Heat Exchangers and Integrated Thermal Systems

Worked Engineering Derivation:

**Given synthetic inputs.** Mass flow 0.2 kg/s, cp=4000 J/(kg K), hot-stream drop 15 K. **Operation.** Use the relationship `Measured hot-stream duty: Q=ṁcpΔT` with all inputs converted to compatible units; the result is **1.2e+04 W**. This is an illustrative arithmetic result, not a measured field or equipment rating. Preserve more precision during intermediate calculations than in the reported answer. If the output is a rate, do not confuse it with accumulated energy. If it is a temperature difference, distinguish it from absolute temperature. If it is nondimensional, the absence of units is a useful audit rather than permission to omit definitions. **Physical interpretation.** Determine how the sign, scale and limiting cases match the original scenario. Recompute the result after raising a physically meaningful driving variable by 10%, but do not automatically assume a proportional change. The formula may contain an exponent, logarithm or regime-dependent coefficient, in which case its sensitivity varies with the operating point. Propose a second measurement that would check the prediction without merely feeding the same data into the same equation. A calibration check tests the measuring instrument, not necessarily the entire heat-transfer model.

Worked Engineering Derivation:

**Given synthetic inputs.** Positive terminal differences ΔT1=40 K, ΔT2=18 K. **Operation.** Use the relationship `LMTD for two terminal differences` with all inputs converted to compatible units; the result is **27.5514 K**. This is an illustrative arithmetic result, not a measured field or equipment rating. Preserve more precision during intermediate calculations than in the reported answer. If the output is a rate, do not confuse it with accumulated energy. If it is a temperature difference, distinguish it from absolute temperature. If it is nondimensional, the absence of units is a useful audit rather than permission to omit definitions. **Physical interpretation.** Determine how the sign, scale and limiting cases match the original scenario. Recompute the result after raising a physically meaningful driving variable by 10%, but do not automatically assume a proportional change. The formula may contain an exponent, logarithm or regime-dependent coefficient, in which case its sensitivity varies with the operating point. Propose a second measurement that would check the prediction without merely feeding the same data into the same equation. A calibration check tests the measuring instrument, not necessarily the entire heat-transfer model.

Worked Engineering Derivation:

**Given synthetic inputs.** Counterflow single-pass, NTU=1.5, Cr=0.5. **Operation.** Use the relationship `Counterflow effectiveness at NTU=1.5, Cr=0.5` with all inputs converted to compatible units; the result is **0.6908 fraction**. This is an illustrative arithmetic result, not a measured field or equipment rating. Preserve more precision during intermediate calculations than in the reported answer. If the output is a rate, do not confuse it with accumulated energy. If it is a temperature difference, distinguish it from absolute temperature. If it is nondimensional, the absence of units is a useful audit rather than permission to omit definitions. **Physical interpretation.** Determine how the sign, scale and limiting cases match the original scenario. Recompute the result after raising a physically meaningful driving variable by 10%, but do not automatically assume a proportional change. The formula may contain an exponent, logarithm or regime-dependent coefficient, in which case its sensitivity varies with the operating point. Propose a second measurement that would check the prediction without merely feeding the same data into the same equation. A calibration check tests the measuring instrument, not necessarily the entire heat-transfer model.

Worked Engineering Derivation:

**Given synthetic inputs.** hi=400, ho=250 W/(m² K); wall δ=0.003 m, k=15; total fouling factor 0.0005 m²K/W. **Operation.** Use the relationship `Common-area resistance U` with all inputs converted to compatible units; the result is **138.8889 W/(m² K)**. This is an illustrative arithmetic result, not a measured field or equipment rating. Preserve more precision during intermediate calculations than in the reported answer. If the output is a rate, do not confuse it with accumulated energy. If it is a temperature difference, distinguish it from absolute temperature. If it is nondimensional, the absence of units is a useful audit rather than permission to omit definitions. **Physical interpretation.** Determine how the sign, scale and limiting cases match the original scenario. Recompute the result after raising a physically meaningful driving variable by 10%, but do not automatically assume a proportional change. The formula may contain an exponent, logarithm or regime-dependent coefficient, in which case its sensitivity varies with the operating point. Propose a second measurement that would check the prediction without merely feeding the same data into the same equation. A calibration check tests the measuring instrument, not necessarily the entire heat-transfer model.

Worked Engineering Derivation:

**Given synthetic inputs.** Thermal power 45 W, overall junction-to-ambient resistance 0.6 K/W. **Operation.** Use the relationship `Heat-sink temperature increase` with all inputs converted to compatible units; the result is **27 K**. This is an illustrative arithmetic result, not a measured field or equipment rating. Preserve more precision during intermediate calculations than in the reported answer. If the output is a rate, do not confuse it with accumulated energy. If it is a temperature difference, distinguish it from absolute temperature. If it is nondimensional, the absence of units is a useful audit rather than permission to omit definitions. **Physical interpretation.** Determine how the sign, scale and limiting cases match the original scenario. Recompute the result after raising a physically meaningful driving variable by 10%, but do not automatically assume a proportional change. The formula may contain an exponent, logarithm or regime-dependent coefficient, in which case its sensitivity varies with the operating point. Propose a second measurement that would check the prediction without merely feeding the same data into the same equation. A calibration check tests the measuring instrument, not necessarily the entire heat-transfer model.