Convert the variables and the empirical constants together—not just the recorded numbers. For an equation such as (C_p=a+bT), the constants (a) and (b) are tied to the units used when the correlation was developed. Accurate pilot-plant processing requires identifying those units, applying the complete variable transformation, and re-deriving the constants so the converted equation remains dimensionally homogeneous.

Core takeaway: Treat an empirical equation as a unit-dependent model. Convert every variable, account for temperature offsets where applicable, and calculate the transformed constants rather than reusing the original numerical coefficients.

Why Empirical Constants Must Be Converted

Constants carry implicit dimensions

In an empirical relation such as

[ y=a+bT ]

the intercept (a) has the units of (y), while the slope (b) has units of (y/T). Their numerical values are therefore valid only for the unit system used to construct the equation.

For example, a coefficient based on temperature in degrees Fahrenheit will not generally have the same numerical value when temperature is entered in kelvins or degrees Celsius.

The equation must remain dimensionally homogeneous

Every additive term must have the same dimensions. In

[ C_p=a+bT ]

both (a) and (bT) must have units of heat capacity.

This requirement is especially important in pilot-plant software, where sensor interfaces may report temperature, flow, pressure, or energy in units different from those used in the original correlation.

Exponents and function arguments require special care

Arguments of exponential, logarithmic, and trigonometric functions must be dimensionless. A relation such as

[ k=k_0\exp\left(-\frac{E}{RT}\right) ]

is valid only when (E/(RT)) is dimensionless and the energy and temperature units are compatible.

Changing units inside such an expression without changing the associated constants can produce numerically plausible but physically incorrect results.

A Reliable Conversion Method

1. Write the original equation with units

Begin by documenting the equation exactly as used, including the units of every variable and parameter.

For example:

[ C_{p,\mathrm{old}}=a_{\mathrm{old}}+ b_{\mathrm{old}}T_{\mathrm{old}} ]

where (C_p) might be reported in Btu/(lbm·°F) and (T) in °F.

Do not begin by changing only the numerical temperature values. First establish what each coefficient represents dimensionally.

2. Write the transformation between variables

For a simple multiplicative conversion, write

[ T_{\mathrm{new}}=\alpha T_{\mathrm{old}}. ]

For temperature scales with different zero points, use an affine transformation:

[ T_{\mathrm{new}}=\alpha T_{\mathrm{old}}+\beta. ]

For example, converting Fahrenheit to kelvins requires

[ T_K=\frac{5}{9}(T_F-32)+273.15. ]

The offset matters because Fahrenheit and kelvin do not share the same zero point. Temperature intervals, however, convert multiplicatively:

[ 1\ ^\circ\mathrm{F}=\frac{5}{9}\ \mathrm{K}. ]

3. Write the output-unit conversion

Suppose the output is converted by

[ y_{\mathrm{new}}=\gamma y_{\mathrm{old}}. ]

Substitute the original equation:

[ y_{\mathrm{new}} =\gamma(a_{\mathrm{old}}+b_{\mathrm{old}}T_{\mathrm{old}}). ]

Using

[ T_{\mathrm{old}}=\frac{T_{\mathrm{new}}-\beta}{\alpha}, ]

gives

[ y_{\mathrm{new}} =\gamma a_{\mathrm{old}} +\gamma b_{\mathrm{old}} \left(\frac{T_{\mathrm{new}}-\beta}{\alpha}\right). ]

Rearranging into the same empirical form,

[ y_{\mathrm{new}}=a_{\mathrm{new}}+ b_{\mathrm{new}}T_{\mathrm{new}}, ]

produces

[ a_{\mathrm{new}} =\gamma\left(a_{\mathrm{old}} -\frac{b_{\mathrm{old}}\beta}{\alpha}\right) ]

and

[ b_{\mathrm{new}}=\frac{\gamma b_{\mathrm{old}}}{\alpha}. ]

This is the general conversion rule for a linear empirical equation when the independent variable may include an offset.

Example: Converting a Heat-Capacity Correlation

Original correlation

Assume a heat-capacity equation is provided as

[ C_p=a_F+b_FT_F ]

with (C_p) in Btu/(lbm·°F) and (T_F) in °F.

To express the result in J/(kg·K) with temperature in kelvins, use

[ T_K=\frac{5}{9}T_F+255.3722. ]

Therefore,

[ T_F=\frac{9}{5}(T_K-255.3722). ]

The heat-capacity output conversion is approximately

[ 1\ \frac{\mathrm{Btu}}{\mathrm{lbm}\cdot{}^\circ\mathrm{F}}

4186.8\ \frac{\mathrm{J}}{\mathrm{kg}\cdot\mathrm{K}}. ]

Thus, (\gamma) is approximately 4186.8, while the Fahrenheit-to-kelvin temperature relationship supplies the values of (\alpha) and (\beta).

The converted equation must be obtained by substitution and collection of terms. It is not correct to multiply both (a_F) and (b_F) by the same heat-capacity conversion factor and then enter kelvins directly into the original formula.

Why the intercept changes

The intercept is the value predicted when the numerical temperature variable equals zero. Zero Fahrenheit and zero kelvin represent different physical temperatures, so the intercept must change when the temperature scale changes.

The slope changes according to the size of a temperature interval and the output-unit conversion. The offset affects the intercept but not the temperature-interval scaling itself.

Converting Compound Pilot-Plant Measurements

Use the dimensional equation method

For a measured quantity, write the starting value and multiply by conversion ratios arranged so that the original units cancel.

For example:

[ Q\left(\frac{\mathrm{cm^3}}{\mathrm{s}}\right) \left(\frac{1\ \mathrm{m}}{100\ \mathrm{cm}}\right)^3

Q\left(\frac{\mathrm{m^3}}{\mathrm{s}}\right). ]

Keeping the units visible exposes errors that may be hidden when only numerical factors are entered into a spreadsheet or data-acquisition system.

Apply powers to the entire conversion factor

Area and volume conversions must be raised to the appropriate power.

For example:

[ 1\ \mathrm{cm^3}

\left(\frac{1\ \mathrm{m}}{100\ \mathrm{cm}}\right)^3 \mathrm{m^3} ]

and an area conversion uses the square of the length factor.

The same principle applies to heat flux, volumetric flow, velocity gradients, and transport-property correlations containing squared or cubed dimensions.

Preserve units in data logging and software

A pilot-plant data pipeline should identify the units at the sensor, conversion, calculation, and model stages. A numerically correct value in the wrong unit system is still incorrect process data.

Store converted values with explicit metadata where possible, rather than relying on a worksheet label or operator memory.

Converting Correlations With Nonlinear Terms

Power-law relationships

For a relation such as

[ \dot m=aT^{1/2}+b, ]

changing the temperature unit changes the numerical value and dimensions of (a). If the temperature transformation includes an offset, such as Fahrenheit to kelvin, the simple square-root form may not retain the same appearance after substitution.

This is because

[ T_F^{1/2} \neq \left(\frac{9}{5}(T_K-255.3722)\right)^{1/2} ]

as a constant multiple of (T_K^{1/2}).

Linearizing experimental data

If the proposed model is

[ \dot m=aT^{1/2}+b, ]

calculate (T^{1/2}) consistently in one unit system, then plot (\dot m) against (T^{1/2}). A straight-line trend supports the proposed model and allows the slope and intercept to be estimated.

For reliable pilot-plant data analysis, use regression across the measured data range rather than relying only on two selected points. The fitted coefficients must still be reported with their corresponding units and valid operating range.

Understanding the Trade-offs

Direct conversion versus refitting

Direct algebraic conversion preserves the original empirical relationship exactly, provided the transformation is performed correctly.

Refitting the model after converting the data can be useful when measurements contain noise, but the resulting coefficients may differ because the fit is being re-estimated rather than merely transformed.

Absolute temperature versus temperature difference

Absolute temperatures require offsets when converting between scales. Temperature differences do not.

Confusing these two cases is a common source of incorrect heat-capacity, reaction-rate, and heat-transfer calculations.

Dimensional consistency does not guarantee physical validity

An equation can be dimensionally correct and still be unsuitable outside the range where it was developed. Empirical correlations also depend on the physical regime, material, geometry, and operating conditions.

Dimensional analysis cannot replace understanding of quantities such as shear stress, pressure difference, flow direction, or the mass basis used in a correlation.

Conversion errors can look reasonable

A unit error may produce smooth trends and plausible magnitudes, especially in automated systems. Independent checks—such as energy balances, limiting values, and comparison with known operating conditions—should therefore be part of pilot-plant validation.

How to Apply This to Your Project

Use the following workflow before connecting an empirical correlation to pilot-plant data processing:

If your primary focus is equation conversion: Write the original equation with units, substitute the complete variable transformations, and re-derive every constant.
If your primary focus is temperature-dependent equations: Distinguish absolute temperature from temperature intervals and include scale offsets when converting Fahrenheit, Celsius, or kelvin values.
If your primary focus is sensor-data integration: Define units at every data-acquisition and software interface, and carry units through intermediate calculations.
If your primary focus is compound quantities: Raise conversion factors to the correct power for area, volume, and other derived dimensions.
If your primary focus is empirical parameter estimation: Convert the data consistently before fitting, verify dimensional homogeneity, and report the coefficient units and applicable range.
If your primary focus is process-model validation: Check converted results against balances, physical limits, and independent measurements before using them for control or scale-up.

Correct unit conversion turns an empirical correlation into a dependable processing model rather than a source of hidden numerical error.

Summary Table:

Topic	Key requirement	Common risk
Linear equations	Transform variables and re-derive intercepts and slopes	Reusing original coefficients in new units
Temperature scales	Include offsets for absolute temperatures; use interval factors for temperature differences	Confusing Celsius, Fahrenheit, and kelvin values
Compound measurements	Apply conversion factors to the correct power for area and volume	Incorrect flow, flux, or transport-property values
Nonlinear correlations	Substitute the full transformation and assess whether the model form remains valid	Treating offset conversions as simple multipliers
Data integration	Track units from sensors through software and model stages	Numerically plausible but incorrectly scaled data
Model validation	Check balances, physical limits, independent measurements, and operating range	Applying dimensionally correct correlations outside their valid regime

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