The purpose of this article is to show how to calculate adiabatic mixing temperature when heat capacity varies with temperature, using a rigorous enthalpy balance that can be implemented in spreadsheets or code for real process engineering work.
1. What “adiabatic mixing temperature” really means.
Adiabatic mixing means the mixing control volume exchanges no heat with the surroundings and does no shaft work, so the total inlet enthalpy rate equals the total outlet enthalpy rate at steady state.
The mixing temperature is the common outlet temperature after streams combine and reach thermal equilibrium, assuming mixing is fast compared to heat loss and the outlet is well mixed.
When Cp is temperature-dependent, you cannot use a single constant Cp to compute temperature, and you must integrate Cp(T) over temperature to compute sensible enthalpy change.
2. Set up the energy balance in a form that always works.
2.1 Control volume statement for steady adiabatic mixing.
For a steady mixing junction with negligible kinetic and potential energy changes, no shaft work, and adiabatic walls, the balance is written as the equality of total inlet and outlet enthalpy flow rates.
In compact form, the balance is Σ(ṁi·hi,in) = ṁout·hout at the mixing temperature and outlet composition.
If there is no reaction and no phase change, the outlet enthalpy is the sum of species enthalpies at the outlet temperature weighted by mass or molar flow, consistent with your property basis.
2.2 Use an explicit reference temperature so the integral is unambiguous.
Pick a reference temperature Tref, such as 298.15 K, and define each stream’s sensible enthalpy relative to that reference using the integral of Cp(T).
For a pure species on a molar basis, the sensible enthalpy relative to Tref is written as h(T) − h(Tref) = ∫(Tref→T) Cp(T) dT.
For a mixture, you may approximate Cp,mixture(T) as a composition-weighted sum of species Cp,k(T) when ideal mixing of heat capacities is acceptable for your accuracy needs.
3. The practical equation you solve for Tmix when Cp depends on T.
3.1 Two-stream mixing without phase change.
Consider two inlet streams labeled 1 and 2, with flow rates ṁ1 and ṁ2, inlet temperatures T1 and T2, and heat capacities Cp,1(T) and Cp,2(T) defined on the same basis.
The adiabatic mixing temperature Tmix satisfies the implicit equation below when there is no reaction and no phase change.
Find Tmix such that: ṁ1 * ∫(Tref→T1) Cp,1(T) dT + ṁ2 * ∫(Tref→T2) Cp,2(T) dT = (ṁ1 + ṁ2) * ∫(Tref→Tmix) Cp,out(T) dT.If the outlet composition is a mixture of the same species entering, Cp,out(T) is computed from the outlet composition, and the outlet composition is determined from mass or molar flow mixing.
If both streams are the same pure species, then Cp,out(T) equals Cp(T) for that species, and the equation becomes a single-variable root-finding problem with a monotonic left-minus-right function for typical Cp(T) models.
3.2 Multi-stream, multi-component generalization.
For N inlet streams, each containing K components, the same idea holds, but you compute each stream enthalpy using its own composition and temperature.
On a molar basis with molar flows ṅi and component mole fractions yi,k, one convenient arrangement is below.
Compute inlet enthalpy rate: Ḣin = Σ_i ṅi * Σ_k yi,k * ∫(Tref→Ti) Cp,k(T) dT.
Compute outlet molar flow:
ṅout = Σ_i ṅi.
Compute outlet composition:
yout,k = (Σ_i ṅi * yi,k) / ṅout.
Compute outlet Cp model:
Cp,out(T) = Σ_k yout,k * Cp,k(T).
Solve for Tmix from:
Ḣin = ṅout * ∫(Tref→Tmix) Cp,out(T) dT.4. Choosing a Cp(T) model that integrates cleanly.
4.1 Polynomial Cp(T) model.
A common engineering model is a low-order polynomial in temperature, such as Cp(T) = a + bT + cT^2 + dT^3 on a chosen basis.
The integral has a closed form, which avoids numerical integration and improves robustness inside spreadsheets.
If Cp(T) = a + bT + cT^2 + dT^3, then: ∫(Tref→T) Cp(T) dT = a(T − Tref) + (b/2)(T^2 − Tref^2) + (c/3)(T^3 − Tref^3) + (d/4)(T^4 − Tref^4).With this form, the mixing equation becomes a polynomial-like expression in Tmix, but it is still usually solved numerically because Cp,out(T) may depend on temperature through composition rules or because you prefer one general solver for all cases.
4.2 Piecewise Cp(T) tables.
When Cp is available as tabulated data, compute the integral using piecewise linear Cp(T) and trapezoidal integration over temperature intervals.
This is often the most transparent approach for audited calculations, because each segment contribution can be shown explicitly in a table.
| Temperature interval. | Cp model in interval. | Integral contribution ∫Cp dT. | Implementation tip. |
|---|---|---|---|
| Tj to Tj+1. | Linear between Cp(Tj) and Cp(Tj+1). | 0.5·(Cp(Tj)+Cp(Tj+1))·(Tj+1−Tj). | Use trapezoid segments and sum them. |
| Below lowest data point. | Extrapolation is risky. | Do not extrapolate unless justified. | Clamp temperature range or use a validated correlation. |
| Above highest data point. | Property uncertainty grows. | Segment sum may drift. | Check sensitivity to Cp uncertainty. |
5. How to solve for Tmix reliably.
5.1 Convert the problem into a single root of a monotonic function.
Define a function f(T) = Ḣin − Ḣout(T), where Ḣout(T) is the outlet enthalpy rate computed at a trial temperature T.
The solution is the temperature where f(T) = 0.
For typical gases and liquids over moderate temperature ranges where Cp is positive, Ḣout(T) increases with T, so f(T) decreases with T, giving a unique solution.
5.2 Bracketing is the safest strategy in engineering calculations.
Choose lower and upper bounds that definitely bracket the answer, then use bisection or a safeguarded method.
A practical bracket for many cases is between the minimum inlet temperature and the maximum inlet temperature when no phase change occurs and mixing enthalpy is negligible, but you should still verify the sign change of f(T) because Cp,out(T) and compositions can shift the result.
5.3 Minimal implementation pattern for spreadsheets and scripts.
The procedure below works whether you integrate Cp analytically or numerically.
1) Choose Tref and property basis. 2) For each inlet stream i, compute Hi = flow_i * IntegralCp(stream_i, Tref, Ti). 3) Sum Ḣin = Σ Hi. 4) Compute outlet flow and composition by mixing rules. 5) Define Ḣout(T) = flow_out * IntegralCp(outlet, Tref, T). 6) Solve f(T) = Ḣin − Ḣout(T) = 0 with a bracketing solver. 7) Report Tmix and perform a balance check by recomputing both sides.6. Worked example with temperature-dependent Cp(T) for two streams.
6.1 Problem statement on a mass basis.
Stream A is a liquid with mass flow 2.0 kg/s entering at 350 K.
Stream B is the same liquid with mass flow 1.0 kg/s entering at 300 K.
The liquid heat capacity is modeled as Cp(T) = 3900 + 1.2(T − 300) J/kg·K over the range of interest.
Find the adiabatic mixing temperature using Cp(T) integration.
6.2 Integrate Cp(T) relative to Tref = 300 K.
Rewrite Cp(T) = 3900 + 1.2(T − 300) = 3900 + 1.2T − 360.
This simplifies to Cp(T) = 3540 + 1.2T in J/kg·K.
The integral from 300 K to T is ∫(300→T) (3540 + 1.2T) dT = 3540(T − 300) + 0.6(T^2 − 300^2).
6.3 Compute inlet enthalpy rates and solve for Tmix.
Compute HA = 2.0 * [3540(350 − 300) + 0.6(350^2 − 300^2)] in J/s.
Compute HB = 1.0 * [3540(300 − 300) + 0.6(300^2 − 300^2)] in J/s, which is zero by construction because Tref equals 300 K.
The outlet flow is 3.0 kg/s, and the outlet Cp(T) is the same model because the fluid is the same.
Solve 2.0 * [3540(50) + 0.6(350^2 − 300^2)] = 3.0 * [3540(Tmix − 300) + 0.6(Tmix^2 − 300^2)].
This equation is solved numerically in most workflows, but it can also be rearranged into a quadratic form for this specific linear Cp(T) example.
| Quantity. | Expression. | Value meaning. | Common mistake to avoid. |
|---|---|---|---|
| Reference temperature. | Tref = 300 K. | Defines zero of sensible enthalpy. | Using a different Tref for different streams. |
| Sensible enthalpy function. | I(T) = 3540(T − 300) + 0.6(T^2 − 300^2). | Integrated Cp from Tref. | Forgetting the square term when Cp depends on T. |
| Balance equation. | ṁA I(350) + ṁB I(300) = (ṁA+ṁB) I(Tmix). | Adiabatic enthalpy conservation. | Using average Cp instead of integrating Cp(T). |
7. Extensions you must include in high-accuracy industrial calculations.
7.1 Phase change and latent heat.
If one inlet is a vapor and the outlet becomes a two-phase mixture or fully condenses, you must include latent heat terms using enthalpy of vaporization or appropriate saturated property data.
In that case, Tmix may be pinned near a saturation temperature at the mixing pressure until enough energy is available to complete the phase change.
7.2 Pressure effects and real-fluid enthalpy.
For ideal gases, enthalpy is often treated as a function of temperature only, but for real gases at high pressure the enthalpy can depend on both temperature and pressure.
If pressure changes across the mixing point are significant or the fluids are strongly non-ideal, use a property package or an equation of state to compute enthalpy rather than relying only on Cp(T) at a reference pressure.
7.3 Heat of mixing and dissolution.
When liquids mix with strong interactions, such as acids with water or solvent blends with large excess enthalpies, the heat of mixing can dominate the temperature change.
In those cases, Cp(T) integration alone is insufficient, and you must add a mixing enthalpy term consistent with composition and temperature.
8. Verification checklist for adiabatic mixing temperature calculations.
Check unit consistency across flow, Cp, and enthalpy integrals, and confirm the final energy residual is near zero relative to the larger of inlet or outlet enthalpy rates.
Confirm that Cp(T) is valid over the full temperature interval from Tref to each inlet and to the computed Tmix, and confirm that the outlet state remains in the assumed phase region.
Confirm that the computed Tmix lies within a physically reasonable range, and if it falls outside, revisit whether the process is truly adiabatic or whether additional energy terms were omitted.
FAQ
Can I use an average Cp instead of integrating Cp(T) for adiabatic mixing temperature.
You can approximate with an average Cp only when the temperature span is small and Cp changes weakly over that span, and you can quantify the resulting temperature error by comparing against the integrated Cp(T) result.
For large temperature differences or when Cp has strong curvature, integrating Cp(T) is the correct approach because enthalpy is the integral of Cp over temperature.
What reference temperature should I use for the enthalpy integral.
Any convenient reference temperature works because only enthalpy differences matter, but you must use the same reference temperature consistently for every stream and for the outlet calculation.
Using 298.15 K is common, but choosing a value near your operating range can reduce numerical cancellation in some implementations.
Why does my solver return a mixing temperature outside the inlet temperature range.
This can happen if you included phase change or heat of mixing implicitly or explicitly, or if you have inconsistent units, an incorrect Cp(T) correlation range, or a sign error in the enthalpy function.
In purely sensible, single-phase, adiabatic mixing with positive Cp, the solution typically lies between the inlet temperatures, so an out-of-range result is a strong diagnostic signal that assumptions or inputs are inconsistent.
How do I handle multiple components with different Cp(T) correlations.
Compute each component’s enthalpy integral from the same reference temperature using its own Cp,k(T) model, then form each stream enthalpy by composition-weighted summation on a molar or mass basis.
After mixing, compute the outlet composition from mixed flows, build Cp,out(T) from the outlet composition, and solve the single outlet temperature that matches the inlet enthalpy rate.
Is it valid to ignore kinetic energy and pressure drop in an adiabatic mixing temperature calculation.
It is often valid when velocities are modest and pressure changes are small, but it is not universally valid, and high-velocity jets or significant throttling can introduce measurable temperature changes through energy redistribution and real-fluid effects.
When in doubt, estimate the kinetic energy term and compare it to sensible enthalpy changes, and use pressure-dependent enthalpy methods if pressure effects are non-negligible.