Exergy and Investment Thresholds for Campus Heat Recovery Under Academic Calendars

1. Introduction

University computing facilities produce heat while teaching buildings require it intermittently. Their physical proximity creates an attractive recovery opportunity, but annual energy totals cannot establish whether heat is available at a useful temperature and time. Reviews identify temperature quality and conversion requirements as persistent constraints on data-centre heat reuse [1,2,3]. Experimental evidence is already substantial: Aquasar demonstrated hot-water cooling with direct energy reuse [4], while subsequent field and laboratory studies examined heat-pump-assisted recovery and integrated cooling systems [5,6]. The engineering question concerns usable heat under specified operating conditions.

Lower distribution temperatures can improve access to otherwise underused thermal resources. Fourth-generation district heating integrates lower-temperature supply with wider energy-system operation [7], and European fifth-generation networks demonstrate approaches using near-ambient loops [8]. Large heat pumps provide a practical connection between such resources and conventional heating networks [9]. Nevertheless, performance depends on temperature lift, refrigerant and cycle configuration [10]; detailed campus-oriented heat-pump comparisons confirm that architecture matters [11]. A screening model should therefore expose its temperature assumptions without presenting a simplified coefficient of performance as a manufacturer-validated operating map.

Heat availability also depends on the receiving system. London modelling established the relevance of matching data centres with neighbouring heating requirements [12]. District-heating simulations subsequently connected load profiles and heat procurement prices to recovery performance [13], while commercial coordination emerged as a broader implementation issue [14]. Viewing a data centre as an energy prosumer further connects computing, cooling and downstream heat use [15]. Investment analysis already accommodates uncertainty and price sensitivity [16]. These findings caution against treating a technical recovery percentage as sufficient evidence of financial attractiveness.

Integrated assessment itself is established. Luo and colleagues combined source–sink compatibility, exergy, temporal availability, costs and environmental effects within decision support [17]. More recent work investigates recovery mismatch [18] and hourly energy–exergy performance with heat pumps and seasonal storage [19]. Consequently, another integrated framework is insufficient as a research contribution. A more precise opportunity is to make the investment boundary auditable: identify how much recoverable source heat must actually be used, distinguish that quantity from compressor-enhanced delivered heat, and separate the value of recovery from the incremental value of storage.

Universities make this distinction consequential because academic schedules modify occupancy and thermal demand. Calendar effects have been quantified [20], and campus demand estimates depend on occupancy-model choice [21]. Practical deployment is also established: Queen Mary University of London reports recovering computing heat for its Priestley Building and Mile End network [22]. That institutional announcement supplies context, not operating measurements for this study. Dvorak and colleagues already examined university data-centre recovery, aquifer storage and uncertainty using campus evidence [23]. The present analysis therefore makes no claim to be the first dynamic campus heat-recovery assessment.

Storage must be evaluated against the duration of the mismatch it addresses. Norwegian campus analysis distinguishes short-term tanks from borehole storage [24]; ETH Zurich research considers daily and seasonal interactions [25], and recent campus optimisation jointly sizes seasonal heating and cooling storage [26]. These studies motivate a deliberately narrower test of short-duration buffering. Established demand-overlap measures already quantify temporal complementarity [27]. Here, a calendar counterfactual preserves annual demand while changing its chronology, allowing schedule effects to be distinguished from differences in total heat consumption.

This study develops a reproducible hourly experiment coupling a recovered-heat interface, a heat pump, bounded thermal storage and boiler backup. It derives a conditional source-utilisation threshold and evaluates temperature–storage interactions against gas and ambient-source heat-pump benchmarks. Particular attention is paid to three decisions often conflated in feasibility screening: whether the heat can reach the required temperature, whether operation saves variable expenditure, and whether those savings repay the installed equipment over its assumed remaining economic life. Its contribution is an inspectable decision test, rather than a universal optimum: additional recovered heat may coexist with lower net present value. Synthetic inputs and conditional uncertainty are reported explicitly, establishing the limits within which the numerical conclusions can support subsequent campus measurements and engineering design.

2. Methodology

An hourly, 365-day synthetic experiment represents a campus heat-recovery subsystem. No campus telemetry, measured weather or equipment calibration is used. The boundary begins at recoverable heat at 35 °C and ends at an idealised campus delivery interface (Figure 1). Computing electricity is invariant and excluded from incremental costs and emissions. Distribution losses, cooling-electricity credits, embodied impacts and potable-water requirements are outside scope. Table 1 specifies analyst-selected inputs; Appendix A defines every profile. Chronological simulation avoids disconnected representative periods, whose treatment can distort storage operation [28,29,30].

Computing power combines a daily cycle with stationary autoregressive noise. Heating demand couples a sinusoidal outdoor temperature with weekday occupancy and 240 synthetic term days. A counterfactual removes term closures and rescales the entire resulting demand series to the original annual total. Nine reference designs combine three delivery temperatures with three storage capacities. One hundred independently seeded computing trajectories repeat every design; their fifth–ninety-fifth percentiles describe conditional numerical variability, not confidence intervals or population risk. Additional sweeps vary computing scale and delivery temperature.

Equations (1)–(3) enforce heat-pump and storage energy balances. Absolute temperatures are in kelvin; thermal rates are in kilowatts and the time step is one hour. The heat-pump surrogate includes approach temperatures and a capacity limit. Unaided direct delivery fails the prescribed pinch constraint; preheating a colder return is outside this model. Dispatch first serves demand, then charges available storage, then discharges against residual demand; unused source heat is rejected and a boiler supplies the remainder. Heat-pump operation is screened against the variable cost of boiler heat. This causal heuristic is not predictive economic optimisation; campus dispatch research illustrates more sophisticated alternatives [31].

Equation (4) measures recovery-only exergy under isothermal source and delivery reservoir approximations, with a fixed 288.15 K reference. It does not resolve refrigerant states, water-return temperatures or tank stratification. Equations (5)–(7) calculate discounted investment value, operational emissions and conditional source-utilisation thresholds. Annual costs repeat in real terms for twenty years, with no escalation, salvage or replacement expenditure. Such simplifications differ from stochastic storage planning [32] and lifetime network optimisation [33,34]. Installed-capacity costs and emission factors are scenario assumptions, not quotations or current national statistics. Repeating each annual trajectory to periodic storage convergence prevents free initial energy. Conservation, capacity and nonnegative exergy-destruction checks are complemented by half-hour dispatch using unchanged hourly inputs; these are numerical verification, not physical validation.

All inputs are assumptions for a synthetic screening experiment, not fitted campus values, market quotations or national emission factors. K = K₀ + 30 Emax. Existing boiler capital is sunk; boiler replacement and fixed maintenance are excluded. The ambient-source benchmark omits defrost and part-load effects.

2.1. Coupled Model Equations

$${\mathrm{S}}_{\mathrm{t}}={\mathsf{\eta }}_{\mathrm{c}\mathrm{a}\mathrm{p}}{\mathrm{P}}_{\mathrm{t}},\mathsf{\kappa }=\mathrm{m}\mathrm{i}\mathrm{n}[8,0.45{\displaystyle \frac{{\mathrm{T}}_{\mathrm{h}}+\mathsf{\delta }}{{\mathrm{T}}_{\mathrm{h}}-{\mathrm{T}}_{\mathrm{s}}+2\mathsf{\delta }}}]$$

$${\mathrm{Q}}_{\mathrm{m}\mathrm{a}\mathrm{x},\mathrm{t}}=\mathrm{m}\mathrm{i}\mathrm{n}[{\mathrm{Q}}_{\mathrm{c}\mathrm{a}\mathrm{p}},{\displaystyle \frac{{\mathrm{S}}_{\mathrm{t}}}{1-1/\mathsf{\kappa }}}]$$

(1)

$${\mathrm{a}}_{\mathrm{t}}=\mathrm{m}\mathrm{i}\mathrm{n}[{\mathrm{D}}_{\mathrm{t}},{\mathrm{Q}}_{\mathrm{m}\mathrm{a}\mathrm{x},\mathrm{t}}],\mathrm{E}{\mathrm{\text{'}}}_{\mathrm{t}}=(1-\mathsf{\lambda }){\mathrm{E}}_{\mathrm{t}}$$

$${\mathrm{c}}_{\mathrm{t}}=\mathrm{m}\mathrm{i}\mathrm{n}[{\mathrm{Q}}_{\mathrm{m}\mathrm{a}\mathrm{x},\mathrm{t}}-{\mathrm{a}}_{\mathrm{t}},400,{\displaystyle \frac{{\mathrm{E}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-\mathrm{E}{\mathrm{\text{'}}}_{\mathrm{t}}}{{\mathsf{\eta }}_{\mathrm{c}}\mathsf{\Delta }\mathrm{t}}}]$$

$${\mathrm{d}}_{\mathrm{t}}=\mathrm{m}\mathrm{i}\mathrm{n}[{\mathrm{D}}_{\mathrm{t}}-{\mathrm{a}}_{\mathrm{t}},400,{\displaystyle \frac{{\mathsf{\eta }}_{\mathrm{d}}(\mathrm{E}{\mathrm{\text{'}}}_{\mathrm{t}}+{\mathsf{\eta }}_{\mathrm{c}}{\mathrm{c}}_{\mathrm{t}}\mathsf{\Delta }\mathrm{t})}{\mathsf{\Delta }\mathrm{t}}}]$$

(2)

$${\mathrm{E}}_{\mathrm{t}+1}=\mathrm{E}{\mathrm{\text{'}}}_{\mathrm{t}}+{\mathsf{\eta }}_{\mathrm{c}}{\mathrm{c}}_{\mathrm{t}}\mathsf{\Delta }\mathrm{t}-{\displaystyle \frac{{\mathrm{d}}_{\mathrm{t}}\mathsf{\Delta }\mathrm{t}}{{\mathsf{\eta }}_{\mathrm{d}}}},{\mathrm{E}}_{8760}={\mathrm{E}}_0$$

$${\mathrm{Q}}_{\mathrm{h}\mathrm{p},\mathrm{t}}={\mathrm{a}}_{\mathrm{t}}+{\mathrm{c}}_{\mathrm{t}},{\mathrm{H}}_{\mathrm{t}}={\mathrm{a}}_{\mathrm{t}}+{\mathrm{d}}_{\mathrm{t}},{\mathrm{Q}}_{\mathrm{s},\mathrm{t}}={\mathrm{Q}}_{\mathrm{h}\mathrm{p},\mathrm{t}}(1-1/\mathsf{\kappa })$$

$${\mathrm{W}}_{\mathrm{t}}={\mathrm{Q}}_{\mathrm{h}\mathrm{p},\mathrm{t}}(1/\mathsf{\kappa }+\mathsf{\alpha }),{\mathrm{G}}_{\mathrm{t}}={\displaystyle \frac{{\mathrm{D}}_{\mathrm{t}}-{\mathrm{H}}_{\mathrm{t}}}{{\mathsf{\eta }}_{\mathrm{b}}}}$$

(3)

$${\mathrm{B}}_{\mathrm{s}}={\sum }_{\mathrm{t}}{\mathrm{Q}}_{\mathrm{s},\mathrm{t}}\mathsf{\Delta }\mathrm{t}(1-{\mathrm{T}}_0/{\mathrm{T}}_{\mathrm{s}}),{\mathrm{B}}_{\mathrm{h}}={\sum }_{\mathrm{t}}{\mathrm{H}}_{\mathrm{t}}\mathsf{\Delta }\mathrm{t}(1-{\mathrm{T}}_0/{\mathrm{T}}_{\mathrm{h}})$$

$${\mathsf{\eta }}_{\mathrm{e}\mathrm{x}}={\displaystyle \frac{{\mathrm{B}}_{\mathrm{h}}}{{\mathrm{B}}_{\mathrm{s}}+\mathrm{W}}},{\mathrm{X}}_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{t}}={\mathrm{B}}_{\mathrm{s}}+\mathrm{W}-{\mathrm{B}}_{\mathrm{h}},\mathrm{W}={\sum }_{\mathrm{t}}{\mathrm{W}}_{\mathrm{t}}\mathsf{\Delta }\mathrm{t}$$

(4)

$$\mathrm{A}={\displaystyle \frac{1-(1+\mathrm{r}{)}^{-\mathrm{n}}}{\mathrm{r}}},\mathrm{N}\mathrm{P}\mathrm{V}=-\mathrm{K}+\mathrm{A}[\mathrm{H}{\mathrm{c}}_{\mathrm{g}}/{\mathsf{\eta }}_{\mathrm{b}}-\mathrm{W}{\mathrm{c}}_{\mathrm{e}}-\mathrm{F}]$$

$$\mathrm{L}\mathrm{C}\mathrm{O}\mathrm{H}={\displaystyle \frac{\mathrm{K}/\mathrm{A}+\mathrm{F}+\mathrm{W}{\mathrm{c}}_{\mathrm{e}}+\mathrm{G}{\mathrm{c}}_{\mathrm{g}}}{\mathrm{D}}},\mathrm{M}=\mathrm{G}{\mathrm{f}}_{\mathrm{g}}+\mathrm{W}{\mathrm{f}}_{\mathrm{e}}$$

(5)

$$\mathrm{S}={\sum }_{\mathrm{t}}{\mathrm{S}}_{\mathrm{t}}\mathsf{\Delta }\mathrm{t},{\mathrm{Q}}_{\mathrm{s}}={\sum }_{\mathrm{t}}{\mathrm{Q}}_{\mathrm{s},\mathrm{t}}\mathsf{\Delta }\mathrm{t},\mathsf{\rho }={\displaystyle \frac{{\mathrm{Q}}_{\mathrm{s}}}{\mathrm{S}}},\mathrm{y}={\displaystyle \frac{\mathrm{H}}{{\mathrm{Q}}_{\mathrm{s}}}},\mathrm{e}={\displaystyle \frac{\mathrm{W}}{{\mathrm{Q}}_{\mathrm{s}}}}$$

(6)

$$\mathrm{m}=\mathrm{y}{\mathrm{c}}_{\mathrm{g}}/{\mathsf{\eta }}_{\mathrm{b}}-\mathrm{e}{\mathrm{c}}_{\mathrm{e}},{\mathsf{\rho }}^{\mathrm{*}}={\displaystyle \frac{\mathrm{K}/\mathrm{A}+\mathrm{F}}{\mathrm{S}\mathrm{m}}},\mathrm{N}\mathrm{P}\mathrm{V}\ge 0\iff \mathsf{\rho }\ge {\mathsf{\rho }}^{\mathrm{*}}(\mathrm{m}>0)$$

$${\displaystyle \frac{\partial \mathrm{N}\mathrm{P}\mathrm{V}}{\partial \mathsf{\rho }}}=\mathrm{A}\mathrm{S}\mathrm{m}(\mathrm{S},\mathrm{m},\mathrm{K},\mathrm{F}\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d})$$

(7)

Rates a, c and d denote direct delivery, charging and discharging. Unsuffixed H, D and G are annual sums. All energy sums use Δt. M is operational emissions. Equation (7) is prospective without storage; with storage, y and e depend on realised dispatch. Appendix A supplies implementation details.

3. Numerical Results and Discussion

The reference simulation produces 1,999.7 MWh of annual demand and 2,365.7 MWh of recoverable computing heat. Annual source availability therefore exceeds demand, yet considerable source heat remains unused. Figure 2 exposes the chronological constraint, while Table 2 reports all nine designs. At delivery temperatures of 45, 55 and 70 °C, modelled heat-pump coefficients of performance are 7.271, 4.997 and 3.482. These are consequences of the specified surrogate, not measured seasonal performance. The source-to-delivery difference must also accommodate two 5 K approaches.

Without storage, delivered heat rises from 1,661.1 MWh at 45 °C to 1,743.0 MWh at 70 °C. Judged by demand coverage alone, the higher-temperature design appears better: 87.2% rather than 83.1%. However, source utilisation falls from 60.6% to 52.5%, and incremental electricity increases from 245.1 to 518.1 MWh. The additional compressor contribution explains the apparent improvement. Heat delivered divided by available source heat is consequently unsuitable as a recovery fraction. Source extraction and purchased electricity must remain separate ledger entries.

Reference seed 2026; source use = extracted recovered heat / available captured heat. Electricity includes compressor and circulation. Each recovery design uses gas backup to serve the same 1,999.7 MWh annual demand. NPV is relative to continued operation of existing gas heating.

The no-storage investment thresholds in Figure 3 are 32.0%, 49.7% and 710.4% source utilisation at 45, 55 and 70 °C, respectively. The last exceeds the physical maximum, excluding capital recovery under these assumptions even if all available source heat were used. Nevertheless, its direct operating margin remains slightly positive, approximately £0.00128 per delivered kilowatt-hour before fixed maintenance. Physical feasibility, positive variable savings and positive investment value are therefore distinct tests. Reference net present values are £333,374, £57,349 and −£347,066, respectively.

Figure 3 shows curve uses annual available source heat of 2.366 GWh, £300,000 investment, annual maintenance of 2% of capital, electricity at £0.22/kWh, gas at £0.06/kWh fuel, 90% boiler efficiency, and a 20-year horizon discounted at 5%. A point above the curve yields positive NPV under these assumptions. Required utilization exceeds 100% above approximately 63.4 °C; the curve is truncated beyond the physical utilization range. The curve is a conditional investment identity, not a universal technology limit.

Storage improves heat supply without consistently improving investment value (Figure 4). At 55 °C, 0.6 MWh adds 67.9 MWh of annual heat delivery but reduces net present value by £9,351. Increasing storage to 2.4 MWh adds 121.3 MWh relative to no storage while reducing value by £71,282, making the project negative. At 45 °C, the smaller store produces a modest £1,737 incremental gain; the larger store loses £55,015. These capacities represent only 1.5 and 6 hours at rated charging power.

Calendar effects are conditional rather than universally adverse. At equal annual demand, replacing term closures with year-round weekday teaching reduces 55 °C no-storage delivery from 1,696.2 to 1,680.1 MWh and net present value from £57,349 to £53,258. With 2.4 MWh storage, the same counterfactual increases delivery from 1,817.5 to 1,838.6 MWh and improves value from −£13,933 to −£9,344. Rescaling changes load concentration, which interacts with capacity and storage. This sign reversal qualifies any simple assumption that removing holiday shutdowns necessarily improves recovery.

Across 100 computing-load realisations, the 55 °C no-storage net present value has a median of £57,699 and a fifth–ninety-fifth percentile range of £56,774–£58,535. The corresponding 2.4 MWh design remains negative throughout that interval, from −£14,419 to −£11,852. Paired incremental value for the 45 °C, 0.6 MWh store has a median of £1,794 and a range of £1,004–£2,730. These narrow ranges reflect stochastic workload variation around a fixed mean; they omit uncertain tariffs, capital costs, weather, degradation and behavioural responses.

Markers and connecting lines show the synthetic reference year (seed 2026); vertical whiskers show the conditional 5th–95th percentiles across 100 computing-load realizations, with all other assumptions fixed. These intervals describe simulated workload variability and are not confidence intervals for measured performance. Many whiskers are narrower than the marker size. NPV includes storage cost of £30/kWh, maintenance at 2% of total capital, and the same tariffs and discounting as Figure 3. The heat axis is intentionally narrowed because points encode values rather than bar lengths.

Financial sensitivity is larger than workload uncertainty. At the realised no-storage heat output, the electricity tariff compatible with zero net present value is £0.3292/kWh at 45 °C, £0.2329/kWh at 55 °C and £0.1662/kWh at 70 °C. The 55 °C case therefore has limited headroom above the assumed £0.22/kWh. Likewise, the 45 °C, 0.6 MWh store breaks even at an installed storage cost of £32.32/kWh, only modestly above the assumed £30/kWh. These thresholds support prioritising verified quotations and operating tariffs before refining stochastic workload forecasts; apparent numerical precision should not imply investment certainty.

Exergy and carbon provide complementary assessments. No-storage subsystem exergy destruction increases from 181.4 to 319.3 MWh of exergy between 45 and 70 °C. Its exergy efficiency nevertheless changes only from 46.3% to 46.7%, because the useful heat also has higher thermodynamic quality. At the assumed emission factors, annual emissions fall from 406.6 tonnes for gas heating to 133.0 tonnes for 55 °C recovery without storage. The idealised ambient-source comparator emits 159.7 tonnes but has higher discounted heat cost (Table 3).

Numerical closure is satisfactory: energy residuals remain below 0.000001 kWh, and halving the dispatch step changes annual delivered heat by less than 0.017 MWh and investment value by less than £6. These checks establish implementation consistency, not agreement with equipment measurements. Moreover, the energy-priority controller can charge storage uneconomically at 70 °C because cycling losses exceed its narrow variable margin. Predictive control could avoid such charging. The broader conclusion is conditional: storage warrants incremental appraisal, and neither annual heat abundance nor increased heat coverage alone establishes a sound investment. Their transferability remains contingent on site-specific engineering.

All systems meet identical demand. Heat cost includes annualised new capital, electricity, residual gas and fixed maintenance, where applicable. Common campus services and existing boiler capital are excluded. The ambient-source heat pump is a simplified full-demand benchmark, not a calibrated equipment comparison.

4. Conclusions

A chronological energy–exergy screening model shows that campus heat recovery requires separate tests of thermal feasibility, operating margin and capital recovery. Under the stated synthetic assumptions, increasing delivery temperature raises the proportion of demand supplied while reducing the proportion of computing heat actually recovered. This divergence results from additional compressor electricity and demonstrates why heat coverage cannot substitute for source utilisation. The derived break-even identity makes the investment requirement explicit and identifies cases where the required utilisation exceeds physical availability.

Short-duration storage increases supplied heat but often reduces net present value. At 55 °C, the 2.4 MWh store changes an otherwise positive investment into a negative one. An equal-demand calendar comparison also reverses the direction of the schedule effect as storage increases, showing that academic closures interact with load concentration and capacity constraints.

These conclusions are conditional on assumed costs, fixed-temperature reservoirs and heuristic dispatch. Future validation should combine metered computing power, source and return temperatures, campus demand, equipment performance maps and installed-cost evidence. Testing predictive control and tariff uncertainty would establish whether the small storage benefit survives realistic operating conditions and whether the screening thresholds transfer to a specific estate.