The Archimedes number evaluates buoyancy relative to forced-flow inertia. For a thermal unit-operations pilot plant, use the measured or prescribed wall condition to calculate a dimensionless Archimedes number. For a prescribed wall temperature difference, use (\mathrm{Ar} = B_R (T_w-T_R) g l/V^2); for a prescribed wall heat flux, use (\mathrm{Ar} = B_R g l^2 q_w/(V^2\lambda_R)).

The two boundary conditions produce different but comparable Archimedes-number definitions. The temperature-difference form uses the wall-to-reference-fluid temperature difference directly, while the heat-flux form first converts heat flux into its associated temperature-gradient scale through (q_w/\lambda_R).

What the Archimedes Number Represents

Buoyancy Compared with Forced Flow

The Archimedes number measures the influence of buoyancy forces relative to inertial forces in the flowing thermal system. It is therefore useful when temperature-dependent density changes affect circulation, velocity profiles, or heat transfer.

Here:

(B_R) is the relevant buoyancy or thermal-expansion parameter evaluated at the reference condition.
(g) is gravitational acceleration.
(l) is the characteristic length.
(V) is the characteristic flow velocity.
(T_w) is wall temperature.
(T_R) is the reference or bulk-fluid temperature.
(q_w) is wall heat flux.
(\lambda_R) is fluid thermal conductivity at the reference condition.

The characteristic length and velocity must be defined consistently for the equipment and correlation being used.

Why It Matters in Pilot Equipment

Temperature gradients can change fluid density, especially in vertical piping, heated tubes, reboilers, and other equipment with substantial elevation or wall-to-fluid temperature differences. These density changes introduce buoyancy that can modify the forced-flow heat-transfer behavior.

Using (\mathrm{Ar}) allows results from different operating conditions and different thermal boundary conditions to be compared on a common dimensionless basis.

Evaluating a Prescribed Wall Temperature Difference

Required Measurements

For a constant-wall-temperature experiment, determine or measure:

The wall temperature, (T_w).
The reference or bulk-fluid temperature, (T_R).
The characteristic length, (l).
The flow velocity, (V).
The reference buoyancy parameter, (B_R).
Gravitational acceleration, (g).

The temperature difference is:

[ \Delta T_w = T_w-T_R ]

Archimedes Number Calculation

Substitute the temperature difference into:

[ \boxed{\mathrm{Ar}_{\Delta T}

\frac{B_R (T_w-T_R) g l}{V^2}} ]

The result is dimensionless when the parameters are expressed in a consistent unit system.

A larger wall-to-fluid temperature difference increases (\mathrm{Ar}), indicating stronger buoyancy influence. A higher flow velocity decreases (\mathrm{Ar}), indicating that forced-flow inertia is becoming more dominant.

Determining the Wall Temperature

In practical pilot equipment, (T_w) may be controlled directly or inferred from heat-transfer measurements. The heat flux can be related to the process-side film coefficient through:

[ \frac{Q}{A}

h_{\text{process}}(T_w-T_{\text{process}}) ]

If (h_{\text{process}}) depends on velocity or temperature, calculating the wall temperature may require an iterative estimate. The resulting (T_w-T_R) is then used in the Archimedes-number calculation.

Evaluating a Prescribed Wall Heat Flux

Required Measurements

For a uniform-heat-flux experiment, determine or measure:

The imposed wall heat flux, (q_w).
The fluid thermal conductivity, (\lambda_R).
The reference buoyancy parameter, (B_R).
The characteristic length, (l).
The flow velocity, (V).
Gravitational acceleration, (g).

The heat flux is commonly obtained from measured heating duty divided by the heated surface area:

[ q_w=\frac{Q}{A} ]

Archimedes Number Calculation

Use the heat-flux form:

[ \boxed{\mathrm{Ar}_{q}

\frac{B_R g l^2 q_w}{V^2\lambda_R}} ]

The factor (q_w/\lambda_R) has the dimensions of a temperature gradient. Consequently, this definition represents the buoyancy effect associated with a prescribed thermal gradient rather than a directly specified wall-to-fluid temperature difference.

A higher heat flux increases (\mathrm{Ar}_q), while higher thermal conductivity reduces it for the same heat flux. Increasing flow velocity again reduces the relative importance of buoyancy.

Relating Heat Flux to Wall Temperature

Although the boundary condition specifies (q_w), the resulting wall temperature is generally an outcome of the experiment. It can be estimated from:

[ q_w

h_{\text{process}}(T_w-T_{\text{process}}) ]

Thus, changes in flow velocity or fluid properties can change (h_{\text{process}}), which changes the wall temperature required to sustain the imposed heat flux. The heat-flux Archimedes number is calculated from the prescribed (q_w), while the resulting wall temperature can be used separately to assess thermal constraints.

Applying the Calculation to Pilot-Plant Data

Instrumentation Inputs

A pilot installation should provide reliable values for:

Wall or heating-surface temperature.
Bulk or reference-fluid temperature.
Flow rate or velocity.
Heating duty and heated area.
Fluid thermal conductivity.
Fluid properties used to evaluate (B_R).

For constant-wall-temperature equipment, the temperature sensors establish (T_w-T_R). For uniform-heat-flux equipment, power or duty measurements establish (q_w), while surface area determines the corresponding heat flux.

Comparing Different Equipment

Constant-wall-temperature heat exchangers and uniform-heat-flux heating surfaces should not be compared by inserting their raw boundary-condition values into the same expression. Their Archimedes numbers must first be calculated using the appropriate definition.

The resulting dimensionless values can then be used alongside heat-transfer measures such as the Nusselt number, overall coefficient (U), or process-side film coefficient (h_{\text{process}}).

Accounting for Property Variation

Wall heating can produce large differences between wall and bulk-fluid viscosity and density. These property changes affect both the flow field and the heat-transfer coefficient, so reference properties should be selected consistently with the correlation and operating condition.

In vertical equipment, density variation can also generate buoyancy-driven changes in the flow profile. This is the physical reason that the Archimedes number is relevant to interpreting heat-transfer performance rather than merely being a reporting parameter.

Understanding the Trade-offs

The Number Depends on the Characteristic Scales

The Archimedes number depends on the selected (l) and (V). A result based on tube diameter and mean velocity is not automatically interchangeable with one based on equipment height and another velocity scale.

Comparisons are meaningful only when the characteristic scales and reference-property definitions are consistent.

Wall Temperature Is Not Always Directly Known

In a heat-flux experiment, the wall temperature is not prescribed. It depends on the process-side heat-transfer coefficient, which itself may vary with flow velocity, temperature, viscosity, and density.

Using an estimated wall temperature in place of the prescribed heat flux would change the boundary-condition model and could lead to an inconsistent evaluation.

Archimedes Number Does Not Fully Predict Heat Transfer

The Archimedes number identifies the relative importance of buoyancy, but it does not replace a complete heat-transfer correlation. Reynolds number, Prandtl number, Nusselt number, geometry, flow orientation, and property variation may also be required.

For turbulent flow, strong mixing can flatten the core temperature profile while concentrating much of the temperature gradient near the wall. That behavior must be interpreted through the relevant heat-transfer correlation in addition to (\mathrm{Ar}).

Making the Right Choice for Your Goal

Use the boundary condition that the pilot equipment actually controls, then calculate the corresponding Archimedes number with consistent reference properties and characteristic scales.

If your primary focus is constant-wall-temperature comparison: Calculate (\mathrm{Ar}_{\Delta T}=B_R(T_w-T_R)gl/V^2) from the measured wall-to-reference-fluid temperature difference.
If your primary focus is uniform-heat-flux comparison: Calculate (\mathrm{Ar}_{q}=B_Rgl^2q_w/(V^2\lambda_R)) from the imposed heat flux and reference thermal conductivity.
If your primary focus is heat-transfer performance: Report the Archimedes number together with (U), (h_{\text{process}}), or Nusselt number so buoyancy effects are not separated from the actual thermal response.
If your primary focus is temperature-sensitive processing: Use the measured or estimated wall temperature to verify thermal limits, even when the experiment is operated under prescribed heat flux.

With consistent definitions, the Archimedes number provides a practical basis for judging when buoyancy may influence pilot-scale heat-transfer results.

Summary Table:

Boundary condition	Archimedes number	Required inputs	Main interpretation
Prescribed wall temperature difference	Ar = B_R(T_w - T_R)gl/V^2	Wall and reference temperatures, B_R, g, l, V	Buoyancy caused by the wall-to-fluid temperature difference
Prescribed wall heat flux	Ar = B_Rgl^2q_w/(V^2lambda_R)	Heat flux, lambda_R, B_R, g, l, V	Buoyancy associated with the imposed thermal gradient
Pilot-plant comparison	Use the appropriate definition for the controlled boundary condition	Consistent scales and reference properties	Enables comparison of buoyancy effects across operating conditions

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