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For the cyclic two-side energy balance in refrigerator, define the dimensionless cooling and heating coefficients byThis identity applies even to irreversible machines, provided both coefficients refer to the same operating point, work input and boundary with no additional energy transfer or storage. It does not compare two separately operated appliances. Unlike thermal efficiency of a heat engine, a COP can exceed one: its numerator counts heat moved, including energy taken from another region. ^[ref]

To obtain temperature-only bounds, impose stronger assumptions: a cyclic machine exchanges heat only with two ideal thermal reservoirs maintained at uniform, fixed absolute thermodynamic temperatures $0<T_c<T_h$, and receives work through an ideal input carrying no entropy. Include the machine and the heat transfer interfaces in the entropy production accounting. Because its own entropy returns to its initial value each cycle, the Second law of thermodynamics givesCombining this inequality with $Q_h=Q_c+W$ givesEquality requires a reversible thermodynamic process for the entire cycle and its contacts, as in a reversed Carnot cycle. These are Carnot's theorem bounds independent of the working substance. Use kelvin values, not degree Celsius readings, in the numerators; the temperature difference can be expressed in either unit. ^{[ref] [ref]}

The diagram fixes $T_c=280\mathrm{K}$ and sets $T_h=320\mathrm{K}$, so the reversible bounds are $280/40=7$ and $320/40=8$. The stipulated machine with ${\mathrm{COP}}_{\mathrm{r}}=3$ and ${\mathrm{COP}}_{\mathrm{heat}}=4$ satisfies both. Its reservoir entropy production per cycle is $4000/320-3000/280=25/14\mathrm{J}{\mathrm{K}}^{-1}>0$. Increasing the temperature lift $\Delta T=T_h-T_c$ at fixed $T_c$ lowers both bounds. Equivalently, delivering fixed $Q_h$ needs at leastThese expressions reveal why a small temperature lift is attractive. Their divergence as $T_h-T_c\to 0^{+}$ is an ideal reversible limit, not a promise of infinite heating or cooling power. For finite heat transfer rates, the working fluid must be colder than the cold bath and hotter than the hot bath; these contact differences produce entropy and increase the internal temperature lift. Dissipative compression and the pressure-reducing restriction of vapor-compression refrigeration supply further departures from reversibility. Thus bath temperatures set an outer bound, not a prediction of real COP. ^[ref]

A motivating bench experiment asks whether a hot bath receives more heat than the electrical work supplied to a heat pump. PHYWE's authored operating instructions, section 4, describe a sealed vapor-compression refrigeration apparatus with its cold and hot coils immersed in separate water vessels. The quantitative procedure calls for insulated vessels, stirring to reduce temperature gradients, timed temperature readings, and a wattmeter for the input. It waits about five minutes for operating conditions to settle before measurements and explicitly includes the glass vessel and immersed copper pipe when calculating the heated body's heat capacity. One bath cooling while the other warms makes the origin of the delivered energy observable. ^[ref]

The following analysis extends that procedure to simultaneous measurements of both finite baths. Weigh the water in each vessel; determine the effective heat capacities $C_c$ and $C_h$, including immersed hardware, by a separate known electrical-input calibration over the working temperature range. A matched pair of calibrated thermometers placed in stirred water measures bulk $T_c(t)$ and $T_h(t)$; keeping probes away from the coils avoids mistaking a local contact temperature for the whole bath. Record the room temperature and synchronise the electrical meter and thermometry. Use short intervals after startup, with no boiling or freezing and small enough temperature changes that constant effective heat capacities are a reasonable approximation. These are proposed measurement controls, rather than reported results from the manual.

Over a common interval $[t_0,t_1]$, the uncorrected bath estimates and measured electrical input areHere the wattmeter records real electrical power $P_{\mathrm{el}}$. The integral handles changing input, and the signs make both estimated useful heat quantities positive during heating and cooling. The effective heat capacity includes water mass times its specific heat capacity plus the response of the vessel and immersed hardware. Without that correction, equal temperature changes need not imply equal heat transfers.

Insulation reduces heat leaks but does not establish an isolated system. A warmer bath can lose heat to the room while a colder bath gains it. As a proposed control, measure pump-off drift at matched bath and room temperatures, with the same stirring and insulation. Writing $L_h>0$ for outward hot-bath leak power and $L_c>0$ for inward cold-bath leak power, the bath balances giveIntegrate these rates over the same interval before dividing by $W_{\mathrm{el}}$. For example, in a pump-off hot-bath control, $L_h=-C_{h}d{T}_h/dt$; in the cold-bath control, $L_c=C_{c}d{T}_c/dt$. Check that a stopped machine does not create a different leak path from the running one; otherwise use an independent bath calibration and keep that discrepancy as an uncertainty. Repeat at the same starting temperatures to distinguish drift from a change in COP as the baths separate.

In the illustrated toy model, $C_h=C_c=4500\mathrm{J}{\mathrm{K}}^{-1}$, $P_{\mathrm{el}}=50\mathrm{W}$, ${\dot{Q}}_h=150\mathrm{W}$, ${\dot{Q}}_c=100\mathrm{W}$, $L_h=15\mathrm{W}$, and $L_c=5\mathrm{W}$ are stipulated constants for $180\mathrm{s}$. The slopes are $d{T}_h/dt=0.030\mathrm{K}{\mathrm{s}}^{-1}$ and $d{T}_c/dt=-19/900\mathrm{K}{\mathrm{s}}^{-1}$. Consequently,The raw heating and cooling ratios are $2.7$ and $1.9$, whose difference is $0.8$. Correcting the bath leaks gives $3$ and $2$, whose difference is one. The apparent failure of $Q_h=Q_c+W$ was incomplete measurement of heat, not a violation of the first law of thermodynamics.

Real apparatus can also release motor heat and pipe heat directly to the room, and store internal energy during startup. For a machine boundary that receives electrical work, an additional outward heat transfer $Q_{\mathrm{room}}$ gives $Q_h+Q_{\mathrm{room}}=Q_c+W_{\mathrm{el}}-\Delta U_{\mathrm{machine}}$. The residual $Q_h-Q_c-W_{\mathrm{el}}$ therefore tests the boundary and corrections; do not force it to zero. Include any auxiliary electrical work in the input if quoting whole-apparatus COP, and account separately for stirring work deposited in a bath. Ignoring vessel heat capacity, sensor lag or room leaks can bias the inferred result. These limitations follow from extending the measured energy budget, and the manual's explicit hardware correction illustrates why such boundaries matter.

Finally, these vessels are finite bodies whose temperatures change; they are not fixed thermal reservoirs. Their ratio of integrated useful heat to integrated input work is an interval COP. A Carnot's theorem comparison using simultaneous bath temperatures is a local approximation only when the machine settles much faster than the baths change. An exact reversible comparison for changing bath temperatures instead uses the bath entropy changes, for example $C_h\ln (T_{h,1}/T_{h,0})+C_c\ln (T_{c,1}/T_{c,0})$ for constant heat capacities, negligible net machine entropy change and no room leaks. The experiment's beauty is that two thermometers, known heat capacities and an electrical-energy measurement expose both conservation of energy and the practical gap from reversibility.