln(ε) using both points, which gives the same result when measurement noise is low.