Physics and mechanics
Heat Transfer Equilibrium Calculator
Solve the steady temperature of one heat-generating node between hot and cold reservoirs, preserve signed branch flows, and prove the node energy balance closes.
CURRENT MODEL
Enter the declared physical case
Electronics, process, and thermal-design teams screening a component coupled to two reservoirs through known total thermal resistances.
Decision supportedDetermine the steady node temperature, identify whether internal generation reverses the hot-side branch, and test whether the result stays within a component temperature limit outside this calculator.
LIVE PHYSICAL ANALYSIS
Signed heat paths around the current node
The live thermal map positions the solved node between the reservoir temperatures and scales each branch arrow from the current signed rate.
| Quantity | Expression | Current value | Unit |
|---|
How to use
Close one steady thermal-node balance
- Define one component or junction that can reasonably be represented by a single uniform steady temperature.
- Enter fixed hot and cold reservoir temperatures with the hot label strictly warmer.
- Enter the complete K/W resistance from the node to each reservoir, including only justified series elements.
- Enter non-negative heat generated inside the node, not heat entering through either branch.
- Read the signed hot and cold branch rates; do not replace a negative branch with an absolute value.
- Verify Q_hot + Qgen - Q_cold closes and compare the node temperature with an independently defined limit.
Equilibrium fundamentals
Five ideas behind the one-node balance
- Thermal node
- An isothermal control point where branch heat rates and internal generation are balanced.
- Reservoir
- A boundary whose temperature is treated as fixed despite the modeled heat flow.
- Thermal resistance
- A linear K/W relation between node-to-reservoir temperature difference and branch heat rate.
- Conductance weighting
- The reciprocal-resistance weighting that determines the passive node temperature.
- Internal generation
- Heat created at the node, such as electrical dissipation, rather than imported through a branch.
- Steady balance
- Zero accumulation: signed inflows plus generation equal signed outflows.
Calculation method
Solve temperature from the branch energy equation
The node balance is written as (Th - Tnode)/Rh + Qgen = (Tnode - Tc)/Rc. Collecting Tnode terms gives a conductance-weighted expression that remains valid when generation drives the node above Th.
The page then recomputes each signed branch from the solved temperature. Their residual is an independent algebraic closure check, while the generation crossover identifies when the nominal hot branch reverses direction.
Branch reversal is physical
If Qgen exceeds (Th - Tc)/Rc, the node rises above the hot reservoir. Heat then leaves through both branches, so Q_hot is negative under the declared sign convention.
Equivalent resistance boundaries
A K/W value can hide convection, conduction, contacts, spreading, and radiation linearization. Preserve the derivation because only compatible series elements can be summed directly.
Steady state versus peak temperature
A steady node may underpredict transient overshoot, startup gradients, or control failures. Thermal capacitance and time-dependent generation require a separate dynamic network.
Detailed calculation process
Symbols, current substitution, intermediate quantities, and reconciliation
T_node = (T_h/R_h + T_c/R_c + Qgen)/(1/R_h + 1/R_c); Q_h + Qgen - Q_c = 0The conductance-weighted node equation uses unrounded values. Temperatures display to 0.01 deg C, rates to 0.001 W, and the balance residual in scientific notation.
| Symbol | Meaning | Default | Unit |
|---|---|---|---|
| T_h | Hot-reservoir temperature | 120 | deg C |
| T_c | Cold-reservoir temperature | 20 | deg C |
| R_h | Node-to-hot resistance | 0.8 | K/W |
| R_c | Node-to-cold resistance | 1.2 | K/W |
| Qgen | Internal heat generation | 30 | W |
| T_node | Solved equilibrium temperature | calculated | deg C |
| Q_hot | Signed rate from hot reservoir into node | calculated | W |
| Q_cold | Signed rate from node to cold reservoir | calculated | W |
Waiting for valid inputs.
Evidence to retain
Keep the branch derivation and dissipation basis
Record geometry, materials, contact conditions, flow states, area definitions, test temperatures, and correlations behind each resistance. Preserve the electrical or chemical basis for internal generation, duty cycle, tolerances, reservoir-control capability, and the component temperature limit used after calculation.
Scope and limitations
What this equilibrium excludes
- Thermal capacitance, startup, cycling, and transient overshoot
- Nonlinear radiation unless already linearized within a resistance
- Temperature-dependent resistance or contact-state changes
- Distributed node temperature and spreading resistance
- Parallel branches beyond the two explicitly modeled
- Reliability, derating, code, or safety certification
One isothermal node, two linear constant K/W resistances, fixed reservoir temperatures, constant internal heat generation, and steady state. No capacitance, radiation nonlinearity, distributed generation, contact changes, or temperature-dependent properties.
Key terminology
Thermal-node glossary
- Thermal reservoir
- A boundary idealized as maintaining a fixed temperature while exchanging heat.
- Thermal node
- A lumped location with one modeled temperature and an energy balance.
- Branch conductance
- 1/R, the heat-rate response per kelvin of node-to-reservoir difference.
- Internal generation
- Heat produced inside the control node, such as electrical loss.
- Flow reversal
- A sign change indicating heat crosses a branch opposite its nominal direction.
- Balance residual
- The remaining watts after adding signed branch rates and generation.
Practical cases
Two node balances with different regimes
Power device on a warm enclosure
A transistor node connects through one path to a warm chassis and another to a cooler liquid plate. Moderate dissipation leaves the node below the chassis temperature, so both the chassis and device feed the cold branch.
Sensor heater near a hot process wall
A self-heated sensor has enough generation to exceed the process-wall temperature. The hot-side branch reverses, and the balance quantifies heat rejected toward both wall and ambient rather than forcing a misleading hot-to-cold narrative.
Important note
Use temperature limits outside the algebra
Energy-balance closure is necessary but not sufficient. Apply component derating, uncertainty, contact variation, worst-case reservoir temperatures, transient peaks, fault conditions, and qualified design review before safety or reliability decisions.
Frequently asked questions
Can the node temperature exceed the hot reservoir?
Yes. Internal generation can raise the node above both reservoirs. The hot-side rate then becomes negative, meaning the node rejects heat toward the nominal hot reservoir as well as the cold one.
Why are the resistances entered in K/W rather than W/(m2 K)?
This model needs total branch resistance. A coefficient becomes K/W only after area and any series layers or contacts are incorporated consistently.
What does a negative hot-branch rate mean?
The calculator defines positive Q_hot as flow from the hot reservoir into the node. A negative value is not an error; it signals reversal from the node toward that reservoir.
Is zero balance residual proof the physical model is correct?
No. It proves only that the implemented algebra closes. Incorrect resistances, omitted radiation, spreading, or temperature-dependent behavior can still make the model boundary wrong.
Can I use the page for transient warm-up?
No. Steady equilibrium omits thermal capacitance and time. Use a transient network or trajectory model when warm-up time or overshoot matters.
How should I combine several layers on one branch?
Series resistances may be summed when the same heat rate crosses each element. Parallel paths require separate conductances and cannot be hidden in a simple series sum without a justified equivalent.
Authority and follow-on work
Reliable sources and related calculators
- U.S. DOE Fundamentals Handbook: Heat TransferCovers heat flow, thermal resistance, and steady energy-balance foundations.
- MIT 2.051 heat-transfer equation sheetProvides conduction and convection resistance relations used to construct branch K/W values.
- OpenStax: Temperature and Thermal EquilibriumDefines thermal equilibrium and the absence of further net heat flow at equal temperature.
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